LMFDB - Embedded newform 150.2.e.a.107.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [150,2,Mod(107,150)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(150, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

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

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

chi := DirichletCharacter("150.107");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 150 = 2 \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	150.e (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:	$1.19775603032$
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:	no (minimal twist has level 30)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			107.2
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	150.107
Dual form			150.2.e.a.143.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{2} +(0.292893 - 1.70711i) q^{3} -1.00000i q^{4} +(-1.00000 - 1.41421i) q^{6} +(1.00000 + 1.00000i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-2.82843 - 1.00000i) q^{9} +O(q^{10})$	\(q+(0.707107 - 0.707107i) q^{2} +(0.292893 - 1.70711i) q^{3} -1.00000i q^{4} +(-1.00000 - 1.41421i) q^{6} +(1.00000 + 1.00000i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-2.82843 - 1.00000i) q^{9} +1.41421i q^{11} +(-1.70711 - 0.292893i) q^{12} +1.41421 q^{14} -1.00000 q^{16} +(1.41421 - 1.41421i) q^{17} +(-2.70711 + 1.29289i) q^{18} +4.00000i q^{19} +(2.00000 - 1.41421i) q^{21} +(1.00000 + 1.00000i) q^{22} +(2.82843 + 2.82843i) q^{23} +(-1.41421 + 1.00000i) q^{24} +(-2.53553 + 4.53553i) q^{27} +(1.00000 - 1.00000i) q^{28} +7.07107 q^{29} -2.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} +(2.41421 + 0.414214i) q^{33} -2.00000i q^{34} +(-1.00000 + 2.82843i) q^{36} +(-6.00000 - 6.00000i) q^{37} +(2.82843 + 2.82843i) q^{38} +5.65685i q^{41} +(0.414214 - 2.41421i) q^{42} +(-6.00000 + 6.00000i) q^{43} +1.41421 q^{44} +4.00000 q^{46} +(-0.292893 + 1.70711i) q^{48} -5.00000i q^{49} +(-2.00000 - 2.82843i) q^{51} +(-2.82843 - 2.82843i) q^{53} +(1.41421 + 5.00000i) q^{54} -1.41421i q^{56} +(6.82843 + 1.17157i) q^{57} +(5.00000 - 5.00000i) q^{58} -9.89949 q^{59} -6.00000 q^{61} +(-1.41421 + 1.41421i) q^{62} +(-1.82843 - 3.82843i) q^{63} +1.00000i q^{64} +(2.00000 - 1.41421i) q^{66} +(4.00000 + 4.00000i) q^{67} +(-1.41421 - 1.41421i) q^{68} +(5.65685 - 4.00000i) q^{69} -14.1421i q^{71} +(1.29289 + 2.70711i) q^{72} +(5.00000 - 5.00000i) q^{73} -8.48528 q^{74} +4.00000 q^{76} +(-1.41421 + 1.41421i) q^{77} +6.00000i q^{79} +(7.00000 + 5.65685i) q^{81} +(4.00000 + 4.00000i) q^{82} +(-8.48528 - 8.48528i) q^{83} +(-1.41421 - 2.00000i) q^{84} +8.48528i q^{86} +(2.07107 - 12.0711i) q^{87} +(1.00000 - 1.00000i) q^{88} -2.82843 q^{89} +(2.82843 - 2.82843i) q^{92} +(-0.585786 + 3.41421i) q^{93} +(1.00000 + 1.41421i) q^{96} +(-3.00000 - 3.00000i) q^{97} +(-3.53553 - 3.53553i) q^{98} +(1.41421 - 4.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{6} + 4 q^{7}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^6 + 4 * q^7	$ 4 q + 4 q^{3} - 4 q^{6} + 4 q^{7} - 4 q^{12} - 4 q^{16} - 8 q^{18} + 8 q^{21} + 4 q^{22} + 4 q^{27} + 4 q^{28} - 8 q^{31} + 4 q^{33} - 4 q^{36} - 24 q^{37} - 4 q^{42} - 24 q^{43} + 16 q^{46} - 4 q^{48} - 8 q^{51} + 16 q^{57} + 20 q^{58} - 24 q^{61} + 4 q^{63} + 8 q^{66} + 16 q^{67} + 8 q^{72} + 20 q^{73} + 16 q^{76} + 28 q^{81} + 16 q^{82} - 20 q^{87} + 4 q^{88} - 8 q^{93} + 4 q^{96} - 12 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^6 + 4 * q^7 - 4 * q^12 - 4 * q^16 - 8 * q^18 + 8 * q^21 + 4 * q^22 + 4 * q^27 + 4 * q^28 - 8 * q^31 + 4 * q^33 - 4 * q^36 - 24 * q^37 - 4 * q^42 - 24 * q^43 + 16 * q^46 - 4 * q^48 - 8 * q^51 + 16 * q^57 + 20 * q^58 - 24 * q^61 + 4 * q^63 + 8 * q^66 + 16 * q^67 + 8 * q^72 + 20 * q^73 + 16 * q^76 + 28 * q^81 + 16 * q^82 - 20 * q^87 + 4 * q^88 - 8 * q^93 + 4 * q^96 - 12 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/150\mathbb{Z}\right)^\times$.

$n$	$101$	$127$
$\chi(n)$	$-1$	$e\left(\frac{1}{4}\right)$

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
$3$	0.292893	−	1.70711i	0.169102	−	0.985599i
$3$	0.292893	−	1.70711i	0.169102	−	0.985599i
$4$		−	1.00000i		−	0.500000i
$5$		0			0
$5$		0			0
$6$	−1.00000	−	1.41421i	−0.408248	−	0.577350i
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	0.870367	−	0.492403i	$-0.163881\pi$
$7$	1.00000	+	1.00000i	0.377964	+	0.377964i	−0.492403	+	0.870367i	$0.663881\pi$
$8$	−0.707107	−	0.707107i	−0.250000	−	0.250000i
$9$	−2.82843	−	1.00000i	−0.942809	−	0.333333i
$10$		0			0
$11$			1.41421i			0.426401i	0.977008	+	0.213201i	$0.0683888\pi$
$11$			1.41421i			0.426401i	−0.977008	+	0.213201i	$0.931611\pi$
$12$	−1.70711	−	0.292893i	−0.492799	−	0.0845510i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$	1.41421			0.377964
$15$		0			0
$16$	−1.00000			−0.250000
$17$	1.41421	−	1.41421i	0.342997	−	0.342997i	−0.514496	−	0.857493i	$-0.672021\pi$
$17$	1.41421	−	1.41421i	0.342997	−	0.342997i	0.857493	+	0.514496i	$0.172021\pi$
$18$	−2.70711	+	1.29289i	−0.638071	+	0.304738i
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$	2.00000	−	1.41421i	0.436436	−	0.308607i
$22$	1.00000	+	1.00000i	0.213201	+	0.213201i
$23$	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	$0.613071\pi$
$24$	−1.41421	+	1.00000i	−0.288675	+	0.204124i
$25$		0			0
$26$		0			0
$27$	−2.53553	+	4.53553i	−0.487964	+	0.872864i
$28$	1.00000	−	1.00000i	0.188982	−	0.188982i
$29$	7.07107			1.31306			0.656532	−	0.754298i	$-0.272023\pi$
$29$	7.07107			1.31306			0.656532	+	0.754298i	$0.272023\pi$
$30$		0			0
$31$	−2.00000			−0.359211			−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000			−0.359211			−0.179605	+	0.983739i	$0.557482\pi$
$32$	−0.707107	+	0.707107i	−0.125000	+	0.125000i
$33$	2.41421	+	0.414214i	0.420261	+	0.0721053i
$34$		−	2.00000i		−	0.342997i
$35$		0			0
$36$	−1.00000	+	2.82843i	−0.166667	+	0.471405i
$37$	−6.00000	−	6.00000i	−0.986394	−	0.986394i	0.0135147	−	0.999909i	$-0.495698\pi$
$37$	−6.00000	−	6.00000i	−0.986394	−	0.986394i	−0.999909	+	0.0135147i	$0.995698\pi$
$38$	2.82843	+	2.82843i	0.458831	+	0.458831i
$39$		0			0
$40$		0			0
$41$			5.65685i			0.883452i	0.897150	+	0.441726i	$0.145634\pi$
$41$			5.65685i			0.883452i	−0.897150	+	0.441726i	$0.854366\pi$
$42$	0.414214	−	2.41421i	0.0639145	−	0.372521i
$43$	−6.00000	+	6.00000i	−0.914991	+	0.914991i	−0.996660	−	0.0816682i	$-0.973975\pi$
$43$	−6.00000	+	6.00000i	−0.914991	+	0.914991i	0.0816682	+	0.996660i	$0.473975\pi$
$44$	1.41421			0.213201
$45$		0			0
$46$	4.00000			0.589768
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$	−0.292893	+	1.70711i	−0.0422755	+	0.246400i
$49$		−	5.00000i		−	0.714286i
$50$		0			0
$51$	−2.00000	−	2.82843i	−0.280056	−	0.396059i
$52$		0			0
$53$	−2.82843	−	2.82843i	−0.388514	−	0.388514i	0.485643	−	0.874157i	$-0.338586\pi$
$53$	−2.82843	−	2.82843i	−0.388514	−	0.388514i	−0.874157	+	0.485643i	$0.838586\pi$
$54$	1.41421	+	5.00000i	0.192450	+	0.680414i
$55$		0			0
$56$		−	1.41421i		−	0.188982i
$57$	6.82843	+	1.17157i	0.904447	+	0.155179i
$58$	5.00000	−	5.00000i	0.656532	−	0.656532i
$59$	−9.89949			−1.28880			−0.644402	−	0.764687i	$-0.722894\pi$
$59$	−9.89949			−1.28880			−0.644402	+	0.764687i	$0.722894\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$	−1.41421	+	1.41421i	−0.179605	+	0.179605i
$63$	−1.82843	−	3.82843i	−0.230360	−	0.482336i
$64$			1.00000i			0.125000i
$65$		0			0
$66$	2.00000	−	1.41421i	0.246183	−	0.174078i
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	0.907889	−	0.419211i	$-0.137693\pi$
