LMFDB - Embedded newform 3751.2.a.p.1.3

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			1.3
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)

$f(q)$	$=$	\(q-2.59478 q^{2} -1.09142 q^{3} +4.73286 q^{4} +3.97879 q^{5} +2.83200 q^{6} +0.943415 q^{7} -7.09117 q^{8} -1.80879 q^{9} -10.3241 q^{10} -5.16556 q^{12} -2.80038 q^{13} -2.44795 q^{14} -4.34255 q^{15} +8.93427 q^{16} -2.11107 q^{17} +4.69341 q^{18} -4.72413 q^{19} +18.8311 q^{20} -1.02967 q^{21} -7.83891 q^{23} +7.73948 q^{24} +10.8308 q^{25} +7.26637 q^{26} +5.24843 q^{27} +4.46505 q^{28} +4.01133 q^{29} +11.2680 q^{30} +1.00000 q^{31} -9.00010 q^{32} +5.47775 q^{34} +3.75365 q^{35} -8.56077 q^{36} +5.69890 q^{37} +12.2581 q^{38} +3.05641 q^{39} -28.2143 q^{40} -1.83466 q^{41} +2.67175 q^{42} +7.78390 q^{43} -7.19681 q^{45} +20.3402 q^{46} +12.9036 q^{47} -9.75109 q^{48} -6.10997 q^{49} -28.1035 q^{50} +2.30407 q^{51} -13.2538 q^{52} -3.11074 q^{53} -13.6185 q^{54} -6.68991 q^{56} +5.15603 q^{57} -10.4085 q^{58} -2.78768 q^{59} -20.5527 q^{60} -12.8934 q^{61} -2.59478 q^{62} -1.70644 q^{63} +5.48470 q^{64} -11.1422 q^{65} +2.15735 q^{67} -9.99139 q^{68} +8.55558 q^{69} -9.73989 q^{70} -1.74810 q^{71} +12.8265 q^{72} +4.23766 q^{73} -14.7874 q^{74} -11.8210 q^{75} -22.3587 q^{76} -7.93069 q^{78} +2.12274 q^{79} +35.5476 q^{80} -0.301892 q^{81} +4.76053 q^{82} +7.51015 q^{83} -4.87327 q^{84} -8.39950 q^{85} -20.1975 q^{86} -4.37806 q^{87} -10.6723 q^{89} +18.6741 q^{90} -2.64192 q^{91} -37.1005 q^{92} -1.09142 q^{93} -33.4821 q^{94} -18.7963 q^{95} +9.82293 q^{96} -0.345394 q^{97} +15.8540 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.59478	−1.83478	−0.917392	−	0.397985i	$-0.869710\pi$
$2$	−2.59478	−1.83478	−0.917392	+	0.397985i	$0.869710\pi$
$3$	−1.09142	−0.630134	−0.315067	−	0.949069i	$-0.602027\pi$
$3$	−1.09142	−0.630134	−0.315067	+	0.949069i	$0.602027\pi$
$4$	4.73286	2.36643
$5$	3.97879	1.77937	0.889685	−	0.456574i	$-0.150924\pi$
$5$	3.97879	1.77937	0.889685	+	0.456574i	$0.150924\pi$
$6$	2.83200	1.15616
$7$	0.943415	0.356577	0.178289	−	0.983978i	$-0.442944\pi$
$7$	0.943415	0.356577	0.178289	+	0.983978i	$0.442944\pi$
$8$	−7.09117	−2.50711
$9$	−1.80879	−0.602931
$10$	−10.3241	−3.26476
$11$	0	0
$11$	0	0
$12$	−5.16556	−1.49117
$13$	−2.80038	−0.776687	−0.388343	−	0.921515i	$-0.626953\pi$
$13$	−2.80038	−0.776687	−0.388343	+	0.921515i	$0.626953\pi$
$14$	−2.44795	−0.654242
$15$	−4.34255	−1.12124
$16$	8.93427	2.23357
$17$	−2.11107	−0.512009	−0.256004	−	0.966676i	$-0.582406\pi$
$17$	−2.11107	−0.512009	−0.256004	+	0.966676i	$0.582406\pi$
$18$	4.69341	1.10625
$19$	−4.72413	−1.08379	−0.541895	−	0.840446i	$-0.682293\pi$
$19$	−4.72413	−1.08379	−0.541895	+	0.840446i	$0.682293\pi$
$20$	18.8311	4.21076
$21$	−1.02967	−0.224692
$22$	0	0
$23$	−7.83891	−1.63453	−0.817263	−	0.576265i	$-0.804510\pi$
$23$	−7.83891	−1.63453	−0.817263	+	0.576265i	$0.804510\pi$
$24$	7.73948	1.57981
$25$	10.8308	2.16616
$26$	7.26637	1.42505
$27$	5.24843	1.01006
$28$	4.46505	0.843816
$29$	4.01133	0.744885	0.372442	−	0.928055i	$-0.378520\pi$
$29$	4.01133	0.744885	0.372442	+	0.928055i	$0.378520\pi$
$30$	11.2680	2.05724
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−9.00010	−1.59101
$33$	0	0
$34$	5.47775	0.939426
$35$	3.75365	0.634483
$36$	−8.56077	−1.42679
$37$	5.69890	0.936893	0.468446	−	0.883492i	$-0.344814\pi$
$37$	5.69890	0.936893	0.468446	+	0.883492i	$0.344814\pi$
$38$	12.2581	1.98852
$39$	3.05641	0.489417
$40$	−28.2143	−4.46107
$41$	−1.83466	−0.286526	−0.143263	−	0.989685i	$-0.545759\pi$
$41$	−1.83466	−0.286526	−0.143263	+	0.989685i	$0.545759\pi$
$42$	2.67175	0.412260
$43$	7.78390	1.18703	0.593516	−	0.804822i	$-0.297739\pi$
$43$	7.78390	1.18703	0.593516	+	0.804822i	$0.297739\pi$
$44$	0	0
$45$	−7.19681	−1.07284
$46$	20.3402	2.99900
$47$	12.9036	1.88219	0.941094	−	0.338144i	$-0.109799\pi$
$47$	12.9036	1.88219	0.941094	+	0.338144i	$0.109799\pi$
$48$	−9.75109	−1.40745
$49$	−6.10997	−0.872853
$50$	−28.1035	−3.97444
$51$	2.30407	0.322634
$52$	−13.2538	−1.83798
$53$	−3.11074	−0.427293	−0.213646	−	0.976911i	$-0.568534\pi$
$53$	−3.11074	−0.427293	−0.213646	+	0.976911i	$0.568534\pi$
$54$	−13.6185	−1.85324
$55$	0	0
$56$	−6.68991	−0.893977
$57$	5.15603	0.682933
$58$	−10.4085	−1.36670
$59$	−2.78768	−0.362925	−0.181462	−	0.983398i	$-0.558083\pi$
$59$	−2.78768	−0.362925	−0.181462	+	0.983398i	$0.558083\pi$
$60$	−20.5527	−2.65334
$61$	−12.8934	−1.65082	−0.825412	−	0.564530i	$-0.809057\pi$
$61$	−12.8934	−1.65082	−0.825412	+	0.564530i	$0.809057\pi$
$62$	−2.59478	−0.329537
$63$	−1.70644	−0.214991
$64$	5.48470	0.685587
$65$	−11.1422	−1.38201
$66$	0	0
$67$	2.15735	0.263562	0.131781	−	0.991279i	$-0.457930\pi$
$67$	2.15735	0.263562	0.131781	+	0.991279i	$0.457930\pi$
$68$	−9.99139	−1.21163
$69$	8.55558	1.02997
$70$	−9.73989	−1.16414
$71$	−1.74810	−0.207462	−0.103731	−	0.994605i	$-0.533078\pi$
$71$	−1.74810	−0.207462	−0.103731	+	0.994605i	$0.533078\pi$
$72$	12.8265	1.51161
$73$	4.23766	0.495980	0.247990	−	0.968763i	$-0.420230\pi$
$73$	4.23766	0.495980	0.247990	+	0.968763i	$0.420230\pi$
$74$	−14.7874	−1.71900
$75$	−11.8210	−1.36497
$76$	−22.3587	−2.56472
$77$	0	0
$78$	−7.93069	−0.897974
$79$	2.12274	0.238827	0.119414	−	0.992845i	$-0.461899\pi$
$79$	2.12274	0.238827	0.119414	+	0.992845i	$0.461899\pi$
$80$	35.5476	3.97435
$81$	−0.301892	−0.0335436
