LMFDB - Embedded newform 637.2.c.d.246.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [637,2,Mod(246,637)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(637, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("637.246");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.c (of order $2$, degree $1$, 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:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 91)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			246.2
Root			$-0.854638 + 0.854638i$ of defining polynomial
Character	$\chi$	$=$	637.246
Dual form			637.2.c.d.246.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.17009i q^{2} -0.539189 q^{3} +0.630898 q^{4} -0.460811i q^{5} +0.630898i q^{6} -3.07838i q^{8} -2.70928 q^{9} +O(q^{10})$	\(q-1.17009i q^{2} -0.539189 q^{3} +0.630898 q^{4} -0.460811i q^{5} +0.630898i q^{6} -3.07838i q^{8} -2.70928 q^{9} -0.539189 q^{10} -0.829914i q^{11} -0.340173 q^{12} +(2.87936 - 2.17009i) q^{13} +0.248464i q^{15} -2.34017 q^{16} -2.87936 q^{17} +3.17009i q^{18} -4.32684i q^{19} -0.290725i q^{20} -0.971071 q^{22} -5.04945 q^{23} +1.65983i q^{24} +4.78765 q^{25} +(-2.53919 - 3.36910i) q^{26} +3.07838 q^{27} +0.261795 q^{29} +0.290725 q^{30} -6.80098i q^{31} -3.41855i q^{32} +0.447480i q^{33} +3.36910i q^{34} -1.70928 q^{36} -9.51026i q^{37} -5.06278 q^{38} +(-1.55252 + 1.17009i) q^{39} -1.41855 q^{40} +6.68035i q^{41} +0.418551 q^{43} -0.523590i q^{44} +1.24846i q^{45} +5.90829i q^{46} +9.24846i q^{47} +1.26180 q^{48} -5.60197i q^{50} +1.55252 q^{51} +(1.81658 - 1.36910i) q^{52} +1.63090 q^{53} -3.60197i q^{54} -0.382433 q^{55} +2.33299i q^{57} -0.306323i q^{58} -2.78765i q^{59} +0.156755i q^{60} -7.26180 q^{61} -7.95774 q^{62} -8.68035 q^{64} +(-1.00000 - 1.32684i) q^{65} +0.523590 q^{66} -5.07838i q^{67} -1.81658 q^{68} +2.72261 q^{69} +12.7721i q^{71} +8.34017i q^{72} +0.353504i q^{73} -11.1278 q^{74} -2.58145 q^{75} -2.72979i q^{76} +(1.36910 + 1.81658i) q^{78} +2.81658 q^{79} +1.07838i q^{80} +6.46800 q^{81} +7.81658 q^{82} -10.3763i q^{83} +1.32684i q^{85} -0.489741i q^{86} -0.141157 q^{87} -2.55479 q^{88} +5.43188i q^{89} +1.46081 q^{90} -3.18568 q^{92} +3.66701i q^{93} +10.8215 q^{94} -1.99386 q^{95} +1.84324i q^{96} +12.6092i q^{97} +2.24846i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{4} - 2 q^{9}+O(q^{10}) $ 6 * q - 4 * q^4 - 2 * q^9	$ 6 q - 4 q^{4} - 2 q^{9} + 20 q^{12} - 8 q^{13} + 8 q^{16} + 8 q^{17} + 24 q^{22} + 6 q^{23} + 8 q^{25} - 12 q^{26} + 12 q^{27} - 14 q^{29} + 16 q^{30} + 4 q^{36} + 4 q^{38} - 8 q^{39} + 20 q^{40} - 26 q^{43} - 8 q^{48} + 8 q^{51} + 20 q^{52} + 2 q^{53} - 12 q^{55} - 28 q^{61} - 16 q^{62} - 8 q^{64} - 6 q^{65} - 28 q^{66} - 20 q^{68} + 4 q^{69} - 24 q^{74} - 44 q^{75} + 16 q^{78} + 26 q^{79} - 26 q^{81} + 56 q^{82} + 40 q^{87} - 40 q^{88} + 12 q^{90} - 36 q^{92} - 20 q^{94} + 58 q^{95}+O(q^{100}) $ 6 * q - 4 * q^4 - 2 * q^9 + 20 * q^12 - 8 * q^13 + 8 * q^16 + 8 * q^17 + 24 * q^22 + 6 * q^23 + 8 * q^25 - 12 * q^26 + 12 * q^27 - 14 * q^29 + 16 * q^30 + 4 * q^36 + 4 * q^38 - 8 * q^39 + 20 * q^40 - 26 * q^43 - 8 * q^48 + 8 * q^51 + 20 * q^52 + 2 * q^53 - 12 * q^55 - 28 * q^61 - 16 * q^62 - 8 * q^64 - 6 * q^65 - 28 * q^66 - 20 * q^68 + 4 * q^69 - 24 * q^74 - 44 * q^75 + 16 * q^78 + 26 * q^79 - 26 * q^81 + 56 * q^82 + 40 * q^87 - 40 * q^88 + 12 * q^90 - 36 * q^92 - 20 * q^94 + 58 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/637\mathbb{Z}\right)^\times$.

$n$	$197$	$248$
$\chi(n)$	$-1$	$1$

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 $

$2$		−	1.17009i		−	0.827376i	−0.910419	−	0.413688i	$-0.864240\pi$
$2$		−	1.17009i		−	0.827376i	0.910419	−	0.413688i	$-0.135760\pi$
$3$	−0.539189			−0.311301			−0.155650	−	0.987812i	$-0.549747\pi$
$3$	−0.539189			−0.311301			−0.155650	+	0.987812i	$0.549747\pi$
$4$	0.630898			0.315449
$5$		−	0.460811i		−	0.206081i	−0.994677	−	0.103041i	$-0.967143\pi$
$5$		−	0.460811i		−	0.206081i	0.994677	−	0.103041i	$-0.0328571\pi$
$6$			0.630898i			0.257563i
$7$		0			0
$7$		0			0
$8$		−	3.07838i		−	1.08837i
$9$	−2.70928			−0.903092
$10$	−0.539189			−0.170506
$11$		−	0.829914i		−	0.250228i	−0.992142	−	0.125114i	$-0.960070\pi$
$11$		−	0.829914i		−	0.250228i	0.992142	−	0.125114i	$-0.0399297\pi$
$12$	−0.340173			−0.0981995
$13$	2.87936	−	2.17009i	0.798591	−	0.601874i
$13$	2.87936	−	2.17009i	0.798591	−	0.601874i
$14$		0			0
$15$			0.248464i			0.0641532i
$16$	−2.34017			−0.585043
$17$	−2.87936			−0.698348			−0.349174	−	0.937058i	$-0.613538\pi$
$17$	−2.87936			−0.698348			−0.349174	+	0.937058i	$0.613538\pi$
$18$			3.17009i			0.747197i
$19$		−	4.32684i		−	0.992646i	−0.868138	−	0.496323i	$-0.834683\pi$
$19$		−	4.32684i		−	0.992646i	0.868138	−	0.496323i	$-0.165317\pi$
$20$		−	0.290725i		−	0.0650080i
$21$		0			0
$22$	−0.971071			−0.207033
$23$	−5.04945			−1.05288			−0.526441	−	0.850211i	$-0.676474\pi$
$23$	−5.04945			−1.05288			−0.526441	+	0.850211i	$0.676474\pi$
$24$			1.65983i			0.338811i
$25$	4.78765			0.957531
$26$	−2.53919	−	3.36910i	−0.497976	−	0.660735i
$27$	3.07838			0.592434
$28$		0			0
$29$	0.261795			0.0486142			0.0243071	−	0.999705i	$-0.492262\pi$
$29$	0.261795			0.0486142			0.0243071	+	0.999705i	$0.492262\pi$
$30$	0.290725			0.0530788
$31$		−	6.80098i		−	1.22149i	−0.791826	−	0.610746i	$-0.790869\pi$
