LMFDB - Embedded newform 4477.2.a.q.1.18

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4477,2,Mod(1,4477)] 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("4477.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4477, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4477 = 11^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4477.a (trivial)

Newform invariants

comment:select newform

sage:traces = [28,0,-16,20,-12,0] 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:	yes
Analytic conductor:	$35.7490249849$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Character	$\chi$	$=$	4477.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.679267 q^{2} -2.24236 q^{3} -1.53860 q^{4} -0.0910404 q^{5} -1.52316 q^{6} -4.09186 q^{7} -2.40365 q^{8} +2.02819 q^{9} -0.0618407 q^{10} +3.45009 q^{12} +2.37049 q^{13} -2.77947 q^{14} +0.204146 q^{15} +1.44447 q^{16} -1.46710 q^{17} +1.37768 q^{18} -1.64859 q^{19} +0.140074 q^{20} +9.17544 q^{21} -0.126483 q^{23} +5.38986 q^{24} -4.99171 q^{25} +1.61019 q^{26} +2.17915 q^{27} +6.29573 q^{28} +9.57272 q^{29} +0.138669 q^{30} +9.86477 q^{31} +5.78848 q^{32} -0.996553 q^{34} +0.372525 q^{35} -3.12057 q^{36} -1.00000 q^{37} -1.11983 q^{38} -5.31549 q^{39} +0.218829 q^{40} +11.4023 q^{41} +6.23257 q^{42} -6.50183 q^{43} -0.184647 q^{45} -0.0859158 q^{46} -7.79096 q^{47} -3.23903 q^{48} +9.74334 q^{49} -3.39070 q^{50} +3.28977 q^{51} -3.64722 q^{52} -13.2421 q^{53} +1.48023 q^{54} +9.83541 q^{56} +3.69674 q^{57} +6.50243 q^{58} +3.29388 q^{59} -0.314098 q^{60} +1.15020 q^{61} +6.70081 q^{62} -8.29907 q^{63} +1.04298 q^{64} -0.215810 q^{65} +4.51270 q^{67} +2.25728 q^{68} +0.283621 q^{69} +0.253044 q^{70} +10.6665 q^{71} -4.87506 q^{72} -5.30034 q^{73} -0.679267 q^{74} +11.1932 q^{75} +2.53652 q^{76} -3.61064 q^{78} -0.0488526 q^{79} -0.131506 q^{80} -10.9710 q^{81} +7.74519 q^{82} -6.61470 q^{83} -14.1173 q^{84} +0.133566 q^{85} -4.41648 q^{86} -21.4655 q^{87} +5.12755 q^{89} -0.125425 q^{90} -9.69971 q^{91} +0.194607 q^{92} -22.1204 q^{93} -5.29214 q^{94} +0.150089 q^{95} -12.9799 q^{96} +7.44905 q^{97} +6.61832 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 16 q^{3} + 20 q^{4} - 12 q^{5} + 16 q^{9} - 22 q^{12} - 6 q^{14} - 16 q^{15} + 36 q^{16} - 50 q^{20} - 28 q^{23} + 12 q^{25} - 68 q^{26} - 34 q^{27} - 8 q^{31} + 44 q^{34} - 16 q^{36} - 28 q^{37}+ \cdots - 14 q^{97}+O(q^{100}) $ 28 * q - 16 * q^3 + 20 * q^4 - 12 * q^5 + 16 * q^9 - 22 * q^12 - 6 * q^14 - 16 * q^15 + 36 * q^16 - 50 * q^20 - 28 * q^23 + 12 * q^25 - 68 * q^26 - 34 * q^27 - 8 * q^31 + 44 * q^34 - 16 * q^36 - 28 * q^37 - 14 * q^38 - 40 * q^42 + 34 * q^45 - 108 * q^47 - 128 * q^48 + 4 * q^49 - 6 * q^53 + 26 * q^56 - 6 * q^58 - 94 * q^59 + 102 * q^60 + 48 * q^64 - 28 * q^67 + 10 * q^69 - 120 * q^70 - 62 * q^71 - 14 * q^75 + 120 * q^78 - 128 * q^80 + 36 * q^81 - 20 * q^86 - 32 * q^89 - 58 * q^91 - 120 * q^92 - 32 * q^93 - 14 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.679267	0.480314	0.240157	−	0.970734i	$-0.422801\pi$
$2$	0.679267	0.480314	0.240157	+	0.970734i	$0.422801\pi$
$3$	−2.24236	−1.29463	−0.647314	−	0.762223i	$-0.724108\pi$
$3$	−2.24236	−1.29463	−0.647314	+	0.762223i	$0.724108\pi$
$4$	−1.53860	−0.769298
$5$	−0.0910404	−0.0407145	−0.0203573	−	0.999793i	$-0.506480\pi$
$5$	−0.0910404	−0.0407145	−0.0203573	+	0.999793i	$0.506480\pi$
$6$	−1.52316	−0.621828
$7$	−4.09186	−1.54658	−0.773289	−	0.634053i	$-0.781390\pi$
$7$	−4.09186	−1.54658	−0.773289	+	0.634053i	$0.781390\pi$
$8$	−2.40365	−0.849819
$9$	2.02819	0.676063
$10$	−0.0618407	−0.0195557
$11$	0	0
$11$	0	0
$12$	3.45009	0.995956
$13$	2.37049	0.657455	0.328727	−	0.944425i	$-0.393380\pi$
$13$	2.37049	0.657455	0.328727	+	0.944425i	$0.393380\pi$
$14$	−2.77947	−0.742843
$15$	0.204146	0.0527102
$16$	1.44447	0.361119
$17$	−1.46710	−0.355825	−0.177912	−	0.984046i	$-0.556934\pi$
$17$	−1.46710	−0.355825	−0.177912	+	0.984046i	$0.556934\pi$
$18$	1.37768	0.324723
$19$	−1.64859	−0.378213	−0.189107	−	0.981957i	$-0.560559\pi$
$19$	−1.64859	−0.378213	−0.189107	+	0.981957i	$0.560559\pi$
$20$	0.140074	0.0313216
$21$	9.17544	2.00224
$22$	0	0
$23$	−0.126483	−0.0263736	−0.0131868	−	0.999913i	$-0.504198\pi$
$23$	−0.126483	−0.0263736	−0.0131868	+	0.999913i	$0.504198\pi$
$24$	5.38986	1.10020
$25$	−4.99171	−0.998342
$26$	1.61019	0.315785
$27$	2.17915	0.419378
$28$	6.29573	1.18978
$29$	9.57272	1.77761	0.888804	−	0.458287i	$-0.151537\pi$
$29$	9.57272	1.77761	0.888804	+	0.458287i	$0.151537\pi$
$30$	0.138669	0.0253174
$31$	9.86477	1.77177	0.885883	−	0.463909i	$-0.153554\pi$
$31$	9.86477	1.77177	0.885883	+	0.463909i	$0.153554\pi$
$32$	5.78848	1.02327
$33$	0	0
$34$	−0.996553	−0.170907
$35$	0.372525	0.0629682
$36$	−3.12057	−0.520094
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−1.11983	−0.181661
$39$	−5.31549	−0.851160
$40$	0.218829	0.0346000
$41$	11.4023	1.78074	0.890369	−	0.455240i	$-0.150447\pi$
$41$	11.4023	1.78074	0.890369	+	0.455240i	$0.150447\pi$
$42$	6.23257	0.961706
$43$	−6.50183	−0.991520	−0.495760	−	0.868460i	$-0.665110\pi$
$43$	−6.50183	−0.991520	−0.495760	+	0.868460i	$0.665110\pi$
$44$	0	0
$45$	−0.184647	−0.0275256
$46$	−0.0859158	−0.0126676
$47$	−7.79096	−1.13643	−0.568214	−	0.822881i	$-0.692365\pi$
$47$	−7.79096	−1.13643	−0.568214	+	0.822881i	$0.692365\pi$
$48$	−3.23903	−0.467514
$49$	9.74334	1.39191
$50$	−3.39070	−0.479518
$51$	3.28977	0.460661
$52$	−3.64722	−0.505779
$53$	−13.2421	−1.81895	−0.909475	−	0.415760i	$-0.863516\pi$
$53$	−13.2421	−1.81895	−0.909475	+	0.415760i	$0.863516\pi$
$54$	1.48023	0.201433
$55$	0	0
$56$	9.83541	1.31431
$57$	3.69674	0.489646
$58$	6.50243	0.853810
