LMFDB - Embedded newform 1110.2.u.d.401.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1110,2,Mod(191,1110)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1110, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 3]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1110.191");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1110 = 2 \cdot 3 \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1110.u (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$8.86339462436$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			401.1
Root			$-0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1110.401
Dual form			1110.2.u.d.191.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 0.707107i) q^{2} +(-1.41421 + 1.00000i) q^{3} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{5} +(0.292893 - 1.70711i) q^{6} +2.00000 q^{7} +(0.707107 + 0.707107i) q^{8} +(1.00000 - 2.82843i) q^{9} +O(q^{10})$	\(q+(-0.707107 + 0.707107i) q^{2} +(-1.41421 + 1.00000i) q^{3} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{5} +(0.292893 - 1.70711i) q^{6} +2.00000 q^{7} +(0.707107 + 0.707107i) q^{8} +(1.00000 - 2.82843i) q^{9} +1.00000 q^{10} +(1.00000 + 1.41421i) q^{12} +(1.00000 - 1.00000i) q^{13} +(-1.41421 + 1.41421i) q^{14} +(1.70711 + 0.292893i) q^{15} -1.00000 q^{16} +(-4.24264 - 4.24264i) q^{17} +(1.29289 + 2.70711i) q^{18} +(-3.00000 + 3.00000i) q^{19} +(-0.707107 + 0.707107i) q^{20} +(-2.82843 + 2.00000i) q^{21} +(2.82843 + 2.82843i) q^{23} +(-1.70711 - 0.292893i) q^{24} +1.00000i q^{25} +1.41421i q^{26} +(1.41421 + 5.00000i) q^{27} -2.00000i q^{28} +(-4.24264 + 4.24264i) q^{29} +(-1.41421 + 1.00000i) q^{30} +(-7.00000 - 7.00000i) q^{31} +(0.707107 - 0.707107i) q^{32} +6.00000 q^{34} +(-1.41421 - 1.41421i) q^{35} +(-2.82843 - 1.00000i) q^{36} +(-1.00000 + 6.00000i) q^{37} -4.24264i q^{38} +(-0.414214 + 2.41421i) q^{39} -1.00000i q^{40} -5.65685 q^{41} +(0.585786 - 3.41421i) q^{42} +(7.00000 - 7.00000i) q^{43} +(-2.70711 + 1.29289i) q^{45} -4.00000 q^{46} +2.82843i q^{47} +(1.41421 - 1.00000i) q^{48} -3.00000 q^{49} +(-0.707107 - 0.707107i) q^{50} +(10.2426 + 1.75736i) q^{51} +(-1.00000 - 1.00000i) q^{52} -5.65685i q^{53} +(-4.53553 - 2.53553i) q^{54} +(1.41421 + 1.41421i) q^{56} +(1.24264 - 7.24264i) q^{57} -6.00000i q^{58} +(-5.65685 - 5.65685i) q^{59} +(0.292893 - 1.70711i) q^{60} +(-9.00000 - 9.00000i) q^{61} +9.89949 q^{62} +(2.00000 - 5.65685i) q^{63} +1.00000i q^{64} -1.41421 q^{65} +4.00000i q^{67} +(-4.24264 + 4.24264i) q^{68} +(-6.82843 - 1.17157i) q^{69} +2.00000 q^{70} -2.82843i q^{71} +(2.70711 - 1.29289i) q^{72} -6.00000i q^{73} +(-3.53553 - 4.94975i) q^{74} +(-1.00000 - 1.41421i) q^{75} +(3.00000 + 3.00000i) q^{76} +(-1.41421 - 2.00000i) q^{78} +(3.00000 - 3.00000i) q^{79} +(0.707107 + 0.707107i) q^{80} +(-7.00000 - 5.65685i) q^{81} +(4.00000 - 4.00000i) q^{82} -2.82843i q^{83} +(2.00000 + 2.82843i) q^{84} +6.00000i q^{85} +9.89949i q^{86} +(1.75736 - 10.2426i) q^{87} +(1.41421 - 1.41421i) q^{89} +(1.00000 - 2.82843i) q^{90} +(2.00000 - 2.00000i) q^{91} +(2.82843 - 2.82843i) q^{92} +(16.8995 + 2.89949i) q^{93} +(-2.00000 - 2.00000i) q^{94} +4.24264 q^{95} +(-0.292893 + 1.70711i) q^{96} +(7.00000 - 7.00000i) q^{97} +(2.12132 - 2.12132i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{6} + 8 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^6 + 8 * q^7 + 4 * q^9	$ 4 q + 4 q^{6} + 8 q^{7} + 4 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} + 8 q^{18} - 12 q^{19} - 4 q^{24} - 28 q^{31} + 24 q^{34} - 4 q^{37} + 4 q^{39} + 8 q^{42} + 28 q^{43} - 8 q^{45} - 16 q^{46} - 12 q^{49} + 24 q^{51} - 4 q^{52} - 4 q^{54} - 12 q^{57} + 4 q^{60} - 36 q^{61} + 8 q^{63} - 16 q^{69} + 8 q^{70} + 8 q^{72} - 4 q^{75} + 12 q^{76} + 12 q^{79} - 28 q^{81} + 16 q^{82} + 8 q^{84} + 24 q^{87} + 4 q^{90} + 8 q^{91} + 28 q^{93} - 8 q^{94} - 4 q^{96} + 28 q^{97}+O(q^{100}) $ 4 * q + 4 * q^6 + 8 * q^7 + 4 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^13 + 4 * q^15 - 4 * q^16 + 8 * q^18 - 12 * q^19 - 4 * q^24 - 28 * q^31 + 24 * q^34 - 4 * q^37 + 4 * q^39 + 8 * q^42 + 28 * q^43 - 8 * q^45 - 16 * q^46 - 12 * q^49 + 24 * q^51 - 4 * q^52 - 4 * q^54 - 12 * q^57 + 4 * q^60 - 36 * q^61 + 8 * q^63 - 16 * q^69 + 8 * q^70 + 8 * q^72 - 4 * q^75 + 12 * q^76 + 12 * q^79 - 28 * q^81 + 16 * q^82 + 8 * q^84 + 24 * q^87 + 4 * q^90 + 8 * q^91 + 28 * q^93 - 8 * q^94 - 4 * q^96 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times$.

$n$	$371$	$631$	$667$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−0.707107	+	0.707107i	−0.500000	+	0.500000i
$2$	−0.707107	+	0.707107i	−0.500000	+	0.500000i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$4$		−	1.00000i		−	0.500000i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$	0.292893	−	1.70711i	0.119573	−	0.696923i
$7$	2.00000			0.755929			0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000			0.755929			0.377964	+	0.925820i	$0.376624\pi$
$8$	0.707107	+	0.707107i	0.250000	+	0.250000i
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$	1.00000			0.316228
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$	1.00000	+	1.41421i	0.288675	+	0.408248i
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	1.00000	−	1.00000i	0.277350	−	0.277350i	0.832050	+	0.554700i	$0.187167\pi$
$14$	−1.41421	+	1.41421i	−0.377964	+	0.377964i
$15$	1.70711	+	0.292893i	0.440773	+	0.0756247i
$16$	−1.00000			−0.250000
$17$	−4.24264	−	4.24264i	−1.02899	−	1.02899i	−0.999567	−	0.0294245i	$-0.990633\pi$
$17$	−4.24264	−	4.24264i	−1.02899	−	1.02899i	−0.0294245	−	0.999567i	$-0.509367\pi$
$18$	1.29289	+	2.70711i	0.304738	+	0.638071i
$19$	−3.00000	+	3.00000i	−0.688247	+	0.688247i	−0.961844	−	0.273597i	$-0.911786\pi$
$19$	−3.00000	+	3.00000i	−0.688247	+	0.688247i	0.273597	+	0.961844i	$0.411786\pi$
$20$	−0.707107	+	0.707107i	−0.158114	+	0.158114i
$21$	−2.82843	+	2.00000i	−0.617213	+	0.436436i
$22$		0			0
$23$	2.82843	+	2.82843i	0.589768	+	0.589768i	0.937568	−	0.347801i	$-0.113071\pi$
$23$	2.82843	+	2.82843i	0.589768	+	0.589768i	−0.347801	+	0.937568i	$0.613071\pi$
$24$	−1.70711	−	0.292893i	−0.348462	−	0.0597866i
$25$			1.00000i			0.200000i
$26$			1.41421i			0.277350i
$27$	1.41421	+	5.00000i	0.272166	+	0.962250i
$28$		−	2.00000i		−	0.377964i
$29$	−4.24264	+	4.24264i	−0.787839	+	0.787839i	−0.981140	−	0.193301i	$-0.938081\pi$
$29$	−4.24264	+	4.24264i	−0.787839	+	0.787839i	0.193301	+	0.981140i	$0.438081\pi$
$30$	−1.41421	+	1.00000i	−0.258199	+	0.182574i
$31$	−7.00000	−	7.00000i	−1.25724	−	1.25724i	−0.952407	−	0.304830i	$-0.901400\pi$
$31$	−7.00000	−	7.00000i	−1.25724	−	1.25724i	−0.304830	−	0.952407i	$-0.598600\pi$
$32$	0.707107	−	0.707107i	0.125000	−	0.125000i
$33$		0			0
$34$	6.00000			1.02899
$35$	−1.41421	−	1.41421i	−0.239046	−	0.239046i
$36$	−2.82843	−	1.00000i	−0.471405	−	0.166667i
$37$	−1.00000	+	6.00000i	−0.164399	+	0.986394i
$37$	−1.00000	+	6.00000i	−0.164399	+	0.986394i
$38$		−	4.24264i		−	0.688247i
$39$	−0.414214	+	2.41421i	−0.0663273	+	0.386584i
$40$		−	1.00000i		−	0.158114i
$41$	−5.65685			−0.883452			−0.441726	−	0.897150i	$-0.645634\pi$
$41$	−5.65685			−0.883452			−0.441726	+	0.897150i	$0.645634\pi$
$42$	0.585786	−	3.41421i	0.0903888	−	0.526825i
$43$	7.00000	−	7.00000i	1.06749	−	1.06749i	0.0699387	−	0.997551i	$-0.477720\pi$
$43$	7.00000	−	7.00000i	1.06749	−	1.06749i	0.997551	−	0.0699387i	$-0.0222804\pi$
$44$		0			0
$45$	−2.70711	+	1.29289i	−0.403552	+	0.192733i
$46$	−4.00000			−0.589768
$47$			2.82843i			0.412568i	0.978492	+	0.206284i	$0.0661372\pi$
$47$			2.82843i			0.412568i	−0.978492	+	0.206284i	$0.933863\pi$
$48$	1.41421	−	1.00000i	0.204124	−	0.144338i
$49$	−3.00000			−0.428571
$50$	−0.707107	−	0.707107i	−0.100000	−	0.100000i
$51$	10.2426	+	1.75736i	1.43426	+	0.246080i
$52$	−1.00000	−	1.00000i	−0.138675	−	0.138675i
$53$		−	5.65685i		−	0.777029i	−0.921443	−	0.388514i	$-0.872988\pi$
$53$		−	5.65685i		−	0.777029i	0.921443	−	0.388514i	$-0.127012\pi$
$54$	−4.53553	−	2.53553i	−0.617208	−	0.345042i
$55$		0			0
$56$	1.41421	+	1.41421i	0.188982	+	0.188982i
$57$	1.24264	−	7.24264i	0.164592	−	0.959311i
$58$		−	6.00000i		−	0.787839i