$82$	4.76053	0.525713
$83$	7.51015	0.824346	0.412173	−	0.911106i	$-0.364770\pi$
$83$	7.51015	0.824346	0.412173	+	0.911106i	$0.364770\pi$
$84$	−4.87327	−0.531717
$85$	−8.39950	−0.911054
$86$	−20.1975	−2.17795
$87$	−4.37806	−0.469377
$88$	0	0
$89$	−10.6723	−1.13126	−0.565628	−	0.824660i	$-0.691366\pi$
$89$	−10.6723	−1.13126	−0.565628	+	0.824660i	$0.691366\pi$
$90$	18.6741	1.96843
$91$	−2.64192	−0.276949
$92$	−37.1005	−3.86800
$93$	−1.09142	−0.113175
$94$	−33.4821	−3.45341
$95$	−18.7963	−1.92846
$96$	9.82293	1.00255
$97$	−0.345394	−0.0350694	−0.0175347	−	0.999846i	$-0.505582\pi$
$97$	−0.345394	−0.0350694	−0.0175347	+	0.999846i	$0.505582\pi$
$98$	15.8540	1.60150
$99$	0	0
$100$	51.2607	5.12607
$101$	−18.7992	−1.87059	−0.935296	−	0.353868i	$-0.884866\pi$
$101$	−18.7992	−1.87059	−0.935296	+	0.353868i	$0.884866\pi$
$102$	−5.97855	−0.591964
$103$	−5.80601	−0.572083	−0.286041	−	0.958217i	$-0.592339\pi$
$103$	−5.80601	−0.572083	−0.286041	+	0.958217i	$0.592339\pi$
$104$	19.8580	1.94724
$105$	−4.09683	−0.399810
$106$	8.07167	0.783990
$107$	10.2364	0.989588	0.494794	−	0.869010i	$-0.335243\pi$
$107$	10.2364	0.989588	0.494794	+	0.869010i	$0.335243\pi$
$108$	24.8401	2.39024
$109$	6.48908	0.621541	0.310771	−	0.950485i	$-0.399413\pi$
$109$	6.48908	0.621541	0.310771	+	0.950485i	$0.399413\pi$
$110$	0	0
$111$	−6.21992	−0.590368
$112$	8.42873	0.796440
$113$	−10.0149	−0.942119	−0.471060	−	0.882101i	$-0.656128\pi$
$113$	−10.0149	−0.942119	−0.471060	+	0.882101i	$0.656128\pi$
$114$	−13.3788	−1.25303
$115$	−31.1894	−2.90843
$116$	18.9851	1.76272
$117$	5.06531	0.468288
$118$	7.23340	0.665888
$119$	−1.99161	−0.182571
$120$	30.7938	2.81108
$121$	0	0
$122$	33.4554	3.02891
$123$	2.00239	0.180550
$124$	4.73286	0.425024
$125$	23.1996	2.07503
$126$	4.42783	0.394463
$127$	−7.68397	−0.681842	−0.340921	−	0.940092i	$-0.610739\pi$
$127$	−7.68397	−0.681842	−0.340921	+	0.940092i	$0.610739\pi$
$128$	3.76863	0.333103
$129$	−8.49553	−0.747990
$130$	28.9114	2.53570
$131$	−17.9971	−1.57241	−0.786207	−	0.617963i	$-0.787958\pi$
$131$	−17.9971	−1.57241	−0.786207	+	0.617963i	$0.787958\pi$
$132$	0	0
$133$	−4.45682	−0.386455
$134$	−5.59783	−0.483579
$135$	20.8824	1.79727
$136$	14.9699	1.28366
$137$	−6.61758	−0.565378	−0.282689	−	0.959212i	$-0.591226\pi$
$137$	−6.61758	−0.565378	−0.282689	+	0.959212i	$0.591226\pi$
$138$	−22.1998	−1.88977
$139$	4.16620	0.353373	0.176686	−	0.984267i	$-0.443462\pi$
$139$	4.16620	0.353373	0.176686	+	0.984267i	$0.443462\pi$
$140$	17.7655	1.50146
$141$	−14.0833	−1.18603
$142$	4.53594	0.380648
$143$	0	0
$144$	−16.1602	−1.34669
$145$	15.9602	1.32543
$146$	−10.9958	−0.910017
$147$	6.66857	0.550014
$148$	26.9721	2.21709
$149$	−19.0217	−1.55832	−0.779159	−	0.626826i	$-0.784354\pi$
$149$	−19.0217	−1.55832	−0.779159	+	0.626826i	$0.784354\pi$
$150$	30.6729	2.50443
$151$	14.4260	1.17397	0.586984	−	0.809598i	$-0.300315\pi$
$151$	14.4260	1.17397	0.586984	+	0.809598i	$0.300315\pi$
$152$	33.4996	2.71718
$153$	3.81848	0.308706
$154$	0	0
$155$	3.97879	0.319584
$156$	14.4656	1.15817
$157$	−19.8622	−1.58517	−0.792586	−	0.609760i	$-0.791266\pi$
$157$	−19.8622	−1.58517	−0.792586	+	0.609760i	$0.791266\pi$
$158$	−5.50804	−0.438196
$159$	3.39514	0.269252
$160$	−35.8095	−2.83099
$161$	−7.39535	−0.582835
$162$	0.783343	0.0615452
$163$	−16.6976	−1.30785	−0.653927	−	0.756557i	$-0.726880\pi$
$163$	−16.6976	−1.30785	−0.653927	+	0.756557i	$0.726880\pi$
$164$	−8.68320	−0.678044
$165$	0	0
$166$	−19.4872	−1.51250
$167$	10.0361	0.776614	0.388307	−	0.921530i	$-0.373060\pi$
$167$	10.0361	0.776614	0.388307	+	0.921530i	$0.373060\pi$
$168$	7.30154	0.563326
$169$	−5.15785	−0.396758
$170$	21.7948	1.67159
$171$	8.54497	0.653450
$172$	36.8401	2.80903
$173$	1.89812	0.144311	0.0721557	−	0.997393i	$-0.477012\pi$
$173$	1.89812	0.144311	0.0721557	+	0.997393i	$0.477012\pi$
$174$	11.3601	0.861206
$175$	10.2179	0.772404
$176$	0	0
$177$	3.04254	0.228691
$178$	27.6921	2.07561
$179$	−15.2085	−1.13674	−0.568369	−	0.822774i	$-0.692425\pi$
$179$	−15.2085	−1.13674	−0.568369	+	0.822774i	$0.692425\pi$
$180$	−34.0615	−2.53880
$181$	15.6361	1.16223	0.581113	−	0.813823i	$-0.302618\pi$
$181$	15.6361	1.16223	0.581113	+	0.813823i	$0.302618\pi$
$182$	6.85520	0.508141
$183$	14.0721	1.04024
$184$	55.5871	4.09793
$185$	22.6747	1.66708
$186$	2.83200	0.207652
$187$	0	0
$188$	61.0712	4.45407
$189$	4.95145	0.360165
$190$	48.7723	3.53832
$191$	−12.4822	−0.903179	−0.451589	−	0.892226i	$-0.649143\pi$
$191$	−12.4822	−0.903179	−0.451589	+	0.892226i	$0.649143\pi$
$192$	−5.98613	−0.432012
$193$	−5.06516	−0.364598	−0.182299	−	0.983243i	$-0.558354\pi$
$193$	−5.06516	−0.364598	−0.182299	+	0.983243i	$0.558354\pi$
$194$	0.896220	0.0643448
$195$	12.1608	0.870854
$196$	−28.9177	−2.06555
$197$	−4.08140	−0.290788	−0.145394	−	0.989374i	$-0.546445\pi$
$197$	−4.08140	−0.290788	−0.145394	+	0.989374i	$0.546445\pi$
$198$	0	0
$199$	−12.3299	−0.874046	−0.437023	−	0.899450i	$-0.643967\pi$
$199$	−12.3299	−0.874046	−0.437023	+	0.899450i	$0.643967\pi$
$200$	−76.8031	−5.43080
$201$	−2.35458	−0.166079
$202$	48.7797	3.43213
$203$	3.78435	0.265609
$204$	10.9048	0.763492
$205$	−7.29974	−0.509836
$206$	15.0653	1.04965
$207$	14.1790	0.985506
$208$	−25.0194	−1.73478
$209$	0	0
$210$	10.6304	0.733564
$211$	17.7867	1.22449	0.612244	−	0.790669i	$-0.290267\pi$
$211$	17.7867	1.22449	0.612244	+	0.790669i	$0.290267\pi$
