LMFDB - Embedded newform 1503.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1503,4,Mod(1,1503)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1503, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1503.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1503 = 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1503.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$88.6798707386$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Character	$\chi$	$=$	1503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.99247 q^{2} +16.9247 q^{4} +7.44760 q^{5} -9.19941 q^{7} -44.5565 q^{8} +O(q^{10})$	\(q-4.99247 q^{2} +16.9247 q^{4} +7.44760 q^{5} -9.19941 q^{7} -44.5565 q^{8} -37.1819 q^{10} -63.8158 q^{11} +74.5441 q^{13} +45.9277 q^{14} +87.0489 q^{16} -71.4437 q^{17} -82.9693 q^{19} +126.049 q^{20} +318.598 q^{22} +150.620 q^{23} -69.5333 q^{25} -372.159 q^{26} -155.698 q^{28} +0.607477 q^{29} +183.444 q^{31} -78.1370 q^{32} +356.680 q^{34} -68.5135 q^{35} +341.637 q^{37} +414.222 q^{38} -331.839 q^{40} +72.6264 q^{41} +474.676 q^{43} -1080.07 q^{44} -751.966 q^{46} +143.681 q^{47} -258.371 q^{49} +347.143 q^{50} +1261.64 q^{52} +335.960 q^{53} -475.274 q^{55} +409.893 q^{56} -3.03281 q^{58} -807.109 q^{59} -736.732 q^{61} -915.836 q^{62} -306.295 q^{64} +555.174 q^{65} -908.155 q^{67} -1209.17 q^{68} +342.051 q^{70} +710.028 q^{71} -695.867 q^{73} -1705.61 q^{74} -1404.23 q^{76} +587.067 q^{77} +654.632 q^{79} +648.305 q^{80} -362.585 q^{82} +668.629 q^{83} -532.084 q^{85} -2369.81 q^{86} +2843.41 q^{88} -568.035 q^{89} -685.761 q^{91} +2549.20 q^{92} -717.322 q^{94} -617.922 q^{95} -391.260 q^{97} +1289.91 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 11 q^{2} + 103 q^{4} - 48 q^{5} + 28 q^{7} - 147 q^{8}+O(q^{10}) $ 23 * q - 11 * q^2 + 103 * q^4 - 48 * q^5 + 28 * q^7 - 147 * q^8	$ 23 q - 11 q^{2} + 103 q^{4} - 48 q^{5} + 28 q^{7} - 147 q^{8} + 43 q^{10} - 204 q^{11} + 34 q^{13} - 101 q^{14} + 587 q^{16} - 368 q^{17} + 448 q^{19} - 543 q^{20} + 307 q^{22} - 406 q^{23} + 863 q^{25} - 335 q^{26} + 115 q^{28} - 556 q^{29} + 776 q^{31} - 1402 q^{32} + 672 q^{34} - 730 q^{35} + 270 q^{37} - 238 q^{38} + 694 q^{40} - 816 q^{41} - 84 q^{43} - 2035 q^{44} - 327 q^{46} - 1420 q^{47} + 1131 q^{49} - 430 q^{50} - 828 q^{52} - 1430 q^{53} - 758 q^{55} + 433 q^{56} - 2233 q^{58} - 3110 q^{59} + 278 q^{61} - 2044 q^{62} + 859 q^{64} - 2332 q^{65} + 802 q^{67} - 1384 q^{68} - 4411 q^{70} - 1696 q^{71} - 1048 q^{73} - 225 q^{74} - 2804 q^{76} - 986 q^{77} + 460 q^{79} - 4026 q^{80} - 3401 q^{82} - 1886 q^{83} + 360 q^{85} - 633 q^{86} - 116 q^{88} - 6404 q^{89} + 3304 q^{91} - 5135 q^{92} - 1860 q^{94} - 796 q^{95} + 1040 q^{97} - 5905 q^{98}+O(q^{100}) $ 23 * q - 11 * q^2 + 103 * q^4 - 48 * q^5 + 28 * q^7 - 147 * q^8 + 43 * q^10 - 204 * q^11 + 34 * q^13 - 101 * q^14 + 587 * q^16 - 368 * q^17 + 448 * q^19 - 543 * q^20 + 307 * q^22 - 406 * q^23 + 863 * q^25 - 335 * q^26 + 115 * q^28 - 556 * q^29 + 776 * q^31 - 1402 * q^32 + 672 * q^34 - 730 * q^35 + 270 * q^37 - 238 * q^38 + 694 * q^40 - 816 * q^41 - 84 * q^43 - 2035 * q^44 - 327 * q^46 - 1420 * q^47 + 1131 * q^49 - 430 * q^50 - 828 * q^52 - 1430 * q^53 - 758 * q^55 + 433 * q^56 - 2233 * q^58 - 3110 * q^59 + 278 * q^61 - 2044 * q^62 + 859 * q^64 - 2332 * q^65 + 802 * q^67 - 1384 * q^68 - 4411 * q^70 - 1696 * q^71 - 1048 * q^73 - 225 * q^74 - 2804 * q^76 - 986 * q^77 + 460 * q^79 - 4026 * q^80 - 3401 * q^82 - 1886 * q^83 + 360 * q^85 - 633 * q^86 - 116 * q^88 - 6404 * q^89 + 3304 * q^91 - 5135 * q^92 - 1860 * q^94 - 796 * q^95 + 1040 * q^97 - 5905 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	−4.99247	−1.76510	−0.882552	−	0.470215i	$-0.844176\pi$
$2$	−4.99247	−1.76510	−0.882552	+	0.470215i	$0.844176\pi$
$3$	0	0
$3$	0	0
$4$	16.9247	2.11559
$5$	7.44760	0.666133	0.333067	−	0.942903i	$-0.391917\pi$
$5$	7.44760	0.666133	0.333067	+	0.942903i	$0.391917\pi$
$6$	0	0
$7$	−9.19941	−0.496721	−0.248361	−	0.968668i	$-0.579892\pi$
$7$	−9.19941	−0.496721	−0.248361	+	0.968668i	$0.579892\pi$
$8$	−44.5565	−1.96914
$9$	0	0
$10$	−37.1819	−1.17579
$11$	−63.8158	−1.74920	−0.874599	−	0.484847i	$-0.838875\pi$
$11$	−63.8158	−1.74920	−0.874599	+	0.484847i	$0.838875\pi$
$12$	0	0
$13$	74.5441	1.59037	0.795185	−	0.606367i	$-0.207374\pi$
$13$	74.5441	1.59037	0.795185	+	0.606367i	$0.207374\pi$
$14$	45.9277	0.876765
$15$	0	0
$16$	87.0489	1.36014
$17$	−71.4437	−1.01927	−0.509636	−	0.860390i	$-0.670220\pi$
$17$	−71.4437	−1.01927	−0.509636	+	0.860390i	$0.670220\pi$
$18$	0	0
$19$	−82.9693	−1.00181	−0.500907	−	0.865501i	$-0.667000\pi$
$19$	−82.9693	−1.00181	−0.500907	+	0.865501i	$0.667000\pi$
$20$	126.049	1.40927
$21$	0	0
$22$	318.598	3.08752
$23$	150.620	1.36550	0.682749	−	0.730653i	$-0.260784\pi$
$23$	150.620	1.36550	0.682749	+	0.730653i	$0.260784\pi$
$24$	0	0
$25$	−69.5333	−0.556266
$26$	−372.159	−2.80717
$27$	0	0
$28$	−155.698	−1.05086
$29$	0.607477	0.00388985	0.00194493	−	0.999998i	$-0.499381\pi$
$29$	0.607477	0.00388985	0.00194493	+	0.999998i	$0.499381\pi$
$30$	0	0
$31$	183.444	1.06282	0.531410	−	0.847115i	$-0.321662\pi$
$31$	183.444	1.06282	0.531410	+	0.847115i	$0.321662\pi$
$32$	−78.1370	−0.431650
$33$	0	0
$34$	356.680	1.79912
$35$	−68.5135	−0.330883
$36$	0	0
$37$	341.637	1.51796	0.758982	−	0.651111i	$-0.225697\pi$
$37$	341.637	1.51796	0.758982	+	0.651111i	$0.225697\pi$
$38$	414.222	1.76831
$39$	0	0
$40$	−331.839	−1.31171
$41$	72.6264	0.276643	0.138321	−	0.990387i	$-0.455829\pi$
$41$	72.6264	0.276643	0.138321	+	0.990387i	$0.455829\pi$
$42$	0	0
$43$	474.676	1.68343	0.841715	−	0.539922i	$-0.181546\pi$
$43$	474.676	1.68343	0.841715	+	0.539922i	$0.181546\pi$
$44$	−1080.07	−3.70059
