LMFDB - Embedded newform 425.2.b.f.324.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(324,425)] 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("425.324"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-22,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	no
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.229451239931904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{7} + 64x^{6} - 30x^{5} + 2x^{4} + 136x^{3} + 324x^{2} + 180x + 50 $ x^10 - 2x^7 + 64x^6 - 30x^5 + 2x^4 + 136x^3 + 324x^2 + 180*x + 50
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			324.10
Root			$1.44796 - 1.44796i$ of defining polynomial
Character	$\chi$	$=$	425.324
Dual form			425.2.b.f.324.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.80107i q^{2} +2.60789i q^{3} -5.84602 q^{4} -7.30489 q^{6} +1.33298i q^{7} -10.7730i q^{8} -3.80107 q^{9} +1.09485 q^{11} -15.2458i q^{12} +3.17083i q^{13} -3.73378 q^{14} +18.4839 q^{16} -1.00000i q^{17} -10.6471i q^{18} -2.75613 q^{19} -3.47626 q^{21} +3.06675i q^{22} +3.57111i q^{23} +28.0947 q^{24} -8.88173 q^{26} -2.08911i q^{27} -7.79262i q^{28} +0.180058 q^{29} +0.816039 q^{31} +30.2288i q^{32} +2.85524i q^{33} +2.80107 q^{34} +22.2211 q^{36} -8.44817i q^{37} -7.72013i q^{38} -8.26917 q^{39} -7.97191 q^{41} -9.73726i q^{42} +6.54798i q^{43} -6.40051 q^{44} -10.0029 q^{46} -0.576074i q^{47} +48.2039i q^{48} +5.22317 q^{49} +2.60789 q^{51} -18.5367i q^{52} +7.84602i q^{53} +5.85176 q^{54} +14.3602 q^{56} -7.18768i q^{57} +0.504355i q^{58} +9.76375 q^{59} -5.21577 q^{61} +2.28579i q^{62} -5.06675i q^{63} -47.7053 q^{64} -7.99775 q^{66} +2.64963i q^{67} +5.84602i q^{68} -9.31305 q^{69} +13.4114 q^{71} +40.9489i q^{72} -12.3299i q^{73} +23.6639 q^{74} +16.1124 q^{76} +1.45941i q^{77} -23.1626i q^{78} -3.74745 q^{79} -5.95506 q^{81} -22.3299i q^{82} -4.66596i q^{83} +20.3223 q^{84} -18.3414 q^{86} +0.469570i q^{87} -11.7948i q^{88} -3.00946 q^{89} -4.22665 q^{91} -20.8768i q^{92} +2.12814i q^{93} +1.61363 q^{94} -78.8332 q^{96} +12.2681i q^{97} +14.6305i q^{98} -4.16160 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9} + 8 q^{11} + 14 q^{14} + 54 q^{16} - 12 q^{19} - 10 q^{21} + 38 q^{24} - 10 q^{26} - 4 q^{29} + 42 q^{31} + 2 q^{34} + 44 q^{36} - 46 q^{39} - 16 q^{41} + 8 q^{44} - 12 q^{46}+ \cdots + 28 q^{99}+O(q^{100}) $ 10 * q - 22 * q^4 + 6 * q^6 - 12 * q^9 + 8 * q^11 + 14 * q^14 + 54 * q^16 - 12 * q^19 - 10 * q^21 + 38 * q^24 - 10 * q^26 - 4 * q^29 + 42 * q^31 + 2 * q^34 + 44 * q^36 - 46 * q^39 - 16 * q^41 + 8 * q^44 - 12 * q^46 - 20 * q^49 + 2 * q^51 + 50 * q^54 - 58 * q^56 - 24 * q^59 - 4 * q^61 - 86 * q^64 - 56 * q^66 + 42 * q^71 + 100 * q^74 - 28 * q^76 - 82 * q^79 - 70 * q^81 + 142 * q^84 - 72 * q^86 + 20 * q^89 + 26 * q^91 + 20 * q^94 - 170 * q^96 + 28 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/425\mathbb{Z}\right)^\times$.

$n$	$52$	$326$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.80107i	1.98066i	0.138737	+	0.990329i	$0.455696\pi$
$2$	2.80107i	1.98066i	−0.138737	+	0.990329i	$0.544304\pi$
$3$	2.60789i	1.50566i	0.658213	+	0.752832i	$0.271313\pi$
$3$	2.60789i	1.50566i	−0.658213	+	0.752832i	$0.728687\pi$
$4$	−5.84602	−2.92301
$5$	0	0
$5$	0	0
$6$	−7.30489	−2.98221
$7$	1.33298i	0.503819i	0.967751	+	0.251909i	$0.0810585\pi$
$7$	1.33298i	0.503819i	−0.967751	+	0.251909i	$0.918941\pi$
$8$	− 10.7730i	− 3.80882i
$9$	−3.80107	−1.26702
$10$	0	0
$11$	1.09485	0.330110	0.165055	−	0.986284i	$-0.447220\pi$
$11$	1.09485	0.330110	0.165055	+	0.986284i	$0.447220\pi$
$12$	− 15.2458i	− 4.40107i
$13$	3.17083i	0.879430i	0.898137	+	0.439715i	$0.144921\pi$
$13$	3.17083i	0.879430i	−0.898137	+	0.439715i	$0.855079\pi$
$14$	−3.73378	−0.997893
$15$	0	0
$16$	18.4839	4.62097
$17$	− 1.00000i	− 0.242536i
$17$	− 1.00000i	− 0.242536i
$18$	− 10.6471i	− 2.50954i
$19$	−2.75613	−0.632300	−0.316150	−	0.948709i	$-0.602390\pi$
$19$	−2.75613	−0.632300	−0.316150	+	0.948709i	$0.602390\pi$
$20$	0	0
$21$	−3.47626	−0.758582
$22$	3.06675i	0.653834i
$23$	3.57111i	0.744628i	0.928107	+	0.372314i	$0.121436\pi$
$23$	3.57111i	0.744628i	−0.928107	+	0.372314i	$0.878564\pi$
$24$	28.0947	5.73481
$25$	0	0
$26$	−8.88173	−1.74185
$27$	− 2.08911i	− 0.402050i
$28$	− 7.79262i	− 1.47267i
$29$	0.180058	0.0334359	0.0167179	−	0.999860i	$-0.494678\pi$
$29$	0.180058	0.0334359	0.0167179	+	0.999860i	$0.494678\pi$
$30$	0	0
$31$	0.816039	0.146565	0.0732825	−	0.997311i	$-0.476653\pi$
$31$	0.816039	0.146565	0.0732825	+	0.997311i	$0.476653\pi$
$32$	30.2288i	5.34374i
$33$	2.85524i	0.497034i
$34$	2.80107	0.480380
$35$	0	0
$36$	22.2211	3.70352
$37$	− 8.44817i	− 1.38887i	−0.719555	−	0.694435i	$-0.755654\pi$
$37$	− 8.44817i	− 1.38887i	0.719555	−	0.694435i	$-0.244346\pi$
$38$	− 7.72013i	− 1.25237i
$39$	−8.26917	−1.32413
$40$	0	0
$41$	−7.97191	−1.24500	−0.622501	−	0.782619i	$-0.713883\pi$
$41$	−7.97191	−1.24500	−0.622501	+	0.782619i	$0.713883\pi$
$42$	− 9.73726i	− 1.50249i
$43$	6.54798i	0.998557i	0.866441	+	0.499279i	$0.166402\pi$
$43$	6.54798i	0.998557i	−0.866441	+	0.499279i	$0.833598\pi$
$44$	−6.40051	−0.964913
$45$	0	0
$46$	−10.0029	−1.47485
$47$	− 0.576074i	− 0.0840290i	−0.999117	−	0.0420145i	$-0.986622\pi$
$47$	− 0.576074i	− 0.0840290i	0.999117	−	0.0420145i	$-0.0133776\pi$
$48$	48.2039i	6.95763i
$49$	5.22317	0.746166
$50$	0	0
$51$	2.60789	0.365177
$52$	− 18.5367i	− 2.57058i
$53$	7.84602i	1.07773i	0.842391	+	0.538867i	$0.181147\pi$
$53$	7.84602i	1.07773i	−0.842391	+	0.538867i	$0.818853\pi$
$54$	5.85176	0.796323
$55$	0	0
$56$	14.3602	1.91896
$57$	− 7.18768i	− 0.952031i
$58$	0.504355i	0.0662251i
