LMFDB - Embedded newform 637.2.a.j.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [637,2,Mod(1,637)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("637.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(637, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 637 = 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	637.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,1,2,3,-2,-4,0,3,7,8,2,12,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.08647060876$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	637.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.34292 q^{2} +1.14637 q^{3} +3.48929 q^{4} +1.34292 q^{5} +2.68585 q^{6} +3.48929 q^{8} -1.68585 q^{9} +3.14637 q^{10} +1.14637 q^{11} +4.00000 q^{12} -1.00000 q^{13} +1.53948 q^{15} +1.19656 q^{16} -5.83221 q^{17} -3.94981 q^{18} +3.34292 q^{19} +4.68585 q^{20} +2.68585 q^{22} -3.17513 q^{23} +4.00000 q^{24} -3.19656 q^{25} -2.34292 q^{26} -5.37169 q^{27} +10.4893 q^{29} +3.60688 q^{30} -1.63565 q^{31} -4.17513 q^{32} +1.31415 q^{33} -13.6644 q^{34} -5.88240 q^{36} +8.51806 q^{37} +7.83221 q^{38} -1.14637 q^{39} +4.68585 q^{40} +0.292731 q^{41} -8.15371 q^{43} +4.00000 q^{44} -2.26396 q^{45} -7.43910 q^{46} +10.6142 q^{47} +1.37169 q^{48} -7.48929 q^{50} -6.68585 q^{51} -3.48929 q^{52} -0.782020 q^{53} -12.5855 q^{54} +1.53948 q^{55} +3.83221 q^{57} +24.5756 q^{58} -12.6430 q^{59} +5.37169 q^{60} +2.00000 q^{61} -3.83221 q^{62} -12.1751 q^{64} -1.34292 q^{65} +3.07896 q^{66} -6.10038 q^{67} -20.3503 q^{68} -3.63986 q^{69} +1.53948 q^{71} -5.88240 q^{72} +15.3001 q^{73} +19.9572 q^{74} -3.66442 q^{75} +11.6644 q^{76} -2.68585 q^{78} +0.882404 q^{79} +1.60688 q^{80} -1.10038 q^{81} +0.685846 q^{82} +12.1292 q^{83} -7.83221 q^{85} -19.1035 q^{86} +12.0246 q^{87} +4.00000 q^{88} -5.73604 q^{89} -5.30429 q^{90} -11.0790 q^{92} -1.87506 q^{93} +24.8683 q^{94} +4.48929 q^{95} -4.78623 q^{96} +5.34292 q^{97} -1.93260 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q + q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - 4 * q^6 + 3 * q^8 + 7 * q^9 + 8 * q^10 + 2 * q^11 + 12 * q^12 - 3 * q^13 - 6 * q^15 - q^16 - 4 * q^17 - 15 * q^18 + 4 * q^19 + 2 * q^20 - 4 * q^22 + 10 * q^23 + 12 * q^24 - 5 * q^25 - q^26 + 8 * q^27 + 24 * q^29 + 20 * q^30 + 4 * q^31 + 7 * q^32 + 16 * q^33 - 14 * q^34 - q^36 + 10 * q^38 - 2 * q^39 + 2 * q^40 - 2 * q^41 + 10 * q^43 + 12 * q^44 - 22 * q^45 - 18 * q^46 + 8 * q^47 - 20 * q^48 - 15 * q^50 - 8 * q^51 - 3 * q^52 + 8 * q^53 - 32 * q^54 - 6 * q^55 - 2 * q^57 + 12 * q^58 + 4 * q^59 - 8 * q^60 + 6 * q^61 + 2 * q^62 - 17 * q^64 + 2 * q^65 - 12 * q^66 - 12 * q^67 - 22 * q^68 + 6 * q^69 - 6 * q^71 - q^72 + 10 * q^73 + 30 * q^74 + 16 * q^75 + 8 * q^76 + 4 * q^78 - 14 * q^79 + 14 * q^80 + 3 * q^81 - 10 * q^82 + 12 * q^83 - 10 * q^85 - 26 * q^86 + 26 * q^87 + 12 * q^88 - 2 * q^89 + 28 * q^90 - 12 * q^92 - 22 * q^93 + 10 * q^94 + 6 * q^95 + 4 * q^96 + 10 * q^97 + 14 * q^99

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 $

$2$	2.34292	1.65670	0.828348	−	0.560213i	$-0.189281\pi$
$2$	2.34292	1.65670	0.828348	+	0.560213i	$0.189281\pi$
$3$	1.14637	0.661854	0.330927	−	0.943656i	$-0.392639\pi$
$3$	1.14637	0.661854	0.330927	+	0.943656i	$0.392639\pi$
$4$	3.48929	1.74464
$5$	1.34292	0.600573	0.300287	−	0.953849i	$-0.402918\pi$
$5$	1.34292	0.600573	0.300287	+	0.953849i	$0.402918\pi$
$6$	2.68585	1.09649
$7$	0	0
$7$	0	0
$8$	3.48929	1.23365
$9$	−1.68585	−0.561949
$10$	3.14637	0.994968
$11$	1.14637	0.345642	0.172821	−	0.984953i	$-0.444712\pi$
$11$	1.14637	0.345642	0.172821	+	0.984953i	$0.444712\pi$
$12$	4.00000	1.15470
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	1.53948	0.397492
$16$	1.19656	0.299139
$17$	−5.83221	−1.41452	−0.707260	−	0.706954i	$-0.750069\pi$
$17$	−5.83221	−1.41452	−0.707260	+	0.706954i	$0.750069\pi$
$18$	−3.94981	−0.930979
$19$	3.34292	0.766919	0.383460	−	0.923558i	$-0.374733\pi$
$19$	3.34292	0.766919	0.383460	+	0.923558i	$0.374733\pi$
$20$	4.68585	1.04779
$21$	0	0
$22$	2.68585	0.572624
$23$	−3.17513	−0.662061	−0.331031	−	0.943620i	$-0.607396\pi$
$23$	−3.17513	−0.662061	−0.331031	+	0.943620i	$0.607396\pi$
$24$	4.00000	0.816497
$25$	−3.19656	−0.639312
$26$	−2.34292	−0.459485
$27$	−5.37169	−1.03378
$28$	0	0
$29$	10.4893	1.94781	0.973906	−	0.226952i	$-0.0728760\pi$
$29$	10.4893	1.94781	0.973906	+	0.226952i	$0.0728760\pi$
$30$	3.60688	0.658524
$31$	−1.63565	−0.293772	−0.146886	−	0.989153i	$-0.546925\pi$
$31$	−1.63565	−0.293772	−0.146886	+	0.989153i	$0.546925\pi$
$32$	−4.17513	−0.738067
$33$	1.31415	0.228765
$34$	−13.6644	−2.34343
$35$	0	0
$36$	−5.88240	−0.980401
$37$	8.51806	1.40036	0.700180	−	0.713966i	$-0.253103\pi$
$37$	8.51806	1.40036	0.700180	+	0.713966i	$0.253103\pi$
$38$	7.83221	1.27055
$39$	−1.14637	−0.183565
$40$	4.68585	0.740897
$41$	0.292731	0.0457169	0.0228584	−	0.999739i	$-0.492723\pi$
$41$	0.292731	0.0457169	0.0228584	+	0.999739i	$0.492723\pi$
$42$	0	0
$43$	−8.15371	−1.24343	−0.621715	−	0.783244i	$-0.713564\pi$
$43$	−8.15371	−1.24343	−0.621715	+	0.783244i	$0.713564\pi$
$44$	4.00000	0.603023
$45$	−2.26396	−0.337491
$46$	−7.43910	−1.09683
$47$	10.6142	1.54824	0.774122	−	0.633036i	$-0.218191\pi$
$47$	10.6142	1.54824	0.774122	+	0.633036i	$0.218191\pi$
$48$	1.37169	0.197987
$49$	0	0
$50$	−7.48929	−1.05915
$51$	−6.68585	−0.936206
$52$	−3.48929	−0.483877
$53$	−0.782020	−0.107419	−0.0537093	−	0.998557i	$-0.517104\pi$
$53$	−0.782020	−0.107419	−0.0537093	+	0.998557i	$0.517104\pi$
$54$	−12.5855	−1.71266
$55$	1.53948	0.207584
$56$	0	0
$57$	3.83221	0.507589
$58$	24.5756	3.22693
$59$	−12.6430	−1.64598	−0.822989	−	0.568057i	$-0.807695\pi$
