LMFDB - Embedded newform 3751.2.a.p.1.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3751,2,Mod(1,3751)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3751.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 3751 = 11^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3751.a (trivial)

Newform invariants

comment:select newform

sage:traces = [30,-9,-1,29,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$29.9518857982$
Analytic rank:	$1$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 341)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	3751.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.18669 q^{2} -1.13252 q^{3} -0.591757 q^{4} +1.29761 q^{5} +1.34395 q^{6} +3.67025 q^{7} +3.07562 q^{8} -1.71740 q^{9} -1.53987 q^{10} +0.670175 q^{12} +6.11622 q^{13} -4.35547 q^{14} -1.46957 q^{15} -2.46631 q^{16} -2.44407 q^{17} +2.03803 q^{18} +3.43054 q^{19} -0.767870 q^{20} -4.15663 q^{21} -6.88465 q^{23} -3.48320 q^{24} -3.31620 q^{25} -7.25808 q^{26} +5.34254 q^{27} -2.17190 q^{28} -10.1661 q^{29} +1.74393 q^{30} +1.00000 q^{31} -3.22449 q^{32} +2.90037 q^{34} +4.76256 q^{35} +1.01628 q^{36} -0.348078 q^{37} -4.07100 q^{38} -6.92673 q^{39} +3.99096 q^{40} -7.25096 q^{41} +4.93264 q^{42} -7.12769 q^{43} -2.22852 q^{45} +8.16997 q^{46} -5.47399 q^{47} +2.79314 q^{48} +6.47074 q^{49} +3.93532 q^{50} +2.76796 q^{51} -3.61931 q^{52} +4.42926 q^{53} -6.33997 q^{54} +11.2883 q^{56} -3.88515 q^{57} +12.0640 q^{58} +0.570768 q^{59} +0.869627 q^{60} -7.17115 q^{61} -1.18669 q^{62} -6.30330 q^{63} +8.75910 q^{64} +7.93647 q^{65} -5.96971 q^{67} +1.44630 q^{68} +7.79699 q^{69} -5.65170 q^{70} -11.8183 q^{71} -5.28208 q^{72} +10.3057 q^{73} +0.413062 q^{74} +3.75566 q^{75} -2.03004 q^{76} +8.21991 q^{78} +7.80045 q^{79} -3.20031 q^{80} -0.898323 q^{81} +8.60468 q^{82} -14.2120 q^{83} +2.45971 q^{84} -3.17146 q^{85} +8.45839 q^{86} +11.5133 q^{87} +0.0753162 q^{89} +2.64457 q^{90} +22.4480 q^{91} +4.07403 q^{92} -1.13252 q^{93} +6.49596 q^{94} +4.45150 q^{95} +3.65179 q^{96} +15.1304 q^{97} -7.67879 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 30 q - 9 q^{2} - q^{3} + 29 q^{4} - q^{5} - 11 q^{6} - 11 q^{7} - 27 q^{8} + 25 q^{9} - 8 q^{10} - 16 q^{12} - 2 q^{13} - 3 q^{14} - 11 q^{15} + 35 q^{16} - 32 q^{17} - 4 q^{18} - 17 q^{19} - 4 q^{20}+ \cdots - 40 q^{98}+O(q^{100}) $ 30 * q - 9 * q^2 - q^3 + 29 * q^4 - q^5 - 11 * q^6 - 11 * q^7 - 27 * q^8 + 25 * q^9 - 8 * q^10 - 16 * q^12 - 2 * q^13 - 3 * q^14 - 11 * q^15 + 35 * q^16 - 32 * q^17 - 4 * q^18 - 17 * q^19 - 4 * q^20 - 30 * q^21 - 11 * q^23 + 7 * q^24 + 27 * q^25 - 21 * q^26 + 5 * q^27 - 17 * q^28 - 41 * q^29 - 12 * q^30 + 30 * q^31 - 68 * q^32 - 21 * q^34 - 33 * q^35 + 22 * q^36 - 7 * q^37 + 87 * q^38 - 36 * q^39 - 20 * q^40 - 79 * q^41 + 42 * q^42 - 10 * q^43 - 36 * q^45 - 8 * q^46 - 67 * q^48 + 31 * q^49 - 47 * q^50 - 26 * q^51 - 8 * q^52 + 4 * q^53 - 70 * q^54 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 33 * q^59 + 64 * q^60 - 19 * q^61 - 9 * q^62 - 38 * q^63 + 39 * q^64 - 93 * q^65 - 13 * q^67 - 32 * q^68 - 6 * q^69 - 40 * q^70 - 13 * q^71 - 32 * q^72 - 27 * q^73 - 82 * q^74 + 64 * q^75 - 65 * q^76 - 16 * q^78 - 29 * q^79 - 18 * q^80 - 42 * q^81 + 81 * q^82 - 32 * q^83 + 16 * q^84 - 12 * q^85 - 12 * q^86 - 48 * q^87 - 8 * q^89 + 51 * q^90 - 27 * q^91 - 49 * q^92 - q^93 - 36 * q^94 - 56 * q^95 + 50 * q^96 + 16 * q^97 - 40 * q^98

Coefficient data

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

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

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

$2$	−1.18669	−0.839120	−0.419560	−	0.907728i	$-0.637816\pi$
$2$	−1.18669	−0.839120	−0.419560	+	0.907728i	$0.637816\pi$
$3$	−1.13252	−0.653860	−0.326930	−	0.945049i	$-0.606014\pi$
$3$	−1.13252	−0.653860	−0.326930	+	0.945049i	$0.606014\pi$
$4$	−0.591757	−0.295878
$5$	1.29761	0.580310	0.290155	−	0.956980i	$-0.406293\pi$
$5$	1.29761	0.580310	0.290155	+	0.956980i	$0.406293\pi$
$6$	1.34395	0.548667
$7$	3.67025	1.38722	0.693612	−	0.720349i	$-0.256018\pi$
$7$	3.67025	1.38722	0.693612	+	0.720349i	$0.256018\pi$
$8$	3.07562	1.08740
$9$	−1.71740	−0.572467
$10$	−1.53987	−0.486949
$11$	0	0
$11$	0	0
$12$	0.670175	0.193463
$13$	6.11622	1.69633	0.848167	−	0.529729i	$-0.177707\pi$
$13$	6.11622	1.69633	0.848167	+	0.529729i	$0.177707\pi$
$14$	−4.35547	−1.16405
$15$	−1.46957	−0.379441
$16$	−2.46631	−0.616578
$17$	−2.44407	−0.592775	−0.296387	−	0.955068i	$-0.595782\pi$
$17$	−2.44407	−0.592775	−0.296387	+	0.955068i	$0.595782\pi$
$18$	2.03803	0.480369
$19$	3.43054	0.787019	0.393510	−	0.919321i	$-0.371261\pi$
$19$	3.43054	0.787019	0.393510	+	0.919321i	$0.371261\pi$
$20$	−0.767870	−0.171701
$21$	−4.15663	−0.907050
$22$	0	0
$23$	−6.88465	−1.43555	−0.717774	−	0.696276i	$-0.754839\pi$
$23$	−6.88465	−1.43555	−0.717774	+	0.696276i	$0.754839\pi$
$24$	−3.48320	−0.711005
$25$	−3.31620	−0.663241
$26$	−7.25808	−1.42343
$27$	5.34254	1.02817
$28$	−2.17190	−0.410450
$29$	−10.1661	−1.88779	−0.943897	−	0.330240i	$-0.892870\pi$
$29$	−10.1661	−1.88779	−0.943897	+	0.330240i	$0.892870\pi$
$30$	1.74393	0.318396
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.22449	−0.570014
$33$	0	0
$34$	2.90037	0.497409
$35$	4.76256	0.805020
$36$	1.01628	0.169381
$37$	−0.348078	−0.0572236	−0.0286118	−	0.999591i	$-0.509109\pi$
$37$	−0.348078	−0.0572236	−0.0286118	+	0.999591i	$0.509109\pi$
$38$	−4.07100	−0.660403
$39$	−6.92673	−1.10916
$40$	3.99096	0.631027
$41$	−7.25096	−1.13241	−0.566205	−	0.824264i	$-0.691589\pi$
$41$	−7.25096	−1.13241	−0.566205	+	0.824264i	$0.691589\pi$
$42$	4.93264	0.761124
$43$	−7.12769	−1.08696	−0.543482	−	0.839421i	$-0.682894\pi$
$43$	−7.12769	−1.08696	−0.543482	+	0.839421i	$0.682894\pi$
$44$	0	0
$45$	−2.22852	−0.332208