$59$	3.29388	0.428826	0.214413	−	0.976743i	$-0.431216\pi$
$59$	3.29388	0.428826	0.214413	+	0.976743i	$0.431216\pi$
$60$	−0.314098	−0.0405498
$61$	1.15020	0.147268	0.0736340	−	0.997285i	$-0.476540\pi$
$61$	1.15020	0.147268	0.0736340	+	0.997285i	$0.476540\pi$
$62$	6.70081	0.851004
$63$	−8.29907	−1.04558
$64$	1.04298	0.130372
$65$	−0.215810	−0.0267680
$66$	0	0
$67$	4.51270	0.551314	0.275657	−	0.961256i	$-0.411105\pi$
$67$	4.51270	0.551314	0.275657	+	0.961256i	$0.411105\pi$
$68$	2.25728	0.273735
$69$	0.283621	0.0341440
$70$	0.253044	0.0302445
$71$	10.6665	1.26588	0.632941	−	0.774200i	$-0.281848\pi$
$71$	10.6665	1.26588	0.632941	+	0.774200i	$0.281848\pi$
$72$	−4.87506	−0.574531
$73$	−5.30034	−0.620358	−0.310179	−	0.950678i	$-0.600389\pi$
$73$	−5.30034	−0.620358	−0.310179	+	0.950678i	$0.600389\pi$
$74$	−0.679267	−0.0789631
$75$	11.1932	1.29248
$76$	2.53652	0.290959
$77$	0	0
$78$	−3.61064	−0.408824
$79$	−0.0488526	−0.00549635	−0.00274818	−	0.999996i	$-0.500875\pi$
$79$	−0.0488526	−0.00549635	−0.00274818	+	0.999996i	$0.500875\pi$
$80$	−0.131506	−0.0147028
$81$	−10.9710	−1.21900
$82$	7.74519	0.855313
$83$	−6.61470	−0.726058	−0.363029	−	0.931778i	$-0.618257\pi$
$83$	−6.61470	−0.726058	−0.363029	+	0.931778i	$0.618257\pi$
$84$	−14.1173	−1.54032
$85$	0.133566	0.0144872
$86$	−4.41648	−0.476241
$87$	−21.4655	−2.30134
$88$	0	0
$89$	5.12755	0.543519	0.271760	−	0.962365i	$-0.412394\pi$
$89$	5.12755	0.543519	0.271760	+	0.962365i	$0.412394\pi$
$90$	−0.125425	−0.0132209
$91$	−9.69971	−1.01681
$92$	0.194607	0.0202892
$93$	−22.1204	−2.29378
$94$	−5.29214	−0.545842
$95$	0.150089	0.0153988
$96$	−12.9799	−1.32475
$97$	7.44905	0.756336	0.378168	−	0.925737i	$-0.376554\pi$
$97$	7.44905	0.756336	0.378168	+	0.925737i	$0.376554\pi$
$98$	6.61832	0.668552
$99$	0	0
$100$	7.68023	0.768023
$101$	−8.43270	−0.839085	−0.419543	−	0.907736i	$-0.637810\pi$
$101$	−8.43270	−0.839085	−0.419543	+	0.907736i	$0.637810\pi$
$102$	2.23463	0.221262
$103$	−7.64854	−0.753633	−0.376817	−	0.926288i	$-0.622981\pi$
$103$	−7.64854	−0.753633	−0.376817	+	0.926288i	$0.622981\pi$
$104$	−5.69782	−0.558718
$105$	−0.835336	−0.0815204
$106$	−8.99495	−0.873667
$107$	5.82594	0.563215	0.281607	−	0.959530i	$-0.409132\pi$
$107$	5.82594	0.563215	0.281607	+	0.959530i	$0.409132\pi$
$108$	−3.35284	−0.322627
$109$	12.3776	1.18556	0.592779	−	0.805365i	$-0.298031\pi$
$109$	12.3776	1.18556	0.592779	+	0.805365i	$0.298031\pi$
$110$	0	0
$111$	2.24236	0.212836
$112$	−5.91059	−0.558498
$113$	10.8817	1.02367	0.511833	−	0.859085i	$-0.328967\pi$
$113$	10.8817	1.02367	0.511833	+	0.859085i	$0.328967\pi$
$114$	2.51107	0.235184
$115$	0.0115151	0.00107379
$116$	−14.7286	−1.36751
$117$	4.80780	0.444481
$118$	2.23742	0.205971
$119$	6.00318	0.550311
$120$	−0.490695	−0.0447941
$121$	0	0
$122$	0.781293	0.0707349
$123$	−25.5681	−2.30539
$124$	−15.1779	−1.36302
$125$	0.909650	0.0813615
$126$	−5.63728	−0.502209
$127$	−21.2218	−1.88313	−0.941564	−	0.336833i	$-0.890644\pi$
$127$	−21.2218	−1.88313	−0.941564	+	0.336833i	$0.890644\pi$
$128$	−10.8685	−0.960650
$129$	14.5795	1.28365
$130$	−0.146593	−0.0128570
$131$	−1.61633	−0.141220	−0.0706099	−	0.997504i	$-0.522495\pi$
$131$	−1.61633	−0.141220	−0.0706099	+	0.997504i	$0.522495\pi$
$132$	0	0
$133$	6.74582	0.584936
$134$	3.06533	0.264804
$135$	−0.198391	−0.0170748
$136$	3.52640	0.302386
$137$	−0.877454	−0.0749659	−0.0374830	−	0.999297i	$-0.511934\pi$
$137$	−0.877454	−0.0749659	−0.0374830	+	0.999297i	$0.511934\pi$
$138$	0.192654	0.0163998
$139$	18.7575	1.59099	0.795496	−	0.605959i	$-0.207210\pi$
$139$	18.7575	1.59099	0.795496	+	0.605959i	$0.207210\pi$
$140$	−0.573166	−0.0484413
$141$	17.4702	1.47125
$142$	7.24540	0.608021
$143$	0	0
$144$	2.92967	0.244139
$145$	−0.871504	−0.0723745
$146$	−3.60034	−0.297966
$147$	−21.8481	−1.80200
$148$	1.53860	0.126472
$149$	2.70491	0.221595	0.110797	−	0.993843i	$-0.464660\pi$
$149$	2.70491	0.221595	0.110797	+	0.993843i	$0.464660\pi$
$150$	7.60318	0.620797
$151$	2.56991	0.209136	0.104568	−	0.994518i	$-0.466654\pi$
$151$	2.56991	0.209136	0.104568	+	0.994518i	$0.466654\pi$
$152$	3.96264	0.321413
$153$	−2.97556	−0.240560
$154$	0	0
$155$	−0.898093	−0.0721366
$156$	8.17840	0.654796
$157$	−0.0901508	−0.00719482	−0.00359741	−	0.999994i	$-0.501145\pi$
$157$	−0.0901508	−0.00719482	−0.00359741	+	0.999994i	$0.501145\pi$
$158$	−0.0331840	−0.00263998
$159$	29.6937	2.35486
$160$	−0.526986	−0.0416619
$161$	0.517552	0.0407888
$162$	−7.45224	−0.585504
$163$	−23.4651	−1.83793	−0.918965	−	0.394339i	$-0.870974\pi$
$163$	−23.4651	−1.83793	−0.918965	+	0.394339i	$0.870974\pi$
$164$	−17.5435	−1.36992
$165$	0	0
$166$	−4.49315	−0.348736
$167$	−7.13287	−0.551958	−0.275979	−	0.961164i	$-0.589002\pi$
$167$	−7.13287	−0.551958	−0.275979	+	0.961164i	$0.589002\pi$
$168$	−22.0545	−1.70155
$169$	−7.38079	−0.567753
$170$	0.0907266	0.00695841
$171$	−3.34366	−0.255696
$172$	10.0037	0.762775
$173$	7.98849	0.607354	0.303677	−	0.952775i	$-0.401786\pi$
$173$	7.98849	0.607354	0.303677	+	0.952775i	$0.401786\pi$
$174$	−14.5808	−1.10537
$175$	20.4254	1.54401
$176$	0	0
$177$	−7.38607	−0.555171
$178$	3.48297	0.261060
$179$	−2.19085	−0.163752	−0.0818759	−	0.996643i	$-0.526091\pi$
$179$	−2.19085	−0.163752	−0.0818759	+	0.996643i	$0.526091\pi$
$180$	0.284098	0.0211754
$181$	−1.67068	−0.124181	−0.0620904	−	0.998071i	$-0.519777\pi$
$181$	−1.67068	−0.124181	−0.0620904	+	0.998071i	$0.519777\pi$
$182$	−6.58869	−0.488386
$183$	−2.57917	−0.190657
$184$	0.304022	0.0224128
$185$	0.0910404	0.00669342
$186$	−15.0256	−1.10173
$187$	0	0
$188$	11.9871	0.874252
$189$	−8.91679	−0.648601
$190$	0.101950	0.00739624
$191$	−12.2479	−0.886224	−0.443112	−	0.896466i	$-0.646126\pi$