$59$	9.76375	1.27113	0.635566	−	0.772046i	$-0.280767\pi$
$59$	9.76375	1.27113	0.635566	+	0.772046i	$0.280767\pi$
$60$	0	0
$61$	−5.21577	−0.667811	−0.333906	−	0.942606i	$-0.608367\pi$
$61$	−5.21577	−0.667811	−0.333906	+	0.942606i	$0.608367\pi$
$62$	2.28579i	0.290295i
$63$	− 5.06675i	− 0.638351i
$64$	−47.7053	−5.96316
$65$	0	0
$66$	−7.99775	−0.984455
$67$	2.64963i	0.323704i	0.986815	+	0.161852i	$0.0517466\pi$
$67$	2.64963i	0.323704i	−0.986815	+	0.161852i	$0.948253\pi$
$68$	5.84602i	0.708934i
$69$	−9.31305	−1.12116
$70$	0	0
$71$	13.4114	1.59164	0.795820	−	0.605534i	$-0.207040\pi$
$71$	13.4114	1.59164	0.795820	+	0.605534i	$0.207040\pi$
$72$	40.9489i	4.82587i
$73$	− 12.3299i	− 1.44311i	−0.692359	−	0.721553i	$-0.743429\pi$
$73$	− 12.3299i	− 1.44311i	0.692359	−	0.721553i	$-0.256571\pi$
$74$	23.6639	2.75088
$75$	0	0
$76$	16.1124	1.84822
$77$	1.45941i	0.166315i
$78$	− 23.1626i	− 2.62264i
$79$	−3.74745	−0.421621	−0.210810	−	0.977527i	$-0.567610\pi$
$79$	−3.74745	−0.421621	−0.210810	+	0.977527i	$0.567610\pi$
$80$	0	0
$81$	−5.95506	−0.661673
$82$	− 22.3299i	− 2.46592i
$83$	− 4.66596i	− 0.512156i	−0.966656	−	0.256078i	$-0.917570\pi$
$83$	− 4.66596i	− 0.512156i	0.966656	−	0.256078i	$-0.0824303\pi$
$84$	20.3223	2.21734
$85$	0	0
$86$	−18.3414	−1.97780
$87$	0.469570i	0.0503432i
$88$	− 11.7948i	− 1.25733i
$89$	−3.00946	−0.319002	−0.159501	−	0.987198i	$-0.550988\pi$
$89$	−3.00946	−0.319002	−0.159501	+	0.987198i	$0.550988\pi$
$90$	0	0
$91$	−4.22665	−0.443074
$92$	− 20.8768i	− 2.17655i
$93$	2.12814i	0.220678i
$94$	1.61363	0.166433
$95$	0	0
$96$	−78.8332	−8.04588
$97$	12.2681i	1.24564i	0.782366	+	0.622819i	$0.214013\pi$
$97$	12.2681i	1.24564i	−0.782366	+	0.622819i	$0.785987\pi$
$98$	14.6305i	1.47790i
$99$	−4.16160	−0.418257
$100$	0	0
$101$	2.98113	0.296634	0.148317	−	0.988940i	$-0.452614\pi$
$101$	2.98113	0.296634	0.148317	+	0.988940i	$0.452614\pi$
$102$	7.30489i	0.723291i
$103$	− 13.8440i	− 1.36409i	−0.731310	−	0.682045i	$-0.761091\pi$
$103$	− 13.8440i	− 1.36409i	0.731310	−	0.682045i	$-0.238909\pi$
$104$	34.1593	3.34959
$105$	0	0
$106$	−21.9773	−2.13462
$107$	7.87103i	0.760921i	0.924797	+	0.380461i	$0.124235\pi$
$107$	7.87103i	0.760921i	−0.924797	+	0.380461i	$0.875765\pi$
$108$	12.2130i	1.17519i
$109$	−13.6739	−1.30972	−0.654859	−	0.755751i	$-0.727272\pi$
$109$	−13.6739	−1.30972	−0.654859	+	0.755751i	$0.727272\pi$
$110$	0	0
$111$	22.0319	2.09117
$112$	24.6386i	2.32813i
$113$	19.5722i	1.84120i	0.390508	+	0.920600i	$0.372300\pi$
$113$	19.5722i	1.84120i	−0.390508	+	0.920600i	$0.627700\pi$
$114$	20.1332	1.88565
$115$	0	0
$116$	−1.05262	−0.0977334
$117$	− 12.0526i	− 1.11426i
$118$	27.3490i	2.51768i
$119$	1.33298	0.122194
$120$	0	0
$121$	−9.80130	−0.891028
$122$	− 14.6098i	− 1.32271i
$123$	− 20.7898i	− 1.87456i
$124$	−4.77058	−0.428411
$125$	0	0
$126$	14.1924	1.26436
$127$	5.00946i	0.444517i	0.974988	+	0.222259i	$0.0713430\pi$
$127$	5.00946i	0.444517i	−0.974988	+	0.222259i	$0.928657\pi$
$128$	− 73.1684i	− 6.46724i
$129$	−17.0764	−1.50349
$130$	0	0
$131$	2.63893	0.230564	0.115282	−	0.993333i	$-0.463223\pi$
$131$	2.63893	0.230564	0.115282	+	0.993333i	$0.463223\pi$
$132$	− 16.6918i	− 1.45284i
$133$	− 3.67387i	− 0.318565i
$134$	−7.42180	−0.641146
$135$	0	0
$136$	−10.7730	−0.923775
$137$	3.27064i	0.279430i	0.990192	+	0.139715i	$0.0446186\pi$
$137$	3.27064i	0.279430i	−0.990192	+	0.139715i	$0.955381\pi$
$138$	− 26.0865i	− 2.22063i
$139$	10.3901	0.881276	0.440638	−	0.897685i	$-0.354752\pi$
$139$	10.3901	0.881276	0.440638	+	0.897685i	$0.354752\pi$
$140$	0	0
$141$	1.50234	0.126519
$142$	37.5663i	3.15249i
$143$	3.47158i	0.290308i
$144$	−70.2586	−5.85488
$145$	0	0
$146$	34.5370	2.85830
$147$	13.6214i	1.12348i
$148$	49.3881i	4.05968i
$149$	11.2566	0.922179	0.461090	−	0.887354i	$-0.347459\pi$
$149$	11.2566	0.922179	0.461090	+	0.887354i	$0.347459\pi$
$150$	0	0
$151$	7.13736	0.580831	0.290415	−	0.956901i	$-0.406207\pi$
$151$	7.13736	0.580831	0.290415	+	0.956901i	$0.406207\pi$
$152$	29.6917i	2.40832i
$153$	3.80107i	0.307299i
$154$	−4.08792	−0.329414
$155$	0	0
$156$	48.3417	3.87043
$157$	8.30595i	0.662887i	0.943475	+	0.331443i	$0.107536\pi$
$157$	8.30595i	0.662887i	−0.943475	+	0.331443i	$0.892464\pi$
$158$	− 10.4969i	− 0.835087i
$159$	−20.4615	−1.62270
$160$	0	0
$161$	−4.76022	−0.375158
$162$	− 16.6806i	− 1.31055i
$163$	11.6059i	0.909042i	0.890736	+	0.454521i	$0.150189\pi$
$163$	11.6059i	0.909042i	−0.890736	+	0.454521i	$0.849811\pi$
$164$	46.6039	3.63915
$165$	0	0
$166$	13.0697	1.01441
$167$	19.6832i	1.52313i	0.648087	+	0.761566i	$0.275569\pi$
$167$	19.6832i	1.52313i	−0.648087	+	0.761566i	$0.724431\pi$
$168$	37.4497i	2.88931i
$169$	2.94583	0.226602
$170$	0	0
$171$	10.4763	0.801140
$172$	− 38.2796i	− 2.91879i
$173$	19.8536i	1.50944i	0.656045	+	0.754722i	$0.272228\pi$
$173$	19.8536i	1.50944i	−0.656045	+	0.754722i	$0.727772\pi$
$174$	−1.31530	−0.0997127
$175$	0	0
$176$	20.2371	1.52543
$177$	25.4628i	1.91390i
$178$	− 8.42971i	− 0.631834i
$179$	6.71142	0.501635	0.250817	−	0.968034i	$-0.419301\pi$
$179$	6.71142	0.501635	0.250817	+	0.968034i	$0.419301\pi$
$180$	0	0
$181$	1.49564	0.111170	0.0555852	−	0.998454i	$-0.482298\pi$
$181$	1.49564	0.111170	0.0555852	+	0.998454i	$0.482298\pi$
$182$	− 11.8392i	− 0.877578i
$183$	− 13.6021i	− 1.00550i
$184$	38.4715	2.83616
$185$	0	0
$186$	−5.96107	−0.437087
$187$	− 1.09485i	− 0.0800633i
$188$	3.36774i	0.245617i
$189$	2.78474	0.202560
$190$	0	0
$191$	−15.9320	−1.15280	−0.576401	−	0.817167i	$-0.695543\pi$
$191$	−15.9320	−1.15280	−0.576401	+	0.817167i	$0.695543\pi$
$192$	− 124.410i	− 8.97851i