$67$	4.00000	+	4.00000i	0.488678	+	0.488678i	−0.419211	+	0.907889i	$0.637693\pi$
$68$	−1.41421	−	1.41421i	−0.171499	−	0.171499i
$69$	5.65685	−	4.00000i	0.681005	−	0.481543i
$70$		0			0
$71$		−	14.1421i		−	1.67836i	−0.543852	−	0.839181i	$-0.683035\pi$
$71$		−	14.1421i		−	1.67836i	0.543852	−	0.839181i	$-0.316965\pi$
$72$	1.29289	+	2.70711i	0.152369	+	0.319036i
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	5.00000	−	5.00000i	0.585206	−	0.585206i	0.936329	+	0.351123i	$0.114200\pi$
$74$	−8.48528			−0.986394
$75$		0			0
$76$	4.00000			0.458831
$77$	−1.41421	+	1.41421i	−0.161165	+	0.161165i
$78$		0			0
$79$			6.00000i			0.675053i	0.941316	+	0.337526i	$0.109590\pi$
$79$			6.00000i			0.675053i	−0.941316	+	0.337526i	$0.890410\pi$
$80$		0			0
$81$	7.00000	+	5.65685i	0.777778	+	0.628539i
$82$	4.00000	+	4.00000i	0.441726	+	0.441726i
$83$	−8.48528	−	8.48528i	−0.931381	−	0.931381i	0.0664117	−	0.997792i	$-0.478845\pi$
$83$	−8.48528	−	8.48528i	−0.931381	−	0.931381i	−0.997792	+	0.0664117i	$0.978845\pi$
$84$	−1.41421	−	2.00000i	−0.154303	−	0.218218i
$85$		0			0
$86$			8.48528i			0.914991i
$87$	2.07107	−	12.0711i	0.222042	−	1.29415i
$88$	1.00000	−	1.00000i	0.106600	−	0.106600i
$89$	−2.82843			−0.299813			−0.149906	−	0.988700i	$-0.547897\pi$
$89$	−2.82843			−0.299813			−0.149906	+	0.988700i	$0.547897\pi$
$90$		0			0
$91$		0			0
$92$	2.82843	−	2.82843i	0.294884	−	0.294884i
$93$	−0.585786	+	3.41421i	−0.0607432	+	0.354037i
$94$		0			0
$95$		0			0
$96$	1.00000	+	1.41421i	0.102062	+	0.144338i
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	0.538208	−	0.842812i	$-0.319101\pi$
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	−0.842812	+	0.538208i	$0.819101\pi$
$98$	−3.53553	−	3.53553i	−0.357143	−	0.357143i
$99$	1.41421	−	4.00000i	0.142134	−	0.402015i
$100$		0			0
$101$		−	9.89949i		−	0.985037i	−0.870302	−	0.492518i	$-0.836076\pi$
$101$		−	9.89949i		−	0.985037i	0.870302	−	0.492518i	$-0.163924\pi$
$102$	−3.41421	−	0.585786i	−0.338058	−	0.0580015i
$103$	−1.00000	+	1.00000i	−0.0985329	+	0.0985329i	−0.754655	−	0.656122i	$-0.772196\pi$
$103$	−1.00000	+	1.00000i	−0.0985329	+	0.0985329i	0.656122	+	0.754655i	$0.272196\pi$
$104$		0			0
$105$		0			0
$106$	−4.00000			−0.388514
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	−0.830481	−	0.557047i	$-0.811934\pi$
$107$	−2.82843	+	2.82843i	−0.273434	+	0.273434i	0.557047	+	0.830481i	$0.311934\pi$
$108$	4.53553	+	2.53553i	0.436432	+	0.243982i
$109$		−	10.0000i		−	0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$		−	10.0000i		−	0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$		0			0
$111$	−12.0000	+	8.48528i	−1.13899	+	0.805387i
$112$	−1.00000	−	1.00000i	−0.0944911	−	0.0944911i
$113$	9.89949	+	9.89949i	0.931266	+	0.931266i	0.997785	−	0.0665190i	$-0.0211893\pi$
$113$	9.89949	+	9.89949i	0.931266	+	0.931266i	−0.0665190	+	0.997785i	$0.521189\pi$
$114$	5.65685	−	4.00000i	0.529813	−	0.374634i
$115$		0			0
$116$		−	7.07107i		−	0.656532i
$117$		0			0
$118$	−7.00000	+	7.00000i	−0.644402	+	0.644402i
$119$	2.82843			0.259281
$120$		0			0
$121$	9.00000			0.818182
$122$	−4.24264	+	4.24264i	−0.384111	+	0.384111i
$123$	9.65685	+	1.65685i	0.870729	+	0.149394i
$124$			2.00000i			0.179605i
$125$		0			0
$126$	−4.00000	−	1.41421i	−0.356348	−	0.125988i
$127$	7.00000	+	7.00000i	0.621150	+	0.621150i	0.945825	−	0.324676i	$-0.105255\pi$
$127$	7.00000	+	7.00000i	0.621150	+	0.621150i	−0.324676	+	0.945825i	$0.605255\pi$
$128$	0.707107	+	0.707107i	0.0625000	+	0.0625000i
$129$	8.48528	+	12.0000i	0.747087	+	1.05654i
$130$		0			0
$131$			18.3848i			1.60629i	0.595787	+	0.803143i	$0.296840\pi$
$131$			18.3848i			1.60629i	−0.595787	+	0.803143i	$0.703160\pi$
$132$	0.414214	−	2.41421i	0.0360527	−	0.210130i
$133$	−4.00000	+	4.00000i	−0.346844	+	0.346844i
$134$	5.65685			0.488678
$135$		0			0
$136$	−2.00000			−0.171499
$137$	−4.24264	+	4.24264i	−0.362473	+	0.362473i	−0.864723	−	0.502249i	$-0.832506\pi$
$137$	−4.24264	+	4.24264i	−0.362473	+	0.362473i	0.502249	+	0.864723i	$0.332506\pi$
$138$	1.17157	−	6.82843i	0.0997309	−	0.581274i
$139$			8.00000i			0.678551i	0.940687	+	0.339276i	$0.110182\pi$
$139$			8.00000i			0.678551i	−0.940687	+	0.339276i	$0.889818\pi$
$140$		0			0
$141$		0			0
$142$	−10.0000	−	10.0000i	−0.839181	−	0.839181i
$143$		0			0
$144$	2.82843	+	1.00000i	0.235702	+	0.0833333i
$145$		0			0
$146$		−	7.07107i		−	0.585206i
$147$	−8.53553	−	1.46447i	−0.703999	−	0.120787i
$148$	−6.00000	+	6.00000i	−0.493197	+	0.493197i
$149$	−12.7279			−1.04271			−0.521356	−	0.853339i	$-0.674574\pi$
$149$	−12.7279			−1.04271			−0.521356	+	0.853339i	$0.674574\pi$
$150$		0			0
$151$	16.0000			1.30206			0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000			1.30206			0.651031	+	0.759051i	$0.274337\pi$
$152$	2.82843	−	2.82843i	0.229416	−	0.229416i
$153$	−5.41421	+	2.58579i	−0.437713	+	0.209048i
$154$			2.00000i			0.161165i
$155$		0			0
$156$		0			0
$157$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$157$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$158$	4.24264	+	4.24264i	0.337526	+	0.337526i
$159$	−5.65685	+	4.00000i	−0.448618	+	0.317221i
$160$		0			0
$161$			5.65685i			0.445823i
$162$	8.94975	−	0.949747i	0.703159	−	0.0746192i
$163$	4.00000	−	4.00000i	0.313304	−	0.313304i	−0.532884	−	0.846188i	$-0.678892\pi$
$163$	4.00000	−	4.00000i	0.313304	−	0.313304i	0.846188	+	0.532884i	$0.178892\pi$
$164$	5.65685			0.441726
$165$		0			0
$166$	−12.0000			−0.931381
$167$	−5.65685	+	5.65685i	−0.437741	+	0.437741i	−0.891251	−	0.453510i	$-0.850171\pi$
$167$	−5.65685	+	5.65685i	−0.437741	+	0.437741i	0.453510	+	0.891251i	$0.350171\pi$
$168$	−2.41421	−	0.414214i	−0.186261	−	0.0319573i
$169$			13.0000i			1.00000i
$170$		0			0
$171$	4.00000	−	11.3137i	0.305888	−	0.865181i
$172$	6.00000	+	6.00000i	0.457496	+	0.457496i
$173$	−1.41421	−	1.41421i	−0.107521	−	0.107521i	0.651300	−	0.758820i	$-0.274224\pi$
$173$	−1.41421	−	1.41421i	−0.107521	−	0.107521i	−0.758820	+	0.651300i	$0.774224\pi$
$174$	−7.07107	−	10.0000i	−0.536056	−	0.758098i
$175$		0			0
$176$		−	1.41421i		−	0.106600i
$177$	−2.89949	+	16.8995i	−0.217939	+	1.27024i
$178$	−2.00000	+	2.00000i	−0.149906	+	0.149906i
$179$	18.3848			1.37414			0.687071	−	0.726590i	$-0.258896\pi$
$179$	18.3848			1.37414			0.687071	+	0.726590i	$0.258896\pi$
$180$		0			0
$181$	−22.0000			−1.63525			−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000			−1.63525			−0.817624	+	0.575753i	$0.804709\pi$
$182$		0			0
$183$	−1.75736	+	10.2426i	−0.129908	+	0.757158i
$184$		−	4.00000i		−	0.294884i
$185$		0			0
$186$	2.00000	+	2.82843i	0.146647	+	0.207390i
$187$	2.00000	+	2.00000i	0.146254	+	0.146254i
$188$		0			0
$189$	−7.07107	+	2.00000i	−0.514344	+	0.145479i
$190$		0			0
$191$			2.82843i			0.204658i	0.994751	+	0.102329i	$0.0326294\pi$
$191$			2.82843i			0.204658i	−0.994751	+	0.102329i	$0.967371\pi$
$192$	1.70711	+	0.292893i	0.123200	+	0.0211377i
$193$	15.0000	−	15.0000i	1.07972	−	1.07972i	0.0831899	−	0.996534i	$-0.473489\pi$
$193$	15.0000	−	15.0000i	1.07972	−	1.07972i	0.996534	−	0.0831899i	$-0.0265108\pi$
$194$	−4.24264			−0.304604
$195$		0			0
$196$	−5.00000			−0.357143
$197$	16.9706	−	16.9706i	1.20910	−	1.20910i	0.237785	−	0.971318i	$-0.423579\pi$
$197$	16.9706	−	16.9706i	1.20910	−	1.20910i	0.971318	−	0.237785i	$-0.0764212\pi$
$198$	−1.82843	−	3.82843i	−0.129941	−	0.272074i
$199$		−	24.0000i		−	1.70131i	−0.525720	−	0.850657i	$-0.676204\pi$
$199$		−	24.0000i		−	1.70131i	0.525720	−	0.850657i	$-0.323796\pi$
$200$		0			0
$201$	8.00000	−	5.65685i	0.564276	−	0.399004i
$202$	−7.00000	−	7.00000i	−0.492518	−	0.492518i