$191$	−12.2479	−0.886224	−0.443112	+	0.896466i	$0.646126\pi$
$192$	−2.33873	−0.168783
$193$	−0.443410	−0.0319173	−0.0159587	−	0.999873i	$-0.505080\pi$
$193$	−0.443410	−0.0319173	−0.0159587	+	0.999873i	$0.505080\pi$
$194$	5.05989	0.363279
$195$	0.483925	0.0346546
$196$	−14.9911	−1.07079
$197$	5.70776	0.406661	0.203330	−	0.979110i	$-0.434823\pi$
$197$	5.70776	0.406661	0.203330	+	0.979110i	$0.434823\pi$
$198$	0	0
$199$	−20.5636	−1.45771	−0.728856	−	0.684667i	$-0.759948\pi$
$199$	−20.5636	−1.45771	−0.728856	+	0.684667i	$0.759948\pi$
$200$	11.9983	0.848410
$201$	−10.1191	−0.713747
$202$	−5.72805	−0.403024
$203$	−39.1702	−2.74921
$204$	−5.06164	−0.354385
$205$	−1.03807	−0.0725019
$206$	−5.19540	−0.361981
$207$	−0.256532	−0.0178302
$208$	3.42411	0.237419
$209$	0	0
$210$	−0.567416	−0.0391554
$211$	−25.1631	−1.73230	−0.866148	−	0.499787i	$-0.833412\pi$
$211$	−25.1631	−1.73230	−0.866148	+	0.499787i	$0.833412\pi$
$212$	20.3743	1.39931
$213$	−23.9182	−1.63885
$214$	3.95737	0.270520
$215$	0.591929	0.0403693
$216$	−5.23792	−0.356395
$217$	−40.3653	−2.74017
$218$	8.40769	0.569441
$219$	11.8853	0.803133
$220$	0	0
$221$	−3.47775	−0.233939
$222$	1.52316	0.102228
$223$	−2.29586	−0.153742	−0.0768709	−	0.997041i	$-0.524493\pi$
$223$	−2.29586	−0.153742	−0.0768709	+	0.997041i	$0.524493\pi$
$224$	−23.6857	−1.58257
$225$	−10.1241	−0.674942
$226$	7.39159	0.491681
$227$	4.67177	0.310076	0.155038	−	0.987908i	$-0.450450\pi$
$227$	4.67177	0.310076	0.155038	+	0.987908i	$0.450450\pi$
$228$	−5.68780	−0.376684
$229$	19.8758	1.31343	0.656714	−	0.754140i	$-0.271946\pi$
$229$	19.8758	1.31343	0.656714	+	0.754140i	$0.271946\pi$
$230$	0.00782181	0.000515755 0
$231$	0	0
$232$	−23.0095	−1.51065
$233$	−18.9071	−1.23864	−0.619321	−	0.785137i	$-0.712592\pi$
$233$	−18.9071	−1.23864	−0.619321	+	0.785137i	$0.712592\pi$
$234$	3.26578	0.213490
$235$	0.709292	0.0462691
$236$	−5.06795	−0.329895
$237$	0.109545	0.00711573
$238$	4.07776	0.264322
$239$	9.02526	0.583795	0.291898	−	0.956450i	$-0.405713\pi$
$239$	9.02526	0.583795	0.291898	+	0.956450i	$0.405713\pi$
$240$	0.294883	0.0190346
$241$	19.6205	1.26387	0.631935	−	0.775021i	$-0.282261\pi$
$241$	19.6205	1.26387	0.631935	+	0.775021i	$0.282261\pi$
$242$	0	0
$243$	18.0635	1.15878
$244$	−1.76969	−0.113293
$245$	−0.887037	−0.0566707
$246$	−17.3675	−1.10731
$247$	−3.90797	−0.248658
$248$	−23.7115	−1.50568
$249$	14.8326	0.939975
$250$	0.617895	0.0390791
$251$	−18.7486	−1.18340	−0.591701	−	0.806158i	$-0.701543\pi$
$251$	−18.7486	−1.18340	−0.591701	+	0.806158i	$0.701543\pi$
$252$	12.7689	0.804367
$253$	0	0
$254$	−14.4152	−0.904493
$255$	−0.299502	−0.0187556
$256$	−9.46857	−0.591785
$257$	21.5946	1.34703	0.673516	−	0.739173i	$-0.264783\pi$
$257$	21.5946	1.34703	0.673516	+	0.739173i	$0.264783\pi$
$258$	9.90334	0.616555
$259$	4.09186	0.254256
$260$	0.332045	0.0205925
$261$	19.4153	1.20178
$262$	−1.09792	−0.0678298
$263$	−1.95591	−0.120606	−0.0603031	−	0.998180i	$-0.519207\pi$
$263$	−1.95591	−0.120606	−0.0603031	+	0.998180i	$0.519207\pi$
$264$	0	0
$265$	1.20557	0.0740576
$266$	4.58221	0.280953
$267$	−11.4978	−0.703656
$268$	−6.94323	−0.424125
$269$	−10.0265	−0.611329	−0.305665	−	0.952139i	$-0.598879\pi$
$269$	−10.0265	−0.611329	−0.305665	+	0.952139i	$0.598879\pi$
$270$	−0.134760	−0.00820125
$271$	−14.5231	−0.882215	−0.441107	−	0.897454i	$-0.645414\pi$
$271$	−14.5231	−0.882215	−0.441107	+	0.897454i	$0.645414\pi$
$272$	−2.11919	−0.128495
$273$	21.7503	1.31639
$274$	−0.596025	−0.0360072
$275$	0	0
$276$	−0.436379	−0.0262669
$277$	19.4591	1.16919	0.584593	−	0.811327i	$-0.301254\pi$
$277$	19.4591	1.16919	0.584593	+	0.811327i	$0.301254\pi$
$278$	12.7414	0.764176
$279$	20.0076	1.19783
$280$	−0.895420	−0.0535116
$281$	12.9311	0.771403	0.385701	−	0.922624i	$-0.373960\pi$
$281$	12.9311	0.771403	0.385701	+	0.922624i	$0.373960\pi$
$282$	11.8669	0.706663
$283$	13.8891	0.825619	0.412810	−	0.910817i	$-0.364547\pi$
$283$	13.8891	0.825619	0.412810	+	0.910817i	$0.364547\pi$
$284$	−16.4115	−0.973841
$285$	−0.336553	−0.0199357
$286$	0	0
$287$	−46.6566	−2.75405
$288$	11.7401	0.691794
$289$	−14.8476	−0.873389
$290$	−0.591984	−0.0347625
$291$	−16.7035	−0.979175
$292$	8.15508	0.477240
$293$	17.0318	0.995011	0.497505	−	0.867461i	$-0.334249\pi$
$293$	17.0318	0.995011	0.497505	+	0.867461i	$0.334249\pi$
$294$	−14.8407	−0.865526
$295$	−0.299876	−0.0174595
$296$	2.40365	0.139709
$297$	0	0
$298$	1.83736	0.106435
$299$	−0.299827	−0.0173394
$300$	−17.2219	−0.994305
$301$	26.6046	1.53346
$302$	1.74566	0.100451
$303$	18.9092	1.08630
$304$	−2.38135	−0.136580
$305$	−0.104715	−0.00599595
$306$	−2.02120	−0.115544
$307$	−16.8128	−0.959555	−0.479778	−	0.877390i	$-0.659283\pi$
$307$	−16.8128	−0.959555	−0.479778	+	0.877390i	$0.659283\pi$
$308$	0	0
$309$	17.1508	0.975675
$310$	−0.610045	−0.0346482
$311$	−8.91522	−0.505536	−0.252768	−	0.967527i	$-0.581341\pi$
$311$	−8.91522	−0.505536	−0.252768	+	0.967527i	$0.581341\pi$
$312$	12.7766	0.723332
$313$	−28.7083	−1.62269	−0.811344	−	0.584570i	$-0.801263\pi$
$313$	−28.7083	−1.62269	−0.811344	+	0.584570i	$0.801263\pi$
$314$	−0.0612364	−0.00345577
$315$	0.755551	0.0425705
$316$	0.0751645	0.00422834
$317$	−8.34318	−0.468600	−0.234300	−	0.972164i	$-0.575280\pi$
$317$	−8.34318	−0.468600	−0.234300	+	0.972164i	$0.575280\pi$
$318$	20.1699	1.13107
$319$	0	0
$320$	−0.0949529	−0.00530803
$321$	−13.0639	−0.729154
$322$	0.351556	0.0195914
$323$	2.41865	0.134578
$324$	16.8800	0.937776
$325$	−11.8328	−0.656365
$326$	−15.9391	−0.882784
$327$	−27.7551	−1.53486
$328$	−27.4071	−1.51330
$329$	31.8795	1.75758
$330$	0	0
$331$	−12.3288	−0.677651	−0.338825	−	0.940849i	$-0.610030\pi$