$193$	− 17.4673i	− 1.25732i	−0.777680	−	0.628661i	$-0.783603\pi$
$193$	− 17.4673i	− 1.25732i	0.777680	−	0.628661i	$-0.216397\pi$
$194$	−34.3639	−2.46718
$195$	0	0
$196$	−30.5347	−2.18105
$197$	14.0436i	1.00057i	0.865862	+	0.500283i	$0.166771\pi$
$197$	14.0436i	1.00057i	−0.865862	+	0.500283i	$0.833229\pi$
$198$	− 11.6570i	− 0.828424i
$199$	4.58571	0.325072	0.162536	−	0.986703i	$-0.448033\pi$
$199$	4.58571	0.325072	0.162536	+	0.986703i	$0.448033\pi$
$200$	0	0
$201$	−6.90993	−0.487389
$202$	8.35037i	0.587530i
$203$	0.240013i	0.0168456i
$204$	−15.2458	−1.06742
$205$	0	0
$206$	38.7781	2.70180
$207$	− 13.5741i	− 0.943462i
$208$	58.6093i	4.06382i
$209$	−3.01755	−0.208728
$210$	0	0
$211$	7.23345	0.497971	0.248986	−	0.968507i	$-0.419903\pi$
$211$	7.23345	0.497971	0.248986	+	0.968507i	$0.419903\pi$
$212$	− 45.8680i	− 3.15022i
$213$	34.9754i	2.39647i
$214$	−22.0473	−1.50713
$215$	0	0
$216$	−22.5060	−1.53134
$217$	1.08776i	0.0738422i
$218$	− 38.3015i	− 2.59411i
$219$	32.1550	2.17283
$220$	0	0
$221$	3.17083	0.213293
$222$	61.7129i	4.14190i
$223$	− 22.2566i	− 1.49041i	−0.666833	−	0.745207i	$-0.732351\pi$
$223$	− 22.2566i	− 1.49041i	0.666833	−	0.745207i	$-0.267649\pi$
$224$	−40.2943	−2.69228
$225$	0	0
$226$	−54.8232	−3.64679
$227$	− 13.3541i	− 0.886342i	−0.896437	−	0.443171i	$-0.853853\pi$
$227$	− 13.3541i	− 0.886342i	0.896437	−	0.443171i	$-0.146147\pi$
$228$	42.0193i	2.78280i
$229$	20.4647	1.35235	0.676174	−	0.736742i	$-0.263637\pi$
$229$	20.4647	1.35235	0.676174	+	0.736742i	$0.263637\pi$
$230$	0	0
$231$	−3.80598	−0.250415
$232$	− 1.93976i	− 0.127351i
$233$	− 16.1239i	− 1.05631i	−0.849148	−	0.528155i	$-0.822884\pi$
$233$	− 16.1239i	− 1.05631i	0.849148	−	0.528155i	$-0.177116\pi$
$234$	33.7601	2.20697
$235$	0	0
$236$	−57.0791	−3.71553
$237$	− 9.77292i	− 0.634820i
$238$	3.73378i	0.242025i
$239$	4.38637	0.283731	0.141865	−	0.989886i	$-0.454690\pi$
$239$	4.38637	0.283731	0.141865	+	0.989886i	$0.454690\pi$
$240$	0	0
$241$	−14.8343	−0.955558	−0.477779	−	0.878480i	$-0.658558\pi$
$241$	−14.8343	−0.955558	−0.477779	+	0.878480i	$0.658558\pi$
$242$	− 27.4542i	− 1.76482i
$243$	− 21.7974i	− 1.39831i
$244$	30.4915	1.95202
$245$	0	0
$246$	58.2339	3.71285
$247$	− 8.73923i	− 0.556064i
$248$	− 8.79118i	− 0.558240i
$249$	12.1683	0.771134
$250$	0	0
$251$	14.5998	0.921534	0.460767	−	0.887521i	$-0.347574\pi$
$251$	14.5998	0.921534	0.460767	+	0.887521i	$0.347574\pi$
$252$	29.6203i	1.86591i
$253$	3.90983i	0.245809i
$254$	−14.0319	−0.880437
$255$	0	0
$256$	109.540	6.84623
$257$	− 5.34722i	− 0.333550i	−0.985995	−	0.166775i	$-0.946665\pi$
$257$	− 5.34722i	− 0.333550i	0.985995	−	0.166775i	$-0.0533354\pi$
$258$	− 47.8322i	− 2.97790i
$259$	11.2612	0.699739
$260$	0	0
$261$	−0.684413	−0.0423641
$262$	7.39183i	0.456669i
$263$	18.3754i	1.13307i	0.824037	+	0.566537i	$0.191717\pi$
$263$	18.3754i	1.13307i	−0.824037	+	0.566537i	$0.808283\pi$
$264$	30.7595	1.89312
$265$	0	0
$266$	10.2908	0.630968
$267$	− 7.84832i	− 0.480310i
$268$	− 15.4898i	− 0.946188i
$269$	15.0812	0.919516	0.459758	−	0.888044i	$-0.347936\pi$
$269$	15.0812	0.919516	0.459758	+	0.888044i	$0.347936\pi$
$270$	0	0
$271$	20.6031	1.25155	0.625774	−	0.780004i	$-0.284783\pi$
$271$	20.6031	1.25155	0.625774	+	0.780004i	$0.284783\pi$
$272$	− 18.4839i	− 1.12075i
$273$	− 11.0226i	− 0.667120i
$274$	−9.16132	−0.553455
$275$	0	0
$276$	54.4443	3.27716
$277$	3.93417i	0.236381i	0.992991	+	0.118191i	$0.0377094\pi$
$277$	3.93417i	0.236381i	−0.992991	+	0.118191i	$0.962291\pi$
$278$	29.1034i	1.74551i
$279$	−3.10183	−0.185701
$280$	0	0
$281$	24.7878	1.47872	0.739358	−	0.673312i	$-0.235129\pi$
$281$	24.7878	1.47872	0.739358	+	0.673312i	$0.235129\pi$
$282$	4.20815i	0.250592i
$283$	10.2892i	0.611631i	0.952091	+	0.305815i	$0.0989290\pi$
$283$	10.2892i	0.611631i	−0.952091	+	0.305815i	$0.901071\pi$
$284$	−78.4032	−4.65237
$285$	0	0
$286$	−9.72416	−0.575002
$287$	− 10.6264i	− 0.627256i
$288$	− 114.902i	− 6.77065i
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	−31.9938	−1.87551
$292$	72.0808i	4.21821i
$293$	− 4.20448i	− 0.245628i	−0.992430	−	0.122814i	$-0.960808\pi$
$293$	− 4.20448i	− 0.245628i	0.992430	−	0.122814i	$-0.0391919\pi$
$294$	−38.1546	−2.22522
$295$	0	0
$296$	−91.0119	−5.28996
$297$	− 2.28726i	− 0.132720i
$298$	31.5307i	1.82652i
$299$	−11.3234	−0.654848
$300$	0	0
$301$	−8.72832	−0.503092
$302$	19.9923i	1.15043i
$303$	7.77446i	0.446631i
$304$	−50.9440	−2.92184
$305$	0	0
$306$	−10.6471	−0.608654
$307$	− 20.2969i	− 1.15841i	−0.815183	−	0.579204i	$-0.803363\pi$
$307$	− 20.2969i	− 1.15841i	0.815183	−	0.579204i	$-0.196637\pi$
$308$	− 8.53175i	− 0.486141i
$309$	36.1036	2.05386
$310$	0	0
$311$	8.03850	0.455822	0.227911	−	0.973682i	$-0.426811\pi$
$311$	8.03850	0.455822	0.227911	+	0.973682i	$0.426811\pi$
$312$	89.0836i	5.04337i
$313$	− 29.6525i	− 1.67606i	−0.545627	−	0.838028i	$-0.683708\pi$
$313$	− 29.6525i	− 1.67606i	0.545627	−	0.838028i	$-0.316292\pi$
$314$	−23.2656	−1.31295
$315$	0	0
$316$	21.9077	1.23240
$317$	− 3.06410i	− 0.172097i	−0.996291	−	0.0860484i	$-0.972576\pi$
$317$	− 3.06410i	− 0.172097i	0.996291	−	0.0860484i	$-0.0274240\pi$
$318$	− 57.3143i	− 3.21402i
$319$	0.197136	0.0110375
$320$	0	0
$321$	−20.5268	−1.14569
$322$	− 13.3337i	− 0.743059i
$323$	2.75613i	0.153355i
$324$	34.8134	1.93408
$325$	0	0
$326$	−32.5089	−1.80050
$327$	− 35.6599i	− 1.97200i
$328$	85.8812i	4.74199i
$329$	0.767895	0.0423354
$330$	0	0
$331$	11.6185	0.638609	0.319305	−	0.947652i	$-0.396551\pi$
$331$	11.6185	0.638609	0.319305	+	0.947652i	$0.396551\pi$
$332$	27.2773i	1.49704i
$333$	32.1121i	1.75973i
$334$	−55.1341	−3.01681
