LMFDB - Embedded newform 9065.2.a.v.1.15

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9065,2,Mod(1,9065)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9065 = 5 \cdot 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9065.a (trivial)

Newform invariants

comment:select newform

sage:traces = [21,-3,-1,13,21,-6,0,-9,12,-3,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.3843894323$
Analytic rank:	$1$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 1295)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	9065.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.872576 q^{2} +2.26649 q^{3} -1.23861 q^{4} +1.00000 q^{5} +1.97769 q^{6} -2.82593 q^{8} +2.13699 q^{9} +0.872576 q^{10} -5.40482 q^{11} -2.80731 q^{12} +5.49607 q^{13} +2.26649 q^{15} +0.0113829 q^{16} +1.44877 q^{17} +1.86469 q^{18} -4.90666 q^{19} -1.23861 q^{20} -4.71611 q^{22} -3.85047 q^{23} -6.40496 q^{24} +1.00000 q^{25} +4.79574 q^{26} -1.95600 q^{27} +7.76996 q^{29} +1.97769 q^{30} -5.05646 q^{31} +5.66180 q^{32} -12.2500 q^{33} +1.26417 q^{34} -2.64690 q^{36} -1.00000 q^{37} -4.28143 q^{38} +12.4568 q^{39} -2.82593 q^{40} +4.23454 q^{41} -5.75256 q^{43} +6.69447 q^{44} +2.13699 q^{45} -3.35982 q^{46} -7.03986 q^{47} +0.0257993 q^{48} +0.872576 q^{50} +3.28364 q^{51} -6.80750 q^{52} +2.32608 q^{53} -1.70676 q^{54} -5.40482 q^{55} -11.1209 q^{57} +6.77988 q^{58} -11.5027 q^{59} -2.80731 q^{60} -13.6375 q^{61} -4.41214 q^{62} +4.91758 q^{64} +5.49607 q^{65} -10.6890 q^{66} +3.60950 q^{67} -1.79447 q^{68} -8.72706 q^{69} -1.09597 q^{71} -6.03900 q^{72} +12.8366 q^{73} -0.872576 q^{74} +2.26649 q^{75} +6.07745 q^{76} +10.8695 q^{78} -9.47305 q^{79} +0.0113829 q^{80} -10.8442 q^{81} +3.69495 q^{82} +8.06502 q^{83} +1.44877 q^{85} -5.01955 q^{86} +17.6106 q^{87} +15.2736 q^{88} -15.4238 q^{89} +1.86469 q^{90} +4.76924 q^{92} -11.4604 q^{93} -6.14281 q^{94} -4.90666 q^{95} +12.8324 q^{96} -18.1098 q^{97} -11.5501 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q - 3 q^{2} - q^{3} + 13 q^{4} + 21 q^{5} - 6 q^{6} - 9 q^{8} + 12 q^{9} - 3 q^{10} - 5 q^{11} - 4 q^{12} + q^{13} - q^{15} + q^{16} + 2 q^{17} - 11 q^{18} - 12 q^{19} + 13 q^{20} - 10 q^{22} - 20 q^{23}+ \cdots - 7 q^{99}+O(q^{100}) $ 21 * q - 3 * q^2 - q^3 + 13 * q^4 + 21 * q^5 - 6 * q^6 - 9 * q^8 + 12 * q^9 - 3 * q^10 - 5 * q^11 - 4 * q^12 + q^13 - q^15 + q^16 + 2 * q^17 - 11 * q^18 - 12 * q^19 + 13 * q^20 - 10 * q^22 - 20 * q^23 - 3 * q^24 + 21 * q^25 - 13 * q^26 - 7 * q^27 - 11 * q^29 - 6 * q^30 - 8 * q^31 - 16 * q^32 - 10 * q^33 + 6 * q^34 - 11 * q^36 - 21 * q^37 - 2 * q^38 - 37 * q^39 - 9 * q^40 - 20 * q^41 - 59 * q^43 - 23 * q^44 + 12 * q^45 - 11 * q^46 - 2 * q^47 + q^48 - 3 * q^50 - 47 * q^51 + 22 * q^52 - 2 * q^53 + 4 * q^54 - 5 * q^55 - 20 * q^57 - 35 * q^58 - 54 * q^59 - 4 * q^60 + 4 * q^61 + 30 * q^62 - 21 * q^64 + q^65 - 44 * q^66 - 15 * q^67 - 15 * q^68 + 2 * q^69 - 13 * q^71 - 61 * q^72 - 4 * q^73 + 3 * q^74 - q^75 - 13 * q^76 + 37 * q^78 - 91 * q^79 + q^80 - 7 * q^81 + 11 * q^82 + 12 * q^83 + 2 * q^85 + 19 * q^86 + 16 * q^87 + 10 * q^88 - 38 * q^89 - 11 * q^90 + 25 * q^92 + 7 * q^93 - 22 * q^94 - 12 * q^95 + 4 * q^96 - 4 * q^97 - 7 * q^99

Coefficient data

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

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

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

$2$	0.872576	0.617004	0.308502	−	0.951224i	$-0.400172\pi$
$2$	0.872576	0.617004	0.308502	+	0.951224i	$0.400172\pi$
$3$	2.26649	1.30856	0.654280	−	0.756252i	$-0.272972\pi$
$3$	2.26649	1.30856	0.654280	+	0.756252i	$0.272972\pi$
$4$	−1.23861	−0.619306
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.97769	0.807387
$7$	0	0
$7$	0	0
$8$	−2.82593	−0.999118
$9$	2.13699	0.712331
$10$	0.872576	0.275933
$11$	−5.40482	−1.62961	−0.814807	−	0.579733i	$-0.803157\pi$
$11$	−5.40482	−1.62961	−0.814807	+	0.579733i	$0.803157\pi$
$12$	−2.80731	−0.810399
$13$	5.49607	1.52434	0.762168	−	0.647379i	$-0.224135\pi$
$13$	5.49607	1.52434	0.762168	+	0.647379i	$0.224135\pi$
$14$	0	0
$15$	2.26649	0.585206
$16$	0.0113829	0.00284572
$17$	1.44877	0.351379	0.175690	−	0.984446i	$-0.443784\pi$
$17$	1.44877	0.351379	0.175690	+	0.984446i	$0.443784\pi$
$18$	1.86469	0.439511
$19$	−4.90666	−1.12567	−0.562833	−	0.826571i	$-0.690289\pi$
$19$	−4.90666	−1.12567	−0.562833	+	0.826571i	$0.690289\pi$
$20$	−1.23861	−0.276962
$21$	0	0
$22$	−4.71611	−1.00548
$23$	−3.85047	−0.802878	−0.401439	−	0.915886i	$-0.631490\pi$
$23$	−3.85047	−0.802878	−0.401439	+	0.915886i	$0.631490\pi$
$24$	−6.40496	−1.30741
$25$	1.00000	0.200000
$26$	4.79574	0.940521
$27$	−1.95600	−0.376432
$28$	0	0
$29$	7.76996	1.44284	0.721422	−	0.692495i	$-0.243489\pi$
$29$	7.76996	1.44284	0.721422	+	0.692495i	$0.243489\pi$
$30$	1.97769	0.361075
$31$	−5.05646	−0.908167	−0.454084	−	0.890959i	$-0.650033\pi$
$31$	−5.05646	−0.908167	−0.454084	+	0.890959i	$0.650033\pi$
$32$	5.66180	1.00087
$33$	−12.2500	−2.13245
$34$	1.26417	0.216803
$35$	0	0
$36$	−2.64690	−0.441151
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−4.28143	−0.694540
$39$	12.4568	1.99469
$40$	−2.82593	−0.446819
$41$	4.23454	0.661324	0.330662	−	0.943749i	$-0.392728\pi$
$41$	4.23454	0.661324	0.330662	+	0.943749i	$0.392728\pi$
$42$	0	0
$43$	−5.75256	−0.877258	−0.438629	−	0.898668i	$-0.644536\pi$
$43$	−5.75256	−0.877258	−0.438629	+	0.898668i	$0.644536\pi$
$44$	6.69447	1.00923
$45$	2.13699	0.318564
$46$	−3.35982	−0.495379
$47$	−7.03986	−1.02687	−0.513435	−	0.858129i	$-0.671627\pi$
$47$	−7.03986	−1.02687	−0.513435	+	0.858129i	$0.671627\pi$
$48$	0.0257993	0.00372380
$49$	0	0