$59$	−5.65685	−	5.65685i	−0.736460	−	0.736460i	0.235431	−	0.971891i	$-0.424350\pi$
$59$	−5.65685	−	5.65685i	−0.736460	−	0.736460i	−0.971891	+	0.235431i	$0.924350\pi$
$60$	0.292893	−	1.70711i	0.0378124	−	0.220387i
$61$	−9.00000	−	9.00000i	−1.15233	−	1.15233i	−0.986084	−	0.166248i	$-0.946835\pi$
$61$	−9.00000	−	9.00000i	−1.15233	−	1.15233i	−0.166248	−	0.986084i	$-0.553165\pi$
$62$	9.89949			1.25724
$63$	2.00000	−	5.65685i	0.251976	−	0.712697i
$64$			1.00000i			0.125000i
$65$	−1.41421			−0.175412
$66$		0			0
$67$			4.00000i			0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$			4.00000i			0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	−4.24264	+	4.24264i	−0.514496	+	0.514496i
$69$	−6.82843	−	1.17157i	−0.822046	−	0.141041i
$70$	2.00000			0.239046
$71$		−	2.82843i		−	0.335673i	−0.985815	−	0.167836i	$-0.946322\pi$
$71$		−	2.82843i		−	0.335673i	0.985815	−	0.167836i	$-0.0536780\pi$
$72$	2.70711	−	1.29289i	0.319036	−	0.152369i
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	−3.53553	−	4.94975i	−0.410997	−	0.575396i
$75$	−1.00000	−	1.41421i	−0.115470	−	0.163299i
$76$	3.00000	+	3.00000i	0.344124	+	0.344124i
$77$		0			0
$78$	−1.41421	−	2.00000i	−0.160128	−	0.226455i
$79$	3.00000	−	3.00000i	0.337526	−	0.337526i	−0.517909	−	0.855436i	$-0.673290\pi$
$79$	3.00000	−	3.00000i	0.337526	−	0.337526i	0.855436	+	0.517909i	$0.173290\pi$
$80$	0.707107	+	0.707107i	0.0790569	+	0.0790569i
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$	4.00000	−	4.00000i	0.441726	−	0.441726i
$83$		−	2.82843i		−	0.310460i	−0.987878	−	0.155230i	$-0.950388\pi$
$83$		−	2.82843i		−	0.310460i	0.987878	−	0.155230i	$-0.0496119\pi$
$84$	2.00000	+	2.82843i	0.218218	+	0.308607i
$85$			6.00000i			0.650791i
$86$			9.89949i			1.06749i
$87$	1.75736	−	10.2426i	0.188409	−	1.09813i
$88$		0			0
$89$	1.41421	−	1.41421i	0.149906	−	0.149906i	−0.628170	−	0.778076i	$-0.716196\pi$
$89$	1.41421	−	1.41421i	0.149906	−	0.149906i	0.778076	+	0.628170i	$0.216196\pi$
$90$	1.00000	−	2.82843i	0.105409	−	0.298142i
$91$	2.00000	−	2.00000i	0.209657	−	0.209657i
$92$	2.82843	−	2.82843i	0.294884	−	0.294884i
$93$	16.8995	+	2.89949i	1.75240	+	0.300664i
$94$	−2.00000	−	2.00000i	−0.206284	−	0.206284i
$95$	4.24264			0.435286
$96$	−0.292893	+	1.70711i	−0.0298933	+	0.174231i
$97$	7.00000	−	7.00000i	0.710742	−	0.710742i	−0.255948	−	0.966691i	$-0.582388\pi$
$97$	7.00000	−	7.00000i	0.710742	−	0.710742i	0.966691	+	0.255948i	$0.0823876\pi$
$98$	2.12132	−	2.12132i	0.214286	−	0.214286i
$99$		0			0
$100$	1.00000			0.100000
$101$	2.82843			0.281439			0.140720	−	0.990050i	$-0.455058\pi$
$101$	2.82843			0.281439			0.140720	+	0.990050i	$0.455058\pi$
$102$	−8.48528	+	6.00000i	−0.840168	+	0.594089i
$103$	−5.00000	−	5.00000i	−0.492665	−	0.492665i	0.416480	−	0.909145i	$-0.363264\pi$
$103$	−5.00000	−	5.00000i	−0.492665	−	0.492665i	−0.909145	+	0.416480i	$0.863264\pi$
$104$	1.41421			0.138675
$105$	3.41421	+	0.585786i	0.333193	+	0.0571669i
$106$	4.00000	+	4.00000i	0.388514	+	0.388514i
$107$			11.3137i			1.09374i	0.837218	+	0.546869i	$0.184180\pi$
$107$			11.3137i			1.09374i	−0.837218	+	0.546869i	$0.815820\pi$
$108$	5.00000	−	1.41421i	0.481125	−	0.136083i
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	−0.836031	−	0.548683i	$-0.815129\pi$
$109$	−3.00000	+	3.00000i	−0.287348	+	0.287348i	0.548683	+	0.836031i	$0.315129\pi$
$110$		0			0
$111$	−4.58579	−	9.48528i	−0.435264	−	0.900303i
$112$	−2.00000			−0.188982
$113$	1.41421	−	1.41421i	0.133038	−	0.133038i	−0.637452	−	0.770490i	$-0.720012\pi$
$113$	1.41421	−	1.41421i	0.133038	−	0.133038i	0.770490	+	0.637452i	$0.220012\pi$
$114$	4.24264	+	6.00000i	0.397360	+	0.561951i
$115$		−	4.00000i		−	0.373002i
$116$	4.24264	+	4.24264i	0.393919	+	0.393919i
$117$	−1.82843	−	3.82843i	−0.169038	−	0.353938i
$118$	8.00000			0.736460
$119$	−8.48528	−	8.48528i	−0.777844	−	0.777844i
$120$	1.00000	+	1.41421i	0.0912871	+	0.129099i
$121$	−11.0000			−1.00000
$122$	12.7279			1.15233
$123$	8.00000	−	5.65685i	0.721336	−	0.510061i
$124$	−7.00000	+	7.00000i	−0.628619	+	0.628619i
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$	2.58579	+	5.41421i	0.230360	+	0.482336i
$127$	−16.0000			−1.41977			−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000			−1.41977			−0.709885	+	0.704317i	$0.751253\pi$
$128$	−0.707107	−	0.707107i	−0.0625000	−	0.0625000i
$129$	−2.89949	+	16.8995i	−0.255286	+	1.48792i
$130$	1.00000	−	1.00000i	0.0877058	−	0.0877058i
$131$	11.3137	−	11.3137i	0.988483	−	0.988483i	−0.0114511	−	0.999934i	$-0.503645\pi$
$131$	11.3137	−	11.3137i	0.988483	−	0.988483i	0.999934	+	0.0114511i	$0.00364509\pi$
$132$		0			0
$133$	−6.00000	+	6.00000i	−0.520266	+	0.520266i
$134$	−2.82843	−	2.82843i	−0.244339	−	0.244339i
$135$	2.53553	−	4.53553i	0.218224	−	0.390357i
$136$		−	6.00000i		−	0.514496i
$137$		−	2.82843i		−	0.241649i	−0.992674	−	0.120824i	$-0.961446\pi$
$137$		−	2.82843i		−	0.241649i	0.992674	−	0.120824i	$-0.0385538\pi$
$138$	5.65685	−	4.00000i	0.481543	−	0.340503i
$139$			4.00000i			0.339276i	0.985506	+	0.169638i	$0.0542598\pi$
$139$			4.00000i			0.339276i	−0.985506	+	0.169638i	$0.945740\pi$
$140$	−1.41421	+	1.41421i	−0.119523	+	0.119523i
$141$	−2.82843	−	4.00000i	−0.238197	−	0.336861i
$142$	2.00000	+	2.00000i	0.167836	+	0.167836i
$143$		0			0
$144$	−1.00000	+	2.82843i	−0.0833333	+	0.235702i
$145$	6.00000			0.498273
$146$	4.24264	+	4.24264i	0.351123	+	0.351123i
$147$	4.24264	−	3.00000i	0.349927	−	0.247436i
$148$	6.00000	+	1.00000i	0.493197	+	0.0821995i
$149$			19.7990i			1.62200i	0.585049	+	0.810998i	$0.301075\pi$
$149$			19.7990i			1.62200i	−0.585049	+	0.810998i	$0.698925\pi$
$150$	1.70711	+	0.292893i	0.139385	+	0.0239146i
$151$		−	18.0000i		−	1.46482i	−0.680864	−	0.732410i	$-0.738396\pi$
$151$		−	18.0000i		−	1.46482i	0.680864	−	0.732410i	$-0.261604\pi$
$152$	−4.24264			−0.344124
$153$	−16.2426	+	7.75736i	−1.31314	+	0.627145i
$154$		0			0
$155$			9.89949i			0.795147i
$156$	2.41421	+	0.414214i	0.193292	+	0.0331636i
$157$	−4.00000			−0.319235			−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000			−0.319235			−0.159617	+	0.987179i	$0.551026\pi$
$158$			4.24264i			0.337526i
$159$	5.65685	+	8.00000i	0.448618	+	0.634441i
$160$	−1.00000			−0.0790569
$161$	5.65685	+	5.65685i	0.445823	+	0.445823i
$162$	8.94975	−	0.949747i	0.703159	−	0.0746192i
$163$	−15.0000	−	15.0000i	−1.17489	−	1.17489i	−0.981029	−	0.193862i	$-0.937899\pi$
$163$	−15.0000	−	15.0000i	−1.17489	−	1.17489i	−0.193862	−	0.981029i	$-0.562101\pi$
$164$			5.65685i			0.441726i
$165$		0			0
$166$	2.00000	+	2.00000i	0.155230	+	0.155230i
$167$	−11.3137	−	11.3137i	−0.875481	−	0.875481i	0.117582	−	0.993063i	$-0.462486\pi$
$167$	−11.3137	−	11.3137i	−0.875481	−	0.875481i	−0.993063	+	0.117582i	$0.962486\pi$
$168$	−3.41421	−	0.585786i	−0.263412	−	0.0451944i
$169$			11.0000i			0.846154i
$170$	−4.24264	−	4.24264i	−0.325396	−	0.325396i
$171$	5.48528	+	11.4853i	0.419470	+	0.878301i
$172$	−7.00000	−	7.00000i	−0.533745	−	0.533745i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$	6.00000	+	8.48528i	0.454859	+	0.643268i
$175$			2.00000i			0.151186i
$176$		0			0
$177$	13.6569	+	2.34315i	1.02651	+	0.176122i
$178$			2.00000i			0.149906i
$179$	−11.3137	+	11.3137i	−0.845626	+	0.845626i	−0.989584	−	0.143958i	$-0.954017\pi$
$179$	−11.3137	+	11.3137i	−0.845626	+	0.845626i	0.143958	+	0.989584i	$0.454017\pi$
$180$	1.29289	+	2.70711i	0.0963666	+	0.201776i
$181$		0			0			−	1.00000i	$-0.5\pi$
$181$		0			0				1.00000i	$0.5\pi$
$182$			2.82843i			0.209657i
$183$	21.7279	+	3.72792i	1.60617	+	0.275576i
$184$			4.00000i			0.294884i
$185$	4.94975	−	3.53553i	0.363913	−	0.259938i
$186$	−14.0000	+	9.89949i	−1.02653	+	0.725866i
$187$		0			0
$188$	2.82843			0.206284
$189$	2.82843	+	10.0000i	0.205738	+	0.727393i
$190$	−3.00000	+	3.00000i	−0.217643	+	0.217643i
$191$	5.65685	+	5.65685i	0.409316	+	0.409316i	0.881500	−	0.472184i	$-0.156534\pi$
$191$	5.65685	+	5.65685i	0.409316	+	0.409316i	−0.472184	+	0.881500i	$0.656534\pi$
$192$	−1.00000	−	1.41421i	−0.0721688	−	0.102062i
$193$	−1.00000	+	1.00000i	−0.0719816	+	0.0719816i	−0.742181	−	0.670199i	$-0.766209\pi$
$193$	−1.00000	+	1.00000i	−0.0719816	+	0.0719816i	0.670199	+	0.742181i	$0.266209\pi$
$194$			9.89949i			0.710742i