$335$	0	0
$336$	−64.2548	−3.50539
$337$	7.19732i	0.392063i	0.980598	+	0.196032i	$0.0628055\pi$
$337$	7.19732i	0.392063i	−0.980598	+	0.196032i	$0.937195\pi$
$338$	8.25149i	0.448822i
$339$	−51.0421	−2.77223
$340$	0	0
$341$	0.893440	0.0483825
$342$	29.3448i	1.58678i
$343$	16.2932i	0.879752i
$344$	70.5412	3.80333
$345$	0	0
$346$	−55.6115	−2.98969
$347$	− 8.49365i	− 0.455963i	−0.973665	−	0.227982i	$-0.926787\pi$
$347$	− 8.49365i	− 0.455963i	0.973665	−	0.227982i	$-0.0732126\pi$
$348$	− 2.74512i	− 0.147154i
$349$	0.281605	0.0150739	0.00753697	−	0.999972i	$-0.497601\pi$
$349$	0.281605	0.0150739	0.00753697	+	0.999972i	$0.497601\pi$
$350$	0	0
$351$	6.62422	0.353575
$352$	33.0960i	1.76402i
$353$	7.56460i	0.402623i	0.979527	+	0.201311i	$0.0645203\pi$
$353$	7.56460i	0.402623i	−0.979527	+	0.201311i	$0.935480\pi$
$354$	−71.3231	−3.79078
$355$	0	0
$356$	17.5933	0.932445
$357$	3.47626i	0.183983i
$358$	18.7992i	0.993568i
$359$	−4.24214	−0.223891	−0.111946	−	0.993714i	$-0.535708\pi$
$359$	−4.24214	−0.223891	−0.111946	+	0.993714i	$0.535708\pi$
$360$	0	0
$361$	−11.4037	−0.600197
$362$	4.18941i	0.220191i
$363$	− 25.5607i	− 1.34159i
$364$	24.7091	1.29511
$365$	0	0
$366$	38.1006	1.99155
$367$	27.9623i	1.45962i	0.683650	+	0.729810i	$0.260392\pi$
$367$	27.9623i	1.45962i	−0.683650	+	0.729810i	$0.739608\pi$
$368$	66.0080i	3.44090i
$369$	30.3018	1.57745
$370$	0	0
$371$	−10.4586	−0.542982
$372$	− 12.4411i	− 0.645043i
$373$	− 9.39715i	− 0.486566i	−0.969955	−	0.243283i	$-0.921776\pi$
$373$	− 9.39715i	− 0.486566i	0.969955	−	0.243283i	$-0.0782244\pi$
$374$	3.06675	0.158578
$375$	0	0
$376$	−6.20603	−0.320052
$377$	0.570933i	0.0294045i
$378$	7.80027i	0.401203i
$379$	−23.9952	−1.23255	−0.616275	−	0.787531i	$-0.711359\pi$
$379$	−23.9952	−1.23255	−0.616275	+	0.787531i	$0.711359\pi$
$380$	0	0
$381$	−13.0641	−0.669294
$382$	− 44.6268i	− 2.28331i
$383$	1.97888i	0.101116i	0.998721	+	0.0505581i	$0.0161000\pi$
$383$	1.97888i	0.101116i	−0.998721	+	0.0505581i	$0.983900\pi$
$384$	190.815	9.73749
$385$	0	0
$386$	48.9271	2.49032
$387$	− 24.8894i	− 1.26520i
$388$	− 71.7196i	− 3.64101i
$389$	26.5739	1.34735	0.673674	−	0.739028i	$-0.264715\pi$
$389$	26.5739	1.34735	0.673674	+	0.739028i	$0.264715\pi$
$390$	0	0
$391$	3.57111	0.180599
$392$	− 56.2691i	− 2.84202i
$393$	6.88202i	0.347152i
$394$	−39.3372	−1.98178
$395$	0	0
$396$	24.3288	1.22257
$397$	− 6.32026i	− 0.317205i	−0.987343	−	0.158602i	$-0.949301\pi$
$397$	− 6.32026i	− 0.317205i	0.987343	−	0.158602i	$-0.0506988\pi$
$398$	12.8449i	0.643857i
$399$	9.58103	0.479651
$400$	0	0
$401$	−14.9995	−0.749041	−0.374520	−	0.927219i	$-0.622193\pi$
$401$	−14.9995	−0.749041	−0.374520	+	0.927219i	$0.622193\pi$
$402$	− 19.3552i	− 0.965351i
$403$	2.58752i	0.128894i
$404$	−17.4277	−0.867063
$405$	0	0
$406$	−0.672295	−0.0333654
$407$	− 9.24947i	− 0.458479i
$408$	− 28.0947i	− 1.39090i
$409$	−18.0132	−0.890697	−0.445348	−	0.895357i	$-0.646920\pi$
$409$	−18.0132	−0.890697	−0.445348	+	0.895357i	$0.646920\pi$
$410$	0	0
$411$	−8.52947	−0.420728
$412$	80.9322i	3.98725i
$413$	13.0149i	0.640421i
$414$	38.0219	1.86868
$415$	0	0
$416$	−95.8503	−4.69945
$417$	27.0962i	1.32691i
$418$	− 8.45238i	− 0.413419i
$419$	11.2294	0.548594	0.274297	−	0.961645i	$-0.411555\pi$
$419$	11.2294	0.548594	0.274297	+	0.961645i	$0.411555\pi$
$420$	0	0
$421$	37.8927	1.84678	0.923389	−	0.383866i	$-0.125408\pi$
$421$	37.8927	1.84678	0.923389	+	0.383866i	$0.125408\pi$
$422$	20.2614i	0.986311i
$423$	2.18970i	0.106467i
$424$	84.5250	4.10490
$425$	0	0
$426$	−97.9687	−4.74660
$427$	− 6.95252i	− 0.336456i
$428$	− 46.0142i	− 2.22418i
$429$	−9.05350	−0.437107
$430$	0	0
$431$	14.7151	0.708803	0.354402	−	0.935093i	$-0.384685\pi$
$431$	14.7151	0.708803	0.354402	+	0.935093i	$0.384685\pi$
$432$	− 38.6149i	− 1.85786i
$433$	− 37.5179i	− 1.80299i	−0.432786	−	0.901497i	$-0.642469\pi$
$433$	− 37.5179i	− 1.80299i	0.432786	−	0.901497i	$-0.357531\pi$
$434$	−3.04691	−0.146256
$435$	0	0
$436$	79.9377	3.82832
$437$	− 9.84245i	− 0.470828i
$438$	90.0685i	4.30364i
$439$	−34.8342	−1.66255	−0.831273	−	0.555864i	$-0.812388\pi$
$439$	−34.8342	−1.66255	−0.831273	+	0.555864i	$0.812388\pi$
$440$	0	0
$441$	−19.8536	−0.945411
$442$	8.88173i	0.422461i
$443$	9.84325i	0.467667i	0.972277	+	0.233833i	$0.0751271\pi$
$443$	9.84325i	0.467667i	−0.972277	+	0.233833i	$0.924873\pi$
$444$	−128.799	−6.11251
$445$	0	0
$446$	62.3425	2.95200
$447$	29.3560i	1.38849i
$448$	− 63.5901i	− 3.00435i
$449$	−27.7439	−1.30932	−0.654658	−	0.755925i	$-0.727187\pi$
$449$	−27.7439	−1.30932	−0.654658	+	0.755925i	$0.727187\pi$
$450$	0	0
$451$	−8.72804	−0.410987
$452$	− 114.420i	− 5.38184i
$453$	18.6134i	0.874536i
$454$	37.4058	1.75554
$455$	0	0
$456$	−77.4327	−3.62612
$457$	− 28.0953i	− 1.31424i	−0.753786	−	0.657120i	$-0.771774\pi$
$457$	− 28.0953i	− 1.31424i	0.753786	−	0.657120i	$-0.228226\pi$
$458$	57.3232i	2.67854i
$459$	−2.08911	−0.0975114
$460$	0	0
$461$	5.20430	0.242388	0.121194	−	0.992629i	$-0.461328\pi$
$461$	5.20430	0.242388	0.121194	+	0.992629i	$0.461328\pi$
$462$	− 10.6608i	− 0.495987i
$463$	− 1.28264i	− 0.0596092i	−0.999556	−	0.0298046i	$-0.990511\pi$
$463$	− 1.28264i	− 0.0596092i	0.999556	−	0.0298046i	$-0.00948851\pi$
$464$	3.32817	0.154506
$465$	0	0
$466$	45.1642	2.09219
$467$	− 24.5006i	− 1.13375i	−0.823803	−	0.566876i	$-0.808152\pi$
$467$	− 24.5006i	− 1.13375i	0.823803	−	0.566876i	$-0.191848\pi$
$468$	70.4595i	3.25699i
$469$	−3.53190	−0.163088
$470$	0	0
$471$	−21.6610	−0.998085
$472$	− 105.185i	− 4.84152i
$473$	7.16905i	0.329633i
$474$	27.3747	1.25736
$475$	0	0
$476$	−7.79262	−0.357174
