LMFDB - Embedded newform 1520.2.q.o.961.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(881,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("1520.881");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.q (of order $3$, degree $2$, not 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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.4601315889.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 $ x^8 - x^7 + 6x^6 - 3x^5 + 26x^4 - 14x^3 + 31x^2 + 12x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			961.3
Root			$-0.245959 + 0.426014i$ of defining polynomial
Character	$\chi$	$=$	1520.961
Dual form			1520.2.q.o.881.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.745959 - 1.29204i) q^{3} +(-0.500000 + 0.866025i) q^{5} +2.84864 q^{7} +(0.387090 + 0.670459i) q^{9} +O(q^{10})$	\(q+(0.745959 - 1.29204i) q^{3} +(-0.500000 + 0.866025i) q^{5} +2.84864 q^{7} +(0.387090 + 0.670459i) q^{9} +0.864801 q^{11} +(-0.321640 - 0.557098i) q^{13} +(0.745959 + 1.29204i) q^{15} +(-1.87093 + 3.24054i) q^{17} +(3.36069 - 2.77592i) q^{19} +(2.12497 - 3.68055i) q^{21} +(0.208730 + 0.361531i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.63077 q^{27} +(4.85261 + 8.40497i) q^{29} -4.93349 q^{31} +(0.645106 - 1.11736i) q^{33} +(-1.42432 + 2.46699i) q^{35} +6.36467 q^{37} -0.959723 q^{39} +(2.00686 - 3.47598i) q^{41} +(-1.02915 + 1.78254i) q^{43} -0.774179 q^{45} +(-1.97698 - 3.42423i) q^{47} +1.11474 q^{49} +(2.79127 + 4.83462i) q^{51} +(5.49374 + 9.51544i) q^{53} +(-0.432400 + 0.748939i) q^{55} +(-1.07966 - 6.41287i) q^{57} +(1.22980 - 2.13007i) q^{59} +(-3.16740 - 5.48609i) q^{61} +(1.10268 + 1.90989i) q^{63} +0.643281 q^{65} +(-1.26610 - 2.19295i) q^{67} +0.622817 q^{69} +(-0.891065 + 1.54337i) q^{71} +(3.56545 - 6.17554i) q^{73} -1.49192 q^{75} +2.46350 q^{77} +(0.912262 - 1.58008i) q^{79} +(3.03905 - 5.26380i) q^{81} +7.43913 q^{83} +(-1.87093 - 3.24054i) q^{85} +14.4794 q^{87} +(-2.22294 - 3.85024i) q^{89} +(-0.916237 - 1.58697i) q^{91} +(-3.68018 + 6.37427i) q^{93} +(0.723670 + 4.29841i) q^{95} +(5.42707 - 9.39996i) q^{97} +(0.334755 + 0.579813i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 3 q^{3} - 4 q^{5} + 8 q^{7} - q^{9}+O(q^{10}) $ 8 * q + 3 * q^3 - 4 * q^5 + 8 * q^7 - q^9	$ 8 q + 3 q^{3} - 4 q^{5} + 8 q^{7} - q^{9} + 4 q^{11} - 7 q^{13} + 3 q^{15} + q^{17} - 5 q^{19} + 4 q^{21} + 2 q^{23} - 4 q^{25} - 24 q^{27} + q^{29} - 19 q^{33} - 4 q^{35} - 4 q^{37} - 30 q^{39} + 8 q^{41} + q^{43} + 2 q^{45} - 12 q^{47} - 20 q^{49} + 22 q^{51} + 5 q^{53} - 2 q^{55} + 7 q^{57} - 5 q^{59} - 3 q^{63} + 14 q^{65} + 4 q^{67} - 18 q^{69} + 20 q^{71} + 20 q^{73} - 6 q^{75} + 28 q^{77} + 17 q^{79} - 12 q^{81} - 2 q^{83} + q^{85} + 32 q^{87} - 11 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - q^{97} + 38 q^{99}+O(q^{100}) $ 8 * q + 3 * q^3 - 4 * q^5 + 8 * q^7 - q^9 + 4 * q^11 - 7 * q^13 + 3 * q^15 + q^17 - 5 * q^19 + 4 * q^21 + 2 * q^23 - 4 * q^25 - 24 * q^27 + q^29 - 19 * q^33 - 4 * q^35 - 4 * q^37 - 30 * q^39 + 8 * q^41 + q^43 + 2 * q^45 - 12 * q^47 - 20 * q^49 + 22 * q^51 + 5 * q^53 - 2 * q^55 + 7 * q^57 - 5 * q^59 - 3 * q^63 + 14 * q^65 + 4 * q^67 - 18 * q^69 + 20 * q^71 + 20 * q^73 - 6 * q^75 + 28 * q^77 + 17 * q^79 - 12 * q^81 - 2 * q^83 + q^85 + 32 * q^87 - 11 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - q^97 + 38 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	0.745959	−	1.29204i	0.430680	−	0.745959i	−0.566252	−	0.824232i	$-0.691607\pi$
$3$	0.745959	−	1.29204i	0.430680	−	0.745959i	0.996932	+	0.0782728i	$0.0249405\pi$
$4$		0			0
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$5$	−0.500000	+	0.866025i	−0.223607	+	0.387298i
$6$		0			0
$7$	2.84864			1.07668			0.538342	−	0.842727i	$-0.319051\pi$
$7$	2.84864			1.07668			0.538342	+	0.842727i	$0.319051\pi$
$8$		0			0
$9$	0.387090	+	0.670459i	0.129030	+	0.223486i
$10$		0			0
$11$	0.864801			0.260747			0.130374	−	0.991465i	$-0.458382\pi$
$11$	0.864801			0.260747			0.130374	+	0.991465i	$0.458382\pi$
$12$		0			0
$13$	−0.321640	−	0.557098i	−0.0892070	−	0.154511i	0.817969	−	0.575262i	$-0.195100\pi$
$13$	−0.321640	−	0.557098i	−0.0892070	−	0.154511i	−0.907176	+	0.420751i	$0.861767\pi$
$14$		0			0
$15$	0.745959	+	1.29204i	0.192606	+	0.333603i
$16$		0			0
$17$	−1.87093	+	3.24054i	−0.453766	+	0.785946i	−0.998616	−	0.0525872i	$-0.983253\pi$
$17$	−1.87093	+	3.24054i	−0.453766	+	0.785946i	0.544850	+	0.838534i	$0.316587\pi$
$18$		0			0
$19$	3.36069	−	2.77592i	0.770996	−	0.636840i
$19$	3.36069	−	2.77592i	0.770996	−	0.636840i
$20$		0			0
$21$	2.12497	−	3.68055i	0.463706	−	0.803162i
$22$		0			0
$23$	0.208730	+	0.361531i	0.0435233	+	0.0753845i	0.886966	−	0.461834i	$-0.152808\pi$
$23$	0.208730	+	0.361531i	0.0435233	+	0.0753845i	−0.843443	+	0.537218i	$0.819475\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	5.63077			1.08364
$28$		0			0
$29$	4.85261	+	8.40497i	0.901108	+	1.56076i	0.826059	+	0.563584i	$0.190578\pi$
$29$	4.85261	+	8.40497i	0.901108	+	1.56076i	0.0750490	+	0.997180i	$0.476089\pi$
$30$		0			0
$31$	−4.93349			−0.886081			−0.443041	−	0.896501i	$-0.646100\pi$
$31$	−4.93349			−0.886081			−0.443041	+	0.896501i	$0.646100\pi$
$32$		0			0
$33$	0.645106	−	1.11736i	0.112299	−	0.194507i
$34$		0			0
$35$	−1.42432	+	2.46699i	−0.240754	+	0.416998i
$36$		0			0
$37$	6.36467			1.04635			0.523173	−	0.852227i	$-0.324748\pi$
$37$	6.36467			1.04635			0.523173	+	0.852227i	$0.324748\pi$