$31$		−	6.80098i		−	1.22149i	0.791826	−	0.610746i	$-0.209131\pi$
$32$		−	3.41855i		−	0.604320i
$33$			0.447480i			0.0778963i
$34$			3.36910i			0.577796i
$35$		0			0
$36$	−1.70928			−0.284879
$37$		−	9.51026i		−	1.56348i	−0.623606	−	0.781739i	$-0.714333\pi$
$37$		−	9.51026i		−	1.56348i	0.623606	−	0.781739i	$-0.285667\pi$
$38$	−5.06278			−0.821291
$39$	−1.55252	+	1.17009i	−0.248602	+	0.187364i
$40$	−1.41855			−0.224293
$41$			6.68035i			1.04329i	0.853161	+	0.521647i	$0.174682\pi$
$41$			6.68035i			1.04329i	−0.853161	+	0.521647i	$0.825318\pi$
$42$		0			0
$43$	0.418551			0.0638284			0.0319142	−	0.999491i	$-0.489840\pi$
$43$	0.418551			0.0638284			0.0319142	+	0.999491i	$0.489840\pi$
$44$		−	0.523590i		−	0.0789342i
$45$			1.24846i			0.186110i
$46$			5.90829i			0.871130i
$47$			9.24846i			1.34903i	0.738262	+	0.674514i	$0.235647\pi$
$47$			9.24846i			1.34903i	−0.738262	+	0.674514i	$0.764353\pi$
$48$	1.26180			0.182124
$49$		0			0
$50$		−	5.60197i		−	0.792238i
$51$	1.55252			0.217396
$52$	1.81658	−	1.36910i	0.251915	−	0.189860i
$53$	1.63090			0.224021			0.112011	−	0.993707i	$-0.464271\pi$
$53$	1.63090			0.224021			0.112011	+	0.993707i	$0.464271\pi$
$54$		−	3.60197i		−	0.490166i
$55$	−0.382433			−0.0515673
$56$		0			0
$57$			2.33299i			0.309011i
$58$		−	0.306323i		−	0.0402222i
$59$		−	2.78765i		−	0.362922i	−0.983398	−	0.181461i	$-0.941917\pi$
$59$		−	2.78765i		−	0.362922i	0.983398	−	0.181461i	$-0.0580825\pi$
$60$			0.156755i			0.0202370i
$61$	−7.26180			−0.929778			−0.464889	−	0.885369i	$-0.653906\pi$
$61$	−7.26180			−0.929778			−0.464889	+	0.885369i	$0.653906\pi$
$62$	−7.95774			−1.01063
$63$		0			0
$64$	−8.68035			−1.08504
$65$	−1.00000	−	1.32684i	−0.124035	−	0.164574i
$66$	0.523590			0.0644495
$67$		−	5.07838i		−	0.620423i	−0.950668	−	0.310211i	$-0.899600\pi$
$67$		−	5.07838i		−	0.620423i	0.950668	−	0.310211i	$-0.100400\pi$
$68$	−1.81658			−0.220293
$69$	2.72261			0.327763
$70$		0			0
$71$			12.7721i			1.51576i	0.652392	+	0.757882i	$0.273766\pi$
$71$			12.7721i			1.51576i	−0.652392	+	0.757882i	$0.726234\pi$
$72$			8.34017i			0.982899i
$73$			0.353504i			0.0413745i	0.999786	+	0.0206873i	$0.00658543\pi$
$73$			0.353504i			0.0413745i	−0.999786	+	0.0206873i	$0.993415\pi$
$74$	−11.1278			−1.29358
$75$	−2.58145			−0.298080
$76$		−	2.72979i		−	0.313129i
$77$		0			0
$78$	1.36910	+	1.81658i	0.155020	+	0.205687i
$79$	2.81658			0.316890			0.158445	−	0.987368i	$-0.449352\pi$
$79$	2.81658			0.316890			0.158445	+	0.987368i	$0.449352\pi$
$80$			1.07838i			0.120566i
$81$	6.46800			0.718667
$82$	7.81658			0.863197
$83$		−	10.3763i		−	1.13895i	−0.822010	−	0.569473i	$-0.807147\pi$
$83$		−	10.3763i		−	1.13895i	0.822010	−	0.569473i	$-0.192853\pi$
$84$		0			0
$85$			1.32684i			0.143916i
$86$		−	0.489741i		−	0.0528101i
$87$	−0.141157			−0.0151336
$88$	−2.55479			−0.272341
$89$			5.43188i			0.575778i	0.957664	+	0.287889i	$0.0929535\pi$
$89$			5.43188i			0.575778i	−0.957664	+	0.287889i	$0.907047\pi$
$90$	1.46081			0.153983
$91$		0			0
$92$	−3.18568			−0.332131
$93$			3.66701i			0.380252i
$94$	10.8215			1.11615
$95$	−1.99386			−0.204565
$96$			1.84324i			0.188125i
$97$			12.6092i			1.28027i	0.768264	+	0.640133i	$0.221121\pi$
$97$			12.6092i			1.28027i	−0.768264	+	0.640133i	$0.778879\pi$
$98$		0			0
$99$			2.24846i			0.225979i
$100$	3.02052			0.302052
$101$	3.64423			0.362614			0.181307	−	0.983427i	$-0.441967\pi$
$101$	3.64423			0.362614			0.181307	+	0.983427i	$0.441967\pi$
$102$		−	1.81658i		−	0.179868i
$103$	17.9421			1.76789			0.883946	−	0.467589i	$-0.154877\pi$
$103$	17.9421			1.76789			0.883946	+	0.467589i	$0.154877\pi$
$104$	−6.68035	−	8.86376i	−0.655062	−	0.869164i
$105$		0			0
$106$		−	1.90829i		−	0.185350i
$107$	2.94441			0.284647			0.142323	−	0.989820i	$-0.454543\pi$
$107$	2.94441			0.284647			0.142323	+	0.989820i	$0.454543\pi$
$108$	1.94214			0.186883
$109$			8.43188i			0.807628i	0.914841	+	0.403814i	$0.132316\pi$
$109$			8.43188i			0.807628i	−0.914841	+	0.403814i	$0.867684\pi$
$110$			0.447480i			0.0426656i
$111$			5.12783i			0.486712i
$112$		0			0
$113$	12.8082			1.20489			0.602446	−	0.798160i	$-0.294193\pi$
$113$	12.8082			1.20489			0.602446	+	0.798160i	$0.294193\pi$
$114$	2.72979			0.255669
$115$			2.32684i			0.216979i
$116$	0.165166			0.0153353
$117$	−7.80098	+	5.87936i	−0.721201	+	0.543547i
$118$	−3.26180			−0.300273
$119$		0			0
$120$	0.764867			0.0698225
$121$	10.3112			0.937386
$122$			8.49693i			0.769276i
$123$		−	3.60197i		−	0.324779i
$124$		−	4.29072i		−	0.385318i
$125$		−	4.51026i		−	0.403410i
$126$		0			0
$127$	13.4680			1.19509			0.597546	−	0.801835i	$-0.296143\pi$
$127$	13.4680			1.19509			0.597546	+	0.801835i	$0.296143\pi$
$128$			3.31965i			0.293419i
$129$	−0.225678			−0.0198698
$130$	−1.55252	+	1.17009i	−0.136165	+	0.102623i
$131$	14.4391			1.26155			0.630774	−	0.775967i	$-0.282738\pi$
$131$	14.4391			1.26155			0.630774	+	0.775967i	$0.282738\pi$
$132$			0.282314i			0.0245723i
$133$		0			0
$134$	−5.94214			−0.513323
$135$		−	1.41855i		−	0.122089i
$136$			8.86376i			0.760061i
$137$		−	16.4319i		−	1.40387i	−0.712241	−	0.701935i	$-0.752320\pi$
$137$		−	16.4319i		−	1.40387i	0.712241	−	0.701935i	$-0.247680\pi$
$138$		−	3.18568i		−	0.271184i
$139$	−13.2195			−1.12127			−0.560633	−	0.828064i	$-0.689442\pi$
