LMFDB - Embedded newform 3751.2.a.s.1.28

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:	$0$
Dimension:	$30$
Twist minimal:	no (minimal twist has level 341)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.28
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	$-0.130290\pi$
$2$	2.59478	1.83478	0.917392	+	0.397985i	$0.130290\pi$
$3$	−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	$-0.557056\pi$
$7$	−0.943415	−0.356577	−0.178289	+	0.983978i	$0.557056\pi$
$8$	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	$-0.373047\pi$
$13$	2.80038	0.776687	0.388343	+	0.921515i	$0.373047\pi$
$14$	−2.44795	−0.654242
$15$	−4.34255	−1.12124
$16$	8.93427	2.23357
$17$	2.11107	0.512009	0.256004	−	0.966676i	$-0.417594\pi$
$17$	2.11107	0.512009	0.256004	+	0.966676i	$0.417594\pi$
$18$	−4.69341	−1.10625
$19$	4.72413	1.08379	0.541895	−	0.840446i	$-0.317707\pi$
$19$	4.72413	1.08379	0.541895	+	0.840446i	$0.317707\pi$
$20$	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	$-0.621480\pi$
$29$	−4.01133	−0.744885	−0.372442	+	0.928055i	$0.621480\pi$
$30$	−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	$-0.454241\pi$
$41$	1.83466	0.286526	0.143263	+	0.989685i	$0.454241\pi$
$42$	2.67175	0.412260
$43$	−7.78390	−1.18703	−0.593516	−	0.804822i	$-0.702261\pi$
$43$	−7.78390	−1.18703	−0.593516	+	0.804822i	$0.702261\pi$
$44$	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	$-0.190943\pi$
$61$	12.8934	1.65082	0.825412	+	0.564530i	$0.190943\pi$
$62$	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	$-0.579770\pi$
$73$	−4.23766	−0.495980	−0.247990	+	0.968763i	$0.579770\pi$
$74$	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	$-0.538101\pi$
$79$	−2.12274	−0.238827	−0.119414	+	0.992845i	$0.538101\pi$
$80$	35.5476	3.97435
$81$	−0.301892	−0.0335436
$82$	4.76053	0.525713
$83$	−7.51015	−0.824346	−0.412173	−	0.911106i	$-0.635230\pi$
$83$	−7.51015	−0.824346	−0.412173	+	0.911106i	$0.635230\pi$
$84$	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	$-0.115134\pi$
$101$	18.7992	1.87059	0.935296	+	0.353868i	$0.115134\pi$
$102$	−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	$-0.664757\pi$
$107$	−10.2364	−0.989588	−0.494794	+	0.869010i	$0.664757\pi$
$108$	24.8401	2.39024
$109$	−6.48908	−0.621541	−0.310771	−	0.950485i	$-0.600587\pi$
$109$	−6.48908	−0.621541	−0.310771	+	0.950485i	$0.600587\pi$
$110$	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	$-0.389261\pi$
$127$	7.68397	0.681842	0.340921	+	0.940092i	$0.389261\pi$
$128$	−3.76863	−0.333103
$129$	8.49553	0.747990
$130$	28.9114	2.53570
$131$	17.9971	1.57241	0.786207	−	0.617963i	$-0.212042\pi$
$131$	17.9971	1.57241	0.786207	+	0.617963i	$0.212042\pi$
$132$	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	$-0.556538\pi$
$139$	−4.16620	−0.353373	−0.176686	+	0.984267i	$0.556538\pi$
$140$	−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	$-0.215646\pi$
$149$	19.0217	1.55832	0.779159	+	0.626826i	$0.215646\pi$
$150$	−30.6729	−2.50443
$151$	−14.4260	−1.17397	−0.586984	−	0.809598i	$-0.699685\pi$
$151$	−14.4260	−1.17397	−0.586984	+	0.809598i	$0.699685\pi$
$152$	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	$-0.626940\pi$
$167$	−10.0361	−0.776614	−0.388307	+	0.921530i	$0.626940\pi$
$168$	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	$-0.522988\pi$
$173$	−1.89812	−0.144311	−0.0721557	+	0.997393i	$0.522988\pi$
$174$	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	$-0.441646\pi$
$193$	5.06516	0.364598	0.182299	+	0.983243i	$0.441646\pi$
$194$	−0.896220	−0.0643448
$195$	−12.1608	−0.870854
$196$	−28.9177	−2.06555
$197$	4.08140	0.290788	0.145394	−	0.989374i	$-0.453555\pi$
$197$	4.08140	0.290788	0.145394	+	0.989374i	$0.453555\pi$
$198$	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	$-0.709733\pi$
$211$	−17.7867	−1.22449	−0.612244	+	0.790669i	$0.709733\pi$
$212$	−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	$-0.454154\pi$
$227$	4.32502	0.287062	0.143531	+	0.989646i	$0.454154\pi$
$228$	−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	$-0.221410\pi$
$233$	23.4363	1.53536	0.767681	+	0.640832i	$0.221410\pi$
$234$	−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	$-0.365163\pi$
$239$	12.7093	0.822096	0.411048	+	0.911614i	$0.365163\pi$
$240$	−38.7976	−2.50437