$212$	−14.7227	−1.01116
$213$	1.90792	0.130729
$214$	−26.5611	−1.81568
$215$	30.9705	2.11217
$216$	−37.2175	−2.53233
$217$	0.943415	0.0640432
$218$	−16.8377	−1.14039
$219$	−4.62508	−0.312534
$220$	0	0
$221$	5.91180	0.397670
$222$	16.1393	1.08320
$223$	5.82635	0.390161	0.195081	−	0.980787i	$-0.437503\pi$
$223$	5.82635	0.390161	0.195081	+	0.980787i	$0.437503\pi$
$224$	−8.49083	−0.567317
$225$	−19.5907	−1.30605
$226$	25.9863	1.72858
$227$	−4.32502	−0.287062	−0.143531	−	0.989646i	$-0.545846\pi$
$227$	−4.32502	−0.287062	−0.143531	+	0.989646i	$0.545846\pi$
$228$	24.4028	1.61612
$229$	7.46753	0.493468	0.246734	−	0.969083i	$-0.420643\pi$
$229$	7.46753	0.493468	0.246734	+	0.969083i	$0.420643\pi$
$230$	80.9296	5.33634
$231$	0	0
$232$	−28.4450	−1.86751
$233$	−23.4363	−1.53536	−0.767681	−	0.640832i	$-0.778590\pi$
$233$	−23.4363	−1.53536	−0.767681	+	0.640832i	$0.778590\pi$
$234$	−13.1434	−0.859208
$235$	51.3409	3.34911
$236$	−13.1937	−0.858836
$237$	−2.31681	−0.150493
$238$	5.16779	0.334978
$239$	−12.7093	−0.822096	−0.411048	−	0.911614i	$-0.634837\pi$
$239$	−12.7093	−0.822096	−0.411048	+	0.911614i	$0.634837\pi$
$240$	−38.7976	−2.50437
$241$	0.104023	0.00670070	0.00335035	−	0.999994i	$-0.498934\pi$
$241$	0.104023	0.00670070	0.00335035	+	0.999994i	$0.498934\pi$
$242$	0	0
$243$	−15.4158	−0.988925
$244$	−61.0225	−3.90656
$245$	−24.3103	−1.55313
$246$	−5.19576	−0.331270
$247$	13.2294	0.841765
$248$	−7.09117	−0.450290
$249$	−8.19676	−0.519449
$250$	−60.1977	−3.80724
$251$	4.07418	0.257160	0.128580	−	0.991699i	$-0.458958\pi$
$251$	4.07418	0.257160	0.128580	+	0.991699i	$0.458958\pi$
$252$	−8.07636	−0.508763
$253$	0	0
$254$	19.9382	1.25103
$255$	9.16742	0.574086
$256$	−20.7482	−1.29676
$257$	−12.1047	−0.755072	−0.377536	−	0.925995i	$-0.623228\pi$
$257$	−12.1047	−0.755072	−0.377536	+	0.925995i	$0.623228\pi$
$258$	22.0440	1.37240
$259$	5.37642	0.334075
$260$	−52.7343	−3.27044
$261$	−7.25566	−0.449114
$262$	46.6985	2.88504
$263$	−7.76333	−0.478708	−0.239354	−	0.970932i	$-0.576936\pi$
$263$	−7.76333	−0.478708	−0.239354	+	0.970932i	$0.576936\pi$
$264$	0	0
$265$	−12.3770	−0.760312
$266$	11.5644	0.709061
$267$	11.6480	0.712844
$268$	10.2104	0.623701
$269$	−8.13513	−0.496008	−0.248004	−	0.968759i	$-0.579775\pi$
$269$	−8.13513	−0.496008	−0.248004	+	0.968759i	$0.579775\pi$
$270$	−54.1853	−3.29761
$271$	11.5528	0.701782	0.350891	−	0.936416i	$-0.385879\pi$
$271$	11.5528	0.701782	0.350891	+	0.936416i	$0.385879\pi$
$272$	−18.8608	−1.14361
$273$	2.88346	0.174515
$274$	17.1711	1.03735
$275$	0	0
$276$	40.4924	2.43736
$277$	2.27333	0.136591	0.0682956	−	0.997665i	$-0.478244\pi$
$277$	2.27333	0.136591	0.0682956	+	0.997665i	$0.478244\pi$
$278$	−10.8104	−0.648362
$279$	−1.80879	−0.108290
$280$	−26.6178	−1.59072
$281$	−21.4312	−1.27848	−0.639238	−	0.769009i	$-0.720750\pi$
$281$	−21.4312	−1.27848	−0.639238	+	0.769009i	$0.720750\pi$
$282$	36.5431	2.17611
$283$	−26.4702	−1.57349	−0.786745	−	0.617278i	$-0.788235\pi$
$283$	−26.4702	−1.57349	−0.786745	+	0.617278i	$0.788235\pi$
$284$	−8.27354	−0.490944
$285$	20.5148	1.21519
$286$	0	0
$287$	−1.73085	−0.102169
$288$	16.2793	0.959268
$289$	−12.5434	−0.737847
$290$	−41.4133	−2.43187
$291$	0.376971	0.0220984
$292$	20.0563	1.17370
$293$	−27.0896	−1.58259	−0.791294	−	0.611436i	$-0.790592\pi$
$293$	−27.0896	−1.58259	−0.791294	+	0.611436i	$0.790592\pi$
$294$	−17.3034	−1.00916
$295$	−11.0916	−0.645777
$296$	−40.4118	−2.34889
$297$	0	0
$298$	49.3571	2.85918
$299$	21.9520	1.26951
$300$	−55.9472	−3.23011
$301$	7.34344	0.423269
$302$	−37.4322	−2.15398
$303$	20.5179	1.17872
$304$	−42.2067	−2.42072
$305$	−51.3000	−2.93743
$306$	−9.90810	−0.566409
$307$	9.15990	0.522783	0.261392	−	0.965233i	$-0.415819\pi$
$307$	9.15990	0.522783	0.261392	+	0.965233i	$0.415819\pi$
$308$	0	0
$309$	6.33682	0.360489
$310$	−10.3241	−0.586368
$311$	2.85040	0.161632	0.0808158	−	0.996729i	$-0.474247\pi$
$311$	2.85040	0.161632	0.0808158	+	0.996729i	$0.474247\pi$
$312$	−21.6735	−1.22702
$313$	20.8258	1.17714	0.588571	−	0.808446i	$-0.299691\pi$
$313$	20.8258	1.17714	0.588571	+	0.808446i	$0.299691\pi$
$314$	51.5379	2.90845
$315$	−6.78958	−0.382549
$316$	10.0467	0.565168
$317$	17.8779	1.00413	0.502063	−	0.864831i	$-0.332575\pi$
$317$	17.8779	1.00413	0.502063	+	0.864831i	$0.332575\pi$
$318$	−8.80962	−0.494019
$319$	0	0
$320$	21.8225	1.21991
$321$	−11.1722	−0.623573
$322$	19.1893	1.06938
$323$	9.97296	0.554910
$324$	−1.42881	−0.0793786
$325$	−30.3304	−1.68243
$326$	43.3265	2.39963
$327$	−7.08234	−0.391654
$328$	13.0099	0.718351
$329$	12.1735	0.671146
$330$	0	0
$331$	−27.8924	−1.53310	−0.766552	−	0.642182i	$-0.778029\pi$
$331$	−27.8924	−1.53310	−0.766552	+	0.642182i	$0.778029\pi$
$332$	35.5445	1.95076
$333$	−10.3081	−0.564882
$334$	−26.0413	−1.42492
$335$	8.58364	0.468974
$336$	−9.19932	−0.501864
$337$	11.8451	0.645244	0.322622	−	0.946528i	$-0.395436\pi$
$337$	11.8451	0.645244	0.322622	+	0.946528i	$0.395436\pi$
$338$	13.3835	0.727965
$339$	10.9305	0.593661
$340$	−39.7537	−2.15595
$341$	0	0
$342$	−22.1723	−1.19894
$343$	−12.3681	−0.667817
$344$	−55.1969	−2.97602
$345$	34.0409	1.83270
$346$	−4.92519	−0.264780
$347$	−3.15873	−0.169570	−0.0847848	−	0.996399i	$-0.527020\pi$
$347$	−3.15873	−0.169570	−0.0847848	+	0.996399i	$0.527020\pi$
$348$	−20.7208	−1.11075
$349$	−0.910133	−0.0487183	−0.0243592	−	0.999703i	$-0.507755\pi$