$203$	7.07107	+	7.07107i	0.496292	+	0.496292i
$204$	−2.82843	+	2.00000i	−0.198030	+	0.140028i
$205$		0			0
$206$			1.41421i			0.0985329i
$207$	−5.17157	−	10.8284i	−0.359449	−	0.752628i
$208$		0			0
$209$	−5.65685			−0.391293
$210$		0			0
$211$	8.00000			0.550743			0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000			0.550743			0.275371	+	0.961338i	$0.411199\pi$
$212$	−2.82843	+	2.82843i	−0.194257	+	0.194257i
$213$	−24.1421	−	4.14214i	−1.65419	−	0.283814i
$214$			4.00000i			0.273434i
$215$		0			0
$216$	5.00000	−	1.41421i	0.340207	−	0.0962250i
$217$	−2.00000	−	2.00000i	−0.135769	−	0.135769i
$218$	−7.07107	−	7.07107i	−0.478913	−	0.478913i
$219$	−7.07107	−	10.0000i	−0.477818	−	0.675737i
$220$		0			0
$221$		0			0
$222$	−2.48528	+	14.4853i	−0.166801	+	0.972188i
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	−0.338340	−	0.941024i	$-0.609865\pi$
$223$	9.00000	−	9.00000i	0.602685	−	0.602685i	0.941024	+	0.338340i	$0.109865\pi$
$224$	−1.41421			−0.0944911
$225$		0			0
$226$	14.0000			0.931266
$227$	15.5563	−	15.5563i	1.03251	−	1.03251i	0.0330577	−	0.999453i	$-0.489475\pi$
$227$	15.5563	−	15.5563i	1.03251	−	1.03251i	0.999453	−	0.0330577i	$-0.0105245\pi$
$228$	1.17157	−	6.82843i	0.0775893	−	0.452224i
$229$		−	6.00000i		−	0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$		−	6.00000i		−	0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$		0			0
$231$	2.00000	+	2.82843i	0.131590	+	0.186097i
$232$	−5.00000	−	5.00000i	−0.328266	−	0.328266i
$233$	−12.7279	−	12.7279i	−0.833834	−	0.833834i	0.154205	−	0.988039i	$-0.450718\pi$
$233$	−12.7279	−	12.7279i	−0.833834	−	0.833834i	−0.988039	+	0.154205i	$0.950718\pi$
$234$		0			0
$235$		0			0
$236$			9.89949i			0.644402i
$237$	10.2426	+	1.75736i	0.665331	+	0.114153i
$238$	2.00000	−	2.00000i	0.129641	−	0.129641i
$239$	8.48528			0.548867			0.274434	−	0.961606i	$-0.411510\pi$
$239$	8.48528			0.548867			0.274434	+	0.961606i	$0.411510\pi$
$240$		0			0
$241$	4.00000			0.257663			0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000			0.257663			0.128831	+	0.991667i	$0.458877\pi$
$242$	6.36396	−	6.36396i	0.409091	−	0.409091i
$243$	11.7071	−	10.2929i	0.751011	−	0.660289i
$244$			6.00000i			0.384111i
$245$		0			0
$246$	8.00000	−	5.65685i	0.510061	−	0.360668i
$247$		0			0
$248$	1.41421	+	1.41421i	0.0898027	+	0.0898027i
$249$	−16.9706	+	12.0000i	−1.07547	+	0.760469i
$250$		0			0
$251$		−	12.7279i		−	0.803379i	−0.915776	−	0.401690i	$-0.868423\pi$
$251$		−	12.7279i		−	0.803379i	0.915776	−	0.401690i	$-0.131577\pi$
$252$	−3.82843	+	1.82843i	−0.241168	+	0.115180i
$253$	−4.00000	+	4.00000i	−0.251478	+	0.251478i
$254$	9.89949			0.621150
$255$		0			0
$256$	1.00000			0.0625000
$257$	−9.89949	+	9.89949i	−0.617514	+	0.617514i	−0.944893	−	0.327379i	$-0.893834\pi$
$257$	−9.89949	+	9.89949i	−0.617514	+	0.617514i	0.327379	+	0.944893i	$0.393834\pi$
$258$	14.4853	+	2.48528i	0.901814	+	0.154727i
$259$		−	12.0000i		−	0.745644i
$260$		0			0
$261$	−20.0000	−	7.07107i	−1.23797	−	0.437688i
$262$	13.0000	+	13.0000i	0.803143	+	0.803143i
$263$	5.65685	+	5.65685i	0.348817	+	0.348817i	0.859669	−	0.510852i	$-0.170670\pi$
$263$	5.65685	+	5.65685i	0.348817	+	0.348817i	−0.510852	+	0.859669i	$0.670670\pi$
$264$	−1.41421	−	2.00000i	−0.0870388	−	0.123091i
$265$		0			0
$266$			5.65685i			0.346844i
$267$	−0.828427	+	4.82843i	−0.0506989	+	0.295495i
$268$	4.00000	−	4.00000i	0.244339	−	0.244339i
$269$	15.5563			0.948487			0.474244	−	0.880394i	$-0.342722\pi$
$269$	15.5563			0.948487			0.474244	+	0.880394i	$0.342722\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−1.41421	+	1.41421i	−0.0857493	+	0.0857493i
$273$		0			0
$274$			6.00000i			0.362473i
$275$		0			0
$276$	−4.00000	−	5.65685i	−0.240772	−	0.340503i
$277$	6.00000	+	6.00000i	0.360505	+	0.360505i	0.863999	−	0.503494i	$-0.167952\pi$
$277$	6.00000	+	6.00000i	0.360505	+	0.360505i	−0.503494	+	0.863999i	$0.667952\pi$
$278$	5.65685	+	5.65685i	0.339276	+	0.339276i
$279$	5.65685	+	2.00000i	0.338667	+	0.119737i
$280$		0			0
$281$		−	8.48528i		−	0.506189i	−0.967442	−	0.253095i	$-0.918552\pi$
$281$		−	8.48528i		−	0.506189i	0.967442	−	0.253095i	$-0.0814484\pi$
$282$		0			0
$283$	−20.0000	+	20.0000i	−1.18888	+	1.18888i	−0.211498	+	0.977378i	$0.567834\pi$
$283$	−20.0000	+	20.0000i	−1.18888	+	1.18888i	−0.977378	+	0.211498i	$0.932166\pi$
$284$	−14.1421			−0.839181
$285$		0			0
$286$		0			0
$287$	−5.65685	+	5.65685i	−0.333914	+	0.333914i
$288$	2.70711	−	1.29289i	0.159518	−	0.0761845i
$289$			13.0000i			0.764706i
$290$		0			0
$291$	−6.00000	+	4.24264i	−0.351726	+	0.248708i
$292$	−5.00000	−	5.00000i	−0.292603	−	0.292603i
$293$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$293$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$294$	−7.07107	+	5.00000i	−0.412393	+	0.291606i
$295$		0			0
$296$			8.48528i			0.493197i
$297$	−6.41421	−	3.58579i	−0.372190	−	0.208068i
$298$	−9.00000	+	9.00000i	−0.521356	+	0.521356i
$299$		0			0
$300$		0			0
$301$	−12.0000			−0.691669
$302$	11.3137	−	11.3137i	0.651031	−	0.651031i
$303$	−16.8995	−	2.89949i	−0.970851	−	0.166572i
$304$		−	4.00000i		−	0.229416i
$305$		0			0
$306$	−2.00000	+	5.65685i	−0.114332	+	0.323381i
$307$	−18.0000	−	18.0000i	−1.02731	−	1.02731i	−0.999616	−	0.0276979i	$-0.991182\pi$
$307$	−18.0000	−	18.0000i	−1.02731	−	1.02731i	−0.0276979	−	0.999616i	$-0.508818\pi$
$308$	1.41421	+	1.41421i	0.0805823	+	0.0805823i
$309$	1.41421	+	2.00000i	0.0804518	+	0.113776i
$310$		0			0
$311$			19.7990i			1.12270i	0.827579	+	0.561349i	$0.189717\pi$
$311$			19.7990i			1.12270i	−0.827579	+	0.561349i	$0.810283\pi$
$312$		0			0
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	−0.914130	−	0.405420i	$-0.867125\pi$
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	0.405420	+	0.914130i	$0.367125\pi$
$314$		0			0
$315$		0			0
$316$	6.00000			0.337526
$317$	−12.7279	+	12.7279i	−0.714871	+	0.714871i	−0.967550	−	0.252679i	$-0.918688\pi$
$317$	−12.7279	+	12.7279i	−0.714871	+	0.714871i	0.252679	+	0.967550i	$0.418688\pi$
$318$	−1.17157	+	6.82843i	−0.0656985	+	0.382919i
$319$			10.0000i			0.559893i
$320$		0			0
$321$	4.00000	+	5.65685i	0.223258	+	0.315735i
$322$	4.00000	+	4.00000i	0.222911	+	0.222911i
$323$	5.65685	+	5.65685i	0.314756	+	0.314756i
$324$	5.65685	−	7.00000i	0.314270	−	0.388889i
$325$		0			0
$326$		−	5.65685i		−	0.313304i
$327$	−17.0711	−	2.92893i	−0.944032	−	0.161970i
$328$	4.00000	−	4.00000i	0.220863	−	0.220863i
$329$		0			0
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$	−8.48528	+	8.48528i	−0.465690	+	0.465690i
$333$	10.9706	+	22.9706i	0.601183	+	1.25878i
$334$			8.00000i			0.437741i
$335$		0			0
$336$	−2.00000	+	1.41421i	−0.109109	+	0.0771517i
$337$	9.00000	+	9.00000i	0.490261	+	0.490261i	0.908388	−	0.418127i	$-0.137313\pi$
$337$	9.00000	+	9.00000i	0.490261	+	0.490261i	−0.418127	+	0.908388i	$0.637313\pi$
$338$	9.19239	+	9.19239i	0.500000	+	0.500000i
$339$	19.7990	−	14.0000i	1.07533	−	0.760376i
$340$		0			0
$341$		−	2.82843i		−	0.153168i
$342$	−5.17157	−	10.8284i	−0.279647	−	0.585534i
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$	8.48528			0.457496
$345$		0			0
$346$	−2.00000			−0.107521
$347$	9.89949	−	9.89949i	0.531433	−	0.531433i	−0.389566	−	0.920999i	$-0.627375\pi$
$347$	9.89949	−	9.89949i	0.531433	−	0.531433i	0.920999	+	0.389566i	$0.127375\pi$
$348$	−12.0711	−	2.07107i	−0.647077	−	0.111021i
$349$		−	2.00000i		−	0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$		−	2.00000i		−	0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$		0			0
$351$		0			0
$352$	−1.00000	−	1.00000i	−0.0533002	−	0.0533002i