$139$	−13.2195			−1.12127			−0.560633	+	0.828064i	$0.689442\pi$
$140$		0			0
$141$		−	4.98667i		−	0.419953i
$142$	14.9444			1.25411
$143$	−1.80098	−	2.38962i	−0.150606	−	0.199830i
$144$	6.34017			0.528348
$145$		−	0.120638i		−	0.0100185i
$146$	0.413630			0.0342323
$147$		0			0
$148$		−	6.00000i		−	0.493197i
$149$			20.5041i			1.67976i	0.542770	+	0.839881i	$0.317376\pi$
$149$			20.5041i			1.67976i	−0.542770	+	0.839881i	$0.682624\pi$
$150$			3.02052i			0.246624i
$151$			1.59478i			0.129781i	0.997892	+	0.0648907i	$0.0206699\pi$
$151$			1.59478i			0.129781i	−0.997892	+	0.0648907i	$0.979330\pi$
$152$	−13.3197			−1.08037
$153$	7.80098			0.630672
$154$		0			0
$155$	−3.13397			−0.251726
$156$	−0.979481	+	0.738205i	−0.0784212	+	0.0591037i
$157$	8.29791			0.662246			0.331123	−	0.943588i	$-0.392573\pi$
$157$	8.29791			0.662246			0.331123	+	0.943588i	$0.392573\pi$
$158$		−	3.29565i		−	0.262187i
$159$	−0.879362			−0.0697379
$160$	−1.57531			−0.124539
$161$		0			0
$162$		−	7.56812i		−	0.594608i
$163$		−	0.398032i		−	0.0311763i	−0.999878	−	0.0155881i	$-0.995038\pi$
$163$		−	0.398032i		−	0.0311763i	0.999878	−	0.0155881i	$-0.00496206\pi$
$164$			4.21461i			0.329106i
$165$	0.206204			0.0160529
$166$	−12.1412			−0.942337
$167$		−	23.8710i		−	1.84719i	−0.383370	−	0.923595i	$-0.625237\pi$
$167$		−	23.8710i		−	1.84719i	0.383370	−	0.923595i	$-0.374763\pi$
$168$		0			0
$169$	3.58145	−	12.4969i	0.275496	−	0.961302i
$170$	1.55252			0.119073
$171$			11.7226i			0.896450i
$172$	0.264063			0.0201346
$173$	25.8999			1.96913			0.984566	−	0.175015i	$-0.0559975\pi$
$173$	25.8999			1.96913			0.984566	+	0.175015i	$0.0559975\pi$
$174$			0.165166i			0.0125212i
$175$		0			0
$176$			1.94214i			0.146394i
$177$			1.50307i			0.112978i
$178$	6.35577			0.476385
$179$	−20.6742			−1.54526			−0.772631	−	0.634855i	$-0.781060\pi$
$179$	−20.6742			−1.54526			−0.772631	+	0.634855i	$0.781060\pi$
$180$			0.787653i			0.0587082i
$181$	15.7165			1.16820			0.584098	−	0.811683i	$-0.301448\pi$
$181$	15.7165			1.16820			0.584098	+	0.811683i	$0.301448\pi$
$182$		0			0
$183$	3.91548			0.289441
$184$			15.5441i			1.14593i
$185$	−4.38243			−0.322203
$186$	4.29072			0.314611
$187$			2.38962i			0.174746i
$188$			5.83483i			0.425549i
$189$		0			0
$190$			2.33299i			0.169253i
$191$	−16.7877			−1.21471			−0.607356	−	0.794430i	$-0.707770\pi$
$191$	−16.7877			−1.21471			−0.607356	+	0.794430i	$0.707770\pi$
$192$	4.68035			0.337775
$193$			20.0144i			1.44067i	0.693628	+	0.720333i	$0.256011\pi$
$193$			20.0144i			1.44067i	−0.693628	+	0.720333i	$0.743989\pi$
$194$	14.7538			1.05926
$195$	0.539189	+	0.715418i	0.0386121	+	0.0512322i
$196$		0			0
$197$			17.5174i			1.24807i	0.781398	+	0.624033i	$0.214507\pi$
$197$			17.5174i			1.24807i	−0.781398	+	0.624033i	$0.785493\pi$
$198$	2.63090			0.186970
$199$	−16.2979			−1.15533			−0.577664	−	0.816275i	$-0.696036\pi$
$199$	−16.2979			−1.15533			−0.577664	+	0.816275i	$0.696036\pi$
$200$		−	14.7382i		−	1.04215i
$201$			2.73820i			0.193138i
$202$		−	4.26406i		−	0.300018i
$203$		0			0
$204$	0.979481			0.0685774
$205$	3.07838			0.215003
$206$		−	20.9939i		−	1.46271i
$207$	13.6803			0.950850
$208$	−6.73820	+	5.07838i	−0.467210	+	0.352122i
$209$	−3.59090			−0.248388
$210$		0			0
$211$	2.70928			0.186514			0.0932571	−	0.995642i	$-0.470272\pi$
$211$	2.70928			0.186514			0.0932571	+	0.995642i	$0.470272\pi$
$212$	1.02893			0.0706672
$213$		−	6.88655i		−	0.471859i
$214$		−	3.44521i		−	0.235510i
$215$		−	0.192873i		−	0.0131538i
$216$		−	9.47641i		−	0.644788i
$217$		0			0
$218$	9.86603			0.668212
$219$		−	0.190605i		−	0.0128799i
$220$	−0.241276			−0.0162668
$221$	−8.29072	+	6.24846i	−0.557694	+	0.420317i
$222$	6.00000			0.402694
$223$			19.7948i			1.32556i	0.748814	+	0.662780i	$0.230624\pi$
$223$			19.7948i			1.32556i	−0.748814	+	0.662780i	$0.769376\pi$
$224$		0			0
$225$	−12.9711			−0.864738
$226$		−	14.9867i		−	0.996898i
$227$		−	13.2618i		−	0.880216i	−0.897945	−	0.440108i	$-0.854940\pi$
$227$		−	13.2618i		−	0.880216i	0.897945	−	0.440108i	$-0.145060\pi$
$228$			1.47187i			0.0974773i
$229$		−	27.8082i		−	1.83762i	−0.394705	−	0.918808i	$-0.629153\pi$
$229$		−	27.8082i		−	1.83762i	0.394705	−	0.918808i	$-0.370847\pi$
$230$	2.72261			0.179523
$231$		0			0
$232$		−	0.805905i		−	0.0529102i
$233$	13.3668			0.875690			0.437845	−	0.899050i	$-0.355742\pi$
$233$	13.3668			0.875690			0.437845	+	0.899050i	$0.355742\pi$
$234$	6.87936	+	9.12783i	0.449718	+	0.596705i
$235$	4.26180			0.278009
$236$		−	1.75872i		−	0.114483i
$237$	−1.51867			−0.0986482
$238$		0			0
$239$		−	8.70701i		−	0.563210i	−0.959530	−	0.281605i	$-0.909133\pi$
$239$		−	8.70701i		−	0.563210i	0.959530	−	0.281605i	$-0.0908667\pi$
$240$		−	0.581449i		−	0.0375324i
$241$			16.9311i			1.09063i	0.838232	+	0.545313i	$0.183589\pi$
$241$			16.9311i			1.09063i	−0.838232	+	0.545313i	$0.816411\pi$
$242$		−	12.0650i		−	0.775571i
$243$	−12.7226			−0.816156
$244$	−4.58145			−0.293297
$245$		0			0
$246$	−4.21461			−0.268714
$247$	−9.38962	−	12.4585i	−0.597447	−	0.792718i
$248$	−20.9360			−1.32944
$249$			5.59478i			0.354555i
$250$	−5.27739			−0.333772
$251$	−8.82150			−0.556808			−0.278404	−	0.960464i	$-0.589805\pi$
$251$	−8.82150			−0.556808			−0.278404	+	0.960464i	$0.589805\pi$
$252$		0			0
$253$			4.19061i			0.263461i