$46$	8.16997	1.20460
$47$	−5.47399	−0.798464	−0.399232	−	0.916850i	$-0.630723\pi$
$47$	−5.47399	−0.798464	−0.399232	+	0.916850i	$0.630723\pi$
$48$	2.79314	0.403155
$49$	6.47074	0.924391
$50$	3.93532	0.556538
$51$	2.76796	0.387592
$52$	−3.61931	−0.501908
$53$	4.42926	0.608405	0.304203	−	0.952607i	$-0.401610\pi$
$53$	4.42926	0.608405	0.304203	+	0.952607i	$0.401610\pi$
$54$	−6.33997	−0.862760
$55$	0	0
$56$	11.2883	1.50846
$57$	−3.88515	−0.514600
$58$	12.0640	1.58408
$59$	0.570768	0.0743077	0.0371538	−	0.999310i	$-0.488171\pi$
$59$	0.570768	0.0743077	0.0371538	+	0.999310i	$0.488171\pi$
$60$	0.869627	0.112268
$61$	−7.17115	−0.918172	−0.459086	−	0.888392i	$-0.651823\pi$
$61$	−7.17115	−0.918172	−0.459086	+	0.888392i	$0.651823\pi$
$62$	−1.18669	−0.150710
$63$	−6.30330	−0.794141
$64$	8.75910	1.09489
$65$	7.93647	0.984398
$66$	0	0
$67$	−5.96971	−0.729316	−0.364658	−	0.931141i	$-0.618814\pi$
$67$	−5.96971	−0.729316	−0.364658	+	0.931141i	$0.618814\pi$
$68$	1.44630	0.175389
$69$	7.79699	0.938647
$70$	−5.65170	−0.675508
$71$	−11.8183	−1.40257	−0.701287	−	0.712879i	$-0.747391\pi$
$71$	−11.8183	−1.40257	−0.701287	+	0.712879i	$0.747391\pi$
$72$	−5.28208	−0.622499
$73$	10.3057	1.20619	0.603097	−	0.797668i	$-0.293933\pi$
$73$	10.3057	1.20619	0.603097	+	0.797668i	$0.293933\pi$
$74$	0.413062	0.0480175
$75$	3.75566	0.433666
$76$	−2.03004	−0.232862
$77$	0	0
$78$	8.21991	0.930721
$79$	7.80045	0.877620	0.438810	−	0.898580i	$-0.355400\pi$
$79$	7.80045	0.877620	0.438810	+	0.898580i	$0.355400\pi$
$80$	−3.20031	−0.357806
$81$	−0.898323	−0.0998137
$82$	8.60468	0.950227
$83$	−14.2120	−1.55997	−0.779983	−	0.625800i	$-0.784773\pi$
$83$	−14.2120	−1.55997	−0.779983	+	0.625800i	$0.784773\pi$
$84$	2.45971	0.268376
$85$	−3.17146	−0.343993
$86$	8.45839	0.912092
$87$	11.5133	1.23435
$88$	0	0
$89$	0.0753162	0.00798350	0.00399175	−	0.999992i	$-0.498729\pi$
$89$	0.0753162	0.00798350	0.00399175	+	0.999992i	$0.498729\pi$
$90$	2.64457	0.278763
$91$	22.4480	2.35319
$92$	4.07403	0.424748
$93$	−1.13252	−0.117437
$94$	6.49596	0.670007
$95$	4.45150	0.456715
$96$	3.65179	0.372710
$97$	15.1304	1.53626	0.768132	−	0.640291i	$-0.221186\pi$
$97$	15.1304	1.53626	0.768132	+	0.640291i	$0.221186\pi$
$98$	−7.67879	−0.775675
$99$	0	0
$100$	1.96239	0.196239
$101$	5.75155	0.572301	0.286150	−	0.958185i	$-0.407624\pi$
$101$	5.75155	0.572301	0.286150	+	0.958185i	$0.407624\pi$
$102$	−3.28472	−0.325236
$103$	−9.89872	−0.975350	−0.487675	−	0.873025i	$-0.662155\pi$
$103$	−9.89872	−0.975350	−0.487675	+	0.873025i	$0.662155\pi$
$104$	18.8112	1.84459
$105$	−5.39369	−0.526370
$106$	−5.25618	−0.510525
$107$	−12.5814	−1.21629	−0.608147	−	0.793824i	$-0.708087\pi$
$107$	−12.5814	−1.21629	−0.608147	+	0.793824i	$0.708087\pi$
$108$	−3.16149	−0.304214
$109$	−2.59825	−0.248867	−0.124433	−	0.992228i	$-0.539711\pi$
$109$	−2.59825	−0.248867	−0.124433	+	0.992228i	$0.539711\pi$
$110$	0	0
$111$	0.394204	0.0374162
$112$	−9.05198	−0.855332
$113$	−14.8808	−1.39987	−0.699936	−	0.714206i	$-0.746788\pi$
$113$	−14.8808	−1.39987	−0.699936	+	0.714206i	$0.746788\pi$
$114$	4.61048	0.431811
$115$	−8.93360	−0.833062
$116$	6.01585	0.558557
$117$	−10.5040	−0.971096
$118$	−0.677327	−0.0623530
$119$	−8.97036	−0.822312
$120$	−4.51984	−0.412603
$121$	0	0
$122$	8.50997	0.770456
$123$	8.21185	0.740437
$124$	−0.591757	−0.0531413
$125$	−10.7912	−0.965195
$126$	7.48009	0.666379
$127$	−1.28268	−0.113819	−0.0569097	−	0.998379i	$-0.518125\pi$
$127$	−1.28268	−0.113819	−0.0569097	+	0.998379i	$0.518125\pi$
$128$	−3.94540	−0.348728
$129$	8.07224	0.710721
$130$	−9.41817	−0.826028
$131$	3.00127	0.262222	0.131111	−	0.991368i	$-0.458146\pi$
$131$	3.00127	0.262222	0.131111	+	0.991368i	$0.458146\pi$
$132$	0	0
$133$	12.5909	1.09177
$134$	7.08422	0.611984
$135$	6.93255	0.596659
$136$	−7.51705	−0.644582
$137$	3.58114	0.305958	0.152979	−	0.988229i	$-0.451113\pi$
$137$	3.58114	0.305958	0.152979	+	0.988229i	$0.451113\pi$
$138$	−9.25264	−0.787637
$139$	−21.3030	−1.80690	−0.903449	−	0.428695i	$-0.858973\pi$
$139$	−21.3030	−1.80690	−0.903449	+	0.428695i	$0.858973\pi$
$140$	−2.81828	−0.238188
$141$	6.19940	0.522084
$142$	14.0247	1.17693
$143$	0	0
$144$	4.23565	0.352971
$145$	−13.1916	−1.09550
$146$	−12.2297	−1.01214
$147$	−7.32823	−0.604422
$148$	0.205977	0.0169312
$149$	11.0757	0.907353	0.453676	−	0.891167i	$-0.350112\pi$
$149$	11.0757	0.907353	0.453676	+	0.891167i	$0.350112\pi$
$150$	−4.45682	−0.363898
$151$	21.2495	1.72926	0.864631	−	0.502407i	$-0.167552\pi$
$151$	21.2495	1.72926	0.864631	+	0.502407i	$0.167552\pi$
$152$	10.5510	0.855802
$153$	4.19746	0.339344
$154$	0	0
$155$	1.29761	0.104227
$156$	4.09894	0.328178
$157$	0.627936	0.0501147	0.0250574	−	0.999686i	$-0.492023\pi$
$157$	0.627936	0.0501147	0.0250574	+	0.999686i	$0.492023\pi$
$158$	−9.25676	−0.736428
$159$	−5.01622	−0.397812
$160$	−4.18413	−0.330785
$161$	−25.2684	−1.99143
$162$	1.06604	0.0837556
$163$	−4.57143	−0.358062	−0.179031	−	0.983843i	$-0.557296\pi$
$163$	−4.57143	−0.358062	−0.179031	+	0.983843i	$0.557296\pi$
$164$	4.29081	0.335056
$165$	0	0
$166$	16.8653	1.30900
$167$	16.0662	1.24324	0.621619	−	0.783320i	$-0.286475\pi$
$167$	16.0662	1.24324	0.621619	+	0.783320i	$0.286475\pi$
$168$	−12.7842	−0.986324
$169$	24.4081	1.87755
$170$	3.76355	0.288651
$171$	−5.89161	−0.450543
$172$	4.21786	0.321609
$173$	9.63625	0.732630	0.366315	−	0.930491i	$-0.380619\pi$
$173$	9.63625	0.732630	0.366315	+	0.930491i	$0.380619\pi$
$174$	−13.6627	−1.03577
$175$	−12.1713	−0.920064
$176$	0	0
$177$	−0.646405	−0.0485868
$178$	−0.0893773	−0.00669911
$179$	13.9372	1.04172	0.520858	−	0.853643i	$-0.325612\pi$
$179$	13.9372	1.04172	0.520858	+	0.853643i	$0.325612\pi$
$180$	1.31874	0.0982932
$181$	−1.02584	−0.0762499	−0.0381250	−	0.999273i	$-0.512138\pi$