$331$	−12.3288	−0.677651	−0.338825	+	0.940849i	$0.610030\pi$
$332$	10.1774	0.558555
$333$	−2.02819	−0.111144
$334$	−4.84512	−0.265113
$335$	−0.410838	−0.0224465
$336$	13.2537	0.723048
$337$	19.8148	1.07938	0.539690	−	0.841864i	$-0.318542\pi$
$337$	19.8148	1.07938	0.539690	+	0.841864i	$0.318542\pi$
$338$	−5.01352	−0.272700
$339$	−24.4008	−1.32527
$340$	−0.205504	−0.0111450
$341$	0	0
$342$	−2.27124	−0.122814
$343$	−11.2254	−0.606112
$344$	15.6281	0.842612
$345$	−0.0258210	−0.00139016
$346$	5.42631	0.291720
$347$	22.3646	1.20059	0.600297	−	0.799777i	$-0.295049\pi$
$347$	22.3646	1.20059	0.600297	+	0.799777i	$0.295049\pi$
$348$	33.0268	1.77042
$349$	−13.4598	−0.720489	−0.360244	−	0.932858i	$-0.617307\pi$
$349$	−13.4598	−0.720489	−0.360244	+	0.932858i	$0.617307\pi$
$350$	13.8743	0.741612
$351$	5.16565	0.275722
$352$	0	0
$353$	−2.54736	−0.135582	−0.0677912	−	0.997700i	$-0.521595\pi$
$353$	−2.54736	−0.135582	−0.0677912	+	0.997700i	$0.521595\pi$
$354$	−5.01711	−0.266656
$355$	−0.971083	−0.0515397
$356$	−7.88924	−0.418129
$357$	−13.4613	−0.712448
$358$	−1.48817	−0.0786523
$359$	−18.4374	−0.973090	−0.486545	−	0.873655i	$-0.661743\pi$
$359$	−18.4374	−0.973090	−0.486545	+	0.873655i	$0.661743\pi$
$360$	0.443827	0.0233918
$361$	−16.2821	−0.856955
$362$	−1.13484	−0.0596457
$363$	0	0
$364$	14.9239	0.782227
$365$	0.482545	0.0252576
$366$	−1.75194	−0.0915754
$367$	34.7411	1.81347	0.906735	−	0.421701i	$-0.138567\pi$
$367$	34.7411	1.81347	0.906735	+	0.421701i	$0.138567\pi$
$368$	−0.182702	−0.00952399
$369$	23.1260	1.20389
$370$	0.0618407	0.00321495
$371$	54.1850	2.81315
$372$	34.0344	1.76460
$373$	31.3924	1.62544	0.812719	−	0.582656i	$-0.197987\pi$
$373$	31.3924	1.62544	0.812719	+	0.582656i	$0.197987\pi$
$374$	0	0
$375$	−2.03976	−0.105333
$376$	18.7267	0.965758
$377$	22.6920	1.16870
$378$	−6.05688	−0.311532
$379$	−7.91087	−0.406354	−0.203177	−	0.979142i	$-0.565127\pi$
$379$	−7.91087	−0.406354	−0.203177	+	0.979142i	$0.565127\pi$
$380$	−0.230926	−0.0118462
$381$	47.5869	2.43795
$382$	−8.31956	−0.425666
$383$	−34.6180	−1.76890	−0.884449	−	0.466636i	$-0.845466\pi$
$383$	−34.6180	−1.76890	−0.884449	+	0.466636i	$0.845466\pi$
$384$	24.3711	1.24368
$385$	0	0
$386$	−0.301194	−0.0153303
$387$	−13.1869	−0.670330
$388$	−11.4611	−0.581848
$389$	29.0141	1.47107	0.735535	−	0.677486i	$-0.236931\pi$
$389$	29.0141	1.47107	0.735535	+	0.677486i	$0.236931\pi$
$390$	0.328714	0.0166451
$391$	0.185564	0.00938437
$392$	−23.4196	−1.18287
$393$	3.62441	0.182827
$394$	3.87709	0.195325
$395$	0.00444757	0.000223781 0
$396$	0	0
$397$	−8.28118	−0.415620	−0.207810	−	0.978169i	$-0.566634\pi$
$397$	−8.28118	−0.415620	−0.207810	+	0.978169i	$0.566634\pi$
$398$	−13.9681	−0.700159
$399$	−15.1266	−0.757275
$400$	−7.21040	−0.360520
$401$	−22.4555	−1.12137	−0.560686	−	0.828028i	$-0.689463\pi$
$401$	−22.4555	−1.12137	−0.560686	+	0.828028i	$0.689463\pi$
$402$	−6.87357	−0.342823
$403$	23.3843	1.16486
$404$	12.9745	0.645507
$405$	0.998806	0.0496311
$406$	−26.6070	−1.32048
$407$	0	0
$408$	−7.90747	−0.391478
$409$	−10.1966	−0.504191	−0.252096	−	0.967702i	$-0.581120\pi$
$409$	−10.1966	−0.504191	−0.252096	+	0.967702i	$0.581120\pi$
$410$	−0.705125	−0.0348237
$411$	1.96757	0.0970530
$412$	11.7680	0.579769
$413$	−13.4781	−0.663214
$414$	−0.174254	−0.00856410
$415$	0.602205	0.0295611
$416$	13.7215	0.672753
$417$	−42.0611	−2.05974
$418$	0	0
$419$	7.04555	0.344198	0.172099	−	0.985080i	$-0.444945\pi$
$419$	7.04555	0.344198	0.172099	+	0.985080i	$0.444945\pi$
$420$	1.28524	0.0627135
$421$	22.4606	1.09466	0.547331	−	0.836916i	$-0.315644\pi$
$421$	22.4606	1.09466	0.547331	+	0.836916i	$0.315644\pi$
$422$	−17.0924	−0.832046
$423$	−15.8015	−0.768297
$424$	31.8295	1.54578
$425$	7.32335	0.355235
$426$	−16.2468	−0.787161
$427$	−4.70646	−0.227762
$428$	−8.96377	−0.433280
$429$	0	0
$430$	0.402078	0.0193899
$431$	25.7105	1.23843	0.619217	−	0.785220i	$-0.287450\pi$
$431$	25.7105	1.23843	0.619217	+	0.785220i	$0.287450\pi$
$432$	3.14773	0.151445
$433$	19.6670	0.945134	0.472567	−	0.881295i	$-0.343328\pi$
$433$	19.6670	0.945134	0.472567	+	0.881295i	$0.343328\pi$
$434$	−27.4188	−1.31614
$435$	1.95423	0.0936981
$436$	−19.0441	−0.912049
$437$	0.208519	0.00997484
$438$	8.07327	0.385756
$439$	−3.18019	−0.151782	−0.0758911	−	0.997116i	$-0.524180\pi$
$439$	−3.18019	−0.151782	−0.0758911	+	0.997116i	$0.524180\pi$
$440$	0	0
$441$	19.7613	0.941016
$442$	−2.36232	−0.112364
$443$	−23.7695	−1.12932	−0.564661	−	0.825323i	$-0.690993\pi$
$443$	−23.7695	−1.12932	−0.564661	+	0.825323i	$0.690993\pi$
$444$	−3.45009	−0.163734
$445$	−0.466814	−0.0221291
$446$	−1.55950	−0.0738444
$447$	−6.06539	−0.286883
$448$	−4.26771	−0.201630
$449$	−9.49379	−0.448040	−0.224020	−	0.974585i	$-0.571918\pi$
$449$	−9.49379	−0.448040	−0.224020	+	0.974585i	$0.571918\pi$
$450$	−6.87699	−0.324184
$451$	0	0
$452$	−16.7426	−0.787505
$453$	−5.76267	−0.270754
$454$	3.17338	0.148934
$455$	0.883065	0.0413987
$456$	−8.88568	−0.416110
$457$	−21.7140	−1.01574	−0.507869	−	0.861435i	$-0.669566\pi$
$457$	−21.7140	−1.01574	−0.507869	+	0.861435i	$0.669566\pi$
$458$	13.5009	0.630858
$459$	−3.19704	−0.149225
$460$	−0.0177171	−0.000826063 0
$461$	−5.39837	−0.251427	−0.125714	−	0.992067i	$-0.540122\pi$
$461$	−5.39837	−0.251427	−0.125714	+	0.992067i	$0.540122\pi$
$462$	0	0
$463$	18.3678	0.853626	0.426813	−	0.904340i	$-0.359636\pi$
$463$	18.3678	0.853626	0.426813	+	0.904340i	$0.359636\pi$
$464$	13.8275	0.641928
$465$	2.01385	0.0933901
$466$	−12.8429	−0.594938
$467$	17.6351	0.816053	0.408026	−	0.912970i	$-0.366217\pi$
$467$	17.6351	0.816053	0.408026	+	0.912970i	$0.366217\pi$
$468$	−7.39726	−0.341939
$469$	−18.4653	−0.852651
$470$	0.481798	0.0222237