$254$		−	15.7587i		−	0.988790i
$255$		−	0.715418i		−	0.0448012i
$256$	−13.4764			−0.842276
$257$	−13.6442			−0.851104			−0.425552	−	0.904934i	$-0.639920\pi$
$257$	−13.6442			−0.851104			−0.425552	+	0.904934i	$0.639920\pi$
$258$			0.264063i			0.0164398i
$259$		0			0
$260$	−0.630898	−	0.837101i	−0.0391266	−	0.0519148i
$261$	−0.709275			−0.0439030
$262$		−	16.8950i		−	1.04377i
$263$	−12.1773			−0.750883			−0.375441	−	0.926846i	$-0.622509\pi$
$263$	−12.1773			−0.750883			−0.375441	+	0.926846i	$0.622509\pi$
$264$	1.37751			0.0847801
$265$		−	0.751536i		−	0.0461665i
$266$		0			0
$267$		−	2.92881i		−	0.179240i
$268$		−	3.20394i		−	0.195712i
$269$	11.0472			0.673559			0.336779	−	0.941584i	$-0.390662\pi$
$269$	11.0472			0.673559			0.336779	+	0.941584i	$0.390662\pi$
$270$	−1.65983			−0.101014
$271$			10.9360i			0.664315i	0.943224	+	0.332157i	$0.107776\pi$
$271$			10.9360i			0.664315i	−0.943224	+	0.332157i	$0.892224\pi$
$272$	6.73820			0.408564
$273$		0			0
$274$	−19.2267			−1.16153
$275$		−	3.97334i		−	0.239601i
$276$	1.71769			0.103393
$277$	26.1194			1.56936			0.784682	−	0.619899i	$-0.212826\pi$
$277$	26.1194			1.56936			0.784682	+	0.619899i	$0.212826\pi$
$278$			15.4680i			0.927709i
$279$			18.4257i			1.10312i
$280$		0			0
$281$			4.80325i			0.286538i	0.989684	+	0.143269i	$0.0457614\pi$
$281$			4.80325i			0.286538i	−0.989684	+	0.143269i	$0.954239\pi$
$282$	−5.83483			−0.347459
$283$	0.979481			0.0582241			0.0291121	−	0.999576i	$-0.490732\pi$
$283$	0.979481			0.0582241			0.0291121	+	0.999576i	$0.490732\pi$
$284$			8.05786i			0.478146i
$285$	1.07507			0.0636814
$286$	−2.79606	+	2.10731i	−0.165335	+	0.124608i
$287$		0			0
$288$			9.26180i			0.545757i
$289$	−8.70928			−0.512310
$290$	−0.141157			−0.00828903
$291$		−	6.79872i		−	0.398548i
$292$			0.223025i			0.0130515i
$293$			24.3268i			1.42119i	0.703602	+	0.710595i	$0.251574\pi$
$293$			24.3268i			1.42119i	−0.703602	+	0.710595i	$0.748426\pi$
$294$		0			0
$295$	−1.28458			−0.0747912
$296$	−29.2762			−1.70164
$297$		−	2.55479i		−	0.148244i
$298$	23.9916			1.38980
$299$	−14.5392	+	10.9577i	−0.840823	+	0.633702i
$300$	−1.62863			−0.0940290
$301$		0			0
$302$	1.86603			0.107378
$303$	−1.96493			−0.112882
$304$			10.1256i			0.580741i
$305$			3.34632i			0.191609i
$306$		−	9.12783i		−	0.521803i
$307$		−	6.21953i		−	0.354968i	−0.984124	−	0.177484i	$-0.943204\pi$
$307$		−	6.21953i		−	0.354968i	0.984124	−	0.177484i	$-0.0567957\pi$
$308$		0			0
$309$	−9.67420			−0.550346
$310$			3.66701i			0.208272i
$311$	−25.1084			−1.42376			−0.711882	−	0.702299i	$-0.752157\pi$
$311$	−25.1084			−1.42376			−0.711882	+	0.702299i	$0.752157\pi$
$312$	3.60197	+	4.77924i	0.203921	+	0.270571i
$313$	−2.24005			−0.126615			−0.0633077	−	0.997994i	$-0.520165\pi$
$313$	−2.24005			−0.126615			−0.0633077	+	0.997994i	$0.520165\pi$
$314$		−	9.70928i		−	0.547926i
$315$		0			0
$316$	1.77698			0.0999627
$317$		−	2.89496i		−	0.162597i	−0.996690	−	0.0812986i	$-0.974093\pi$
$317$		−	2.89496i		−	0.162597i	0.996690	−	0.0812986i	$-0.0259067\pi$
$318$			1.02893i			0.0576995i
$319$		−	0.217267i		−	0.0121646i
$320$			4.00000i			0.223607i
$321$	−1.58759			−0.0886108
$322$		0			0
$323$			12.4585i			0.693212i
$324$	4.08065			0.226703
$325$	13.7854	−	10.3896i	0.764676	−	0.576312i
$326$	−0.465732			−0.0257945
$327$		−	4.54638i		−	0.251415i
$328$	20.5646			1.13549
$329$		0			0
$330$		−	0.241276i		−	0.0132818i
$331$		−	4.25565i		−	0.233912i	−0.993137	−	0.116956i	$-0.962686\pi$
$331$		−	4.25565i		−	0.233912i	0.993137	−	0.116956i	$-0.0373136\pi$
$332$		−	6.54638i		−	0.359279i
$333$			25.7659i			1.41196i
$334$	−27.9311			−1.52832
$335$	−2.34017			−0.127857
$336$		0			0
$337$	−15.5464			−0.846865			−0.423433	−	0.905928i	$-0.639175\pi$
$337$	−15.5464			−0.846865			−0.423433	+	0.905928i	$0.639175\pi$
$338$	−14.6225	−	4.19061i	−0.795358	−	0.227939i
$339$	−6.90602			−0.375084
$340$			0.837101i			0.0453982i
$341$	−5.64423			−0.305652
$342$	13.7165			0.741701
$343$		0			0
$344$		−	1.28846i		−	0.0694690i
$345$		−	1.25461i		−	0.0675458i
$346$		−	30.3051i		−	1.62921i
$347$	−20.3812			−1.09412			−0.547060	−	0.837093i	$-0.684253\pi$
$347$	−20.3812			−1.09412			−0.547060	+	0.837093i	$0.684253\pi$
$348$	−0.0890557			−0.00477388
$349$		−	7.16394i		−	0.383477i	−0.981446	−	0.191739i	$-0.938587\pi$
$349$		−	7.16394i		−	0.383477i	0.981446	−	0.191739i	$-0.0614126\pi$
$350$		0			0
$351$	8.86376	−	6.68035i	0.473113	−	0.356570i
$352$	−2.83710			−0.151218
$353$			5.38348i			0.286534i	0.989684	+	0.143267i	$0.0457607\pi$
$353$			5.38348i			0.286534i	−0.989684	+	0.143267i	$0.954239\pi$
$354$	1.75872			0.0934751
$355$	5.88550			0.312370
$356$			3.42696i			0.181629i
$357$		0			0
$358$			24.1906i			1.27851i
$359$		−	15.6937i		−	0.828281i	−0.910213	−	0.414140i	$-0.864082\pi$
$359$		−	15.6937i		−	0.828281i	0.910213	−	0.414140i	$-0.135918\pi$
$360$	3.84324			0.202557
$361$	0.278438			0.0146547
$362$		−	18.3896i		−	0.966537i
$363$	−5.55971			−0.291809
$364$		0			0
$365$	0.162899			0.00852650
$366$		−	4.58145i		−	0.239476i
$367$	−0.324575			−0.0169427			−0.00847133	−	0.999964i	$-0.502697\pi$
$367$	−0.324575			−0.0169427			−0.00847133	+	0.999964i	$0.502697\pi$
$368$	11.8166			0.615982
$369$		−	18.0989i		−	0.942191i
$370$			5.12783i			0.266583i
$371$		0			0
$372$			2.31351i			0.119950i