$477$	− 29.8233i	− 1.36551i
$478$	12.2866i	0.561974i
$479$	31.1171	1.42178	0.710888	−	0.703306i	$-0.248293\pi$
$479$	31.1171	1.42178	0.710888	+	0.703306i	$0.248293\pi$
$480$	0	0
$481$	26.7877	1.22141
$482$	− 41.5518i	− 1.89263i
$483$	− 12.4141i	− 0.564861i
$484$	57.2986	2.60448
$485$	0	0
$486$	61.0563	2.76957
$487$	− 19.8828i	− 0.900978i	−0.892782	−	0.450489i	$-0.851250\pi$
$487$	− 19.8828i	− 0.900978i	0.892782	−	0.450489i	$-0.148750\pi$
$488$	56.1894i	2.54358i
$489$	−30.2668	−1.36871
$490$	0	0
$491$	19.8895	0.897602	0.448801	−	0.893632i	$-0.351851\pi$
$491$	19.8895	0.897602	0.448801	+	0.893632i	$0.351851\pi$
$492$	121.538i	5.47934i
$493$	− 0.180058i	− 0.00810939i
$494$	24.4792	1.10137
$495$	0	0
$496$	15.0836	0.677272
$497$	17.8771i	0.801898i
$498$	34.0843i	1.52735i
$499$	−4.68299	−0.209639	−0.104820	−	0.994491i	$-0.533427\pi$
$499$	−4.68299	−0.209639	−0.104820	+	0.994491i	$0.533427\pi$
$500$	0	0
$501$	−51.3316	−2.29333
$502$	40.8952i	1.82524i
$503$	39.2233i	1.74888i	0.485134	+	0.874440i	$0.338771\pi$
$503$	39.2233i	1.74888i	−0.485134	+	0.874440i	$0.661229\pi$
$504$	−54.5840	−2.43137
$505$	0	0
$506$	−10.9517	−0.486863
$507$	7.68239i	0.341187i
$508$	− 29.2854i	− 1.29933i
$509$	−16.9770	−0.752494	−0.376247	−	0.926519i	$-0.622786\pi$
$509$	−16.9770	−0.752494	−0.376247	+	0.926519i	$0.622786\pi$
$510$	0	0
$511$	16.4355	0.727064
$512$	160.492i	7.09281i
$513$	5.75786i	0.254216i
$514$	14.9780	0.660649
$515$	0	0
$516$	99.8289	4.39472
$517$	− 0.630714i	− 0.0277388i
$518$	31.5436i	1.38594i
$519$	−51.7760	−2.27272
$520$	0	0
$521$	−24.6927	−1.08181	−0.540903	−	0.841085i	$-0.681917\pi$
$521$	−24.6927	−1.08181	−0.540903	+	0.841085i	$0.681917\pi$
$522$	− 1.91709i	− 0.0839088i
$523$	− 41.2117i	− 1.80206i	−0.433753	−	0.901032i	$-0.642811\pi$
$523$	− 41.2117i	− 1.80206i	0.433753	−	0.901032i	$-0.357189\pi$
$524$	−15.4272	−0.673941
$525$	0	0
$526$	−51.4707	−2.24423
$527$	− 0.816039i	− 0.0355472i
$528$	52.7760i	2.29678i
$529$	10.2472	0.445529
$530$	0	0
$531$	−37.1128	−1.61056
$532$	21.4775i	0.931167i
$533$	− 25.2776i	− 1.09489i
$534$	21.9837	0.951329
$535$	0	0
$536$	28.5444	1.23293
$537$	17.5026i	0.755294i
$538$	42.2435i	1.82125i
$539$	5.71858	0.246317
$540$	0	0
$541$	−3.70168	−0.159147	−0.0795737	−	0.996829i	$-0.525356\pi$
$541$	−3.70168	−0.159147	−0.0795737	+	0.996829i	$0.525356\pi$
$542$	57.7108i	2.47889i
$543$	3.90047i	0.167385i
$544$	30.2288	1.29605
$545$	0	0
$546$	30.8752	1.32134
$547$	− 31.0162i	− 1.32616i	−0.748550	−	0.663079i	$-0.769250\pi$
$547$	− 31.0162i	− 1.32616i	0.748550	−	0.663079i	$-0.230750\pi$
$548$	− 19.1202i	− 0.816776i
$549$	19.8255	0.846134
$550$	0	0
$551$	−0.496263	−0.0211415
$552$	100.329i	4.27030i
$553$	− 4.99527i	− 0.212421i
$554$	−11.0199	−0.468191
$555$	0	0
$556$	−60.7407	−2.57598
$557$	40.6170i	1.72100i	0.509453	+	0.860498i	$0.329848\pi$
$557$	40.6170i	1.72100i	−0.509453	+	0.860498i	$0.670152\pi$
$558$	− 8.68845i	− 0.367811i
$559$	−20.7625	−0.878162
$560$	0	0
$561$	2.85524	0.120548
$562$	69.4325i	2.92883i
$563$	− 6.63255i	− 0.279529i	−0.990185	−	0.139764i	$-0.955366\pi$
$563$	− 6.63255i	− 0.279529i	0.990185	−	0.139764i	$-0.0446345\pi$
$564$	−8.78268	−0.369817
$565$	0	0
$566$	−28.8209	−1.21143
$567$	− 7.93797i	− 0.333363i
$568$	− 144.481i	− 6.06227i
$569$	20.9921	0.880035	0.440017	−	0.897989i	$-0.354972\pi$
$569$	20.9921	0.880035	0.440017	+	0.897989i	$0.354972\pi$
$570$	0	0
$571$	9.22867	0.386208	0.193104	−	0.981178i	$-0.438145\pi$
$571$	9.22867	0.386208	0.193104	+	0.981178i	$0.438145\pi$
$572$	− 20.2949i	− 0.848574i
$573$	− 41.5490i	− 1.73573i
$574$	29.7653	1.24238
$575$	0	0
$576$	181.331	7.55547
$577$	1.17031i	0.0487208i	0.999703	+	0.0243604i	$0.00775493\pi$
$577$	1.17031i	0.0487208i	−0.999703	+	0.0243604i	$0.992245\pi$
$578$	− 2.80107i	− 0.116509i
$579$	45.5526	1.89310
$580$	0	0
$581$	6.21963	0.258034
$582$	− 89.6171i	− 3.71475i
$583$	8.59021i	0.355770i
$584$	−132.830	−5.49653
$585$	0	0
$586$	11.7771	0.486506
$587$	− 32.1314i	− 1.32620i	−0.748529	−	0.663102i	$-0.769239\pi$
$587$	− 32.1314i	− 1.32620i	0.748529	−	0.663102i	$-0.230761\pi$
$588$	− 79.6311i	− 3.28393i
$589$	−2.24911	−0.0926730
$590$	0	0
$591$	−36.6242	−1.50652
$592$	− 156.155i	− 6.41793i
$593$	− 24.9096i	− 1.02291i	−0.859309	−	0.511456i	$-0.829106\pi$
$593$	− 24.9096i	− 1.02291i	0.859309	−	0.511456i	$-0.170894\pi$
$594$	6.40679	0.262874
$595$	0	0
$596$	−65.8065	−2.69554
$597$	11.9590i	0.489450i
$598$	− 31.7176i	− 1.29703i
$599$	30.4971	1.24608	0.623039	−	0.782191i	$-0.285898\pi$
$599$	30.4971	1.24608	0.623039	+	0.782191i	$0.285898\pi$
$600$	0	0
$601$	−4.43000	−0.180703	−0.0903517	−	0.995910i	$-0.528799\pi$
$601$	−4.43000	−0.180703	−0.0903517	+	0.995910i	$0.528799\pi$
$602$	− 24.4487i	− 0.996454i
$603$	− 10.0714i	− 0.410140i
$604$	−41.7252	−1.69777
$605$	0	0
$606$	−21.7768	−0.884623
$607$	− 28.4929i	− 1.15649i	−0.815862	−	0.578246i	$-0.803737\pi$
$607$	− 28.4929i	− 1.15649i	0.815862	−	0.578246i	$-0.196263\pi$
$608$	− 83.3145i	− 3.37885i
$609$	−0.625927	−0.0253639
$610$	0	0
$611$	1.82663	0.0738976
$612$	− 22.2211i	− 0.898237i
$613$	− 5.63970i	− 0.227785i	−0.993493	−	0.113893i	$-0.963668\pi$
$613$	− 5.63970i	− 0.227785i	0.993493	−	0.113893i	$-0.0363320\pi$
$614$	56.8533	2.29441
$615$	0	0
$616$	15.7222	0.633466
$617$	− 5.49860i	− 0.221365i	−0.993856	−	0.110683i	$-0.964696\pi$
$617$	− 5.49860i	− 0.221365i	0.993856	−	0.110683i	$-0.0353037\pi$
$618$	101.129i	4.06800i
$619$	20.7770	0.835096	0.417548	−	0.908655i	$-0.362889\pi$
$619$	20.7770	0.835096	0.417548	+	0.908655i	$0.362889\pi$
$620$	0	0
$621$	7.46045	0.299377