$195$	2.00000	−	1.41421i	0.143223	−	0.101274i
$196$			3.00000i			0.214286i
$197$		−	2.82843i		−	0.201517i	−0.994911	−	0.100759i	$-0.967873\pi$
$197$		−	2.82843i		−	0.201517i	0.994911	−	0.100759i	$-0.0321270\pi$
$198$		0			0
$199$	1.00000	+	1.00000i	0.0708881	+	0.0708881i	0.741662	−	0.670774i	$-0.234038\pi$
$199$	1.00000	+	1.00000i	0.0708881	+	0.0708881i	−0.670774	+	0.741662i	$0.734038\pi$
$200$	−0.707107	+	0.707107i	−0.0500000	+	0.0500000i
$201$	−4.00000	−	5.65685i	−0.282138	−	0.399004i
$202$	−2.00000	+	2.00000i	−0.140720	+	0.140720i
$203$	−8.48528	+	8.48528i	−0.595550	+	0.595550i
$204$	1.75736	−	10.2426i	0.123040	−	0.717128i
$205$	4.00000	+	4.00000i	0.279372	+	0.279372i
$206$	7.07107			0.492665
$207$	10.8284	−	5.17157i	0.752628	−	0.359449i
$208$	−1.00000	+	1.00000i	−0.0693375	+	0.0693375i
$209$		0			0
$210$	−2.82843	+	2.00000i	−0.195180	+	0.138013i
$211$	20.0000			1.37686			0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000			1.37686			0.688428	+	0.725304i	$0.258301\pi$
$212$	−5.65685			−0.388514
$213$	2.82843	+	4.00000i	0.193801	+	0.274075i
$214$	−8.00000	−	8.00000i	−0.546869	−	0.546869i
$215$	−9.89949			−0.675140
$216$	−2.53553	+	4.53553i	−0.172521	+	0.308604i
$217$	−14.0000	−	14.0000i	−0.950382	−	0.950382i
$218$		−	4.24264i		−	0.287348i
$219$	6.00000	+	8.48528i	0.405442	+	0.573382i
$220$		0			0
$221$	−8.48528			−0.570782
$222$	9.94975	+	3.46447i	0.667783	+	0.232520i
$223$	−24.0000			−1.60716			−0.803579	−	0.595198i	$-0.797074\pi$
$223$	−24.0000			−1.60716			−0.803579	+	0.595198i	$0.797074\pi$
$224$	1.41421	−	1.41421i	0.0944911	−	0.0944911i
$225$	2.82843	+	1.00000i	0.188562	+	0.0666667i
$226$			2.00000i			0.133038i
$227$	16.9706	+	16.9706i	1.12638	+	1.12638i	0.990762	+	0.135614i	$0.0433007\pi$
$227$	16.9706	+	16.9706i	1.12638	+	1.12638i	0.135614	+	0.990762i	$0.456699\pi$
$228$	−7.24264	−	1.24264i	−0.479656	−	0.0822959i
$229$	−26.0000			−1.71813			−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000			−1.71813			−0.859064	+	0.511868i	$0.828954\pi$
$230$	2.82843	+	2.82843i	0.186501	+	0.186501i
$231$		0			0
$232$	−6.00000			−0.393919
$233$	−28.2843			−1.85296			−0.926482	−	0.376339i	$-0.877183\pi$
$233$	−28.2843			−1.85296			−0.926482	+	0.376339i	$0.877183\pi$
$234$	4.00000	+	1.41421i	0.261488	+	0.0924500i
$235$	2.00000	−	2.00000i	0.130466	−	0.130466i
$236$	−5.65685	+	5.65685i	−0.368230	+	0.368230i
$237$	−1.24264	+	7.24264i	−0.0807182	+	0.470460i
$238$	12.0000			0.777844
$239$	−14.1421	−	14.1421i	−0.914779	−	0.914779i	0.0818647	−	0.996643i	$-0.473912\pi$
$239$	−14.1421	−	14.1421i	−0.914779	−	0.914779i	−0.996643	+	0.0818647i	$0.973912\pi$
$240$	−1.70711	−	0.292893i	−0.110193	−	0.0189062i
$241$	−11.0000	+	11.0000i	−0.708572	+	0.708572i	−0.966235	−	0.257663i	$-0.917048\pi$
$241$	−11.0000	+	11.0000i	−0.708572	+	0.708572i	0.257663	+	0.966235i	$0.417048\pi$
$242$	7.77817	−	7.77817i	0.500000	−	0.500000i
$243$	15.5563	+	1.00000i	0.997940	+	0.0641500i
$244$	−9.00000	+	9.00000i	−0.576166	+	0.576166i
$245$	2.12132	+	2.12132i	0.135526	+	0.135526i
$246$	−1.65685	+	9.65685i	−0.105637	+	0.615699i
$247$			6.00000i			0.381771i
$248$		−	9.89949i		−	0.628619i
$249$	2.82843	+	4.00000i	0.179244	+	0.253490i
$250$			1.00000i			0.0632456i
$251$	8.48528	−	8.48528i	0.535586	−	0.535586i	−0.386643	−	0.922229i	$-0.626365\pi$
$251$	8.48528	−	8.48528i	0.535586	−	0.535586i	0.922229	+	0.386643i	$0.126365\pi$
$252$	−5.65685	−	2.00000i	−0.356348	−	0.125988i
$253$		0			0
$254$	11.3137	−	11.3137i	0.709885	−	0.709885i
$255$	−6.00000	−	8.48528i	−0.375735	−	0.531369i
$256$	1.00000			0.0625000
$257$	9.89949	+	9.89949i	0.617514	+	0.617514i	0.944893	−	0.327379i	$-0.106166\pi$
$257$	9.89949	+	9.89949i	0.617514	+	0.617514i	−0.327379	+	0.944893i	$0.606166\pi$
$258$	−9.89949	−	14.0000i	−0.616316	−	0.871602i
$259$	−2.00000	+	12.0000i	−0.124274	+	0.745644i
$260$			1.41421i			0.0877058i
$261$	7.75736	+	16.2426i	0.480168	+	1.00539i
$262$			16.0000i			0.988483i
$263$	31.1127			1.91849			0.959246	−	0.282574i	$-0.0911882\pi$
$263$	31.1127			1.91849			0.959246	+	0.282574i	$0.0911882\pi$
$264$		0			0
$265$	−4.00000	+	4.00000i	−0.245718	+	0.245718i
$266$		−	8.48528i		−	0.520266i
$267$	−0.585786	+	3.41421i	−0.0358495	+	0.208946i
$268$	4.00000			0.244339
$269$		−	19.7990i		−	1.20717i	−0.797300	−	0.603583i	$-0.793739\pi$
$269$		−	19.7990i		−	1.20717i	0.797300	−	0.603583i	$-0.206261\pi$
$270$	1.41421	+	5.00000i	0.0860663	+	0.304290i
$271$	10.0000			0.607457			0.303728	−	0.952759i	$-0.401768\pi$
$271$	10.0000			0.607457			0.303728	+	0.952759i	$0.401768\pi$
$272$	4.24264	+	4.24264i	0.257248	+	0.257248i
$273$	−0.828427	+	4.82843i	−0.0501387	+	0.292230i
$274$	2.00000	+	2.00000i	0.120824	+	0.120824i
$275$		0			0
$276$	−1.17157	+	6.82843i	−0.0705204	+	0.411023i
$277$	11.0000	+	11.0000i	0.660926	+	0.660926i	0.955598	−	0.294672i	$-0.0952105\pi$
$277$	11.0000	+	11.0000i	0.660926	+	0.660926i	−0.294672	+	0.955598i	$0.595211\pi$
$278$	−2.82843	−	2.82843i	−0.169638	−	0.169638i
$279$	−26.7990	+	12.7990i	−1.60441	+	0.766255i
$280$		−	2.00000i		−	0.119523i
$281$	−12.7279	−	12.7279i	−0.759284	−	0.759284i	0.216908	−	0.976192i	$-0.430403\pi$
$281$	−12.7279	−	12.7279i	−0.759284	−	0.759284i	−0.976192	+	0.216908i	$0.930403\pi$
$282$	4.82843	+	0.828427i	0.287529	+	0.0493321i
$283$	3.00000	+	3.00000i	0.178331	+	0.178331i	0.790628	−	0.612297i	$-0.209754\pi$
$283$	3.00000	+	3.00000i	0.178331	+	0.178331i	−0.612297	+	0.790628i	$0.709754\pi$
$284$	−2.82843			−0.167836
$285$	−6.00000	+	4.24264i	−0.355409	+	0.251312i
$286$		0			0
$287$	−11.3137			−0.667827
$288$	−1.29289	−	2.70711i	−0.0761845	−	0.159518i
$289$			19.0000i			1.11765i
$290$	−4.24264	+	4.24264i	−0.249136	+	0.249136i
$291$	−2.89949	+	16.8995i	−0.169971	+	0.990666i
$292$	−6.00000			−0.351123
$293$			22.6274i			1.32191i	0.750426	+	0.660954i	$0.229848\pi$
$293$			22.6274i			1.32191i	−0.750426	+	0.660954i	$0.770152\pi$
$294$	−0.878680	+	5.12132i	−0.0512456	+	0.298681i
$295$			8.00000i			0.465778i
$296$	−4.94975	+	3.53553i	−0.287698	+	0.205499i
$297$		0			0
$298$	−14.0000	−	14.0000i	−0.810998	−	0.810998i
$299$	5.65685			0.327144
$300$	−1.41421	+	1.00000i	−0.0816497	+	0.0577350i
$301$	14.0000	−	14.0000i	0.806947	−	0.806947i
$302$	12.7279	+	12.7279i	0.732410	+	0.732410i
$303$	−4.00000	+	2.82843i	−0.229794	+	0.162489i
$304$	3.00000	−	3.00000i	0.172062	−	0.172062i
$305$			12.7279i			0.728799i
$306$	6.00000	−	16.9706i	0.342997	−	0.970143i
$307$		−	10.0000i		−	0.570730i	−0.958419	−	0.285365i	$-0.907885\pi$
$307$		−	10.0000i		−	0.570730i	0.958419	−	0.285365i	$-0.0921148\pi$
$308$		0			0
$309$	12.0711	+	2.07107i	0.686699	+	0.117819i
$310$	−7.00000	−	7.00000i	−0.397573	−	0.397573i
$311$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$311$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$312$	−2.00000	+	1.41421i	−0.113228	+	0.0800641i
$313$	1.00000	−	1.00000i	0.0565233	−	0.0565233i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	1.00000	−	1.00000i	0.0565233	−	0.0565233i	0.734803	+	0.678280i	$0.237274\pi$
$314$	2.82843	−	2.82843i	0.159617	−	0.159617i
$315$	−5.41421	+	2.58579i	−0.305056	+	0.145693i
$316$	−3.00000	−	3.00000i	−0.168763	−	0.168763i
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$	−9.65685	−	1.65685i	−0.541529	−	0.0929118i
$319$		0			0
$320$	0.707107	−	0.707107i	0.0395285	−	0.0395285i
$321$	−11.3137	−	16.0000i	−0.631470	−	0.893033i
$322$	−8.00000			−0.445823
$323$	25.4558			1.41640
$324$	−5.65685	+	7.00000i	−0.314270	+	0.388889i
$325$	1.00000	+	1.00000i	0.0554700	+	0.0554700i
$326$	21.2132			1.17489
$327$	1.24264	−	7.24264i	0.0687182	−	0.400519i
$328$	−4.00000	−	4.00000i	−0.220863	−	0.220863i
$329$			5.65685i			0.311872i
$330$		0			0
$331$	15.0000	−	15.0000i	0.824475	−	0.824475i	−0.162272	−	0.986746i	$-0.551882\pi$
$331$	15.0000	−	15.0000i	0.824475	−	0.824475i	0.986746	+	0.162272i	$0.0518821\pi$
$332$	−2.82843			−0.155230
$333$	15.9706	+	8.82843i	0.875181	+	0.483795i
$334$	16.0000			0.875481
$335$	2.82843	−	2.82843i	0.154533	−	0.154533i
$336$	2.82843	−	2.00000i	0.154303	−	0.109109i
$337$			20.0000i			1.08947i	0.838608	+	0.544735i	$0.183370\pi$