$373$	−25.3112			−1.31057			−0.655283	−	0.755383i	$-0.727451\pi$
$373$	−25.3112			−1.31057			−0.655283	+	0.755383i	$0.727451\pi$
$374$	2.79606			0.144581
$375$			2.43188i			0.125582i
$376$	28.4703			1.46824
$377$	0.753803	−	0.568118i	0.0388228	−	0.0292596i
$378$		0			0
$379$		−	16.7187i		−	0.858784i	−0.903118	−	0.429392i	$-0.858728\pi$
$379$		−	16.7187i		−	0.858784i	0.903118	−	0.429392i	$-0.141272\pi$
$380$	−1.25792			−0.0645299
$381$	−7.26180			−0.372033
$382$			19.6430i			1.00502i
$383$		−	13.1629i		−	0.672593i	−0.941756	−	0.336296i	$-0.890826\pi$
$383$		−	13.1629i		−	0.672593i	0.941756	−	0.336296i	$-0.109174\pi$
$384$		−	1.78992i		−	0.0913415i
$385$		0			0
$386$	23.4186			1.19197
$387$	−1.13397			−0.0576429
$388$			7.95509i			0.403858i
$389$	10.0761			0.510879			0.255440	−	0.966825i	$-0.417780\pi$
$389$	10.0761			0.510879			0.255440	+	0.966825i	$0.417780\pi$
$390$	0.837101	−	0.630898i	0.0423883	−	0.0319467i
$391$	14.5392			0.735278
$392$		0			0
$393$	−7.78539			−0.392721
$394$	20.4969			1.03262
$395$		−	1.29791i		−	0.0653051i
$396$			1.41855i			0.0712849i
$397$			8.07119i			0.405081i	0.979274	+	0.202541i	$0.0649198\pi$
$397$			8.07119i			0.405081i	−0.979274	+	0.202541i	$0.935080\pi$
$398$			19.0700i			0.955891i
$399$		0			0
$400$	−11.2039			−0.560197
$401$		−	12.8371i		−	0.641054i	−0.947239	−	0.320527i	$-0.896140\pi$
$401$		−	12.8371i		−	0.641054i	0.947239	−	0.320527i	$-0.103860\pi$
$402$	3.20394			0.159798
$403$	−14.7587	−	19.5825i	−0.735184	−	0.975474i
$404$	2.29914			0.114386
$405$		−	2.98053i		−	0.148104i
$406$		0			0
$407$	−7.89269			−0.391226
$408$		−	4.77924i		−	0.236608i
$409$			18.7515i			0.927204i	0.886044	+	0.463602i	$0.153443\pi$
$409$			18.7515i			0.927204i	−0.886044	+	0.463602i	$0.846557\pi$
$410$		−	3.60197i		−	0.177889i
$411$			8.85989i			0.437026i
$412$	11.3197			0.557679
$413$		0			0
$414$		−	16.0072i		−	0.786710i
$415$	−4.78151			−0.234715
$416$	−7.41855	−	9.84324i	−0.363724	−	0.482605i
$417$	7.12783			0.349051
$418$			4.20167i			0.205510i
$419$	−23.6319			−1.15450			−0.577248	−	0.816569i	$-0.695873\pi$
$419$	−23.6319			−1.15450			−0.577248	+	0.816569i	$0.695873\pi$
$420$		0			0
$421$		−	19.4524i		−	0.948052i	−0.880511	−	0.474026i	$-0.842800\pi$
$421$		−	19.4524i		−	0.948052i	0.880511	−	0.474026i	$-0.157200\pi$
$422$		−	3.17009i		−	0.154317i
$423$		−	25.0566i		−	1.21830i
$424$		−	5.02052i		−	0.243818i
$425$	−13.7854			−0.668689
$426$	−8.05786			−0.390405
$427$		0			0
$428$	1.85762			0.0897915
$429$	0.971071	+	1.28846i	0.0468837	+	0.0622073i
$430$	−0.225678			−0.0108832
$431$		−	18.0989i		−	0.871793i	−0.899997	−	0.435897i	$-0.856431\pi$
$431$		−	18.0989i		−	0.871793i	0.899997	−	0.435897i	$-0.143569\pi$
$432$	−7.20394			−0.346600
$433$	20.5380			0.986992			0.493496	−	0.869748i	$-0.335719\pi$
$433$	20.5380			0.986992			0.493496	+	0.869748i	$0.335719\pi$
$434$		0			0
$435$			0.0650468i			0.00311875i
$436$			5.31965i			0.254765i
$437$			21.8482i			1.04514i
$438$	−0.223025			−0.0106565
$439$	−26.9627			−1.28686			−0.643429	−	0.765506i	$-0.722489\pi$
$439$	−26.9627			−1.28686			−0.643429	+	0.765506i	$0.722489\pi$
$440$			1.17727i			0.0561244i
$441$		0			0
$442$	7.31124	+	9.70086i	0.347760	+	0.461423i
$443$	26.4101			1.25478			0.627392	−	0.778704i	$-0.284122\pi$
$443$	26.4101			1.25478			0.627392	+	0.778704i	$0.284122\pi$
$444$			3.23513i			0.153533i
$445$	2.50307			0.118657
$446$	23.1617			1.09674
$447$		−	11.0556i		−	0.522912i
$448$		0			0
$449$		−	31.4329i		−	1.48341i	−0.670725	−	0.741706i	$-0.734017\pi$
$449$		−	31.4329i		−	1.48341i	0.670725	−	0.741706i	$-0.265983\pi$
$450$			15.1773i			0.715464i
$451$	5.54411			0.261062
$452$	8.08065			0.380082
$453$		−	0.859888i		−	0.0404011i
$454$	−15.5174			−0.728270
$455$		0			0
$456$	7.18181			0.336319
$457$			18.0650i			0.845047i	0.906352	+	0.422524i	$0.138856\pi$
$457$			18.0650i			0.845047i	−0.906352	+	0.422524i	$0.861144\pi$
$458$	−32.5380			−1.52040
$459$	−8.86376			−0.413725
$460$			1.46800i			0.0684458i
$461$		−	23.0784i		−	1.07487i	−0.843306	−	0.537434i	$-0.819394\pi$
$461$		−	23.0784i		−	1.07487i	0.843306	−	0.537434i	$-0.180606\pi$
$462$		0			0
$463$		−	4.54760i		−	0.211345i	−0.994401	−	0.105672i	$-0.966301\pi$
$463$		−	4.54760i		−	0.211345i	0.994401	−	0.105672i	$-0.0336995\pi$
$464$	−0.612646			−0.0284414
$465$	1.68980			0.0783627
$466$		−	15.6404i		−	0.724525i
$467$	37.8154			1.74989			0.874943	−	0.484226i	$-0.160899\pi$
$467$	37.8154			1.74989			0.874943	+	0.484226i	$0.160899\pi$
$468$	−4.92162	+	3.70928i	−0.227502	+	0.171461i
$469$		0			0
$470$		−	4.98667i		−	0.230018i
$471$	−4.47414			−0.206158
$472$	−8.58145			−0.394993
$473$		−	0.347361i		−	0.0159717i
$474$			1.77698i			0.0816192i
$475$		−	20.7154i		−	0.950489i
$476$		0			0
$477$	−4.41855			−0.202312
$478$	−10.1880			−0.465986
$479$			9.24005i			0.422189i	0.977466	+	0.211094i	$0.0677028\pi$
$479$			9.24005i			0.422189i	−0.977466	+	0.211094i	$0.932297\pi$
$480$	0.849388			0.0387691
$481$	−20.6381	−	27.3835i	−0.941016	−	1.24858i
$482$	19.8108			0.902358
$483$		0			0
$484$	6.50534			0.295697
$485$	5.81044			0.263838
$486$			14.8865i			0.675268i
$487$		−	13.6598i		−	0.618986i	−0.950902	−	0.309493i	$-0.899841\pi$
$487$		−	13.6598i		−	0.618986i	0.950902	−	0.309493i	$-0.100159\pi$