$353$	12.7279	+	12.7279i	0.677439	+	0.677439i	0.959420	−	0.281981i	$-0.0909915\pi$
$353$	12.7279	+	12.7279i	0.677439	+	0.677439i	−0.281981	+	0.959420i	$0.590992\pi$
$354$	9.89949	+	14.0000i	0.526152	+	0.744092i
$355$		0			0
$356$			2.82843i			0.149906i
$357$	0.828427	−	4.82843i	0.0438450	−	0.255547i
$358$	13.0000	−	13.0000i	0.687071	−	0.687071i
$359$	−11.3137			−0.597115			−0.298557	−	0.954392i	$-0.596505\pi$
$359$	−11.3137			−0.597115			−0.298557	+	0.954392i	$0.596505\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−15.5563	+	15.5563i	−0.817624	+	0.817624i
$363$	2.63604	−	15.3640i	0.138356	−	0.806399i
$364$		0			0
$365$		0			0
$366$	6.00000	+	8.48528i	0.313625	+	0.443533i
$367$	19.0000	+	19.0000i	0.991792	+	0.991792i	0.999967	−	0.00817466i	$-0.00260210\pi$
$367$	19.0000	+	19.0000i	0.991792	+	0.991792i	−0.00817466	+	0.999967i	$0.502602\pi$
$368$	−2.82843	−	2.82843i	−0.147442	−	0.147442i
$369$	5.65685	−	16.0000i	0.294484	−	0.832927i
$370$		0			0
$371$		−	5.65685i		−	0.293689i
$372$	3.41421	+	0.585786i	0.177019	+	0.0303716i
$373$	−4.00000	+	4.00000i	−0.207112	+	0.207112i	−0.803039	−	0.595927i	$-0.796785\pi$
$373$	−4.00000	+	4.00000i	−0.207112	+	0.207112i	0.595927	+	0.803039i	$0.296785\pi$
$374$	2.82843			0.146254
$375$		0			0
$376$		0			0
$377$		0			0
$378$	−3.58579	+	6.41421i	−0.184433	+	0.329912i
$379$			16.0000i			0.821865i	0.911666	+	0.410932i	$0.134797\pi$
$379$			16.0000i			0.821865i	−0.911666	+	0.410932i	$0.865203\pi$
$380$		0			0
$381$	14.0000	−	9.89949i	0.717242	−	0.507166i
$382$	2.00000	+	2.00000i	0.102329	+	0.102329i
$383$	−16.9706	−	16.9706i	−0.867155	−	0.867155i	0.125001	−	0.992157i	$-0.460106\pi$
$383$	−16.9706	−	16.9706i	−0.867155	−	0.867155i	−0.992157	+	0.125001i	$0.960106\pi$
$384$	1.41421	−	1.00000i	0.0721688	−	0.0510310i
$385$		0			0
$386$		−	21.2132i		−	1.07972i
$387$	22.9706	−	10.9706i	1.16766	−	0.557665i
$388$	−3.00000	+	3.00000i	−0.152302	+	0.152302i
$389$	−4.24264			−0.215110			−0.107555	−	0.994199i	$-0.534302\pi$
$389$	−4.24264			−0.215110			−0.107555	+	0.994199i	$0.534302\pi$
$390$		0			0
$391$	8.00000			0.404577
$392$	−3.53553	+	3.53553i	−0.178571	+	0.178571i
$393$	31.3848	+	5.38478i	1.58315	+	0.271626i
$394$		−	24.0000i		−	1.20910i
$395$		0			0
$396$	−4.00000	−	1.41421i	−0.201008	−	0.0710669i
$397$	−22.0000	−	22.0000i	−1.10415	−	1.10415i	−0.993905	−	0.110244i	$-0.964837\pi$
$397$	−22.0000	−	22.0000i	−1.10415	−	1.10415i	−0.110244	−	0.993905i	$-0.535163\pi$
$398$	−16.9706	−	16.9706i	−0.850657	−	0.850657i
$399$	5.65685	+	8.00000i	0.283197	+	0.400501i
$400$		0			0
$401$			8.48528i			0.423735i	0.977298	+	0.211867i	$0.0679545\pi$
$401$			8.48528i			0.423735i	−0.977298	+	0.211867i	$0.932046\pi$
$402$	1.65685	−	9.65685i	0.0826364	−	0.481640i
$403$		0			0
$404$	−9.89949			−0.492518
$405$		0			0
$406$	10.0000			0.496292
$407$	8.48528	−	8.48528i	0.420600	−	0.420600i
$408$	−0.585786	+	3.41421i	−0.0290008	+	0.169029i
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$	6.00000	+	8.48528i	0.295958	+	0.418548i
$412$	1.00000	+	1.00000i	0.0492665	+	0.0492665i
$413$	−9.89949	−	9.89949i	−0.487122	−	0.487122i
$414$	−11.3137	−	4.00000i	−0.556038	−	0.196589i
$415$		0			0
$416$		0			0
$417$	13.6569	+	2.34315i	0.668779	+	0.114744i
$418$	−4.00000	+	4.00000i	−0.195646	+	0.195646i
$419$	−21.2132			−1.03633			−0.518166	−	0.855280i	$-0.673385\pi$
$419$	−21.2132			−1.03633			−0.518166	+	0.855280i	$0.673385\pi$
$420$		0			0
$421$	14.0000			0.682318			0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000			0.682318			0.341159	+	0.940006i	$0.389181\pi$
$422$	5.65685	−	5.65685i	0.275371	−	0.275371i
$423$		0			0
$424$			4.00000i			0.194257i
$425$		0			0
$426$	−20.0000	+	14.1421i	−0.969003	+	0.685189i
$427$	−6.00000	−	6.00000i	−0.290360	−	0.290360i
$428$	2.82843	+	2.82843i	0.136717	+	0.136717i
$429$		0			0
$430$		0			0
$431$		−	11.3137i		−	0.544962i	−0.962161	−	0.272481i	$-0.912156\pi$
$431$		−	11.3137i		−	0.544962i	0.962161	−	0.272481i	$-0.0878442\pi$
$432$	2.53553	−	4.53553i	0.121991	−	0.218216i
$433$	1.00000	−	1.00000i	0.0480569	−	0.0480569i	−0.682670	−	0.730727i	$-0.739181\pi$
$433$	1.00000	−	1.00000i	0.0480569	−	0.0480569i	0.730727	+	0.682670i	$0.239181\pi$
$434$	−2.82843			−0.135769
$435$		0			0
$436$	−10.0000			−0.478913
$437$	−11.3137	+	11.3137i	−0.541208	+	0.541208i
$438$	−12.0711	−	2.07107i	−0.576778	−	0.0989594i
$439$		−	16.0000i		−	0.763638i	−0.924237	−	0.381819i	$-0.875298\pi$
$439$		−	16.0000i		−	0.763638i	0.924237	−	0.381819i	$-0.124702\pi$
$440$		0			0
$441$	−5.00000	+	14.1421i	−0.238095	+	0.673435i
$442$		0			0
$443$	−4.24264	−	4.24264i	−0.201574	−	0.201574i	0.599100	−	0.800674i	$-0.295525\pi$
$443$	−4.24264	−	4.24264i	−0.201574	−	0.201574i	−0.800674	+	0.599100i	$0.795525\pi$
$444$	8.48528	+	12.0000i	0.402694	+	0.569495i
$445$		0			0
$446$		−	12.7279i		−	0.602685i
$447$	−3.72792	+	21.7279i	−0.176325	+	1.02770i
$448$	−1.00000	+	1.00000i	−0.0472456	+	0.0472456i
$449$	−14.1421			−0.667409			−0.333704	−	0.942678i	$-0.608299\pi$
$449$	−14.1421			−0.667409			−0.333704	+	0.942678i	$0.608299\pi$
$450$		0			0
$451$	−8.00000			−0.376705
$452$	9.89949	−	9.89949i	0.465633	−	0.465633i
$453$	4.68629	−	27.3137i	0.220181	−	1.28331i
$454$		−	22.0000i		−	1.03251i
$455$		0			0
$456$	−4.00000	−	5.65685i	−0.187317	−	0.264906i
$457$	15.0000	+	15.0000i	0.701670	+	0.701670i	0.964769	−	0.263099i	$-0.0847444\pi$
$457$	15.0000	+	15.0000i	0.701670	+	0.701670i	−0.263099	+	0.964769i	$0.584744\pi$
$458$	−4.24264	−	4.24264i	−0.198246	−	0.198246i
$459$	2.82843	+	10.0000i	0.132020	+	0.466760i
$460$		0			0
$461$		−	7.07107i		−	0.329332i	−0.986349	−	0.164666i	$-0.947345\pi$
$461$		−	7.07107i		−	0.329332i	0.986349	−	0.164666i	$-0.0526547\pi$
$462$	3.41421	+	0.585786i	0.158844	+	0.0272533i
$463$	−5.00000	+	5.00000i	−0.232370	+	0.232370i	−0.813681	−	0.581311i	$-0.802540\pi$
$463$	−5.00000	+	5.00000i	−0.232370	+	0.232370i	0.581311	+	0.813681i	$0.302540\pi$
$464$	−7.07107			−0.328266
$465$		0			0
$466$	−18.0000			−0.833834
$467$	−19.7990	+	19.7990i	−0.916188	+	0.916188i	−0.996750	−	0.0805616i	$-0.974329\pi$
$467$	−19.7990	+	19.7990i	−0.916188	+	0.916188i	0.0805616	+	0.996750i	$0.474329\pi$
$468$		0			0
$469$			8.00000i			0.369406i
$470$		0			0
$471$		0			0
$472$	7.00000	+	7.00000i	0.322201	+	0.322201i
$473$	−8.48528	−	8.48528i	−0.390154	−	0.390154i
$474$	8.48528	−	6.00000i	0.389742	−	0.275589i
$475$		0			0
$476$		−	2.82843i		−	0.129641i
$477$	5.17157	+	10.8284i	0.236790	+	0.495800i
$478$	6.00000	−	6.00000i	0.274434	−	0.274434i
$479$	−31.1127			−1.42158			−0.710788	−	0.703407i	$-0.751661\pi$
$479$	−31.1127			−1.42158			−0.710788	+	0.703407i	$0.751661\pi$
$480$		0			0
$481$		0			0
$482$	2.82843	−	2.82843i	0.128831	−	0.128831i
$483$	9.65685	+	1.65685i	0.439402	+	0.0753895i
$484$		−	9.00000i		−	0.409091i
$485$		0			0
$486$	1.00000	−	15.5563i	0.0453609	−	0.705650i
$487$	−9.00000	−	9.00000i	−0.407829	−	0.407829i	0.473152	−	0.880981i	$-0.343116\pi$
$487$	−9.00000	−	9.00000i	−0.407829	−	0.407829i	−0.880981	+	0.473152i	$0.843116\pi$
$488$	4.24264	+	4.24264i	0.192055	+	0.192055i
$489$	−5.65685	−	8.00000i	−0.255812	−	0.361773i
$490$		0			0
$491$		−	26.8701i		−	1.21263i	−0.795225	−	0.606314i	$-0.792647\pi$
$491$		−	26.8701i		−	1.21263i	0.795225	−	0.606314i	$-0.207353\pi$
$492$	1.65685	−	9.65685i	0.0746968	−	0.435365i
$493$	10.0000	−	10.0000i	0.450377	−	0.450377i
$494$		0			0
$495$		0			0
$496$	2.00000			0.0898027
$497$	14.1421	−	14.1421i	0.634361	−	0.634361i
$498$	−3.51472	+	20.4853i	−0.157498	+	0.917967i