$181$	−1.02584	−0.0762499	−0.0381250	+	0.999273i	$0.512138\pi$
$182$	−26.6390	−1.97461
$183$	8.12146	0.600356
$184$	−21.1746	−1.56101
$185$	−0.451670	−0.0332074
$186$	1.34395	0.0985434
$187$	0	0
$188$	3.23927	0.236248
$189$	19.6085	1.42631
$190$	−5.28258	−0.383238
$191$	14.3642	1.03936	0.519680	−	0.854361i	$-0.326051\pi$
$191$	14.3642	1.03936	0.519680	+	0.854361i	$0.326051\pi$
$192$	−9.91985	−0.715903
$193$	−16.1079	−1.15947	−0.579736	−	0.814804i	$-0.696844\pi$
$193$	−16.1079	−1.15947	−0.579736	+	0.814804i	$0.696844\pi$
$194$	−17.9552	−1.28911
$195$	−8.98820	−0.643659
$196$	−3.82910	−0.273507
$197$	−14.7182	−1.04863	−0.524314	−	0.851525i	$-0.675678\pi$
$197$	−14.7182	−1.04863	−0.524314	+	0.851525i	$0.675678\pi$
$198$	0	0
$199$	7.58154	0.537441	0.268720	−	0.963218i	$-0.413399\pi$
$199$	7.58154	0.537441	0.268720	+	0.963218i	$0.413399\pi$
$200$	−10.1994	−0.721206
$201$	6.76081	0.476871
$202$	−6.82533	−0.480229
$203$	−37.3121	−2.61879
$204$	−1.63796	−0.114680
$205$	−9.40893	−0.657148
$206$	11.7468	0.818435
$207$	11.8237	0.821804
$208$	−15.0845	−1.04592
$209$	0	0
$210$	6.40066	0.441687
$211$	−18.7948	−1.29388	−0.646942	−	0.762539i	$-0.723953\pi$
$211$	−18.7948	−1.29388	−0.646942	+	0.762539i	$0.723953\pi$
$212$	−2.62104	−0.180014
$213$	13.3844	0.917087
$214$	14.9303	1.02062
$215$	−9.24898	−0.630775
$216$	16.4317	1.11803
$217$	3.67025	0.249153
$218$	3.08332	0.208829
$219$	−11.6714	−0.788681
$220$	0	0
$221$	−14.9485	−1.00554
$222$	−0.467800	−0.0313967
$223$	−18.5306	−1.24090	−0.620449	−	0.784246i	$-0.713050\pi$
$223$	−18.5306	−1.24090	−0.620449	+	0.784246i	$0.713050\pi$
$224$	−11.8347	−0.790738
$225$	5.69526	0.379684
$226$	17.6590	1.17466
$227$	15.5681	1.03329	0.516646	−	0.856199i	$-0.327180\pi$
$227$	15.5681	1.03329	0.516646	+	0.856199i	$0.327180\pi$
$228$	2.29906	0.152259
$229$	3.53114	0.233344	0.116672	−	0.993170i	$-0.462777\pi$
$229$	3.53114	0.233344	0.116672	+	0.993170i	$0.462777\pi$
$230$	10.6014	0.699039
$231$	0	0
$232$	−31.2670	−2.05278
$233$	−5.69038	−0.372789	−0.186395	−	0.982475i	$-0.559680\pi$
$233$	−5.69038	−0.372789	−0.186395	+	0.982475i	$0.559680\pi$
$234$	12.4650	0.814865
$235$	−7.10312	−0.463356
$236$	−0.337756	−0.0219860
$237$	−8.83416	−0.573840
$238$	10.6451	0.690018
$239$	−8.94256	−0.578446	−0.289223	−	0.957262i	$-0.593397\pi$
$239$	−8.94256	−0.578446	−0.289223	+	0.957262i	$0.593397\pi$
$240$	3.62441	0.233955
$241$	−22.0465	−1.42014	−0.710070	−	0.704131i	$-0.751337\pi$
$241$	−22.0465	−1.42014	−0.710070	+	0.704131i	$0.751337\pi$
$242$	0	0
$243$	−15.0103	−0.962909
$244$	4.24358	0.271667
$245$	8.39651	0.536433
$246$	−9.74495	−0.621316
$247$	20.9819	1.33505
$248$	3.07562	0.195302
$249$	16.0953	1.02000
$250$	12.8059	0.809914
$251$	5.12195	0.323295	0.161647	−	0.986849i	$-0.448319\pi$
$251$	5.12195	0.323295	0.161647	+	0.986849i	$0.448319\pi$
$252$	3.73002	0.234969
$253$	0	0
$254$	1.52215	0.0955080
$255$	3.59173	0.224923
$256$	−12.8362	−0.802264
$257$	−18.4105	−1.14842	−0.574209	−	0.818709i	$-0.694690\pi$
$257$	−18.4105	−1.14842	−0.574209	+	0.818709i	$0.694690\pi$
$258$	−9.57929	−0.596380
$259$	−1.27753	−0.0793820
$260$	−4.69646	−0.291262
$261$	17.4593	1.08070
$262$	−3.56159	−0.220036
$263$	14.1228	0.870852	0.435426	−	0.900225i	$-0.356598\pi$
$263$	14.1228	0.870852	0.435426	+	0.900225i	$0.356598\pi$
$264$	0	0
$265$	5.74746	0.353064
$266$	−14.9416	−0.916127
$267$	−0.0852970	−0.00522009
$268$	3.53262	0.215789
$269$	9.00512	0.549052	0.274526	−	0.961580i	$-0.411479\pi$
$269$	9.00512	0.549052	0.274526	+	0.961580i	$0.411479\pi$
$270$	−8.22682	−0.500668
$271$	0.706167	0.0428966	0.0214483	−	0.999770i	$-0.493172\pi$
$271$	0.706167	0.0428966	0.0214483	+	0.999770i	$0.493172\pi$
$272$	6.02784	0.365492
$273$	−25.4228	−1.53866
$274$	−4.24972	−0.256735
$275$	0	0
$276$	−4.61392	−0.277725
$277$	−18.1276	−1.08918	−0.544591	−	0.838702i	$-0.683315\pi$
$277$	−18.1276	−1.08918	−0.544591	+	0.838702i	$0.683315\pi$
$278$	25.2802	1.51620
$279$	−1.71740	−0.102818
$280$	14.6478	0.875376
$281$	18.8161	1.12247	0.561237	−	0.827655i	$-0.310326\pi$
$281$	18.8161	1.12247	0.561237	+	0.827655i	$0.310326\pi$
$282$	−7.35679	−0.438091
$283$	7.36357	0.437719	0.218859	−	0.975756i	$-0.429766\pi$
$283$	7.36357	0.437719	0.218859	+	0.975756i	$0.429766\pi$
$284$	6.99356	0.414991
$285$	−5.04141	−0.298627
$286$	0	0
$287$	−26.6129	−1.57091
$288$	5.53774	0.326315
$289$	−11.0265	−0.648618
$290$	15.6544	0.919260
$291$	−17.1355	−1.00450
$292$	−6.09848	−0.356886
$293$	−27.1689	−1.58722	−0.793611	−	0.608425i	$-0.791802\pi$
$293$	−27.1689	−1.58722	−0.793611	+	0.608425i	$0.791802\pi$
$294$	8.69637	0.507183
$295$	0.740635	0.0431215
$296$	−1.07056	−0.0622248
$297$	0	0
$298$	−13.1434	−0.761377
$299$	−42.1080	−2.43517
$300$	−2.22244	−0.128313
$301$	−26.1604	−1.50786
$302$	−25.2167	−1.45106
$303$	−6.51374	−0.374204
$304$	−8.46077	−0.485258
$305$	−9.30537	−0.532824
$306$	−4.98110	−0.284750
$307$	−30.8870	−1.76282	−0.881408	−	0.472356i	$-0.843404\pi$
$307$	−30.8870	−1.76282	−0.881408	+	0.472356i	$0.843404\pi$
$308$	0	0
$309$	11.2105	0.637742
$310$	−1.53987	−0.0874586
$311$	7.95778	0.451244	0.225622	−	0.974215i	$-0.427559\pi$
$311$	7.95778	0.451244	0.225622	+	0.974215i	$0.427559\pi$
$312$	−21.3040	−1.20610
$313$	−23.8830	−1.34995	−0.674975	−	0.737841i	$-0.735846\pi$
$313$	−23.8830	−1.34995	−0.674975	+	0.737841i	$0.735846\pi$
$314$	−0.745168	−0.0420523
$315$	−8.17923	−0.460847
$316$	−4.61597	−0.259669
$317$	0.507853	0.0285239	0.0142619	−	0.999898i	$-0.495460\pi$
$317$	0.507853	0.0285239	0.0142619	+	0.999898i	$0.495460\pi$
$318$	5.95272	0.333812
$319$	0	0
$320$	11.3659	0.635374
$321$	14.2487	0.795286
$322$	29.9858	1.67105
$323$	−8.38448	−0.466525
$324$	0.531589	0.0295327
$325$	−20.2826	−1.12508
$326$	5.42489	0.300457
$327$	2.94256	0.162724