$622$	22.5164i	0.902827i
$623$	− 4.01154i	− 0.160719i
$624$	−152.846	−6.11875
$625$	0	0
$626$	83.0588	3.31970
$627$	− 7.86943i	− 0.314275i
$628$	− 48.5567i	− 1.93762i
$629$	−8.44817	−0.336850
$630$	0	0
$631$	−9.55265	−0.380285	−0.190142	−	0.981757i	$-0.560895\pi$
$631$	−9.55265	−0.380285	−0.190142	+	0.981757i	$0.560895\pi$
$632$	40.3712i	1.60588i
$633$	18.8640i	0.749778i
$634$	8.58276	0.340865
$635$	0	0
$636$	119.618	4.74318
$637$	16.5618i	0.656201i
$638$	0.552193i	0.0218615i
$639$	−50.9777	−2.01665
$640$	0	0
$641$	−38.2638	−1.51133	−0.755664	−	0.654959i	$-0.772686\pi$
$641$	−38.2638	−1.51133	−0.755664	+	0.654959i	$0.772686\pi$
$642$	− 57.4970i	− 2.26922i
$643$	2.28036i	0.0899286i	0.998989	+	0.0449643i	$0.0143174\pi$
$643$	2.28036i	0.0899286i	−0.998989	+	0.0449643i	$0.985683\pi$
$644$	27.8283	1.09659
$645$	0	0
$646$	−7.72013	−0.303744
$647$	− 15.3848i	− 0.604840i	−0.953175	−	0.302420i	$-0.902206\pi$
$647$	− 15.3848i	− 0.604840i	0.953175	−	0.302420i	$-0.0977944\pi$
$648$	64.1537i	2.52020i
$649$	10.6898	0.419613
$650$	0	0
$651$	−2.83677	−0.111182
$652$	− 67.8481i	− 2.65714i
$653$	24.7373i	0.968046i	0.875055	+	0.484023i	$0.160825\pi$
$653$	24.7373i	0.968046i	−0.875055	+	0.484023i	$0.839175\pi$
$654$	99.8860	3.90585
$655$	0	0
$656$	−147.352	−5.75312
$657$	46.8669i	1.82845i
$658$	2.15093i	0.0838520i
$659$	17.3744	0.676812	0.338406	−	0.941000i	$-0.390112\pi$
$659$	17.3744	0.676812	0.338406	+	0.941000i	$0.390112\pi$
$660$	0	0
$661$	4.18538	0.162793	0.0813963	−	0.996682i	$-0.474062\pi$
$661$	4.18538	0.162793	0.0813963	+	0.996682i	$0.474062\pi$
$662$	32.5442i	1.26487i
$663$	8.26917i	0.321148i
$664$	−50.2663	−1.95071
$665$	0	0
$666$	−89.9484	−3.48543
$667$	0.643006i	0.0248973i
$668$	− 115.068i	− 4.45213i
$669$	58.0428	2.24406
$670$	0	0
$671$	−5.71049	−0.220451
$672$	− 105.083i	− 4.05367i
$673$	16.8948i	0.651246i	0.945500	+	0.325623i	$0.105574\pi$
$673$	16.8948i	0.651246i	−0.945500	+	0.325623i	$0.894426\pi$
$674$	−20.1602	−0.776543
$675$	0	0
$676$	−17.2214	−0.662361
$677$	− 6.13351i	− 0.235730i	−0.993030	−	0.117865i	$-0.962395\pi$
$677$	− 6.13351i	− 0.235730i	0.993030	−	0.117865i	$-0.0376050\pi$
$678$	− 142.973i	− 5.49084i
$679$	−16.3531	−0.627576
$680$	0	0
$681$	34.8260	1.33453
$682$	2.50259i	0.0958292i
$683$	− 26.7588i	− 1.02390i	−0.859016	−	0.511949i	$-0.828924\pi$
$683$	− 26.7588i	− 1.02390i	0.859016	−	0.511949i	$-0.171076\pi$
$684$	−61.2444	−2.34174
$685$	0	0
$686$	−45.6386	−1.74249
$687$	53.3697i	2.03618i
$688$	121.032i	4.61430i
$689$	−24.8784	−0.947791
$690$	0	0
$691$	23.1367	0.880160	0.440080	−	0.897959i	$-0.354950\pi$
$691$	23.1367	0.880160	0.440080	+	0.897959i	$0.354950\pi$
$692$	− 116.065i	− 4.41212i
$693$	− 5.54733i	− 0.210726i
$694$	23.7914	0.903107
$695$	0	0
$696$	5.05867	0.191748
$697$	7.97191i	0.301957i
$698$	0.788795i	0.0298563i
$699$	42.0492	1.59045
$700$	0	0
$701$	41.0017	1.54861	0.774307	−	0.632810i	$-0.218098\pi$
$701$	41.0017	1.54861	0.774307	+	0.632810i	$0.218098\pi$
$702$	18.5549i	0.700311i
$703$	23.2843i	0.878182i
$704$	−52.2301	−1.96850
$705$	0	0
$706$	−21.1890	−0.797458
$707$	3.97379i	0.149450i
$708$	− 148.856i	− 5.59434i
$709$	10.7096	0.402207	0.201103	−	0.979570i	$-0.435547\pi$
$709$	10.7096	0.402207	0.201103	+	0.979570i	$0.435547\pi$
$710$	0	0
$711$	14.2443	0.534204
$712$	32.4208i	1.21502i
$713$	2.91417i	0.109136i
$714$	−9.73726	−0.364408
$715$	0	0
$716$	−39.2351	−1.46628
$717$	11.4392i	0.427204i
$718$	− 11.8825i	− 0.443452i
$719$	6.17046	0.230119	0.115060	−	0.993359i	$-0.463294\pi$
$719$	6.17046	0.230119	0.115060	+	0.993359i	$0.463294\pi$
$720$	0	0
$721$	18.4538	0.687254
$722$	− 31.9427i	− 1.18878i
$723$	− 38.6861i	− 1.43875i
$724$	−8.74357	−0.324952
$725$	0	0
$726$	71.5974	2.65723
$727$	38.9277i	1.44375i	0.692024	+	0.721875i	$0.256719\pi$
$727$	38.9277i	1.44375i	−0.692024	+	0.721875i	$0.743281\pi$
$728$	45.5337i	1.68759i
$729$	38.9801	1.44371
$730$	0	0
$731$	6.54798	0.242186
$732$	79.5184i	2.93908i
$733$	31.8680i	1.17707i	0.808472	+	0.588535i	$0.200295\pi$
$733$	31.8680i	1.17707i	−0.808472	+	0.588535i	$0.799705\pi$
$734$	−78.3245	−2.89101
$735$	0	0
$736$	−107.950	−3.97910
$737$	2.90094i	0.106858i
$738$	84.8776i	3.12439i
$739$	−29.7808	−1.09551	−0.547753	−	0.836640i	$-0.684517\pi$
$739$	−29.7808	−1.09551	−0.547753	+	0.836640i	$0.684517\pi$
$740$	0	0
$741$	22.7909	0.837245
$742$	− 29.2953i	− 1.07546i
$743$	33.4684i	1.22784i	0.789370	+	0.613918i	$0.210408\pi$
$743$	33.4684i	1.22784i	−0.789370	+	0.613918i	$0.789592\pi$
$744$	22.9264	0.840522
$745$	0	0
$746$	26.3221	0.963721
$747$	17.7357i	0.648914i
$748$	6.40051i	0.234026i
$749$	−10.4919	−0.383367
$750$	0	0
$751$	15.9989	0.583808	0.291904	−	0.956448i	$-0.405711\pi$
$751$	15.9989	0.583808	0.291904	+	0.956448i	$0.405711\pi$
$752$	− 10.6481i	− 0.388295i
$753$	38.0747i	1.38752i
$754$	−1.59922	−0.0582403
$755$	0	0
$756$	−16.2797	−0.592085
$757$	20.6978i	0.752275i	0.926564	+	0.376138i	$0.122748\pi$
$757$	20.6978i	0.752275i	−0.926564	+	0.376138i	$0.877252\pi$
$758$	− 67.2123i	− 2.44126i
$759$	−10.1964	−0.370105
$760$	0	0
$761$	−22.4458	−0.813660	−0.406830	−	0.913504i	$-0.633366\pi$
$761$	−22.4458	−0.813660	−0.406830	+	0.913504i	$0.633366\pi$
$762$	− 36.5935i	− 1.32564i
$763$	− 18.2270i	− 0.659861i
$764$	93.1390	3.36965
$765$	0	0
$766$	−5.54300	−0.200277
$767$	30.9592i	1.11787i
$768$	285.667i	10.3081i
$769$	−30.6260	−1.10440	−0.552202	−	0.833711i	$-0.686212\pi$
$769$	−30.6260	−1.10440	−0.552202	+	0.833711i	$0.686212\pi$
$770$	0	0
$771$	13.9449	0.502215
$772$	102.114i	3.67516i
$773$	− 10.9578i	− 0.394126i	−0.980391	−	0.197063i	$-0.936860\pi$