$337$			20.0000i			1.08947i	−0.838608	+	0.544735i	$0.816630\pi$
$338$	−7.77817	−	7.77817i	−0.423077	−	0.423077i
$339$	−0.585786	+	3.41421i	−0.0318156	+	0.185435i
$340$	6.00000			0.325396
$341$		0			0
$342$	−12.0000	−	4.24264i	−0.648886	−	0.229416i
$343$	−20.0000			−1.07990
$344$	9.89949			0.533745
$345$	4.00000	+	5.65685i	0.215353	+	0.304555i
$346$		0			0
$347$	5.65685	−	5.65685i	0.303676	−	0.303676i	−0.538774	−	0.842450i	$-0.681112\pi$
$347$	5.65685	−	5.65685i	0.303676	−	0.303676i	0.842450	+	0.538774i	$0.181112\pi$
$348$	−10.2426	−	1.75736i	−0.549063	−	0.0942043i
$349$	32.0000			1.71292			0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000			1.71292			0.856460	+	0.516213i	$0.172659\pi$
$350$	−1.41421	−	1.41421i	−0.0755929	−	0.0755929i
$351$	6.41421	+	3.58579i	0.342365	+	0.191395i
$352$		0			0
$353$	18.3848	−	18.3848i	0.978523	−	0.978523i	−0.0212513	−	0.999774i	$-0.506765\pi$
$353$	18.3848	−	18.3848i	0.978523	−	0.978523i	0.999774	+	0.0212513i	$0.00676499\pi$
$354$	−11.3137	+	8.00000i	−0.601317	+	0.425195i
$355$	−2.00000	+	2.00000i	−0.106149	+	0.106149i
$356$	−1.41421	−	1.41421i	−0.0749532	−	0.0749532i
$357$	20.4853	+	3.51472i	1.08420	+	0.186019i
$358$		−	16.0000i		−	0.845626i
$359$		−	5.65685i		−	0.298557i	−0.988795	−	0.149279i	$-0.952305\pi$
$359$		−	5.65685i		−	0.298557i	0.988795	−	0.149279i	$-0.0476951\pi$
$360$	−2.82843	−	1.00000i	−0.149071	−	0.0527046i
$361$			1.00000i			0.0526316i
$362$		0			0
$363$	15.5563	−	11.0000i	0.816497	−	0.577350i
$364$	−2.00000	−	2.00000i	−0.104828	−	0.104828i
$365$	−4.24264	+	4.24264i	−0.222070	+	0.222070i
$366$	−18.0000	+	12.7279i	−0.940875	+	0.665299i
$367$	32.0000			1.67039			0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000			1.67039			0.835193	+	0.549957i	$0.185356\pi$
$368$	−2.82843	−	2.82843i	−0.147442	−	0.147442i
$369$	−5.65685	+	16.0000i	−0.294484	+	0.832927i
$370$	−1.00000	+	6.00000i	−0.0519875	+	0.311925i
$371$		−	11.3137i		−	0.587378i
$372$	2.89949	−	16.8995i	0.150332	−	0.876198i
$373$			10.0000i			0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$			10.0000i			0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$		0			0
$375$	−0.292893	+	1.70711i	−0.0151249	+	0.0881546i
$376$	−2.00000	+	2.00000i	−0.103142	+	0.103142i
$377$			8.48528i			0.437014i
$378$	−9.07107	−	5.07107i	−0.466565	−	0.260828i
$379$	−6.00000			−0.308199			−0.154100	−	0.988055i	$-0.549248\pi$
$379$	−6.00000			−0.308199			−0.154100	+	0.988055i	$0.549248\pi$
$380$		−	4.24264i		−	0.217643i
$381$	22.6274	−	16.0000i	1.15924	−	0.819705i
$382$	−8.00000			−0.409316
$383$	25.4558	+	25.4558i	1.30073	+	1.30073i	0.927898	+	0.372835i	$0.121614\pi$
$383$	25.4558	+	25.4558i	1.30073	+	1.30073i	0.372835	+	0.927898i	$0.378386\pi$
$384$	1.70711	+	0.292893i	0.0871154	+	0.0149466i
$385$		0			0
$386$		−	1.41421i		−	0.0719816i
$387$	−12.7990	−	26.7990i	−0.650609	−	1.36227i
$388$	−7.00000	−	7.00000i	−0.355371	−	0.355371i
$389$	−18.3848	−	18.3848i	−0.932145	−	0.932145i	0.0656946	−	0.997840i	$-0.479074\pi$
$389$	−18.3848	−	18.3848i	−0.932145	−	0.932145i	−0.997840	+	0.0656946i	$0.979074\pi$
$390$	−0.414214	+	2.41421i	−0.0209745	+	0.122248i
$391$		−	24.0000i		−	1.21373i
$392$	−2.12132	−	2.12132i	−0.107143	−	0.107143i
$393$	−4.68629	+	27.3137i	−0.236392	+	1.37779i
$394$	2.00000	+	2.00000i	0.100759	+	0.100759i
$395$	−4.24264			−0.213470
$396$		0			0
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$	−1.41421			−0.0708881
$399$	2.48528	−	14.4853i	0.124420	−	0.725171i
$400$		−	1.00000i		−	0.0500000i
$401$	−12.7279	+	12.7279i	−0.635602	+	0.635602i	−0.949467	−	0.313865i	$-0.898376\pi$
$401$	−12.7279	+	12.7279i	−0.635602	+	0.635602i	0.313865	+	0.949467i	$0.398376\pi$
$402$	6.82843	+	1.17157i	0.340571	+	0.0584327i
$403$	−14.0000			−0.697390
$404$		−	2.82843i		−	0.140720i
$405$	0.949747	+	8.94975i	0.0471933	+	0.444717i
$406$		−	12.0000i		−	0.595550i
$407$		0			0
$408$	6.00000	+	8.48528i	0.297044	+	0.420084i
$409$	−5.00000	−	5.00000i	−0.247234	−	0.247234i	0.572600	−	0.819835i	$-0.305935\pi$
$409$	−5.00000	−	5.00000i	−0.247234	−	0.247234i	−0.819835	+	0.572600i	$0.805935\pi$
$410$	−5.65685			−0.279372
$411$	2.82843	+	4.00000i	0.139516	+	0.197305i
$412$	−5.00000	+	5.00000i	−0.246332	+	0.246332i
$413$	−11.3137	−	11.3137i	−0.556711	−	0.556711i
$414$	−4.00000	+	11.3137i	−0.196589	+	0.556038i
$415$	−2.00000	+	2.00000i	−0.0981761	+	0.0981761i
$416$		−	1.41421i		−	0.0693375i
$417$	−4.00000	−	5.65685i	−0.195881	−	0.277017i
$418$		0			0
$419$		−	25.4558i		−	1.24360i	−0.783176	−	0.621800i	$-0.786402\pi$
$419$		−	25.4558i		−	1.24360i	0.783176	−	0.621800i	$-0.213598\pi$
$420$	0.585786	−	3.41421i	0.0285835	−	0.166597i
$421$	9.00000	+	9.00000i	0.438633	+	0.438633i	0.891552	−	0.452919i	$-0.149617\pi$
$421$	9.00000	+	9.00000i	0.438633	+	0.438633i	−0.452919	+	0.891552i	$0.649617\pi$
$422$	−14.1421	+	14.1421i	−0.688428	+	0.688428i
$423$	8.00000	+	2.82843i	0.388973	+	0.137523i
$424$	4.00000	−	4.00000i	0.194257	−	0.194257i
$425$	4.24264	−	4.24264i	0.205798	−	0.205798i
$426$	−4.82843	−	0.828427i	−0.233938	−	0.0401374i
$427$	−18.0000	−	18.0000i	−0.871081	−	0.871081i
$428$	11.3137			0.546869
$429$		0			0
$430$	7.00000	−	7.00000i	0.337570	−	0.337570i
$431$	−5.65685	+	5.65685i	−0.272481	+	0.272481i	−0.830098	−	0.557617i	$-0.811716\pi$
$431$	−5.65685	+	5.65685i	−0.272481	+	0.272481i	0.557617	+	0.830098i	$0.311716\pi$
$432$	−1.41421	−	5.00000i	−0.0680414	−	0.240563i
$433$	−24.0000			−1.15337			−0.576683	−	0.816968i	$-0.695653\pi$
$433$	−24.0000			−1.15337			−0.576683	+	0.816968i	$0.695653\pi$
$434$	19.7990			0.950382
$435$	−8.48528	+	6.00000i	−0.406838	+	0.287678i
$436$	3.00000	+	3.00000i	0.143674	+	0.143674i
$437$	−16.9706			−0.811812
$438$	−10.2426	−	1.75736i	−0.489412	−	0.0839699i
$439$	27.0000	+	27.0000i	1.28864	+	1.28864i	0.935613	+	0.353026i	$0.114847\pi$
$439$	27.0000	+	27.0000i	1.28864	+	1.28864i	0.353026	+	0.935613i	$0.385153\pi$
$440$		0			0
$441$	−3.00000	+	8.48528i	−0.142857	+	0.404061i
$442$	6.00000	−	6.00000i	0.285391	−	0.285391i
$443$	−14.1421			−0.671913			−0.335957	−	0.941877i	$-0.609060\pi$
$443$	−14.1421			−0.671913			−0.335957	+	0.941877i	$0.609060\pi$
$444$	−9.48528	+	4.58579i	−0.450152	+	0.217632i
$445$	−2.00000			−0.0948091
$446$	16.9706	−	16.9706i	0.803579	−	0.803579i
$447$	−19.7990	−	28.0000i	−0.936460	−	1.32435i
$448$			2.00000i			0.0944911i
$449$	−18.3848	−	18.3848i	−0.867631	−	0.867631i	0.124579	−	0.992210i	$-0.460242\pi$
$449$	−18.3848	−	18.3848i	−0.867631	−	0.867631i	−0.992210	+	0.124579i	$0.960242\pi$
$450$	−2.70711	+	1.29289i	−0.127614	+	0.0609476i
$451$		0			0
$452$	−1.41421	−	1.41421i	−0.0665190	−	0.0665190i
$453$	18.0000	+	25.4558i	0.845714	+	1.19602i
$454$	−24.0000			−1.12638
$455$	−2.82843			−0.132599
$456$	6.00000	−	4.24264i	0.280976	−	0.198680i
$457$	−7.00000	+	7.00000i	−0.327446	+	0.327446i	−0.851615	−	0.524168i	$-0.824376\pi$
$457$	−7.00000	+	7.00000i	−0.327446	+	0.327446i	0.524168	+	0.851615i	$0.324376\pi$
$458$	18.3848	−	18.3848i	0.859064	−	0.859064i
$459$	15.2132	−	27.2132i	0.710092	−	1.27020i
$460$	−4.00000			−0.186501
$461$	−21.2132	−	21.2132i	−0.987997	−	0.987997i	0.0119314	−	0.999929i	$-0.496202\pi$
$461$	−21.2132	−	21.2132i	−0.987997	−	0.987997i	−0.999929	+	0.0119314i	$0.996202\pi$
$462$		0			0
$463$	11.0000	−	11.0000i	0.511213	−	0.511213i	−0.403685	−	0.914898i	$-0.632271\pi$
$463$	11.0000	−	11.0000i	0.511213	−	0.511213i	0.914898	+	0.403685i	$0.132271\pi$
$464$	4.24264	−	4.24264i	0.196960	−	0.196960i
$465$	−9.89949	−	14.0000i	−0.459078	−	0.649234i
$466$	20.0000	−	20.0000i	0.926482	−	0.926482i
$467$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$467$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$468$	−3.82843	+	1.82843i	−0.176969	+	0.0845191i
$469$			8.00000i			0.369406i
$470$			2.82843i			0.130466i
$471$	5.65685	−	4.00000i	0.260654	−	0.184310i
$472$		−	8.00000i		−	0.368230i
$473$		0			0
$474$	−4.24264	−	6.00000i	−0.194871	−	0.275589i
$475$	−3.00000	−	3.00000i	−0.137649	−	0.137649i
$476$	−8.48528	+	8.48528i	−0.388922	+	0.388922i
$477$	−16.0000	−	5.65685i	−0.732590	−	0.259010i
$478$	20.0000			0.914779
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$	1.41421	−	1.00000i	0.0645497	−	0.0456435i