$38$		0			0
$39$	−0.959723			−0.153679
$40$		0			0
$41$	2.00686	−	3.47598i	0.313419	−	0.542857i	−0.665681	−	0.746236i	$-0.731859\pi$
$41$	2.00686	−	3.47598i	0.313419	−	0.542857i	0.979100	+	0.203379i	$0.0651924\pi$
$42$		0			0
$43$	−1.02915	+	1.78254i	−0.156944	+	0.271834i	−0.933765	−	0.357887i	$-0.883497\pi$
$43$	−1.02915	+	1.78254i	−0.156944	+	0.271834i	0.776821	+	0.629721i	$0.216831\pi$
$44$		0			0
$45$	−0.774179			−0.115408
$46$		0			0
$47$	−1.97698	−	3.42423i	−0.288372	−	0.499475i	0.685049	−	0.728497i	$-0.259781\pi$
$47$	−1.97698	−	3.42423i	−0.288372	−	0.499475i	−0.973421	+	0.229022i	$0.926447\pi$
$48$		0			0
$49$	1.11474			0.159248
$50$		0			0
$51$	2.79127	+	4.83462i	0.390856	+	0.676982i
$52$		0			0
$53$	5.49374	+	9.51544i	0.754624	+	1.30705i	0.945561	+	0.325444i	$0.105514\pi$
$53$	5.49374	+	9.51544i	0.754624	+	1.30705i	−0.190937	+	0.981602i	$0.561153\pi$
$54$		0			0
$55$	−0.432400	+	0.748939i	−0.0583048	+	0.100987i
$56$		0			0
$57$	−1.07966	−	6.41287i	−0.143004	−	0.849406i
$58$		0			0
$59$	1.22980	−	2.13007i	0.160106	−	0.277311i	−0.774801	−	0.632206i	$-0.782150\pi$
$59$	1.22980	−	2.13007i	0.160106	−	0.277311i	0.934906	+	0.354894i	$0.115483\pi$
$60$		0			0
$61$	−3.16740	−	5.48609i	−0.405543	−	0.702422i	0.588841	−	0.808249i	$-0.299584\pi$
$61$	−3.16740	−	5.48609i	−0.405543	−	0.702422i	−0.994385	+	0.105827i	$0.966251\pi$
$62$		0			0
$63$	1.10268	+	1.90989i	0.138924	+	0.240624i
$64$		0			0
$65$	0.643281			0.0797892
$66$		0			0
$67$	−1.26610	−	2.19295i	−0.154678	−	0.267911i	0.778263	−	0.627938i	$-0.216101\pi$
$67$	−1.26610	−	2.19295i	−0.154678	−	0.267911i	−0.932942	+	0.360027i	$0.882767\pi$
$68$		0			0
$69$	0.622817			0.0749783
$70$		0			0
$71$	−0.891065	+	1.54337i	−0.105750	+	0.183164i	−0.914044	−	0.405614i	$-0.867058\pi$
$71$	−0.891065	+	1.54337i	−0.105750	+	0.183164i	0.808294	+	0.588779i	$0.200391\pi$
$72$		0			0
$73$	3.56545	−	6.17554i	0.417304	−	0.722792i	−0.578363	−	0.815780i	$-0.696308\pi$
$73$	3.56545	−	6.17554i	0.417304	−	0.722792i	0.995667	+	0.0929873i	$0.0296416\pi$
$74$		0			0
$75$	−1.49192			−0.172272
$76$		0			0
$77$	2.46350			0.280742
$78$		0			0
$79$	0.912262	−	1.58008i	0.102637	−	0.177773i	−0.810133	−	0.586246i	$-0.800605\pi$
$79$	0.912262	−	1.58008i	0.102637	−	0.177773i	0.912771	+	0.408473i	$0.133939\pi$
$80$		0			0
$81$	3.03905	−	5.26380i	0.337673	−	0.584866i
$82$		0			0
$83$	7.43913			0.816550			0.408275	−	0.912859i	$-0.366130\pi$
$83$	7.43913			0.816550			0.408275	+	0.912859i	$0.366130\pi$
$84$		0			0
$85$	−1.87093	−	3.24054i	−0.202930	−	0.351486i
$86$		0			0
$87$	14.4794			1.55236
$88$		0			0
$89$	−2.22294	−	3.85024i	−0.235631	−	0.408125i	0.723825	−	0.689984i	$-0.242382\pi$
$89$	−2.22294	−	3.85024i	−0.235631	−	0.408125i	−0.959456	+	0.281859i	$0.909049\pi$
$90$		0			0
$91$	−0.916237	−	1.58697i	−0.0960478	−	0.166360i
$92$		0			0
$93$	−3.68018	+	6.37427i	−0.381617	+	0.660981i
$94$		0			0
$95$	0.723670	+	4.29841i	0.0742470	+	0.441007i
$96$		0			0
$97$	5.42707	−	9.39996i	0.551036	−	0.954422i	−0.447165	−	0.894452i	$-0.647566\pi$
$97$	5.42707	−	9.39996i	0.551036	−	0.954422i	0.998200	−	0.0599699i	$-0.0191005\pi$
$98$		0			0
$99$	0.334755	+	0.579813i	0.0336442	+	0.0582734i
$100$		0			0
$101$	2.64799	+	4.58645i	0.263485	+	0.456369i	0.967166	−	0.254147i	$-0.0817948\pi$
$101$	2.64799	+	4.58645i	0.263485	+	0.456369i	−0.703681	+	0.710516i	$0.748461\pi$
$102$		0			0
$103$	0.385134			0.0379484			0.0189742	−	0.999820i	$-0.493960\pi$
$103$	0.385134			0.0379484			0.0189742	+	0.999820i	$0.493960\pi$
$104$		0			0
$105$	2.12497	+	3.68055i	0.207376	+	0.359185i
$106$		0			0
$107$	6.43336			0.621937			0.310968	−	0.950420i	$-0.399347\pi$
$107$	6.43336			0.621937			0.310968	+	0.950420i	$0.399347\pi$
$108$		0			0
$109$	−3.28441	+	5.68877i	−0.314590	+	0.544885i	−0.979350	−	0.202171i	$-0.935200\pi$
$109$	−3.28441	+	5.68877i	−0.314590	+	0.544885i	0.664761	+	0.747056i	$0.268534\pi$
$110$		0			0
$111$	4.74778	−	8.22340i	0.450640	−	0.780531i
$112$		0			0
$113$	0.294513			0.0277054			0.0138527	−	0.999904i	$-0.495590\pi$
$113$	0.294513			0.0277054			0.0138527	+	0.999904i	$0.495590\pi$
$114$		0			0
$115$	−0.417460			−0.0389284
$116$		0			0
$117$	0.249007	−	0.431294i	0.0230207	−	0.0398731i
$118$		0			0
$119$	−5.32959	+	9.23112i	−0.488563	+	0.846216i
$120$		0			0
$121$	−10.2521			−0.932011
$122$		0			0
$123$	−2.99407	−	5.18588i	−0.269966	−	0.467595i
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	4.41746	+	7.65127i	0.391986	+	0.678940i	0.992711	−	0.120516i	$-0.0384548\pi$
$127$	4.41746	+	7.65127i	0.391986	+	0.678940i	−0.600725	+	0.799456i	$0.705121\pi$
$128$		0			0
$129$	1.53540	+	2.65940i	0.135185	+	0.234147i
$130$		0			0
$131$	10.4564	−	18.1110i	0.913578	−	1.58236i	0.104609	−	0.994513i	$-0.466641\pi$
$131$	10.4564	−	18.1110i	0.913578	−	1.58236i	0.808969	−	0.587851i	$-0.200026\pi$
$132$		0			0
$133$	9.57340	−	7.90759i	0.830119	−	0.685675i
$134$		0			0
$135$	−2.81538	+	4.87639i	−0.242310	+	0.419693i
$136$		0			0
$137$	−2.60739	−	4.51613i	−0.222764	−	0.385839i	0.732882	−	0.680356i	$-0.238175\pi$
$137$	−2.60739	−	4.51613i	−0.222764	−	0.385839i	−0.955646	+	0.294517i	$0.904841\pi$
$138$		0			0
$139$	−5.36192	−	9.28711i	−0.454792	−	0.787723i	0.543884	−	0.839160i	$-0.316953\pi$
$139$	−5.36192	−	9.28711i	−0.454792	−	0.787723i	−0.998676	+	0.0514375i	$0.983620\pi$
$140$		0			0
$141$	−5.89898			−0.496784
$142$		0			0