$349$	−0.910133	−0.0487183	−0.0243592	+	0.999703i	$0.507755\pi$
$350$	−26.5133	−1.41719
$351$	−14.6976	−0.784501
$352$	0	0
$353$	−29.5861	−1.57471	−0.787354	−	0.616502i	$-0.788549\pi$
$353$	−29.5861	−1.57471	−0.787354	+	0.616502i	$0.788549\pi$
$354$	−7.89471	−0.419599
$355$	−6.95535	−0.369152
$356$	−50.5103	−2.67704
$357$	2.17369	0.115044
$358$	39.4627	2.08567
$359$	1.47815	0.0780137	0.0390069	−	0.999239i	$-0.487581\pi$
$359$	1.47815	0.0780137	0.0390069	+	0.999239i	$0.487581\pi$
$360$	51.0338	2.68972
$361$	3.31742	0.174601
$362$	−40.5723	−2.13243
$363$	0	0
$364$	−12.5039	−0.655381
$365$	16.8608	0.882533
$366$	−36.5140	−1.90862
$367$	−0.221728	−0.0115741	−0.00578706	−	0.999983i	$-0.501842\pi$
$367$	−0.221728	−0.0115741	−0.00578706	+	0.999983i	$0.501842\pi$
$368$	−70.0350	−3.65083
$369$	3.31852	0.172755
$370$	−58.8359	−3.05873
$371$	−2.93472	−0.152363
$372$	−5.16556	−0.267822
$373$	−13.8927	−0.719338	−0.359669	−	0.933080i	$-0.617110\pi$
$373$	−13.8927	−0.719338	−0.359669	+	0.933080i	$0.617110\pi$
$374$	0	0
$375$	−25.3206	−1.30755
$376$	−91.5019	−4.71885
$377$	−11.2333	−0.578542
$378$	−12.8479	−0.660825
$379$	−15.7486	−0.808950	−0.404475	−	0.914549i	$-0.632546\pi$
$379$	−15.7486	−0.808950	−0.404475	+	0.914549i	$0.632546\pi$
$380$	−88.9606	−4.56358
$381$	8.38648	0.429652
$382$	32.3885	1.65714
$383$	−27.8268	−1.42188	−0.710941	−	0.703251i	$-0.751731\pi$
$383$	−27.8268	−1.42188	−0.710941	+	0.703251i	$0.751731\pi$
$384$	−4.11318	−0.209900
$385$	0	0
$386$	13.1430	0.668959
$387$	−14.0795	−0.715699
$388$	−1.63470	−0.0829894
$389$	−18.8336	−0.954901	−0.477451	−	0.878659i	$-0.658439\pi$
$389$	−18.8336	−0.954901	−0.477451	+	0.878659i	$0.658439\pi$
$390$	−31.5546	−1.59783
$391$	16.5485	0.836892
$392$	43.3268	2.18834
$393$	19.6425	0.990832
$394$	10.5903	0.533533
$395$	8.44595	0.424962
$396$	0	0
$397$	9.19514	0.461491	0.230745	−	0.973014i	$-0.425884\pi$
$397$	9.19514	0.461491	0.230745	+	0.973014i	$0.425884\pi$
$398$	31.9934	1.60369
$399$	4.86428	0.243518
$400$	96.7654	4.83827
$401$	34.9864	1.74714	0.873569	−	0.486700i	$-0.161800\pi$
$401$	34.9864	1.74714	0.873569	+	0.486700i	$0.161800\pi$
$402$	6.10961	0.304720
$403$	−2.80038	−0.139497
$404$	−88.9741	−4.42663
$405$	−1.20117	−0.0596865
$406$	−9.81953	−0.487335
$407$	0	0
$408$	−16.3386	−0.808879
$409$	26.7715	1.32377	0.661883	−	0.749607i	$-0.269758\pi$
$409$	26.7715	1.32377	0.661883	+	0.749607i	$0.269758\pi$
$410$	18.9412	0.935438
$411$	7.22259	0.356264
$412$	−27.4790	−1.35379
$413$	−2.62994	−0.129411
$414$	−36.7912	−1.80819
$415$	29.8813	1.46682
$416$	25.2037	1.23571
$417$	−4.54709	−0.222672
$418$	0	0
$419$	−11.7485	−0.573951	−0.286975	−	0.957938i	$-0.592650\pi$
$419$	−11.7485	−0.573951	−0.286975	+	0.957938i	$0.592650\pi$
$420$	−19.3897	−0.946122
$421$	−16.5030	−0.804309	−0.402155	−	0.915572i	$-0.631739\pi$
$421$	−16.5030	−0.804309	−0.402155	+	0.915572i	$0.631739\pi$
$422$	−46.1526	−2.24667
$423$	−23.3400	−1.13483
$424$	22.0588	1.07127
$425$	−22.8645	−1.10909
$426$	−4.95064	−0.239859
$427$	−12.1638	−0.588646
$428$	48.4474	2.34179
$429$	0	0
$430$	−80.3616	−3.87538
$431$	−27.7178	−1.33512	−0.667559	−	0.744556i	$-0.732661\pi$
$431$	−27.7178	−1.33512	−0.667559	+	0.744556i	$0.732661\pi$
$432$	46.8909	2.25604
$433$	7.81740	0.375680	0.187840	−	0.982200i	$-0.439851\pi$
$433$	7.81740	0.375680	0.187840	+	0.982200i	$0.439851\pi$
$434$	−2.44795	−0.117505
$435$	−17.4194	−0.835197
$436$	30.7119	1.47083
$437$	37.0321	1.77148
$438$	12.0011	0.573433
$439$	39.7818	1.89868	0.949340	−	0.314251i	$-0.101754\pi$
$439$	39.7818	1.89868	0.949340	+	0.314251i	$0.101754\pi$
$440$	0	0
$441$	11.0517	0.526270
$442$	−15.3398	−0.729639
$443$	−8.71872	−0.414239	−0.207119	−	0.978316i	$-0.566409\pi$
$443$	−8.71872	−0.414239	−0.207119	+	0.978316i	$0.566409\pi$
$444$	−29.4380	−1.39707
$445$	−42.4627	−2.01293
$446$	−15.1181	−0.715862
$447$	20.7607	0.981950
$448$	5.17434	0.244465
$449$	−34.0178	−1.60540	−0.802700	−	0.596383i	$-0.796604\pi$
$449$	−34.0178	−1.60540	−0.802700	+	0.596383i	$0.796604\pi$
$450$	50.8334	2.39631
$451$	0	0
$452$	−47.3990	−2.22946
$453$	−15.7449	−0.739758
$454$	11.2225	0.526696
$455$	−10.5117	−0.492795
$456$	−36.5623	−1.71219
$457$	23.4481	1.09686	0.548428	−	0.836198i	$-0.315227\pi$
$457$	23.4481	1.09686	0.548428	+	0.836198i	$0.315227\pi$
$458$	−19.3766	−0.905407
$459$	−11.0798	−0.517160
$460$	−147.615	−6.88260
$461$	−34.1914	−1.59245	−0.796227	−	0.604998i	$-0.793174\pi$
$461$	−34.1914	−1.59245	−0.796227	+	0.604998i	$0.793174\pi$
$462$	0	0
$463$	8.34781	0.387956	0.193978	−	0.981006i	$-0.437861\pi$
$463$	8.34781	0.387956	0.193978	+	0.981006i	$0.437861\pi$
$464$	35.8383	1.66375
$465$	−4.34255	−0.201381
$466$	60.8120	2.81706
$467$	9.68806	0.448310	0.224155	−	0.974554i	$-0.428038\pi$
$467$	9.68806	0.448310	0.224155	+	0.974554i	$0.428038\pi$
$468$	23.9734	1.10817
$469$	2.03527	0.0939801
$470$	−133.218	−6.14490
$471$	21.6780	0.998872
$472$	19.7679	0.909891
$473$	0	0
$474$	6.01161	0.276122
$475$	−51.1661	−2.34766
$476$	−9.42603	−0.432041
$477$	5.62668	0.257628
$478$	32.9778	1.50837
$479$	35.1006	1.60379	0.801895	−	0.597466i	$-0.203826\pi$
$479$	35.1006	1.60379	0.801895	+	0.597466i	$0.203826\pi$
$480$	39.0834	1.78391
$481$	−15.9591	−0.727672
$482$	−0.269916	−0.0122943
$483$	8.07146	0.367264
$484$	0	0
$485$	−1.37425	−0.0624015