$59$	−12.6430	−1.64598	−0.822989	+	0.568057i	$0.807695\pi$
$60$	5.37169	0.693482
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−3.83221	−0.486691
$63$	0	0
$64$	−12.1751	−1.52189
$65$	−1.34292	−0.166569
$66$	3.07896	0.378994
$67$	−6.10038	−0.745281	−0.372640	−	0.927976i	$-0.621547\pi$
$67$	−6.10038	−0.745281	−0.372640	+	0.927976i	$0.621547\pi$
$68$	−20.3503	−2.46783
$69$	−3.63986	−0.438188
$70$	0	0
$71$	1.53948	0.182703	0.0913514	−	0.995819i	$-0.470881\pi$
$71$	1.53948	0.182703	0.0913514	+	0.995819i	$0.470881\pi$
$72$	−5.88240	−0.693248
$73$	15.3001	1.79074	0.895369	−	0.445324i	$-0.146912\pi$
$73$	15.3001	1.79074	0.895369	+	0.445324i	$0.146912\pi$
$74$	19.9572	2.31997
$75$	−3.66442	−0.423131
$76$	11.6644	1.33800
$77$	0	0
$78$	−2.68585	−0.304112
$79$	0.882404	0.0992782	0.0496391	−	0.998767i	$-0.484193\pi$
$79$	0.882404	0.0992782	0.0496391	+	0.998767i	$0.484193\pi$
$80$	1.60688	0.179655
$81$	−1.10038	−0.122265
$82$	0.685846	0.0757390
$83$	12.1292	1.33135	0.665674	−	0.746243i	$-0.268144\pi$
$83$	12.1292	1.33135	0.665674	+	0.746243i	$0.268144\pi$
$84$	0	0
$85$	−7.83221	−0.849523
$86$	−19.1035	−2.05999
$87$	12.0246	1.28917
$88$	4.00000	0.426401
$89$	−5.73604	−0.608019	−0.304009	−	0.952669i	$-0.598325\pi$
$89$	−5.73604	−0.608019	−0.304009	+	0.952669i	$0.598325\pi$
$90$	−5.30429	−0.559121
$91$	0	0
$92$	−11.0790	−1.15506
$93$	−1.87506	−0.194434
$94$	24.8683	2.56497
$95$	4.48929	0.460591
$96$	−4.78623	−0.488493
$97$	5.34292	0.542492	0.271246	−	0.962510i	$-0.412564\pi$
$97$	5.34292	0.542492	0.271246	+	0.962510i	$0.412564\pi$
$98$	0	0
$99$	−1.93260	−0.194233
$100$	−11.1537	−1.11537
$101$	−11.1464	−1.10910	−0.554552	−	0.832149i	$-0.687111\pi$
$101$	−11.1464	−1.10910	−0.554552	+	0.832149i	$0.687111\pi$
$102$	−15.6644	−1.55101
$103$	3.41454	0.336444	0.168222	−	0.985749i	$-0.446197\pi$
$103$	3.41454	0.336444	0.168222	+	0.985749i	$0.446197\pi$
$104$	−3.48929	−0.342153
$105$	0	0
$106$	−1.83221	−0.177960
$107$	4.97858	0.481297	0.240649	−	0.970612i	$-0.422640\pi$
$107$	4.97858	0.481297	0.240649	+	0.970612i	$0.422640\pi$
$108$	−18.7434	−1.80358
$109$	−13.4966	−1.29274	−0.646372	−	0.763023i	$-0.723714\pi$
$109$	−13.4966	−1.29274	−0.646372	+	0.763023i	$0.723714\pi$
$110$	3.60688	0.343903
$111$	9.76481	0.926835
$112$	0	0
$113$	16.4464	1.54715	0.773576	−	0.633704i	$-0.218466\pi$
$113$	16.4464	1.54715	0.773576	+	0.633704i	$0.218466\pi$
$114$	8.97858	0.840921
$115$	−4.26396	−0.397616
$116$	36.6002	3.39824
$117$	1.68585	0.155857
$118$	−29.6216	−2.72689
$119$	0	0
$120$	5.37169	0.490366
$121$	−9.68585	−0.880531
$122$	4.68585	0.424237
$123$	0.335577	0.0302579
$124$	−5.70727	−0.512528
$125$	−11.0073	−0.984527
$126$	0	0
$127$	12.0575	1.06993	0.534967	−	0.844873i	$-0.320324\pi$
$127$	12.0575	1.06993	0.534967	+	0.844873i	$0.320324\pi$
$128$	−20.1751	−1.78325
$129$	−9.34713	−0.822969
$130$	−3.14637	−0.275955
$131$	3.66442	0.320162	0.160081	−	0.987104i	$-0.448824\pi$
$131$	3.66442	0.320162	0.160081	+	0.987104i	$0.448824\pi$
$132$	4.58546	0.399113
$133$	0	0
$134$	−14.2927	−1.23470
$135$	−7.21377	−0.620862
$136$	−20.3503	−1.74502
$137$	−13.1035	−1.11951	−0.559755	−	0.828658i	$-0.689105\pi$
$137$	−13.1035	−1.11951	−0.559755	+	0.828658i	$0.689105\pi$
$138$	−8.52792	−0.725945
$139$	−7.49663	−0.635856	−0.317928	−	0.948115i	$-0.602987\pi$
$139$	−7.49663	−0.635856	−0.317928	+	0.948115i	$0.602987\pi$
$140$	0	0
$141$	12.1678	1.02471
$142$	3.60688	0.302683
$143$	−1.14637	−0.0958639
$144$	−2.01721	−0.168101
$145$	14.0863	1.16980
$146$	35.8469	2.96671
$147$	0	0
$148$	29.7220	2.44313
$149$	2.16779	0.177592	0.0887961	−	0.996050i	$-0.471698\pi$
$149$	2.16779	0.177592	0.0887961	+	0.996050i	$0.471698\pi$
$150$	−8.58546	−0.701000
$151$	14.9112	1.21345	0.606727	−	0.794910i	$-0.292482\pi$
$151$	14.9112	1.21345	0.606727	+	0.794910i	$0.292482\pi$
$152$	11.6644	0.946110
$153$	9.83221	0.794887
$154$	0	0
$155$	−2.19656	−0.176432
$156$	−4.00000	−0.320256
$157$	−22.8683	−1.82509	−0.912546	−	0.408975i	$-0.865886\pi$
$157$	−22.8683	−1.82509	−0.912546	+	0.408975i	$0.865886\pi$
$158$	2.06740	0.164474
$159$	−0.896480	−0.0710955
$160$	−5.60688	−0.443263
$161$	0	0
$162$	−2.57812	−0.202556
$163$	7.07896	0.554467	0.277234	−	0.960803i	$-0.410582\pi$
$163$	7.07896	0.554467	0.277234	+	0.960803i	$0.410582\pi$
$164$	1.02142	0.0797597
$165$	1.76481	0.137390
$166$	28.4177	2.20564
$167$	−2.61423	−0.202295	−0.101148	−	0.994871i	$-0.532251\pi$
$167$	−2.61423	−0.202295	−0.101148	+	0.994871i	$0.532251\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−18.3503	−1.40740
$171$	−5.63565	−0.430969
$172$	−28.4507	−2.16934
$173$	−11.0031	−0.836553	−0.418276	−	0.908320i	$-0.637366\pi$
$173$	−11.0031	−0.836553	−0.418276	+	0.908320i	$0.637366\pi$
$174$	28.1726	2.13576
$175$	0	0
$176$	1.37169	0.103395
$177$	−14.4935	−1.08940
$178$	−13.4391	−1.00730
$179$	23.9614	1.79096	0.895478	−	0.445105i	$-0.146834\pi$
$179$	23.9614	1.79096	0.895478	+	0.445105i	$0.146834\pi$
$180$	−7.89962	−0.588803
$181$	−6.56090	−0.487668	−0.243834	−	0.969817i	$-0.578405\pi$
$181$	−6.56090	−0.487668	−0.243834	+	0.969817i	$0.578405\pi$
$182$	0	0
$183$	2.29273	0.169484
$184$	−11.0790	−0.816752
$185$	11.4391	0.841019
$186$	−4.39312	−0.322119
$187$	−6.68585	−0.488917
$188$	37.0361	2.70114
$189$	0	0
$190$	10.5181	0.763060
$191$	−4.39312	−0.317875	−0.158937	−	0.987289i	$-0.550807\pi$
$191$	−4.39312	−0.317875	−0.158937	+	0.987289i	$0.550807\pi$
$192$	−13.9572	−1.00727
$193$	−8.29273	−0.596924	−0.298462	−	0.954422i	$-0.596474\pi$
$193$	−8.29273	−0.596924	−0.298462	+	0.954422i	$0.596474\pi$
$194$	12.5181	0.898744
$195$	−1.53948	−0.110245
$196$	0	0