$241$	−0.104023	−0.00670070	−0.00335035	−	0.999994i	$-0.501066\pi$
$241$	−0.104023	−0.00670070	−0.00335035	+	0.999994i	$0.501066\pi$
$242$	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	$-0.423064\pi$
$263$	7.76333	0.478708	0.239354	+	0.970932i	$0.423064\pi$
$264$	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	$-0.614121\pi$
$271$	−11.5528	−0.701782	−0.350891	+	0.936416i	$0.614121\pi$
$272$	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	$-0.521756\pi$
$277$	−2.27333	−0.136591	−0.0682956	+	0.997665i	$0.521756\pi$
$278$	−10.8104	−0.648362
$279$	−1.80879	−0.108290
$280$	−26.6178	−1.59072
$281$	21.4312	1.27848	0.639238	−	0.769009i	$-0.279250\pi$
$281$	21.4312	1.27848	0.639238	+	0.769009i	$0.279250\pi$
$282$	−36.5431	−2.17611
$283$	26.4702	1.57349	0.786745	−	0.617278i	$-0.211765\pi$
$283$	26.4702	1.57349	0.786745	+	0.617278i	$0.211765\pi$
$284$	−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	$-0.209408\pi$
$293$	27.0896	1.58259	0.791294	+	0.611436i	$0.209408\pi$
$294$	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	$-0.584181\pi$
$307$	−9.15990	−0.522783	−0.261392	+	0.965233i	$0.584181\pi$
$308$	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	$-0.604564\pi$
$337$	−11.8451	−0.645244	−0.322622	+	0.946528i	$0.604564\pi$
$338$	−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	$-0.472980\pi$
$347$	3.15873	0.169570	0.0847848	+	0.996399i	$0.472980\pi$
$348$	20.7208	1.11075
$349$	0.910133	0.0487183	0.0243592	−	0.999703i	$-0.492245\pi$
$349$	0.910133	0.0487183	0.0243592	+	0.999703i	$0.492245\pi$
$350$	−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	$-0.512419\pi$
$359$	−1.47815	−0.0780137	−0.0390069	+	0.999239i	$0.512419\pi$
$360$	−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	$-0.382890\pi$
$373$	13.8927	0.719338	0.359669	+	0.933080i	$0.382890\pi$
$374$	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	$-0.730242\pi$
$409$	−26.7715	−1.32377	−0.661883	+	0.749607i	$0.730242\pi$
$410$	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	$-0.267339\pi$
$431$	27.7178	1.33512	0.667559	+	0.744556i	$0.267339\pi$
$432$	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	$-0.898246\pi$
$439$	−39.7818	−1.89868	−0.949340	+	0.314251i	$0.898246\pi$
$440$	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	$-0.684773\pi$
$457$	−23.4481	−1.09686	−0.548428	+	0.836198i	$0.684773\pi$
$458$	19.3766	0.905407
$459$	11.0798	0.517160
$460$	−147.615	−6.88260
$461$	34.1914	1.59245	0.796227	−	0.604998i	$-0.206826\pi$
$461$	34.1914	1.59245	0.796227	+	0.604998i	$0.206826\pi$
$462$	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	$-0.796174\pi$
$479$	−35.1006	−1.60379	−0.801895	+	0.597466i	$0.796174\pi$
$480$	−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	$-0.590484\pi$
$491$	−12.4288	−0.560905	−0.280452	+	0.959868i	$0.590484\pi$
$492$	−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	$-0.628134\pi$
$503$	−17.5725	−0.783521	−0.391761	+	0.920067i	$0.628134\pi$
$504$	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	$-0.338900\pi$
$523$	22.1730	0.969557	0.484778	+	0.874637i	$0.338900\pi$
$524$	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	$-0.307551\pi$
$541$	26.4427	1.13686	0.568431	+	0.822731i	$0.307551\pi$
$542$	−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	$-0.479659\pi$
$547$	2.98712	0.127720	0.0638599	+	0.997959i	$0.479659\pi$
$548$	−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	$-0.644835\pi$
$557$	−20.7439	−0.878949	−0.439475	+	0.898255i	$0.644835\pi$
$558$	−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	$-0.618396\pi$
$563$	−17.2469	−0.726868	−0.363434	+	0.931620i	$0.618396\pi$
$564$	−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	$-0.700733\pi$
$569$	−28.1306	−1.17930	−0.589648	+	0.807660i	$0.700733\pi$
$570$	−53.2313	−2.22961
$571$	8.79886	0.368221	0.184110	−	0.982906i	$-0.441060\pi$
$571$	8.79886	0.368221	0.184110	+	0.982906i	$0.441060\pi$
$572$	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	$-0.671151\pi$
$593$	−24.9434	−1.02430	−0.512151	+	0.858895i	$0.671151\pi$
$594$	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	$-0.574995\pi$
$601$	−11.4452	−0.466857	−0.233429	+	0.972374i	$0.574995\pi$