$486$	40.0006	1.81446
$487$	38.6855	1.75301	0.876505	−	0.481393i	$-0.159869\pi$
$487$	38.6855	1.75301	0.876505	+	0.481393i	$0.159869\pi$
$488$	91.4290	4.13879
$489$	18.2241	0.824124
$490$	63.0798	2.84966
$491$	12.4288	0.560905	0.280452	−	0.959868i	$-0.409516\pi$
$491$	12.4288	0.560905	0.280452	+	0.959868i	$0.409516\pi$
$492$	9.47706	0.427259
$493$	−8.46818	−0.381388
$494$	−34.3273	−1.54446
$495$	0	0
$496$	8.93427	0.401161
$497$	−1.64919	−0.0739762
$498$	21.2688	0.953076
$499$	0.696612	0.0311846	0.0155923	−	0.999878i	$-0.495037\pi$
$499$	0.696612	0.0311846	0.0155923	+	0.999878i	$0.495037\pi$
$500$	109.800	4.91042
$501$	−10.9536	−0.489371
$502$	−10.5716	−0.471833
$503$	17.5725	0.783521	0.391761	−	0.920067i	$-0.371866\pi$
$503$	17.5725	0.783521	0.391761	+	0.920067i	$0.371866\pi$
$504$	12.1007	0.539007
$505$	−74.7982	−3.32848
$506$	0	0
$507$	5.62941	0.250011
$508$	−36.3672	−1.61353
$509$	26.6855	1.18281	0.591407	−	0.806373i	$-0.298573\pi$
$509$	26.6855	1.18281	0.591407	+	0.806373i	$0.298573\pi$
$510$	−23.7874	−1.05332
$511$	3.99787	0.176855
$512$	46.2996	2.04617
$513$	−24.7943	−1.09469
$514$	31.4090	1.38539
$515$	−23.1009	−1.01795
$516$	−40.2082	−1.77007
$517$	0	0
$518$	−13.9506	−0.612955
$519$	−2.07165	−0.0909355
$520$	79.0109	3.46486
$521$	−15.6986	−0.687767	−0.343884	−	0.939012i	$-0.611743\pi$
$521$	−15.6986	−0.687767	−0.343884	+	0.939012i	$0.611743\pi$
$522$	18.8268	0.824027
$523$	−22.1730	−0.969557	−0.484778	−	0.874637i	$-0.661100\pi$
$523$	−22.1730	−0.969557	−0.484778	+	0.874637i	$0.661100\pi$
$524$	−85.1779	−3.72101
$525$	−11.1521	−0.486718
$526$	20.1441	0.878325
$527$	−2.11107	−0.0919595
$528$	0	0
$529$	38.4486	1.67168
$530$	32.1155	1.39501
$531$	5.04233	0.218818
$532$	−21.0935	−0.914519
$533$	5.13775	0.222541
$534$	−30.2239	−1.30791
$535$	40.7285	1.76084
$536$	−15.2981	−0.660778
$537$	16.5989	0.716297
$538$	21.1088	0.910067
$539$	0	0
$540$	98.8337	4.25313
$541$	−26.4427	−1.13686	−0.568431	−	0.822731i	$-0.692449\pi$
$541$	−26.4427	−1.13686	−0.568431	+	0.822731i	$0.692449\pi$
$542$	−29.9769	−1.28762
$543$	−17.0657	−0.732358
$544$	18.9998	0.814610
$545$	25.8187	1.10595
$546$	−7.48193	−0.320197
$547$	−2.98712	−0.127720	−0.0638599	−	0.997959i	$-0.520341\pi$
$547$	−2.98712	−0.127720	−0.0638599	+	0.997959i	$0.520341\pi$
$548$	−31.3201	−1.33793
$549$	23.3214	0.995333
$550$	0	0
$551$	−18.9500	−0.807299
$552$	−60.6691	−2.58225
$553$	2.00263	0.0851603
$554$	−5.89879	−0.250615
$555$	−24.7478	−1.05048
$556$	19.7181	0.836232
$557$	20.7439	0.878949	0.439475	−	0.898255i	$-0.355165\pi$
$557$	20.7439	0.878949	0.439475	+	0.898255i	$0.355165\pi$
$558$	4.69341	0.198688
$559$	−21.7979	−0.921953
$560$	33.5362	1.41716
$561$	0	0
$562$	55.6091	2.34573
$563$	17.2469	0.726868	0.363434	−	0.931620i	$-0.381604\pi$
$563$	17.2469	0.726868	0.363434	+	0.931620i	$0.381604\pi$
$564$	−66.6546	−2.80666
$565$	−39.8471	−1.67638
$566$	68.6843	2.88701
$567$	−0.284809	−0.0119609
$568$	12.3961	0.520129
$569$	28.1306	1.17930	0.589648	−	0.807660i	$-0.299267\pi$
$569$	28.1306	1.17930	0.589648	+	0.807660i	$0.299267\pi$
$570$	−53.2313	−2.22961
$571$	−8.79886	−0.368221	−0.184110	−	0.982906i	$-0.558940\pi$
$571$	−8.79886	−0.368221	−0.184110	+	0.982906i	$0.558940\pi$
$572$	0	0
$573$	13.6234	0.569124
$574$	4.49116	0.187457
$575$	−84.9017	−3.54065
$576$	−9.92068	−0.413362
$577$	10.6156	0.441934	0.220967	−	0.975281i	$-0.429079\pi$
$577$	10.6156	0.441934	0.220967	+	0.975281i	$0.429079\pi$
$578$	32.5473	1.35379
$579$	5.52824	0.229746
$580$	75.5377	3.13653
$581$	7.08519	0.293943
$582$	−0.978156	−0.0405459
$583$	0	0
$584$	−30.0499	−1.24348
$585$	20.1538	0.833259
$586$	70.2913	2.90371
$587$	18.7969	0.775831	0.387916	−	0.921695i	$-0.373195\pi$
$587$	18.7969	0.775831	0.387916	+	0.921695i	$0.373195\pi$
$588$	31.5614	1.30157
$589$	−4.72413	−0.194654
$590$	28.7802	1.18486
$591$	4.45454	0.183235
$592$	50.9155	2.09261
$593$	24.9434	1.02430	0.512151	−	0.858895i	$-0.328849\pi$
$593$	24.9434	1.02430	0.512151	+	0.858895i	$0.328849\pi$
$594$	0	0
$595$	−7.92421	−0.324861
$596$	−90.0271	−3.68765
$597$	13.4572	0.550766
$598$	−56.9604	−2.32929
$599$	−35.2146	−1.43883	−0.719415	−	0.694580i	$-0.755590\pi$
$599$	−35.2146	−1.43883	−0.719415	+	0.694580i	$0.755590\pi$
$600$	83.8248	3.42213
$601$	11.4452	0.466857	0.233429	−	0.972374i	$-0.425005\pi$
$601$	11.4452	0.466857	0.233429	+	0.972374i	$0.425005\pi$
$602$	−19.0546	−0.776607
$603$	−3.90219	−0.158910
$604$	68.2761	2.77812
$605$	0	0
$606$	−53.2394	−2.16270
$607$	−12.5735	−0.510341	−0.255170	−	0.966896i	$-0.582132\pi$
$607$	−12.5735	−0.510341	−0.255170	+	0.966896i	$0.582132\pi$
$608$	42.5177	1.72432
$609$	−4.13033	−0.167369
$610$	133.112	5.38955
$611$	−36.1351	−1.46187
$612$	18.0724	0.730532
$613$	39.4093	1.59173	0.795863	−	0.605476i	$-0.207017\pi$
$613$	39.4093	1.59173	0.795863	+	0.605476i	$0.207017\pi$
$614$	−23.7679	−0.959194
$615$	7.96711	0.321265
$616$	0	0
$617$	−10.4788	−0.421862	−0.210931	−	0.977501i	$-0.567650\pi$
$617$	−10.4788	−0.421862	−0.210931	+	0.977501i	$0.567650\pi$
$618$	−16.4426	−0.661419
$619$	41.3087	1.66034	0.830168	−	0.557513i	$-0.188244\pi$
$619$	41.3087	1.66034	0.830168	+	0.557513i	$0.188244\pi$
$620$	18.8311	0.756275
$621$	−41.1420	−1.65097
$622$	−7.39616	−0.296559
$623$	−10.0684	−0.403380
$624$	27.3068	1.09315
$625$	38.1523	1.52609
$626$	−54.0382	−2.15980