$328$	−22.3012	−1.23138
$329$	−20.0909	−1.10765
$330$	0	0
$331$	21.8183	1.19924	0.599621	−	0.800284i	$-0.295318\pi$
$331$	21.8183	1.19924	0.599621	+	0.800284i	$0.295318\pi$
$332$	8.41003	0.461560
$333$	0.597789	0.0327587
$334$	−19.0656	−1.04322
$335$	−7.74637	−0.423229
$336$	10.2515	0.559267
$337$	12.6619	0.689740	0.344870	−	0.938651i	$-0.387923\pi$
$337$	12.6619	0.689740	0.344870	+	0.938651i	$0.387923\pi$
$338$	−28.9650	−1.57549
$339$	16.8528	0.915319
$340$	1.87673	0.101780
$341$	0	0
$342$	6.99154	0.378059
$343$	−1.94252	−0.104886
$344$	−21.9221	−1.18196
$345$	10.1175	0.544706
$346$	−11.4353	−0.614765
$347$	21.9551	1.17861	0.589306	−	0.807910i	$-0.299401\pi$
$347$	21.9551	1.17861	0.589306	+	0.807910i	$0.299401\pi$
$348$	−6.81306	−0.365218
$349$	−13.3765	−0.716027	−0.358014	−	0.933716i	$-0.616546\pi$
$349$	−13.3765	−0.716027	−0.358014	+	0.933716i	$0.616546\pi$
$350$	14.4436	0.772044
$351$	32.6762	1.74412
$352$	0	0
$353$	2.81075	0.149601	0.0748005	−	0.997199i	$-0.476168\pi$
$353$	2.81075	0.149601	0.0748005	+	0.997199i	$0.476168\pi$
$354$	0.767085	0.0407701
$355$	−15.3356	−0.813927
$356$	−0.0445689	−0.00236214
$357$	10.1591	0.537677
$358$	−16.5392	−0.874125
$359$	−13.9707	−0.737345	−0.368673	−	0.929559i	$-0.620188\pi$
$359$	−13.9707	−0.737345	−0.368673	+	0.929559i	$0.620188\pi$
$360$	−6.85409	−0.361242
$361$	−7.23142	−0.380601
$362$	1.21736	0.0639828
$363$	0	0
$364$	−13.2838	−0.696259
$365$	13.3728	0.699965
$366$	−9.63769	−0.503770
$367$	26.1045	1.36264	0.681322	−	0.731984i	$-0.261406\pi$
$367$	26.1045	1.36264	0.681322	+	0.731984i	$0.261406\pi$
$368$	16.9797	0.885127
$369$	12.4528	0.648268
$370$	0.535994	0.0278650
$371$	16.2565	0.843995
$372$	0.670175	0.0347470
$373$	−23.8199	−1.23335	−0.616675	−	0.787218i	$-0.711521\pi$
$373$	−23.8199	−1.23335	−0.616675	+	0.787218i	$0.711521\pi$
$374$	0	0
$375$	12.2212	0.631102
$376$	−16.8359	−0.868247
$377$	−62.1780	−3.20233
$378$	−23.2693	−1.19684
$379$	−20.0732	−1.03109	−0.515546	−	0.856862i	$-0.672411\pi$
$379$	−20.0732	−1.03109	−0.515546	+	0.856862i	$0.672411\pi$
$380$	−2.63421	−0.135132
$381$	1.45266	0.0744219
$382$	−17.0460	−0.872147
$383$	−12.1606	−0.621379	−0.310690	−	0.950511i	$-0.600560\pi$
$383$	−12.1606	−0.621379	−0.310690	+	0.950511i	$0.600560\pi$
$384$	4.46824	0.228019
$385$	0	0
$386$	19.1152	0.972936
$387$	12.2411	0.622251
$388$	−8.95354	−0.454547
$389$	19.6085	0.994188	0.497094	−	0.867697i	$-0.334400\pi$
$389$	19.6085	0.994188	0.497094	+	0.867697i	$0.334400\pi$
$390$	10.6662	0.540107
$391$	16.8266	0.850957
$392$	19.9016	1.00518
$393$	−3.39900	−0.171457
$394$	17.4660	0.879924
$395$	10.1220	0.509291
$396$	0	0
$397$	−5.68717	−0.285431	−0.142715	−	0.989764i	$-0.545583\pi$
$397$	−5.68717	−0.285431	−0.142715	+	0.989764i	$0.545583\pi$
$398$	−8.99696	−0.450977
$399$	−14.2595	−0.713866
$400$	8.17879	0.408939
$401$	−24.4360	−1.22027	−0.610137	−	0.792296i	$-0.708886\pi$
$401$	−24.4360	−1.22027	−0.610137	+	0.792296i	$0.708886\pi$
$402$	−8.02301	−0.400152
$403$	6.11622	0.304670
$404$	−3.40352	−0.169331
$405$	−1.16567	−0.0579228
$406$	44.2780	2.19748
$407$	0	0
$408$	8.51319	0.421466
$409$	−12.5151	−0.618832	−0.309416	−	0.950927i	$-0.600134\pi$
$409$	−12.5151	−0.618832	−0.309416	+	0.950927i	$0.600134\pi$
$410$	11.1655	0.551426
$411$	−4.05571	−0.200054
$412$	5.85763	0.288585
$413$	2.09486	0.103081
$414$	−14.0311	−0.689592
$415$	−18.4416	−0.905264
$416$	−19.7217	−0.966934
$417$	24.1261	1.18146
$418$	0	0
$419$	−5.20715	−0.254386	−0.127193	−	0.991878i	$-0.540597\pi$
$419$	−5.20715	−0.254386	−0.127193	+	0.991878i	$0.540597\pi$
$420$	3.19175	0.155741
$421$	−7.82052	−0.381149	−0.190574	−	0.981673i	$-0.561035\pi$
$421$	−7.82052	−0.381149	−0.190574	+	0.981673i	$0.561035\pi$
$422$	22.3036	1.08572
$423$	9.40105	0.457095
$424$	13.6227	0.661578
$425$	8.10505	0.393152
$426$	−15.8832	−0.769545
$427$	−26.3199	−1.27371
$428$	7.44515	0.359875
$429$	0	0
$430$	10.9757	0.529296
$431$	−3.13037	−0.150785	−0.0753924	−	0.997154i	$-0.524021\pi$
$431$	−3.13037	−0.150785	−0.0753924	+	0.997154i	$0.524021\pi$
$432$	−13.1764	−0.633949
$433$	18.7542	0.901268	0.450634	−	0.892709i	$-0.351198\pi$
$433$	18.7542	0.901268	0.450634	+	0.892709i	$0.351198\pi$
$434$	−4.35547	−0.209069
$435$	14.9398	0.716307
$436$	1.53753	0.0736343
$437$	−23.6180	−1.12980
$438$	13.8504	0.661798
$439$	29.6733	1.41623	0.708115	−	0.706097i	$-0.249546\pi$
$439$	29.6733	1.41623	0.708115	+	0.706097i	$0.249546\pi$
$440$	0	0
$441$	−11.1129	−0.529184
$442$	17.7393	0.843771
$443$	8.56785	0.407071	0.203536	−	0.979068i	$-0.434757\pi$
$443$	8.56785	0.407071	0.203536	+	0.979068i	$0.434757\pi$
$444$	−0.233273	−0.0110707
$445$	0.0977312	0.00463290
$446$	21.9901	1.04126
$447$	−12.5434	−0.593281
$448$	32.1481	1.51886
$449$	27.6683	1.30575	0.652874	−	0.757466i	$-0.273563\pi$
$449$	27.6683	1.30575	0.652874	+	0.757466i	$0.273563\pi$
$450$	−6.75853	−0.318600
$451$	0	0
$452$	8.80583	0.414191
$453$	−24.0655	−1.13070
$454$	−18.4746	−0.867055
$455$	29.1289	1.36558
$456$	−11.9492	−0.559575
$457$	11.7843	0.551247	0.275624	−	0.961266i	$-0.411116\pi$
$457$	11.7843	0.551247	0.275624	+	0.961266i	$0.411116\pi$
$458$	−4.19038	−0.195804
$459$	−13.0576	−0.609475
$460$	5.28652	0.246485
$461$	−12.2033	−0.568364	−0.284182	−	0.958770i	$-0.591722\pi$
$461$	−12.2033	−0.568364	−0.284182	+	0.958770i	$0.591722\pi$
$462$	0	0
$463$	2.42916	0.112893	0.0564463	−	0.998406i	$-0.482023\pi$
$463$	2.42916	0.112893	0.0564463	+	0.998406i	$0.482023\pi$
$464$	25.0727	1.16397
$465$	−1.46957	−0.0681496
$466$	6.75274	0.312815
$467$	28.1598	1.30308	0.651541	−	0.758613i	$-0.274123\pi$
$467$	28.1598	1.30308	0.651541	+	0.758613i	$0.274123\pi$
$468$	6.21581	0.287326
$469$	−21.9103	−1.01173
$470$	8.42923	0.388811
$471$	−0.711149	−0.0327680
$472$	1.75547	0.0808019