$471$	0.202151	0.00931461
$472$	−7.91733	−0.364425
$473$	0	0
$474$	0.0744105	0.00341779
$475$	8.22930	0.377586
$476$	−9.23647	−0.423353
$477$	−26.8576	−1.22972
$478$	6.13056	0.280405
$479$	−34.5759	−1.57981	−0.789906	−	0.613228i	$-0.789871\pi$
$479$	−34.5759	−1.57981	−0.789906	+	0.613228i	$0.789871\pi$
$480$	1.18169	0.0539367
$481$	−2.37049	−0.108085
$482$	13.3276	0.607055
$483$	−1.16054	−0.0528064
$484$	0	0
$485$	−0.678164	−0.0307939
$486$	12.2700	0.556577
$487$	34.3891	1.55832	0.779160	−	0.626825i	$-0.215646\pi$
$487$	34.3891	1.55832	0.779160	+	0.626825i	$0.215646\pi$
$488$	−2.76468	−0.125151
$489$	52.6173	2.37944
$490$	−0.602535	−0.0272197
$491$	32.3024	1.45779	0.728894	−	0.684626i	$-0.240035\pi$
$491$	32.3024	1.45779	0.728894	+	0.684626i	$0.240035\pi$
$492$	39.3389	1.77354
$493$	−14.0442	−0.632517
$494$	−2.65455	−0.119434
$495$	0	0
$496$	14.2494	0.639818
$497$	−43.6459	−1.95779
$498$	10.0753	0.451483
$499$	−13.9532	−0.624630	−0.312315	−	0.949979i	$-0.601105\pi$
$499$	−13.9532	−0.624630	−0.312315	+	0.949979i	$0.601105\pi$
$500$	−1.39958	−0.0625913
$501$	15.9945	0.714580
$502$	−12.7353	−0.568404
$503$	41.8801	1.86734	0.933671	−	0.358132i	$-0.116586\pi$
$503$	41.8801	1.86734	0.933671	+	0.358132i	$0.116586\pi$
$504$	19.9481	0.888557
$505$	0.767717	0.0341629
$506$	0	0
$507$	16.5504	0.735029
$508$	32.6518	1.44869
$509$	−3.04272	−0.134866	−0.0674332	−	0.997724i	$-0.521481\pi$
$509$	−3.04272	−0.134866	−0.0674332	+	0.997724i	$0.521481\pi$
$510$	−0.203442	−0.00900856
$511$	21.6883	0.959432
$512$	15.3053	0.676407
$513$	−3.59254	−0.158614
$514$	14.6685	0.646998
$515$	0.696326	0.0306838
$516$	−22.4319	−0.987510
$517$	0	0
$518$	2.77947	0.122123
$519$	−17.9131	−0.786297
$520$	0.518732	0.0227479
$521$	−8.83937	−0.387260	−0.193630	−	0.981075i	$-0.562026\pi$
$521$	−8.83937	−0.387260	−0.193630	+	0.981075i	$0.562026\pi$
$522$	13.1882	0.577230
$523$	−31.7745	−1.38940	−0.694701	−	0.719299i	$-0.744463\pi$
$523$	−31.7745	−1.38940	−0.694701	+	0.719299i	$0.744463\pi$
$524$	2.48689	0.108640
$525$	−45.8011	−1.99893
$526$	−1.32858	−0.0579289
$527$	−14.4726	−0.630438
$528$	0	0
$529$	−22.9840	−0.999304
$530$	0.818904	0.0355709
$531$	6.68061	0.289914
$532$	−10.3791	−0.449991
$533$	27.0290	1.17075
$534$	−7.81009	−0.337976
$535$	−0.530396	−0.0229310
$536$	−10.8470	−0.468517
$537$	4.91268	0.211998
$538$	−6.81070	−0.293630
$539$	0	0
$540$	0.305244	0.0131356
$541$	−30.8722	−1.32730	−0.663650	−	0.748043i	$-0.730993\pi$
$541$	−30.8722	−1.32730	−0.663650	+	0.748043i	$0.730993\pi$
$542$	−9.86505	−0.423740
$543$	3.74627	0.160768
$544$	−8.49230	−0.364104
$545$	−1.12686	−0.0482695
$546$	14.7742	0.632278
$547$	−11.1535	−0.476889	−0.238445	−	0.971156i	$-0.576638\pi$
$547$	−11.1535	−0.476889	−0.238445	+	0.971156i	$0.576638\pi$
$548$	1.35005	0.0576712
$549$	2.33282	0.0995625
$550$	0	0
$551$	−15.7815	−0.672315
$552$	−0.681726	−0.0290162
$553$	0.199898	0.00850054
$554$	13.2179	0.561576
$555$	−0.204146	−0.00866550
$556$	−28.8603	−1.22395
$557$	31.3991	1.33042	0.665212	−	0.746655i	$-0.268341\pi$
$557$	31.3991	1.33042	0.665212	+	0.746655i	$0.268341\pi$
$558$	13.5905	0.575332
$559$	−15.4125	−0.651880
$560$	0.538103	0.0227390
$561$	0	0
$562$	8.78364	0.370515
$563$	17.1574	0.723097	0.361549	−	0.932353i	$-0.382248\pi$
$563$	17.1574	0.723097	0.361549	+	0.932353i	$0.382248\pi$
$564$	−26.8795	−1.13183
$565$	−0.990676	−0.0416781
$566$	9.43438	0.396557
$567$	44.8919	1.88528
$568$	−25.6386	−1.07577
$569$	−15.2802	−0.640580	−0.320290	−	0.947320i	$-0.603780\pi$
$569$	−15.2802	−0.640580	−0.320290	+	0.947320i	$0.603780\pi$
$570$	−0.228609	−0.00957539
$571$	−6.42163	−0.268737	−0.134368	−	0.990931i	$-0.542901\pi$
$571$	−6.42163	−0.268737	−0.134368	+	0.990931i	$0.542901\pi$
$572$	0	0
$573$	27.4641	1.14733
$574$	−31.6923	−1.32281
$575$	0.631368	0.0263299
$576$	2.11535	0.0881396
$577$	−23.2656	−0.968560	−0.484280	−	0.874913i	$-0.660918\pi$
$577$	−23.2656	−0.968560	−0.484280	+	0.874913i	$0.660918\pi$
$578$	−10.0855	−0.419501
$579$	0.994286	0.0413211
$580$	1.34089	0.0556776
$581$	27.0664	1.12291
$582$	−11.3461	−0.470311
$583$	0	0
$584$	12.7402	0.527192
$585$	−0.437704	−0.0180968
$586$	11.5692	0.477918
$587$	27.9536	1.15377	0.576884	−	0.816826i	$-0.304268\pi$
$587$	27.9536	1.15377	0.576884	+	0.816826i	$0.304268\pi$
$588$	33.6154	1.38628
$589$	−16.2630	−0.670105
$590$	−0.203696	−0.00838602
$591$	−12.7989	−0.526475
$592$	−1.44447	−0.0593675
$593$	22.4514	0.921969	0.460984	−	0.887408i	$-0.347496\pi$
$593$	22.4514	0.921969	0.460984	+	0.887408i	$0.347496\pi$
$594$	0	0
$595$	−0.546532	−0.0224056
$596$	−4.16177	−0.170473
$597$	46.1109	1.88720
$598$	−0.203662	−0.00832838
$599$	−43.9958	−1.79762	−0.898810	−	0.438338i	$-0.855567\pi$
$599$	−43.9958	−1.79762	−0.898810	+	0.438338i	$0.855567\pi$
$600$	−26.9046	−1.09838
$601$	−33.2501	−1.35630	−0.678150	−	0.734923i	$-0.737218\pi$
$601$	−33.2501	−1.35630	−0.678150	+	0.734923i	$0.737218\pi$
$602$	18.0716	0.736544
$603$	9.15261	0.372723
$604$	−3.95406	−0.160888
$605$	0	0
$606$	12.8444	0.521767
$607$	−18.3936	−0.746572	−0.373286	−	0.927716i	$-0.621769\pi$
$607$	−18.3936	−0.746572	−0.373286	+	0.927716i	$0.621769\pi$
$608$	−9.54286	−0.387014
$609$	87.8339	3.55921
$610$	−0.0711292	−0.00287994
$611$	−18.4684	−0.747150
$612$	4.57819	0.185062
$613$	−20.8255	−0.841134	−0.420567	−	0.907261i	$-0.638169\pi$
$613$	−20.8255	−0.841134	−0.420567	+	0.907261i	$0.638169\pi$
$614$	−11.4203	−0.460888
$615$	2.32773	0.0938630
$616$	0	0
$617$	21.7106	0.874034	0.437017	−	0.899453i	$-0.356035\pi$
$617$	21.7106	0.874034	0.437017	+	0.899453i	$0.356035\pi$
$618$	11.6500	0.468630
$619$	−24.8785	−0.999953	−0.499976	−	0.866039i	$-0.666658\pi$