$627$	0	0
$628$	−94.0049	−3.75120
$629$	−12.0308	−0.479697
$630$	17.6174	0.701896
$631$	28.2903	1.12622	0.563109	−	0.826383i	$-0.309605\pi$
$631$	28.2903	1.12622	0.563109	+	0.826383i	$0.309605\pi$
$632$	−15.0527	−0.598765
$633$	−19.4129	−0.771592
$634$	−46.3893	−1.84235
$635$	−30.5730	−1.21325
$636$	16.0687	0.637166
$637$	17.1103	0.677933
$638$	0	0
$639$	3.16196	0.125085
$640$	14.9946	0.592715
$641$	−10.4828	−0.414047	−0.207024	−	0.978336i	$-0.566378\pi$
$641$	−10.4828	−0.414047	−0.207024	+	0.978336i	$0.566378\pi$
$642$	28.9895	1.14412
$643$	9.87595	0.389469	0.194735	−	0.980856i	$-0.437615\pi$
$643$	9.87595	0.389469	0.194735	+	0.980856i	$0.437615\pi$
$644$	−35.0012	−1.37924
$645$	−33.8020	−1.33095
$646$	−25.8776	−1.01814
$647$	−23.0701	−0.906979	−0.453490	−	0.891262i	$-0.649821\pi$
$647$	−23.0701	−0.906979	−0.453490	+	0.891262i	$0.649821\pi$
$648$	2.14077	0.0840973
$649$	0	0
$650$	78.7006	3.08689
$651$	−1.02967	−0.0403558
$652$	−79.0273	−3.09495
$653$	23.4917	0.919302	0.459651	−	0.888100i	$-0.347974\pi$
$653$	23.4917	0.919302	0.459651	+	0.888100i	$0.347974\pi$
$654$	18.3771	0.718601
$655$	−71.6068	−2.79791
$656$	−16.3914	−0.639975
$657$	−7.66504	−0.299042
$658$	−31.5875	−1.23141
$659$	−22.9063	−0.892303	−0.446152	−	0.894957i	$-0.647206\pi$
$659$	−22.9063	−0.892303	−0.446152	+	0.894957i	$0.647206\pi$
$660$	0	0
$661$	41.9465	1.63153	0.815764	−	0.578385i	$-0.196317\pi$
$661$	41.9465	1.63153	0.815764	+	0.578385i	$0.196317\pi$
$662$	72.3745	2.81291
$663$	−6.45228	−0.250586
$664$	−53.2557	−2.06672
$665$	−17.7328	−0.687647
$666$	26.7473	1.03644
$667$	−31.4445	−1.21753
$668$	47.4993	1.83780
$669$	−6.35902	−0.245854
$670$	−22.2726	−0.860466
$671$	0	0
$672$	9.26710	0.357486
$673$	37.2341	1.43527	0.717635	−	0.696419i	$-0.245225\pi$
$673$	37.2341	1.43527	0.717635	+	0.696419i	$0.245225\pi$
$674$	−30.7354	−1.18388
$675$	56.8448	2.18796
$676$	−24.4114	−0.938900
$677$	−30.4392	−1.16987	−0.584937	−	0.811079i	$-0.698881\pi$
$677$	−30.4392	−1.16987	−0.584937	+	0.811079i	$0.698881\pi$
$678$	−28.3621	−1.08924
$679$	−0.325850	−0.0125050
$680$	59.5623	2.28411
$681$	4.72043	0.180887
$682$	0	0
$683$	22.5409	0.862505	0.431252	−	0.902231i	$-0.358072\pi$
$683$	22.5409	0.862505	0.431252	+	0.902231i	$0.358072\pi$
$684$	40.4422	1.54635
$685$	−26.3300	−1.00602
$686$	32.0926	1.22530
$687$	−8.15024	−0.310951
$688$	69.5434	2.65132
$689$	8.71126	0.331873
$690$	−88.3285	−3.36261
$691$	24.7842	0.942834	0.471417	−	0.881910i	$-0.343743\pi$
$691$	24.7842	0.942834	0.471417	+	0.881910i	$0.343743\pi$
$692$	8.98354	0.341503
$693$	0	0
$694$	8.19620	0.311123
$695$	16.5765	0.628781
$696$	31.0456	1.17678
$697$	3.87309	0.146704
$698$	2.36159	0.0893876
$699$	25.5790	0.967485
$700$	48.3601	1.82784
$701$	23.4281	0.884868	0.442434	−	0.896801i	$-0.354115\pi$
$701$	23.4281	0.884868	0.442434	+	0.896801i	$0.354115\pi$
$702$	38.1371	1.43939
$703$	−26.9223	−1.01540
$704$	0	0
$705$	−56.0347	−2.11039
$706$	76.7692	2.88925
$707$	−17.7354	−0.667010
$708$	14.3999	0.541182
$709$	−0.631329	−0.0237101	−0.0118550	−	0.999930i	$-0.503774\pi$
$709$	−0.631329	−0.0237101	−0.0118550	+	0.999930i	$0.503774\pi$
$710$	18.0476	0.677313
$711$	−3.83960	−0.143996
$712$	75.6788	2.83618
$713$	−7.83891	−0.293570
$714$	−5.64025	−0.211081
$715$	0	0
$716$	−71.9798	−2.69001
$717$	13.8712	0.518031
$718$	−3.83547	−0.143138
$719$	13.8351	0.515961	0.257981	−	0.966150i	$-0.416943\pi$
$719$	13.8351	0.515961	0.257981	+	0.966150i	$0.416943\pi$
$720$	−64.2983	−2.39626
$721$	−5.47747	−0.203992
$722$	−8.60796	−0.320355
$723$	−0.113533	−0.00422234
$724$	74.0037	2.75033
$725$	43.4459	1.61354
$726$	0	0
$727$	−42.5296	−1.57734	−0.788669	−	0.614819i	$-0.789229\pi$
$727$	−42.5296	−1.57734	−0.788669	+	0.614819i	$0.789229\pi$
$728$	18.7343	0.694340
$729$	17.7309	0.656699
$730$	−43.7499	−1.61926
$731$	−16.4323	−0.607771
$732$	66.6014	2.46166
$733$	20.8343	0.769531	0.384766	−	0.923014i	$-0.374282\pi$
$733$	20.8343	0.769531	0.384766	+	0.923014i	$0.374282\pi$
$734$	0.575335	0.0212360
$735$	26.5329	0.978679
$736$	70.5510	2.60054
$737$	0	0
$738$	−8.61082	−0.316969
$739$	−40.9275	−1.50554	−0.752772	−	0.658281i	$-0.771284\pi$
$739$	−40.9275	−1.50554	−0.752772	+	0.658281i	$0.771284\pi$
$740$	107.316	3.94503
$741$	−14.4389	−0.530425
$742$	7.61493	0.279553
$743$	37.2039	1.36488	0.682440	−	0.730942i	$-0.260919\pi$
$743$	37.2039	1.36488	0.682440	+	0.730942i	$0.260919\pi$
$744$	7.73948	0.283743
$745$	−75.6834	−2.77283
$746$	36.0485	1.31983
$747$	−13.5843	−0.497024
$748$	0	0
$749$	9.65715	0.352865
$750$	65.7012	2.39907
$751$	7.31210	0.266822	0.133411	−	0.991061i	$-0.457407\pi$
$751$	7.31210	0.266822	0.133411	+	0.991061i	$0.457407\pi$
$752$	115.285	4.20400
$753$	−4.44666	−0.162045
$754$	29.1478	1.06150
$755$	57.3979	2.08893
$756$	23.4345	0.852306
$757$	−14.4499	−0.525191	−0.262595	−	0.964906i	$-0.584578\pi$
$757$	−14.4499	−0.525191	−0.262595	+	0.964906i	$0.584578\pi$
$758$	40.8641	1.48425
$759$	0	0
$760$	133.288	4.83487
$761$	−32.7847	−1.18844	−0.594222	−	0.804301i	$-0.702540\pi$
$761$	−32.7847	−1.18844	−0.594222	+	0.804301i	$0.702540\pi$
$762$	−21.7610	−0.788319
$763$	6.12189	0.221627
$764$	−59.0765	−2.13731
$765$	15.1930	0.549302
$766$	72.2043	2.60885
$767$	7.80656	0.281879
$768$	22.6451	0.817133
$769$	−13.1660	−0.474776	−0.237388	−	0.971415i	$-0.576291\pi$