$143$	−0.278155	−	0.481778i	−0.0232605	−	0.0402883i
$144$		0			0
$145$	−9.70523			−0.805975
$146$		0			0
$147$	0.831547	−	1.44028i	0.0685848	−	0.118792i
$148$		0			0
$149$	7.45578	−	12.9138i	0.610801	−	1.05794i	−0.380304	−	0.924861i	$-0.624181\pi$
$149$	7.45578	−	12.9138i	0.610801	−	1.05794i	0.991106	−	0.133078i	$-0.0424860\pi$
$150$		0			0
$151$	−21.4589			−1.74630			−0.873152	−	0.487448i	$-0.837928\pi$
$151$	−21.4589			−1.74630			−0.873152	+	0.487448i	$0.837928\pi$
$152$		0			0
$153$	−2.89687			−0.234198
$154$		0			0
$155$	2.46675	−	4.27253i	0.198134	−	0.343178i
$156$		0			0
$157$	1.21559	−	2.10546i	0.0970145	−	0.168034i	−0.813433	−	0.581659i	$-0.802404\pi$
$157$	1.21559	−	2.10546i	0.0970145	−	0.168034i	0.910448	+	0.413624i	$0.135737\pi$
$158$		0			0
$159$	16.3924			1.30000
$160$		0			0
$161$	0.594597	+	1.02987i	0.0468608	+	0.0811653i
$162$		0			0
$163$	−17.8175			−1.39558			−0.697788	−	0.716305i	$-0.745832\pi$
$163$	−17.8175			−1.39558			−0.697788	+	0.716305i	$0.745832\pi$
$164$		0			0
$165$	0.645106	+	1.11736i	0.0502214	+	0.0869861i
$166$		0			0
$167$	−0.202799	−	0.351258i	−0.0156931	−	0.0271812i	0.858072	−	0.513529i	$-0.171662\pi$
$167$	−0.202799	−	0.351258i	−0.0156931	−	0.0271812i	−0.873765	+	0.486348i	$0.838329\pi$
$168$		0			0
$169$	6.29309	−	10.9000i	0.484084	−	0.838458i
$170$		0			0
$171$	3.16203	+	1.17868i	0.241807	+	0.0901358i
$172$		0			0
$173$	−9.01051	+	15.6067i	−0.685056	+	1.18655i	0.288363	+	0.957521i	$0.406889\pi$
$173$	−9.01051	+	15.6067i	−0.685056	+	1.18655i	−0.973419	+	0.229031i	$0.926444\pi$
$174$		0			0
$175$	−1.42432	−	2.46699i	−0.107668	−	0.186487i
$176$		0			0
$177$	−1.83476	−	3.17789i	−0.137909	−	0.238865i
$178$		0			0
$179$	−20.1523			−1.50625			−0.753127	−	0.657875i	$-0.771455\pi$
$179$	−20.1523			−1.50625			−0.753127	+	0.657875i	$0.771455\pi$
$180$		0			0
$181$	8.55541	+	14.8184i	0.635919	+	1.10144i	0.986320	+	0.164844i	$0.0527120\pi$
$181$	8.55541	+	14.8184i	0.635919	+	1.10144i	−0.350401	+	0.936600i	$0.613955\pi$
$182$		0			0
$183$	−9.45099			−0.698637
$184$		0			0
$185$	−3.18233	+	5.51197i	−0.233970	+	0.405248i
$186$		0			0
$187$	−1.61798	+	2.80242i	−0.118318	+	0.204933i
$188$		0			0
$189$	16.0400			1.16674
$190$		0			0
$191$	−5.28080			−0.382105			−0.191053	−	0.981580i	$-0.561190\pi$
$191$	−5.28080			−0.382105			−0.191053	+	0.981580i	$0.561190\pi$
$192$		0			0
$193$	−9.00182	+	15.5916i	−0.647966	+	1.12231i	0.335642	+	0.941989i	$0.391047\pi$
$193$	−9.00182	+	15.5916i	−0.647966	+	1.12231i	−0.983608	+	0.180320i	$0.942287\pi$
$194$		0			0
$195$	0.479861	−	0.831144i	0.0343636	−	0.0595195i
$196$		0			0
$197$	8.07785			0.575523			0.287761	−	0.957702i	$-0.407089\pi$
$197$	8.07785			0.575523			0.287761	+	0.957702i	$0.407089\pi$
$198$		0			0
$199$	0.701872	+	1.21568i	0.0497544	+	0.0861771i	0.889830	−	0.456292i	$-0.150823\pi$
$199$	0.701872	+	1.21568i	0.0497544	+	0.0861771i	−0.840076	+	0.542469i	$0.817489\pi$
$200$		0			0
$201$	−3.77783			−0.266468
$202$		0			0
$203$	13.8233	+	23.9427i	0.970208	+	1.68045i
$204$		0			0
$205$	2.00686	+	3.47598i	0.140165	+	0.242773i
$206$		0			0
$207$	−0.161595	+	0.279890i	−0.0112316	+	0.0194537i
$208$		0			0
$209$	2.90633	−	2.40062i	0.201035	−	0.166054i
$210$		0			0
$211$	9.45817	−	16.3820i	0.651128	−	1.12779i	−0.331722	−	0.943377i	$-0.607630\pi$
$211$	9.45817	−	16.3820i	0.651128	−	1.12779i	0.982850	−	0.184409i	$-0.0590370\pi$
$212$		0			0
$213$	1.32940	+	2.30258i	0.0910888	+	0.157770i
$214$		0			0
$215$	−1.02915	−	1.78254i	−0.0701873	−	0.121568i
$216$		0			0
$217$	−14.0537			−0.954030
$218$		0			0
$219$	−5.31936	−	9.21340i	−0.359449	−	0.622584i
$220$		0			0
$221$	2.40706			0.161917
$222$		0			0
$223$	−8.07400	+	13.9846i	−0.540675	+	0.936477i	0.458190	+	0.888854i	$0.348498\pi$
$223$	−8.07400	+	13.9846i	−0.540675	+	0.936477i	−0.998865	+	0.0476227i	$0.984835\pi$
$224$		0			0
$225$	0.387090	−	0.670459i	0.0258060	−	0.0446973i
$226$		0			0
$227$	−26.3186			−1.74683			−0.873414	−	0.486978i	$-0.838099\pi$
$227$	−26.3186			−1.74683			−0.873414	+	0.486978i	$0.838099\pi$
$228$		0			0
$229$	−13.3323			−0.881026			−0.440513	−	0.897746i	$-0.645203\pi$
$229$	−13.3323			−0.881026			−0.440513	+	0.897746i	$0.645203\pi$
$230$		0			0
$231$	1.83767	−	3.18294i	0.120910	−	0.209422i
$232$		0			0
$233$	−12.6547	+	21.9186i	−0.829038	+	1.43594i	0.0697556	+	0.997564i	$0.477778\pi$
$233$	−12.6547	+	21.9186i	−0.829038	+	1.43594i	−0.898794	+	0.438372i	$0.855555\pi$
$234$		0			0
$235$	3.95396			0.257928
$236$		0			0
$237$	−1.36102	−	2.35736i	−0.0884077	−	0.153127i
$238$		0			0
$239$	23.5500			1.52332			0.761660	−	0.647977i	$-0.224385\pi$
$239$	23.5500			1.52332			0.761660	+	0.647977i	$0.224385\pi$
$240$		0			0
$241$	−4.19208	−	7.26089i	−0.270035	−	0.467715i	0.698835	−	0.715283i	$-0.253702\pi$
$241$	−4.19208	−	7.26089i	−0.270035	−	0.467715i	−0.968871	+	0.247568i	$0.920369\pi$
$242$		0			0
$243$	3.91213	+	6.77601i	0.250963	+	0.434681i
$244$		0			0
$245$	−0.557368	+	0.965389i	−0.0356089	+	0.0616764i
$246$		0			0
$247$	−2.62739	−	0.979387i	−0.167177	−	0.0623169i
$248$		0			0
$249$	5.54929	−	9.61165i	0.351672	−	0.609113i
$250$		0			0
$251$	9.12391	+	15.8031i	0.575896	+	0.997481i	0.995944	+	0.0899792i	$0.0286800\pi$
$251$	9.12391	+	15.8031i	0.575896	+	0.997481i	−0.420048	+	0.907502i	$0.637987\pi$
$252$		0			0
$253$	0.180510	+	0.312652i	0.0113486	+	0.0196563i
$254$		0			0
$255$	−5.58254			−0.349592
$256$		0			0