$619$	−24.8785	−0.999953	−0.499976	+	0.866039i	$0.666658\pi$
$620$	1.38180	0.0554946
$621$	−0.275626	−0.0110605
$622$	−6.05581	−0.242816
$623$	−20.9812	−0.840595
$624$	−7.67809	−0.307370
$625$	24.8757	0.995030
$626$	−19.5006	−0.779399
$627$	0	0
$628$	0.138706	0.00553496
$629$	1.46710	0.0584972
$630$	0.513220	0.0204472
$631$	−22.5492	−0.897669	−0.448835	−	0.893615i	$-0.648161\pi$
$631$	−22.5492	−0.897669	−0.448835	+	0.893615i	$0.648161\pi$
$632$	0.117425	0.00467090
$633$	56.4247	2.24268
$634$	−5.66724	−0.225075
$635$	1.93204	0.0766707
$636$	−45.6866	−1.81159
$637$	23.0965	0.915115
$638$	0	0
$639$	21.6337	0.855816
$640$	0.989474	0.0391124
$641$	−15.4310	−0.609488	−0.304744	−	0.952434i	$-0.598571\pi$
$641$	−15.4310	−0.609488	−0.304744	+	0.952434i	$0.598571\pi$
$642$	−8.87385	−0.350223
$643$	−1.81774	−0.0716847	−0.0358423	−	0.999357i	$-0.511411\pi$
$643$	−1.81774	−0.0716847	−0.0358423	+	0.999357i	$0.511411\pi$
$644$	−0.796304	−0.0313788
$645$	−1.32732	−0.0522632
$646$	1.64291	0.0646395
$647$	6.57818	0.258615	0.129307	−	0.991605i	$-0.458725\pi$
$647$	6.57818	0.258615	0.129307	+	0.991605i	$0.458725\pi$
$648$	26.3705	1.03593
$649$	0	0
$650$	−8.03762	−0.315261
$651$	90.5136	3.54751
$652$	36.1034	1.41392
$653$	−32.7826	−1.28288	−0.641441	−	0.767172i	$-0.721663\pi$
$653$	−32.7826	−1.28288	−0.641441	+	0.767172i	$0.721663\pi$
$654$	−18.8531	−0.737214
$655$	0.147152	0.00574969
$656$	16.4703	0.643057
$657$	−10.7501	−0.419401
$658$	21.6547	0.844188
$659$	−37.7397	−1.47013	−0.735066	−	0.677996i	$-0.762849\pi$
$659$	−37.7397	−1.47013	−0.735066	+	0.677996i	$0.762849\pi$
$660$	0	0
$661$	−5.12478	−0.199331	−0.0996654	−	0.995021i	$-0.531777\pi$
$661$	−5.12478	−0.199331	−0.0996654	+	0.995021i	$0.531777\pi$
$662$	−8.37452	−0.325485
$663$	7.79837	0.302864
$664$	15.8994	0.617018
$665$	−0.614142	−0.0238154
$666$	−1.37768	−0.0533841
$667$	−1.21079	−0.0468819
$668$	10.9746	0.424620
$669$	5.14814	0.199039
$670$	−0.279069	−0.0107814
$671$	0	0
$672$	53.1119	2.04884
$673$	13.1163	0.505596	0.252798	−	0.967519i	$-0.418649\pi$
$673$	13.1163	0.505596	0.252798	+	0.967519i	$0.418649\pi$
$674$	13.4595	0.518441
$675$	−10.8777	−0.418683
$676$	11.3561	0.436772
$677$	−2.91718	−0.112116	−0.0560582	−	0.998428i	$-0.517853\pi$
$677$	−2.91718	−0.112116	−0.0560582	+	0.998428i	$0.517853\pi$
$678$	−16.5746	−0.636544
$679$	−30.4805	−1.16973
$680$	−0.321045	−0.0123115
$681$	−10.4758	−0.401434
$682$	0	0
$683$	18.2190	0.697130	0.348565	−	0.937285i	$-0.386669\pi$
$683$	18.2190	0.697130	0.348565	+	0.937285i	$0.386669\pi$
$684$	5.14454	0.196707
$685$	0.0798838	0.00305220
$686$	−7.62501	−0.291124
$687$	−44.5687	−1.70040
$688$	−9.39173	−0.358056
$689$	−31.3903	−1.19588
$690$	−0.0175393	−0.000667711 0
$691$	−12.6948	−0.482932	−0.241466	−	0.970409i	$-0.577628\pi$
$691$	−12.6948	−0.482932	−0.241466	+	0.970409i	$0.577628\pi$
$692$	−12.2911	−0.467236
$693$	0	0
$694$	15.1915	0.576662
$695$	−1.70769	−0.0647764
$696$	51.5956	1.95572
$697$	−16.7283	−0.633630
$698$	−9.14282	−0.346061
$699$	42.3965	1.60358
$700$	−31.4265	−1.18781
$701$	34.9651	1.32061	0.660307	−	0.750996i	$-0.270426\pi$
$701$	34.9651	1.32061	0.660307	+	0.750996i	$0.270426\pi$
$702$	3.50886	0.132433
$703$	1.64859	0.0621779
$704$	0	0
$705$	−1.59049	−0.0599013
$706$	−1.73034	−0.0651221
$707$	34.5055	1.29771
$708$	11.3642	0.427092
$709$	5.10749	0.191816	0.0959079	−	0.995390i	$-0.469425\pi$
$709$	5.10749	0.191816	0.0959079	+	0.995390i	$0.469425\pi$
$710$	−0.659624	−0.0247553
$711$	−0.0990824	−0.00371588
$712$	−12.3248	−0.461893
$713$	−1.24773	−0.0467278
$714$	−9.14381	−0.342199
$715$	0	0
$716$	3.37084	0.125974
$717$	−20.2379	−0.755798
$718$	−12.5239	−0.467389
$719$	−0.537410	−0.0200420	−0.0100210	−	0.999950i	$-0.503190\pi$
$719$	−0.537410	−0.0200420	−0.0100210	+	0.999950i	$0.503190\pi$
$720$	−0.266718	−0.00994000
$721$	31.2968	1.16555
$722$	−11.0599	−0.411607
$723$	−43.9964	−1.63624
$724$	2.57050	0.0955320
$725$	−47.7842	−1.77466
$726$	0	0
$727$	−43.6667	−1.61951	−0.809754	−	0.586769i	$-0.800400\pi$
$727$	−43.6667	−1.61951	−0.809754	+	0.586769i	$0.800400\pi$
$728$	23.3147	0.864101
$729$	−7.59195	−0.281183
$730$	0.327777	0.0121316
$731$	9.53885	0.352807
$732$	3.96830	0.146672
$733$	−12.1855	−0.450081	−0.225040	−	0.974349i	$-0.572251\pi$
$733$	−12.1855	−0.450081	−0.225040	+	0.974349i	$0.572251\pi$
$734$	23.5985	0.871035
$735$	1.98906	0.0733676
$736$	−0.732146	−0.0269873
$737$	0	0
$738$	15.7087	0.578246
$739$	28.8363	1.06076	0.530380	−	0.847760i	$-0.322049\pi$
$739$	28.8363	1.06076	0.530380	+	0.847760i	$0.322049\pi$
$740$	−0.140074	−0.00514924
$741$	8.76308	0.321920
$742$	36.8061	1.35119
$743$	−37.2323	−1.36592	−0.682960	−	0.730455i	$-0.739308\pi$
$743$	−37.2323	−1.36592	−0.682960	+	0.730455i	$0.739308\pi$
$744$	53.1697	1.94930
$745$	−0.246256	−0.00902213
$746$	21.3238	0.780720
$747$	−13.4159	−0.490861
$748$	0	0
$749$	−23.8389	−0.871056
$750$	−1.38554	−0.0505929
$751$	31.8432	1.16197	0.580987	−	0.813913i	$-0.302667\pi$
$751$	31.8432	1.16197	0.580987	+	0.813913i	$0.302667\pi$
$752$	−11.2538	−0.410385
$753$	42.0412	1.53206
$754$	15.4139	0.561342
$755$	−0.233966	−0.00851489
$756$	13.7193	0.498968
$757$	−43.9081	−1.59587	−0.797934	−	0.602745i	$-0.794073\pi$
$757$	−43.9081	−1.59587	−0.797934	+	0.602745i	$0.794073\pi$
$758$	−5.37359	−0.195178
$759$	0	0
$760$	−0.360761	−0.0130862
$761$	53.3303	1.93322	0.966611	−	0.256250i	$-0.0824869\pi$
$761$	53.3303	1.93322	0.966611	+	0.256250i	$0.0824869\pi$
$762$	32.3242	1.17098
$763$	−50.6474	−1.83356
$764$	18.8445	0.681771
$765$	0.270896	0.00979427
$766$	−23.5149	−0.849627
$767$	7.80810	0.281934
$768$	21.2320	0.766142