$499$			20.0000i			0.895323i	0.894203	+	0.447661i	$0.147743\pi$
$499$			20.0000i			0.895323i	−0.894203	+	0.447661i	$0.852257\pi$
$500$		0			0
$501$	8.00000	+	11.3137i	0.357414	+	0.505459i
$502$	−9.00000	−	9.00000i	−0.401690	−	0.401690i
$503$	8.48528	+	8.48528i	0.378340	+	0.378340i	0.870503	−	0.492163i	$-0.163794\pi$
$503$	8.48528	+	8.48528i	0.378340	+	0.378340i	−0.492163	+	0.870503i	$0.663794\pi$
$504$	−1.41421	+	4.00000i	−0.0629941	+	0.178174i
$505$		0			0
$506$			5.65685i			0.251478i
$507$	22.1924	+	3.80761i	0.985599	+	0.169102i
$508$	7.00000	−	7.00000i	0.310575	−	0.310575i
$509$	24.0416			1.06563			0.532813	−	0.846233i	$-0.321135\pi$
$509$	24.0416			1.06563			0.532813	+	0.846233i	$0.321135\pi$
$510$		0			0
$511$	10.0000			0.442374
$512$	0.707107	−	0.707107i	0.0312500	−	0.0312500i
$513$	−18.1421	−	10.1421i	−0.800995	−	0.447786i
$514$			14.0000i			0.617514i
$515$		0			0
$516$	12.0000	−	8.48528i	0.528271	−	0.373544i
$517$		0			0
$518$	−8.48528	−	8.48528i	−0.372822	−	0.372822i
$519$	−2.82843	+	2.00000i	−0.124154	+	0.0877903i
$520$		0			0
$521$			25.4558i			1.11524i	0.830096	+	0.557620i	$0.188286\pi$
$521$			25.4558i			1.11524i	−0.830096	+	0.557620i	$0.811714\pi$
$522$	−19.1421	+	9.14214i	−0.837829	+	0.400140i
$523$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$523$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$524$	18.3848			0.803143
$525$		0			0
$526$	8.00000			0.348817
$527$	−2.82843	+	2.82843i	−0.123208	+	0.123208i
$528$	−2.41421	−	0.414214i	−0.105065	−	0.0180263i
$529$		−	7.00000i		−	0.304348i
$530$		0			0
$531$	28.0000	+	9.89949i	1.21510	+	0.429601i
$532$	4.00000	+	4.00000i	0.173422	+	0.173422i
$533$		0			0
$534$	2.82843	+	4.00000i	0.122398	+	0.173097i
$535$		0			0
$536$		−	5.65685i		−	0.244339i
$537$	5.38478	−	31.3848i	0.232370	−	1.35435i
$538$	11.0000	−	11.0000i	0.474244	−	0.474244i
$539$	7.07107			0.304572
$540$		0			0
$541$	−2.00000			−0.0859867			−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000			−0.0859867			−0.0429934	+	0.999075i	$0.513689\pi$
$542$		0			0
$543$	−6.44365	+	37.5563i	−0.276524	+	1.61170i
$544$			2.00000i			0.0857493i
$545$		0			0
$546$		0			0
$547$	−6.00000	−	6.00000i	−0.256541	−	0.256541i	0.567104	−	0.823646i	$-0.308064\pi$
$547$	−6.00000	−	6.00000i	−0.256541	−	0.256541i	−0.823646	+	0.567104i	$0.808064\pi$
$548$	4.24264	+	4.24264i	0.181237	+	0.181237i
$549$	16.9706	+	6.00000i	0.724286	+	0.256074i
$550$		0			0
$551$			28.2843i			1.20495i
$552$	−6.82843	−	1.17157i	−0.290637	−	0.0498655i
$553$	−6.00000	+	6.00000i	−0.255146	+	0.255146i
$554$	8.48528			0.360505
$555$		0			0
$556$	8.00000			0.339276
$557$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$557$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$558$	5.41421	−	2.58579i	0.229202	−	0.109465i
$559$		0			0
$560$		0			0
$561$	4.00000	−	2.82843i	0.168880	−	0.119416i
$562$	−6.00000	−	6.00000i	−0.253095	−	0.253095i
$563$	21.2132	+	21.2132i	0.894030	+	0.894030i	0.994900	−	0.100870i	$-0.0321625\pi$
$563$	21.2132	+	21.2132i	0.894030	+	0.894030i	−0.100870	+	0.994900i	$0.532163\pi$
$564$		0			0
$565$		0			0
$566$			28.2843i			1.18888i
$567$	1.34315	+	12.6569i	0.0564068	+	0.531538i
$568$	−10.0000	+	10.0000i	−0.419591	+	0.419591i
$569$	14.1421			0.592869			0.296435	−	0.955053i	$-0.404202\pi$
$569$	14.1421			0.592869			0.296435	+	0.955053i	$0.404202\pi$
$570$		0			0
$571$	−40.0000			−1.67395			−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000			−1.67395			−0.836974	+	0.547243i	$0.815677\pi$
$572$		0			0
$573$	4.82843	+	0.828427i	0.201710	+	0.0346080i
$574$			8.00000i			0.333914i
$575$		0			0
$576$	1.00000	−	2.82843i	0.0416667	−	0.117851i
$577$	−13.0000	−	13.0000i	−0.541197	−	0.541197i	0.382683	−	0.923880i	$-0.375000\pi$
$577$	−13.0000	−	13.0000i	−0.541197	−	0.541197i	−0.923880	+	0.382683i	$0.875000\pi$
$578$	9.19239	+	9.19239i	0.382353	+	0.382353i
$579$	−21.2132	−	30.0000i	−0.881591	−	1.24676i
$580$		0			0
$581$		−	16.9706i		−	0.704058i
$582$	−1.24264	+	7.24264i	−0.0515091	+	0.300217i
$583$	4.00000	−	4.00000i	0.165663	−	0.165663i
$584$	−7.07107			−0.292603
$585$		0			0
$586$		0			0
$587$	−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	$0.983431\pi$
$588$	−1.46447	+	8.53553i	−0.0603936	+	0.351999i
$589$		−	8.00000i		−	0.329634i
$590$		0			0
$591$	−24.0000	−	33.9411i	−0.987228	−	1.39615i
$592$	6.00000	+	6.00000i	0.246598	+	0.246598i
$593$	15.5563	+	15.5563i	0.638823	+	0.638823i	0.950265	−	0.311442i	$-0.100812\pi$
$593$	15.5563	+	15.5563i	0.638823	+	0.638823i	−0.311442	+	0.950265i	$0.600812\pi$
$594$	−7.07107	+	2.00000i	−0.290129	+	0.0820610i
$595$		0			0
$596$			12.7279i			0.521356i
$597$	−40.9706	−	7.02944i	−1.67681	−	0.287696i
$598$		0			0
$599$	45.2548			1.84906			0.924531	−	0.381106i	$-0.124457\pi$
$599$	45.2548			1.84906			0.924531	+	0.381106i	$0.124457\pi$
$600$		0			0
$601$	−10.0000			−0.407909			−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000			−0.407909			−0.203954	+	0.978980i	$0.565379\pi$
$602$	−8.48528	+	8.48528i	−0.345834	+	0.345834i
$603$	−7.31371	−	15.3137i	−0.297837	−	0.623622i
$604$		−	16.0000i		−	0.651031i
$605$		0			0
$606$	−14.0000	+	9.89949i	−0.568711	+	0.402139i
$607$	−3.00000	−	3.00000i	−0.121766	−	0.121766i	0.643598	−	0.765364i	$-0.277441\pi$
$607$	−3.00000	−	3.00000i	−0.121766	−	0.121766i	−0.765364	+	0.643598i	$0.777441\pi$
$608$	−2.82843	−	2.82843i	−0.114708	−	0.114708i
$609$	14.1421	−	10.0000i	0.573068	−	0.405220i
$610$		0			0
$611$		0			0
$612$	2.58579	+	5.41421i	0.104524	+	0.218857i
$613$	−18.0000	+	18.0000i	−0.727013	+	0.727013i	−0.970024	−	0.243011i	$-0.921865\pi$
$613$	−18.0000	+	18.0000i	−0.727013	+	0.727013i	0.243011	+	0.970024i	$0.421865\pi$
$614$	−25.4558			−1.02731
$615$		0			0
$616$	2.00000			0.0805823
$617$	21.2132	−	21.2132i	0.854011	−	0.854011i	−0.136613	−	0.990624i	$-0.543622\pi$
$617$	21.2132	−	21.2132i	0.854011	−	0.854011i	0.990624	+	0.136613i	$0.0436217\pi$
$618$	2.41421	+	0.414214i	0.0971139	+	0.0166621i
$619$			24.0000i			0.964641i	0.875995	+	0.482321i	$0.160206\pi$
$619$			24.0000i			0.964641i	−0.875995	+	0.482321i	$0.839794\pi$
$620$		0			0
$621$	−20.0000	+	5.65685i	−0.802572	+	0.227002i
$622$	14.0000	+	14.0000i	0.561349	+	0.561349i
$623$	−2.82843	−	2.82843i	−0.113319	−	0.113319i
$624$		0			0
$625$		0			0
$626$			12.7279i			0.508710i
$627$	−1.65685	+	9.65685i	−0.0661684	+	0.385658i
$628$		0			0
$629$	−16.9706			−0.676661
$630$		0			0
$631$	−14.0000			−0.557331			−0.278666	−	0.960388i	$-0.589892\pi$
$631$	−14.0000			−0.557331			−0.278666	+	0.960388i	$0.589892\pi$
$632$	4.24264	−	4.24264i	0.168763	−	0.168763i
$633$	2.34315	−	13.6569i	0.0931317	−	0.542811i
$634$			18.0000i			0.714871i
$635$		0			0
$636$	4.00000	+	5.65685i	0.158610	+	0.224309i
$637$		0			0
$638$	7.07107	+	7.07107i	0.279946	+	0.279946i
$639$	−14.1421	+	40.0000i	−0.559454	+	1.58238i
$640$		0			0
$641$		−	5.65685i		−	0.223432i	−0.993740	−	0.111716i	$-0.964365\pi$
$641$		−	5.65685i		−	0.223432i	0.993740	−	0.111716i	$-0.0356347\pi$
$642$	6.82843	+	1.17157i	0.269497	+	0.0462383i
$643$	−24.0000	+	24.0000i	−0.946468	+	0.946468i	−0.998638	−	0.0521706i	$-0.983386\pi$
$643$	−24.0000	+	24.0000i	−0.946468	+	0.946468i	0.0521706	+	0.998638i	$0.483386\pi$
$644$	5.65685			0.222911
$645$		0			0
$646$	8.00000			0.314756
$647$	19.7990	−	19.7990i	0.778379	−	0.778379i	−0.201176	−	0.979555i	$-0.564476\pi$
$647$	19.7990	−	19.7990i	0.778379	−	0.778379i	0.979555	+	0.201176i	$0.0644765\pi$
$648$	−0.949747	−	8.94975i	−0.0373096	−	0.351579i
$649$		−	14.0000i		−	0.549548i
$650$		0			0
$651$	−4.00000	+	2.82843i	−0.156772	+	0.110855i