$473$	0	0
$474$	10.4834	0.481521
$475$	−11.3764	−0.521983
$476$	5.30827	0.243304
$477$	−7.60682	−0.348292
$478$	10.6121	0.485386
$479$	−1.58402	−0.0723757	−0.0361879	−	0.999345i	$-0.511521\pi$
$479$	−1.58402	−0.0723757	−0.0361879	+	0.999345i	$0.511521\pi$
$480$	4.73861	0.216287
$481$	−2.12892	−0.0970703
$482$	26.1625	1.19167
$483$	28.6169	1.30211
$484$	0	0
$485$	19.6334	0.891509
$486$	17.8126	0.807996
$487$	19.7180	0.893509	0.446754	−	0.894657i	$-0.352580\pi$
$487$	19.7180	0.893509	0.446754	+	0.894657i	$0.352580\pi$
$488$	−22.0558	−0.998418
$489$	5.17722	0.234122
$490$	−9.96409	−0.450132
$491$	19.0122	0.858008	0.429004	−	0.903303i	$-0.358865\pi$
$491$	19.0122	0.858008	0.429004	+	0.903303i	$0.358865\pi$
$492$	−4.85942	−0.219079
$493$	24.8466	1.11904
$494$	−24.8991	−1.12026
$495$	0	0
$496$	−2.46631	−0.110741
$497$	−43.3761	−1.94568
$498$	−19.1002	−0.855902
$499$	33.6808	1.50776	0.753880	−	0.657012i	$-0.228180\pi$
$499$	33.6808	1.50776	0.753880	+	0.657012i	$0.228180\pi$
$500$	6.38577	0.285580
$501$	−18.1952	−0.812903
$502$	−6.07819	−0.271283
$503$	9.78776	0.436415	0.218207	−	0.975902i	$-0.429979\pi$
$503$	9.78776	0.436415	0.218207	+	0.975902i	$0.429979\pi$
$504$	−19.3866	−0.863546
$505$	7.46328	0.332112
$506$	0	0
$507$	−27.6426	−1.22765
$508$	0.759034	0.0336767
$509$	−29.4262	−1.30429	−0.652146	−	0.758093i	$-0.726131\pi$
$509$	−29.4262	−1.30429	−0.652146	+	0.758093i	$0.726131\pi$
$510$	−4.26229	−0.188737
$511$	37.8246	1.67326
$512$	23.1235	1.02192
$513$	18.3278	0.809192
$514$	21.8477	0.963660
$515$	−12.8447	−0.566005
$516$	−4.77680	−0.210287
$517$	0	0
$518$	1.51604	0.0666110
$519$	−10.9132	−0.479038
$520$	24.4096	1.07043
$521$	18.4233	0.807141	0.403570	−	0.914949i	$-0.367769\pi$
$521$	18.4233	0.807141	0.403570	+	0.914949i	$0.367769\pi$
$522$	−20.7188	−0.906837
$523$	2.63526	0.115232	0.0576160	−	0.998339i	$-0.481650\pi$
$523$	2.63526	0.115232	0.0576160	+	0.998339i	$0.481650\pi$
$524$	−1.77602	−0.0775859
$525$	13.7842	0.601593
$526$	−16.7595	−0.730749
$527$	−2.44407	−0.106466
$528$	0	0
$529$	24.3984	1.06080
$530$	−6.82048	−0.296263
$531$	−0.980238	−0.0425387
$532$	−7.45077	−0.323032
$533$	−44.3485	−1.92094
$534$	0.101221	0.00438028
$535$	−16.3258	−0.705827
$536$	−18.3606	−0.793056
$537$	−15.7842	−0.681137
$538$	−10.6863	−0.460720
$539$	0	0
$540$	−4.10238	−0.176538
$541$	−3.48839	−0.149978	−0.0749888	−	0.997184i	$-0.523892\pi$
$541$	−3.48839	−0.149978	−0.0749888	+	0.997184i	$0.523892\pi$
$542$	−0.838005	−0.0359954
$543$	1.16178	0.0498567
$544$	7.88089	0.337890
$545$	−3.37151	−0.144420
$546$	30.1691	1.29112
$547$	28.6797	1.22626	0.613128	−	0.789984i	$-0.289911\pi$
$547$	28.6797	1.22626	0.613128	+	0.789984i	$0.289911\pi$
$548$	−2.11917	−0.0905263
$549$	12.3158	0.525624
$550$	0	0
$551$	−34.8751	−1.48573
$552$	23.9806	1.02068
$553$	28.6296	1.21746
$554$	21.5119	0.913953
$555$	0.511524	0.0217130
$556$	12.6062	0.534622
$557$	34.9732	1.48186	0.740931	−	0.671581i	$-0.234384\pi$
$557$	34.9732	1.48186	0.740931	+	0.671581i	$0.234384\pi$
$558$	2.03803	0.0862767
$559$	−43.5945	−1.84385
$560$	−11.7460	−0.496357
$561$	0	0
$562$	−22.3289	−0.941890
$563$	3.98675	0.168021	0.0840107	−	0.996465i	$-0.473227\pi$
$563$	3.98675	0.168021	0.0840107	+	0.996465i	$0.473227\pi$
$564$	−3.66853	−0.154473
$565$	−19.3095	−0.812359
$566$	−8.73830	−0.367298
$567$	−3.29707	−0.138464
$568$	−36.3486	−1.52515
$569$	−40.8723	−1.71346	−0.856728	−	0.515769i	$-0.827506\pi$
$569$	−40.8723	−1.71346	−0.856728	+	0.515769i	$0.827506\pi$
$570$	5.98261	0.250584
$571$	−1.26770	−0.0530516	−0.0265258	−	0.999648i	$-0.508444\pi$
$571$	−1.26770	−0.0530516	−0.0265258	+	0.999648i	$0.508444\pi$
$572$	0	0
$573$	−16.2678	−0.679596
$574$	31.5813	1.31818
$575$	22.8309	0.952114
$576$	−15.0429	−0.626788
$577$	30.6866	1.27750	0.638749	−	0.769415i	$-0.279452\pi$
$577$	30.6866	1.27750	0.638749	+	0.769415i	$0.279452\pi$
$578$	13.0851	0.544268
$579$	18.2425	0.758133
$580$	7.80623	0.324136
$581$	−52.1615	−2.16402
$582$	20.3346	0.842897
$583$	0	0
$584$	31.6965	1.31161
$585$	−13.6301	−0.563536
$586$	32.2411	1.33187
$587$	6.46235	0.266730	0.133365	−	0.991067i	$-0.457422\pi$
$587$	6.46235	0.266730	0.133365	+	0.991067i	$0.457422\pi$
$588$	4.33653	0.178835
$589$	3.43054	0.141353
$590$	−0.878908	−0.0361841
$591$	16.6686	0.685656
$592$	0.858468	0.0352828
$593$	−26.3139	−1.08058	−0.540290	−	0.841479i	$-0.681686\pi$
$593$	−26.3139	−1.08058	−0.540290	+	0.841479i	$0.681686\pi$
$594$	0	0
$595$	−11.6400	−0.477195
$596$	−6.55409	−0.268466
$597$	−8.58623	−0.351411
$598$	49.9693	2.04340
$599$	−22.6761	−0.926519	−0.463260	−	0.886223i	$-0.653320\pi$
$599$	−22.6761	−0.926519	−0.463260	+	0.886223i	$0.653320\pi$
$600$	11.5510	0.471568
$601$	−32.8414	−1.33963	−0.669815	−	0.742528i	$-0.733627\pi$
$601$	−32.8414	−1.33963	−0.669815	+	0.742528i	$0.733627\pi$
$602$	31.0444	1.26528
$603$	10.2524	0.417510
$604$	−12.5746	−0.511651
$605$	0	0
$606$	7.72982	0.314002
$607$	33.1530	1.34564	0.672819	−	0.739807i	$-0.265083\pi$
$607$	33.1530	1.34564	0.672819	+	0.739807i	$0.265083\pi$
$608$	−11.0617	−0.448612
$609$	42.2566	1.71232
$610$	11.0426	0.447103
$611$	−33.4801	−1.35446
$612$	−2.48387	−0.100405
$613$	−26.4390	−1.06786	−0.533931	−	0.845528i	$-0.679286\pi$
$613$	−26.4390	−1.06786	−0.533931	+	0.845528i	$0.679286\pi$
$614$	36.6535	1.47921
$615$	10.6558	0.429683
$616$	0	0
$617$	−3.96590	−0.159661	−0.0798305	−	0.996808i	$-0.525438\pi$
$617$	−3.96590	−0.159661	−0.0798305	+	0.996808i	$0.525438\pi$
$618$	−13.3034	−0.535142
$619$	27.9867	1.12488	0.562439	−	0.826838i	$-0.309863\pi$
$619$	27.9867	1.12488	0.562439	+	0.826838i	$0.309863\pi$
$620$	−0.767870	−0.0308384
$621$	−36.7815	−1.47599
$622$	−9.44345	−0.378648
$623$	0.276429	0.0110749
$624$	17.0835	0.683886
$625$	2.57823	0.103129