$769$	0.949944	0.0342559	0.0171279	−	0.999853i	$-0.494548\pi$
$769$	0.949944	0.0342559	0.0171279	+	0.999853i	$0.494548\pi$
$770$	0	0
$771$	−48.4228	−1.74391
$772$	0.682229	0.0245540
$773$	−22.8365	−0.821371	−0.410685	−	0.911777i	$-0.634711\pi$
$773$	−22.8365	−0.821371	−0.410685	+	0.911777i	$0.634711\pi$
$774$	−8.95745	−0.321969
$775$	−49.2421	−1.76883
$776$	−17.9049	−0.642749
$777$	−9.17544	−0.329167
$778$	19.7083	0.706576
$779$	−18.7977	−0.673499
$780$	−0.744565	−0.0266597
$781$	0	0
$782$	0.126047	0.00450744
$783$	20.8604	0.745490
$784$	14.0740	0.502643
$785$	0.00820736	0.000292933 0
$786$	2.46194	0.0878144
$787$	52.7838	1.88154	0.940769	−	0.339048i	$-0.110105\pi$
$787$	52.7838	1.88154	0.940769	+	0.339048i	$0.110105\pi$
$788$	−8.78194	−0.312844
$789$	4.38585	0.156140
$790$	0.00302108	0.000107485 0
$791$	−44.5265	−1.58318
$792$	0	0
$793$	2.72654	0.0968221
$794$	−5.62513	−0.199628
$795$	−2.70333	−0.0958771
$796$	31.6390	1.12142
$797$	15.2189	0.539081	0.269541	−	0.962989i	$-0.413128\pi$
$797$	15.2189	0.539081	0.269541	+	0.962989i	$0.413128\pi$
$798$	−10.2750	−0.363730
$799$	11.4301	0.404369
$800$	−28.8944	−1.02157
$801$	10.3996	0.367453
$802$	−15.2532	−0.538611
$803$	0	0
$804$	15.5692	0.549084
$805$	−0.0471181	−0.00166070
$806$	15.8842	0.559497
$807$	22.4831	0.791444
$808$	20.2693	0.713070
$809$	45.1467	1.58727	0.793637	−	0.608392i	$-0.208185\pi$
$809$	45.1467	1.58727	0.793637	+	0.608392i	$0.208185\pi$
$810$	0.678455	0.0238385
$811$	5.11786	0.179712	0.0898562	−	0.995955i	$-0.471359\pi$
$811$	5.11786	0.179712	0.0898562	+	0.995955i	$0.471359\pi$
$812$	60.2672	2.11496
$813$	32.5660	1.14214
$814$	0	0
$815$	2.13627	0.0748304
$816$	4.75199	0.166353
$817$	10.7189	0.375006
$818$	−6.92623	−0.242170
$819$	−19.6728	−0.687425
$820$	1.59717	0.0557756
$821$	−20.9638	−0.731642	−0.365821	−	0.930685i	$-0.619212\pi$
$821$	−20.9638	−0.731642	−0.365821	+	0.930685i	$0.619212\pi$
$822$	1.33650	0.0466159
$823$	−55.4994	−1.93459	−0.967294	−	0.253658i	$-0.918366\pi$
$823$	−55.4994	−1.93459	−0.967294	+	0.253658i	$0.918366\pi$
$824$	18.3844	0.640452
$825$	0	0
$826$	−9.15522	−0.318551
$827$	30.6048	1.06423	0.532117	−	0.846671i	$-0.321397\pi$
$827$	30.6048	1.06423	0.532117	+	0.846671i	$0.321397\pi$
$828$	0.394699	0.0137167
$829$	−13.2384	−0.459789	−0.229895	−	0.973216i	$-0.573838\pi$
$829$	−13.2384	−0.459789	−0.229895	+	0.973216i	$0.573838\pi$
$830$	0.409058	0.0141986
$831$	−43.6344	−1.51366
$832$	2.47236	0.0857137
$833$	−14.2945	−0.495274
$834$	−28.5707	−0.989323
$835$	0.649379	0.0224727
$836$	0	0
$837$	21.4968	0.743040
$838$	4.78581	0.165323
$839$	−38.5905	−1.33229	−0.666147	−	0.745821i	$-0.732058\pi$
$839$	−38.5905	−1.33229	−0.666147	+	0.745821i	$0.732058\pi$
$840$	2.00786	0.0692776
$841$	62.6369	2.15989
$842$	15.2567	0.525781
$843$	−28.9961	−0.998680
$844$	38.7158	1.33265
$845$	0.671950	0.0231158
$846$	−10.7335	−0.369024
$847$	0	0
$848$	−19.1279	−0.656856
$849$	−31.1443	−1.06887
$850$	4.97451	0.170624
$851$	0.126483	0.00433579
$852$	36.8004	1.26076
$853$	−0.193288	−0.00661804	−0.00330902	−	0.999995i	$-0.501053\pi$
$853$	−0.193288	−0.00661804	−0.00330902	+	0.999995i	$0.501053\pi$
$854$	−3.19694	−0.109397
$855$	0.304408	0.0104105
$856$	−14.0035	−0.478631
$857$	29.8135	1.01841	0.509205	−	0.860645i	$-0.329939\pi$
$857$	29.8135	1.01841	0.509205	+	0.860645i	$0.329939\pi$
$858$	0	0
$859$	14.3845	0.490793	0.245397	−	0.969423i	$-0.421082\pi$
$859$	14.3845	0.490793	0.245397	+	0.969423i	$0.421082\pi$
$860$	−0.910741	−0.0310560
$861$	104.621	3.56547
$862$	17.4643	0.594837
$863$	18.0437	0.614214	0.307107	−	0.951675i	$-0.400639\pi$
$863$	18.0437	0.614214	0.307107	+	0.951675i	$0.400639\pi$
$864$	12.6140	0.429137
$865$	−0.727275	−0.0247281
$866$	13.3591	0.453961
$867$	33.2937	1.13071
$868$	62.1059	2.10801
$869$	0	0
$870$	1.32744	0.0450045
$871$	10.6973	0.362464
$872$	−29.7514	−1.00751
$873$	15.1081	0.511331
$874$	0.141640	0.00479105
$875$	−3.72216	−0.125832
$876$	−18.2867	−0.617849
$877$	1.79387	0.0605746	0.0302873	−	0.999541i	$-0.490358\pi$
$877$	1.79387	0.0605746	0.0302873	+	0.999541i	$0.490358\pi$
$878$	−2.16020	−0.0729031
$879$	−38.1916	−1.28817
$880$	0	0
$881$	−21.0254	−0.708363	−0.354182	−	0.935177i	$-0.615241\pi$
$881$	−21.0254	−0.708363	−0.354182	+	0.935177i	$0.615241\pi$
$882$	13.4232	0.451983
$883$	−34.3288	−1.15526	−0.577628	−	0.816300i	$-0.696022\pi$
$883$	−34.3288	−1.15526	−0.577628	+	0.816300i	$0.696022\pi$
$884$	5.35085	0.179969
$885$	0.672431	0.0226035
$886$	−16.1458	−0.542429
$887$	−38.0606	−1.27795	−0.638975	−	0.769227i	$-0.720641\pi$
$887$	−38.0606	−1.27795	−0.638975	+	0.769227i	$0.720641\pi$
$888$	−5.38986	−0.180872
$889$	86.8366	2.91241
$890$	−0.317091	−0.0106289
$891$	0	0
$892$	3.53240	0.118273
$893$	12.8441	0.429812
$894$	−4.12002	−0.137794
$895$	0.199456	0.00666708
$896$	44.4724	1.48572
$897$	0.672321	0.0224481
$898$	−6.44881	−0.215200
$899$	94.4327	3.14951
$900$	15.5770	0.519232
$901$	19.4276	0.647227
$902$	0	0
$903$	−59.6572	−1.98527
$904$	−26.1558	−0.869931
$905$	0.152099	0.00505596
$906$	−3.91439	−0.130047
$907$	−19.4935	−0.647272	−0.323636	−	0.946182i	$-0.604905\pi$
$907$	−19.4935	−0.647272	−0.323636	+	0.946182i	$0.604905\pi$
$908$	−7.18797	−0.238541
$909$	−17.1031	−0.567274
$910$	0.599837	0.0198844
$911$	50.8978	1.68632	0.843159	−	0.537664i	$-0.180693\pi$
$911$	50.8978	1.68632	0.843159	+	0.537664i	$0.180693\pi$
$912$	5.33985	0.176820
$913$	0	0
$914$	−14.7496	−0.487873
$915$	0.234808	0.00776252
$916$	−30.5808	−1.01042
$917$	6.61381	0.218407
$918$	−2.17164	−0.0716749
$919$	−6.41575	−0.211636	−0.105818	−	0.994386i	$-0.533746\pi$
$919$	−6.41575	−0.211636	−0.105818	+	0.994386i	$0.533746\pi$
$920$	−0.0276782	−0.000912525 0