$488$			22.3545i			1.01194i
$489$			0.214614i			0.00970519i
$490$		0			0
$491$	29.6514			1.33815			0.669075	−	0.743195i	$-0.266691\pi$
$491$	29.6514			1.33815			0.669075	+	0.743195i	$0.266691\pi$
$492$		−	2.27247i		−	0.102451i
$493$	−0.753803			−0.0339496
$494$	−14.5776	+	10.9867i	−0.655876	+	0.494314i
$495$	1.03612			0.0465700
$496$			15.9155i			0.714626i
$497$		0			0
$498$	6.54638			0.293350
$499$			35.5246i			1.59030i	0.606412	+	0.795151i	$0.292608\pi$
$499$			35.5246i			1.59030i	−0.606412	+	0.795151i	$0.707392\pi$
$500$		−	2.84551i		−	0.127255i
$501$			12.8710i			0.575032i
$502$			10.3219i			0.460690i
$503$	−34.4124			−1.53437			−0.767187	−	0.641424i	$-0.778344\pi$
$503$	−34.4124			−1.53437			−0.767187	+	0.641424i	$0.778344\pi$
$504$		0			0
$505$		−	1.67930i		−	0.0747279i
$506$	4.90337			0.217981
$507$	−1.93108	+	6.73820i	−0.0857622	+	0.299254i
$508$	8.49693			0.376990
$509$		−	11.1184i		−	0.492813i	−0.969167	−	0.246407i	$-0.920750\pi$
$509$		−	11.1184i		−	0.492813i	0.969167	−	0.246407i	$-0.0792498\pi$
$510$	−0.837101			−0.0370675
$511$		0			0
$512$			22.4079i			0.990297i
$513$		−	13.3197i		−	0.588077i
$514$			15.9649i			0.704183i
$515$		−	8.26794i		−	0.364329i
$516$	−0.142380			−0.00626791
$517$	7.67543			0.337565
$518$		0			0
$519$	−13.9649			−0.612992
$520$	−4.08452	+	3.07838i	−0.179118	+	0.134996i
$521$	17.8888			0.783723			0.391862	−	0.920024i	$-0.371831\pi$
$521$	17.8888			0.783723			0.391862	+	0.920024i	$0.371831\pi$
$522$			0.829914i			0.0363243i
$523$	23.9877			1.04891			0.524455	−	0.851438i	$-0.324269\pi$
$523$	23.9877			1.04891			0.524455	+	0.851438i	$0.324269\pi$
$524$	9.10957			0.397954
$525$		0			0
$526$			14.2485i			0.621263i
$527$			19.5825i			0.853027i
$528$		−	1.04718i		−	0.0455727i
$529$	2.49693			0.108562
$530$	−0.879362			−0.0381970
$531$			7.55252i			0.327751i
$532$		0			0
$533$	14.4969	+	19.2351i	0.627932	+	0.833166i
$534$	−3.42696			−0.148299
$535$		−	1.35682i		−	0.0586603i
$536$	−15.6332			−0.675250
$537$	11.1473			0.481042
$538$		−	12.9262i		−	0.557286i
$539$		0			0
$540$		−	0.894960i		−	0.0385130i
$541$			19.4114i			0.834560i	0.908778	+	0.417280i	$0.137017\pi$
$541$			19.4114i			0.834560i	−0.908778	+	0.417280i	$0.862983\pi$
$542$	12.7961			0.549638
$543$	−8.47414			−0.363660
$544$			9.84324i			0.422026i
$545$	3.88550			0.166437
$546$		0			0
$547$	−1.81432			−0.0775745			−0.0387873	−	0.999247i	$-0.512349\pi$
$547$	−1.81432			−0.0775745			−0.0387873	+	0.999247i	$0.512349\pi$
$548$		−	10.3668i		−	0.442849i
$549$	19.6742			0.839675
$550$	−4.64915			−0.198240
$551$		−	1.13275i		−	0.0482566i
$552$		−	8.38121i		−	0.356728i
$553$		0			0
$554$		−	30.5620i		−	1.29845i
$555$	2.36296			0.100302
$556$	−8.34017			−0.353702
$557$		−	23.4186i		−	0.992276i	−0.868244	−	0.496138i	$-0.834751\pi$
$557$		−	23.4186i		−	0.992276i	0.868244	−	0.496138i	$-0.165249\pi$
$558$	21.5597			0.912695
$559$	1.20516	−	0.908291i	0.0509728	−	0.0384166i
$560$		0			0
$561$		−	1.28846i		−	0.0543987i
$562$	5.62022			0.237075
$563$	26.6947			1.12505			0.562524	−	0.826781i	$-0.309830\pi$
$563$	26.6947			1.12505			0.562524	+	0.826781i	$0.309830\pi$
$564$		−	3.14608i		−	0.132474i
$565$		−	5.90215i		−	0.248305i
$566$		−	1.14608i		−	0.0481732i
$567$		0			0
$568$	39.3172			1.64971
$569$	26.5897			1.11470			0.557349	−	0.830279i	$-0.311819\pi$
$569$	26.5897			1.11470			0.557349	+	0.830279i	$0.311819\pi$
$570$		−	1.25792i		−	0.0526885i
$571$	3.43310			0.143671			0.0718355	−	0.997416i	$-0.477114\pi$
$571$	3.43310			0.143671			0.0718355	+	0.997416i	$0.477114\pi$
$572$	−1.13624	−	1.50761i	−0.0475084	−	0.0630362i
$573$	9.05172			0.378141
$574$		0			0
$575$	−24.1750			−1.00817
$576$	23.5174			0.979894
$577$		−	30.4657i		−	1.26831i	−0.773208	−	0.634153i	$-0.781349\pi$
$577$		−	30.4657i		−	1.26831i	0.773208	−	0.634153i	$-0.218651\pi$
$578$			10.1906i			0.423873i
$579$		−	10.7915i		−	0.448481i
$580$		−	0.0761103i		−	0.00316031i
$581$		0			0
$582$	−7.95509			−0.329749
$583$		−	1.35350i		−	0.0560564i
$584$	1.08822			0.0450308
$585$	2.70928	+	3.59478i	0.112015	+	0.148626i
$586$	28.4645			1.17586
$587$		−	26.2606i		−	1.08389i	−0.840414	−	0.541945i	$-0.817688\pi$
$587$		−	26.2606i		−	1.08389i	0.840414	−	0.541945i	$-0.182312\pi$
$588$		0			0
$589$	−29.4268			−1.21251
$590$			1.50307i			0.0618805i
$591$		−	9.44521i		−	0.388524i
$592$			22.2557i			0.914702i
$593$			40.8420i			1.67718i	0.544762	+	0.838590i	$0.316620\pi$
$593$			40.8420i			1.67718i	−0.544762	+	0.838590i	$0.683380\pi$
$594$	−2.98932			−0.122653
$595$		0			0
$596$			12.9360i			0.529879i
$597$	8.78765			0.359655
$598$	12.8215	+	17.0121i	0.524310	+	0.695677i
$599$	−27.3545			−1.11768			−0.558838	−	0.829277i	$-0.688753\pi$
$599$	−27.3545			−1.11768			−0.558838	+	0.829277i	$0.688753\pi$
$600$			7.94668i			0.324422i
$601$	−25.4908			−1.03979			−0.519895	−	0.854230i	$-0.674029\pi$
$601$	−25.4908			−1.03979			−0.519895	+	0.854230i	$0.674029\pi$
$602$		0			0
$603$			13.7587i			0.560299i
$604$			1.00614i			0.0409394i
$605$		−	4.75154i		−	0.193177i
$606$			2.29914i			0.0933960i
$607$	−19.9311			−0.808977			−0.404489	−	0.914543i	$-0.632551\pi$
$607$	−19.9311			−0.808977			−0.404489	+	0.914543i	$0.632551\pi$
$608$	−14.7915			−0.599876
$609$		0			0
$610$	3.91548			0.158533
$611$	20.0700	+	26.6297i	0.811944	+	1.07732i