$626$	28.3419	1.13277
$627$	0	0
$628$	−0.371585	−0.0148279
$629$	0.850728	0.0339207
$630$	9.70625	0.386706
$631$	−6.01910	−0.239617	−0.119808	−	0.992797i	$-0.538228\pi$
$631$	−6.01910	−0.239617	−0.119808	+	0.992797i	$0.538228\pi$
$632$	23.9913	0.954321
$633$	21.2854	0.846019
$634$	−0.602666	−0.0239349
$635$	−1.66442	−0.0660505
$636$	2.96838	0.117704
$637$	39.5764	1.56808
$638$	0	0
$639$	20.2968	0.802928
$640$	−5.11960	−0.202370
$641$	−15.1573	−0.598678	−0.299339	−	0.954147i	$-0.596766\pi$
$641$	−15.1573	−0.598678	−0.299339	+	0.954147i	$0.596766\pi$
$642$	−16.9089	−0.667340
$643$	8.46032	0.333642	0.166821	−	0.985987i	$-0.446650\pi$
$643$	8.46032	0.333642	0.166821	+	0.985987i	$0.446650\pi$
$644$	14.9527	0.589220
$645$	10.4746	0.412438
$646$	9.94982	0.391470
$647$	5.71391	0.224637	0.112319	−	0.993672i	$-0.464172\pi$
$647$	5.71391	0.224637	0.112319	+	0.993672i	$0.464172\pi$
$648$	−2.76290	−0.108537
$649$	0	0
$650$	24.0693	0.944075
$651$	−4.15663	−0.162911
$652$	2.70517	0.105943
$653$	−10.2965	−0.402933	−0.201466	−	0.979495i	$-0.564571\pi$
$653$	−10.2965	−0.402933	−0.201466	+	0.979495i	$0.564571\pi$
$654$	−3.49192	−0.136545
$655$	3.89449	0.152170
$656$	17.8831	0.698219
$657$	−17.6991	−0.690506
$658$	23.8418	0.929450
$659$	−40.3482	−1.57174	−0.785872	−	0.618390i	$-0.787785\pi$
$659$	−40.3482	−1.57174	−0.785872	+	0.618390i	$0.787785\pi$
$660$	0	0
$661$	14.1287	0.549544	0.274772	−	0.961509i	$-0.411398\pi$
$661$	14.1287	0.549544	0.274772	+	0.961509i	$0.411398\pi$
$662$	−25.8916	−1.00631
$663$	16.9294	0.657485
$664$	−43.7107	−1.69630
$665$	16.3381	0.633566
$666$	−0.709393	−0.0274884
$667$	69.9899	2.71002
$668$	−9.50726	−0.367847
$669$	20.9862	0.811374
$670$	9.19257	0.355140
$671$	0	0
$672$	13.4030	0.517032
$673$	22.9673	0.885326	0.442663	−	0.896688i	$-0.354034\pi$
$673$	22.9673	0.885326	0.442663	+	0.896688i	$0.354034\pi$
$674$	−15.0258	−0.578774
$675$	−17.7170	−0.681926
$676$	−14.4437	−0.555525
$677$	−32.3905	−1.24487	−0.622434	−	0.782672i	$-0.713856\pi$
$677$	−32.3905	−1.24487	−0.622434	+	0.782672i	$0.713856\pi$
$678$	−19.9991	−0.768062
$679$	55.5325	2.13114
$680$	−9.75421	−0.374057
$681$	−17.6312	−0.675628
$682$	0	0
$683$	−28.8520	−1.10399	−0.551996	−	0.833847i	$-0.686134\pi$
$683$	−28.8520	−1.10399	−0.551996	+	0.833847i	$0.686134\pi$
$684$	3.48640	0.133306
$685$	4.64694	0.177550
$686$	2.30517	0.0880119
$687$	−3.99908	−0.152574
$688$	17.5791	0.670197
$689$	27.0903	1.03206
$690$	−12.0063	−0.457073
$691$	−8.64368	−0.328821	−0.164411	−	0.986392i	$-0.552572\pi$
$691$	−8.64368	−0.328821	−0.164411	+	0.986392i	$0.552572\pi$
$692$	−5.70231	−0.216769
$693$	0	0
$694$	−26.0540	−0.988997
$695$	−27.6430	−1.04856
$696$	35.4105	1.34223
$697$	17.7219	0.671264
$698$	15.8738	0.600833
$699$	6.44446	0.243752
$700$	7.20245	0.272227
$701$	40.3484	1.52394	0.761970	−	0.647613i	$-0.224233\pi$
$701$	40.3484	1.52394	0.761970	+	0.647613i	$0.224233\pi$
$702$	−38.7766	−1.46353
$703$	−1.19409	−0.0450361
$704$	0	0
$705$	8.04441	0.302970
$706$	−3.33550	−0.125533
$707$	21.1096	0.793910
$708$	0.382515	0.0143758
$709$	20.0297	0.752229	0.376115	−	0.926573i	$-0.377260\pi$
$709$	20.0297	0.752229	0.376115	+	0.926573i	$0.377260\pi$
$710$	18.1986	0.682982
$711$	−13.3965	−0.502409
$712$	0.231644	0.00868123
$713$	−6.88465	−0.257832
$714$	−12.0557	−0.451175
$715$	0	0
$716$	−8.24744	−0.308221
$717$	10.1276	0.378223
$718$	16.5790	0.618721
$719$	−35.1647	−1.31142	−0.655710	−	0.755013i	$-0.727631\pi$
$719$	−35.1647	−1.31142	−0.655710	+	0.755013i	$0.727631\pi$
$720$	5.49623	0.204832
$721$	−36.3308	−1.35303
$722$	8.58148	0.319370
$723$	24.9681	0.928573
$724$	0.607046	0.0225607
$725$	33.7128	1.25206
$726$	0	0
$727$	−15.1753	−0.562820	−0.281410	−	0.959588i	$-0.590802\pi$
$727$	−15.1753	−0.562820	−0.281410	+	0.959588i	$0.590802\pi$
$728$	69.0417	2.55886
$729$	19.6944	0.729421
$730$	−15.8695	−0.587355
$731$	17.4206	0.644324
$732$	−4.80593	−0.177632
$733$	−13.7050	−0.506206	−0.253103	−	0.967439i	$-0.581451\pi$
$733$	−13.7050	−0.506206	−0.253103	+	0.967439i	$0.581451\pi$
$734$	−30.9780	−1.14342
$735$	−9.50920	−0.350752
$736$	22.1995	0.818283
$737$	0	0
$738$	−14.7777	−0.543974
$739$	34.8712	1.28276	0.641379	−	0.767225i	$-0.278363\pi$
$739$	34.8712	1.28276	0.641379	+	0.767225i	$0.278363\pi$
$740$	0.267279	0.00982536
$741$	−23.7624	−0.872933
$742$	−19.2915	−0.708213
$743$	27.1718	0.996837	0.498419	−	0.866936i	$-0.333914\pi$
$743$	27.1718	0.996837	0.498419	+	0.866936i	$0.333914\pi$
$744$	−3.48320	−0.127700
$745$	14.3719	0.526545
$746$	28.2670	1.03493
$747$	24.4077	0.893030
$748$	0	0
$749$	−46.1771	−1.68727
$750$	−14.5029	−0.529570
$751$	−4.23379	−0.154493	−0.0772466	−	0.997012i	$-0.524613\pi$
$751$	−4.23379	−0.154493	−0.0772466	+	0.997012i	$0.524613\pi$
$752$	13.5006	0.492315
$753$	−5.80070	−0.211389
$754$	73.7862	2.68714
$755$	27.5737	1.00351
$756$	−11.6034	−0.422013
$757$	−4.63065	−0.168304	−0.0841519	−	0.996453i	$-0.526818\pi$
$757$	−4.63065	−0.168304	−0.0841519	+	0.996453i	$0.526818\pi$
$758$	23.8208	0.865210
$759$	0	0
$760$	13.6911	0.496630
$761$	12.0204	0.435741	0.217870	−	0.975978i	$-0.430089\pi$
$761$	12.0204	0.435741	0.217870	+	0.975978i	$0.430089\pi$
$762$	−1.72386	−0.0624489
$763$	−9.53621	−0.345234
$764$	−8.50014	−0.307524
$765$	5.44667	0.196925
$766$	14.4310	0.521412
$767$	3.49094	0.126051
$768$	14.5373	0.524568
$769$	−2.92278	−0.105398	−0.0526991	−	0.998610i	$-0.516782\pi$
$769$	−2.92278	−0.105398	−0.0526991	+	0.998610i	$0.516782\pi$
$770$	0	0
$771$	20.8503	0.750904
$772$	9.53196	0.343063
$773$	−14.7636	−0.531009	−0.265505	−	0.964110i	$-0.585539\pi$
$773$	−14.7636	−0.531009	−0.265505	+	0.964110i	$0.585539\pi$
$774$	−14.5265	−0.522143
$775$	−3.31620	−0.119122
$776$	46.5355	1.67053
$777$	1.44683	0.0519047
$778$	−23.2692	−0.834243