$652$	−4.00000	−	4.00000i	−0.156652	−	0.156652i
$653$	4.24264	+	4.24264i	0.166027	+	0.166027i	0.785231	−	0.619203i	$-0.212544\pi$
$653$	4.24264	+	4.24264i	0.166027	+	0.166027i	−0.619203	+	0.785231i	$0.712544\pi$
$654$	−14.1421	+	10.0000i	−0.553001	+	0.391031i
$655$		0			0
$656$		−	5.65685i		−	0.220863i
$657$	−19.1421	+	9.14214i	−0.746806	+	0.356669i
$658$		0			0
$659$	35.3553			1.37725			0.688624	−	0.725118i	$-0.258215\pi$
$659$	35.3553			1.37725			0.688624	+	0.725118i	$0.258215\pi$
$660$		0			0
$661$	50.0000			1.94477			0.972387	−	0.233373i	$-0.0749763\pi$
$661$	50.0000			1.94477			0.972387	+	0.233373i	$0.0749763\pi$
$662$	−5.65685	+	5.65685i	−0.219860	+	0.219860i
$663$		0			0
$664$			12.0000i			0.465690i
$665$		0			0
$666$	24.0000	+	8.48528i	0.929981	+	0.328798i
$667$	20.0000	+	20.0000i	0.774403	+	0.774403i
$668$	5.65685	+	5.65685i	0.218870	+	0.218870i
$669$	−12.7279	−	18.0000i	−0.492090	−	0.695920i
$670$		0			0
$671$		−	8.48528i		−	0.327571i
$672$	−0.414214	+	2.41421i	−0.0159786	+	0.0931303i
$673$	−13.0000	+	13.0000i	−0.501113	+	0.501113i	−0.911784	−	0.410671i	$-0.865295\pi$
$673$	−13.0000	+	13.0000i	−0.501113	+	0.501113i	0.410671	+	0.911784i	$0.365295\pi$
$674$	12.7279			0.490261
$675$		0			0
$676$	13.0000			0.500000
$677$	18.3848	−	18.3848i	0.706584	−	0.706584i	−0.259231	−	0.965815i	$-0.583469\pi$
$677$	18.3848	−	18.3848i	0.706584	−	0.706584i	0.965815	+	0.259231i	$0.0834691\pi$
$678$	4.10051	−	23.8995i	0.157479	−	0.917855i
$679$		−	6.00000i		−	0.230259i
$680$		0			0
$681$	−22.0000	−	31.1127i	−0.843042	−	1.19224i
$682$	−2.00000	−	2.00000i	−0.0765840	−	0.0765840i
$683$	−25.4558	−	25.4558i	−0.974041	−	0.974041i	0.0256307	−	0.999671i	$-0.491841\pi$
$683$	−25.4558	−	25.4558i	−0.974041	−	0.974041i	−0.999671	+	0.0256307i	$0.991841\pi$
$684$	−11.3137	−	4.00000i	−0.432590	−	0.152944i
$685$		0			0
$686$		−	16.9706i		−	0.647939i
$687$	−10.2426	−	1.75736i	−0.390781	−	0.0670474i
$688$	6.00000	−	6.00000i	0.228748	−	0.228748i
$689$		0			0
$690$		0			0
$691$	12.0000			0.456502			0.228251	−	0.973602i	$-0.426699\pi$
$691$	12.0000			0.456502			0.228251	+	0.973602i	$0.426699\pi$
$692$	−1.41421	+	1.41421i	−0.0537603	+	0.0537603i
$693$	5.41421	−	2.58579i	0.205669	−	0.0982259i
$694$		−	14.0000i		−	0.531433i
$695$		0			0
$696$	−10.0000	+	7.07107i	−0.379049	+	0.268028i
$697$	8.00000	+	8.00000i	0.303022	+	0.303022i
$698$	−1.41421	−	1.41421i	−0.0535288	−	0.0535288i
$699$	−25.4558	+	18.0000i	−0.962828	+	0.680823i
$700$		0			0
$701$			26.8701i			1.01487i	0.861691	+	0.507434i	$0.169406\pi$
$701$			26.8701i			1.01487i	−0.861691	+	0.507434i	$0.830594\pi$
$702$		0			0
$703$	24.0000	−	24.0000i	0.905177	−	0.905177i
$704$	−1.41421			−0.0533002
$705$		0			0
$706$	18.0000			0.677439
$707$	9.89949	−	9.89949i	0.372309	−	0.372309i
$708$	16.8995	+	2.89949i	0.635122	+	0.108970i
$709$			10.0000i			0.375558i	0.982211	+	0.187779i	$0.0601289\pi$
$709$			10.0000i			0.375558i	−0.982211	+	0.187779i	$0.939871\pi$
$710$		0			0
$711$	6.00000	−	16.9706i	0.225018	−	0.636446i
$712$	2.00000	+	2.00000i	0.0749532	+	0.0749532i
$713$	−5.65685	−	5.65685i	−0.211851	−	0.211851i
$714$	−2.82843	−	4.00000i	−0.105851	−	0.149696i
$715$		0			0
$716$		−	18.3848i		−	0.687071i
$717$	2.48528	−	14.4853i	0.0928145	−	0.540963i
$718$	−8.00000	+	8.00000i	−0.298557	+	0.298557i
$719$	−22.6274			−0.843860			−0.421930	−	0.906628i	$-0.638647\pi$
$719$	−22.6274			−0.843860			−0.421930	+	0.906628i	$0.638647\pi$
$720$		0			0
$721$	−2.00000			−0.0744839
$722$	2.12132	−	2.12132i	0.0789474	−	0.0789474i
$723$	1.17157	−	6.82843i	0.0435713	−	0.253952i
$724$			22.0000i			0.817624i
$725$		0			0
$726$	−9.00000	−	12.7279i	−0.334021	−	0.472377i
$727$	3.00000	+	3.00000i	0.111264	+	0.111264i	0.760547	−	0.649283i	$-0.224931\pi$
$727$	3.00000	+	3.00000i	0.111264	+	0.111264i	−0.649283	+	0.760547i	$0.724931\pi$
$728$		0			0
$729$	−14.1421	−	23.0000i	−0.523783	−	0.851852i
$730$		0			0
$731$			16.9706i			0.627679i
$732$	10.2426	+	1.75736i	0.378579	+	0.0649539i
$733$	26.0000	−	26.0000i	0.960332	−	0.960332i	−0.0389108	−	0.999243i	$-0.512389\pi$
$733$	26.0000	−	26.0000i	0.960332	−	0.960332i	0.999243	+	0.0389108i	$0.0123888\pi$
$734$	26.8701			0.991792
$735$		0			0
$736$	−4.00000			−0.147442
$737$	−5.65685	+	5.65685i	−0.208373	+	0.208373i
$738$	−7.31371	−	15.3137i	−0.269221	−	0.563705i
$739$		−	40.0000i		−	1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$		−	40.0000i		−	1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$		0			0
$741$		0			0
$742$	−4.00000	−	4.00000i	−0.146845	−	0.146845i
$743$	−36.7696	−	36.7696i	−1.34894	−	1.34894i	−0.886810	−	0.462134i	$-0.847084\pi$
$743$	−36.7696	−	36.7696i	−1.34894	−	1.34894i	−0.462134	−	0.886810i	$-0.652916\pi$
$744$	2.82843	−	2.00000i	0.103695	−	0.0733236i
$745$		0			0
$746$			5.65685i			0.207112i
$747$	15.5147	+	32.4853i	0.567654	+	1.18857i
$748$	2.00000	−	2.00000i	0.0731272	−	0.0731272i
$749$	−5.65685			−0.206697
$750$		0			0
$751$	−30.0000			−1.09472			−0.547358	−	0.836899i	$-0.684366\pi$
$751$	−30.0000			−1.09472			−0.547358	+	0.836899i	$0.684366\pi$
$752$		0			0
$753$	−21.7279	−	3.72792i	−0.791809	−	0.135853i
$754$		0			0
$755$		0			0
$756$	2.00000	+	7.07107i	0.0727393	+	0.257172i
$757$	30.0000	+	30.0000i	1.09037	+	1.09037i	0.995489	+	0.0948798i	$0.0302467\pi$
$757$	30.0000	+	30.0000i	1.09037	+	1.09037i	0.0948798	+	0.995489i	$0.469753\pi$
$758$	11.3137	+	11.3137i	0.410932	+	0.410932i
$759$	5.65685	+	8.00000i	0.205331	+	0.290382i
$760$		0			0
$761$		−	36.7696i		−	1.33290i	−0.745552	−	0.666448i	$-0.767814\pi$
$761$		−	36.7696i		−	1.33290i	0.745552	−	0.666448i	$-0.232186\pi$
$762$	2.89949	−	16.8995i	0.105038	−	0.612204i
$763$	10.0000	−	10.0000i	0.362024	−	0.362024i
$764$	2.82843			0.102329
$765$		0			0
$766$	−24.0000			−0.867155
$767$		0			0
$768$	0.292893	−	1.70711i	0.0105689	−	0.0615999i
$769$		−	22.0000i		−	0.793340i	−0.917961	−	0.396670i	$-0.870166\pi$
$769$		−	22.0000i		−	0.793340i	0.917961	−	0.396670i	$-0.129834\pi$
$770$		0			0
$771$	14.0000	+	19.7990i	0.504198	+	0.713043i
$772$	−15.0000	−	15.0000i	−0.539862	−	0.539862i
$773$	29.6985	+	29.6985i	1.06818	+	1.06818i	0.997499	+	0.0706813i	$0.0225173\pi$
$773$	29.6985	+	29.6985i	1.06818	+	1.06818i	0.0706813	+	0.997499i	$0.477483\pi$
$774$	8.48528	−	24.0000i	0.304997	−	0.862662i
$775$		0			0
$776$			4.24264i			0.152302i
$777$	−20.4853	−	3.51472i	−0.734905	−	0.126090i
$778$	−3.00000	+	3.00000i	−0.107555	+	0.107555i
$779$	−22.6274			−0.810711
$780$		0			0
$781$	20.0000			0.715656
$782$	5.65685	−	5.65685i	0.202289	−	0.202289i
$783$	−17.9289	+	32.0711i	−0.640728	+	1.14613i
$784$			5.00000i			0.178571i
$785$		0			0
$786$	26.0000	−	18.3848i	0.927389	−	0.655763i
$787$	−28.0000	−	28.0000i	−0.998092	−	0.998092i	0.00190598	−	0.999998i	$-0.499393\pi$
$787$	−28.0000	−	28.0000i	−0.998092	−	0.998092i	−0.999998	+	0.00190598i	$0.999393\pi$
$788$	−16.9706	−	16.9706i	−0.604551	−	0.604551i
$789$	11.3137	−	8.00000i	0.402779	−	0.284808i
$790$		0			0
$791$			19.7990i			0.703971i
$792$	−3.82843	+	1.82843i	−0.136037	+	0.0649703i
$793$		0			0
$794$	−31.1127			−1.10415
$795$		0			0
$796$	−24.0000			−0.850657
$797$	12.7279	−	12.7279i	0.450846	−	0.450846i	−0.444789	−	0.895635i	$-0.646721\pi$
$797$	12.7279	−	12.7279i	0.450846	−	0.450846i	0.895635	+	0.444789i	$0.146721\pi$
$798$	9.65685	+	1.65685i	0.341849	+	0.0586520i
$799$		0			0
$800$		0			0
$801$	8.00000	+	2.82843i	0.282666	+	0.0999376i
$802$	6.00000	+	6.00000i	0.211867	+	0.211867i
$803$	7.07107	+	7.07107i	0.249533	+	0.249533i
$804$	−5.65685	−	8.00000i	−0.199502	−	0.282138i