$257$	−7.04989	−	12.2108i	−0.439760	−	0.761687i	0.557911	−	0.829901i	$-0.311603\pi$
$257$	−7.04989	−	12.2108i	−0.439760	−	0.761687i	−0.997671	+	0.0682144i	$0.978270\pi$
$258$		0			0
$259$	18.1306			1.12658
$260$		0			0
$261$	−3.75679	+	6.50696i	−0.232540	+	0.402770i
$262$		0			0
$263$	−3.20536	+	5.55184i	−0.197651	+	0.342341i	−0.947766	−	0.318966i	$-0.896665\pi$
$263$	−3.20536	+	5.55184i	−0.197651	+	0.342341i	0.750116	+	0.661307i	$0.229998\pi$
$264$		0			0
$265$	−10.9875			−0.674956
$266$		0			0
$267$	−6.63288			−0.405926
$268$		0			0
$269$	−8.99557	+	15.5808i	−0.548469	+	0.949977i	0.449910	+	0.893074i	$0.351456\pi$
$269$	−8.99557	+	15.5808i	−0.548469	+	0.949977i	−0.998380	+	0.0569032i	$0.981877\pi$
$270$		0			0
$271$	5.94095	−	10.2900i	0.360887	−	0.625075i	−0.627220	−	0.778842i	$-0.715807\pi$
$271$	5.94095	−	10.2900i	0.360887	−	0.625075i	0.988107	+	0.153767i	$0.0491406\pi$
$272$		0			0
$273$	−2.73390			−0.165463
$274$		0			0
$275$	−0.432400	−	0.748939i	−0.0260747	−	0.0451627i
$276$		0			0
$277$	23.6240			1.41943			0.709715	−	0.704489i	$-0.248824\pi$
$277$	23.6240			1.41943			0.709715	+	0.704489i	$0.248824\pi$
$278$		0			0
$279$	−1.90970	−	3.30770i	−0.114331	−	0.198027i
$280$		0			0
$281$	−6.90465	−	11.9592i	−0.411897	−	0.713426i	0.583200	−	0.812328i	$-0.301800\pi$
$281$	−6.90465	−	11.9592i	−0.411897	−	0.713426i	−0.995097	+	0.0989020i	$0.968467\pi$
$282$		0			0
$283$	−5.87868	+	10.1822i	−0.349451	+	0.605268i	−0.986152	−	0.165843i	$-0.946965\pi$
$283$	−5.87868	+	10.1822i	−0.349451	+	0.605268i	0.636701	+	0.771111i	$0.280299\pi$
$284$		0			0
$285$	6.09354	+	2.27143i	0.360950	+	0.134548i
$286$		0			0
$287$	5.71681	−	9.90181i	0.337453	−	0.584485i
$288$		0			0
$289$	1.49927	+	2.59681i	0.0881922	+	0.152753i
$290$		0			0
$291$	−8.09675	−	14.0240i	−0.474640	−	0.822100i
$292$		0			0
$293$	−27.0576			−1.58072			−0.790362	−	0.612640i	$-0.790108\pi$
$293$	−27.0576			−1.58072			−0.790362	+	0.612640i	$0.790108\pi$
$294$		0			0
$295$	1.22980	+	2.13007i	0.0716015	+	0.124017i
$296$		0			0
$297$	4.86949			0.282557
$298$		0			0
$299$	0.134272	−	0.232566i	0.00776516	−	0.0134497i
$300$		0			0
$301$	−2.93167	+	5.07780i	−0.168979	+	0.292679i
$302$		0			0
$303$	7.90117			0.453910
$304$		0			0
$305$	6.33479			0.362729
$306$		0			0
$307$	4.41912	−	7.65414i	0.252212	−	0.436845i	−0.711922	−	0.702258i	$-0.752175\pi$
$307$	4.41912	−	7.65414i	0.252212	−	0.436845i	0.964135	+	0.265414i	$0.0855085\pi$
$308$		0			0
$309$	0.287294	−	0.497608i	0.0163436	−	0.0283080i
$310$		0			0
$311$	0.651493			0.0369428			0.0184714	−	0.999829i	$-0.494120\pi$
$311$	0.651493			0.0369428			0.0184714	+	0.999829i	$0.494120\pi$
$312$		0			0
$313$	1.48278	+	2.56825i	0.0838116	+	0.145166i	0.904884	−	0.425658i	$-0.139957\pi$
$313$	1.48278	+	2.56825i	0.0838116	+	0.145166i	−0.821073	+	0.570824i	$0.806624\pi$
$314$		0			0
$315$	−2.20536			−0.124258
$316$		0			0
$317$	5.18993	+	8.98921i	0.291495	+	0.504885i	0.974164	−	0.225844i	$-0.0725139\pi$
$317$	5.18993	+	8.98921i	0.291495	+	0.504885i	−0.682668	+	0.730728i	$0.739181\pi$
$318$		0			0
$319$	4.19654	+	7.26862i	0.234961	+	0.406965i
$320$		0			0
$321$	4.79903	−	8.31216i	0.267856	−	0.463939i
$322$		0			0
$323$	2.70787	+	16.0840i	0.150670	+	0.894938i
$324$		0			0
$325$	−0.321640	+	0.557098i	−0.0178414	+	0.0309022i
$326$		0			0
$327$	4.90007	+	8.48718i	0.270975	+	0.469342i
$328$		0			0
$329$	−5.63170	−	9.75438i	−0.310485	−	0.537776i
$330$		0			0
$331$	−15.0922			−0.829543			−0.414772	−	0.909926i	$-0.636139\pi$
$331$	−15.0922			−0.829543			−0.414772	+	0.909926i	$0.636139\pi$
$332$		0			0
$333$	2.46370	+	4.26725i	0.135010	+	0.233844i
$334$		0			0
$335$	2.53220			0.138349
$336$		0			0
$337$	−7.89872	+	13.6810i	−0.430271	+	0.745251i	−0.996896	−	0.0787246i	$-0.974915\pi$
$337$	−7.89872	+	13.6810i	−0.430271	+	0.745251i	0.566626	+	0.823975i	$0.308249\pi$
$338$		0			0
$339$	0.219695	−	0.380522i	0.0119322	−	0.0206671i
$340$		0			0
$341$	−4.26649			−0.231043
$342$		0			0
$343$	−16.7650			−0.905224
$344$		0			0
$345$	−0.311408	+	0.539375i	−0.0167657	+	0.0290390i
$346$		0			0
$347$	−10.6761	+	18.4915i	−0.573122	+	0.992676i	0.423121	+	0.906073i	$0.360935\pi$
$347$	−10.6761	+	18.4915i	−0.573122	+	0.992676i	−0.996243	+	0.0866031i	$0.972399\pi$
$348$		0			0
$349$	−32.3897			−1.73378			−0.866891	−	0.498497i	$-0.833885\pi$
$349$	−32.3897			−1.73378			−0.866891	+	0.498497i	$0.833885\pi$
$350$		0			0
$351$	−1.81108	−	3.13689i	−0.0966685	−	0.167435i
$352$		0			0
$353$	0.730583			0.0388850			0.0194425	−	0.999811i	$-0.493811\pi$
$353$	0.730583			0.0388850			0.0194425	+	0.999811i	$0.493811\pi$
$354$		0			0
$355$	−0.891065	−	1.54337i	−0.0472928	−	0.0819136i
$356$		0			0
$357$	7.95132	+	13.7721i	0.420828	+	0.728896i
$358$		0			0
$359$	−13.4248	+	23.2524i	−0.708533	+	1.22722i	0.256868	+	0.966447i	$0.417309\pi$
$359$	−13.4248	+	23.2524i	−0.708533	+	1.22722i	−0.965401	+	0.260769i	$0.916024\pi$
$360$		0			0
$361$	3.58853	−	18.6580i	0.188870	−	0.982002i
$362$		0			0
$363$	−7.64766	+	13.2461i	−0.401398	+	0.695242i
$364$		0			0
$365$	3.56545	+	6.17554i	0.186624	+	0.323242i
$366$		0			0
$367$	−11.4822	−	19.8877i	−0.599364	−	1.03813i	−0.992915	−	0.118826i	$-0.962087\pi$
$367$	−11.4822	−	19.8877i	−0.599364	−	1.03813i	0.393551	−	0.919303i	$-0.371246\pi$
$368$		0			0
$369$	3.10734			0.161761
$370$		0			0
$371$	15.6497	+	27.1060i	0.812491	+	1.40728i
$372$		0			0