$921$	37.7003	1.24227
$922$	−3.66693	−0.120764
$923$	25.2848	0.832260
$924$	0	0
$925$	4.99171	0.164126
$926$	12.4767	0.410008
$927$	−15.5127	−0.509503
$928$	55.4115	1.81897
$929$	29.1854	0.957543	0.478771	−	0.877940i	$-0.341082\pi$
$929$	29.1854	0.957543	0.478771	+	0.877940i	$0.341082\pi$
$930$	1.36794	0.0448566
$931$	−16.0628	−0.526437
$932$	29.0904	0.952886
$933$	19.9912	0.654481
$934$	11.9789	0.391962
$935$	0	0
$936$	−11.5563	−0.377728
$937$	50.2542	1.64173	0.820866	−	0.571121i	$-0.193491\pi$
$937$	50.2542	1.64173	0.820866	+	0.571121i	$0.193491\pi$
$938$	−12.5429	−0.409540
$939$	64.3743	2.10078
$940$	−1.09131	−0.0355948
$941$	18.3180	0.597148	0.298574	−	0.954386i	$-0.403489\pi$
$941$	18.3180	0.597148	0.298574	+	0.954386i	$0.403489\pi$
$942$	0.137314	0.00447394
$943$	−1.44220	−0.0469644
$944$	4.75792	0.154857
$945$	0.811788	0.0264075
$946$	0	0
$947$	30.0741	0.977276	0.488638	−	0.872487i	$-0.337494\pi$
$947$	30.0741	0.977276	0.488638	+	0.872487i	$0.337494\pi$
$948$	−0.168546	−0.00547412
$949$	−12.5644	−0.407857
$950$	5.58989	0.181360
$951$	18.7084	0.606663
$952$	−14.4295	−0.467664
$953$	18.7370	0.606950	0.303475	−	0.952839i	$-0.401853\pi$
$953$	18.7370	0.606950	0.303475	+	0.952839i	$0.401853\pi$
$954$	−18.2435	−0.590654
$955$	1.11505	0.0360822
$956$	−13.8862	−0.449113
$957$	0	0
$958$	−23.4862	−0.758806
$959$	3.59042	0.115941
$960$	0.212919	0.00687193
$961$	66.3138	2.13915
$962$	−1.61019	−0.0519147
$963$	11.8161	0.380769
$964$	−30.1881	−0.972293
$965$	0.0403682	0.00129950
$966$	−0.788315	−0.0253636
$967$	41.3119	1.32850	0.664250	−	0.747510i	$-0.268751\pi$
$967$	41.3119	1.32850	0.664250	+	0.747510i	$0.268751\pi$
$968$	0	0
$969$	−5.42350	−0.174228
$970$	−0.460654	−0.0147907
$971$	−58.5971	−1.88047	−0.940236	−	0.340524i	$-0.889396\pi$
$971$	−58.5971	−1.88047	−0.940236	+	0.340524i	$0.889396\pi$
$972$	−27.7925	−0.891445
$973$	−76.7532	−2.46059
$974$	23.3594	0.748483
$975$	26.5334	0.849749
$976$	1.66143	0.0531812
$977$	−1.24941	−0.0399721	−0.0199860	−	0.999800i	$-0.506362\pi$
$977$	−1.24941	−0.0399721	−0.0199860	+	0.999800i	$0.506362\pi$
$978$	35.7412	1.14288
$979$	0	0
$980$	1.36479	0.0435967
$981$	25.1041	0.801513
$982$	21.9420	0.700196
$983$	4.59589	0.146586	0.0732930	−	0.997310i	$-0.476649\pi$
$983$	4.59589	0.146586	0.0732930	+	0.997310i	$0.476649\pi$
$984$	61.4567	1.95917
$985$	−0.519637	−0.0165570
$986$	−9.53972	−0.303807
$987$	−71.4854	−2.27541
$988$	6.01279	0.191292
$989$	0.822373	0.0261499
$990$	0	0
$991$	41.1835	1.30824	0.654119	−	0.756392i	$-0.273040\pi$
$991$	41.1835	1.30824	0.654119	+	0.756392i	$0.273040\pi$
$992$	57.1021	1.81299
$993$	27.6456	0.877306
$994$	−29.6472	−0.940352
$995$	1.87211	0.0593500
$996$	−22.8213	−0.723121
$997$	−46.9642	−1.48737	−0.743686	−	0.668529i	$-0.766924\pi$
$997$	−46.9642	−1.48737	−0.743686	+	0.668529i	$0.766924\pi$
$998$	−9.47793	−0.300019
$999$	−2.17915	−0.0689453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4477.2.a.q.1.18	yes	28
11.10	odd	2	inner	4477.2.a.q.1.11	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4477.2.a.q.1.11	✓	28	11.10	odd	2	inner
4477.2.a.q.1.18	yes	28	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4477 = 11^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4477.a (trivial)

\(f(q)\)	\(=\)	\(q+0.679267 q^{2} -2.24236 q^{3} -1.53860 q^{4} -0.0910404 q^{5} -1.52316 q^{6} -4.09186 q^{7} -2.40365 q^{8} +2.02819 q^{9} -0.0618407 q^{10} +3.45009 q^{12} +2.37049 q^{13} -2.77947 q^{14} +0.204146 q^{15} +1.44447 q^{16} -1.46710 q^{17} +1.37768 q^{18} -1.64859 q^{19} +0.140074 q^{20} +9.17544 q^{21} -0.126483 q^{23} +5.38986 q^{24} -4.99171 q^{25} +1.61019 q^{26} +2.17915 q^{27} +6.29573 q^{28} +9.57272 q^{29} +0.138669 q^{30} +9.86477 q^{31} +5.78848 q^{32} -0.996553 q^{34} +0.372525 q^{35} -3.12057 q^{36} -1.00000 q^{37} -1.11983 q^{38} -5.31549 q^{39} +0.218829 q^{40} +11.4023 q^{41} +6.23257 q^{42} -6.50183 q^{43} -0.184647 q^{45} -0.0859158 q^{46} -7.79096 q^{47} -3.23903 q^{48} +9.74334 q^{49} -3.39070 q^{50} +3.28977 q^{51} -3.64722 q^{52} -13.2421 q^{53} +1.48023 q^{54} +9.83541 q^{56} +3.69674 q^{57} +6.50243 q^{58} +3.29388 q^{59} -0.314098 q^{60} +1.15020 q^{61} +6.70081 q^{62} -8.29907 q^{63} +1.04298 q^{64} -0.215810 q^{65} +4.51270 q^{67} +2.25728 q^{68} +0.283621 q^{69} +0.253044 q^{70} +10.6665 q^{71} -4.87506 q^{72} -5.30034 q^{73} -0.679267 q^{74} +11.1932 q^{75} +2.53652 q^{76} -3.61064 q^{78} -0.0488526 q^{79} -0.131506 q^{80} -10.9710 q^{81} +7.74519 q^{82} -6.61470 q^{83} -14.1173 q^{84} +0.133566 q^{85} -4.41648 q^{86} -21.4655 q^{87} +5.12755 q^{89} -0.125425 q^{90} -9.69971 q^{91} +0.194607 q^{92} -22.1204 q^{93} -5.29214 q^{94} +0.150089 q^{95} -12.9799 q^{96} +7.44905 q^{97} +6.61832 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 16 q^{3} + 20 q^{4} - 12 q^{5} + 16 q^{9} - 22 q^{12} - 6 q^{14} - 16 q^{15} + 36 q^{16} - 50 q^{20} - 28 q^{23} + 12 q^{25} - 68 q^{26} - 34 q^{27} - 8 q^{31} + 44 q^{34} - 16 q^{36} - 28 q^{37}+ \cdots - 14 q^{97}+O(q^{100}) \) 28 * q - 16 * q^3 + 20 * q^4 - 12 * q^5 + 16 * q^9 - 22 * q^12 - 6 * q^14 - 16 * q^15 + 36 * q^16 - 50 * q^20 - 28 * q^23 + 12 * q^25 - 68 * q^26 - 34 * q^27 - 8 * q^31 + 44 * q^34 - 16 * q^36 - 28 * q^37 - 14 * q^38 - 40 * q^42 + 34 * q^45 - 108 * q^47 - 128 * q^48 + 4 * q^49 - 6 * q^53 + 26 * q^56 - 6 * q^58 - 94 * q^59 + 102 * q^60 + 48 * q^64 - 28 * q^67 + 10 * q^69 - 120 * q^70 - 62 * q^71 - 14 * q^75 + 120 * q^78 - 128 * q^80 + 36 * q^81 - 20 * q^86 - 32 * q^89 - 58 * q^91 - 120 * q^92 - 32 * q^93 - 14 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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