$773$	− 10.9578i	− 0.394126i	0.980391	−	0.197063i	$-0.0631403\pi$
$774$	69.7169	2.50592
$775$	0	0
$776$	132.164	4.74441
$777$	29.3680i	1.05357i
$778$	74.4354i	2.66864i
$779$	21.9716	0.787215
$780$	0	0
$781$	14.6835	0.525415
$782$	10.0029i	0.357705i
$783$	− 0.376161i	− 0.0134429i
$784$	96.5444	3.44801
$785$	0	0
$786$	−19.2770	−0.687590
$787$	− 31.1539i	− 1.11052i	−0.831677	−	0.555259i	$-0.812619\pi$
$787$	− 31.1539i	− 1.11052i	0.831677	−	0.555259i	$-0.187381\pi$
$788$	− 82.0993i	− 2.92467i
$789$	−47.9209	−1.70603
$790$	0	0
$791$	−26.0894	−0.927631
$792$	44.8329i	1.59307i
$793$	− 16.5383i	− 0.587294i
$794$	17.7035	0.628274
$795$	0	0
$796$	−26.8081	−0.950189
$797$	− 15.2518i	− 0.540246i	−0.962826	−	0.270123i	$-0.912936\pi$
$797$	− 15.2518i	− 0.540246i	0.962826	−	0.270123i	$-0.0870643\pi$
$798$	26.8372i	0.950026i
$799$	−0.576074	−0.0203800
$800$	0	0
$801$	11.4392	0.404183
$802$	− 42.0148i	− 1.48359i
$803$	− 13.4994i	− 0.476383i
$804$	40.3956	1.42464
$805$	0	0
$806$	−7.24784	−0.255294
$807$	39.3300i	1.38448i
$808$	− 32.1157i	− 1.12983i
$809$	2.66539	0.0937100	0.0468550	−	0.998902i	$-0.485080\pi$
$809$	2.66539	0.0937100	0.0468550	+	0.998902i	$0.485080\pi$
$810$	0	0
$811$	5.54482	0.194705	0.0973525	−	0.995250i	$-0.468963\pi$
$811$	5.54482	0.194705	0.0973525	+	0.995250i	$0.468963\pi$
$812$	− 1.40312i	− 0.0492399i
$813$	53.7305i	1.88441i
$814$	25.9085	0.908091
$815$	0	0
$816$	48.2039	1.68747
$817$	− 18.0471i	− 0.631388i
$818$	− 50.4564i	− 1.76417i
$819$	16.0658	0.561385
$820$	0	0
$821$	31.2022	1.08896	0.544482	−	0.838773i	$-0.316726\pi$
$821$	31.2022	1.08896	0.544482	+	0.838773i	$0.316726\pi$
$822$	− 23.8917i	− 0.833318i
$823$	50.4682i	1.75921i	0.475703	+	0.879606i	$0.342194\pi$
$823$	50.4682i	1.75921i	−0.475703	+	0.879606i	$0.657806\pi$
$824$	−149.141	−5.19558
$825$	0	0
$826$	−36.4557	−1.26845
$827$	25.7704i	0.896123i	0.894003	+	0.448062i	$0.147886\pi$
$827$	25.7704i	0.896123i	−0.894003	+	0.448062i	$0.852114\pi$
$828$	79.3542i	2.75775i
$829$	−4.34228	−0.150814	−0.0754068	−	0.997153i	$-0.524026\pi$
$829$	−4.34228	−0.150814	−0.0754068	+	0.997153i	$0.524026\pi$
$830$	0	0
$831$	−10.2599	−0.355911
$832$	− 151.265i	− 5.24418i
$833$	− 5.22317i	− 0.180972i
$834$	−75.8984	−2.62815
$835$	0	0
$836$	17.6406	0.610114
$837$	− 1.70480i	− 0.0589264i
$838$	31.4545i	1.08658i
$839$	−44.9482	−1.55178	−0.775892	−	0.630866i	$-0.782700\pi$
$839$	−44.9482	−1.55178	−0.775892	+	0.630866i	$0.782700\pi$
$840$	0	0
$841$	−28.9676	−0.998882
$842$	106.140i	3.65784i
$843$	64.6438i	2.22645i
$844$	−42.2869	−1.45557
$845$	0	0
$846$	−6.13351	−0.210874
$847$	− 13.0649i	− 0.448917i
$848$	145.025i	4.98017i
$849$	−26.8331	−0.920910
$850$	0	0
$851$	30.1693	1.03419
$852$	− 204.467i	− 7.00491i
$853$	− 15.7009i	− 0.537590i	−0.963197	−	0.268795i	$-0.913375\pi$
$853$	− 15.7009i	− 0.537590i	0.963197	−	0.268795i	$-0.0866254\pi$
$854$	19.4745	0.666405
$855$	0	0
$856$	84.7945	2.89821
$857$	− 6.77401i	− 0.231396i	−0.993284	−	0.115698i	$-0.963090\pi$
$857$	− 6.77401i	− 0.231396i	0.993284	−	0.115698i	$-0.0369105\pi$
$858$	− 25.3595i	− 0.865760i
$859$	−36.0071	−1.22855	−0.614273	−	0.789094i	$-0.710551\pi$
$859$	−36.0071	−1.22855	−0.614273	+	0.789094i	$0.710551\pi$
$860$	0	0
$861$	27.7124	0.944437
$862$	41.2182i	1.40390i
$863$	− 35.5716i	− 1.21087i	−0.795894	−	0.605436i	$-0.792999\pi$
$863$	− 35.5716i	− 1.21087i	0.795894	−	0.605436i	$-0.207001\pi$
$864$	63.1513	2.14845
$865$	0	0
$866$	105.090	3.57111
$867$	− 2.60789i	− 0.0885685i
$868$	− 6.35909i	− 0.215841i
$869$	−4.10289	−0.139181
$870$	0	0
$871$	−8.40152	−0.284675
$872$	147.308i	4.98849i
$873$	− 46.6320i	− 1.57825i
$874$	27.5694	0.932550
$875$	0	0
$876$	−187.979	−6.35121
$877$	24.1451i	0.815321i	0.913134	+	0.407660i	$0.133655\pi$
$877$	24.1451i	0.815321i	−0.913134	+	0.407660i	$0.866345\pi$
$878$	− 97.5733i	− 3.29294i
$879$	10.9648	0.369834
$880$	0	0
$881$	32.9124	1.10885	0.554423	−	0.832235i	$-0.312939\pi$
$881$	32.9124	1.10885	0.554423	+	0.832235i	$0.312939\pi$
$882$	− 55.6115i	− 1.87254i
$883$	15.5885i	0.524594i	0.964987	+	0.262297i	$0.0844800\pi$
$883$	15.5885i	0.524594i	−0.964987	+	0.262297i	$0.915520\pi$
$884$	−18.5367	−0.623458
$885$	0	0
$886$	−27.5717	−0.926289
$887$	− 8.59951i	− 0.288743i	−0.989524	−	0.144372i	$-0.953884\pi$
$887$	− 8.59951i	− 0.288743i	0.989524	−	0.144372i	$-0.0461161\pi$
$888$	− 237.349i	− 7.96490i
$889$	−6.67750	−0.223956
$890$	0	0
$891$	−6.51989	−0.218425
$892$	130.113i	4.35649i
$893$	1.58773i	0.0531315i
$894$	−82.2284	−2.75013
$895$	0	0
$896$	97.5320	3.25832
$897$	− 29.5301i	− 0.985982i
$898$	− 77.7127i	− 2.59331i
$899$	0.146934	0.00490053
$900$	0	0
$901$	7.84602	0.261389
$902$	− 24.4479i	− 0.814025i
$903$	− 22.7625i	− 0.757488i
$904$	210.851	7.01280
$905$	0	0
$906$	−52.1376	−1.73216
$907$	44.9819i	1.49360i	0.665049	+	0.746800i	$0.268411\pi$
$907$	44.9819i	1.49360i	−0.665049	+	0.746800i	$0.731589\pi$
$908$	78.0682i	2.59079i
$909$	−11.3315	−0.375842
$910$	0	0
$911$	50.9800	1.68904	0.844522	−	0.535521i	$-0.179885\pi$
$911$	50.9800	1.68904	0.844522	+	0.535521i	$0.179885\pi$
$912$	− 132.856i	− 4.39931i
$913$	− 5.10852i	− 0.169067i
$914$	78.6969	2.60306
$915$	0	0
$916$	−119.637	−3.95292
$917$	3.51763i	0.116163i
$918$	− 5.85176i	− 0.193137i
$919$	13.0346	0.429973	0.214986	−	0.976617i	$-0.431029\pi$
$919$	13.0346	0.429973	0.214986	+	0.976617i	$0.431029\pi$
$920$	0	0
$921$	52.9321	1.74417
$922$	14.5776i	0.480088i
$923$	42.5252i	1.39974i
$924$	22.2498	0.731966
$925$	0	0
$926$	3.59276	0.118066
$927$	52.6221i	1.72834i
$928$	5.44292i	0.178673i
$929$	−37.8748	−1.24263	−0.621316	−	0.783560i	$-0.713402\pi$