$50$	0.872576	0.123401
$51$	3.28364	0.459801
$52$	−6.80750	−0.944030
$53$	2.32608	0.319511	0.159756	−	0.987157i	$-0.448929\pi$
$53$	2.32608	0.319511	0.159756	+	0.987157i	$0.448929\pi$
$54$	−1.70676	−0.232260
$55$	−5.40482	−0.728785
$56$	0	0
$57$	−11.1209	−1.47300
$58$	6.77988	0.890241
$59$	−11.5027	−1.49752	−0.748759	−	0.662843i	$-0.769350\pi$
$59$	−11.5027	−1.49752	−0.748759	+	0.662843i	$0.769350\pi$
$60$	−2.80731	−0.362422
$61$	−13.6375	−1.74610	−0.873051	−	0.487628i	$-0.837862\pi$
$61$	−13.6375	−1.74610	−0.873051	+	0.487628i	$0.837862\pi$
$62$	−4.41214	−0.560343
$63$	0	0
$64$	4.91758	0.614698
$65$	5.49607	0.681704
$66$	−10.6890	−1.31573
$67$	3.60950	0.440971	0.220486	−	0.975390i	$-0.429236\pi$
$67$	3.60950	0.440971	0.220486	+	0.975390i	$0.429236\pi$
$68$	−1.79447	−0.217611
$69$	−8.72706	−1.05061
$70$	0	0
$71$	−1.09597	−0.130068	−0.0650340	−	0.997883i	$-0.520716\pi$
$71$	−1.09597	−0.130068	−0.0650340	+	0.997883i	$0.520716\pi$
$72$	−6.03900	−0.711703
$73$	12.8366	1.50241	0.751203	−	0.660071i	$-0.229474\pi$
$73$	12.8366	1.50241	0.751203	+	0.660071i	$0.229474\pi$
$74$	−0.872576	−0.101435
$75$	2.26649	0.261712
$76$	6.07745	0.697131
$77$	0	0
$78$	10.8695	1.23073
$79$	−9.47305	−1.06580	−0.532901	−	0.846178i	$-0.678898\pi$
$79$	−9.47305	−1.06580	−0.532901	+	0.846178i	$0.678898\pi$
$80$	0.0113829	0.00127265
$81$	−10.8442	−1.20492
$82$	3.69495	0.408039
$83$	8.06502	0.885251	0.442626	−	0.896707i	$-0.354047\pi$
$83$	8.06502	0.885251	0.442626	+	0.896707i	$0.354047\pi$
$84$	0	0
$85$	1.44877	0.157142
$86$	−5.01955	−0.541272
$87$	17.6106	1.88805
$88$	15.2736	1.62818
$89$	−15.4238	−1.63492	−0.817461	−	0.575984i	$-0.804619\pi$
$89$	−15.4238	−1.63492	−0.817461	+	0.575984i	$0.804619\pi$
$90$	1.86469	0.196555
$91$	0	0
$92$	4.76924	0.497227
$93$	−11.4604	−1.18839
$94$	−6.14281	−0.633583
$95$	−4.90666	−0.503413
$96$	12.8324	1.30970
$97$	−18.1098	−1.83877	−0.919386	−	0.393356i	$-0.871314\pi$
$97$	−18.1098	−1.83877	−0.919386	+	0.393356i	$0.871314\pi$
$98$	0	0
$99$	−11.5501	−1.16082
$100$	−1.23861	−0.123861
$101$	16.2339	1.61534	0.807669	−	0.589636i	$-0.200729\pi$
$101$	16.2339	1.61534	0.807669	+	0.589636i	$0.200729\pi$
$102$	2.86522	0.283699
$103$	11.2969	1.11311	0.556557	−	0.830810i	$-0.312122\pi$
$103$	11.2969	1.11311	0.556557	+	0.830810i	$0.312122\pi$
$104$	−15.5315	−1.52299
$105$	0	0
$106$	2.02968	0.197140
$107$	−6.19187	−0.598591	−0.299295	−	0.954160i	$-0.596752\pi$
$107$	−6.19187	−0.598591	−0.299295	+	0.954160i	$0.596752\pi$
$108$	2.42272	0.233127
$109$	−11.2632	−1.07882	−0.539410	−	0.842044i	$-0.681353\pi$
$109$	−11.2632	−1.07882	−0.539410	+	0.842044i	$0.681353\pi$
$110$	−4.71611	−0.449663
$111$	−2.26649	−0.215126
$112$	0	0
$113$	−14.8020	−1.39245	−0.696227	−	0.717821i	$-0.745139\pi$
$113$	−14.8020	−1.39245	−0.696227	+	0.717821i	$0.745139\pi$
$114$	−9.70384	−0.908848
$115$	−3.85047	−0.359058
$116$	−9.62396	−0.893562
$117$	11.7451	1.08583
$118$	−10.0369	−0.923975
$119$	0	0
$120$	−6.40496	−0.584690
$121$	18.2120	1.65564
$122$	−11.8997	−1.07735
$123$	9.59755	0.865382
$124$	6.26299	0.562433
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.39298	0.389814	0.194907	−	0.980822i	$-0.437560\pi$
$127$	4.39298	0.389814	0.194907	+	0.980822i	$0.437560\pi$
$128$	−7.03264	−0.621603
$129$	−13.0381	−1.14794
$130$	4.79574	0.420614
$131$	−15.7079	−1.37241	−0.686203	−	0.727410i	$-0.740724\pi$
$131$	−15.7079	−1.37241	−0.686203	+	0.727410i	$0.740724\pi$
$132$	15.1730	1.32064
$133$	0	0
$134$	3.14957	0.272081
$135$	−1.95600	−0.168346
$136$	−4.09414	−0.351070
$137$	−0.773751	−0.0661060	−0.0330530	−	0.999454i	$-0.510523\pi$
$137$	−0.773751	−0.0661060	−0.0330530	+	0.999454i	$0.510523\pi$
$138$	−7.61502	−0.648234
$139$	−9.68350	−0.821344	−0.410672	−	0.911783i	$-0.634706\pi$
$139$	−9.68350	−0.821344	−0.410672	+	0.911783i	$0.634706\pi$
$140$	0	0
$141$	−15.9558	−1.34372
$142$	−0.956318	−0.0802524
$143$	−29.7052	−2.48408
$144$	0.0243252	0.00202710
$145$	7.76996	0.645260
$146$	11.2009	0.926991
$147$	0	0
$148$	1.23861	0.101813
$149$	4.53348	0.371397	0.185699	−	0.982607i	$-0.440545\pi$
$149$	4.53348	0.371397	0.185699	+	0.982607i	$0.440545\pi$
$150$	1.97769	0.161477
$151$	3.68593	0.299956	0.149978	−	0.988689i	$-0.452080\pi$
$151$	3.68593	0.299956	0.149978	+	0.988689i	$0.452080\pi$
$152$	13.8659	1.12467
$153$	3.09602	0.250298
$154$	0	0
$155$	−5.05646	−0.406145
$156$	−15.4292	−1.23532
$157$	2.15240	0.171780	0.0858901	−	0.996305i	$-0.472627\pi$
$157$	2.15240	0.171780	0.0858901	+	0.996305i	$0.472627\pi$
$158$	−8.26595	−0.657604
$159$	5.27204	0.418100
$160$	5.66180	0.447605
$161$	0	0
$162$	−9.46242	−0.743438
$163$	14.2599	1.11692	0.558462	−	0.829530i	$-0.311391\pi$
$163$	14.2599	1.11692	0.558462	+	0.829530i	$0.311391\pi$
$164$	−5.24495	−0.409562
$165$	−12.2500	−0.953660
$166$	7.03734	0.546204
$167$	−14.0933	−1.09057	−0.545287	−	0.838250i	$-0.683579\pi$
$167$	−14.0933	−1.09057	−0.545287	+	0.838250i	$0.683579\pi$
$168$	0	0
$169$	17.2068	1.32360
$170$	1.26417	0.0969570
$171$	−10.4855	−0.801847
$172$	7.12519	0.543291
$173$	−16.7586	−1.27413	−0.637066	−	0.770809i	$-0.719852\pi$
$173$	−16.7586	−1.27413	−0.637066	+	0.770809i	$0.719852\pi$
$174$	15.3665	1.16493
$175$	0	0
$176$	−0.0615224	−0.00463743
$177$	−26.0707	−1.95959
$178$	−13.4585	−1.00875
$179$	−0.214819	−0.0160564	−0.00802818	−	0.999968i	$-0.502555\pi$
$179$	−0.214819	−0.0160564	−0.00802818	+	0.999968i	$0.502555\pi$
$180$	−2.64690	−0.197289
$181$	−10.8076	−0.803325	−0.401662	−	0.915788i	$-0.631567\pi$
$181$	−10.8076	−0.803325	−0.401662	+	0.915788i	$0.631567\pi$
$182$	0	0
$183$	−30.9093	−2.28488
$184$	10.8812	0.802170
$185$	−1.00000	−0.0735215