$779$	−24.8747	−0.891228
$780$	5.31883	0.190445
$781$	0	0
$782$	−19.9680	−0.714054
$783$	−54.3127	−1.94098
$784$	−15.9589	−0.569959
$785$	0.814817	0.0290821
$786$	4.03357	0.143873
$787$	15.0364	0.535989	0.267994	−	0.963420i	$-0.413639\pi$
$787$	15.0364	0.535989	0.267994	+	0.963420i	$0.413639\pi$
$788$	8.70959	0.310266
$789$	−15.9944	−0.569415
$790$	−12.0117	−0.427356
$791$	−54.6164	−1.94194
$792$	0	0
$793$	−43.8603	−1.55753
$794$	6.74893	0.239511
$795$	−6.50910	−0.230854
$796$	−4.48642	−0.159017
$797$	−23.7690	−0.841942	−0.420971	−	0.907074i	$-0.638311\pi$
$797$	−23.7690	−0.841942	−0.420971	+	0.907074i	$0.638311\pi$
$798$	16.9216	0.599019
$799$	13.3788	0.473309
$800$	10.6931	0.378057
$801$	−0.129348	−0.00457029
$802$	28.9980	1.02396
$803$	0	0
$804$	−4.00075	−0.141096
$805$	−32.7885	−1.15564
$806$	−7.25808	−0.255655
$807$	−10.1985	−0.359003
$808$	17.6896	0.622318
$809$	2.70697	0.0951719	0.0475859	−	0.998867i	$-0.484847\pi$
$809$	2.70697	0.0951719	0.0475859	+	0.998867i	$0.484847\pi$
$810$	1.38330	0.0486042
$811$	−20.9352	−0.735135	−0.367567	−	0.929997i	$-0.619809\pi$
$811$	−20.9352	−0.735135	−0.367567	+	0.929997i	$0.619809\pi$
$812$	22.0797	0.774844
$813$	−0.799748	−0.0280484
$814$	0	0
$815$	−5.93194	−0.207787
$816$	−6.82664	−0.238980
$817$	−24.4518	−0.855461
$818$	14.8516	0.519274
$819$	−38.5523	−1.34713
$820$	5.56780	0.194436
$821$	−4.32663	−0.151000	−0.0755002	−	0.997146i	$-0.524055\pi$
$821$	−4.32663	−0.151000	−0.0755002	+	0.997146i	$0.524055\pi$
$822$	4.81289	0.167869
$823$	−33.9974	−1.18507	−0.592537	−	0.805543i	$-0.701874\pi$
$823$	−33.9974	−1.18507	−0.592537	+	0.805543i	$0.701874\pi$
$824$	−30.4447	−1.06059
$825$	0	0
$826$	−2.48596	−0.0864976
$827$	46.2735	1.60909	0.804544	−	0.593893i	$-0.202410\pi$
$827$	46.2735	1.60909	0.804544	+	0.593893i	$0.202410\pi$
$828$	−6.99676	−0.243154
$829$	8.83594	0.306885	0.153442	−	0.988158i	$-0.450964\pi$
$829$	8.83594	0.306885	0.153442	+	0.988158i	$0.450964\pi$
$830$	21.8846	0.759624
$831$	20.5298	0.712172
$832$	53.5726	1.85729
$833$	−15.8150	−0.547956
$834$	−28.6303	−0.991385
$835$	20.8476	0.721462
$836$	0	0
$837$	5.34254	0.184665
$838$	6.17929	0.213460
$839$	−10.0551	−0.347142	−0.173571	−	0.984821i	$-0.555531\pi$
$839$	−10.0551	−0.347142	−0.173571	+	0.984821i	$0.555531\pi$
$840$	−16.5889	−0.572373
$841$	74.3492	2.56377
$842$	9.28056	0.319829
$843$	−21.3096	−0.733940
$844$	11.1219	0.382832
$845$	31.6722	1.08956
$846$	−11.1562	−0.383557
$847$	0	0
$848$	−10.9239	−0.375129
$849$	−8.33937	−0.286207
$850$	−9.61821	−0.329902
$851$	2.39639	0.0821473
$852$	−7.92033	−0.271346
$853$	12.6313	0.432487	0.216244	−	0.976339i	$-0.430619\pi$
$853$	12.6313	0.432487	0.216244	+	0.976339i	$0.430619\pi$
$854$	31.2337	1.06880
$855$	−7.64502	−0.261454
$856$	−38.6958	−1.32259
$857$	7.29551	0.249210	0.124605	−	0.992206i	$-0.460234\pi$
$857$	7.29551	0.249210	0.124605	+	0.992206i	$0.460234\pi$
$858$	0	0
$859$	−45.0890	−1.53842	−0.769208	−	0.638999i	$-0.779349\pi$
$859$	−45.0890	−1.53842	−0.769208	+	0.638999i	$0.779349\pi$
$860$	5.47314	0.186633
$861$	30.1395	1.02715
$862$	3.71480	0.126526
$863$	18.7496	0.638245	0.319122	−	0.947714i	$-0.396612\pi$
$863$	18.7496	0.638245	0.319122	+	0.947714i	$0.396612\pi$
$864$	−17.2270	−0.586074
$865$	12.5041	0.425152
$866$	−22.2555	−0.756271
$867$	12.4877	0.424105
$868$	−2.17190	−0.0737189
$869$	0	0
$870$	−17.7289	−0.601067
$871$	−36.5121	−1.23716
$872$	−7.99122	−0.270617
$873$	−25.9851	−0.879461
$874$	28.0274	0.948040
$875$	−39.6064	−1.33894
$876$	6.90664	0.233354
$877$	−17.4129	−0.587993	−0.293996	−	0.955807i	$-0.594985\pi$
$877$	−17.4129	−0.587993	−0.293996	+	0.955807i	$0.594985\pi$
$878$	−35.2131	−1.18839
$879$	30.7692	1.03782
$880$	0	0
$881$	−24.5842	−0.828262	−0.414131	−	0.910217i	$-0.635915\pi$
$881$	−24.5842	−0.828262	−0.414131	+	0.910217i	$0.635915\pi$
$882$	13.1876	0.444049
$883$	21.0545	0.708541	0.354271	−	0.935143i	$-0.384729\pi$
$883$	21.0545	0.708541	0.354271	+	0.935143i	$0.384729\pi$
$884$	8.84586	0.297519
$885$	−0.838783	−0.0281954
$886$	−10.1674	−0.341581
$887$	−33.5138	−1.12528	−0.562641	−	0.826701i	$-0.690215\pi$
$887$	−33.5138	−1.12528	−0.562641	+	0.826701i	$0.690215\pi$
$888$	1.21242	0.0406863
$889$	−4.70775	−0.157893
$890$	−0.115977	−0.00388756
$891$	0	0
$892$	10.9656	0.367155
$893$	−18.7787	−0.628406
$894$	14.8852	0.497834
$895$	18.0851	0.604518
$896$	−14.4806	−0.483763
$897$	47.6881	1.59226
$898$	−32.8338	−1.09568
$899$	−10.1661	−0.339058
$900$	−3.37021	−0.112340
$901$	−10.8254	−0.360647
$902$	0	0
$903$	29.6272	0.985930
$904$	−45.7678	−1.52222
$905$	−1.33114	−0.0442486
$906$	28.5584	0.948789
$907$	−39.5520	−1.31330	−0.656651	−	0.754194i	$-0.728028\pi$
$907$	−39.5520	−1.31330	−0.656651	+	0.754194i	$0.728028\pi$
$908$	−9.21253	−0.305729
$909$	−9.87773	−0.327624
$910$	−34.5670	−1.14589
$911$	−0.714896	−0.0236856	−0.0118428	−	0.999930i	$-0.503770\pi$
$911$	−0.714896	−0.0236856	−0.0118428	+	0.999930i	$0.503770\pi$
$912$	9.58198	0.317291
$913$	0	0
$914$	−13.9844	−0.462562
$915$	10.5385	0.348392
$916$	−2.08957	−0.0690415
$917$	11.0154	0.363761
$918$	15.4953	0.511423
$919$	−14.6434	−0.483041	−0.241520	−	0.970396i	$-0.577646\pi$
$919$	−14.6434	−0.483041	−0.241520	+	0.970396i	$0.577646\pi$
$920$	−27.4764	−0.905869
$921$	34.9801	1.15263
$922$	14.4816	0.476925
$923$	−72.2833	−2.37923
$924$	0	0
$925$	1.15430	0.0379530
$926$	−2.88267	−0.0947304
$927$	17.0001	0.558356
$928$	32.7804	1.07607
$929$	46.2664	1.51795	0.758976	−	0.651118i	$-0.225700\pi$
$929$	46.2664	1.51795	0.758976	+	0.651118i	$0.225700\pi$
$930$	1.74393	0.0571857
$931$	22.1981	0.727514
$932$	3.36732	0.110300
$933$	−9.01233	−0.295050
$934$	−33.4171	−1.09344
$935$	0	0
$936$	−32.3064	−1.05597
$937$	10.8711	0.355142	0.177571	−	0.984108i	$-0.443176\pi$