$929$	−37.8748	−1.24263	−0.621316	+	0.783560i	$0.713402\pi$
$930$	0	0
$931$	−14.3957	−0.471801
$932$	94.2604i	3.08760i
$933$	20.9635i	0.686314i
$934$	68.6280	2.24558
$935$	0	0
$936$	−129.842	−4.24402
$937$	− 22.9509i	− 0.749772i	−0.927071	−	0.374886i	$-0.877682\pi$
$937$	− 22.9509i	− 0.749772i	0.927071	−	0.374886i	$-0.122318\pi$
$938$	− 9.89311i	− 0.323022i
$939$	77.3303	2.52358
$940$	0	0
$941$	−33.8175	−1.10242	−0.551209	−	0.834367i	$-0.685833\pi$
$941$	−33.8175	−1.10242	−0.551209	+	0.834367i	$0.685833\pi$
$942$	− 60.6740i	− 1.97687i
$943$	− 28.4685i	− 0.927064i
$944$	180.472	5.87387
$945$	0	0
$946$	−20.0810	−0.652891
$947$	− 10.0614i	− 0.326953i	−0.986547	−	0.163476i	$-0.947729\pi$
$947$	− 10.0614i	− 0.326953i	0.986547	−	0.163476i	$-0.0522707\pi$
$948$	57.1327i	1.85558i
$949$	39.0960	1.26911
$950$	0	0
$951$	7.99082	0.259120
$952$	− 14.3602i	− 0.465416i
$953$	38.8829i	1.25954i	0.776781	+	0.629770i	$0.216851\pi$
$953$	38.8829i	1.25954i	−0.776781	+	0.629770i	$0.783149\pi$
$954$	83.5373	2.70462
$955$	0	0
$956$	−25.6428	−0.829348
$957$	0.514109i	0.0166188i
$958$	87.1612i	2.81605i
$959$	−4.35970	−0.140782
$960$	0	0
$961$	−30.3341	−0.978519
$962$	75.0344i	2.41920i
$963$	− 29.9184i	− 0.964106i
$964$	86.7213	2.79311
$965$	0	0
$966$	34.7728	1.11880
$967$	5.40033i	0.173663i	0.996223	+	0.0868315i	$0.0276742\pi$
$967$	5.40033i	0.173663i	−0.996223	+	0.0868315i	$0.972326\pi$
$968$	105.589i	3.39377i
$969$	−7.18768	−0.230902
$970$	0	0
$971$	10.1469	0.325629	0.162815	−	0.986657i	$-0.447943\pi$
$971$	10.1469	0.325629	0.162815	+	0.986657i	$0.447943\pi$
$972$	127.428i	4.08726i
$973$	13.8498i	0.444004i
$974$	55.6933	1.78453
$975$	0	0
$976$	−96.4077	−3.08594
$977$	9.09855i	0.291088i	0.989352	+	0.145544i	$0.0464933\pi$
$977$	9.09855i	0.291088i	−0.989352	+	0.145544i	$0.953507\pi$
$978$	− 84.7795i	− 2.71095i
$979$	−3.29490	−0.105306
$980$	0	0
$981$	51.9754	1.65945
$982$	55.7121i	1.77784i
$983$	49.1437i	1.56744i	0.621113	+	0.783721i	$0.286681\pi$
$983$	49.1437i	1.56744i	−0.621113	+	0.783721i	$0.713319\pi$
$984$	−223.968	−7.13985
$985$	0	0
$986$	0.504355	0.0160619
$987$	2.00258i	0.0637429i
$988$	51.0897i	1.62538i
$989$	−23.3836	−0.743554
$990$	0	0
$991$	43.2065	1.37250	0.686249	−	0.727366i	$-0.259256\pi$
$991$	43.2065	1.37250	0.686249	+	0.727366i	$0.259256\pi$
$992$	24.6679i	0.783205i
$993$	30.2997i	0.961531i
$994$	−50.0751	−1.58829
$995$	0	0
$996$	−71.1361	−2.25403
$997$	− 48.5452i	− 1.53744i	−0.639584	−	0.768721i	$-0.720893\pi$
$997$	− 48.5452i	− 1.53744i	0.639584	−	0.768721i	$-0.279107\pi$
$998$	− 13.1174i	− 0.415224i
$999$	−17.6492	−0.558395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.b.f.324.10		10
5.2	odd	4		425.2.a.i.1.1	✓	5
5.3	odd	4		425.2.a.j.1.5	yes	5
5.4	even	2	inner	425.2.b.f.324.1		10
15.2	even	4		3825.2.a.bq.1.5		5
15.8	even	4		3825.2.a.bl.1.1		5
20.3	even	4		6800.2.a.cd.1.5		5
20.7	even	4		6800.2.a.bz.1.1		5
85.33	odd	4		7225.2.a.y.1.5		5
85.67	odd	4		7225.2.a.x.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.1	✓	5	5.2	odd	4
425.2.a.j.1.5	yes	5	5.3	odd	4
425.2.b.f.324.1		10	5.4	even	2	inner
425.2.b.f.324.10		10	1.1	even	1	trivial
3825.2.a.bl.1.1		5	15.8	even	4
3825.2.a.bq.1.5		5	15.2	even	4
6800.2.a.bz.1.1		5	20.7	even	4
6800.2.a.cd.1.5		5	20.3	even	4
7225.2.a.x.1.1		5	85.67	odd	4
7225.2.a.y.1.5		5	85.33	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.80107i q^{2} +2.60789i q^{3} -5.84602 q^{4} -7.30489 q^{6} +1.33298i q^{7} -10.7730i q^{8} -3.80107 q^{9} +1.09485 q^{11} -15.2458i q^{12} +3.17083i q^{13} -3.73378 q^{14} +18.4839 q^{16} -1.00000i q^{17} -10.6471i q^{18} -2.75613 q^{19} -3.47626 q^{21} +3.06675i q^{22} +3.57111i q^{23} +28.0947 q^{24} -8.88173 q^{26} -2.08911i q^{27} -7.79262i q^{28} +0.180058 q^{29} +0.816039 q^{31} +30.2288i q^{32} +2.85524i q^{33} +2.80107 q^{34} +22.2211 q^{36} -8.44817i q^{37} -7.72013i q^{38} -8.26917 q^{39} -7.97191 q^{41} -9.73726i q^{42} +6.54798i q^{43} -6.40051 q^{44} -10.0029 q^{46} -0.576074i q^{47} +48.2039i q^{48} +5.22317 q^{49} +2.60789 q^{51} -18.5367i q^{52} +7.84602i q^{53} +5.85176 q^{54} +14.3602 q^{56} -7.18768i q^{57} +0.504355i q^{58} +9.76375 q^{59} -5.21577 q^{61} +2.28579i q^{62} -5.06675i q^{63} -47.7053 q^{64} -7.99775 q^{66} +2.64963i q^{67} +5.84602i q^{68} -9.31305 q^{69} +13.4114 q^{71} +40.9489i q^{72} -12.3299i q^{73} +23.6639 q^{74} +16.1124 q^{76} +1.45941i q^{77} -23.1626i q^{78} -3.74745 q^{79} -5.95506 q^{81} -22.3299i q^{82} -4.66596i q^{83} +20.3223 q^{84} -18.3414 q^{86} +0.469570i q^{87} -11.7948i q^{88} -3.00946 q^{89} -4.22665 q^{91} -20.8768i q^{92} +2.12814i q^{93} +1.61363 q^{94} -78.8332 q^{96} +12.2681i q^{97} +14.6305i q^{98} -4.16160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 22 q^{4} + 6 q^{6} - 12 q^{9} + 8 q^{11} + 14 q^{14} + 54 q^{16} - 12 q^{19} - 10 q^{21} + 38 q^{24} - 10 q^{26} - 4 q^{29} + 42 q^{31} + 2 q^{34} + 44 q^{36} - 46 q^{39} - 16 q^{41} + 8 q^{44} - 12 q^{46}+ \cdots + 28 q^{99}+O(q^{100}) \) 10 * q - 22 * q^4 + 6 * q^6 - 12 * q^9 + 8 * q^11 + 14 * q^14 + 54 * q^16 - 12 * q^19 - 10 * q^21 + 38 * q^24 - 10 * q^26 - 4 * q^29 + 42 * q^31 + 2 * q^34 + 44 * q^36 - 46 * q^39 - 16 * q^41 + 8 * q^44 - 12 * q^46 - 20 * q^49 + 2 * q^51 + 50 * q^54 - 58 * q^56 - 24 * q^59 - 4 * q^61 - 86 * q^64 - 56 * q^66 + 42 * q^71 + 100 * q^74 - 28 * q^76 - 82 * q^79 - 70 * q^81 + 142 * q^84 - 72 * q^86 + 20 * q^89 + 26 * q^91 + 20 * q^94 - 170 * q^96 + 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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