$186$	−10.0001	−0.733243
$187$	−7.83036	−0.572612
$188$	8.71966	0.635946
$189$	0	0
$190$	−4.28143	−0.310608
$191$	−10.3541	−0.749195	−0.374598	−	0.927187i	$-0.622219\pi$
$191$	−10.3541	−0.749195	−0.374598	+	0.927187i	$0.622219\pi$
$192$	11.1457	0.804369
$193$	−12.4078	−0.893135	−0.446567	−	0.894750i	$-0.647354\pi$
$193$	−12.4078	−0.893135	−0.446567	+	0.894750i	$0.647354\pi$
$194$	−15.8022	−1.13453
$195$	12.4568	0.892051
$196$	0	0
$197$	8.67524	0.618085	0.309042	−	0.951048i	$-0.399991\pi$
$197$	8.67524	0.618085	0.309042	+	0.951048i	$0.399991\pi$
$198$	−10.0783	−0.716233
$199$	21.8456	1.54859	0.774295	−	0.632825i	$-0.218105\pi$
$199$	21.8456	1.54859	0.774295	+	0.632825i	$0.218105\pi$
$200$	−2.82593	−0.199824
$201$	8.18092	0.577038
$202$	14.1653	0.996670
$203$	0	0
$204$	−4.06715	−0.284758
$205$	4.23454	0.295753
$206$	9.85737	0.686796
$207$	−8.22843	−0.571915
$208$	0.0625612	0.00433784
$209$	26.5196	1.83440
$210$	0	0
$211$	−14.7366	−1.01451	−0.507255	−	0.861796i	$-0.669340\pi$
$211$	−14.7366	−1.01451	−0.507255	+	0.861796i	$0.669340\pi$
$212$	−2.88111	−0.197875
$213$	−2.48401	−0.170202
$214$	−5.40288	−0.369333
$215$	−5.75256	−0.392322
$216$	5.52752	0.376100
$217$	0	0
$218$	−9.82800	−0.665636
$219$	29.0940	1.96599
$220$	6.69447	0.451341
$221$	7.96257	0.535620
$222$	−1.97769	−0.132734
$223$	−2.29486	−0.153675	−0.0768376	−	0.997044i	$-0.524482\pi$
$223$	−2.29486	−0.153675	−0.0768376	+	0.997044i	$0.524482\pi$
$224$	0	0
$225$	2.13699	0.142466
$226$	−12.9159	−0.859150
$227$	9.51861	0.631772	0.315886	−	0.948797i	$-0.397698\pi$
$227$	9.51861	0.631772	0.315886	+	0.948797i	$0.397698\pi$
$228$	13.7745	0.912239
$229$	6.04664	0.399573	0.199787	−	0.979839i	$-0.435975\pi$
$229$	6.04664	0.399573	0.199787	+	0.979839i	$0.435975\pi$
$230$	−3.35982	−0.221540
$231$	0	0
$232$	−21.9574	−1.44157
$233$	15.9764	1.04665	0.523323	−	0.852134i	$-0.324692\pi$
$233$	15.9764	1.04665	0.523323	+	0.852134i	$0.324692\pi$
$234$	10.2485	0.669963
$235$	−7.03986	−0.459230
$236$	14.2473	0.927421
$237$	−21.4706	−1.39467
$238$	0	0
$239$	26.1360	1.69060	0.845298	−	0.534295i	$-0.179423\pi$
$239$	26.1360	1.69060	0.845298	+	0.534295i	$0.179423\pi$
$240$	0.0257993	0.00166533
$241$	−21.0790	−1.35782	−0.678910	−	0.734222i	$-0.737547\pi$
$241$	−21.0790	−1.35782	−0.678910	+	0.734222i	$0.737547\pi$
$242$	15.8914	1.02154
$243$	−18.7104	−1.20027
$244$	16.8916	1.08137
$245$	0	0
$246$	8.37459	0.533944
$247$	−26.9674	−1.71589
$248$	14.2892	0.907367
$249$	18.2793	1.15840
$250$	0.872576	0.0551865
$251$	−13.2932	−0.839057	−0.419528	−	0.907742i	$-0.637805\pi$
$251$	−13.2932	−0.839057	−0.419528	+	0.907742i	$0.637805\pi$
$252$	0	0
$253$	20.8111	1.30838
$254$	3.83320	0.240517
$255$	3.28364	0.205629
$256$	−15.9717	−0.998230
$257$	4.70785	0.293668	0.146834	−	0.989161i	$-0.453092\pi$
$257$	4.70785	0.293668	0.146834	+	0.989161i	$0.453092\pi$
$258$	−11.3768	−0.708287
$259$	0	0
$260$	−6.80750	−0.422183
$261$	16.6043	1.02778
$262$	−13.7063	−0.846781
$263$	−26.4699	−1.63221	−0.816103	−	0.577907i	$-0.803870\pi$
$263$	−26.4699	−1.63221	−0.816103	+	0.577907i	$0.803870\pi$
$264$	34.6176	2.13057
$265$	2.32608	0.142890
$266$	0	0
$267$	−34.9580	−2.13940
$268$	−4.47078	−0.273096
$269$	24.8461	1.51489	0.757445	−	0.652898i	$-0.226447\pi$
$269$	24.8461	1.51489	0.757445	+	0.652898i	$0.226447\pi$
$270$	−1.70676	−0.103870
$271$	15.7458	0.956489	0.478244	−	0.878227i	$-0.341273\pi$
$271$	15.7458	0.956489	0.478244	+	0.878227i	$0.341273\pi$
$272$	0.0164912	0.000999929 0
$273$	0	0
$274$	−0.675157	−0.0407877
$275$	−5.40482	−0.325923
$276$	10.8094	0.650652
$277$	−2.58291	−0.155192	−0.0775961	−	0.996985i	$-0.524724\pi$
$277$	−2.58291	−0.155192	−0.0775961	+	0.996985i	$0.524724\pi$
$278$	−8.44958	−0.506772
$279$	−10.8056	−0.646916
$280$	0	0
$281$	26.0538	1.55424	0.777120	−	0.629353i	$-0.216680\pi$
$281$	26.0538	1.55424	0.777120	+	0.629353i	$0.216680\pi$
$282$	−13.9226	−0.829082
$283$	−14.0353	−0.834313	−0.417156	−	0.908835i	$-0.636973\pi$
$283$	−14.0353	−0.834313	−0.417156	+	0.908835i	$0.636973\pi$
$284$	1.35748	0.0805518
$285$	−11.1209	−0.658746
$286$	−25.9201	−1.53269
$287$	0	0
$288$	12.0992	0.712954
$289$	−14.9011	−0.876533
$290$	6.77988	0.398128
$291$	−41.0458	−2.40615
$292$	−15.8995	−0.930449
$293$	9.29101	0.542787	0.271393	−	0.962469i	$-0.412516\pi$
$293$	9.29101	0.542787	0.271393	+	0.962469i	$0.412516\pi$
$294$	0	0
$295$	−11.5027	−0.669710
$296$	2.82593	0.164254
$297$	10.5718	0.613439
$298$	3.95581	0.229154
$299$	−21.1624	−1.22386
$300$	−2.80731	−0.162080
$301$	0	0
$302$	3.21625	0.185074
$303$	36.7941	2.11377
$304$	−0.0558520	−0.00320333
$305$	−13.6375	−0.780881
$306$	2.70151	0.154435
$307$	18.2744	1.04297	0.521486	−	0.853260i	$-0.325378\pi$
$307$	18.2744	1.04297	0.521486	+	0.853260i	$0.325378\pi$
$308$	0	0
$309$	25.6043	1.45658
$310$	−4.41214	−0.250593
$311$	−28.8779	−1.63752	−0.818758	−	0.574138i	$-0.805337\pi$
$311$	−28.8779	−1.63752	−0.818758	+	0.574138i	$0.805337\pi$
$312$	−35.2021	−1.99293
$313$	−10.0722	−0.569314	−0.284657	−	0.958629i	$-0.591880\pi$
$313$	−10.0722	−0.569314	−0.284657	+	0.958629i	$0.591880\pi$
$314$	1.87813	0.105989
$315$	0	0
$316$	11.7334	0.660057
$317$	−11.7781	−0.661524	−0.330762	−	0.943714i	$-0.607306\pi$
$317$	−11.7781	−0.661524	−0.330762	+	0.943714i	$0.607306\pi$
$318$	4.60025	0.257969
$319$	−41.9952	−2.35128
$320$	4.91758	0.274901
$321$	−14.0338	−0.783292
$322$	0	0
$323$	−7.10865	−0.395536
$324$	13.4318	0.746211
$325$	5.49607	0.304867
$326$	12.4429	0.689147
$327$	−25.5280	−1.41170
$328$	−11.9665	−0.660741
$329$	0	0
$330$	−10.6890	−0.588412
$331$	−16.4942	−0.906604	−0.453302	−	0.891357i	$-0.649754\pi$