$612$	4.92162			0.198945
$613$			22.7214i			0.917708i	0.888512	+	0.458854i	$0.151740\pi$
$613$			22.7214i			0.917708i	−0.888512	+	0.458854i	$0.848260\pi$
$614$	−7.27739			−0.293692
$615$	−1.65983			−0.0669307
$616$		0			0
$617$		−	14.3545i		−	0.577892i	−0.957345	−	0.288946i	$-0.906695\pi$
$617$		−	14.3545i		−	0.577892i	0.957345	−	0.288946i	$-0.0933049\pi$
$618$			11.3197i			0.455343i
$619$			27.9832i			1.12474i	0.826886	+	0.562369i	$0.190110\pi$
$619$			27.9832i			1.12474i	−0.826886	+	0.562369i	$0.809890\pi$
$620$	−1.97721			−0.0794068
$621$	−15.5441			−0.623764
$622$			29.3789i			1.17799i
$623$		0			0
$624$	3.63317	−	2.73820i	0.145443	−	0.109616i
$625$	21.8599			0.874396
$626$			2.62106i			0.104758i
$627$	1.93618			0.0773234
$628$	5.23513			0.208905
$629$			27.3835i			1.09185i
$630$		0			0
$631$			21.4186i			0.852659i	0.904568	+	0.426330i	$0.140194\pi$
$631$			21.4186i			0.852659i	−0.904568	+	0.426330i	$0.859806\pi$
$632$		−	8.67050i		−	0.344894i
$633$	−1.46081			−0.0580620
$634$	−3.38735			−0.134529
$635$		−	6.20620i		−	0.246286i
$636$	−0.554787			−0.0219987
$637$		0			0
$638$	−0.254222			−0.0100647
$639$		−	34.6030i		−	1.36887i
$640$	1.52973			0.0604680
$641$	37.9360			1.49838			0.749191	−	0.662354i	$-0.230443\pi$
$641$	37.9360			1.49838			0.749191	+	0.662354i	$0.230443\pi$
$642$			1.85762i			0.0733144i
$643$		−	36.2122i		−	1.42807i	−0.700111	−	0.714034i	$-0.746866\pi$
$643$		−	36.2122i		−	1.42807i	0.700111	−	0.714034i	$-0.253134\pi$
$644$		0			0
$645$			0.103995i			0.00409479i
$646$	14.5776			0.573547
$647$	42.7948			1.68244			0.841219	−	0.540694i	$-0.181838\pi$
$647$	42.7948			1.68244			0.841219	+	0.540694i	$0.181838\pi$
$648$		−	19.9109i		−	0.782176i
$649$	−2.31351			−0.0908132
$650$	−12.1568	−	16.1301i	−0.476827	−	0.632674i
$651$		0			0
$652$		−	0.251117i		−	0.00983451i
$653$	5.54864			0.217135			0.108568	−	0.994089i	$-0.465374\pi$
$653$	5.54864			0.217135			0.108568	+	0.994089i	$0.465374\pi$
$654$	−5.31965			−0.208015
$655$		−	6.65368i		−	0.259981i
$656$		−	15.6332i		−	0.610373i
$657$		−	0.957740i		−	0.0373650i
$658$		0			0
$659$	−23.1529			−0.901908			−0.450954	−	0.892547i	$-0.648916\pi$
$659$	−23.1529			−0.901908			−0.450954	+	0.892547i	$0.648916\pi$
$660$	0.130094			0.00506388
$661$			47.2544i			1.83798i	0.394276	+	0.918992i	$0.370995\pi$
$661$			47.2544i			1.83798i	−0.394276	+	0.918992i	$0.629005\pi$
$662$	−4.97948			−0.193533
$663$	4.47027	−	3.36910i	0.173611	−	0.130845i
$664$	−31.9421			−1.23960
$665$		0			0
$666$	30.1483			1.16822
$667$	−1.32192			−0.0511850
$668$		−	15.0601i		−	0.582694i
$669$		−	10.6732i		−	0.412648i
$670$			2.73820i			0.105786i
$671$			6.02666i			0.232657i
$672$		0			0
$673$	3.42082			0.131863			0.0659314	−	0.997824i	$-0.478998\pi$
$673$	3.42082			0.131863			0.0659314	+	0.997824i	$0.478998\pi$
$674$			18.1906i			0.700676i
$675$	14.7382			0.567274
$676$	2.25953	−	7.88428i	0.0869049	−	0.303242i
$677$	−10.6959			−0.411079			−0.205539	−	0.978649i	$-0.565895\pi$
$677$	−10.6959			−0.411079			−0.205539	+	0.978649i	$0.565895\pi$
$678$			8.08065i			0.310335i
$679$		0			0
$680$	4.08452			0.156634
$681$			7.15061i			0.274012i
$682$			6.60424i			0.252889i
$683$		−	6.59583i		−	0.252382i	−0.992006	−	0.126191i	$-0.959725\pi$
$683$		−	6.59583i		−	0.252382i	0.992006	−	0.126191i	$-0.0402753\pi$
$684$			7.39576i			0.282784i
$685$	−7.57199			−0.289311
$686$		0			0
$687$			14.9939i			0.572051i
$688$	−0.979481			−0.0373424
$689$	4.69594	−	3.53919i	0.178901	−	0.134832i
$690$	−1.46800			−0.0558858
$691$			49.4729i			1.88204i	0.338353	+	0.941019i	$0.390130\pi$
$691$			49.4729i			1.88204i	−0.338353	+	0.941019i	$0.609870\pi$
$692$	16.3402			0.621160
$693$		0			0
$694$			23.8478i			0.905249i
$695$			6.09171i			0.231072i
$696$			0.434535i			0.0164710i
$697$		−	19.2351i		−	0.728583i
$698$	−8.38243			−0.317280
$699$	−7.20725			−0.272603
$700$		0			0
$701$	−20.8166			−0.786231			−0.393116	−	0.919489i	$-0.628603\pi$
$701$	−20.8166			−0.786231			−0.393116	+	0.919489i	$0.628603\pi$
$702$	−7.81658	−	10.3714i	−0.295018	−	0.391442i
$703$	−41.1494			−1.55198
$704$			7.20394i			0.271509i
$705$	−2.29791			−0.0865444
$706$	6.29914			0.237071
$707$		0			0
$708$			0.948284i			0.0356387i
$709$		−	5.95386i		−	0.223602i	−0.993731	−	0.111801i	$-0.964338\pi$
$709$		−	5.95386i		−	0.223602i	0.993731	−	0.111801i	$-0.0356619\pi$
$710$		−	6.88655i		−	0.258448i
$711$	−7.63090			−0.286181
$712$	16.7214			0.626660
$713$			34.3412i			1.28609i
$714$		0			0
$715$	−1.10116	+	0.829914i	−0.0411812	+	0.0310370i
$716$	−13.0433			−0.487451
$717$			4.69472i			0.175328i
$718$	−18.3630			−0.685300
$719$	−1.68980			−0.0630190			−0.0315095	−	0.999503i	$-0.510031\pi$
$719$	−1.68980			−0.0630190			−0.0315095	+	0.999503i	$0.510031\pi$
$720$		−	2.92162i		−	0.108882i
$721$		0			0
$722$		−	0.325797i		−	0.0121249i
$723$		−	9.12905i		−	0.339513i
$724$	9.91548			0.368506
$725$	1.25338			0.0465495
$726$			6.50534i			0.241436i
$727$	14.0722			0.521910			0.260955	−	0.965351i	$-0.415963\pi$
$727$	14.0722			0.521910			0.260955	+	0.965351i	$0.415963\pi$
$728$		0			0
$729$	−12.5441			−0.464597
$730$		−	0.190605i		−	0.00705462i
$731$	−1.20516			−0.0445744
$732$	2.47027			0.0913037
$733$			10.9311i			0.403749i	0.979411	+	0.201874i	$0.0647032\pi$