$373$	29.5305			1.52903			0.764515	−	0.644606i	$-0.222979\pi$
$373$	29.5305			1.52903			0.764515	+	0.644606i	$0.222979\pi$
$374$		0			0
$375$	0.745959	−	1.29204i	0.0385212	−	0.0667206i
$376$		0			0
$377$	3.12159	−	5.40676i	0.160770	−	0.278462i
$378$		0			0
$379$	−17.5117			−0.899517			−0.449759	−	0.893150i	$-0.648490\pi$
$379$	−17.5117			−0.899517			−0.449759	+	0.893150i	$0.648490\pi$
$380$		0			0
$381$	13.1810			0.675282
$382$		0			0
$383$	4.05326	−	7.02045i	0.207112	−	0.358728i	−0.743692	−	0.668523i	$-0.766927\pi$
$383$	4.05326	−	7.02045i	0.207112	−	0.358728i	0.950804	+	0.309794i	$0.100260\pi$
$384$		0			0
$385$	−1.23175	+	2.13346i	−0.0627759	+	0.108731i
$386$		0			0
$387$	−1.59349			−0.0810016
$388$		0			0
$389$	−8.65392	−	14.9890i	−0.438771	−	0.759974i	0.558824	−	0.829286i	$-0.311253\pi$
$389$	−8.65392	−	14.9890i	−0.438771	−	0.759974i	−0.997595	+	0.0693125i	$0.977919\pi$
$390$		0			0
$391$	−1.56208			−0.0789976
$392$		0			0
$393$	−15.6001	−	27.0201i	−0.786920	−	1.36298i
$394$		0			0
$395$	0.912262	+	1.58008i	0.0459009	+	0.0795026i
$396$		0			0
$397$	5.69472	−	9.86354i	0.285810	−	0.495037i	−0.686996	−	0.726662i	$-0.741071\pi$
$397$	5.69472	−	9.86354i	0.285810	−	0.495037i	0.972805	+	0.231625i	$0.0744041\pi$
$398$		0			0
$399$	−3.07555	−	18.2679i	−0.153970	−	0.914541i
$400$		0			0
$401$	4.46930	−	7.74106i	0.223186	−	0.386570i	−0.732587	−	0.680673i	$-0.761687\pi$
$401$	4.46930	−	7.74106i	0.223186	−	0.386570i	0.955774	+	0.294103i	$0.0950208\pi$
$402$		0			0
$403$	1.58681	+	2.74844i	0.0790447	+	0.136909i
$404$		0			0
$405$	3.03905	+	5.26380i	0.151012	+	0.261560i
$406$		0			0
$407$	5.50417			0.272832
$408$		0			0
$409$	3.27235	+	5.66788i	0.161808	+	0.280259i	0.935517	−	0.353282i	$-0.114934\pi$
$409$	3.27235	+	5.66788i	0.161808	+	0.280259i	−0.773709	+	0.633541i	$0.781601\pi$
$410$		0			0
$411$	−7.78001			−0.383760
$412$		0			0
$413$	3.50324	−	6.06780i	0.172383	−	0.298577i
$414$		0			0
$415$	−3.71956	+	6.44247i	−0.182586	+	0.316249i
$416$		0			0
$417$	−15.9991			−0.783479
$418$		0			0
$419$	−21.8441			−1.06715			−0.533576	−	0.845752i	$-0.679152\pi$
$419$	−21.8441			−1.06715			−0.533576	+	0.845752i	$0.679152\pi$
$420$		0			0
$421$	14.6717	−	25.4121i	0.715054	−	1.23851i	−0.247885	−	0.968789i	$-0.579736\pi$
$421$	14.6717	−	25.4121i	0.715054	−	1.23851i	0.962939	−	0.269720i	$-0.0869311\pi$
$422$		0			0
$423$	1.53054	−	2.65097i	0.0744172	−	0.128894i
$424$		0			0
$425$	3.74185			0.181507
$426$		0			0
$427$	−9.02276	−	15.6279i	−0.436642	−	0.756286i
$428$		0			0
$429$	−0.829969			−0.0400713
$430$		0			0
$431$	−6.44336	−	11.1602i	−0.310366	−	0.537570i	0.668076	−	0.744093i	$-0.267118\pi$
$431$	−6.44336	−	11.1602i	−0.310366	−	0.537570i	−0.978442	+	0.206524i	$0.933785\pi$
$432$		0			0
$433$	6.92144	+	11.9883i	0.332623	+	0.576120i	0.983025	−	0.183470i	$-0.0587330\pi$
$433$	6.92144	+	11.9883i	0.332623	+	0.576120i	−0.650402	+	0.759590i	$0.725400\pi$
$434$		0			0
$435$	−7.23970	+	12.5395i	−0.347117	+	0.601225i
$436$		0			0
$437$	1.70506	+	0.635578i	0.0815641	+	0.0304038i
$438$		0			0
$439$	−0.0354040	+	0.0613216i	−0.00168974	+	0.00292672i	−0.866869	−	0.498536i	$-0.833871\pi$
$439$	−0.0354040	+	0.0613216i	−0.00168974	+	0.00292672i	0.865179	+	0.501463i	$0.167205\pi$
$440$		0			0
$441$	0.431503	+	0.747384i	0.0205477	+	0.0355897i
$442$		0			0
$443$	−1.89457	−	3.28149i	−0.0900137	−	0.155908i	0.817503	−	0.575924i	$-0.195358\pi$
$443$	−1.89457	−	3.28149i	−0.0900137	−	0.155908i	−0.907517	+	0.420016i	$0.862024\pi$
$444$		0			0
$445$	4.44588			0.210755
$446$		0			0
$447$	−11.1234	−	19.2663i	−0.526120	−	0.911266i
$448$		0			0
$449$	−26.5765			−1.25422			−0.627112	−	0.778929i	$-0.715763\pi$
$449$	−26.5765			−1.25422			−0.627112	+	0.778929i	$0.715763\pi$
$450$		0			0
$451$	1.73553	−	3.00603i	0.0817230	−	0.141548i
$452$		0			0
$453$	−16.0075	+	27.7258i	−0.752098	+	1.30267i
$454$		0			0
$455$	1.83247			0.0859077
$456$		0			0
$457$	−33.1523			−1.55080			−0.775400	−	0.631471i	$-0.782452\pi$
$457$	−33.1523			−1.55080			−0.775400	+	0.631471i	$0.782452\pi$
$458$		0			0
$459$	−10.5348	+	18.2467i	−0.491720	+	0.851684i
$460$		0			0
$461$	9.62679	−	16.6741i	0.448364	−	0.776590i	−0.549915	−	0.835220i	$-0.685340\pi$
$461$	9.62679	−	16.6741i	0.448364	−	0.776590i	0.998280	+	0.0586304i	$0.0186734\pi$
$462$		0			0
$463$	−39.1713			−1.82044			−0.910222	−	0.414120i	$-0.864089\pi$
$463$	−39.1713			−1.82044			−0.910222	+	0.414120i	$0.864089\pi$
$464$		0			0
$465$	−3.68018	−	6.37427i	−0.170664	−	0.295600i
$466$		0			0
$467$	39.0650			1.80771			0.903856	−	0.427836i	$-0.140724\pi$
$467$	39.0650			1.80771			0.903856	+	0.427836i	$0.140724\pi$
$468$		0			0
$469$	−3.60665	−	6.24691i	−0.166540	−	0.288455i
$470$		0			0
$471$	−1.81356	−	3.14118i	−0.0835644	−	0.144738i
$472$		0			0
$473$	−0.890007	+	1.54154i	−0.0409226	+	0.0708800i
$474$		0			0
$475$	−4.08436	−	1.52249i	−0.187404	−	0.0698565i
$476$		0			0
$477$	−4.25314	+	7.36666i	−0.194738	+	0.337296i
$478$		0			0
$479$	−12.3775	−	21.4385i	−0.565543	−	0.979550i	−0.996999	−	0.0774158i	$-0.975333\pi$
$479$	−12.3775	−	21.4385i	−0.565543	−	0.979550i	0.431455	−	0.902134i	$-0.358000\pi$
$480$		0			0
$481$	−2.04714	−	3.54574i	−0.0933413	−	0.161672i
$482$		0			0
$483$	1.77418			0.0807280
$484$		0			0
$485$	5.42707	+	9.39996i	0.246431	+	0.426830i
$486$		0			0
$487$	−21.8871			−0.991797			−0.495899	−	0.868380i	$-0.665161\pi$
$487$	−21.8871			−0.991797			−0.495899	+	0.868380i	$0.665161\pi$