$331$	−16.4942	−0.906604	−0.453302	+	0.891357i	$0.649754\pi$
$332$	−9.98943	−0.548241
$333$	−2.13699	−0.117107
$334$	−12.2975	−0.672888
$335$	3.60950	0.197208
$336$	0	0
$337$	9.15608	0.498763	0.249382	−	0.968405i	$-0.419773\pi$
$337$	9.15608	0.498763	0.249382	+	0.968405i	$0.419773\pi$
$338$	15.0142	0.816666
$339$	−33.5486	−1.82211
$340$	−1.79447	−0.0973187
$341$	27.3292	1.47996
$342$	−9.14940	−0.494743
$343$	0	0
$344$	16.2564	0.876484
$345$	−8.72706	−0.469849
$346$	−14.6231	−0.786145
$347$	−12.4922	−0.670616	−0.335308	−	0.942109i	$-0.608840\pi$
$347$	−12.4922	−0.670616	−0.335308	+	0.942109i	$0.608840\pi$
$348$	−21.8126	−1.16928
$349$	19.1322	1.02412	0.512062	−	0.858948i	$-0.328882\pi$
$349$	19.1322	1.02412	0.512062	+	0.858948i	$0.328882\pi$
$350$	0	0
$351$	−10.7503	−0.573809
$352$	−30.6010	−1.63104
$353$	−18.3652	−0.977481	−0.488740	−	0.872429i	$-0.662543\pi$
$353$	−18.3652	−0.977481	−0.488740	+	0.872429i	$0.662543\pi$
$354$	−22.7486	−1.20908
$355$	−1.09597	−0.0581681
$356$	19.1041	1.01252
$357$	0	0
$358$	−0.187446	−0.00990684
$359$	0.493019	0.0260206	0.0130103	−	0.999915i	$-0.495859\pi$
$359$	0.493019	0.0260206	0.0130103	+	0.999915i	$0.495859\pi$
$360$	−6.03900	−0.318283
$361$	5.07534	0.267123
$362$	−9.43048	−0.495655
$363$	41.2774	2.16650
$364$	0	0
$365$	12.8366	0.671896
$366$	−26.9707	−1.40978
$367$	25.9495	1.35456	0.677278	−	0.735728i	$-0.263160\pi$
$367$	25.9495	1.35456	0.677278	+	0.735728i	$0.263160\pi$
$368$	−0.0438295	−0.00228477
$369$	9.04918	0.471081
$370$	−0.872576	−0.0453630
$371$	0	0
$372$	14.1950	0.735978
$373$	2.25019	0.116510	0.0582551	−	0.998302i	$-0.481446\pi$
$373$	2.25019	0.116510	0.0582551	+	0.998302i	$0.481446\pi$
$374$	−6.83258	−0.353304
$375$	2.26649	0.117041
$376$	19.8942	1.02596
$377$	42.7042	2.19938
$378$	0	0
$379$	−6.14943	−0.315875	−0.157937	−	0.987449i	$-0.550484\pi$
$379$	−6.14943	−0.315875	−0.157937	+	0.987449i	$0.550484\pi$
$380$	6.07745	0.311767
$381$	9.95665	0.510095
$382$	−9.03472	−0.462257
$383$	−4.74296	−0.242354	−0.121177	−	0.992631i	$-0.538667\pi$
$383$	−4.74296	−0.242354	−0.121177	+	0.992631i	$0.538667\pi$
$384$	−15.9394	−0.813405
$385$	0	0
$386$	−10.8268	−0.551068
$387$	−12.2932	−0.624898
$388$	22.4310	1.13876
$389$	36.1324	1.83199	0.915993	−	0.401194i	$-0.131405\pi$
$389$	36.1324	1.83199	0.915993	+	0.401194i	$0.131405\pi$
$390$	10.8695	0.550399
$391$	−5.57846	−0.282115
$392$	0	0
$393$	−35.6019	−1.79588
$394$	7.56980	0.381361
$395$	−9.47305	−0.476641
$396$	14.3060	0.718905
$397$	−8.53534	−0.428377	−0.214188	−	0.976792i	$-0.568711\pi$
$397$	−8.53534	−0.428377	−0.214188	+	0.976792i	$0.568711\pi$
$398$	19.0619	0.955486
$399$	0	0
$400$	0.0113829	0.000569145 0
$401$	7.71974	0.385505	0.192753	−	0.981247i	$-0.438258\pi$
$401$	7.71974	0.385505	0.192753	+	0.981247i	$0.438258\pi$
$402$	7.13847	0.356035
$403$	−27.7907	−1.38435
$404$	−20.1076	−1.00039
$405$	−10.8442	−0.538855
$406$	0	0
$407$	5.40482	0.267907
$408$	−9.27934	−0.459396
$409$	15.4958	0.766220	0.383110	−	0.923703i	$-0.374853\pi$
$409$	15.4958	0.766220	0.383110	+	0.923703i	$0.374853\pi$
$410$	3.69495	0.182481
$411$	−1.75370	−0.0865038
$412$	−13.9924	−0.689358
$413$	0	0
$414$	−7.17992	−0.352874
$415$	8.06502	0.395896
$416$	31.1177	1.52567
$417$	−21.9476	−1.07478
$418$	23.1404	1.13183
$419$	23.3708	1.14174	0.570868	−	0.821042i	$-0.306607\pi$
$419$	23.3708	1.14174	0.570868	+	0.821042i	$0.306607\pi$
$420$	0	0
$421$	−32.9042	−1.60365	−0.801826	−	0.597558i	$-0.796138\pi$
$421$	−32.9042	−1.60365	−0.801826	+	0.597558i	$0.796138\pi$
$422$	−12.8588	−0.625957
$423$	−15.0441	−0.731471
$424$	−6.57334	−0.319230
$425$	1.44877	0.0702759
$426$	−2.16749	−0.105015
$427$	0	0
$428$	7.66933	0.370711
$429$	−67.3268	−3.25057
$430$	−5.01955	−0.242064
$431$	23.3272	1.12363	0.561815	−	0.827263i	$-0.310103\pi$
$431$	23.3272	1.12363	0.561815	+	0.827263i	$0.310103\pi$
$432$	−0.0222649	−0.00107122
$433$	−27.3135	−1.31260	−0.656302	−	0.754498i	$-0.727880\pi$
$433$	−27.3135	−1.31260	−0.656302	+	0.754498i	$0.727880\pi$
$434$	0	0
$435$	17.6106	0.844362
$436$	13.9507	0.668119
$437$	18.8930	0.903772
$438$	25.3867	1.21302
$439$	38.1754	1.82201	0.911005	−	0.412394i	$-0.135307\pi$
$439$	38.1754	1.82201	0.911005	+	0.412394i	$0.135307\pi$
$440$	15.2736	0.728143
$441$	0	0
$442$	6.94794	0.330480
$443$	−28.0017	−1.33040	−0.665199	−	0.746666i	$-0.731654\pi$
$443$	−28.0017	−1.33040	−0.665199	+	0.746666i	$0.731654\pi$
$444$	2.80731	0.133229
$445$	−15.4238	−0.731160
$446$	−2.00244	−0.0948182
$447$	10.2751	0.485996
$448$	0	0
$449$	−6.19759	−0.292482	−0.146241	−	0.989249i	$-0.546718\pi$
$449$	−6.19759	−0.292482	−0.146241	+	0.989249i	$0.546718\pi$
$450$	1.86469	0.0879022
$451$	−22.8869	−1.07770
$452$	18.3339	0.862355
$453$	8.35413	0.392511
$454$	8.30570	0.389806
$455$	0	0
$456$	31.4270	1.47170
$457$	−16.6654	−0.779574	−0.389787	−	0.920905i	$-0.627451\pi$
$457$	−16.6654	−0.779574	−0.389787	+	0.920905i	$0.627451\pi$
$458$	5.27615	0.246538
$459$	−2.83380	−0.132271
$460$	4.76924	0.222367
$461$	−0.732995	−0.0341390	−0.0170695	−	0.999854i	$-0.505434\pi$
$461$	−0.732995	−0.0341390	−0.0170695	+	0.999854i	$0.505434\pi$
$462$	0	0
$463$	21.1445	0.982670	0.491335	−	0.870971i	$-0.336509\pi$
$463$	21.1445	0.982670	0.491335	+	0.870971i	$0.336509\pi$
$464$	0.0884446	0.00410594
$465$	−11.4604	−0.531465
$466$	13.9406	0.645785
$467$	−19.5651	−0.905366	−0.452683	−	0.891671i	$-0.649533\pi$
$467$	−19.5651	−0.905366	−0.452683	+	0.891671i	$0.649533\pi$
$468$	−14.5476	−0.672462
$469$	0	0
$470$	−6.14281	−0.283347
$471$	4.87840	0.224785
$472$	32.5057	1.49620
$473$	31.0915	1.42959
$474$	−18.7347	−0.860515