$197$	−3.17092	−0.225919	−0.112959	−	0.993600i	$-0.536033\pi$
$197$	−3.17092	−0.225919	−0.112959	+	0.993600i	$0.536033\pi$
$198$	−4.52792	−0.321786
$199$	13.5970	0.963867	0.481934	−	0.876208i	$-0.339935\pi$
$199$	13.5970	0.963867	0.481934	+	0.876208i	$0.339935\pi$
$200$	−11.1537	−0.788687
$201$	−6.99327	−0.493267
$202$	−26.1151	−1.83745
$203$	0	0
$204$	−23.3288	−1.63335
$205$	0.393115	0.0274564
$206$	8.00000	0.557386
$207$	5.35279	0.372045
$208$	−1.19656	−0.0829663
$209$	3.83221	0.265080
$210$	0	0
$211$	9.27552	0.638553	0.319277	−	0.947662i	$-0.396560\pi$
$211$	9.27552	0.638553	0.319277	+	0.947662i	$0.396560\pi$
$212$	−2.72869	−0.187407
$213$	1.76481	0.120923
$214$	11.6644	0.797364
$215$	−10.9498	−0.746771
$216$	−18.7434	−1.27533
$217$	0	0
$218$	−31.6216	−2.14168
$219$	17.5395	1.18521
$220$	5.37169	0.362159
$221$	5.83221	0.392317
$222$	22.8782	1.53548
$223$	19.5928	1.31203	0.656016	−	0.754747i	$-0.272240\pi$
$223$	19.5928	1.31203	0.656016	+	0.754747i	$0.272240\pi$
$224$	0	0
$225$	5.38890	0.359260
$226$	38.5328	2.56316
$227$	19.6644	1.30517	0.652587	−	0.757714i	$-0.273684\pi$
$227$	19.6644	1.30517	0.652587	+	0.757714i	$0.273684\pi$
$228$	13.3717	0.885562
$229$	−7.76481	−0.513113	−0.256556	−	0.966529i	$-0.582588\pi$
$229$	−7.76481	−0.513113	−0.256556	+	0.966529i	$0.582588\pi$
$230$	−9.99013	−0.658730
$231$	0	0
$232$	36.6002	2.40292
$233$	12.1966	0.799023	0.399512	−	0.916728i	$-0.369180\pi$
$233$	12.1966	0.799023	0.399512	+	0.916728i	$0.369180\pi$
$234$	3.94981	0.258207
$235$	14.2541	0.929835
$236$	−44.1151	−2.87165
$237$	1.01156	0.0657077
$238$	0	0
$239$	−10.2927	−0.665781	−0.332891	−	0.942965i	$-0.608024\pi$
$239$	−10.2927	−0.665781	−0.332891	+	0.942965i	$0.608024\pi$
$240$	1.84208	0.118906
$241$	4.02877	0.259516	0.129758	−	0.991546i	$-0.458580\pi$
$241$	4.02877	0.259516	0.129758	+	0.991546i	$0.458580\pi$
$242$	−22.6932	−1.45877
$243$	14.8536	0.952861
$244$	6.97858	0.446758
$245$	0	0
$246$	0.786230	0.0501282
$247$	−3.34292	−0.212705
$248$	−5.70727	−0.362412
$249$	13.9044	0.881158
$250$	−25.7894	−1.63106
$251$	−2.91117	−0.183752	−0.0918758	−	0.995770i	$-0.529286\pi$
$251$	−2.91117	−0.183752	−0.0918758	+	0.995770i	$0.529286\pi$
$252$	0	0
$253$	−3.63986	−0.228836
$254$	28.2499	1.77256
$255$	−8.97858	−0.562260
$256$	−22.9185	−1.43241
$257$	19.5970	1.22243	0.611214	−	0.791465i	$-0.290681\pi$
$257$	19.5970	1.22243	0.611214	+	0.791465i	$0.290681\pi$
$258$	−21.8996	−1.36341
$259$	0	0
$260$	−4.68585	−0.290604
$261$	−17.6833	−1.09457
$262$	8.58546	0.530412
$263$	7.56825	0.466678	0.233339	−	0.972395i	$-0.425035\pi$
$263$	7.56825	0.466678	0.233339	+	0.972395i	$0.425035\pi$
$264$	4.58546	0.282216
$265$	−1.05019	−0.0645128
$266$	0	0
$267$	−6.57560	−0.402420
$268$	−21.2860	−1.30025
$269$	9.47208	0.577523	0.288761	−	0.957401i	$-0.406757\pi$
$269$	9.47208	0.577523	0.288761	+	0.957401i	$0.406757\pi$
$270$	−16.9013	−1.02858
$271$	−29.3717	−1.78420	−0.892102	−	0.451835i	$-0.850770\pi$
$271$	−29.3717	−1.78420	−0.892102	+	0.451835i	$0.850770\pi$
$272$	−6.97858	−0.423138
$273$	0	0
$274$	−30.7005	−1.85469
$275$	−3.66442	−0.220973
$276$	−12.7005	−0.764483
$277$	−1.90383	−0.114390	−0.0571949	−	0.998363i	$-0.518216\pi$
$277$	−1.90383	−0.114390	−0.0571949	+	0.998363i	$0.518216\pi$
$278$	−17.5640	−1.05342
$279$	2.75746	0.165085
$280$	0	0
$281$	−20.5756	−1.22744	−0.613719	−	0.789525i	$-0.710327\pi$
$281$	−20.5756	−1.22744	−0.613719	+	0.789525i	$0.710327\pi$
$282$	28.5082	1.69764
$283$	26.9933	1.60458	0.802292	−	0.596932i	$-0.203614\pi$
$283$	26.9933	1.60458	0.802292	+	0.596932i	$0.203614\pi$
$284$	5.37169	0.318751
$285$	5.14637	0.304844
$286$	−2.68585	−0.158817
$287$	0	0
$288$	7.03863	0.414756
$289$	17.0147	1.00086
$290$	33.0031	1.93801
$291$	6.12494	0.359050
$292$	53.3864	3.12420
$293$	14.9070	0.870874	0.435437	−	0.900219i	$-0.356594\pi$
$293$	14.9070	0.870874	0.435437	+	0.900219i	$0.356594\pi$
$294$	0	0
$295$	−16.9786	−0.988531
$296$	29.7220	1.72755
$297$	−6.15792	−0.357319
$298$	5.07896	0.294216
$299$	3.17513	0.183623
$300$	−12.7862	−0.738213
$301$	0	0
$302$	34.9357	2.01033
$303$	−12.7778	−0.734066
$304$	4.00000	0.229416
$305$	2.68585	0.153791
$306$	23.0361	1.31689
$307$	−26.0288	−1.48554	−0.742770	−	0.669546i	$-0.766488\pi$
$307$	−26.0288	−1.48554	−0.742770	+	0.669546i	$0.766488\pi$
$308$	0	0
$309$	3.91431	0.222677
$310$	−5.14637	−0.292294
$311$	−19.4966	−1.10555	−0.552776	−	0.833330i	$-0.686432\pi$
$311$	−19.4966	−1.10555	−0.552776	+	0.833330i	$0.686432\pi$
$312$	−4.00000	−0.226455
$313$	3.48194	0.196811	0.0984055	−	0.995146i	$-0.468626\pi$
$313$	3.48194	0.196811	0.0984055	+	0.995146i	$0.468626\pi$
$314$	−53.5787	−3.02362
$315$	0	0
$316$	3.07896	0.173205
$317$	5.02142	0.282031	0.141016	−	0.990007i	$-0.454963\pi$
$317$	5.02142	0.282031	0.141016	+	0.990007i	$0.454963\pi$
$318$	−2.10038	−0.117784
$319$	12.0246	0.673246
$320$	−16.3503	−0.914008
$321$	5.70727	0.318549
$322$	0	0
$323$	−19.4966	−1.08482
$324$	−3.83956	−0.213309
$325$	3.19656	0.177313
$326$	16.5855	0.918584
$327$	−15.4721	−0.855608
$328$	1.02142	0.0563986
$329$	0	0
$330$	4.13481	0.227614
$331$	−6.14950	−0.338007	−0.169004	−	0.985615i	$-0.554055\pi$
$331$	−6.14950	−0.338007	−0.169004	+	0.985615i	$0.554055\pi$
$332$	42.3221	2.32273
$333$	−14.3601	−0.786931
$334$	−6.12494	−0.335142
$335$	−8.19235	−0.447596
$336$	0	0
$337$	−25.6258	−1.39593	−0.697963	−	0.716134i	$-0.745910\pi$
$337$	−25.6258	−1.39593	−0.697963	+	0.716134i	$0.745910\pi$
$338$	2.34292	0.127438
$339$	18.8536	1.02399
$340$	−27.3288	−1.48211
$341$	−1.87506	−0.101540
$342$	−13.2039	−0.713985
$343$	0	0
$344$	−28.4507	−1.53396
$345$	−4.88806	−0.263164
$346$	−25.7795	−1.38591