$481$	5.00000	+	7.00000i	0.227980	+	0.319173i
$482$		−	15.5563i		−	0.708572i
$483$	−13.6569	−	2.34315i	−0.621408	−	0.106617i
$484$			11.0000i			0.500000i
$485$	−9.89949			−0.449513
$486$	−11.7071	+	10.2929i	−0.531045	+	0.466895i
$487$	17.0000	−	17.0000i	0.770344	−	0.770344i	−0.207823	−	0.978166i	$-0.566638\pi$
$487$	17.0000	−	17.0000i	0.770344	−	0.770344i	0.978166	+	0.207823i	$0.0666378\pi$
$488$		−	12.7279i		−	0.576166i
$489$	36.2132	+	6.21320i	1.63762	+	0.280971i
$490$	−3.00000			−0.135526
$491$			14.1421i			0.638226i	0.947717	+	0.319113i	$0.103385\pi$
$491$			14.1421i			0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$492$	−5.65685	−	8.00000i	−0.255031	−	0.360668i
$493$	36.0000			1.62136
$494$	−4.24264	−	4.24264i	−0.190885	−	0.190885i
$495$		0			0
$496$	7.00000	+	7.00000i	0.314309	+	0.314309i
$497$		−	5.65685i		−	0.253745i
$498$	−4.82843	−	0.828427i	−0.216367	−	0.0371227i
$499$	3.00000	+	3.00000i	0.134298	+	0.134298i	0.771060	−	0.636762i	$-0.219727\pi$
$499$	3.00000	+	3.00000i	0.134298	+	0.134298i	−0.636762	+	0.771060i	$0.719727\pi$
$500$	−0.707107	−	0.707107i	−0.0316228	−	0.0316228i
$501$	27.3137	+	4.68629i	1.22029	+	0.209368i
$502$			12.0000i			0.535586i
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$	5.41421	−	2.58579i	0.241168	−	0.115180i
$505$	−2.00000	−	2.00000i	−0.0889988	−	0.0889988i
$506$		0			0
$507$	−11.0000	−	15.5563i	−0.488527	−	0.690882i
$508$			16.0000i			0.709885i
$509$	5.65685			0.250736			0.125368	−	0.992110i	$-0.459989\pi$
$509$	5.65685			0.250736			0.125368	+	0.992110i	$0.459989\pi$
$510$	10.2426	+	1.75736i	0.453552	+	0.0778172i
$511$		−	12.0000i		−	0.530849i
$512$	−0.707107	+	0.707107i	−0.0312500	+	0.0312500i
$513$	−19.2426	−	10.7574i	−0.849583	−	0.474949i
$514$	−14.0000			−0.617514
$515$			7.07107i			0.311588i
$516$	16.8995	+	2.89949i	0.743959	+	0.127643i
$517$		0			0
$518$	−7.07107	−	9.89949i	−0.310685	−	0.434959i
$519$		0			0
$520$	−1.00000	−	1.00000i	−0.0438529	−	0.0438529i
$521$	16.9706			0.743494			0.371747	−	0.928334i	$-0.378759\pi$
$521$	16.9706			0.743494			0.371747	+	0.928334i	$0.378759\pi$
$522$	−16.9706	−	6.00000i	−0.742781	−	0.262613i
$523$	3.00000	−	3.00000i	0.131181	−	0.131181i	−0.638468	−	0.769649i	$-0.720431\pi$
$523$	3.00000	−	3.00000i	0.131181	−	0.131181i	0.769649	+	0.638468i	$0.220431\pi$
$524$	−11.3137	−	11.3137i	−0.494242	−	0.494242i
$525$	−2.00000	−	2.82843i	−0.0872872	−	0.123443i
$526$	−22.0000	+	22.0000i	−0.959246	+	0.959246i
$527$			59.3970i			2.58737i
$528$		0			0
$529$		−	7.00000i		−	0.304348i
$530$		−	5.65685i		−	0.245718i
$531$	−21.6569	+	10.3431i	−0.939827	+	0.448854i
$532$	6.00000	+	6.00000i	0.260133	+	0.260133i
$533$	−5.65685	+	5.65685i	−0.245026	+	0.245026i
$534$	−2.00000	−	2.82843i	−0.0865485	−	0.122398i
$535$	8.00000	−	8.00000i	0.345870	−	0.345870i
$536$	−2.82843	+	2.82843i	−0.122169	+	0.122169i
$537$	4.68629	−	27.3137i	0.202228	−	1.17867i
$538$	14.0000	+	14.0000i	0.603583	+	0.603583i
$539$		0			0
$540$	−4.53553	−	2.53553i	−0.195178	−	0.109112i
$541$	21.0000	−	21.0000i	0.902861	−	0.902861i	−0.0928222	−	0.995683i	$-0.529589\pi$
$541$	21.0000	−	21.0000i	0.902861	−	0.902861i	0.995683	+	0.0928222i	$0.0295888\pi$
$542$	−7.07107	+	7.07107i	−0.303728	+	0.303728i
$543$		0			0
$544$	−6.00000			−0.257248
$545$	4.24264			0.181735
$546$	−2.82843	−	4.00000i	−0.121046	−	0.171184i
$547$	7.00000	+	7.00000i	0.299298	+	0.299298i	0.840739	−	0.541441i	$-0.182121\pi$
$547$	7.00000	+	7.00000i	0.299298	+	0.299298i	−0.541441	+	0.840739i	$0.682121\pi$
$548$	−2.82843			−0.120824
$549$	−34.4558	+	16.4558i	−1.47054	+	0.702318i
$550$		0			0
$551$		−	25.4558i		−	1.08446i
$552$	−4.00000	−	5.65685i	−0.170251	−	0.240772i
$553$	6.00000	−	6.00000i	0.255146	−	0.255146i
$554$	−15.5563			−0.660926
$555$	−3.46447	+	9.94975i	−0.147058	+	0.422343i
$556$	4.00000			0.169638
$557$	−7.07107	+	7.07107i	−0.299611	+	0.299611i	−0.840861	−	0.541251i	$-0.817951\pi$
$557$	−7.07107	+	7.07107i	−0.299611	+	0.299611i	0.541251	+	0.840861i	$0.317951\pi$
$558$	9.89949	−	28.0000i	0.419079	−	1.18533i
$559$		−	14.0000i		−	0.592137i
$560$	1.41421	+	1.41421i	0.0597614	+	0.0597614i
$561$		0			0
$562$	18.0000			0.759284
$563$	8.48528	+	8.48528i	0.357612	+	0.357612i	0.862932	−	0.505320i	$-0.168626\pi$
$563$	8.48528	+	8.48528i	0.357612	+	0.357612i	−0.505320	+	0.862932i	$0.668626\pi$
$564$	−4.00000	+	2.82843i	−0.168430	+	0.119098i
$565$	−2.00000			−0.0841406
$566$	−4.24264			−0.178331
$567$	−14.0000	−	11.3137i	−0.587945	−	0.475131i
$568$	2.00000	−	2.00000i	0.0839181	−	0.0839181i
$569$	−32.5269	+	32.5269i	−1.36360	+	1.36360i	−0.494318	+	0.869281i	$0.664582\pi$
$569$	−32.5269	+	32.5269i	−1.36360	+	1.36360i	−0.869281	+	0.494318i	$0.835418\pi$
$570$	1.24264	−	7.24264i	0.0520485	−	0.303361i
$571$	30.0000			1.25546			0.627730	−	0.778431i	$-0.283984\pi$
$571$	30.0000			1.25546			0.627730	+	0.778431i	$0.283984\pi$
$572$		0			0
$573$	−13.6569	−	2.34315i	−0.570523	−	0.0978863i
$574$	8.00000	−	8.00000i	0.333914	−	0.333914i
$575$	−2.82843	+	2.82843i	−0.117954	+	0.117954i
$576$	2.82843	+	1.00000i	0.117851	+	0.0416667i
$577$	27.0000	−	27.0000i	1.12402	−	1.12402i	0.132895	−	0.991130i	$-0.457573\pi$
$577$	27.0000	−	27.0000i	1.12402	−	1.12402i	0.991130	−	0.132895i	$-0.0424272\pi$
$578$	−13.4350	−	13.4350i	−0.558824	−	0.558824i
$579$	0.414214	−	2.41421i	0.0172141	−	0.100331i
$580$		−	6.00000i		−	0.249136i
$581$		−	5.65685i		−	0.234686i
$582$	−9.89949	−	14.0000i	−0.410347	−	0.580319i
$583$		0			0
$584$	4.24264	−	4.24264i	0.175562	−	0.175562i
$585$	−1.41421	+	4.00000i	−0.0584705	+	0.165380i
$586$	−16.0000	−	16.0000i	−0.660954	−	0.660954i
$587$	−25.4558	+	25.4558i	−1.05068	+	1.05068i	−0.0520296	+	0.998646i	$0.516569\pi$
$587$	−25.4558	+	25.4558i	−1.05068	+	1.05068i	−0.998646	+	0.0520296i	$0.983431\pi$
$588$	−3.00000	−	4.24264i	−0.123718	−	0.174964i
$589$	42.0000			1.73058
$590$	−5.65685	−	5.65685i	−0.232889	−	0.232889i
$591$	2.82843	+	4.00000i	0.116346	+	0.164538i
$592$	1.00000	−	6.00000i	0.0410997	−	0.246598i
$593$		−	39.5980i		−	1.62609i	−0.582198	−	0.813047i	$-0.697807\pi$
$593$		−	39.5980i		−	1.62609i	0.582198	−	0.813047i	$-0.302193\pi$
$594$		0			0
$595$			12.0000i			0.491952i
$596$	19.7990			0.810998
$597$	−2.41421	−	0.414214i	−0.0988072	−	0.0169526i
$598$	−4.00000	+	4.00000i	−0.163572	+	0.163572i
$599$		−	8.48528i		−	0.346699i	−0.984860	−	0.173350i	$-0.944541\pi$
$599$		−	8.48528i		−	0.346699i	0.984860	−	0.173350i	$-0.0554591\pi$
$600$	0.292893	−	1.70711i	0.0119573	−	0.0696923i
$601$	12.0000			0.489490			0.244745	−	0.969587i	$-0.421296\pi$
$601$	12.0000			0.489490			0.244745	+	0.969587i	$0.421296\pi$
$602$			19.7990i			0.806947i
$603$	11.3137	+	4.00000i	0.460730	+	0.162893i
$604$	−18.0000			−0.732410
$605$	7.77817	+	7.77817i	0.316228	+	0.316228i
$606$	0.828427	−	4.82843i	0.0336526	−	0.196141i
$607$	−9.00000	−	9.00000i	−0.365299	−	0.365299i	0.500461	−	0.865759i	$-0.333164\pi$
$607$	−9.00000	−	9.00000i	−0.365299	−	0.365299i	−0.865759	+	0.500461i	$0.833164\pi$
$608$			4.24264i			0.172062i
$609$	3.51472	−	20.4853i	0.142424	−	0.830105i
$610$	−9.00000	−	9.00000i	−0.364399	−	0.364399i
$611$	2.82843	+	2.82843i	0.114426	+	0.114426i
$612$	7.75736	+	16.2426i	0.313573	+	0.656570i
$613$		−	4.00000i		−	0.161558i	−0.996732	−	0.0807792i	$-0.974259\pi$
$613$		−	4.00000i		−	0.161558i	0.996732	−	0.0807792i	$-0.0257409\pi$
$614$	7.07107	+	7.07107i	0.285365	+	0.285365i
$615$	−9.65685	−	1.65685i	−0.389402	−	0.0668108i
$616$		0			0
$617$	14.1421			0.569341			0.284670	−	0.958625i	$-0.408116\pi$
$617$	14.1421			0.569341			0.284670	+	0.958625i	$0.408116\pi$
$618$	−10.0000	+	7.07107i	−0.402259	+	0.284440i
$619$			14.0000i			0.562708i	0.959604	+	0.281354i	$0.0907834\pi$
$619$			14.0000i			0.562708i	−0.959604	+	0.281354i	$0.909217\pi$
$620$	9.89949			0.397573
$621$	−10.1421	+	18.1421i	−0.406990	+	0.728019i
$622$		0			0
$623$	2.82843	−	2.82843i	0.113319	−	0.113319i
$624$	0.414214	−	2.41421i	0.0165818	−	0.0966459i
$625$	−1.00000			−0.0400000
$626$			1.41421i			0.0565233i
$627$		0			0
$628$			4.00000i			0.159617i
$629$	29.6985	−	21.2132i	1.18416	−	0.845826i
$630$	2.00000	−	5.65685i	0.0796819	−	0.225374i