$769$	−13.1660	−0.474776	−0.237388	+	0.971415i	$0.576291\pi$
$770$	0	0
$771$	13.2114	0.475796
$772$	−23.9727	−0.862797
$773$	21.7279	0.781498	0.390749	−	0.920497i	$-0.372216\pi$
$773$	21.7279	0.781498	0.390749	+	0.920497i	$0.372216\pi$
$774$	36.5330	1.31315
$775$	10.8308	0.389054
$776$	2.44925	0.0879228
$777$	−5.86796	−0.210512
$778$	48.8690	1.75204
$779$	8.66718	0.310534
$780$	57.5555	2.06082
$781$	0	0
$782$	−42.9396	−1.53552
$783$	21.0532	0.752380
$784$	−54.5881	−1.94958
$785$	−79.0274	−2.82061
$786$	−50.9679	−1.81796
$787$	−8.24564	−0.293925	−0.146963	−	0.989142i	$-0.546950\pi$
$787$	−8.24564	−0.293925	−0.146963	+	0.989142i	$0.546950\pi$
$788$	−19.3167	−0.688130
$789$	8.47309	0.301650
$790$	−21.9154	−0.779714
$791$	−9.44817	−0.335938
$792$	0	0
$793$	36.1063	1.28217
$794$	−23.8593	−0.846736
$795$	13.5085	0.479099
$796$	−58.3559	−2.06837
$797$	−30.1405	−1.06763	−0.533816	−	0.845601i	$-0.679243\pi$
$797$	−30.1405	−1.06763	−0.533816	+	0.845601i	$0.679243\pi$
$798$	−12.6217	−0.446804
$799$	−27.2404	−0.963697
$800$	−97.4783	−3.44638
$801$	19.3039	0.682070
$802$	−90.7819	−3.20562
$803$	0	0
$804$	−11.1439	−0.393015
$805$	−29.4246	−1.03708
$806$	7.26637	0.255947
$807$	8.87888	0.312551
$808$	133.308	4.68977
$809$	−11.4464	−0.402433	−0.201216	−	0.979547i	$-0.564489\pi$
$809$	−11.4464	−0.402433	−0.201216	+	0.979547i	$0.564489\pi$
$810$	3.11676	0.109512
$811$	26.4982	0.930477	0.465238	−	0.885185i	$-0.345969\pi$
$811$	26.4982	0.930477	0.465238	+	0.885185i	$0.345969\pi$
$812$	17.9108	0.628546
$813$	−12.6090	−0.442217
$814$	0	0
$815$	−66.4362	−2.32716
$816$	20.5852	0.720626
$817$	−36.7721	−1.28649
$818$	−69.4661	−2.42882
$819$	4.77869	0.166981
$820$	−34.5487	−1.20649
$821$	−12.4000	−0.432762	−0.216381	−	0.976309i	$-0.569425\pi$
$821$	−12.4000	−0.432762	−0.216381	+	0.976309i	$0.569425\pi$
$822$	−18.7410	−0.653668
$823$	−46.6157	−1.62492	−0.812460	−	0.583016i	$-0.801872\pi$
$823$	−46.6157	−1.62492	−0.812460	+	0.583016i	$0.801872\pi$
$824$	41.1714	1.43427
$825$	0	0
$826$	6.82409	0.237441
$827$	−14.2837	−0.496693	−0.248347	−	0.968671i	$-0.579887\pi$
$827$	−14.2837	−0.496693	−0.248347	+	0.968671i	$0.579887\pi$
$828$	67.1071	2.33213
$829$	−9.70944	−0.337223	−0.168611	−	0.985683i	$-0.553928\pi$
$829$	−9.70944	−0.337223	−0.168611	+	0.985683i	$0.553928\pi$
$830$	−77.5354	−2.69129
$831$	−2.48117	−0.0860708
$832$	−15.3593	−0.532486
$833$	12.8986	0.446908
$834$	11.7987	0.408555
$835$	39.9314	1.38188
$836$	0	0
$837$	5.24843	0.181412
$838$	30.4847	1.05308
$839$	1.45901	0.0503705	0.0251852	−	0.999683i	$-0.491982\pi$
$839$	1.45901	0.0503705	0.0251852	+	0.999683i	$0.491982\pi$
$840$	29.0513	1.00237
$841$	−12.9092	−0.445147
$842$	42.8217	1.47573
$843$	23.3905	0.805611
$844$	84.1821	2.89767
$845$	−20.5220	−0.705979
$846$	60.5621	2.08217
$847$	0	0
$848$	−27.7922	−0.954388
$849$	28.8902	0.991510
$850$	59.3284	2.03495
$851$	−44.6732	−1.53138
$852$	9.02995	0.309361
$853$	−42.5950	−1.45842	−0.729212	−	0.684288i	$-0.760113\pi$
$853$	−42.5950	−1.45842	−0.729212	+	0.684288i	$0.760113\pi$
$854$	31.5623	1.08004
$855$	33.9987	1.16273
$856$	−72.5879	−2.48100
$857$	−10.9396	−0.373689	−0.186844	−	0.982390i	$-0.559826\pi$
$857$	−10.9396	−0.373689	−0.186844	+	0.982390i	$0.559826\pi$
$858$	0	0
$859$	−10.8166	−0.369059	−0.184530	−	0.982827i	$-0.559076\pi$
$859$	−10.8166	−0.369059	−0.184530	+	0.982827i	$0.559076\pi$
$860$	146.579	4.99831
$861$	1.88909	0.0643799
$862$	71.9215	2.44965
$863$	37.4530	1.27491	0.637457	−	0.770486i	$-0.279986\pi$
$863$	37.4530	1.27491	0.637457	+	0.770486i	$0.279986\pi$
$864$	−47.2364	−1.60702
$865$	7.55223	0.256783
$866$	−20.2844	−0.689292
$867$	13.6902	0.464943
$868$	4.46505	0.151554
$869$	0	0
$870$	45.1995	1.53241
$871$	−6.04140	−0.204705
$872$	−46.0152	−1.55827
$873$	0.624746	0.0211444
$874$	−96.0899	−3.25029
$875$	21.8868	0.739909
$876$	−21.8899	−0.739591
$877$	2.62187	0.0885342	0.0442671	−	0.999020i	$-0.485905\pi$
$877$	2.62187	0.0885342	0.0442671	+	0.999020i	$0.485905\pi$
$878$	−103.225	−3.48367
$879$	29.5662	0.997243
$880$	0	0
$881$	2.04370	0.0688540	0.0344270	−	0.999407i	$-0.489039\pi$
$881$	2.04370	0.0688540	0.0344270	+	0.999407i	$0.489039\pi$
$882$	−28.6766	−0.965591
$883$	7.07464	0.238081	0.119040	−	0.992889i	$-0.462018\pi$
$883$	7.07464	0.238081	0.119040	+	0.992889i	$0.462018\pi$
$884$	27.9797	0.941060
$885$	12.1056	0.406926
$886$	22.6231	0.760039
$887$	−13.8271	−0.464270	−0.232135	−	0.972684i	$-0.574571\pi$
$887$	−13.8271	−0.464270	−0.232135	+	0.972684i	$0.574571\pi$
$888$	44.1065	1.48012
$889$	−7.24917	−0.243130
$890$	110.181	3.69328
$891$	0	0
$892$	27.5753	0.923291
$893$	−60.9585	−2.03990
$894$	−53.8695	−1.80167
$895$	−60.5115	−2.02268
$896$	3.55539	0.118777
$897$	−23.9589	−0.799965
$898$	88.2686	2.94556
$899$	4.01133	0.133785
$900$	−92.7200	−3.09067
$901$	6.56698	0.218778
$902$	0	0
$903$	−8.01481	−0.266716
$904$	71.0171	2.36199
$905$	62.2130	2.06803
$906$	40.8544	1.35730
$907$	8.47212	0.281312	0.140656	−	0.990059i	$-0.455079\pi$
$907$	8.47212	0.281312	0.140656	+	0.990059i	$0.455079\pi$
$908$	−20.4697	−0.679312
$909$	34.0039	1.12784
$910$	27.2754	0.904172
$911$	−5.85873	−0.194108	−0.0970541	−	0.995279i	$-0.530942\pi$
$911$	−5.85873	−0.194108	−0.0970541	+	0.995279i	$0.530942\pi$
$912$	46.0654	1.52538
$913$	0	0
$914$	−60.8425	−2.01249
$915$	55.9901	1.85097
$916$	35.3428	1.16776
$917$	−16.9787	−0.560687