$602$	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	$-0.417868\pi$
$607$	12.5735	0.510341	0.255170	+	0.966896i	$0.417868\pi$
$608$	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	$-0.792983\pi$
$613$	−39.4093	−1.59173	−0.795863	+	0.605476i	$0.792983\pi$
$614$	−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	$-0.352794\pi$
$659$	22.9063	0.892303	0.446152	+	0.894957i	$0.352794\pi$
$660$	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	$-0.754775\pi$
$673$	−37.2341	−1.43527	−0.717635	+	0.696419i	$0.754775\pi$
$674$	−30.7354	−1.18388
$675$	56.8448	2.18796
$676$	−24.4114	−0.938900
$677$	30.4392	1.16987	0.584937	−	0.811079i	$-0.301119\pi$
$677$	30.4392	1.16987	0.584937	+	0.811079i	$0.301119\pi$
$678$	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	$-0.645885\pi$
$701$	−23.4281	−0.884868	−0.442434	+	0.896801i	$0.645885\pi$
$702$	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	$-0.625718\pi$
$733$	−20.8343	−0.769531	−0.384766	+	0.923014i	$0.625718\pi$
$734$	−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	$-0.228716\pi$
$739$	40.9275	1.50554	0.752772	+	0.658281i	$0.228716\pi$
$740$	107.316	3.94503
$741$	−14.4389	−0.530425
$742$	7.61493	0.279553
$743$	−37.2039	−1.36488	−0.682440	−	0.730942i	$-0.739081\pi$
$743$	−37.2039	−1.36488	−0.682440	+	0.730942i	$0.739081\pi$
$744$	−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	$-0.297460\pi$
$761$	32.7847	1.18844	0.594222	+	0.804301i	$0.297460\pi$
$762$	−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	$-0.423709\pi$
$769$	13.1660	0.474776	0.237388	+	0.971415i	$0.423709\pi$
$770$	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	$-0.453050\pi$
$787$	8.24564	0.293925	0.146963	+	0.989142i	$0.453050\pi$
$788$	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	$-0.435511\pi$
$809$	11.4464	0.402433	0.201216	+	0.979547i	$0.435511\pi$
$810$	−3.11676	−0.109512
$811$	−26.4982	−0.930477	−0.465238	−	0.885185i	$-0.654031\pi$
$811$	−26.4982	−0.930477	−0.465238	+	0.885185i	$0.654031\pi$
$812$	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	$-0.430575\pi$
$821$	12.4000	0.432762	0.216381	+	0.976309i	$0.430575\pi$
$822$	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	$-0.420113\pi$
$827$	14.2837	0.496693	0.248347	+	0.968671i	$0.420113\pi$
$828$	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	$-0.239887\pi$
$853$	42.5950	1.45842	0.729212	+	0.684288i	$0.239887\pi$
$854$	−31.5623	−1.08004
$855$	−33.9987	−1.16273
$856$	−72.5879	−2.48100
$857$	10.9396	0.373689	0.186844	−	0.982390i	$-0.440174\pi$
$857$	10.9396	0.373689	0.186844	+	0.982390i	$0.440174\pi$
$858$	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	$-0.514095\pi$
$877$	−2.62187	−0.0885342	−0.0442671	+	0.999020i	$0.514095\pi$
$878$	−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	$-0.425429\pi$
$887$	13.8271	0.464270	0.232135	+	0.972684i	$0.425429\pi$
$888$	−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	$-0.413226\pi$
$919$	16.3243	0.538488	0.269244	+	0.963072i	$0.413226\pi$
$920$	−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	$-0.479578\pi$
$937$	3.92506	0.128226	0.0641131	+	0.997943i	$0.479578\pi$
$938$	−5.28108	−0.172433
$939$	−22.7297	−0.741757
$940$	242.990	7.92545
$941$	38.1921	1.24503	0.622513	−	0.782609i	$-0.286112\pi$
$941$	38.1921	1.24503	0.622513	+	0.782609i	$0.286112\pi$
$942$	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	$-0.571828\pi$
$953$	−13.8143	−0.447490	−0.223745	+	0.974648i	$0.571828\pi$
$954$	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	$-0.601295\pi$
$967$	−19.4593	−0.625768	−0.312884	+	0.949791i	$0.601295\pi$
$968$	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	$-0.427477\pi$
$997$	14.2638	0.451740	0.225870	+	0.974157i	$0.427477\pi$
$998$	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.s.1.28		30
11.5	even	5		341.2.h.a.311.2	yes	60
11.9	even	5		341.2.h.a.125.2	✓	60
11.10	odd	2		3751.2.a.p.1.3		30

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
341.2.h.a.125.2	✓	60	11.9	even	5
341.2.h.a.311.2	yes	60	11.5	even	5
3751.2.a.p.1.3		30	11.10	odd	2
3751.2.a.s.1.28		30	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3751 = 11^{2} \cdot 31 \)
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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