$937$	10.8711	0.355142	0.177571	+	0.984108i	$0.443176\pi$
$938$	26.0009	0.848959
$939$	27.0480	0.882678
$940$	4.20332	0.137097
$941$	−35.8378	−1.16828	−0.584139	−	0.811654i	$-0.698568\pi$
$941$	−35.8378	−1.16828	−0.584139	+	0.811654i	$0.698568\pi$
$942$	0.843917	0.0274963
$943$	49.9203	1.62563
$944$	−1.40769	−0.0458165
$945$	25.4442	0.827700
$946$	0	0
$947$	20.0843	0.652651	0.326326	−	0.945257i	$-0.394189\pi$
$947$	20.0843	0.652651	0.326326	+	0.945257i	$0.394189\pi$
$948$	5.22767	0.169787
$949$	63.0320	2.04611
$950$	13.5003	0.438006
$951$	−0.575153	−0.0186506
$952$	−27.5895	−0.894179
$953$	−12.6965	−0.411280	−0.205640	−	0.978628i	$-0.565928\pi$
$953$	−12.6965	−0.411280	−0.205640	+	0.978628i	$0.565928\pi$
$954$	9.02697	0.292259
$955$	18.6392	0.603151
$956$	5.29182	0.171150
$957$	0	0
$958$	1.87975	0.0607319
$959$	13.1437	0.424432
$960$	−12.8721	−0.415446
$961$	1.00000	0.0322581
$962$	2.52638	0.0814536
$963$	21.6074	0.696289
$964$	13.0462	0.420189
$965$	−20.9018	−0.672853
$966$	−33.9595	−1.09263
$967$	2.29916	0.0739359	0.0369679	−	0.999316i	$-0.488230\pi$
$967$	2.29916	0.0739359	0.0369679	+	0.999316i	$0.488230\pi$
$968$	0	0
$969$	9.49558	0.305042
$970$	−23.2989	−0.748082
$971$	3.28171	0.105315	0.0526575	−	0.998613i	$-0.483231\pi$
$971$	3.28171	0.105315	0.0526575	+	0.998613i	$0.483231\pi$
$972$	8.88242	0.284904
$973$	−78.1874	−2.50657
$974$	−23.3993	−0.749761
$975$	22.9704	0.735643
$976$	17.6863	0.566124
$977$	56.5807	1.81018	0.905088	−	0.425224i	$-0.139805\pi$
$977$	56.5807	1.81018	0.905088	+	0.425224i	$0.139805\pi$
$978$	−6.14378	−0.196457
$979$	0	0
$980$	−4.96869	−0.158719
$981$	4.46223	0.142468
$982$	−22.5616	−0.719971
$983$	−13.2851	−0.423727	−0.211864	−	0.977299i	$-0.567953\pi$
$983$	−13.2851	−0.423727	−0.211864	+	0.977299i	$0.567953\pi$
$984$	25.2565	0.805149
$985$	−19.0985	−0.608529
$986$	−29.4854	−0.939006
$987$	22.7533	0.724247
$988$	−12.4162	−0.395011
$989$	49.0717	1.56039
$990$	0	0
$991$	4.55607	0.144728	0.0723642	−	0.997378i	$-0.476946\pi$
$991$	4.55607	0.144728	0.0723642	+	0.997378i	$0.476946\pi$
$992$	−3.22449	−0.102378
$993$	−24.7096	−0.784135
$994$	51.4742	1.63266
$995$	9.83789	0.311882
$996$	−9.52451	−0.301796
$997$	−50.8238	−1.60961	−0.804804	−	0.593541i	$-0.797729\pi$
$997$	−50.8238	−1.60961	−0.804804	+	0.593541i	$0.797729\pi$
$998$	−39.9688	−1.26519
$999$	−1.85962	−0.0588358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3751.2.a.p.1.12		30
11.2	odd	10		341.2.h.a.125.7	✓	60
11.6	odd	10		341.2.h.a.311.7	yes	60
11.10	odd	2		3751.2.a.s.1.19		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
341.2.h.a.125.7	✓	60	11.2	odd	10
341.2.h.a.311.7	yes	60	11.6	odd	10
3751.2.a.p.1.12		30	1.1	even	1	trivial
3751.2.a.s.1.19		30	11.10	odd	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:	\( N \)	\(=\)	\( 3751 = 11^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3751.a (trivial)

\(f(q)\)	\(=\)	\(q-1.18669 q^{2} -1.13252 q^{3} -0.591757 q^{4} +1.29761 q^{5} +1.34395 q^{6} +3.67025 q^{7} +3.07562 q^{8} -1.71740 q^{9} -1.53987 q^{10} +0.670175 q^{12} +6.11622 q^{13} -4.35547 q^{14} -1.46957 q^{15} -2.46631 q^{16} -2.44407 q^{17} +2.03803 q^{18} +3.43054 q^{19} -0.767870 q^{20} -4.15663 q^{21} -6.88465 q^{23} -3.48320 q^{24} -3.31620 q^{25} -7.25808 q^{26} +5.34254 q^{27} -2.17190 q^{28} -10.1661 q^{29} +1.74393 q^{30} +1.00000 q^{31} -3.22449 q^{32} +2.90037 q^{34} +4.76256 q^{35} +1.01628 q^{36} -0.348078 q^{37} -4.07100 q^{38} -6.92673 q^{39} +3.99096 q^{40} -7.25096 q^{41} +4.93264 q^{42} -7.12769 q^{43} -2.22852 q^{45} +8.16997 q^{46} -5.47399 q^{47} +2.79314 q^{48} +6.47074 q^{49} +3.93532 q^{50} +2.76796 q^{51} -3.61931 q^{52} +4.42926 q^{53} -6.33997 q^{54} +11.2883 q^{56} -3.88515 q^{57} +12.0640 q^{58} +0.570768 q^{59} +0.869627 q^{60} -7.17115 q^{61} -1.18669 q^{62} -6.30330 q^{63} +8.75910 q^{64} +7.93647 q^{65} -5.96971 q^{67} +1.44630 q^{68} +7.79699 q^{69} -5.65170 q^{70} -11.8183 q^{71} -5.28208 q^{72} +10.3057 q^{73} +0.413062 q^{74} +3.75566 q^{75} -2.03004 q^{76} +8.21991 q^{78} +7.80045 q^{79} -3.20031 q^{80} -0.898323 q^{81} +8.60468 q^{82} -14.2120 q^{83} +2.45971 q^{84} -3.17146 q^{85} +8.45839 q^{86} +11.5133 q^{87} +0.0753162 q^{89} +2.64457 q^{90} +22.4480 q^{91} +4.07403 q^{92} -1.13252 q^{93} +6.49596 q^{94} +4.45150 q^{95} +3.65179 q^{96} +15.1304 q^{97} -7.67879 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{2} - q^{3} + 29 q^{4} - q^{5} - 11 q^{6} - 11 q^{7} - 27 q^{8} + 25 q^{9} - 8 q^{10} - 16 q^{12} - 2 q^{13} - 3 q^{14} - 11 q^{15} + 35 q^{16} - 32 q^{17} - 4 q^{18} - 17 q^{19} - 4 q^{20}+ \cdots - 40 q^{98}+O(q^{100}) \) 30 * q - 9 * q^2 - q^3 + 29 * q^4 - q^5 - 11 * q^6 - 11 * q^7 - 27 * q^8 + 25 * q^9 - 8 * q^10 - 16 * q^12 - 2 * q^13 - 3 * q^14 - 11 * q^15 + 35 * q^16 - 32 * q^17 - 4 * q^18 - 17 * q^19 - 4 * q^20 - 30 * q^21 - 11 * q^23 + 7 * q^24 + 27 * q^25 - 21 * q^26 + 5 * q^27 - 17 * q^28 - 41 * q^29 - 12 * q^30 + 30 * q^31 - 68 * q^32 - 21 * q^34 - 33 * q^35 + 22 * q^36 - 7 * q^37 + 87 * q^38 - 36 * q^39 - 20 * q^40 - 79 * q^41 + 42 * q^42 - 10 * q^43 - 36 * q^45 - 8 * q^46 - 67 * q^48 + 31 * q^49 - 47 * q^50 - 26 * q^51 - 8 * q^52 + 4 * q^53 - 70 * q^54 - 17 * q^56 - 34 * q^57 + 18 * q^58 - 33 * q^59 + 64 * q^60 - 19 * q^61 - 9 * q^62 - 38 * q^63 + 39 * q^64 - 93 * q^65 - 13 * q^67 - 32 * q^68 - 6 * q^69 - 40 * q^70 - 13 * q^71 - 32 * q^72 - 27 * q^73 - 82 * q^74 + 64 * q^75 - 65 * q^76 - 16 * q^78 - 29 * q^79 - 18 * q^80 - 42 * q^81 + 81 * q^82 - 32 * q^83 + 16 * q^84 - 12 * q^85 - 12 * q^86 - 48 * q^87 - 8 * q^89 + 51 * q^90 - 27 * q^91 - 49 * q^92 - q^93 - 36 * q^94 - 56 * q^95 + 50 * q^96 + 16 * q^97 - 40 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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