$918$	28.7496	0.948878
$919$	−16.3243	−0.538488	−0.269244	−	0.963072i	$-0.586774\pi$
$919$	−16.3243	−0.538488	−0.269244	+	0.963072i	$0.586774\pi$
$920$	221.170	7.29174
$921$	−9.99734	−0.329424
$922$	88.7191	2.92181
$923$	4.89536	0.161133
$924$	0	0
$925$	61.7236	2.02946
$926$	−21.6607	−0.711815
$927$	10.5019	0.344926
$928$	−36.1023	−1.18512
$929$	−35.3946	−1.16126	−0.580630	−	0.814167i	$-0.697194\pi$
$929$	−35.3946	−1.16126	−0.580630	+	0.814167i	$0.697194\pi$
$930$	11.2680	0.369491
$931$	28.8643	0.945989
$932$	−110.921	−3.63333
$933$	−3.11100	−0.101850
$934$	−25.1383	−0.822552
$935$	0	0
$936$	−35.9190	−1.17405
$937$	−3.92506	−0.128226	−0.0641131	−	0.997943i	$-0.520422\pi$
$937$	−3.92506	−0.128226	−0.0641131	+	0.997943i	$0.520422\pi$
$938$	−5.28108	−0.172433
$939$	−22.7297	−0.741757
$940$	242.990	7.92545
$941$	−38.1921	−1.24503	−0.622513	−	0.782609i	$-0.713888\pi$
$941$	−38.1921	−1.24503	−0.622513	+	0.782609i	$0.713888\pi$
$942$	−56.2497	−1.83271
$943$	14.3817	0.468334
$944$	−24.9059	−0.810617
$945$	19.7008	0.640867
$946$	0	0
$947$	39.5964	1.28671	0.643355	−	0.765568i	$-0.277542\pi$
$947$	39.5964	1.28671	0.643355	+	0.765568i	$0.277542\pi$
$948$	−10.9652	−0.356132
$949$	−11.8671	−0.385221
$950$	132.765	4.30746
$951$	−19.5124	−0.632734
$952$	14.1229	0.457724
$953$	13.8143	0.447490	0.223745	−	0.974648i	$-0.428172\pi$
$953$	13.8143	0.447490	0.223745	+	0.974648i	$0.428172\pi$
$954$	−14.6000	−0.472692
$955$	−49.6640	−1.60709
$956$	−60.1514	−1.94543
$957$	0	0
$958$	−91.0783	−2.94261
$959$	−6.24313	−0.201601
$960$	−23.8176	−0.768710
$961$	1.00000	0.0322581
$962$	41.4103	1.33512
$963$	−18.5155	−0.596653
$964$	0.492326	0.0158567
$965$	−20.1532	−0.648756
$966$	−20.9436	−0.673851
$967$	19.4593	0.625768	0.312884	−	0.949791i	$-0.398705\pi$
$967$	19.4593	0.625768	0.312884	+	0.949791i	$0.398705\pi$
$968$	0	0
$969$	−10.8847	−0.349668
$970$	3.56587	0.114493
$971$	−5.20244	−0.166954	−0.0834771	−	0.996510i	$-0.526603\pi$
$971$	−5.20244	−0.166954	−0.0834771	+	0.996510i	$0.526603\pi$
$972$	−72.9609	−2.34022
$973$	3.93046	0.126005
$974$	−100.380	−3.21639
$975$	33.1033	1.06016
$976$	−115.193	−3.68723
$977$	−26.5606	−0.849748	−0.424874	−	0.905253i	$-0.639682\pi$
$977$	−26.5606	−0.849748	−0.424874	+	0.905253i	$0.639682\pi$
$978$	−47.2876	−1.51209
$979$	0	0
$980$	−115.057	−3.67537
$981$	−11.7374	−0.374746
$982$	−32.2500	−1.02914
$983$	48.0759	1.53338	0.766691	−	0.642016i	$-0.221902\pi$
$983$	48.0759	1.53338	0.766691	+	0.642016i	$0.221902\pi$
$984$	−14.1993	−0.452658
$985$	−16.2391	−0.517420
$986$	21.9730	0.699764
$987$	−13.2864	−0.422912
$988$	62.6129	1.99198
$989$	−61.0173	−1.94024
$990$	0	0
$991$	17.5197	0.556531	0.278265	−	0.960504i	$-0.410240\pi$
$991$	17.5197	0.556531	0.278265	+	0.960504i	$0.410240\pi$
$992$	−9.00010	−0.285753
$993$	30.4424	0.966061
$994$	4.27927	0.135730
$995$	−49.0583	−1.55525
$996$	−38.7942	−1.22924
$997$	−14.2638	−0.451740	−0.225870	−	0.974157i	$-0.572523\pi$
$997$	−14.2638	−0.451740	−0.225870	+	0.974157i	$0.572523\pi$
$998$	−1.80755	−0.0572170
$999$	29.9103	0.946319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3751.2.a.p.1.3		30
11.2	odd	10		341.2.h.a.125.2	✓	60
11.6	odd	10		341.2.h.a.311.2	yes	60
11.10	odd	2		3751.2.a.s.1.28		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
341.2.h.a.125.2	✓	60	11.2	odd	10
341.2.h.a.311.2	yes	60	11.6	odd	10
3751.2.a.p.1.3		30	1.1	even	1	trivial
3751.2.a.s.1.28		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)

\(f(q)\)	\(=\)	\(q-2.59478 q^{2} -1.09142 q^{3} +4.73286 q^{4} +3.97879 q^{5} +2.83200 q^{6} +0.943415 q^{7} -7.09117 q^{8} -1.80879 q^{9} -10.3241 q^{10} -5.16556 q^{12} -2.80038 q^{13} -2.44795 q^{14} -4.34255 q^{15} +8.93427 q^{16} -2.11107 q^{17} +4.69341 q^{18} -4.72413 q^{19} +18.8311 q^{20} -1.02967 q^{21} -7.83891 q^{23} +7.73948 q^{24} +10.8308 q^{25} +7.26637 q^{26} +5.24843 q^{27} +4.46505 q^{28} +4.01133 q^{29} +11.2680 q^{30} +1.00000 q^{31} -9.00010 q^{32} +5.47775 q^{34} +3.75365 q^{35} -8.56077 q^{36} +5.69890 q^{37} +12.2581 q^{38} +3.05641 q^{39} -28.2143 q^{40} -1.83466 q^{41} +2.67175 q^{42} +7.78390 q^{43} -7.19681 q^{45} +20.3402 q^{46} +12.9036 q^{47} -9.75109 q^{48} -6.10997 q^{49} -28.1035 q^{50} +2.30407 q^{51} -13.2538 q^{52} -3.11074 q^{53} -13.6185 q^{54} -6.68991 q^{56} +5.15603 q^{57} -10.4085 q^{58} -2.78768 q^{59} -20.5527 q^{60} -12.8934 q^{61} -2.59478 q^{62} -1.70644 q^{63} +5.48470 q^{64} -11.1422 q^{65} +2.15735 q^{67} -9.99139 q^{68} +8.55558 q^{69} -9.73989 q^{70} -1.74810 q^{71} +12.8265 q^{72} +4.23766 q^{73} -14.7874 q^{74} -11.8210 q^{75} -22.3587 q^{76} -7.93069 q^{78} +2.12274 q^{79} +35.5476 q^{80} -0.301892 q^{81} +4.76053 q^{82} +7.51015 q^{83} -4.87327 q^{84} -8.39950 q^{85} -20.1975 q^{86} -4.37806 q^{87} -10.6723 q^{89} +18.6741 q^{90} -2.64192 q^{91} -37.1005 q^{92} -1.09142 q^{93} -33.4821 q^{94} -18.7963 q^{95} +9.82293 q^{96} -0.345394 q^{97} +15.8540 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 30 q - 9 q^{2} - q^{3} + 29 q^{4} - q^{5} - 11 q^{6} - 11 q^{7} - 27 q^{8} + 25 q^{9} - 8 q^{10} - 16 q^{12} - 2 q^{13} - 3 q^{14} - 11 q^{15} + 35 q^{16} - 32 q^{17} - 4 q^{18} - 17 q^{19} - 4 q^{20}+ \cdots - 40 q^{98}+O(q^{100}) \) 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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