$347$	−16.7005	−0.896532	−0.448266	−	0.893900i	$-0.647958\pi$
$347$	−16.7005	−0.896532	−0.448266	+	0.893900i	$0.647958\pi$
$348$	41.9572	2.24914
$349$	23.5500	1.26060	0.630300	−	0.776351i	$-0.282932\pi$
$349$	23.5500	1.26060	0.630300	+	0.776351i	$0.282932\pi$
$350$	0	0
$351$	5.37169	0.286720
$352$	−4.78623	−0.255107
$353$	7.64973	0.407154	0.203577	−	0.979059i	$-0.434743\pi$
$353$	7.64973	0.407154	0.203577	+	0.979059i	$0.434743\pi$
$354$	−33.9572	−1.80480
$355$	2.06740	0.109726
$356$	−20.0147	−1.06078
$357$	0	0
$358$	56.1396	2.96707
$359$	18.3748	0.969786	0.484893	−	0.874573i	$-0.338858\pi$
$359$	18.3748	0.969786	0.484893	+	0.874573i	$0.338858\pi$
$360$	−7.89962	−0.416346
$361$	−7.82487	−0.411835
$362$	−15.3717	−0.807918
$363$	−11.1035	−0.582784
$364$	0	0
$365$	20.5468	1.07547
$366$	5.37169	0.280783
$367$	−5.33871	−0.278679	−0.139339	−	0.990245i	$-0.544498\pi$
$367$	−5.33871	−0.278679	−0.139339	+	0.990245i	$0.544498\pi$
$368$	−3.79923	−0.198049
$369$	−0.493499	−0.0256906
$370$	26.8009	1.39331
$371$	0	0
$372$	−6.54262	−0.339219
$373$	21.5212	1.11433	0.557163	−	0.830403i	$-0.311890\pi$
$373$	21.5212	1.11433	0.557163	+	0.830403i	$0.311890\pi$
$374$	−15.6644	−0.809988
$375$	−12.6184	−0.651614
$376$	37.0361	1.90999
$377$	−10.4893	−0.540226
$378$	0	0
$379$	4.61002	0.236801	0.118400	−	0.992966i	$-0.462223\pi$
$379$	4.61002	0.236801	0.118400	+	0.992966i	$0.462223\pi$
$380$	15.6644	0.803568
$381$	13.8223	0.708140
$382$	−10.2927	−0.526622
$383$	8.33558	0.425928	0.212964	−	0.977060i	$-0.431688\pi$
$383$	8.33558	0.425928	0.212964	+	0.977060i	$0.431688\pi$
$384$	−23.1281	−1.18025
$385$	0	0
$386$	−19.4292	−0.988922
$387$	13.7459	0.698744
$388$	18.6430	0.946455
$389$	−6.44223	−0.326634	−0.163317	−	0.986574i	$-0.552219\pi$
$389$	−6.44223	−0.326634	−0.163317	+	0.986574i	$0.552219\pi$
$390$	−3.60688	−0.182642
$391$	18.5181	0.936498
$392$	0	0
$393$	4.20077	0.211901
$394$	−7.42923	−0.374279
$395$	1.18500	0.0596238
$396$	−6.74338	−0.338868
$397$	−1.40046	−0.0702872	−0.0351436	−	0.999382i	$-0.511189\pi$
$397$	−1.40046	−0.0702872	−0.0351436	+	0.999382i	$0.511189\pi$
$398$	31.8568	1.59684
$399$	0	0
$400$	−3.82487	−0.191243
$401$	−6.97858	−0.348494	−0.174247	−	0.984702i	$-0.555749\pi$
$401$	−6.97858	−0.348494	−0.174247	+	0.984702i	$0.555749\pi$
$402$	−16.3847	−0.817194
$403$	1.63565	0.0814777
$404$	−38.8929	−1.93499
$405$	−1.47773	−0.0734291
$406$	0	0
$407$	9.76481	0.484024
$408$	−23.3288	−1.15495
$409$	18.3790	0.908785	0.454392	−	0.890802i	$-0.349856\pi$
$409$	18.3790	0.908785	0.454392	+	0.890802i	$0.349856\pi$
$410$	0.921039	0.0454869
$411$	−15.0214	−0.740952
$412$	11.9143	0.586976
$413$	0	0
$414$	12.5412	0.616365
$415$	16.2885	0.799572
$416$	4.17513	0.204703
$417$	−8.59388	−0.420844
$418$	8.97858	0.439157
$419$	30.0393	1.46751	0.733757	−	0.679412i	$-0.237765\pi$
$419$	30.0393	1.46751	0.733757	+	0.679412i	$0.237765\pi$
$420$	0	0
$421$	−8.31729	−0.405360	−0.202680	−	0.979245i	$-0.564965\pi$
$421$	−8.31729	−0.405360	−0.202680	+	0.979245i	$0.564965\pi$
$422$	21.7318	1.05789
$423$	−17.8940	−0.870034
$424$	−2.72869	−0.132517
$425$	18.6430	0.904318
$426$	4.13481	0.200332
$427$	0	0
$428$	17.3717	0.839692
$429$	−1.31415	−0.0634479
$430$	−25.6546	−1.23717
$431$	9.64973	0.464811	0.232406	−	0.972619i	$-0.425340\pi$
$431$	9.64973	0.464811	0.232406	+	0.972619i	$0.425340\pi$
$432$	−6.42754	−0.309245
$433$	−26.3074	−1.26425	−0.632127	−	0.774865i	$-0.717818\pi$
$433$	−26.3074	−1.26425	−0.632127	+	0.774865i	$0.717818\pi$
$434$	0	0
$435$	16.1481	0.774240
$436$	−47.0937	−2.25538
$437$	−10.6142	−0.507748
$438$	41.0937	1.96353
$439$	−33.8139	−1.61385	−0.806925	−	0.590654i	$-0.798870\pi$
$439$	−33.8139	−1.61385	−0.806925	+	0.590654i	$0.798870\pi$
$440$	5.37169	0.256085
$441$	0	0
$442$	13.6644	0.649950
$443$	26.4464	1.25651	0.628254	−	0.778008i	$-0.283770\pi$
$443$	26.4464	1.25651	0.628254	+	0.778008i	$0.283770\pi$
$444$	34.0722	1.61700
$445$	−7.70306	−0.365160
$446$	45.9044	2.17364
$447$	2.48508	0.117540
$448$	0	0
$449$	2.64300	0.124731	0.0623655	−	0.998053i	$-0.480136\pi$
$449$	2.64300	0.124731	0.0623655	+	0.998053i	$0.480136\pi$
$450$	12.6258	0.595185
$451$	0.335577	0.0158017
$452$	57.3864	2.69923
$453$	17.0937	0.803130
$454$	46.0722	2.16228
$455$	0	0
$456$	13.3717	0.626187
$457$	−33.6890	−1.57590	−0.787952	−	0.615737i	$-0.788859\pi$
$457$	−33.6890	−1.57590	−0.787952	+	0.615737i	$0.788859\pi$
$458$	−18.1923	−0.850073
$459$	31.3288	1.46231
$460$	−14.8782	−0.693699
$461$	−33.0790	−1.54064	−0.770320	−	0.637657i	$-0.779904\pi$
$461$	−33.0790	−1.54064	−0.770320	+	0.637657i	$0.779904\pi$
$462$	0	0
$463$	−2.51806	−0.117024	−0.0585120	−	0.998287i	$-0.518636\pi$
$463$	−2.51806	−0.117024	−0.0585120	+	0.998287i	$0.518636\pi$
$464$	12.5510	0.582667
$465$	−2.51806	−0.116772
$466$	28.5756	1.32374
$467$	−2.57560	−0.119184	−0.0595922	−	0.998223i	$-0.518980\pi$
$467$	−2.57560	−0.119184	−0.0595922	+	0.998223i	$0.518980\pi$
$468$	5.88240	0.271914
$469$	0	0
$470$	33.3963	1.54045
$471$	−26.2155	−1.20794
$472$	−44.1151	−2.03056
$473$	−9.34713	−0.429782
$474$	2.37000	0.108858
$475$	−10.6858	−0.490300
$476$	0	0
$477$	1.31836	0.0603638
$478$	−24.1151	−1.10300
$479$	−0.513847	−0.0234783	−0.0117391	−	0.999931i	$-0.503737\pi$
$479$	−0.513847	−0.0234783	−0.0117391	+	0.999931i	$0.503737\pi$
$480$	−6.42754	−0.293376
$481$	−8.51806	−0.388390
$482$	9.43910	0.429939
$483$	0	0
$484$	−33.7967	−1.53621
$485$	7.17513	0.325806
$486$	34.8009	1.57860
$487$	36.0575	1.63392	0.816962	−	0.576692i	$-0.195657\pi$
$487$	36.0575	1.63392	0.816962	+	0.576692i	$0.195657\pi$
$488$	6.97858	0.315905
$489$	8.11508	0.366976
$490$	0	0