$805$		0			0
$806$		0			0
$807$	4.55635	−	26.5563i	0.160391	−	0.934828i
$808$	−7.00000	+	7.00000i	−0.246259	+	0.246259i
$809$	22.6274			0.795538			0.397769	−	0.917486i	$-0.369785\pi$
$809$	22.6274			0.795538			0.397769	+	0.917486i	$0.369785\pi$
$810$		0			0
$811$	−4.00000			−0.140459			−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000			−0.140459			−0.0702295	+	0.997531i	$0.522373\pi$
$812$	7.07107	−	7.07107i	0.248146	−	0.248146i
$813$		0			0
$814$		−	12.0000i		−	0.420600i
$815$		0			0
$816$	2.00000	+	2.82843i	0.0700140	+	0.0990148i
$817$	−24.0000	−	24.0000i	−0.839654	−	0.839654i
$818$		0			0
$819$		0			0
$820$		0			0
$821$			43.8406i			1.53005i	0.644002	+	0.765024i	$0.277273\pi$
$821$			43.8406i			1.53005i	−0.644002	+	0.765024i	$0.722727\pi$
$822$	10.2426	+	1.75736i	0.357253	+	0.0612949i
$823$	25.0000	−	25.0000i	0.871445	−	0.871445i	−0.121185	−	0.992630i	$-0.538669\pi$
$823$	25.0000	−	25.0000i	0.871445	−	0.871445i	0.992630	+	0.121185i	$0.0386693\pi$
$824$	1.41421			0.0492665
$825$		0			0
$826$	−14.0000			−0.487122
$827$	−12.7279	+	12.7279i	−0.442593	+	0.442593i	−0.892883	−	0.450289i	$-0.851321\pi$
$827$	−12.7279	+	12.7279i	−0.442593	+	0.442593i	0.450289	+	0.892883i	$0.351321\pi$
$828$	−10.8284	+	5.17157i	−0.376314	+	0.179725i
$829$			14.0000i			0.486240i	0.969996	+	0.243120i	$0.0781709\pi$
$829$			14.0000i			0.486240i	−0.969996	+	0.243120i	$0.921829\pi$
$830$		0			0
$831$	12.0000	−	8.48528i	0.416275	−	0.294351i
$832$		0			0
$833$	−7.07107	−	7.07107i	−0.244998	−	0.244998i
$834$	11.3137	−	8.00000i	0.391762	−	0.277017i
$835$		0			0
$836$			5.65685i			0.195646i
$837$	5.07107	−	9.07107i	0.175282	−	0.313542i
$838$	−15.0000	+	15.0000i	−0.518166	+	0.518166i
$839$	5.65685			0.195296			0.0976481	−	0.995221i	$-0.468868\pi$
$839$	5.65685			0.195296			0.0976481	+	0.995221i	$0.468868\pi$
$840$		0			0
$841$	21.0000			0.724138
$842$	9.89949	−	9.89949i	0.341159	−	0.341159i
$843$	−14.4853	−	2.48528i	−0.498900	−	0.0855976i
$844$		−	8.00000i		−	0.275371i
$845$		0			0
$846$		0			0
$847$	9.00000	+	9.00000i	0.309244	+	0.309244i
$848$	2.82843	+	2.82843i	0.0971286	+	0.0971286i
$849$	28.2843	+	40.0000i	0.970714	+	1.37280i
$850$		0			0
$851$		−	33.9411i		−	1.16349i
$852$	−4.14214	+	24.1421i	−0.141907	+	0.827096i
$853$	2.00000	−	2.00000i	0.0684787	−	0.0684787i	−0.672038	−	0.740517i	$-0.734581\pi$
$853$	2.00000	−	2.00000i	0.0684787	−	0.0684787i	0.740517	+	0.672038i	$0.234581\pi$
$854$	−8.48528			−0.290360
$855$		0			0
$856$	4.00000			0.136717
$857$	−38.1838	+	38.1838i	−1.30433	+	1.30433i	−0.378892	+	0.925441i	$0.623695\pi$
$857$	−38.1838	+	38.1838i	−1.30433	+	1.30433i	−0.925441	+	0.378892i	$0.876305\pi$
$858$		0			0
$859$			32.0000i			1.09183i	0.837842	+	0.545913i	$0.183817\pi$
$859$			32.0000i			1.09183i	−0.837842	+	0.545913i	$0.816183\pi$
$860$		0			0
$861$	8.00000	+	11.3137i	0.272639	+	0.385570i
$862$	−8.00000	−	8.00000i	−0.272481	−	0.272481i
$863$	36.7696	+	36.7696i	1.25165	+	1.25165i	0.954980	+	0.296670i	$0.0958762\pi$
$863$	36.7696	+	36.7696i	1.25165	+	1.25165i	0.296670	+	0.954980i	$0.404124\pi$
$864$	−1.41421	−	5.00000i	−0.0481125	−	0.170103i
$865$		0			0
$866$		−	1.41421i		−	0.0480569i
$867$	22.1924	+	3.80761i	0.753693	+	0.129313i
$868$	−2.00000	+	2.00000i	−0.0678844	+	0.0678844i
$869$	−8.48528			−0.287843
$870$		0			0
$871$		0			0
$872$	−7.07107	+	7.07107i	−0.239457	+	0.239457i
$873$	5.48528	+	11.4853i	0.185649	+	0.388718i
$874$			16.0000i			0.541208i
$875$		0			0
$876$	−10.0000	+	7.07107i	−0.337869	+	0.238909i
$877$	36.0000	+	36.0000i	1.21563	+	1.21563i	0.969146	+	0.246488i	$0.0792765\pi$
$877$	36.0000	+	36.0000i	1.21563	+	1.21563i	0.246488	+	0.969146i	$0.420724\pi$
$878$	−11.3137	−	11.3137i	−0.381819	−	0.381819i
$879$		0			0
$880$		0			0
$881$		−	19.7990i		−	0.667045i	−0.942742	−	0.333522i	$-0.891763\pi$
$881$		−	19.7990i		−	0.667045i	0.942742	−	0.333522i	$-0.108237\pi$
$882$	6.46447	+	13.5355i	0.217670	+	0.455765i
$883$	−4.00000	+	4.00000i	−0.134611	+	0.134611i	−0.771202	−	0.636591i	$-0.780344\pi$
$883$	−4.00000	+	4.00000i	−0.134611	+	0.134611i	0.636591	+	0.771202i	$0.280344\pi$
$884$		0			0
$885$		0			0
$886$	−6.00000			−0.201574
$887$	8.48528	−	8.48528i	0.284908	−	0.284908i	−0.550155	−	0.835063i	$-0.685431\pi$
$887$	8.48528	−	8.48528i	0.284908	−	0.284908i	0.835063	+	0.550155i	$0.185431\pi$
$888$	14.4853	+	2.48528i	0.486094	+	0.0834006i
$889$			14.0000i			0.469545i
$890$		0			0
$891$	−8.00000	+	9.89949i	−0.268010	+	0.331646i
$892$	−9.00000	−	9.00000i	−0.301342	−	0.301342i
$893$		0			0
$894$	12.7279	+	18.0000i	0.425685	+	0.602010i
$895$		0			0
$896$			1.41421i			0.0472456i
$897$		0			0
$898$	−10.0000	+	10.0000i	−0.333704	+	0.333704i
$899$	−14.1421			−0.471667
$900$		0			0
$901$	−8.00000			−0.266519
$902$	−5.65685	+	5.65685i	−0.188353	+	0.188353i
$903$	−3.51472	+	20.4853i	−0.116963	+	0.681707i
$904$		−	14.0000i		−	0.465633i
$905$		0			0
$906$	−16.0000	−	22.6274i	−0.531564	−	0.751746i
$907$	−22.0000	−	22.0000i	−0.730498	−	0.730498i	0.240220	−	0.970718i	$-0.422780\pi$
$907$	−22.0000	−	22.0000i	−0.730498	−	0.730498i	−0.970718	+	0.240220i	$0.922780\pi$
$908$	−15.5563	−	15.5563i	−0.516256	−	0.516256i
$909$	−9.89949	+	28.0000i	−0.328346	+	0.928701i
$910$		0			0
$911$		−	39.5980i		−	1.31194i	−0.754787	−	0.655970i	$-0.772260\pi$
$911$		−	39.5980i		−	1.31194i	0.754787	−	0.655970i	$-0.227740\pi$
$912$	−6.82843	−	1.17157i	−0.226112	−	0.0387947i
$913$	12.0000	−	12.0000i	0.397142	−	0.397142i
$914$	21.2132			0.701670
$915$		0			0
$916$	−6.00000			−0.198246
$917$	−18.3848	+	18.3848i	−0.607119	+	0.607119i
$918$	9.07107	+	5.07107i	0.299390	+	0.167370i
$919$		−	34.0000i		−	1.12156i	−0.827966	−	0.560778i	$-0.810502\pi$
$919$		−	34.0000i		−	1.12156i	0.827966	−	0.560778i	$-0.189498\pi$
$920$		0			0
$921$	−36.0000	+	25.4558i	−1.18624	+	0.838799i
$922$	−5.00000	−	5.00000i	−0.164666	−	0.164666i
$923$		0			0
$924$	2.82843	−	2.00000i	0.0930484	−	0.0657952i
$925$		0			0
$926$			7.07107i			0.232370i
$927$	3.82843	−	1.82843i	0.125742	−	0.0600534i
$928$	−5.00000	+	5.00000i	−0.164133	+	0.164133i
$929$	2.82843			0.0927977			0.0463988	−	0.998923i	$-0.485225\pi$
$929$	2.82843			0.0927977			0.0463988	+	0.998923i	$0.485225\pi$
$930$		0			0
$931$	20.0000			0.655474
$932$	−12.7279	+	12.7279i	−0.416917	+	0.416917i
$933$	33.7990	+	5.79899i	1.10653	+	0.189850i
$934$			28.0000i			0.916188i
$935$		0			0
$936$		0			0
$937$	5.00000	+	5.00000i	0.163343	+	0.163343i	0.784046	−	0.620703i	$-0.213153\pi$
$937$	5.00000	+	5.00000i	0.163343	+	0.163343i	−0.620703	+	0.784046i	$0.713153\pi$
$938$	5.65685	+	5.65685i	0.184703	+	0.184703i
$939$	12.7279	+	18.0000i	0.415360	+	0.587408i
$940$		0			0
$941$			12.7279i			0.414918i	0.978244	+	0.207459i	$0.0665194\pi$
$941$			12.7279i			0.414918i	−0.978244	+	0.207459i	$0.933481\pi$
$942$		0			0
$943$	−16.0000	+	16.0000i	−0.521032	+	0.521032i
$944$	9.89949			0.322201
$945$		0			0
$946$	−12.0000			−0.390154
$947$	−18.3848	+	18.3848i	−0.597425	+	0.597425i	−0.939627	−	0.342202i	$-0.888827\pi$
$947$	−18.3848	+	18.3848i	−0.597425	+	0.597425i	0.342202	+	0.939627i	$0.388827\pi$
$948$	1.75736	−	10.2426i	0.0570764	−	0.332666i
$949$		0			0
$950$		0			0
$951$	18.0000	+	25.4558i	0.583690	+	0.825462i
$952$	−2.00000	−	2.00000i	−0.0648204	−	0.0648204i
$953$	−4.24264	−	4.24264i	−0.137433	−	0.137433i	0.635044	−	0.772476i	$-0.280982\pi$
$953$	−4.24264	−	4.24264i	−0.137433	−	0.137433i	−0.772476	+	0.635044i	$0.780982\pi$
$954$	11.3137	+	4.00000i	0.366295	+	0.129505i
$955$		0			0
$956$		−	8.48528i		−	0.274434i