$631$	−19.0000	−	19.0000i	−0.756378	−	0.756378i	0.219283	−	0.975661i	$-0.429628\pi$
$631$	−19.0000	−	19.0000i	−0.756378	−	0.756378i	−0.975661	+	0.219283i	$0.929628\pi$
$632$	4.24264			0.168763
$633$	−28.2843	+	20.0000i	−1.12420	+	0.794929i
$634$		0			0
$635$	11.3137	+	11.3137i	0.448971	+	0.448971i
$636$	8.00000	−	5.65685i	0.317221	−	0.224309i
$637$	−3.00000	+	3.00000i	−0.118864	+	0.118864i
$638$		0			0
$639$	−8.00000	−	2.82843i	−0.316475	−	0.111891i
$640$			1.00000i			0.0395285i
$641$		−	22.6274i		−	0.893729i	−0.894602	−	0.446865i	$-0.852541\pi$
$641$		−	22.6274i		−	0.893729i	0.894602	−	0.446865i	$-0.147459\pi$
$642$	19.3137	+	3.31371i	0.762251	+	0.130782i
$643$	27.0000	+	27.0000i	1.06478	+	1.06478i	0.997751	+	0.0670247i	$0.0213506\pi$
$643$	27.0000	+	27.0000i	1.06478	+	1.06478i	0.0670247	+	0.997751i	$0.478649\pi$
$644$	5.65685	−	5.65685i	0.222911	−	0.222911i
$645$	14.0000	−	9.89949i	0.551249	−	0.389792i
$646$	−18.0000	+	18.0000i	−0.708201	+	0.708201i
$647$	−25.4558	+	25.4558i	−1.00077	+	1.00077i	−0.000772798		1.00000i	$0.500246\pi$
$647$	−25.4558	+	25.4558i	−1.00077	+	1.00077i	−1.00000		0.000772798i	$0.999754\pi$
$648$	−0.949747	−	8.94975i	−0.0373096	−	0.351579i
$649$		0			0
$650$	−1.41421			−0.0554700
$651$	33.7990	+	5.79899i	1.32469	+	0.227280i
$652$	−15.0000	+	15.0000i	−0.587445	+	0.587445i
$653$	−24.0416	+	24.0416i	−0.940822	+	0.940822i	−0.998344	−	0.0575225i	$-0.981680\pi$
$653$	−24.0416	+	24.0416i	−0.940822	+	0.940822i	0.0575225	+	0.998344i	$0.481680\pi$
$654$	4.24264	+	6.00000i	0.165900	+	0.234619i
$655$	−16.0000			−0.625172
$656$	5.65685			0.220863
$657$	−16.9706	−	6.00000i	−0.662085	−	0.234082i
$658$	−4.00000	−	4.00000i	−0.155936	−	0.155936i
$659$	−16.9706			−0.661079			−0.330540	−	0.943792i	$-0.607231\pi$
$659$	−16.9706			−0.661079			−0.330540	+	0.943792i	$0.607231\pi$
$660$		0			0
$661$	11.0000	+	11.0000i	0.427850	+	0.427850i	0.887896	−	0.460045i	$-0.152167\pi$
$661$	11.0000	+	11.0000i	0.427850	+	0.427850i	−0.460045	+	0.887896i	$0.652167\pi$
$662$			21.2132i			0.824475i
$663$	12.0000	−	8.48528i	0.466041	−	0.329541i
$664$	2.00000	−	2.00000i	0.0776151	−	0.0776151i
$665$	8.48528			0.329045
$666$	−17.5355	+	5.05025i	−0.679488	+	0.195693i
$667$	−24.0000			−0.929284
$668$	−11.3137	+	11.3137i	−0.437741	+	0.437741i
$669$	33.9411	−	24.0000i	1.31224	−	0.927894i
$670$			4.00000i			0.154533i
$671$		0			0
$672$	−0.585786	+	3.41421i	−0.0225972	+	0.131706i
$673$	−4.00000			−0.154189			−0.0770943	−	0.997024i	$-0.524564\pi$
$673$	−4.00000			−0.154189			−0.0770943	+	0.997024i	$0.524564\pi$
$674$	−14.1421	−	14.1421i	−0.544735	−	0.544735i
$675$	−5.00000	+	1.41421i	−0.192450	+	0.0544331i
$676$	11.0000			0.423077
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	−2.00000	−	2.82843i	−0.0768095	−	0.108625i
$679$	14.0000	−	14.0000i	0.537271	−	0.537271i
$680$	−4.24264	+	4.24264i	−0.162698	+	0.162698i
$681$	−40.9706	−	7.02944i	−1.57000	−	0.269369i
$682$		0			0
$683$	−2.82843	−	2.82843i	−0.108227	−	0.108227i	0.650920	−	0.759147i	$-0.274383\pi$
$683$	−2.82843	−	2.82843i	−0.108227	−	0.108227i	−0.759147	+	0.650920i	$0.774383\pi$
$684$	11.4853	−	5.48528i	0.439151	−	0.209735i
$685$	−2.00000	+	2.00000i	−0.0764161	+	0.0764161i
$686$	14.1421	−	14.1421i	0.539949	−	0.539949i
$687$	36.7696	−	26.0000i	1.40285	−	0.991962i
$688$	−7.00000	+	7.00000i	−0.266872	+	0.266872i
$689$	−5.65685	−	5.65685i	−0.215509	−	0.215509i
$690$	−6.82843	−	1.17157i	−0.259954	−	0.0446010i
$691$			20.0000i			0.760836i	0.924815	+	0.380418i	$0.124220\pi$
$691$			20.0000i			0.760836i	−0.924815	+	0.380418i	$0.875780\pi$
$692$		0			0
$693$		0			0
$694$			8.00000i			0.303676i
$695$	2.82843	−	2.82843i	0.107288	−	0.107288i
$696$	8.48528	−	6.00000i	0.321634	−	0.227429i
$697$	24.0000	+	24.0000i	0.909065	+	0.909065i
$698$	−22.6274	+	22.6274i	−0.856460	+	0.856460i
$699$	40.0000	−	28.2843i	1.51294	−	1.06981i
$700$	2.00000			0.0755929
$701$	−18.3848	−	18.3848i	−0.694383	−	0.694383i	0.268810	−	0.963193i	$-0.413370\pi$
$701$	−18.3848	−	18.3848i	−0.694383	−	0.694383i	−0.963193	+	0.268810i	$0.913370\pi$
$702$	−7.07107	+	2.00000i	−0.266880	+	0.0754851i
$703$	−15.0000	−	21.0000i	−0.565736	−	0.792030i
$704$		0			0
$705$	−0.828427	+	4.82843i	−0.0312004	+	0.181849i
$706$			26.0000i			0.978523i
$707$	5.65685			0.212748
$708$	2.34315	−	13.6569i	0.0880608	−	0.513256i
$709$	15.0000	−	15.0000i	0.563337	−	0.563337i	−0.366917	−	0.930254i	$-0.619587\pi$
$709$	15.0000	−	15.0000i	0.563337	−	0.563337i	0.930254	+	0.366917i	$0.119587\pi$
$710$		−	2.82843i		−	0.106149i
$711$	−5.48528	−	11.4853i	−0.205714	−	0.430732i
$712$	2.00000			0.0749532
$713$		−	39.5980i		−	1.48296i
$714$	−16.9706	+	12.0000i	−0.635107	+	0.449089i
$715$		0			0
$716$	11.3137	+	11.3137i	0.422813	+	0.422813i
$717$	34.1421	+	5.85786i	1.27506	+	0.218766i
$718$	4.00000	+	4.00000i	0.149279	+	0.149279i
$719$		−	11.3137i		−	0.421930i	−0.977494	−	0.210965i	$-0.932339\pi$
$719$		−	11.3137i		−	0.421930i	0.977494	−	0.210965i	$-0.0676606\pi$
$720$	2.70711	−	1.29289i	0.100888	−	0.0481833i
$721$	−10.0000	−	10.0000i	−0.372419	−	0.372419i
$722$	−0.707107	−	0.707107i	−0.0263158	−	0.0263158i
$723$	4.55635	−	26.5563i	0.169452	−	0.987641i
$724$		0			0
$725$	−4.24264	−	4.24264i	−0.157568	−	0.157568i
$726$	−3.22183	+	18.7782i	−0.119573	+	0.696923i
$727$	−27.0000	−	27.0000i	−1.00137	−	1.00137i	−0.999999	−	0.00137552i	$-0.999562\pi$
$727$	−27.0000	−	27.0000i	−1.00137	−	1.00137i	−0.00137552	−	0.999999i	$-0.500438\pi$
$728$	2.82843			0.104828
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		−	6.00000i		−	0.222070i
$731$	−59.3970			−2.19688
$732$	3.72792	−	21.7279i	0.137788	−	0.803087i
$733$		−	4.00000i		−	0.147743i	−0.997268	−	0.0738717i	$-0.976464\pi$
$733$		−	4.00000i		−	0.147743i	0.997268	−	0.0738717i	$-0.0235355\pi$
$734$	−22.6274	+	22.6274i	−0.835193	+	0.835193i
$735$	−5.12132	−	0.878680i	−0.188903	−	0.0324106i
$736$	4.00000			0.147442
$737$		0			0
$738$	−7.31371	−	15.3137i	−0.269221	−	0.563705i
$739$			22.0000i			0.809283i	0.914475	+	0.404642i	$0.132604\pi$
$739$			22.0000i			0.809283i	−0.914475	+	0.404642i	$0.867396\pi$
$740$	−3.53553	−	4.94975i	−0.129969	−	0.181956i
$741$	−6.00000	−	8.48528i	−0.220416	−	0.311715i
$742$	8.00000	+	8.00000i	0.293689	+	0.293689i
$743$	31.1127			1.14141			0.570707	−	0.821154i	$-0.306669\pi$
$743$	31.1127			1.14141			0.570707	+	0.821154i	$0.306669\pi$
$744$	9.89949	+	14.0000i	0.362933	+	0.513265i
$745$	14.0000	−	14.0000i	0.512920	−	0.512920i
$746$	−7.07107	−	7.07107i	−0.258890	−	0.258890i
$747$	−8.00000	−	2.82843i	−0.292705	−	0.103487i
$748$		0			0
$749$			22.6274i			0.826788i
$750$	−1.00000	−	1.41421i	−0.0365148	−	0.0516398i
$751$			40.0000i			1.45962i	0.683650	+	0.729810i	$0.260392\pi$
$751$			40.0000i			1.45962i	−0.683650	+	0.729810i	$0.739608\pi$
$752$		−	2.82843i		−	0.103142i
$753$	−3.51472	+	20.4853i	−0.128083	+	0.746525i
$754$	−6.00000	−	6.00000i	−0.218507	−	0.218507i
$755$	−12.7279	+	12.7279i	−0.463217	+	0.463217i
$756$	10.0000	−	2.82843i	0.363696	−	0.102869i
$757$	3.00000	−	3.00000i	0.109037	−	0.109037i	−0.650484	−	0.759520i	$-0.725434\pi$
$757$	3.00000	−	3.00000i	0.109037	−	0.109037i	0.759520	+	0.650484i	$0.225434\pi$
$758$	4.24264	−	4.24264i	0.154100	−	0.154100i
$759$		0			0
$760$	3.00000	+	3.00000i	0.108821	+	0.108821i
$761$	−2.82843			−0.102530			−0.0512652	−	0.998685i	$-0.516325\pi$
$761$	−2.82843			−0.102530			−0.0512652	+	0.998685i	$0.516325\pi$
$762$	−4.68629	+	27.3137i	−0.169766	+	0.989471i
$763$	−6.00000	+	6.00000i	−0.217215	+	0.217215i
$764$	5.65685	−	5.65685i	0.204658	−	0.204658i
$765$	16.9706	+	6.00000i	0.613572	+	0.216930i
$766$	−36.0000			−1.30073
$767$	−11.3137			−0.408514
$768$	−1.41421	+	1.00000i	−0.0510310	+	0.0360844i
$769$	3.00000	+	3.00000i	0.108183	+	0.108183i	0.759126	−	0.650943i	$-0.225627\pi$
$769$	3.00000	+	3.00000i	0.108183	+	0.108183i	−0.650943	+	0.759126i	$0.725627\pi$
$770$		0			0
$771$	−23.8995	−	4.10051i	−0.860719	−	0.147676i
$772$	1.00000	+	1.00000i	0.0359908	+	0.0359908i
$773$		−	8.48528i		−	0.305194i	−0.988288	−	0.152597i	$-0.951236\pi$
$773$		−	8.48528i		−	0.305194i	0.988288	−	0.152597i	$-0.0487637\pi$
$774$	28.0000	+	9.89949i	1.00644	+	0.355830i