$491$	−9.22846	−0.416475	−0.208237	−	0.978078i	$-0.566773\pi$
$491$	−9.22846	−0.416475	−0.208237	+	0.978078i	$0.566773\pi$
$492$	1.17092	0.0527893
$493$	−61.1758	−2.75522
$494$	−7.83221	−0.352388
$495$	−2.59533	−0.116651
$496$	−1.95715	−0.0878788
$497$	0	0
$498$	32.5770	1.45981
$499$	1.00314	0.0449065	0.0224533	−	0.999748i	$-0.492852\pi$
$499$	1.00314	0.0449065	0.0224533	+	0.999748i	$0.492852\pi$
$500$	−38.4078	−1.71765
$501$	−2.99686	−0.133890
$502$	−6.82065	−0.304421
$503$	30.3503	1.35325	0.676626	−	0.736327i	$-0.263441\pi$
$503$	30.3503	1.35325	0.676626	+	0.736327i	$0.263441\pi$
$504$	0	0
$505$	−14.9687	−0.666099
$506$	−8.52792	−0.379112
$507$	1.14637	0.0509119
$508$	42.0722	1.86665
$509$	−10.5995	−0.469816	−0.234908	−	0.972018i	$-0.575479\pi$
$509$	−10.5995	−0.469816	−0.234908	+	0.972018i	$0.575479\pi$
$510$	−21.0361	−0.931495
$511$	0	0
$512$	−13.3461	−0.589818
$513$	−17.9572	−0.792828
$514$	45.9143	2.02519
$515$	4.58546	0.202060
$516$	−32.6148	−1.43579
$517$	12.1678	0.535139
$518$	0	0
$519$	−12.6136	−0.553676
$520$	−4.68585	−0.205488
$521$	16.2646	0.712564	0.356282	−	0.934378i	$-0.384044\pi$
$521$	16.2646	0.712564	0.356282	+	0.934378i	$0.384044\pi$
$522$	−41.4307	−1.81337
$523$	−7.22219	−0.315804	−0.157902	−	0.987455i	$-0.550473\pi$
$523$	−7.22219	−0.315804	−0.157902	+	0.987455i	$0.550473\pi$
$524$	12.7862	0.558569
$525$	0	0
$526$	17.7318	0.773144
$527$	9.53948	0.415546
$528$	1.57246	0.0684326
$529$	−12.9185	−0.561675
$530$	−2.46052	−0.106878
$531$	21.3142	0.924955
$532$	0	0
$533$	−0.292731	−0.0126796
$534$	−15.4061	−0.666688
$535$	6.68585	0.289054
$536$	−21.2860	−0.919415
$537$	27.4685	1.18535
$538$	22.1923	0.956780
$539$	0	0
$540$	−25.1709	−1.08318
$541$	17.3534	0.746081	0.373041	−	0.927815i	$-0.378315\pi$
$541$	17.3534	0.746081	0.373041	+	0.927815i	$0.378315\pi$
$542$	−68.8156	−2.95588
$543$	−7.52119	−0.322765
$544$	24.3503	1.04401
$545$	−18.1249	−0.776387
$546$	0	0
$547$	−34.1109	−1.45848	−0.729238	−	0.684261i	$-0.760125\pi$
$547$	−34.1109	−1.45848	−0.729238	+	0.684261i	$0.760125\pi$
$548$	−45.7220	−1.95315
$549$	−3.37169	−0.143900
$550$	−8.58546	−0.366085
$551$	35.0649	1.49381
$552$	−12.7005	−0.540571
$553$	0	0
$554$	−4.46052	−0.189509
$555$	13.1134	0.556632
$556$	−26.1579	−1.10934
$557$	−13.2222	−0.560242	−0.280121	−	0.959965i	$-0.590375\pi$
$557$	−13.2222	−0.560242	−0.280121	+	0.959965i	$0.590375\pi$
$558$	6.46052	0.273496
$559$	8.15371	0.344865
$560$	0	0
$561$	−7.66442	−0.323592
$562$	−48.2070	−2.03349
$563$	41.3717	1.74361	0.871804	−	0.489854i	$-0.162950\pi$
$563$	41.3717	1.74361	0.871804	+	0.489854i	$0.162950\pi$
$564$	42.4569	1.78776
$565$	22.0863	0.929178
$566$	63.2432	2.65831
$567$	0	0
$568$	5.37169	0.225391
$569$	−2.68164	−0.112420	−0.0562100	−	0.998419i	$-0.517902\pi$
$569$	−2.68164	−0.112420	−0.0562100	+	0.998419i	$0.517902\pi$
$570$	12.0575	0.505035
$571$	−28.9315	−1.21075	−0.605373	−	0.795942i	$-0.706976\pi$
$571$	−28.9315	−1.21075	−0.605373	+	0.795942i	$0.706976\pi$
$572$	−4.00000	−0.167248
$573$	−5.03612	−0.210387
$574$	0	0
$575$	10.1495	0.423263
$576$	20.5254	0.855225
$577$	−37.5296	−1.56238	−0.781189	−	0.624294i	$-0.785387\pi$
$577$	−37.5296	−1.56238	−0.781189	+	0.624294i	$0.785387\pi$
$578$	39.8641	1.65813
$579$	−9.50650	−0.395077
$580$	49.1512	2.04089
$581$	0	0
$582$	14.3503	0.594838
$583$	−0.896480	−0.0371284
$584$	53.3864	2.20914
$585$	2.26396	0.0936033
$586$	34.9259	1.44277
$587$	−23.0649	−0.951990	−0.475995	−	0.879448i	$-0.657912\pi$
$587$	−23.0649	−0.951990	−0.475995	+	0.879448i	$0.657912\pi$
$588$	0	0
$589$	−5.46787	−0.225299
$590$	−39.7795	−1.63770
$591$	−3.63504	−0.149525
$592$	10.1923	0.418903
$593$	11.0502	0.453777	0.226889	−	0.973921i	$-0.427145\pi$
$593$	11.0502	0.453777	0.226889	+	0.973921i	$0.427145\pi$
$594$	−14.4275	−0.591969
$595$	0	0
$596$	7.56404	0.309835
$597$	15.5872	0.637940
$598$	7.43910	0.304207
$599$	−44.6044	−1.82248	−0.911242	−	0.411870i	$-0.864876\pi$
$599$	−44.6044	−1.82248	−0.911242	+	0.411870i	$0.864876\pi$
$600$	−12.7862	−0.521996
$601$	−47.8715	−1.95272	−0.976359	−	0.216156i	$-0.930648\pi$
$601$	−47.8715	−1.95272	−0.976359	+	0.216156i	$0.930648\pi$
$602$	0	0
$603$	10.2843	0.418809
$604$	52.0294	2.11705
$605$	−13.0073	−0.528824
$606$	−29.9374	−1.21612
$607$	−45.0691	−1.82930	−0.914649	−	0.404249i	$-0.867533\pi$
$607$	−45.0691	−1.82930	−0.914649	+	0.404249i	$0.867533\pi$
$608$	−13.9572	−0.566037
$609$	0	0
$610$	6.29273	0.254785
$611$	−10.6142	−0.429406
$612$	34.3074	1.38680
$613$	−25.5212	−1.03079	−0.515396	−	0.856952i	$-0.672355\pi$
$613$	−25.5212	−1.03079	−0.515396	+	0.856952i	$0.672355\pi$
$614$	−60.9834	−2.46109
$615$	0.450654	0.0181721
$616$	0	0
$617$	29.2432	1.17729	0.588643	−	0.808393i	$-0.299663\pi$
$617$	29.2432	1.17729	0.588643	+	0.808393i	$0.299663\pi$
$618$	9.17092	0.368909
$619$	−4.78623	−0.192375	−0.0961874	−	0.995363i	$-0.530665\pi$
$619$	−4.78623	−0.192375	−0.0961874	+	0.995363i	$0.530665\pi$
$620$	−7.66442	−0.307811
$621$	17.0558	0.684428
$622$	−45.6791	−1.83157
$623$	0	0
$624$	−1.37169	−0.0549116
$625$	1.20077	0.0480307
$626$	8.15792	0.326056
$627$	4.39312	0.175444
$628$	−79.7942	−3.18413
$629$	−49.6791	−1.98084
$630$	0	0
$631$	−28.3931	−1.13031	−0.565156	−	0.824984i	$-0.691184\pi$
$631$	−28.3931	−1.13031	−0.565156	+	0.824984i	$0.691184\pi$
$632$	3.07896	0.122475
$633$	10.6331	0.422629
$634$	11.7648	0.467240
$635$	16.1923	0.642574
$636$	−3.12808	−0.124036
$637$	0	0
$638$	28.1726	1.11536
$639$	−2.59533	−0.102670
$640$	−27.0937	−1.07097
$641$	5.96137	0.235460	0.117730	−	0.993046i	$-0.462438\pi$
$641$	5.96137	0.235460	0.117730	+	0.993046i	$0.462438\pi$
$642$	13.3717	0.527739