$45$	0	0
$46$	−751.966	−2.41025
$47$	143.681	0.445915	0.222958	−	0.974828i	$-0.428429\pi$
$47$	143.681	0.445915	0.222958	+	0.974828i	$0.428429\pi$
$48$	0	0
$49$	−258.371	−0.753268
$50$	347.143	0.981868
$51$	0	0
$52$	1261.64	3.36457
$53$	335.960	0.870710	0.435355	−	0.900259i	$-0.356623\pi$
$53$	335.960	0.870710	0.435355	+	0.900259i	$0.356623\pi$
$54$	0	0
$55$	−475.274	−1.16520
$56$	409.893	0.978112
$57$	0	0
$58$	−3.03281	−0.00686599
$59$	−807.109	−1.78096	−0.890480	−	0.455023i	$-0.849631\pi$
$59$	−807.109	−1.78096	−0.890480	+	0.455023i	$0.849631\pi$
$60$	0	0
$61$	−736.732	−1.54638	−0.773188	−	0.634177i	$-0.781339\pi$
$61$	−736.732	−1.54638	−0.773188	+	0.634177i	$0.781339\pi$
$62$	−915.836	−1.87599
$63$	0	0
$64$	−306.295	−0.598232
$65$	555.174	1.05940
$66$	0	0
$67$	−908.155	−1.65595	−0.827976	−	0.560763i	$-0.810508\pi$
$67$	−908.155	−1.65595	−0.827976	+	0.560763i	$0.810508\pi$
$68$	−1209.17	−2.15637
$69$	0	0
$70$	342.051	0.584042
$71$	710.028	1.18683	0.593415	−	0.804897i	$-0.297779\pi$
$71$	710.028	1.18683	0.593415	+	0.804897i	$0.297779\pi$
$72$	0	0
$73$	−695.867	−1.11569	−0.557843	−	0.829946i	$-0.688371\pi$
$73$	−695.867	−1.11569	−0.557843	+	0.829946i	$0.688371\pi$
$74$	−1705.61	−2.67937
$75$	0	0
$76$	−1404.23	−2.11943
$77$	587.067	0.868864
$78$	0	0
$79$	654.632	0.932302	0.466151	−	0.884705i	$-0.345640\pi$
$79$	654.632	0.932302	0.466151	+	0.884705i	$0.345640\pi$
$80$	648.305	0.906034
$81$	0	0
$82$	−362.585	−0.488303
$83$	668.629	0.884235	0.442118	−	0.896957i	$-0.354227\pi$
$83$	668.629	0.884235	0.442118	+	0.896957i	$0.354227\pi$
$84$	0	0
$85$	−532.084	−0.678972
$86$	−2369.81	−2.97143
$87$	0	0
$88$	2843.41	3.44441
$89$	−568.035	−0.676534	−0.338267	−	0.941050i	$-0.609841\pi$
$89$	−568.035	−0.676534	−0.338267	+	0.941050i	$0.609841\pi$
$90$	0	0
$91$	−685.761	−0.789971
$92$	2549.20	2.88884
$93$	0	0
$94$	−717.322	−0.787086
$95$	−617.922	−0.667342
$96$	0	0
$97$	−391.260	−0.409550	−0.204775	−	0.978809i	$-0.565646\pi$
$97$	−391.260	−0.409550	−0.204775	+	0.978809i	$0.565646\pi$
$98$	1289.91	1.32960
$99$	0	0
$100$	−1176.83	−1.17683
$101$	1081.71	1.06568	0.532841	−	0.846215i	$-0.321124\pi$
$101$	1081.71	1.06568	0.532841	+	0.846215i	$0.321124\pi$
$102$	0	0
$103$	−259.147	−0.247908	−0.123954	−	0.992288i	$-0.539558\pi$
$103$	−259.147	−0.247908	−0.123954	+	0.992288i	$0.539558\pi$
$104$	−3321.42	−3.13166
$105$	0	0
$106$	−1677.27	−1.53689
$107$	1020.81	0.922294	0.461147	−	0.887324i	$-0.347438\pi$
$107$	1020.81	0.922294	0.461147	+	0.887324i	$0.347438\pi$
$108$	0	0
$109$	−1349.42	−1.18579	−0.592894	−	0.805280i	$-0.702015\pi$
$109$	−1349.42	−1.18579	−0.592894	+	0.805280i	$0.702015\pi$
$110$	2372.79	2.05670
$111$	0	0
$112$	−800.798	−0.675610
$113$	−527.442	−0.439093	−0.219547	−	0.975602i	$-0.570458\pi$
$113$	−527.442	−0.439093	−0.219547	+	0.975602i	$0.570458\pi$
$114$	0	0
$115$	1121.76	0.909604