$475$	−4.90666	−0.225133
$476$	0	0
$477$	4.97081	0.227598
$478$	22.8056	1.04310
$479$	12.8217	0.585838	0.292919	−	0.956137i	$-0.405373\pi$
$479$	12.8217	0.585838	0.292919	+	0.956137i	$0.405373\pi$
$480$	12.8324	0.585718
$481$	−5.49607	−0.250599
$482$	−18.3930	−0.837780
$483$	0	0
$484$	−22.5576	−1.02535
$485$	−18.1098	−0.822324
$486$	−16.3262	−0.740573
$487$	−5.36154	−0.242954	−0.121477	−	0.992594i	$-0.538763\pi$
$487$	−5.36154	−0.242954	−0.121477	+	0.992594i	$0.538763\pi$
$488$	38.5387	1.74456
$489$	32.3201	1.46156
$490$	0	0
$491$	−5.54338	−0.250169	−0.125085	−	0.992146i	$-0.539920\pi$
$491$	−5.54338	−0.250169	−0.125085	+	0.992146i	$0.539920\pi$
$492$	−11.8876	−0.535936
$493$	11.2569	0.506986
$494$	−23.5311	−1.05871
$495$	−11.5501	−0.519136
$496$	−0.0575572	−0.00258439
$497$	0	0
$498$	15.9501	0.714741
$499$	7.32107	0.327736	0.163868	−	0.986482i	$-0.447603\pi$
$499$	7.32107	0.327736	0.163868	+	0.986482i	$0.447603\pi$
$500$	−1.23861	−0.0553924
$501$	−31.9424	−1.42708
$502$	−11.5993	−0.517701
$503$	24.6636	1.09969	0.549847	−	0.835265i	$-0.314686\pi$
$503$	24.6636	1.09969	0.549847	+	0.835265i	$0.314686\pi$
$504$	0	0
$505$	16.2339	0.722401
$506$	18.1592	0.807276
$507$	38.9991	1.73201
$508$	−5.44119	−0.241414
$509$	−17.5381	−0.777363	−0.388681	−	0.921372i	$-0.627069\pi$
$509$	−17.5381	−0.777363	−0.388681	+	0.921372i	$0.627069\pi$
$510$	2.86522	0.126874
$511$	0	0
$512$	0.128782	0.00569143
$513$	9.59743	0.423737
$514$	4.10795	0.181194
$515$	11.2969	0.497799
$516$	16.1492	0.710929
$517$	38.0492	1.67340
$518$	0	0
$519$	−37.9832	−1.66728
$520$	−15.5315	−0.681103
$521$	21.4468	0.939600	0.469800	−	0.882773i	$-0.344326\pi$
$521$	21.4468	0.939600	0.469800	+	0.882773i	$0.344326\pi$
$522$	14.4885	0.634146
$523$	−4.77447	−0.208773	−0.104386	−	0.994537i	$-0.533288\pi$
$523$	−4.77447	−0.208773	−0.104386	+	0.994537i	$0.533288\pi$
$524$	19.4560	0.849940
$525$	0	0
$526$	−23.0970	−1.00708
$527$	−7.32567	−0.319111
$528$	−0.139440	−0.00606836
$529$	−8.17389	−0.355387
$530$	2.02968	0.0881636
$531$	−24.5811	−1.06673
$532$	0	0
$533$	23.2733	1.00808
$534$	−30.5035	−1.32002
$535$	−6.19187	−0.267698
$536$	−10.2002	−0.440582
$537$	−0.486887	−0.0210107
$538$	21.6801	0.934694
$539$	0	0
$540$	2.42272	0.104257
$541$	−10.4077	−0.447461	−0.223731	−	0.974651i	$-0.571824\pi$
$541$	−10.4077	−0.447461	−0.223731	+	0.974651i	$0.571824\pi$
$542$	13.7394	0.590158
$543$	−24.4954	−1.05120
$544$	8.20267	0.351687
$545$	−11.2632	−0.482463
$546$	0	0
$547$	31.5603	1.34942	0.674710	−	0.738083i	$-0.264269\pi$
$547$	31.5603	1.34942	0.674710	+	0.738083i	$0.264269\pi$
$548$	0.958378	0.0409399
$549$	−29.1432	−1.24380
$550$	−4.71611	−0.201096
$551$	−38.1246	−1.62416
$552$	24.6621	1.04969
$553$	0	0
$554$	−2.25379	−0.0957542
$555$	−2.26649	−0.0962073
$556$	11.9941	0.508663
$557$	38.8042	1.64419	0.822094	−	0.569351i	$-0.192806\pi$
$557$	38.8042	1.64419	0.822094	+	0.569351i	$0.192806\pi$
$558$	−9.42872	−0.399150
$559$	−31.6165	−1.33724
$560$	0	0
$561$	−17.7475	−0.749298
$562$	22.7339	0.958972
$563$	−35.9014	−1.51306	−0.756531	−	0.653958i	$-0.773107\pi$
$563$	−35.9014	−1.51306	−0.756531	+	0.653958i	$0.773107\pi$
$564$	19.7630	0.832175
$565$	−14.8020	−0.622725
$566$	−12.2469	−0.514774
$567$	0	0
$568$	3.09714	0.129953
$569$	1.47574	0.0618662	0.0309331	−	0.999521i	$-0.490152\pi$
$569$	1.47574	0.0618662	0.0309331	+	0.999521i	$0.490152\pi$
$570$	−9.70384	−0.406449
$571$	32.3600	1.35422	0.677111	−	0.735881i	$-0.263232\pi$
$571$	32.3600	1.35422	0.677111	+	0.735881i	$0.263232\pi$
$572$	36.7933	1.53840
$573$	−23.4675	−0.980368
$574$	0	0
$575$	−3.85047	−0.160576
$576$	10.5088	0.437868
$577$	23.6419	0.984224	0.492112	−	0.870532i	$-0.336225\pi$
$577$	23.6419	0.984224	0.492112	+	0.870532i	$0.336225\pi$
$578$	−13.0023	−0.540824
$579$	−28.1223	−1.16872
$580$	−9.62396	−0.399613
$581$	0	0
$582$	−35.8155	−1.48460
$583$	−12.5720	−0.520680
$584$	−36.2753	−1.50108
$585$	11.7451	0.485599
$586$	8.10711	0.334902
$587$	32.8786	1.35704	0.678522	−	0.734580i	$-0.262621\pi$
$587$	32.8786	1.35704	0.678522	+	0.734580i	$0.262621\pi$
$588$	0	0
$589$	24.8103	1.02229
$590$	−10.0369	−0.413214
$591$	19.6624	0.808802
$592$	−0.0113829	−0.000467834 0
$593$	−15.8650	−0.651498	−0.325749	−	0.945456i	$-0.605616\pi$
$593$	−15.8650	−0.651498	−0.325749	+	0.945456i	$0.605616\pi$
$594$	9.22471	0.378494
$595$	0	0
$596$	−5.61522	−0.230009
$597$	49.5128	2.02642
$598$	−18.4658	−0.755124
$599$	37.9014	1.54861	0.774304	−	0.632814i	$-0.218100\pi$
$599$	37.9014	1.54861	0.774304	+	0.632814i	$0.218100\pi$
$600$	−6.40496	−0.261481
$601$	−8.42327	−0.343593	−0.171796	−	0.985133i	$-0.554957\pi$
$601$	−8.42327	−0.343593	−0.171796	+	0.985133i	$0.554957\pi$
$602$	0	0
$603$	7.71349	0.314117
$604$	−4.56543	−0.185765
$605$	18.2120	0.740424
$606$	32.1057	1.30420
$607$	−21.7124	−0.881281	−0.440640	−	0.897684i	$-0.645249\pi$
$607$	−21.7124	−0.881281	−0.440640	+	0.897684i	$0.645249\pi$
$608$	−27.7805	−1.12665
$609$	0	0
$610$	−11.8997	−0.481807
$611$	−38.6916	−1.56529
$612$	−3.83477	−0.155011
$613$	20.9592	0.846534	0.423267	−	0.906005i	$-0.360883\pi$
$613$	20.9592	0.846534	0.423267	+	0.906005i	$0.360883\pi$
$614$	15.9458	0.643518
$615$	9.59755	0.387011
$616$	0	0
$617$	−12.7620	−0.513778	−0.256889	−	0.966441i	$-0.582697\pi$
$617$	−12.7620	−0.513778	−0.256889	+	0.966441i	$0.582697\pi$
$618$	22.3417	0.898714
$619$	−10.2303	−0.411189	−0.205595	−	0.978637i	$-0.565913\pi$
$619$	−10.2303	−0.411189	−0.205595	+	0.978637i	$0.565913\pi$
$620$	6.26299	0.251528
$621$	7.53151	0.302229
$622$	−25.1982	−1.01035
$623$	0	0
$624$	0.141795	0.00567632
$625$	1.00000	0.0400000