$775$	7.00000	−	7.00000i	0.251447	−	0.251447i
$776$	9.89949			0.355371
$777$	−9.17157	−	18.9706i	−0.329028	−	0.680565i
$778$	26.0000			0.932145
$779$	16.9706	−	16.9706i	0.608034	−	0.608034i
$780$	−1.41421	−	2.00000i	−0.0506370	−	0.0716115i
$781$		0			0
$782$	16.9706	+	16.9706i	0.606866	+	0.606866i
$783$	−27.2132	−	15.2132i	−0.972521	−	0.543676i
$784$	3.00000			0.107143
$785$	2.82843	+	2.82843i	0.100951	+	0.100951i
$786$	−16.0000	−	22.6274i	−0.570701	−	0.807093i
$787$	−28.0000			−0.998092			−0.499046	−	0.866575i	$-0.666316\pi$
$787$	−28.0000			−0.998092			−0.499046	+	0.866575i	$0.666316\pi$
$788$	−2.82843			−0.100759
$789$	−44.0000	+	31.1127i	−1.56644	+	1.10764i
$790$	3.00000	−	3.00000i	0.106735	−	0.106735i
$791$	2.82843	−	2.82843i	0.100567	−	0.100567i
$792$		0			0
$793$	−18.0000			−0.639199
$794$	−12.7279	−	12.7279i	−0.451697	−	0.451697i
$795$	1.65685	−	9.65685i	0.0587626	−	0.342493i
$796$	1.00000	−	1.00000i	0.0354441	−	0.0354441i
$797$	15.5563	−	15.5563i	0.551034	−	0.551034i	−0.375705	−	0.926739i	$-0.622599\pi$
$797$	15.5563	−	15.5563i	0.551034	−	0.551034i	0.926739	+	0.375705i	$0.122599\pi$
$798$	8.48528	+	12.0000i	0.300376	+	0.424795i
$799$	12.0000	−	12.0000i	0.424529	−	0.424529i
$800$	0.707107	+	0.707107i	0.0250000	+	0.0250000i
$801$	−2.58579	−	5.41421i	−0.0913643	−	0.191302i
$802$		−	18.0000i		−	0.635602i
$803$		0			0
$804$	−5.65685	+	4.00000i	−0.199502	+	0.141069i
$805$		−	8.00000i		−	0.281963i
$806$	9.89949	−	9.89949i	0.348695	−	0.348695i
$807$	19.7990	+	28.0000i	0.696957	+	0.985647i
$808$	2.00000	+	2.00000i	0.0703598	+	0.0703598i
$809$	15.5563	−	15.5563i	0.546932	−	0.546932i	−0.378620	−	0.925552i	$-0.623601\pi$
$809$	15.5563	−	15.5563i	0.546932	−	0.546932i	0.925552	+	0.378620i	$0.123601\pi$
$810$	−7.00000	−	5.65685i	−0.245955	−	0.198762i
$811$	−30.0000			−1.05344			−0.526721	−	0.850038i	$-0.676579\pi$
$811$	−30.0000			−1.05344			−0.526721	+	0.850038i	$0.676579\pi$
$812$	8.48528	+	8.48528i	0.297775	+	0.297775i
$813$	−14.1421	+	10.0000i	−0.495986	+	0.350715i
$814$		0			0
$815$			21.2132i			0.743066i
$816$	−10.2426	−	1.75736i	−0.358564	−	0.0615199i
$817$			42.0000i			1.46939i
$818$	7.07107			0.247234
$819$	−3.65685	−	7.65685i	−0.127781	−	0.267552i
$820$	4.00000	−	4.00000i	0.139686	−	0.139686i
$821$			28.2843i			0.987128i	0.869710	+	0.493564i	$0.164306\pi$
$821$			28.2843i			0.987128i	−0.869710	+	0.493564i	$0.835694\pi$
$822$	−4.82843	−	0.828427i	−0.168411	−	0.0288947i
$823$	32.0000			1.11545			0.557725	−	0.830026i	$-0.311674\pi$
$823$	32.0000			1.11545			0.557725	+	0.830026i	$0.311674\pi$
$824$		−	7.07107i		−	0.246332i
$825$		0			0
$826$	16.0000			0.556711
$827$	2.82843	+	2.82843i	0.0983540	+	0.0983540i	0.754572	−	0.656218i	$-0.227845\pi$
$827$	2.82843	+	2.82843i	0.0983540	+	0.0983540i	−0.656218	+	0.754572i	$0.727845\pi$
$828$	−5.17157	−	10.8284i	−0.179725	−	0.376314i
$829$	21.0000	+	21.0000i	0.729360	+	0.729360i	0.970492	−	0.241132i	$-0.0775187\pi$
$829$	21.0000	+	21.0000i	0.729360	+	0.729360i	−0.241132	+	0.970492i	$0.577519\pi$
$830$		−	2.82843i		−	0.0981761i
$831$	−26.5563	−	4.55635i	−0.921230	−	0.158058i
$832$	1.00000	+	1.00000i	0.0346688	+	0.0346688i
$833$	12.7279	+	12.7279i	0.440996	+	0.440996i
$834$	6.82843	+	1.17157i	0.236449	+	0.0405683i
$835$			16.0000i			0.553703i
$836$		0			0
$837$	25.1005	−	44.8995i	0.867600	−	1.55195i
$838$	18.0000	+	18.0000i	0.621800	+	0.621800i
$839$	36.7696			1.26943			0.634713	−	0.772748i	$-0.281118\pi$
$839$	36.7696			1.26943			0.634713	+	0.772748i	$0.281118\pi$
$840$	2.00000	+	2.82843i	0.0690066	+	0.0975900i
$841$		−	7.00000i		−	0.241379i
$842$	−12.7279			−0.438633
$843$	30.7279	+	5.27208i	1.05833	+	0.181580i
$844$		−	20.0000i		−	0.688428i
$845$	7.77817	−	7.77817i	0.267577	−	0.267577i
$846$	−7.65685	+	3.65685i	−0.263248	+	0.125725i
$847$	−22.0000			−0.755929
$848$			5.65685i			0.194257i
$849$	−7.24264	−	1.24264i	−0.248567	−	0.0426473i
$850$			6.00000i			0.205798i
$851$	−19.7990	+	14.1421i	−0.678701	+	0.484786i
$852$	4.00000	−	2.82843i	0.137038	−	0.0969003i
$853$	21.0000	+	21.0000i	0.719026	+	0.719026i	0.968406	−	0.249380i	$-0.0802267\pi$
$853$	21.0000	+	21.0000i	0.719026	+	0.719026i	−0.249380	+	0.968406i	$0.580227\pi$
$854$	25.4558			0.871081
$855$	4.24264	−	12.0000i	0.145095	−	0.410391i
$856$	−8.00000	+	8.00000i	−0.273434	+	0.273434i
$857$	−29.6985	−	29.6985i	−1.01448	−	1.01448i	−0.999894	−	0.0145873i	$-0.995357\pi$
$857$	−29.6985	−	29.6985i	−1.01448	−	1.01448i	−0.0145873	−	0.999894i	$-0.504643\pi$
$858$		0			0
$859$	−15.0000	+	15.0000i	−0.511793	+	0.511793i	−0.915076	−	0.403282i	$-0.867869\pi$
$859$	−15.0000	+	15.0000i	−0.511793	+	0.511793i	0.403282	+	0.915076i	$0.367869\pi$
$860$			9.89949i			0.337570i
$861$	16.0000	−	11.3137i	0.545279	−	0.385570i
$862$		−	8.00000i		−	0.272481i
$863$			28.2843i			0.962808i	0.876499	+	0.481404i	$0.159873\pi$
$863$			28.2843i			0.962808i	−0.876499	+	0.481404i	$0.840127\pi$
$864$	4.53553	+	2.53553i	0.154302	+	0.0862606i
$865$		0			0
$866$	16.9706	−	16.9706i	0.576683	−	0.576683i
$867$	−19.0000	−	26.8701i	−0.645274	−	0.912555i
$868$	−14.0000	+	14.0000i	−0.475191	+	0.475191i
$869$		0			0
$870$	1.75736	−	10.2426i	0.0595801	−	0.347258i
$871$	4.00000	+	4.00000i	0.135535	+	0.135535i
$872$	−4.24264			−0.143674
$873$	−12.7990	−	26.7990i	−0.433180	−	0.907008i
$874$	12.0000	−	12.0000i	0.405906	−	0.405906i
$875$	1.41421	−	1.41421i	0.0478091	−	0.0478091i
$876$	8.48528	−	6.00000i	0.286691	−	0.202721i
$877$	28.0000			0.945493			0.472746	−	0.881199i	$-0.343263\pi$
$877$	28.0000			0.945493			0.472746	+	0.881199i	$0.343263\pi$
$878$	−38.1838			−1.28864
$879$	−22.6274	−	32.0000i	−0.763204	−	1.07933i
$880$		0			0
$881$	−31.1127			−1.04821			−0.524107	−	0.851653i	$-0.675601\pi$
$881$	−31.1127			−1.04821			−0.524107	+	0.851653i	$0.675601\pi$
$882$	−3.87868	−	8.12132i	−0.130602	−	0.273459i
$883$	35.0000	+	35.0000i	1.17784	+	1.17784i	0.980293	+	0.197551i	$0.0632989\pi$
$883$	35.0000	+	35.0000i	1.17784	+	1.17784i	0.197551	+	0.980293i	$0.436701\pi$
$884$			8.48528i			0.285391i
$885$	−8.00000	−	11.3137i	−0.268917	−	0.380306i
$886$	10.0000	−	10.0000i	0.335957	−	0.335957i
$887$	−2.82843			−0.0949693			−0.0474846	−	0.998872i	$-0.515121\pi$
$887$	−2.82843			−0.0949693			−0.0474846	+	0.998872i	$0.515121\pi$
$888$	3.46447	−	9.94975i	0.116260	−	0.333892i
$889$	−32.0000			−1.07325
$890$	1.41421	−	1.41421i	0.0474045	−	0.0474045i
$891$		0			0
$892$			24.0000i			0.803579i
$893$	−8.48528	−	8.48528i	−0.283949	−	0.283949i
$894$	33.7990	+	5.79899i	1.13041	+	0.193947i
$895$	16.0000			0.534821
$896$	−1.41421	−	1.41421i	−0.0472456	−	0.0472456i
$897$	−8.00000	+	5.65685i	−0.267112	+	0.188877i
$898$	26.0000			0.867631
$899$	59.3970			1.98100
$900$	1.00000	−	2.82843i	0.0333333	−	0.0942809i
$901$	−24.0000	+	24.0000i	−0.799556	+	0.799556i
$902$		0			0
$903$	−5.79899	+	33.7990i	−0.192978	+	1.12476i
$904$	2.00000			0.0665190
$905$		0			0
$906$	−30.7279	−	5.27208i	−1.02087	−	0.175153i
$907$	−13.0000	+	13.0000i	−0.431658	+	0.431658i	−0.889192	−	0.457534i	$-0.848733\pi$
$907$	−13.0000	+	13.0000i	−0.431658	+	0.431658i	0.457534	+	0.889192i	$0.348733\pi$
$908$	16.9706	−	16.9706i	0.563188	−	0.563188i
$909$	2.82843	−	8.00000i	0.0938130	−	0.265343i
$910$	2.00000	−	2.00000i	0.0662994	−	0.0662994i
$911$	25.4558	+	25.4558i	0.843390	+	0.843390i	0.989298	−	0.145908i	$-0.0466105\pi$
$911$	25.4558	+	25.4558i	0.843390	+	0.843390i	−0.145908	+	0.989298i	$0.546610\pi$
$912$	−1.24264	+	7.24264i	−0.0411479	+	0.239828i
$913$		0			0
$914$		−	9.89949i		−	0.327446i
$915$	−12.7279	−	18.0000i	−0.420772	−	0.595062i
$916$			26.0000i			0.859064i
$917$	22.6274	−	22.6274i	0.747223	−	0.747223i
$918$	8.48528	+	30.0000i	0.280056	+	0.990148i
$919$	−37.0000	−	37.0000i	−1.22052	−	1.22052i	−0.967448	−	0.253069i	$-0.918560\pi$
$919$	−37.0000	−	37.0000i	−1.22052	−	1.22052i	−0.253069	−	0.967448i	$-0.581440\pi$
$920$	2.82843	−	2.82843i	0.0932505	−	0.0932505i
$921$	10.0000	+	14.1421i	0.329511	+	0.465999i
$922$	30.0000			0.987997
$923$	−2.82843	−	2.82843i	−0.0930988	−	0.0930988i
$924$		0			0
$925$	−6.00000	−	1.00000i	−0.197279	−	0.0328798i
$926$			15.5563i			0.511213i
$927$	−19.1421	+	9.14214i	−0.628710	+	0.300267i