$116$	10.2814	0.00822934
$117$	0	0
$118$	4029.46	3.14358
$119$	657.240	0.506295
$120$	0	0
$121$	2741.45	2.05969
$122$	3678.11	2.72951
$123$	0	0
$124$	3104.74	2.24850
$125$	−1448.81	−1.03668
$126$	0	0
$127$	1612.41	1.12660	0.563301	−	0.826252i	$-0.309531\pi$
$127$	1612.41	1.12660	0.563301	+	0.826252i	$0.309531\pi$
$128$	2154.26	1.48759
$129$	0	0
$130$	−2771.69	−1.86995
$131$	1712.32	1.14203	0.571015	−	0.820939i	$-0.306550\pi$
$131$	1712.32	1.14203	0.571015	+	0.820939i	$0.306550\pi$
$132$	0	0
$133$	763.268	0.497622
$134$	4533.94	2.92293
$135$	0	0
$136$	3183.28	2.00709
$137$	−2008.36	−1.25245	−0.626225	−	0.779643i	$-0.715401\pi$
$137$	−2008.36	−1.25245	−0.626225	+	0.779643i	$0.715401\pi$
$138$	0	0
$139$	1289.96	0.787146	0.393573	−	0.919293i	$-0.371239\pi$
$139$	1289.96	0.787146	0.393573	+	0.919293i	$0.371239\pi$
$140$	−1159.57	−0.700013
$141$	0	0
$142$	−3544.79	−2.09488
$143$	−4757.09	−2.78187
$144$	0	0
$145$	4.52425	0.00259116
$146$	3474.10	1.96930
$147$	0	0
$148$	5782.11	3.21140
$149$	2353.56	1.29403	0.647017	−	0.762475i	$-0.276016\pi$
$149$	2353.56	1.29403	0.647017	+	0.762475i	$0.276016\pi$
$150$	0	0
$151$	−3001.88	−1.61781	−0.808906	−	0.587938i	$-0.799940\pi$
$151$	−3001.88	−1.61781	−0.808906	+	0.587938i	$0.799940\pi$
$152$	3696.82	1.97271
$153$	0	0
$154$	−2930.91	−1.53364
$155$	1366.21	0.707980
$156$	0	0
$157$	−1894.78	−0.963182	−0.481591	−	0.876396i	$-0.659941\pi$
$157$	−1894.78	−0.963182	−0.481591	+	0.876396i	$0.659941\pi$
$158$	−3268.23	−1.64561
$159$	0	0
$160$	−581.933	−0.287536
$161$	−1385.62	−0.678272
$162$	0	0
$163$	−3455.42	−1.66043	−0.830213	−	0.557446i	$-0.811781\pi$
$163$	−3455.42	−1.66043	−0.830213	+	0.557446i	$0.811781\pi$
$164$	1229.18	0.585263
$165$	0	0
$166$	−3338.11	−1.56077
$167$	−167.000	−0.0773823
$167$	−167.000	−0.0773823
$168$	0	0
$169$	3359.82	1.52928
$170$	2656.41	1.19846
$171$	0	0
$172$	8033.77	3.56145
$173$	−2726.11	−1.19805	−0.599023	−	0.800732i	$-0.704444\pi$
$173$	−2726.11	−1.19805	−0.599023	+	0.800732i	$0.704444\pi$
$174$	0	0
$175$	639.665	0.276309
$176$	−5555.09	−2.37915
$177$	0	0
$178$	2835.89	1.19415
$179$	−1056.88	−0.441313	−0.220657	−	0.975352i	$-0.570820\pi$
$179$	−1056.88	−0.441313	−0.220657	+	0.975352i	$0.570820\pi$
$180$	0	0
$181$	−958.077	−0.393444	−0.196722	−	0.980459i	$-0.563030\pi$
$181$	−958.077	−0.393444	−0.196722	+	0.980459i	$0.563030\pi$
$182$	3423.64	1.39438
$183$	0	0
$184$	−6711.10	−2.68885
$185$	2544.37	1.01117
$186$	0	0
$187$	4559.23	1.78291
$188$	2431.76	0.943374
$189$	0	0
$190$	3084.96	1.17793
$191$	2286.09	0.866052	0.433026	−	0.901381i	$-0.357446\pi$
$191$	2286.09	0.866052	0.433026	+	0.901381i	$0.357446\pi$
$192$	0	0
$193$	−1444.11	−0.538597	−0.269298	−	0.963057i	$-0.586792\pi$
$193$	−1444.11	−0.538597	−0.269298	+	0.963057i	$0.586792\pi$
$194$	1953.35	0.722899
$195$	0	0
$196$	−4372.86	−1.59361