$626$	−8.78874	−0.351269
$627$	60.1065	2.40042
$628$	−2.66599	−0.106385
$629$	−1.44877	−0.0577664
$630$	0	0
$631$	−21.1580	−0.842287	−0.421143	−	0.906994i	$-0.638371\pi$
$631$	−21.1580	−0.842287	−0.421143	+	0.906994i	$0.638371\pi$
$632$	26.7702	1.06486
$633$	−33.4005	−1.32755
$634$	−10.2773	−0.408163
$635$	4.39298	0.174330
$636$	−6.53001	−0.258932
$637$	0	0
$638$	−36.6440	−1.45075
$639$	−2.34208	−0.0926514
$640$	−7.03264	−0.277989
$641$	−10.5859	−0.418117	−0.209059	−	0.977903i	$-0.567040\pi$
$641$	−10.5859	−0.418117	−0.209059	+	0.977903i	$0.567040\pi$
$642$	−12.2456	−0.483295
$643$	9.09491	0.358668	0.179334	−	0.983788i	$-0.442606\pi$
$643$	9.09491	0.358668	0.179334	+	0.983788i	$0.442606\pi$
$644$	0	0
$645$	−13.0381	−0.513377
$646$	−6.20283	−0.244047
$647$	3.83139	0.150627	0.0753137	−	0.997160i	$-0.476004\pi$
$647$	3.83139	0.150627	0.0753137	+	0.997160i	$0.476004\pi$
$648$	30.6451	1.20385
$649$	62.1697	2.44037
$650$	4.79574	0.188104
$651$	0	0
$652$	−17.6625	−0.691718
$653$	6.42255	0.251334	0.125667	−	0.992072i	$-0.459893\pi$
$653$	6.42255	0.251334	0.125667	+	0.992072i	$0.459893\pi$
$654$	−22.2751	−0.871025
$655$	−15.7079	−0.613759
$656$	0.0482013	0.00188194
$657$	27.4316	1.07021
$658$	0	0
$659$	28.9307	1.12698	0.563490	−	0.826123i	$-0.309459\pi$
$659$	28.9307	1.12698	0.563490	+	0.826123i	$0.309459\pi$
$660$	15.1730	0.590607
$661$	−2.75652	−0.107216	−0.0536082	−	0.998562i	$-0.517072\pi$
$661$	−2.75652	−0.107216	−0.0536082	+	0.998562i	$0.517072\pi$
$662$	−14.3924	−0.559378
$663$	18.0471	0.700891
$664$	−22.7912	−0.884471
$665$	0	0
$666$	−1.86469	−0.0722552
$667$	−29.9180	−1.15843
$668$	17.4561	0.675398
$669$	−5.20129	−0.201093
$670$	3.14957	0.121678
$671$	73.7082	2.84547
$672$	0	0
$673$	2.91328	0.112299	0.0561493	−	0.998422i	$-0.482118\pi$
$673$	2.91328	0.112299	0.0561493	+	0.998422i	$0.482118\pi$
$674$	7.98937	0.307739
$675$	−1.95600	−0.0752865
$676$	−21.3125	−0.819713
$677$	2.56979	0.0987650	0.0493825	−	0.998780i	$-0.484275\pi$
$677$	2.56979	0.0987650	0.0493825	+	0.998780i	$0.484275\pi$
$678$	−29.2737	−1.12425
$679$	0	0
$680$	−4.09414	−0.157003
$681$	21.5739	0.826712
$682$	23.8468	0.913142
$683$	−20.3232	−0.777646	−0.388823	−	0.921313i	$-0.627118\pi$
$683$	−20.3232	−0.777646	−0.388823	+	0.921313i	$0.627118\pi$
$684$	12.9875	0.496588
$685$	−0.773751	−0.0295635
$686$	0	0
$687$	13.7047	0.522866
$688$	−0.0654808	−0.00249643
$689$	12.7843	0.487043
$690$	−7.61502	−0.289899
$691$	11.9672	0.455255	0.227627	−	0.973748i	$-0.426903\pi$
$691$	11.9672	0.455255	0.227627	+	0.973748i	$0.426903\pi$
$692$	20.7574	0.789077
$693$	0	0
$694$	−10.9004	−0.413773
$695$	−9.68350	−0.367316
$696$	−49.7663	−1.88639
$697$	6.13489	0.232375
$698$	16.6943	0.631889
$699$	36.2103	1.36960
$700$	0	0
$701$	29.2734	1.10564	0.552821	−	0.833300i	$-0.313551\pi$
$701$	29.2734	1.10564	0.552821	+	0.833300i	$0.313551\pi$
$702$	−9.38046	−0.354043
$703$	4.90666	0.185058
$704$	−26.5786	−1.00172
$705$	−15.9558	−0.600930
$706$	−16.0250	−0.603110
$707$	0	0
$708$	32.2915	1.21359
$709$	26.3528	0.989701	0.494851	−	0.868978i	$-0.335223\pi$
$709$	26.3528	0.989701	0.494851	+	0.868978i	$0.335223\pi$
$710$	−0.956318	−0.0358900
$711$	−20.2438	−0.759203
$712$	43.5867	1.63348
$713$	19.4697	0.729148
$714$	0	0
$715$	−29.7052	−1.11091
$716$	0.266078	0.00994380
$717$	59.2370	2.21225
$718$	0.430197	0.0160548
$719$	31.7692	1.18479	0.592395	−	0.805648i	$-0.298182\pi$
$719$	31.7692	1.18479	0.592395	+	0.805648i	$0.298182\pi$
$720$	0.0243252	0.000906545 0
$721$	0	0
$722$	4.42862	0.164816
$723$	−47.7755	−1.77679
$724$	13.3865	0.497504
$725$	7.76996	0.288569
$726$	36.0177	1.33674
$727$	−34.4510	−1.27772	−0.638859	−	0.769324i	$-0.720593\pi$
$727$	−34.4510	−1.27772	−0.638859	+	0.769324i	$0.720593\pi$
$728$	0	0
$729$	−9.87429	−0.365714
$730$	11.2009	0.414563
$731$	−8.33417	−0.308250
$732$	38.2846	1.41504
$733$	17.1054	0.631801	0.315901	−	0.948792i	$-0.397693\pi$
$733$	17.1054	0.631801	0.315901	+	0.948792i	$0.397693\pi$
$734$	22.6429	0.835766
$735$	0	0
$736$	−21.8006	−0.803580
$737$	−19.5087	−0.718612
$738$	7.89609	0.290659
$739$	−33.9110	−1.24744	−0.623718	−	0.781650i	$-0.714378\pi$
$739$	−33.9110	−1.24744	−0.623718	+	0.781650i	$0.714378\pi$
$740$	1.23861	0.0455323
$741$	−61.1214	−2.24535
$742$	0	0
$743$	−5.27721	−0.193602	−0.0968010	−	0.995304i	$-0.530861\pi$
$743$	−5.27721	−0.193602	−0.0968010	+	0.995304i	$0.530861\pi$
$744$	32.3864	1.18734
$745$	4.53348	0.166094
$746$	1.96346	0.0718873
$747$	17.2349	0.630592
$748$	9.69877	0.354622
$749$	0	0
$750$	1.97769	0.0722149
$751$	−45.8235	−1.67212	−0.836061	−	0.548636i	$-0.815147\pi$
$751$	−45.8235	−1.67212	−0.836061	+	0.548636i	$0.815147\pi$
$752$	−0.0801340	−0.00292219
$753$	−30.1289	−1.09796
$754$	37.2627	1.35703
$755$	3.68593	0.134145
$756$	0	0
$757$	−14.5832	−0.530036	−0.265018	−	0.964243i	$-0.585378\pi$
$757$	−14.5832	−0.530036	−0.265018	+	0.964243i	$0.585378\pi$
$758$	−5.36584	−0.194896
$759$	47.1682	1.71210
$760$	13.8659	0.502969
$761$	38.7629	1.40515	0.702577	−	0.711607i	$-0.252032\pi$
$761$	38.7629	1.40515	0.702577	+	0.711607i	$0.252032\pi$
$762$	8.68793	0.314731
$763$	0	0
$764$	12.8247	0.463981
$765$	3.09602	0.111937
$766$	−4.13859	−0.149533
$767$	−63.2194	−2.28272
$768$	−36.1997	−1.30624
$769$	−7.31877	−0.263921	−0.131961	−	0.991255i	$-0.542127\pi$
$769$	−7.31877	−0.263921	−0.131961	+	0.991255i	$0.542127\pi$
$770$	0	0
$771$	10.6703	0.384282
$772$	15.3685	0.553124
$773$	−7.48899	−0.269360	−0.134680	−	0.990889i	$-0.543001\pi$
$773$	−7.48899	−0.269360	−0.134680	+	0.990889i	$0.543001\pi$
$774$	−10.7267	−0.385565
$775$	−5.05646	−0.181633
$776$	51.1771	1.83715