$733$			10.9311i			0.403749i	−0.979411	+	0.201874i	$0.935297\pi$
$734$			0.379780i			0.0140179i
$735$		0			0
$736$			17.2618i			0.636278i
$737$	−4.21461			−0.155247
$738$	−21.1773			−0.779546
$739$			3.17009i			0.116614i	0.998299	+	0.0583068i	$0.0185701\pi$
$739$			3.17009i			0.116614i	−0.998299	+	0.0583068i	$0.981430\pi$
$740$	−2.76487			−0.101639
$741$	5.06278	+	6.71751i	0.185986	+	0.246774i
$742$		0			0
$743$			9.73206i			0.357035i	0.983937	+	0.178517i	$0.0571301\pi$
$743$			9.73206i			0.357035i	−0.983937	+	0.178517i	$0.942870\pi$
$744$	11.2885			0.413855
$745$	9.44852			0.346167
$746$			29.6163i			1.08433i
$747$			28.1122i			1.02857i
$748$			1.50761i			0.0551235i
$749$		0			0
$750$	2.84551			0.103903
$751$	−7.77924			−0.283869			−0.141934	−	0.989876i	$-0.545332\pi$
$751$	−7.77924			−0.283869			−0.141934	+	0.989876i	$0.545332\pi$
$752$		−	21.6430i		−	0.789239i
$753$	4.75646			0.173335
$754$	−0.664748	−	0.882015i	−0.0242087	−	0.0321211i
$755$	0.734892			0.0267455
$756$		0			0
$757$	10.8576			0.394627			0.197313	−	0.980340i	$-0.436778\pi$
$757$	10.8576			0.394627			0.197313	+	0.980340i	$0.436778\pi$
$758$	−19.5624			−0.710537
$759$		−	2.25953i		−	0.0820157i
$760$			6.13784i			0.222643i
$761$			1.82150i			0.0660294i	0.999455	+	0.0330147i	$0.0105108\pi$
$761$			1.82150i			0.0660294i	−0.999455	+	0.0330147i	$0.989489\pi$
$762$			8.49693i			0.307811i
$763$		0			0
$764$	−10.5913			−0.383179
$765$		−	3.59478i		−	0.129970i
$766$	−15.4017			−0.556487
$767$	−6.04945	−	8.02666i	−0.218433	−	0.289826i
$768$	7.26633			0.262201
$769$		−	39.7948i		−	1.43504i	−0.696539	−	0.717519i	$-0.745277\pi$
$769$		−	39.7948i		−	1.43504i	0.696539	−	0.717519i	$-0.254723\pi$
$770$		0			0
$771$	7.35682			0.264949
$772$			12.6270i			0.454456i
$773$		−	18.3135i		−	0.658691i	−0.944209	−	0.329346i	$-0.893172\pi$
$773$		−	18.3135i		−	0.658691i	0.944209	−	0.329346i	$-0.106828\pi$
$774$			1.32684i			0.0476924i
$775$		−	32.5608i		−	1.16962i
$776$	38.8157			1.39340
$777$		0			0
$778$		−	11.7899i		−	0.422689i
$779$	28.9048			1.03562
$780$	0.340173	+	0.451356i	0.0121801	+	0.0161611i
$781$	10.5997			0.379287
$782$		−	17.0121i		−	0.608352i
$783$	0.805905			0.0288007
$784$		0			0
$785$		−	3.82377i		−	0.136476i
$786$			9.10957i			0.324928i
$787$		−	26.2606i		−	0.936088i	−0.883705	−	0.468044i	$-0.844959\pi$
$787$		−	26.2606i		−	0.936088i	0.883705	−	0.468044i	$-0.155041\pi$

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:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.17009i q^{2} -0.539189 q^{3} +0.630898 q^{4} -0.460811i q^{5} +0.630898i q^{6} -3.07838i q^{8} -2.70928 q^{9} +O(q^{10})\)	\(q-1.17009i q^{2} -0.539189 q^{3} +0.630898 q^{4} -0.460811i q^{5} +0.630898i q^{6} -3.07838i q^{8} -2.70928 q^{9} -0.539189 q^{10} -0.829914i q^{11} -0.340173 q^{12} +(2.87936 - 2.17009i) q^{13} +0.248464i q^{15} -2.34017 q^{16} -2.87936 q^{17} +3.17009i q^{18} -4.32684i q^{19} -0.290725i q^{20} -0.971071 q^{22} -5.04945 q^{23} +1.65983i q^{24} +4.78765 q^{25} +(-2.53919 - 3.36910i) q^{26} +3.07838 q^{27} +0.261795 q^{29} +0.290725 q^{30} -6.80098i q^{31} -3.41855i q^{32} +0.447480i q^{33} +3.36910i q^{34} -1.70928 q^{36} -9.51026i q^{37} -5.06278 q^{38} +(-1.55252 + 1.17009i) q^{39} -1.41855 q^{40} +6.68035i q^{41} +0.418551 q^{43} -0.523590i q^{44} +1.24846i q^{45} +5.90829i q^{46} +9.24846i q^{47} +1.26180 q^{48} -5.60197i q^{50} +1.55252 q^{51} +(1.81658 - 1.36910i) q^{52} +1.63090 q^{53} -3.60197i q^{54} -0.382433 q^{55} +2.33299i q^{57} -0.306323i q^{58} -2.78765i q^{59} +0.156755i q^{60} -7.26180 q^{61} -7.95774 q^{62} -8.68035 q^{64} +(-1.00000 - 1.32684i) q^{65} +0.523590 q^{66} -5.07838i q^{67} -1.81658 q^{68} +2.72261 q^{69} +12.7721i q^{71} +8.34017i q^{72} +0.353504i q^{73} -11.1278 q^{74} -2.58145 q^{75} -2.72979i q^{76} +(1.36910 + 1.81658i) q^{78} +2.81658 q^{79} +1.07838i q^{80} +6.46800 q^{81} +7.81658 q^{82} -10.3763i q^{83} +1.32684i q^{85} -0.489741i q^{86} -0.141157 q^{87} -2.55479 q^{88} +5.43188i q^{89} +1.46081 q^{90} -3.18568 q^{92} +3.66701i q^{93} +10.8215 q^{94} -1.99386 q^{95} +1.84324i q^{96} +12.6092i q^{97} +2.24846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{4} - 2 q^{9}+O(q^{10}) \) 6 * q - 4 * q^4 - 2 * q^9	\( 6 q - 4 q^{4} - 2 q^{9} + 20 q^{12} - 8 q^{13} + 8 q^{16} + 8 q^{17} + 24 q^{22} + 6 q^{23} + 8 q^{25} - 12 q^{26} + 12 q^{27} - 14 q^{29} + 16 q^{30} + 4 q^{36} + 4 q^{38} - 8 q^{39} + 20 q^{40} - 26 q^{43} - 8 q^{48} + 8 q^{51} + 20 q^{52} + 2 q^{53} - 12 q^{55} - 28 q^{61} - 16 q^{62} - 8 q^{64} - 6 q^{65} - 28 q^{66} - 20 q^{68} + 4 q^{69} - 24 q^{74} - 44 q^{75} + 16 q^{78} + 26 q^{79} - 26 q^{81} + 56 q^{82} + 40 q^{87} - 40 q^{88} + 12 q^{90} - 36 q^{92} - 20 q^{94} + 58 q^{95}+O(q^{100}) \) 6 * q - 4 * q^4 - 2 * q^9 + 20 * q^12 - 8 * q^13 + 8 * q^16 + 8 * q^17 + 24 * q^22 + 6 * q^23 + 8 * q^25 - 12 * q^26 + 12 * q^27 - 14 * q^29 + 16 * q^30 + 4 * q^36 + 4 * q^38 - 8 * q^39 + 20 * q^40 - 26 * q^43 - 8 * q^48 + 8 * q^51 + 20 * q^52 + 2 * q^53 - 12 * q^55 - 28 * q^61 - 16 * q^62 - 8 * q^64 - 6 * q^65 - 28 * q^66 - 20 * q^68 + 4 * q^69 - 24 * q^74 - 44 * q^75 + 16 * q^78 + 26 * q^79 - 26 * q^81 + 56 * q^82 + 40 * q^87 - 40 * q^88 + 12 * q^90 - 36 * q^92 - 20 * q^94 + 58 * q^95

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