$643$	−31.1940	−1.23017	−0.615086	−	0.788460i	$-0.710879\pi$
$643$	−31.1940	−1.23017	−0.615086	+	0.788460i	$0.710879\pi$
$644$	0	0
$645$	−12.5525	−0.494253
$646$	−45.6791	−1.79722
$647$	−14.9112	−0.586219	−0.293109	−	0.956079i	$-0.594690\pi$
$647$	−14.9112	−0.586219	−0.293109	+	0.956079i	$0.594690\pi$
$648$	−3.83956	−0.150832
$649$	−14.4935	−0.568920
$650$	7.48929	0.293754
$651$	0	0
$652$	24.7005	0.967348
$653$	−3.57246	−0.139801	−0.0699006	−	0.997554i	$-0.522268\pi$
$653$	−3.57246	−0.139801	−0.0699006	+	0.997554i	$0.522268\pi$
$654$	−36.2499	−1.41748
$655$	4.92104	0.192281
$656$	0.350269	0.0136757
$657$	−25.7936	−1.00630
$658$	0	0
$659$	−3.90383	−0.152071	−0.0760357	−	0.997105i	$-0.524226\pi$
$659$	−3.90383	−0.152071	−0.0760357	+	0.997105i	$0.524226\pi$
$660$	6.15792	0.239697
$661$	−13.7936	−0.536508	−0.268254	−	0.963348i	$-0.586447\pi$
$661$	−13.7936	−0.536508	−0.268254	+	0.963348i	$0.586447\pi$
$662$	−14.4078	−0.559975
$663$	6.68585	0.259657
$664$	42.3221	1.64242
$665$	0	0
$666$	−33.6447	−1.30371
$667$	−33.3049	−1.28957
$668$	−9.12181	−0.352933
$669$	22.4605	0.868374
$670$	−19.1940	−0.741530
$671$	2.29273	0.0885099
$672$	0	0
$673$	5.70306	0.219837	0.109918	−	0.993941i	$-0.464941\pi$
$673$	5.70306	0.219837	0.109918	+	0.993941i	$0.464941\pi$
$674$	−60.0393	−2.31263
$675$	17.1709	0.660909
$676$	3.48929	0.134203
$677$	−35.2614	−1.35521	−0.677604	−	0.735427i	$-0.736981\pi$
$677$	−35.2614	−1.35521	−0.677604	+	0.735427i	$0.736981\pi$
$678$	44.1726	1.69644
$679$	0	0
$680$	−27.3288	−1.04801
$681$	22.5426	0.863835
$682$	−4.39312	−0.168221
$683$	1.03612	0.0396459	0.0198229	−	0.999804i	$-0.493690\pi$
$683$	1.03612	0.0396459	0.0198229	+	0.999804i	$0.493690\pi$
$684$	−19.6644	−0.751888
$685$	−17.5970	−0.672348
$686$	0	0
$687$	−8.90131	−0.339606
$688$	−9.75639	−0.371959
$689$	0.782020	0.0297926
$690$	−11.4523	−0.435983
$691$	−3.67850	−0.139937	−0.0699684	−	0.997549i	$-0.522290\pi$
$691$	−3.67850	−0.139937	−0.0699684	+	0.997549i	$0.522290\pi$
$692$	−38.3931	−1.45949
$693$	0	0
$694$	−39.1281	−1.48528
$695$	−10.0674	−0.381878
$696$	41.9572	1.59038
$697$	−1.70727	−0.0646674
$698$	55.1758	2.08843
$699$	13.9817	0.528837
$700$	0	0
$701$	−0.0617493	−0.00233224	−0.00116612	−	0.999999i	$-0.500371\pi$
$701$	−0.0617493	−0.00233224	−0.00116612	+	0.999999i	$0.500371\pi$
$702$	12.5855	0.475008
$703$	28.4752	1.07396
$704$	−13.9572	−0.526030
$705$	16.3404	0.615415
$706$	17.9227	0.674531
$707$	0	0
$708$	−50.5720	−1.90061
$709$	−42.9834	−1.61428	−0.807138	−	0.590363i	$-0.798985\pi$
$709$	−42.9834	−1.61428	−0.807138	+	0.590363i	$0.798985\pi$
$710$	4.84377	0.181783
$711$	−1.48760	−0.0557892
$712$	−20.0147	−0.750082
$713$	5.19342	0.194495
$714$	0	0
$715$	−1.53948	−0.0575733
$716$	83.6081	3.12458
$717$	−11.7992	−0.440650
$718$	43.0508	1.60664
$719$	−17.6546	−0.658404	−0.329202	−	0.944260i	$-0.606780\pi$
$719$	−17.6546	−0.658404	−0.329202	+	0.944260i	$0.606780\pi$
$720$	−2.70896	−0.100957
$721$	0	0
$722$	−18.3331	−0.682286
$723$	4.61844	0.171762
$724$	−22.8929	−0.850807
$725$	−33.5296	−1.24526
$726$	−26.0147	−0.965496
$727$	23.8077	0.882977	0.441488	−	0.897267i	$-0.354451\pi$
$727$	23.8077	0.882977	0.441488	+	0.897267i	$0.354451\pi$
$728$	0	0
$729$	20.3288	0.752920
$730$	48.1396	1.78173
$731$	47.5542	1.75885
$732$	8.00000	0.295689
$733$	−31.3492	−1.15791	−0.578954	−	0.815360i	$-0.696539\pi$
$733$	−31.3492	−1.15791	−0.578954	+	0.815360i	$0.696539\pi$
$734$	−12.5082	−0.461686
$735$	0	0
$736$	13.2566	0.488645
$737$	−6.99327	−0.257600
$738$	−1.15623	−0.0425615
$739$	−24.8108	−0.912680	−0.456340	−	0.889806i	$-0.650840\pi$
$739$	−24.8108	−0.912680	−0.456340	+	0.889806i	$0.650840\pi$
$740$	39.9143	1.46728
$741$	−3.83221	−0.140780
$742$	0	0
$743$	21.3717	0.784051	0.392026	−	0.919954i	$-0.371774\pi$
$743$	21.3717	0.784051	0.392026	+	0.919954i	$0.371774\pi$
$744$	−6.54262	−0.239864
$745$	2.91117	0.106657
$746$	50.4225	1.84610
$747$	−20.4479	−0.748149
$748$	−23.3288	−0.852987
$749$	0	0
$750$	−29.5640	−1.07953
$751$	19.2243	0.701503	0.350751	−	0.936469i	$-0.385926\pi$
$751$	19.2243	0.701503	0.350751	+	0.936469i	$0.385926\pi$
$752$	12.7005	0.463141
$753$	−3.33727	−0.121617
$754$	−24.5756	−0.894990
$755$	20.0246	0.728768
$756$	0	0
$757$	−19.8610	−0.721860	−0.360930	−	0.932593i	$-0.617541\pi$
$757$	−19.8610	−0.721860	−0.360930	+	0.932593i	$0.617541\pi$
$758$	10.8009	0.392307
$759$	−4.17262	−0.151456
$760$	15.6644	0.568208
$761$	−6.12073	−0.221876	−0.110938	−	0.993827i	$-0.535386\pi$
$761$	−6.12073	−0.221876	−0.110938	+	0.993827i	$0.535386\pi$
$762$	32.3847	1.17317
$763$	0	0
$764$	−15.3288	−0.554578
$765$	13.2039	0.477388
$766$	19.5296	0.705634
$767$	12.6430	0.456512
$768$	−26.2730	−0.948045
$769$	−3.82800	−0.138041	−0.0690206	−	0.997615i	$-0.521987\pi$
$769$	−3.82800	−0.138041	−0.0690206	+	0.997615i	$0.521987\pi$
$770$	0	0
$771$	22.4653	0.809070
$772$	−28.9357	−1.04142
$773$	6.53635	0.235096	0.117548	−	0.993067i	$-0.462497\pi$
$773$	6.53635	0.235096	0.117548	+	0.993067i	$0.462497\pi$
$774$	32.2056	1.15761
$775$	5.22846	0.187812
$776$	18.6430	0.669245
$777$	0	0
$778$	−15.0937	−0.541134
$779$	0.978577	0.0350612
$780$	−5.37169	−0.192337
$781$	1.76481	0.0631498
$782$	43.3864	1.55149
$783$	−56.3452	−2.01361
$784$	0	0
$785$	−30.7104	−1.09610
$786$	9.84208	0.351055
$787$	−30.8066	−1.09814	−0.549068	−	0.835778i	$-0.685017\pi$
$787$	−30.8066	−1.09814	−0.549068	+	0.835778i	$0.685017\pi$
$788$	−11.0643	−0.394148
$789$	8.67598	0.308873
$790$	2.77636	0.0987786
$791$	0	0
$792$	−6.74338	−0.239616
$793$	−2.00000	−0.0710221
$794$	−3.28117	−0.116444
$795$	−1.20390	−0.0426981
$796$	47.4439	1.68161