$928$			6.00000i			0.196960i
$929$	45.2548			1.48476			0.742381	−	0.669977i	$-0.233696\pi$
$929$	45.2548			1.48476			0.742381	+	0.669977i	$0.233696\pi$
$930$	16.8995	+	2.89949i	0.554156	+	0.0950782i
$931$	9.00000	−	9.00000i	0.294963	−	0.294963i
$932$			28.2843i			0.926482i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	1.41421	−	4.00000i	0.0462250	−	0.130744i
$937$	−10.0000			−0.326686			−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000			−0.326686			−0.163343	+	0.986569i	$0.552228\pi$
$938$	−5.65685	−	5.65685i	−0.184703	−	0.184703i
$939$	−0.414214	+	2.41421i	−0.0135173	+	0.0787849i
$940$	−2.00000	−	2.00000i	−0.0652328	−	0.0652328i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$	−1.17157	+	6.82843i	−0.0381719	+	0.222482i
$943$	−16.0000	−	16.0000i	−0.521032	−	0.521032i
$944$	5.65685	+	5.65685i	0.184115	+	0.184115i
$945$	5.07107	−	9.07107i	0.164962	−	0.295082i
$946$		0			0
$947$	36.7696	+	36.7696i	1.19485	+	1.19485i	0.975689	+	0.219161i	$0.0703321\pi$
$947$	36.7696	+	36.7696i	1.19485	+	1.19485i	0.219161	+	0.975689i	$0.429668\pi$
$948$	7.24264	+	1.24264i	0.235230	+	0.0403591i
$949$	−6.00000	−	6.00000i	−0.194768	−	0.194768i
$950$	4.24264			0.137649
$951$		0			0
$952$		−	12.0000i		−	0.388922i
$953$	48.0833			1.55757			0.778785	−	0.627291i	$-0.215836\pi$
$953$	48.0833			1.55757			0.778785	+	0.627291i	$0.215836\pi$
$954$	15.3137	−	7.31371i	0.495800	−	0.236790i
$955$		−	8.00000i		−	0.258874i
$956$	−14.1421	+	14.1421i	−0.457389	+	0.457389i
$957$		0			0
$958$		0			0
$959$		−	5.65685i		−	0.182669i
$960$	−0.292893	+	1.70711i	−0.00945309	+	0.0550966i
$961$			67.0000i			2.16129i
$962$	−8.48528	−	1.41421i	−0.273576	−	0.0455961i
$963$	32.0000	+	11.3137i	1.03119	+	0.364579i
$964$	11.0000	+	11.0000i	0.354286	+	0.354286i
$965$	1.41421			0.0455251
$966$	11.3137	−	8.00000i	0.364013	−	0.257396i
$967$	−5.00000	+	5.00000i	−0.160789	+	0.160789i	−0.782916	−	0.622127i	$-0.786269\pi$
$967$	−5.00000	+	5.00000i	−0.160789	+	0.160789i	0.622127	+	0.782916i	$0.286269\pi$
$968$	−7.77817	−	7.77817i	−0.250000	−	0.250000i
$969$	−36.0000	+	25.4558i	−1.15649	+	0.817760i
$970$	7.00000	−	7.00000i	0.224756	−	0.224756i
$971$			19.7990i			0.635380i	0.948195	+	0.317690i	$0.102907\pi$
$971$			19.7990i			0.635380i	−0.948195	+	0.317690i	$0.897093\pi$
$972$	1.00000	−	15.5563i	0.0320750	−	0.498970i
$973$			8.00000i			0.256468i
$974$			24.0416i			0.770344i
$975$	−2.41421	−	0.414214i	−0.0773167	−	0.0132655i
$976$	9.00000	+	9.00000i	0.288083	+	0.288083i
$977$	18.3848	−	18.3848i	0.588181	−	0.588181i	−0.348957	−	0.937139i	$-0.613464\pi$
$977$	18.3848	−	18.3848i	0.588181	−	0.588181i	0.937139	+	0.348957i	$0.113464\pi$
$978$	−30.0000	+	21.2132i	−0.959294	+	0.678323i
$979$		0			0
$980$	2.12132	−	2.12132i	0.0677631	−	0.0677631i
$981$	5.48528	+	11.4853i	0.175132	+	0.366697i
$982$	−10.0000	−	10.0000i	−0.319113	−	0.319113i
$983$	−48.0833			−1.53362			−0.766809	−	0.641875i	$-0.778157\pi$
$983$	−48.0833			−1.53362			−0.766809	+	0.641875i	$0.778157\pi$
$984$	9.65685	+	1.65685i	0.307849	+	0.0528186i
$985$	−2.00000	+	2.00000i	−0.0637253	+	0.0637253i
$986$	−25.4558	+	25.4558i	−0.810679	+	0.810679i
$987$	−5.65685	−	8.00000i	−0.180060	−	0.254643i
$988$	6.00000			0.190885
$989$	39.5980			1.25914
$990$		0			0
$991$	−9.00000	−	9.00000i	−0.285894	−	0.285894i	0.549560	−	0.835454i	$-0.314795\pi$
$991$	−9.00000	−	9.00000i	−0.285894	−	0.285894i	−0.835454	+	0.549560i	$0.814795\pi$
$992$	−9.89949			−0.314309
$993$	−6.21320	+	36.2132i	−0.197170	+	1.14919i
$994$	4.00000	+	4.00000i	0.126872	+	0.126872i
$995$		−	1.41421i		−	0.0448336i
$996$	4.00000	−	2.82843i	0.126745	−	0.0896221i
$997$	−11.0000	+	11.0000i	−0.348373	+	0.348373i	−0.859503	−	0.511130i	$-0.829227\pi$
$997$	−11.0000	+	11.0000i	−0.348373	+	0.348373i	0.511130	+	0.859503i	$0.329227\pi$
$998$	−4.24264			−0.134298
$999$	−31.4142	+	3.48528i	−0.993902	+	0.110269i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1110.2.u.d.401.1	yes	4
3.2	odd	2	inner	1110.2.u.d.401.2	yes	4
37.6	odd	4	inner	1110.2.u.d.191.2	yes	4
111.80	even	4	inner	1110.2.u.d.191.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1110.2.u.d.191.1	✓	4	111.80	even	4	inner
1110.2.u.d.191.2	yes	4	37.6	odd	4	inner
1110.2.u.d.401.1	yes	4	1.1	even	1	trivial
1110.2.u.d.401.2	yes	4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1110.u (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 0.707107i) q^{2} +(-1.41421 + 1.00000i) q^{3} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{5} +(0.292893 - 1.70711i) q^{6} +2.00000 q^{7} +(0.707107 + 0.707107i) q^{8} +(1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 0.707107i) q^{2} +(-1.41421 + 1.00000i) q^{3} -1.00000i q^{4} +(-0.707107 - 0.707107i) q^{5} +(0.292893 - 1.70711i) q^{6} +2.00000 q^{7} +(0.707107 + 0.707107i) q^{8} +(1.00000 - 2.82843i) q^{9} +1.00000 q^{10} +(1.00000 + 1.41421i) q^{12} +(1.00000 - 1.00000i) q^{13} +(-1.41421 + 1.41421i) q^{14} +(1.70711 + 0.292893i) q^{15} -1.00000 q^{16} +(-4.24264 - 4.24264i) q^{17} +(1.29289 + 2.70711i) q^{18} +(-3.00000 + 3.00000i) q^{19} +(-0.707107 + 0.707107i) q^{20} +(-2.82843 + 2.00000i) q^{21} +(2.82843 + 2.82843i) q^{23} +(-1.70711 - 0.292893i) q^{24} +1.00000i q^{25} +1.41421i q^{26} +(1.41421 + 5.00000i) q^{27} -2.00000i q^{28} +(-4.24264 + 4.24264i) q^{29} +(-1.41421 + 1.00000i) q^{30} +(-7.00000 - 7.00000i) q^{31} +(0.707107 - 0.707107i) q^{32} +6.00000 q^{34} +(-1.41421 - 1.41421i) q^{35} +(-2.82843 - 1.00000i) q^{36} +(-1.00000 + 6.00000i) q^{37} -4.24264i q^{38} +(-0.414214 + 2.41421i) q^{39} -1.00000i q^{40} -5.65685 q^{41} +(0.585786 - 3.41421i) q^{42} +(7.00000 - 7.00000i) q^{43} +(-2.70711 + 1.29289i) q^{45} -4.00000 q^{46} +2.82843i q^{47} +(1.41421 - 1.00000i) q^{48} -3.00000 q^{49} +(-0.707107 - 0.707107i) q^{50} +(10.2426 + 1.75736i) q^{51} +(-1.00000 - 1.00000i) q^{52} -5.65685i q^{53} +(-4.53553 - 2.53553i) q^{54} +(1.41421 + 1.41421i) q^{56} +(1.24264 - 7.24264i) q^{57} -6.00000i q^{58} +(-5.65685 - 5.65685i) q^{59} +(0.292893 - 1.70711i) q^{60} +(-9.00000 - 9.00000i) q^{61} +9.89949 q^{62} +(2.00000 - 5.65685i) q^{63} +1.00000i q^{64} -1.41421 q^{65} +4.00000i q^{67} +(-4.24264 + 4.24264i) q^{68} +(-6.82843 - 1.17157i) q^{69} +2.00000 q^{70} -2.82843i q^{71} +(2.70711 - 1.29289i) q^{72} -6.00000i q^{73} +(-3.53553 - 4.94975i) q^{74} +(-1.00000 - 1.41421i) q^{75} +(3.00000 + 3.00000i) q^{76} +(-1.41421 - 2.00000i) q^{78} +(3.00000 - 3.00000i) q^{79} +(0.707107 + 0.707107i) q^{80} +(-7.00000 - 5.65685i) q^{81} +(4.00000 - 4.00000i) q^{82} -2.82843i q^{83} +(2.00000 + 2.82843i) q^{84} +6.00000i q^{85} +9.89949i q^{86} +(1.75736 - 10.2426i) q^{87} +(1.41421 - 1.41421i) q^{89} +(1.00000 - 2.82843i) q^{90} +(2.00000 - 2.00000i) q^{91} +(2.82843 - 2.82843i) q^{92} +(16.8995 + 2.89949i) q^{93} +(-2.00000 - 2.00000i) q^{94} +4.24264 q^{95} +(-0.292893 + 1.70711i) q^{96} +(7.00000 - 7.00000i) q^{97} +(2.12132 - 2.12132i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{6} + 8 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^6 + 8 * q^7 + 4 * q^9	\( 4 q + 4 q^{6} + 8 q^{7} + 4 q^{9} + 4 q^{10} + 4 q^{12} + 4 q^{13} + 4 q^{15} - 4 q^{16} + 8 q^{18} - 12 q^{19} - 4 q^{24} - 28 q^{31} + 24 q^{34} - 4 q^{37} + 4 q^{39} + 8 q^{42} + 28 q^{43} - 8 q^{45} - 16 q^{46} - 12 q^{49} + 24 q^{51} - 4 q^{52} - 4 q^{54} - 12 q^{57} + 4 q^{60} - 36 q^{61} + 8 q^{63} - 16 q^{69} + 8 q^{70} + 8 q^{72} - 4 q^{75} + 12 q^{76} + 12 q^{79} - 28 q^{81} + 16 q^{82} + 8 q^{84} + 24 q^{87} + 4 q^{90} + 8 q^{91} + 28 q^{93} - 8 q^{94} - 4 q^{96} + 28 q^{97}+O(q^{100}) \) 4 * q + 4 * q^6 + 8 * q^7 + 4 * q^9 + 4 * q^10 + 4 * q^12 + 4 * q^13 + 4 * q^15 - 4 * q^16 + 8 * q^18 - 12 * q^19 - 4 * q^24 - 28 * q^31 + 24 * q^34 - 4 * q^37 + 4 * q^39 + 8 * q^42 + 28 * q^43 - 8 * q^45 - 16 * q^46 - 12 * q^49 + 24 * q^51 - 4 * q^52 - 4 * q^54 - 12 * q^57 + 4 * q^60 - 36 * q^61 + 8 * q^63 - 16 * q^69 + 8 * q^70 + 8 * q^72 - 4 * q^75 + 12 * q^76 + 12 * q^79 - 28 * q^81 + 16 * q^82 + 8 * q^84 + 24 * q^87 + 4 * q^90 + 8 * q^91 + 28 * q^93 - 8 * q^94 - 4 * q^96 + 28 * q^97

Introduction

L-functions

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