$488$		0			0
$489$	−13.2911	+	23.0209i	−0.601046	+	1.04104i
$490$		0			0
$491$	4.69777	−	8.13677i	0.212007	−	0.367207i	−0.740335	−	0.672238i	$-0.765333\pi$
$491$	4.69777	−	8.13677i	0.212007	−	0.367207i	0.952343	+	0.305030i	$0.0986666\pi$
$492$		0			0
$493$	−36.3155			−1.63557
$494$		0			0
$495$	−0.669511			−0.0300923
$496$		0			0
$497$	−2.53832	+	4.39650i	−0.113859	+	0.197210i
$498$		0			0
$499$	12.4558	−	21.5740i	0.557596	−	0.965785i	−0.440100	−	0.897949i	$-0.645057\pi$
$499$	12.4558	−	21.5740i	0.557596	−	0.965785i	0.997696	−	0.0678367i	$-0.0216097\pi$
$500$		0			0
$501$	−0.605119			−0.0270347
$502$		0			0
$503$	−15.6590	−	27.1222i	−0.698200	−	1.20932i	−0.969090	−	0.246707i	$-0.920651\pi$
$503$	−15.6590	−	27.1222i	−0.698200	−	1.20932i	0.270890	−	0.962610i	$-0.412682\pi$
$504$		0			0
$505$	−5.29598			−0.235668
$506$		0			0
$507$	−9.38878	−	16.2619i	−0.416971	−	0.722214i
$508$		0			0
$509$	4.83310	+	8.37117i	0.214223	+	0.371045i	0.953032	−	0.302870i	$-0.0979447\pi$
$509$	4.83310	+	8.37117i	0.214223	+	0.371045i	−0.738809	+	0.673915i	$0.764611\pi$
$510$		0			0
$511$	10.1567	−	17.5919i	0.449305	−	0.778219i
$512$		0			0
$513$	18.9233	−	15.6306i	0.835484	−	0.690106i
$514$		0			0
$515$	−0.192567	+	0.333536i	−0.00848552	+	0.0146973i
$516$		0			0
$517$	−1.70969	−	2.96127i	−0.0751922	−	0.130237i
$518$		0			0
$519$	13.4429	+	23.2839i	0.590080	+	1.02205i
$520$		0			0
$521$	−0.982633			−0.0430499			−0.0215250	−	0.999768i	$-0.506852\pi$
$521$	−0.982633			−0.0430499			−0.0215250	+	0.999768i	$0.506852\pi$
$522$		0			0
$523$	19.8604	+	34.3993i	0.868436	+	1.50418i	0.863594	+	0.504187i	$0.168208\pi$
$523$	19.8604	+	34.3993i	0.868436	+	1.50418i	0.00484172	+	0.999988i	$0.498459\pi$
$524$		0			0
$525$	−4.24993			−0.185482
$526$		0			0
$527$	9.23020	−	15.9872i	0.402074	−	0.696413i
$528$		0			0
$529$	11.4129	−	19.7677i	0.496211	−	0.859463i
$530$		0			0
$531$	1.90417			0.0826337
$532$		0			0
$533$	−2.58195			−0.111837
$534$		0			0
$535$	−3.21668	+	5.57146i	−0.139069	+	0.240875i
$536$		0			0
$537$	−15.0328	+	26.0376i	−0.648713	+	1.12360i
$538$		0			0
$539$	0.964024			0.0415234
$540$		0			0
$541$	−15.3887	−	26.6541i	−0.661614	−	1.14595i	−0.980191	−	0.198052i	$-0.936538\pi$
$541$	−15.3887	−	26.6541i	−0.661614	−	1.14595i	0.318577	−	0.947897i	$-0.396795\pi$
$542$		0			0
$543$	25.5280			1.09551
$544$		0			0
$545$	−3.28441	−	5.68877i	−0.140689	−	0.243680i
$546$		0			0
$547$	−8.93287	−	15.4722i	−0.381942	−	0.661543i	0.609398	−	0.792865i	$-0.291411\pi$
$547$	−8.93287	−	15.4722i	−0.381942	−	0.661543i	−0.991340	+	0.131322i	$0.958078\pi$
$548$		0			0
$549$	2.45213	−	4.24722i	0.104654	−	0.181267i
$550$		0			0
$551$	39.6397	+	14.7761i	1.68871	+	0.629482i
$552$		0			0
$553$	2.59870	−	4.50109i	0.110508	−	0.191406i
$554$		0			0
$555$	4.74778	+	8.22340i	0.201532	+	0.349064i
$556$		0			0
$557$	5.32878	+	9.22971i	0.225787	+	0.391075i	0.956555	−	0.291551i	$-0.0941712\pi$
$557$	5.32878	+	9.22971i	0.225787	+	0.391075i	−0.730768	+	0.682626i	$0.760838\pi$
$558$		0			0
$559$	1.32406			0.0560019
$560$		0			0
$561$	2.41389	+	4.18098i	0.101915	+	0.176521i
$562$		0			0
$563$	−7.75961			−0.327029			−0.163514	−	0.986541i	$-0.552283\pi$
$563$	−7.75961			−0.327029			−0.163514	+	0.986541i	$0.552283\pi$
$564$		0			0
$565$	−0.147256	+	0.255056i	−0.00619513	+	0.0107303i
$566$		0			0
$567$	8.65716	−	14.9946i	0.363567	−	0.629716i
$568$		0			0
$569$	5.72754			0.240111			0.120056	−	0.992767i	$-0.461693\pi$
$569$	5.72754			0.240111			0.120056	+	0.992767i	$0.461693\pi$
$570$		0			0
$571$	20.8347			0.871903			0.435952	−	0.899970i	$-0.356412\pi$
$571$	20.8347			0.871903			0.435952	+	0.899970i	$0.356412\pi$
$572$		0			0
$573$	−3.93926	+	6.82300i	−0.164565	+	0.285035i
$574$		0			0
$575$	0.208730	−	0.361531i	0.00870465	−	0.0150769i
$576$		0			0
$577$	−5.11190			−0.212811			−0.106406	−	0.994323i	$-0.533934\pi$
$577$	−5.11190			−0.212811			−0.106406	+	0.994323i	$0.533934\pi$
$578$		0			0
$579$	13.4300	+	23.2614i	0.558131	+	0.966712i
$580$		0			0
$581$	21.1914			0.879167
$582$		0			0
$583$	4.75099	+	8.22896i	0.196766	+	0.340809i
$584$		0			0
$585$	0.249007	+	0.431294i	0.0102952	+	0.0178318i
$586$		0			0
$587$	−5.33462	+	9.23984i	−0.220184	+	0.381369i	−0.954864	−	0.297045i	$-0.903999\pi$
$587$	−5.33462	+	9.23984i	−0.220184	+	0.381369i	0.734680	+	0.678414i	$0.237332\pi$
$588$		0			0
$589$	−16.5800	+	13.6950i	−0.683165	+	0.564292i
$590$		0			0
$591$	6.02574	−	10.4369i	0.247866	−	0.429316i
$592$		0			0
$593$	8.50133	+	14.7247i	0.349108	+	0.604673i	0.986091	−	0.166205i	$-0.0531513\pi$
$593$	8.50133	+	14.7247i	0.349108	+	0.604673i	−0.636983	+	0.770878i	$0.719818\pi$
$594$		0			0
$595$	−5.32959	−	9.23112i	−0.218492	−	0.378439i
$596$		0			0
$597$	2.09427			0.0857128
$598$		0			0
$599$	14.3375	+	24.8334i	0.585816	+	1.01466i	0.994773	+	0.102110i	$0.0325592\pi$
$599$	14.3375	+	24.8334i	0.585816	+	1.01466i	−0.408957	+	0.912554i	$0.634107\pi$
$600$		0			0
$601$	27.4370			1.11918			0.559590	−	0.828770i	$-0.310959\pi$
$601$	27.4370			1.11918			0.559590	+	0.828770i	$0.310959\pi$
$602$		0			0
$603$	0.980187	−	1.69773i	0.0399163	−	0.0691371i
$604$		0			0
$605$	5.12606	−	8.87860i	0.208404	−	0.360966i
$606$		0			0
$607$	17.7547			0.720639			0.360320	−	0.932829i	$-0.382668\pi$
$607$	17.7547			0.720639			0.360320	+	0.932829i	$0.382668\pi$
$608$		0			0
$609$	41.2466			1.67140
$610$		0			0