$797$	38.8156	1.37492	0.687460	−	0.726222i	$-0.258726\pi$
$797$	38.8156	1.37492	0.687460	+	0.726222i	$0.258726\pi$
$798$	0	0
$799$	−61.9044	−2.19002
$800$	13.3461	0.471854
$801$	9.67008	0.341675
$802$	−16.3503	−0.577348
$803$	17.5395	0.618955
$804$	−24.4015	−0.860576
$805$	0	0
$806$	3.83221	0.134984
$807$	10.8585	0.382236
$808$	−38.8929	−1.36825
$809$	1.04033	0.0365759	0.0182880	−	0.999833i	$-0.494178\pi$
$809$	1.04033	0.0365759	0.0182880	+	0.999833i	$0.494178\pi$
$810$	−3.46221	−0.121650
$811$	−12.6712	−0.444944	−0.222472	−	0.974939i	$-0.571413\pi$
$811$	−12.6712	−0.444944	−0.222472	+	0.974939i	$0.571413\pi$
$812$	0	0
$813$	−33.6707	−1.18088
$814$	22.8782	0.801880
$815$	9.50650	0.332998
$816$	−8.00000	−0.280056
$817$	−27.2572	−0.953610
$818$	43.0607	1.50558
$819$	0	0
$820$	1.37169	0.0479016
$821$	27.0361	0.943567	0.471783	−	0.881714i	$-0.343610\pi$
$821$	27.0361	0.943567	0.471783	+	0.881714i	$0.343610\pi$
$822$	−35.1940	−1.22753
$823$	−31.6363	−1.10277	−0.551386	−	0.834251i	$-0.685901\pi$
$823$	−31.6363	−1.10277	−0.551386	+	0.834251i	$0.685901\pi$
$824$	11.9143	0.415055
$825$	−4.20077	−0.146252
$826$	0	0
$827$	56.4800	1.96400	0.982002	−	0.188872i	$-0.0604832\pi$
$827$	56.4800	1.96400	0.982002	+	0.188872i	$0.0604832\pi$
$828$	18.6774	0.649085
$829$	42.6760	1.48220	0.741099	−	0.671396i	$-0.234305\pi$
$829$	42.6760	1.48220	0.741099	+	0.671396i	$0.234305\pi$
$830$	38.1627	1.32465
$831$	−2.18248	−0.0757094
$832$	12.1751	0.422097
$833$	0	0
$834$	−20.1348	−0.697211
$835$	−3.51071	−0.121493
$836$	13.3717	0.462470
$837$	8.78623	0.303697
$838$	70.3797	2.43122
$839$	40.1642	1.38662	0.693311	−	0.720639i	$-0.256151\pi$
$839$	40.1642	1.38662	0.693311	+	0.720639i	$0.256151\pi$
$840$	0	0
$841$	81.0252	2.79397
$842$	−19.4868	−0.671558
$843$	−23.5872	−0.812385
$844$	32.3650	1.11405
$845$	1.34292	0.0461980
$846$	−41.9242	−1.44138
$847$	0	0
$848$	−0.935731	−0.0321331
$849$	30.9442	1.06200
$850$	43.6791	1.49818
$851$	−27.0460	−0.927124
$852$	6.15792	0.210967
$853$	−19.6932	−0.674282	−0.337141	−	0.941454i	$-0.609460\pi$
$853$	−19.6932	−0.674282	−0.337141	+	0.941454i	$0.609460\pi$
$854$	0	0
$855$	−7.56825	−0.258829
$856$	17.3717	0.593752
$857$	−1.66442	−0.0568556	−0.0284278	−	0.999596i	$-0.509050\pi$
$857$	−1.66442	−0.0568556	−0.0284278	+	0.999596i	$0.509050\pi$
$858$	−3.07896	−0.105114
$859$	41.2944	1.40895	0.704474	−	0.709730i	$-0.251183\pi$
$859$	41.2944	1.40895	0.704474	+	0.709730i	$0.251183\pi$
$860$	−38.2070	−1.30285
$861$	0	0
$862$	22.6086	0.770051
$863$	−31.3288	−1.06645	−0.533223	−	0.845975i	$-0.679019\pi$
$863$	−31.3288	−1.06645	−0.533223	+	0.845975i	$0.679019\pi$
$864$	22.4275	0.763000
$865$	−14.7764	−0.502411
$866$	−61.6363	−2.09449
$867$	19.5051	0.662426
$868$	0	0
$869$	1.01156	0.0343147
$870$	37.8337	1.28268
$871$	6.10038	0.206704
$872$	−47.0937	−1.59479
$873$	−9.00735	−0.304852
$874$	−24.8683	−0.841184
$875$	0	0
$876$	61.2003	2.06777
$877$	−53.8041	−1.81683	−0.908417	−	0.418065i	$-0.862708\pi$
$877$	−53.8041	−1.81683	−0.908417	+	0.418065i	$0.862708\pi$
$878$	−79.2234	−2.67366
$879$	17.0888	0.576392
$880$	1.84208	0.0620964
$881$	−8.09196	−0.272625	−0.136313	−	0.990666i	$-0.543525\pi$
$881$	−8.09196	−0.272625	−0.136313	+	0.990666i	$0.543525\pi$
$882$	0	0
$883$	−27.0705	−0.910996	−0.455498	−	0.890237i	$-0.650539\pi$
$883$	−27.0705	−0.910996	−0.455498	+	0.890237i	$0.650539\pi$
$884$	20.3503	0.684454
$885$	−19.4637	−0.654264
$886$	61.9620	2.08165
$887$	38.4935	1.29249	0.646243	−	0.763132i	$-0.276339\pi$
$887$	38.4935	1.29249	0.646243	+	0.763132i	$0.276339\pi$
$888$	34.0722	1.14339
$889$	0	0
$890$	−18.0477	−0.604959
$891$	−1.26144	−0.0422599
$892$	68.3650	2.28903
$893$	35.4826	1.18738
$894$	5.82235	0.194728
$895$	32.1783	1.07560
$896$	0	0
$897$	3.63986	0.121532
$898$	6.19235	0.206641
$899$	−17.1568	−0.572213
$900$	18.8034	0.626781
$901$	4.56090	0.151946
$902$	0.786230	0.0261786
$903$	0	0
$904$	57.3864	1.90864
$905$	−8.81079	−0.292881
$906$	40.0491	1.33054
$907$	15.7031	0.521411	0.260706	−	0.965418i	$-0.416045\pi$
$907$	15.7031	0.521411	0.260706	+	0.965418i	$0.416045\pi$
$908$	68.6148	2.27706
$909$	18.7911	0.623260
$910$	0	0
$911$	44.9399	1.48893	0.744463	−	0.667663i	$-0.232705\pi$
$911$	44.9399	1.48893	0.744463	+	0.667663i	$0.232705\pi$
$912$	4.58546	0.151840
$913$	13.9044	0.460170
$914$	−78.9307	−2.61080
$915$	3.07896	0.101787
$916$	−27.0937	−0.895200
$917$	0	0
$918$	73.4011	2.42260
$919$	−27.2432	−0.898669	−0.449334	−	0.893364i	$-0.648339\pi$
$919$	−27.2432	−0.898669	−0.449334	+	0.893364i	$0.648339\pi$
$920$	−14.8782	−0.490519
$921$	−29.8385	−0.983211
$922$	−77.5015	−2.55237
$923$	−1.53948	−0.0506726
$924$	0	0
$925$	−27.2285	−0.895266
$926$	−5.89962	−0.193873
$927$	−5.75639	−0.189065
$928$	−43.7942	−1.43761
$929$	17.1422	0.562416	0.281208	−	0.959647i	$-0.409265\pi$
$929$	17.1422	0.562416	0.281208	+	0.959647i	$0.409265\pi$
$930$	−5.89962	−0.193456
$931$	0	0
$932$	42.5573	1.39401
$933$	−22.3503	−0.731715
$934$	−6.03442	−0.197452
$935$	−8.97858	−0.293631
$936$	5.88240	0.192272
$937$	51.5197	1.68308	0.841538	−	0.540197i	$-0.181650\pi$
$937$	51.5197	1.68308	0.841538	+	0.540197i	$0.181650\pi$
$938$	0	0
$939$	3.99158	0.130260
$940$	49.7367	1.62223
$941$	−32.7575	−1.06786	−0.533931	−	0.845528i	$-0.679286\pi$
$941$	−32.7575	−1.06786	−0.533931	+	0.845528i	$0.679286\pi$
$942$	−61.4208	−2.00120
$943$	−0.929460	−0.0302674
$944$	−15.1281	−0.492377
$945$	0	0
$946$	−21.8996	−0.712018
$947$	20.9295	0.680116	0.340058	−	0.940404i	$-0.389553\pi$
$947$	20.9295	0.680116	0.340058	+	0.940404i	$0.389553\pi$
$948$	3.52962	0.114637
$949$	−15.3001	−0.496662
$950$	−25.0361	−0.812279
$951$	5.75639	0.186664
$952$	0	0