$777$	0	0
$778$	31.5283	1.13034
$779$	−20.7774	−0.744429
$780$	−15.4292	−0.552452
$781$	5.92352	0.211960
$782$	−4.86763	−0.174066
$783$	−15.1980	−0.543133
$784$	0	0
$785$	2.15240	0.0768225
$786$	−31.0653	−1.10806
$787$	−45.1825	−1.61058	−0.805291	−	0.592880i	$-0.797991\pi$
$787$	−45.1825	−1.61058	−0.805291	+	0.592880i	$0.797991\pi$
$788$	−10.7452	−0.382784
$789$	−59.9939	−2.13584
$790$	−8.26595	−0.294089
$791$	0	0
$792$	32.6397	1.15980
$793$	−74.9527	−2.66165
$794$	−7.44773	−0.264310
$795$	5.27204	0.186980
$796$	−27.0582	−0.959051
$797$	−4.91928	−0.174250	−0.0871250	−	0.996197i	$-0.527768\pi$
$797$	−4.91928	−0.174250	−0.0871250	+	0.996197i	$0.527768\pi$
$798$	0	0
$799$	−10.1992	−0.360821
$800$	5.66180	0.200175
$801$	−32.9606	−1.16461
$802$	6.73606	0.237858
$803$	−69.3792	−2.44834
$804$	−10.1330	−0.357363
$805$	0	0
$806$	−24.2495	−0.854151
$807$	56.3134	1.98233
$808$	−45.8760	−1.61391
$809$	13.4001	0.471124	0.235562	−	0.971859i	$-0.424307\pi$
$809$	13.4001	0.471124	0.235562	+	0.971859i	$0.424307\pi$
$810$	−9.46242	−0.332476
$811$	24.6127	0.864267	0.432134	−	0.901810i	$-0.357761\pi$
$811$	24.6127	0.864267	0.432134	+	0.901810i	$0.357761\pi$
$812$	0	0
$813$	35.6877	1.25162
$814$	4.71611	0.165300
$815$	14.2599	0.499504
$816$	0.0373773	0.00130847
$817$	28.2259	0.987499
$818$	13.5213	0.472761
$819$	0	0
$820$	−5.24495	−0.183162
$821$	42.7476	1.49190	0.745951	−	0.666001i	$-0.231995\pi$
$821$	42.7476	1.49190	0.745951	+	0.666001i	$0.231995\pi$
$822$	−1.53024	−0.0533732
$823$	−17.2878	−0.602613	−0.301307	−	0.953527i	$-0.597423\pi$
$823$	−17.2878	−0.602613	−0.301307	+	0.953527i	$0.597423\pi$
$824$	−31.9242	−1.11213
$825$	−12.2500	−0.426490
$826$	0	0
$827$	−4.24836	−0.147730	−0.0738650	−	0.997268i	$-0.523533\pi$
$827$	−4.24836	−0.147730	−0.0738650	+	0.997268i	$0.523533\pi$
$828$	10.1918	0.354190
$829$	−24.1597	−0.839101	−0.419551	−	0.907732i	$-0.637812\pi$
$829$	−24.1597	−0.839101	−0.419551	+	0.907732i	$0.637812\pi$
$830$	7.03734	0.244270
$831$	−5.85416	−0.203078
$832$	27.0274	0.937006
$833$	0	0
$834$	−19.1509	−0.663142
$835$	−14.0933	−0.487719
$836$	−32.8475	−1.13605
$837$	9.89043	0.341863
$838$	20.3928	0.704456
$839$	18.3474	0.633424	0.316712	−	0.948522i	$-0.397421\pi$
$839$	18.3474	0.633424	0.316712	+	0.948522i	$0.397421\pi$
$840$	0	0
$841$	31.3722	1.08180
$842$	−28.7114	−0.989460
$843$	59.0508	2.03382
$844$	18.2530	0.628293
$845$	17.2068	0.591932
$846$	−13.1271	−0.451321
$847$	0	0
$848$	0.0264775	0.000909241 0
$849$	−31.8109	−1.09175
$850$	1.26417	0.0433605
$851$	3.85047	0.131992
$852$	3.07673	0.105407
$853$	53.1321	1.81921	0.909604	−	0.415476i	$-0.136385\pi$
$853$	53.1321	1.81921	0.909604	+	0.415476i	$0.136385\pi$
$854$	0	0
$855$	−10.4855	−0.358597
$856$	17.4978	0.598063
$857$	−24.6816	−0.843108	−0.421554	−	0.906803i	$-0.638515\pi$
$857$	−24.6816	−0.843108	−0.421554	+	0.906803i	$0.638515\pi$
$858$	−58.7477	−2.00561
$859$	35.1531	1.19941	0.599704	−	0.800222i	$-0.295285\pi$
$859$	35.1531	1.19941	0.599704	+	0.800222i	$0.295285\pi$
$860$	7.12519	0.242967
$861$	0	0
$862$	20.3547	0.693284
$863$	−16.7311	−0.569532	−0.284766	−	0.958597i	$-0.591916\pi$
$863$	−16.7311	−0.569532	−0.284766	+	0.958597i	$0.591916\pi$
$864$	−11.0745	−0.376761
$865$	−16.7586	−0.569809
$866$	−23.8331	−0.809882
$867$	−33.7731	−1.14700
$868$	0	0
$869$	51.2001	1.73684
$870$	15.3665	0.520975
$871$	19.8381	0.672188
$872$	31.8291	1.07787
$873$	−38.7005	−1.30981
$874$	16.4855	0.557631
$875$	0	0
$876$	−36.0362	−1.21755
$877$	−2.29029	−0.0773374	−0.0386687	−	0.999252i	$-0.512312\pi$
$877$	−2.29029	−0.0773374	−0.0386687	+	0.999252i	$0.512312\pi$
$878$	33.3109	1.12419
$879$	21.0580	0.710269
$880$	−0.0615224	−0.00207392
$881$	−1.51818	−0.0511488	−0.0255744	−	0.999673i	$-0.508141\pi$
$881$	−1.51818	−0.0511488	−0.0255744	+	0.999673i	$0.508141\pi$
$882$	0	0
$883$	−2.62124	−0.0882118	−0.0441059	−	0.999027i	$-0.514044\pi$
$883$	−2.62124	−0.0882118	−0.0441059	+	0.999027i	$0.514044\pi$
$884$	−9.86253	−0.331713
$885$	−26.0707	−0.876356
$886$	−24.4336	−0.820862
$887$	41.3577	1.38866	0.694328	−	0.719659i	$-0.255702\pi$
$887$	41.3577	1.38866	0.694328	+	0.719659i	$0.255702\pi$
$888$	6.40496	0.214936
$889$	0	0
$890$	−13.4585	−0.451128
$891$	58.6111	1.96355
$892$	2.84244	0.0951720
$893$	34.5422	1.15591
$894$	8.96581	0.299861
$895$	−0.214819	−0.00718062
$896$	0	0
$897$	−47.9646	−1.60149
$898$	−5.40786	−0.180463
$899$	−39.2885	−1.31034
$900$	−2.64690	−0.0882302
$901$	3.36996	0.112270
$902$	−19.9705	−0.664946
$903$	0	0
$904$	41.8295	1.39123
$905$	−10.8076	−0.359258
$906$	7.28961	0.242181
$907$	−45.8658	−1.52295	−0.761475	−	0.648194i	$-0.775524\pi$
$907$	−45.8658	−1.52295	−0.761475	+	0.648194i	$0.775524\pi$
$908$	−11.7899	−0.391260
$909$	34.6918	1.15066
$910$	0	0
$911$	32.0100	1.06054	0.530269	−	0.847829i	$-0.322091\pi$
$911$	32.0100	1.06054	0.530269	+	0.847829i	$0.322091\pi$
$912$	−0.126588	−0.00419176
$913$	−43.5900	−1.44262
$914$	−14.5418	−0.481000
$915$	−30.9093	−1.02183
$916$	−7.48944	−0.247458
$917$	0	0
$918$	−2.47271	−0.0816115
$919$	−34.9721	−1.15362	−0.576811	−	0.816877i	$-0.695703\pi$
$919$	−34.9721	−1.15362	−0.576811	+	0.816877i	$0.695703\pi$
$920$	10.8812	0.358742
$921$	41.4187	1.36479
$922$	−0.639594	−0.0210639
$923$	−6.02354	−0.198267
$924$	0	0
$925$	−1.00000	−0.0328798
$926$	18.4502	0.606312
$927$	24.1413	0.792905
$928$	43.9919	1.44411
$929$	21.3491	0.700441	0.350221	−	0.936667i	$-0.386107\pi$
$929$	21.3491	0.700441	0.350221	+	0.936667i	$0.386107\pi$
$930$	−10.0001	−0.327916
$931$	0	0
$932$	−19.7885	−0.648194
$933$	−65.4516	−2.14279
$934$	−17.0721	−0.558615
$935$	−7.83036	−0.256080