$611$	−1.27175	+	2.20274i	−0.0514496	+	0.0891133i
$612$		0			0
$613$	−17.3196	+	29.9983i	−0.699530	+	1.21162i	0.269099	+	0.963112i	$0.413274\pi$
$613$	−17.3196	+	29.9983i	−0.699530	+	1.21162i	−0.968629	+	0.248509i	$0.920059\pi$
$614$		0			0
$615$	5.98814			0.241465
$616$		0			0
$617$	−2.23284	−	3.86740i	−0.0898909	−	0.155696i	0.817574	−	0.575824i	$-0.195319\pi$
$617$	−2.23284	−	3.86740i	−0.0898909	−	0.155696i	−0.907465	+	0.420128i	$0.861985\pi$
$618$		0			0
$619$	17.9112			0.719913			0.359957	−	0.932969i	$-0.382791\pi$
$619$	17.9112			0.719913			0.359957	+	0.932969i	$0.382791\pi$
$620$		0			0
$621$	1.17531	+	2.03570i	0.0471636	+	0.0816898i
$622$		0			0
$623$	−6.33234	−	10.9679i	−0.253700	−	0.439421i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−0.933688	−	5.54586i	−0.0372879	−	0.221480i
$628$		0			0
$629$	−11.9078	+	20.6250i	−0.474796	+	0.822371i
$630$		0			0
$631$	2.48440	+	4.30311i	0.0989026	+	0.171304i	0.911231	−	0.411896i	$-0.135133\pi$
$631$	2.48440	+	4.30311i	0.0989026	+	0.171304i	−0.812328	+	0.583201i	$0.801800\pi$
$632$		0			0
$633$	−14.1108	−	24.4407i	−0.560855	−	0.971429i
$634$		0			0
$635$	−8.83492			−0.350603
$636$		0			0
$637$	−0.358544	−	0.621016i	−0.0142060	−	0.0246056i
$638$		0			0
$639$	−1.37969			−0.0545796
$640$		0			0
$641$	18.9760	−	32.8675i	0.749508	−	1.29819i	−0.198550	−	0.980091i	$-0.563623\pi$
$641$	18.9760	−	32.8675i	0.749508	−	1.29819i	0.948059	−	0.318096i	$-0.103043\pi$
$642$		0			0
$643$	17.6251	−	30.5276i	0.695067	−	1.20389i	−0.275092	−	0.961418i	$-0.588708\pi$
$643$	17.6251	−	30.5276i	0.695067	−	1.20389i	0.970158	−	0.242473i	$-0.0779585\pi$
$644$		0			0
$645$	−3.07081			−0.120913
$646$		0			0
$647$	−35.5219			−1.39651			−0.698254	−	0.715850i	$-0.746040\pi$
$647$	−35.5219			−1.39651			−0.698254	+	0.715850i	$0.746040\pi$
$648$		0			0
$649$	1.06353	−	1.84209i	0.0417471	−	0.0723082i
$650$		0			0
$651$	−10.4835	+	18.1580i	−0.410881	+	0.711667i
$652$		0			0
$653$	8.02411			0.314008			0.157004	−	0.987598i	$-0.449816\pi$
$653$	8.02411			0.314008			0.157004	+	0.987598i	$0.449816\pi$
$654$		0			0
$655$	10.4564	+	18.1110i	0.408565	+	0.707655i
$656$		0			0
$657$	5.52059			0.215379
$658$		0			0
$659$	−23.6098	−	40.8933i	−0.919706	−	1.59298i	−0.799861	−	0.600185i	$-0.795094\pi$
$659$	−23.6098	−	40.8933i	−0.919706	−	1.59298i	−0.119844	−	0.992793i	$-0.538240\pi$
$660$		0			0
$661$	13.0580	+	22.6171i	0.507896	+	0.879702i	0.999958	+	0.00914181i	$0.00290997\pi$
$661$	13.0580	+	22.6171i	0.507896	+	0.879702i	−0.492062	+	0.870560i	$0.663757\pi$
$662$		0			0
$663$	1.79557	−	3.11002i	0.0697342	−	0.120783i
$664$		0			0
$665$	2.06147	+	12.2446i	0.0799405	+	0.474825i
$666$		0			0
$667$	−2.02577	+	3.50874i	−0.0784383	+	0.135859i
$668$		0			0
$669$	12.0458	+	20.8639i	0.465716	+	0.806643i
$670$		0			0
$671$	−2.73917	−	4.74437i	−0.105744	−	0.183154i
$672$		0			0
$673$	15.3820			0.592931			0.296466	−	0.955044i	$-0.404192\pi$
$673$	15.3820			0.592931			0.296466	+	0.955044i	$0.404192\pi$
$674$		0			0
$675$	−2.81538	−	4.87639i	−0.108364	−	0.187692i
$676$		0			0
$677$	24.4763			0.940701			0.470350	−	0.882480i	$-0.344128\pi$
$677$	24.4763			0.940701			0.470350	+	0.882480i	$0.344128\pi$
$678$		0			0
$679$	15.4598	−	26.7771i	0.593291	−	1.02761i
$680$		0			0
$681$	−19.6326	+	34.0047i	−0.752324	+	1.30306i
$682$		0			0
$683$	17.8502			0.683018			0.341509	−	0.939879i	$-0.389062\pi$
$683$	17.8502			0.683018			0.341509	+	0.939879i	$0.389062\pi$
$684$		0			0
$685$	5.21477			0.199246
$686$		0			0
$687$	−9.94538	+	17.2259i	−0.379440	+	0.657210i
$688$		0			0
$689$	3.53402	−	6.12110i	0.134635	−	0.233195i
$690$		0			0
$691$	9.27242			0.352739			0.176370	−	0.984324i	$-0.443565\pi$
$691$	9.27242			0.352739			0.176370	+	0.984324i	$0.443565\pi$
$692$		0			0
$693$	0.953597	+	1.65168i	0.0362241	+	0.0627421i
$694$		0			0
$695$	10.7238			0.406778
$696$		0			0
$697$	7.50937	+	13.0066i	0.284438	+	0.492660i
$698$		0			0
$699$	18.8798	+	32.7008i	0.714100	+	1.23686i
$700$		0			0
$701$	3.84453	−	6.65892i	0.145206	−	0.251504i	−0.784244	−	0.620453i	$-0.786949\pi$
$701$	3.84453	−	6.65892i	0.145206	−	0.251504i	0.929450	+	0.368949i	$0.120282\pi$
$702$		0			0
$703$	21.3897	−	17.6678i	0.806728	−	0.666354i
$704$		0			0
$705$	2.94949	−	5.10867i	0.111084	−	0.192404i
$706$		0			0
$707$	7.54316	+	13.0651i	0.283690	+	0.491365i
$708$		0			0
$709$	−12.2187	−	21.1635i	−0.458885	−	0.794812i	0.540018	−	0.841654i	$-0.318418\pi$
$709$	−12.2187	−	21.1635i	−0.458885	−	0.794812i	−0.998902	+	0.0468421i	$0.985084\pi$
$710$		0			0
$711$	1.41251			0.0529732
$712$		0			0
$713$	−1.02977	−	1.78361i	−0.0385652	−	0.0667968i
$714$		0			0
$715$	0.556310			0.0208048
$716$		0			0
$717$	17.5673	−	30.4275i	0.656063	−	1.13633i
$718$		0			0
$719$	−11.0563	+	19.1501i	−0.412331	+	0.714178i	−0.995144	−	0.0984282i	$-0.968619\pi$
$719$	−11.0563	+	19.1501i	−0.412331	+	0.714178i	0.582813	+	0.812606i	$0.301952\pi$
$720$		0			0
$721$	1.09711			0.0408584
$722$		0			0
$723$	−12.5085			−0.465195
$724$		0			0
$725$	4.85261	−	8.40497i	0.180222	−	0.312153i
$726$		0			0
$727$	−14.5247	+	25.1575i	−0.538692	+	0.933042i	0.460283	+	0.887772i	$0.347748\pi$
$727$	−14.5247	+	25.1575i	−0.538692	+	0.933042i	−0.998975	+	0.0452694i	$0.985585\pi$
$728$		0			0
$729$	29.9075			1.10768
$730$		0			0
$731$	−3.85092	−	6.66999i	−0.142431	−	0.246698i
$732$		0			0
$733$	14.5428			0.537151			0.268576	−	0.963259i	$-0.413447\pi$