$953$	46.4120	1.50343	0.751716	−	0.659487i	$-0.229226\pi$
$953$	46.4120	1.50343	0.751716	+	0.659487i	$0.229226\pi$
$954$	3.08883	0.100004
$955$	−5.89962	−0.190907
$956$	−35.9143	−1.16155
$957$	13.7845	0.445591
$958$	−1.20390	−0.0388964
$959$	0	0
$960$	−18.7434	−0.604940
$961$	−28.3246	−0.913698
$962$	−19.9572	−0.643444
$963$	−8.39312	−0.270464
$964$	14.0575	0.452763
$965$	−11.1365	−0.358497
$966$	0	0
$967$	−23.2186	−0.746660	−0.373330	−	0.927699i	$-0.621784\pi$
$967$	−23.2186	−0.746660	−0.373330	+	0.927699i	$0.621784\pi$
$968$	−33.7967	−1.08627
$969$	−22.3503	−0.717994
$970$	16.8108	0.539762
$971$	14.1004	0.452503	0.226251	−	0.974069i	$-0.427353\pi$
$971$	14.1004	0.452503	0.226251	+	0.974069i	$0.427353\pi$
$972$	51.8286	1.66240
$973$	0	0
$974$	84.4800	2.70692
$975$	3.66442	0.117355
$976$	2.39312	0.0766018
$977$	−2.95402	−0.0945074	−0.0472537	−	0.998883i	$-0.515047\pi$
$977$	−2.95402	−0.0945074	−0.0472537	+	0.998883i	$0.515047\pi$
$978$	19.0130	0.607969
$979$	−6.57560	−0.210157
$980$	0	0
$981$	22.7533	0.726455
$982$	−21.6216	−0.689972
$983$	35.0367	1.11750	0.558749	−	0.829337i	$-0.311281\pi$
$983$	35.0367	1.11750	0.558749	+	0.829337i	$0.311281\pi$
$984$	1.17092	0.0373277
$985$	−4.25831	−0.135681
$986$	−143.330	−4.56456
$987$	0	0
$988$	−11.6644	−0.371095
$989$	25.8891	0.823227
$990$	−6.08065	−0.193256
$991$	47.9718	1.52388	0.761938	−	0.647650i	$-0.224248\pi$
$991$	47.9718	1.52388	0.761938	+	0.647650i	$0.224248\pi$
$992$	6.82908	0.216823
$993$	−7.04958	−0.223712
$994$	0	0
$995$	18.2598	0.578873
$996$	48.5166	1.53731
$997$	38.4422	1.21748	0.608739	−	0.793371i	$-0.291676\pi$
$997$	38.4422	1.21748	0.608739	+	0.793371i	$0.291676\pi$
$998$	2.35027	0.0743965
$999$	−45.7564	−1.44767

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	637.2.a.j.1.3		3
3.2	odd	2		5733.2.a.x.1.1		3
7.2	even	3		637.2.e.i.508.1		6
7.3	odd	6		637.2.e.j.79.1		6
7.4	even	3		637.2.e.i.79.1		6
7.5	odd	6		637.2.e.j.508.1		6
7.6	odd	2		91.2.a.d.1.3	✓	3
13.12	even	2		8281.2.a.bg.1.1		3
21.20	even	2		819.2.a.i.1.1		3
28.27	even	2		1456.2.a.t.1.2		3
35.34	odd	2		2275.2.a.m.1.1		3
56.13	odd	2		5824.2.a.by.1.2		3
56.27	even	2		5824.2.a.bs.1.2		3
91.34	even	4		1183.2.c.f.337.6		6
91.83	even	4		1183.2.c.f.337.1		6
91.90	odd	2		1183.2.a.i.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.a.d.1.3	✓	3	7.6	odd	2
637.2.a.j.1.3		3	1.1	even	1	trivial
637.2.e.i.79.1		6	7.4	even	3
637.2.e.i.508.1		6	7.2	even	3
637.2.e.j.79.1		6	7.3	odd	6
637.2.e.j.508.1		6	7.5	odd	6
819.2.a.i.1.1		3	21.20	even	2
1183.2.a.i.1.1		3	91.90	odd	2
1183.2.c.f.337.1		6	91.83	even	4
1183.2.c.f.337.6		6	91.34	even	4
1456.2.a.t.1.2		3	28.27	even	2
2275.2.a.m.1.1		3	35.34	odd	2
5733.2.a.x.1.1		3	3.2	odd	2
5824.2.a.bs.1.2		3	56.27	even	2
5824.2.a.by.1.2		3	56.13	odd	2
8281.2.a.bg.1.1		3	13.12	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 637 = 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	637.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34292 q^{2} +1.14637 q^{3} +3.48929 q^{4} +1.34292 q^{5} +2.68585 q^{6} +3.48929 q^{8} -1.68585 q^{9} +3.14637 q^{10} +1.14637 q^{11} +4.00000 q^{12} -1.00000 q^{13} +1.53948 q^{15} +1.19656 q^{16} -5.83221 q^{17} -3.94981 q^{18} +3.34292 q^{19} +4.68585 q^{20} +2.68585 q^{22} -3.17513 q^{23} +4.00000 q^{24} -3.19656 q^{25} -2.34292 q^{26} -5.37169 q^{27} +10.4893 q^{29} +3.60688 q^{30} -1.63565 q^{31} -4.17513 q^{32} +1.31415 q^{33} -13.6644 q^{34} -5.88240 q^{36} +8.51806 q^{37} +7.83221 q^{38} -1.14637 q^{39} +4.68585 q^{40} +0.292731 q^{41} -8.15371 q^{43} +4.00000 q^{44} -2.26396 q^{45} -7.43910 q^{46} +10.6142 q^{47} +1.37169 q^{48} -7.48929 q^{50} -6.68585 q^{51} -3.48929 q^{52} -0.782020 q^{53} -12.5855 q^{54} +1.53948 q^{55} +3.83221 q^{57} +24.5756 q^{58} -12.6430 q^{59} +5.37169 q^{60} +2.00000 q^{61} -3.83221 q^{62} -12.1751 q^{64} -1.34292 q^{65} +3.07896 q^{66} -6.10038 q^{67} -20.3503 q^{68} -3.63986 q^{69} +1.53948 q^{71} -5.88240 q^{72} +15.3001 q^{73} +19.9572 q^{74} -3.66442 q^{75} +11.6644 q^{76} -2.68585 q^{78} +0.882404 q^{79} +1.60688 q^{80} -1.10038 q^{81} +0.685846 q^{82} +12.1292 q^{83} -7.83221 q^{85} -19.1035 q^{86} +12.0246 q^{87} +4.00000 q^{88} -5.73604 q^{89} -5.30429 q^{90} -11.0790 q^{92} -1.87506 q^{93} +24.8683 q^{94} +4.48929 q^{95} -4.78623 q^{96} +5.34292 q^{97} -1.93260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{8} + 7 q^{9} + 8 q^{10} + 2 q^{11} + 12 q^{12} - 3 q^{13} - 6 q^{15} - q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{19} + 2 q^{20} - 4 q^{22} + 10 q^{23}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q + q^2 + 2 * q^3 + 3 * q^4 - 2 * q^5 - 4 * q^6 + 3 * q^8 + 7 * q^9 + 8 * q^10 + 2 * q^11 + 12 * q^12 - 3 * q^13 - 6 * q^15 - q^16 - 4 * q^17 - 15 * q^18 + 4 * q^19 + 2 * q^20 - 4 * q^22 + 10 * q^23 + 12 * q^24 - 5 * q^25 - q^26 + 8 * q^27 + 24 * q^29 + 20 * q^30 + 4 * q^31 + 7 * q^32 + 16 * q^33 - 14 * q^34 - q^36 + 10 * q^38 - 2 * q^39 + 2 * q^40 - 2 * q^41 + 10 * q^43 + 12 * q^44 - 22 * q^45 - 18 * q^46 + 8 * q^47 - 20 * q^48 - 15 * q^50 - 8 * q^51 - 3 * q^52 + 8 * q^53 - 32 * q^54 - 6 * q^55 - 2 * q^57 + 12 * q^58 + 4 * q^59 - 8 * q^60 + 6 * q^61 + 2 * q^62 - 17 * q^64 + 2 * q^65 - 12 * q^66 - 12 * q^67 - 22 * q^68 + 6 * q^69 - 6 * q^71 - q^72 + 10 * q^73 + 30 * q^74 + 16 * q^75 + 8 * q^76 + 4 * q^78 - 14 * q^79 + 14 * q^80 + 3 * q^81 - 10 * q^82 + 12 * q^83 - 10 * q^85 - 26 * q^86 + 26 * q^87 + 12 * q^88 - 2 * q^89 + 28 * q^90 - 12 * q^92 - 22 * q^93 + 10 * q^94 + 6 * q^95 + 4 * q^96 + 10 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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