$936$	−33.1908	−1.08487
$937$	−43.2352	−1.41243	−0.706217	−	0.707996i	$-0.749600\pi$
$937$	−43.2352	−1.41243	−0.706217	+	0.707996i	$0.749600\pi$
$938$	0	0
$939$	−22.8285	−0.744981
$940$	8.71966	0.284404
$941$	−30.1810	−0.983873	−0.491936	−	0.870631i	$-0.663711\pi$
$941$	−30.1810	−0.983873	−0.491936	+	0.870631i	$0.663711\pi$
$942$	4.25677	0.138693
$943$	−16.3050	−0.530962
$944$	−0.130933	−0.00426152
$945$	0	0
$946$	27.1297	0.882063
$947$	−17.8343	−0.579536	−0.289768	−	0.957097i	$-0.593578\pi$
$947$	−17.8343	−0.579536	−0.289768	+	0.957097i	$0.593578\pi$
$948$	26.5938	0.863725
$949$	70.5507	2.29017
$950$	−4.28143	−0.138908
$951$	−26.6950	−0.865644
$952$	0	0
$953$	−23.3165	−0.755295	−0.377648	−	0.925949i	$-0.623267\pi$
$953$	−23.3165	−0.755295	−0.377648	+	0.925949i	$0.623267\pi$
$954$	4.33741	0.140429
$955$	−10.3541	−0.335050
$956$	−32.3723	−1.04700
$957$	−95.1818	−3.07679
$958$	11.1879	0.361464
$959$	0	0
$960$	11.1457	0.359725
$961$	−5.43221	−0.175232
$962$	−4.79574	−0.154621
$963$	−13.2320	−0.426395
$964$	26.1087	0.840906
$965$	−12.4078	−0.399422
$966$	0	0
$967$	56.1130	1.80447	0.902237	−	0.431241i	$-0.141924\pi$
$967$	56.1130	1.80447	0.902237	+	0.431241i	$0.141924\pi$
$968$	−51.4660	−1.65418
$969$	−16.1117	−0.517582
$970$	−15.8022	−0.507377
$971$	−19.3788	−0.621896	−0.310948	−	0.950427i	$-0.600647\pi$
$971$	−19.3788	−0.621896	−0.310948	+	0.950427i	$0.600647\pi$
$972$	23.1749	0.743336
$973$	0	0
$974$	−4.67834	−0.149904
$975$	12.4568	0.398937
$976$	−0.155234	−0.00496893
$977$	−37.0688	−1.18594	−0.592969	−	0.805225i	$-0.702044\pi$
$977$	−37.0688	−1.18594	−0.592969	+	0.805225i	$0.702044\pi$
$978$	28.2017	0.901791
$979$	83.3629	2.66429
$980$	0	0
$981$	−24.0694	−0.768476
$982$	−4.83702	−0.154355
$983$	23.0679	0.735753	0.367877	−	0.929875i	$-0.380085\pi$
$983$	23.0679	0.735753	0.367877	+	0.929875i	$0.380085\pi$
$984$	−27.1220	−0.864619
$985$	8.67524	0.276416
$986$	9.82251	0.312812
$987$	0	0
$988$	33.4021	1.06266
$989$	22.1501	0.704331
$990$	−10.0783	−0.320309
$991$	16.0128	0.508663	0.254331	−	0.967117i	$-0.418145\pi$
$991$	16.0128	0.508663	0.254331	+	0.967117i	$0.418145\pi$
$992$	−28.6287	−0.908961
$993$	−37.3840	−1.18635
$994$	0	0
$995$	21.8456	0.692551
$996$	−22.6410	−0.717407
$997$	−2.06423	−0.0653749	−0.0326875	−	0.999466i	$-0.510407\pi$
$997$	−2.06423	−0.0653749	−0.0326875	+	0.999466i	$0.510407\pi$
$998$	6.38819	0.202215
$999$	1.95600	0.0618851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.v.1.15		21
7.3	odd	6		1295.2.j.b.926.7	yes	42
7.5	odd	6		1295.2.j.b.186.7	✓	42
7.6	odd	2		9065.2.a.w.1.15		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1295.2.j.b.186.7	✓	42	7.5	odd	6
1295.2.j.b.926.7	yes	42	7.3	odd	6
9065.2.a.v.1.15		21	1.1	even	1	trivial
9065.2.a.w.1.15		21	7.6	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:	\( N \)	\(=\)	\( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9065.a (trivial)

\(f(q)\)	\(=\)	\(q+0.872576 q^{2} +2.26649 q^{3} -1.23861 q^{4} +1.00000 q^{5} +1.97769 q^{6} -2.82593 q^{8} +2.13699 q^{9} +0.872576 q^{10} -5.40482 q^{11} -2.80731 q^{12} +5.49607 q^{13} +2.26649 q^{15} +0.0113829 q^{16} +1.44877 q^{17} +1.86469 q^{18} -4.90666 q^{19} -1.23861 q^{20} -4.71611 q^{22} -3.85047 q^{23} -6.40496 q^{24} +1.00000 q^{25} +4.79574 q^{26} -1.95600 q^{27} +7.76996 q^{29} +1.97769 q^{30} -5.05646 q^{31} +5.66180 q^{32} -12.2500 q^{33} +1.26417 q^{34} -2.64690 q^{36} -1.00000 q^{37} -4.28143 q^{38} +12.4568 q^{39} -2.82593 q^{40} +4.23454 q^{41} -5.75256 q^{43} +6.69447 q^{44} +2.13699 q^{45} -3.35982 q^{46} -7.03986 q^{47} +0.0257993 q^{48} +0.872576 q^{50} +3.28364 q^{51} -6.80750 q^{52} +2.32608 q^{53} -1.70676 q^{54} -5.40482 q^{55} -11.1209 q^{57} +6.77988 q^{58} -11.5027 q^{59} -2.80731 q^{60} -13.6375 q^{61} -4.41214 q^{62} +4.91758 q^{64} +5.49607 q^{65} -10.6890 q^{66} +3.60950 q^{67} -1.79447 q^{68} -8.72706 q^{69} -1.09597 q^{71} -6.03900 q^{72} +12.8366 q^{73} -0.872576 q^{74} +2.26649 q^{75} +6.07745 q^{76} +10.8695 q^{78} -9.47305 q^{79} +0.0113829 q^{80} -10.8442 q^{81} +3.69495 q^{82} +8.06502 q^{83} +1.44877 q^{85} -5.01955 q^{86} +17.6106 q^{87} +15.2736 q^{88} -15.4238 q^{89} +1.86469 q^{90} +4.76924 q^{92} -11.4604 q^{93} -6.14281 q^{94} -4.90666 q^{95} +12.8324 q^{96} -18.1098 q^{97} -11.5501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q - 3 q^{2} - q^{3} + 13 q^{4} + 21 q^{5} - 6 q^{6} - 9 q^{8} + 12 q^{9} - 3 q^{10} - 5 q^{11} - 4 q^{12} + q^{13} - q^{15} + q^{16} + 2 q^{17} - 11 q^{18} - 12 q^{19} + 13 q^{20} - 10 q^{22} - 20 q^{23}+ \cdots - 7 q^{99}+O(q^{100}) \) 21 * q - 3 * q^2 - q^3 + 13 * q^4 + 21 * q^5 - 6 * q^6 - 9 * q^8 + 12 * q^9 - 3 * q^10 - 5 * q^11 - 4 * q^12 + q^13 - q^15 + q^16 + 2 * q^17 - 11 * q^18 - 12 * q^19 + 13 * q^20 - 10 * q^22 - 20 * q^23 - 3 * q^24 + 21 * q^25 - 13 * q^26 - 7 * q^27 - 11 * q^29 - 6 * q^30 - 8 * q^31 - 16 * q^32 - 10 * q^33 + 6 * q^34 - 11 * q^36 - 21 * q^37 - 2 * q^38 - 37 * q^39 - 9 * q^40 - 20 * q^41 - 59 * q^43 - 23 * q^44 + 12 * q^45 - 11 * q^46 - 2 * q^47 + q^48 - 3 * q^50 - 47 * q^51 + 22 * q^52 - 2 * q^53 + 4 * q^54 - 5 * q^55 - 20 * q^57 - 35 * q^58 - 54 * q^59 - 4 * q^60 + 4 * q^61 + 30 * q^62 - 21 * q^64 + q^65 - 44 * q^66 - 15 * q^67 - 15 * q^68 + 2 * q^69 - 13 * q^71 - 61 * q^72 - 4 * q^73 + 3 * q^74 - q^75 - 13 * q^76 + 37 * q^78 - 91 * q^79 + q^80 - 7 * q^81 + 11 * q^82 + 12 * q^83 + 2 * q^85 + 19 * q^86 + 16 * q^87 + 10 * q^88 - 38 * q^89 - 11 * q^90 + 25 * q^92 + 7 * q^93 - 22 * q^94 - 12 * q^95 + 4 * q^96 - 4 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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