$733$	14.5428			0.537151			0.268576	+	0.963259i	$0.413447\pi$
$734$		0			0
$735$	0.831547	+	1.44028i	0.0306721	+	0.0531256i
$736$		0			0
$737$	−1.09492	−	1.89646i	−0.0403320	−	0.0698570i
$738$		0			0
$739$	−2.37798	+	4.11878i	−0.0874754	+	0.151512i	−0.906443	−	0.422327i	$-0.861213\pi$
$739$	−2.37798	+	4.11878i	−0.0874754	+	0.151512i	0.818968	+	0.573839i	$0.194547\pi$
$740$		0			0
$741$	−3.22533	+	2.66411i	−0.118486	+	0.0978687i
$742$		0			0
$743$	2.93853	−	5.08968i	0.107804	−	0.186722i	−0.807076	−	0.590447i	$-0.798951\pi$
$743$	2.93853	−	5.08968i	0.107804	−	0.186722i	0.914880	+	0.403725i	$0.132285\pi$
$744$		0			0
$745$	7.45578	+	12.9138i	0.273159	+	0.473125i
$746$		0			0
$747$	2.87961	+	4.98763i	0.105359	+	0.182488i
$748$		0			0
$749$	18.3263			0.669629
$750$		0			0
$751$	0.810481	+	1.40379i	0.0295749	+	0.0512252i	0.880434	−	0.474169i	$-0.157251\pi$
$751$	0.810481	+	1.40379i	0.0295749	+	0.0512252i	−0.850859	+	0.525394i	$0.823918\pi$
$752$		0			0
$753$	27.2243			0.992107
$754$		0			0
$755$	10.7295	−	18.5840i	0.390485	−	0.676341i
$756$		0			0
$757$	14.0567	−	24.3470i	0.510901	−	0.884907i	−0.489019	−	0.872273i	$-0.662645\pi$
$757$	14.0567	−	24.3470i	0.510901	−	0.884907i	0.999920	−	0.0126336i	$-0.00402151\pi$
$758$		0			0
$759$	0.538612			0.0195504
$760$		0			0
$761$	20.1663			0.731027			0.365514	−	0.930806i	$-0.380893\pi$
$761$	20.1663			0.731027			0.365514	+	0.930806i	$0.380893\pi$
$762$		0			0
$763$	−9.35610	+	16.2052i	−0.338713	+	0.586669i
$764$		0			0
$765$	1.44843	−	2.50876i	0.0523682	−	0.0907044i
$766$		0			0
$767$	−1.58221			−0.0571303
$768$		0			0
$769$	−22.6524	−	39.2350i	−0.816865	−	1.41485i	−0.907981	−	0.419011i	$-0.862377\pi$
$769$	−22.6524	−	39.2350i	−0.816865	−	1.41485i	0.0911160	−	0.995840i	$-0.470957\pi$
$770$		0			0
$771$	−21.0357			−0.757583
$772$		0			0
$773$	10.0881	+	17.4731i	0.362843	+	0.628462i	0.988428	−	0.151694i	$-0.0484728\pi$
$773$	10.0881	+	17.4731i	0.362843	+	0.628462i	−0.625585	+	0.780156i	$0.715139\pi$
$774$		0			0
$775$	2.46675	+	4.27253i	0.0886081	+	0.153474i
$776$		0			0
$777$	13.5247	−	23.4255i	0.485196	−	0.840385i
$778$		0			0
$779$	−2.90461	−	17.2526i	−0.104068	−	0.618138i
$780$		0			0
$781$	−0.770594	+	1.33471i	−0.0275740	+	0.0477596i
$782$		0			0
$783$	27.3239	+	47.3264i	0.976478	+	1.69131i
$784$		0			0
$785$	1.21559	+	2.10546i	0.0433862	+	0.0751471i
$786$		0			0
$787$	46.1385			1.64466			0.822331	−	0.569010i	$-0.192673\pi$
$787$	46.1385			1.64466			0.822331	+	0.569010i	$0.192673\pi$
$788$		0			0
$789$	4.78213	+	8.28289i	0.170248	+	0.294879i
$790$		0			0
$791$	0.838961			0.0298300
$792$		0			0

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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.745959 - 1.29204i) q^{3} +(-0.500000 + 0.866025i) q^{5} +2.84864 q^{7} +(0.387090 + 0.670459i) q^{9} +O(q^{10})\)	\(q+(0.745959 - 1.29204i) q^{3} +(-0.500000 + 0.866025i) q^{5} +2.84864 q^{7} +(0.387090 + 0.670459i) q^{9} +0.864801 q^{11} +(-0.321640 - 0.557098i) q^{13} +(0.745959 + 1.29204i) q^{15} +(-1.87093 + 3.24054i) q^{17} +(3.36069 - 2.77592i) q^{19} +(2.12497 - 3.68055i) q^{21} +(0.208730 + 0.361531i) q^{23} +(-0.500000 - 0.866025i) q^{25} +5.63077 q^{27} +(4.85261 + 8.40497i) q^{29} -4.93349 q^{31} +(0.645106 - 1.11736i) q^{33} +(-1.42432 + 2.46699i) q^{35} +6.36467 q^{37} -0.959723 q^{39} +(2.00686 - 3.47598i) q^{41} +(-1.02915 + 1.78254i) q^{43} -0.774179 q^{45} +(-1.97698 - 3.42423i) q^{47} +1.11474 q^{49} +(2.79127 + 4.83462i) q^{51} +(5.49374 + 9.51544i) q^{53} +(-0.432400 + 0.748939i) q^{55} +(-1.07966 - 6.41287i) q^{57} +(1.22980 - 2.13007i) q^{59} +(-3.16740 - 5.48609i) q^{61} +(1.10268 + 1.90989i) q^{63} +0.643281 q^{65} +(-1.26610 - 2.19295i) q^{67} +0.622817 q^{69} +(-0.891065 + 1.54337i) q^{71} +(3.56545 - 6.17554i) q^{73} -1.49192 q^{75} +2.46350 q^{77} +(0.912262 - 1.58008i) q^{79} +(3.03905 - 5.26380i) q^{81} +7.43913 q^{83} +(-1.87093 - 3.24054i) q^{85} +14.4794 q^{87} +(-2.22294 - 3.85024i) q^{89} +(-0.916237 - 1.58697i) q^{91} +(-3.68018 + 6.37427i) q^{93} +(0.723670 + 4.29841i) q^{95} +(5.42707 - 9.39996i) q^{97} +(0.334755 + 0.579813i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 3 q^{3} - 4 q^{5} + 8 q^{7} - q^{9}+O(q^{10}) \) 8 * q + 3 * q^3 - 4 * q^5 + 8 * q^7 - q^9	\( 8 q + 3 q^{3} - 4 q^{5} + 8 q^{7} - q^{9} + 4 q^{11} - 7 q^{13} + 3 q^{15} + q^{17} - 5 q^{19} + 4 q^{21} + 2 q^{23} - 4 q^{25} - 24 q^{27} + q^{29} - 19 q^{33} - 4 q^{35} - 4 q^{37} - 30 q^{39} + 8 q^{41} + q^{43} + 2 q^{45} - 12 q^{47} - 20 q^{49} + 22 q^{51} + 5 q^{53} - 2 q^{55} + 7 q^{57} - 5 q^{59} - 3 q^{63} + 14 q^{65} + 4 q^{67} - 18 q^{69} + 20 q^{71} + 20 q^{73} - 6 q^{75} + 28 q^{77} + 17 q^{79} - 12 q^{81} - 2 q^{83} + q^{85} + 32 q^{87} - 11 q^{89} + 6 q^{91} + 8 q^{93} + 4 q^{95} - q^{97} + 38 q^{99}+O(q^{100}) \) 8 * q + 3 * q^3 - 4 * q^5 + 8 * q^7 - q^9 + 4 * q^11 - 7 * q^13 + 3 * q^15 + q^17 - 5 * q^19 + 4 * q^21 + 2 * q^23 - 4 * q^25 - 24 * q^27 + q^29 - 19 * q^33 - 4 * q^35 - 4 * q^37 - 30 * q^39 + 8 * q^41 + q^43 + 2 * q^45 - 12 * q^47 - 20 * q^49 + 22 * q^51 + 5 * q^53 - 2 * q^55 + 7 * q^57 - 5 * q^59 - 3 * q^63 + 14 * q^65 + 4 * q^67 - 18 * q^69 + 20 * q^71 + 20 * q^73 - 6 * q^75 + 28 * q^77 + 17 * q^79 - 12 * q^81 - 2 * q^83 + q^85 + 32 * q^87 - 11 * q^89 + 6 * q^91 + 8 * q^93 + 4 * q^95 - q^97 + 38 * q^99

Introduction

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