$957$	17.0711	+	2.92893i	0.551829	+	0.0946789i
$958$	−22.0000	+	22.0000i	−0.710788	+	0.710788i
$959$	−8.48528			−0.274004
$960$		0			0
$961$	−27.0000			−0.870968
$962$		0			0
$963$	10.8284	−	5.17157i	0.348941	−	0.166652i
$964$		−	4.00000i		−	0.128831i
$965$		0			0
$966$	8.00000	−	5.65685i	0.257396	−	0.182006i
$967$	−19.0000	−	19.0000i	−0.610999	−	0.610999i	0.332208	−	0.943206i	$-0.392207\pi$
$967$	−19.0000	−	19.0000i	−0.610999	−	0.610999i	−0.943206	+	0.332208i	$0.892207\pi$
$968$	−6.36396	−	6.36396i	−0.204545	−	0.204545i
$969$	11.3137	−	8.00000i	0.363449	−	0.256997i
$970$		0			0
$971$			41.0122i			1.31614i	0.752955	+	0.658072i	$0.228628\pi$
$971$			41.0122i			1.31614i	−0.752955	+	0.658072i	$0.771372\pi$
$972$	−10.2929	−	11.7071i	−0.330145	−	0.375506i
$973$	−8.00000	+	8.00000i	−0.256468	+	0.256468i
$974$	−12.7279			−0.407829
$975$		0			0
$976$	6.00000			0.192055
$977$	4.24264	−	4.24264i	0.135734	−	0.135734i	−0.635975	−	0.771709i	$-0.719402\pi$
$977$	4.24264	−	4.24264i	0.135734	−	0.135734i	0.771709	+	0.635975i	$0.219402\pi$
$978$	−9.65685	−	1.65685i	−0.308792	−	0.0529804i
$979$		−	4.00000i		−	0.127841i
$980$		0			0
$981$	−10.0000	+	28.2843i	−0.319275	+	0.903047i
$982$	−19.0000	−	19.0000i	−0.606314	−	0.606314i
$983$	14.1421	+	14.1421i	0.451064	+	0.451064i	0.895708	−	0.444644i	$-0.146670\pi$
$983$	14.1421	+	14.1421i	0.451064	+	0.451064i	−0.444644	+	0.895708i	$0.646670\pi$
$984$	−5.65685	−	8.00000i	−0.180334	−	0.255031i
$985$		0			0
$986$		−	14.1421i		−	0.450377i
$987$		0			0
$988$		0			0
$989$	−33.9411			−1.07927
$990$		0			0
$991$	40.0000			1.27064			0.635321	−	0.772248i	$-0.280868\pi$
$991$	40.0000			1.27064			0.635321	+	0.772248i	$0.280868\pi$
$992$	1.41421	−	1.41421i	0.0449013	−	0.0449013i
$993$	−2.34315	+	13.6569i	−0.0743575	+	0.433387i
$994$		−	20.0000i		−	0.634361i
$995$		0			0
$996$	12.0000	+	16.9706i	0.380235	+	0.537733i
$997$	8.00000	+	8.00000i	0.253363	+	0.253363i	0.822348	−	0.568985i	$-0.192664\pi$
$997$	8.00000	+	8.00000i	0.253363	+	0.253363i	−0.568985	+	0.822348i	$0.692664\pi$
$998$	14.1421	+	14.1421i	0.447661	+	0.447661i
$999$	42.4264	−	12.0000i	1.34231	−	0.379663i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	150.2.e.a.107.2		4
3.2	odd	2	inner	150.2.e.a.107.1		4
4.3	odd	2		1200.2.v.b.257.2		4
5.2	odd	4		30.2.e.a.23.2	yes	4
5.3	odd	4	inner	150.2.e.a.143.1		4
5.4	even	2		30.2.e.a.17.1	✓	4
12.11	even	2		1200.2.v.b.257.1		4
15.2	even	4		30.2.e.a.23.1	yes	4
15.8	even	4	inner	150.2.e.a.143.2		4
15.14	odd	2		30.2.e.a.17.2	yes	4
20.3	even	4		1200.2.v.b.593.1		4
20.7	even	4		240.2.v.e.113.2		4
20.19	odd	2		240.2.v.e.17.1		4
40.19	odd	2		960.2.v.c.257.2		4
40.27	even	4		960.2.v.c.833.1		4
40.29	even	2		960.2.v.k.257.1		4
40.37	odd	4		960.2.v.k.833.2		4
45.2	even	12		810.2.m.f.53.1		8
45.4	even	6		810.2.m.f.107.1		8
45.7	odd	12		810.2.m.f.53.2		8
45.14	odd	6		810.2.m.f.107.2		8
45.22	odd	12		810.2.m.f.593.1		8
45.29	odd	6		810.2.m.f.377.1		8
45.32	even	12		810.2.m.f.593.2		8
45.34	even	6		810.2.m.f.377.2		8
60.23	odd	4		1200.2.v.b.593.2		4
60.47	odd	4		240.2.v.e.113.1		4
60.59	even	2		240.2.v.e.17.2		4
120.29	odd	2		960.2.v.k.257.2		4
120.59	even	2		960.2.v.c.257.1		4
120.77	even	4		960.2.v.k.833.1		4
120.107	odd	4		960.2.v.c.833.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
30.2.e.a.17.1	✓	4	5.4	even	2
30.2.e.a.17.2	yes	4	15.14	odd	2
30.2.e.a.23.1	yes	4	15.2	even	4
30.2.e.a.23.2	yes	4	5.2	odd	4
150.2.e.a.107.1		4	3.2	odd	2	inner
150.2.e.a.107.2		4	1.1	even	1	trivial
150.2.e.a.143.1		4	5.3	odd	4	inner
150.2.e.a.143.2		4	15.8	even	4	inner
240.2.v.e.17.1		4	20.19	odd	2
240.2.v.e.17.2		4	60.59	even	2
240.2.v.e.113.1		4	60.47	odd	4
240.2.v.e.113.2		4	20.7	even	4
810.2.m.f.53.1		8	45.2	even	12
810.2.m.f.53.2		8	45.7	odd	12
810.2.m.f.107.1		8	45.4	even	6
810.2.m.f.107.2		8	45.14	odd	6
810.2.m.f.377.1		8	45.29	odd	6
810.2.m.f.377.2		8	45.34	even	6
810.2.m.f.593.1		8	45.22	odd	12
810.2.m.f.593.2		8	45.32	even	12
960.2.v.c.257.1		4	120.59	even	2
960.2.v.c.257.2		4	40.19	odd	2
960.2.v.c.833.1		4	40.27	even	4
960.2.v.c.833.2		4	120.107	odd	4
960.2.v.k.257.1		4	40.29	even	2
960.2.v.k.257.2		4	120.29	odd	2
960.2.v.k.833.1		4	120.77	even	4
960.2.v.k.833.2		4	40.37	odd	4
1200.2.v.b.257.1		4	12.11	even	2
1200.2.v.b.257.2		4	4.3	odd	2
1200.2.v.b.593.1		4	20.3	even	4
1200.2.v.b.593.2		4	60.23	odd	4

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:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	150.e (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{2} +(0.292893 - 1.70711i) q^{3} -1.00000i q^{4} +(-1.00000 - 1.41421i) q^{6} +(1.00000 + 1.00000i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-2.82843 - 1.00000i) q^{9} +O(q^{10})\)	\(q+(0.707107 - 0.707107i) q^{2} +(0.292893 - 1.70711i) q^{3} -1.00000i q^{4} +(-1.00000 - 1.41421i) q^{6} +(1.00000 + 1.00000i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-2.82843 - 1.00000i) q^{9} +1.41421i q^{11} +(-1.70711 - 0.292893i) q^{12} +1.41421 q^{14} -1.00000 q^{16} +(1.41421 - 1.41421i) q^{17} +(-2.70711 + 1.29289i) q^{18} +4.00000i q^{19} +(2.00000 - 1.41421i) q^{21} +(1.00000 + 1.00000i) q^{22} +(2.82843 + 2.82843i) q^{23} +(-1.41421 + 1.00000i) q^{24} +(-2.53553 + 4.53553i) q^{27} +(1.00000 - 1.00000i) q^{28} +7.07107 q^{29} -2.00000 q^{31} +(-0.707107 + 0.707107i) q^{32} +(2.41421 + 0.414214i) q^{33} -2.00000i q^{34} +(-1.00000 + 2.82843i) q^{36} +(-6.00000 - 6.00000i) q^{37} +(2.82843 + 2.82843i) q^{38} +5.65685i q^{41} +(0.414214 - 2.41421i) q^{42} +(-6.00000 + 6.00000i) q^{43} +1.41421 q^{44} +4.00000 q^{46} +(-0.292893 + 1.70711i) q^{48} -5.00000i q^{49} +(-2.00000 - 2.82843i) q^{51} +(-2.82843 - 2.82843i) q^{53} +(1.41421 + 5.00000i) q^{54} -1.41421i q^{56} +(6.82843 + 1.17157i) q^{57} +(5.00000 - 5.00000i) q^{58} -9.89949 q^{59} -6.00000 q^{61} +(-1.41421 + 1.41421i) q^{62} +(-1.82843 - 3.82843i) q^{63} +1.00000i q^{64} +(2.00000 - 1.41421i) q^{66} +(4.00000 + 4.00000i) q^{67} +(-1.41421 - 1.41421i) q^{68} +(5.65685 - 4.00000i) q^{69} -14.1421i q^{71} +(1.29289 + 2.70711i) q^{72} +(5.00000 - 5.00000i) q^{73} -8.48528 q^{74} +4.00000 q^{76} +(-1.41421 + 1.41421i) q^{77} +6.00000i q^{79} +(7.00000 + 5.65685i) q^{81} +(4.00000 + 4.00000i) q^{82} +(-8.48528 - 8.48528i) q^{83} +(-1.41421 - 2.00000i) q^{84} +8.48528i q^{86} +(2.07107 - 12.0711i) q^{87} +(1.00000 - 1.00000i) q^{88} -2.82843 q^{89} +(2.82843 - 2.82843i) q^{92} +(-0.585786 + 3.41421i) q^{93} +(1.00000 + 1.41421i) q^{96} +(-3.00000 - 3.00000i) q^{97} +(-3.53553 - 3.53553i) q^{98} +(1.41421 - 4.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{6} + 4 q^{7}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^6 + 4 * q^7	\( 4 q + 4 q^{3} - 4 q^{6} + 4 q^{7} - 4 q^{12} - 4 q^{16} - 8 q^{18} + 8 q^{21} + 4 q^{22} + 4 q^{27} + 4 q^{28} - 8 q^{31} + 4 q^{33} - 4 q^{36} - 24 q^{37} - 4 q^{42} - 24 q^{43} + 16 q^{46} - 4 q^{48} - 8 q^{51} + 16 q^{57} + 20 q^{58} - 24 q^{61} + 4 q^{63} + 8 q^{66} + 16 q^{67} + 8 q^{72} + 20 q^{73} + 16 q^{76} + 28 q^{81} + 16 q^{82} - 20 q^{87} + 4 q^{88} - 8 q^{93} + 4 q^{96} - 12 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^6 + 4 * q^7 - 4 * q^12 - 4 * q^16 - 8 * q^18 + 8 * q^21 + 4 * q^22 + 4 * q^27 + 4 * q^28 - 8 * q^31 + 4 * q^33 - 4 * q^36 - 24 * q^37 - 4 * q^42 - 24 * q^43 + 16 * q^46 - 4 * q^48 - 8 * q^51 + 16 * q^57 + 20 * q^58 - 24 * q^61 + 4 * q^63 + 8 * q^66 + 16 * q^67 + 8 * q^72 + 20 * q^73 + 16 * q^76 + 28 * q^81 + 16 * q^82 - 20 * q^87 + 4 * q^88 - 8 * q^93 + 4 * q^96 - 12 * q^97

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