$197$	−3368.12	−1.21812	−0.609058	−	0.793126i	$-0.708452\pi$
$197$	−3368.12	−1.21812	−0.609058	+	0.793126i	$0.708452\pi$
$198$	0	0
$199$	−594.538	−0.211787	−0.105894	−	0.994377i	$-0.533770\pi$
$199$	−594.538	−0.211787	−0.105894	+	0.994377i	$0.533770\pi$
$200$	3098.16	1.09536
$201$	0	0
$202$	−5400.39	−1.88104
$203$	−5.58843	−0.00193217
$204$	0	0
$205$	540.892	0.184281
$206$	1293.78	0.437583
$207$	0	0
$208$	6488.98	2.16312
$209$	5294.75	1.75237
$210$	0	0
$211$	1768.33	0.576951	0.288475	−	0.957487i	$-0.406852\pi$
$211$	1768.33	0.576951	0.288475	+	0.957487i	$0.406852\pi$
$212$	5686.03	1.84207
$213$	0	0
$214$	−5096.36	−1.62794
$215$	3535.20	1.12139
$216$	0	0
$217$	−1687.57	−0.527926
$218$	6736.94	2.09304
$219$	0	0
$220$	−8043.89	−2.46509
$221$	−5325.70	−1.62102
$222$	0	0
$223$	2643.16	0.793719	0.396859	−	0.917879i	$-0.370100\pi$
$223$	2643.16	0.793719	0.396859	+	0.917879i	$0.370100\pi$
$224$	718.814	0.214410
$225$	0	0
$226$	2633.24	0.775045
$227$	−4190.64	−1.22530	−0.612649	−	0.790355i	$-0.709896\pi$
$227$	−4190.64	−1.22530	−0.612649	+	0.790355i	$0.709896\pi$
$228$	0	0
$229$	6540.38	1.88734	0.943670	−	0.330889i	$-0.107348\pi$
$229$	6540.38	1.88734	0.943670	+	0.330889i	$0.107348\pi$
$230$	−5600.34	−1.60555
$231$	0	0
$232$	−27.0670	−0.00765965
$233$	−2031.89	−0.571302	−0.285651	−	0.958334i	$-0.592210\pi$
$233$	−2031.89	−0.571302	−0.285651	+	0.958334i	$0.592210\pi$
$234$	0	0
$235$	1070.08	0.297039
$236$	−13660.1	−3.76778
$237$	0	0
$238$	−3281.25	−0.893663
$239$	−1095.87	−0.296595	−0.148297	−	0.988943i	$-0.547379\pi$
$239$	−1095.87	−0.296595	−0.148297	+	0.988943i	$0.547379\pi$
$240$	0	0
$241$	2831.50	0.756817	0.378409	−	0.925639i	$-0.376471\pi$
$241$	2831.50	0.756817	0.378409	+	0.925639i	$0.376471\pi$
$242$	−13686.6	−3.63557
$243$	0	0
$244$	−12469.0	−3.27150
$245$	−1924.24	−0.501777
$246$	0	0
$247$	−6184.87	−1.59325
$248$	−8173.60	−2.09284
$249$	0	0
$250$	7233.12	1.82985
$251$	−3315.53	−0.833762	−0.416881	−	0.908961i	$-0.636877\pi$
$251$	−3315.53	−0.833762	−0.416881	+	0.908961i	$0.636877\pi$
$252$	0	0
$253$	−9611.93	−2.38853
$254$	−8049.91	−1.98857
$255$	0	0
$256$	−8304.73	−2.02752
$257$	−5994.11	−1.45487	−0.727437	−	0.686175i	$-0.759289\pi$
$257$	−5994.11	−1.45487	−0.727437	+	0.686175i	$0.759289\pi$
$258$	0	0
$259$	−3142.85	−0.754006
$260$	9396.18	2.24126
$261$	0	0
$262$	−8548.70	−2.01580
$263$	−7050.29	−1.65300	−0.826501	−	0.562936i	$-0.809672\pi$
$263$	−7050.29	−1.65300	−0.826501	+	0.562936i	$0.809672\pi$
$264$	0	0
$265$	2502.09	0.580009
$266$	−3810.59	−0.878355
$267$	0	0
$268$	−15370.3	−3.50332
$269$	−2094.48	−0.474731	−0.237365	−	0.971420i	$-0.576284\pi$
$269$	−2094.48	−0.474731	−0.237365	+	0.971420i	$0.576284\pi$
$270$	0	0
$271$	2920.22	0.654579	0.327289	−	0.944924i	$-0.393865\pi$
$271$	2920.22	0.654579	0.327289	+	0.944924i	$0.393865\pi$
$272$	−6219.09	−1.38635
$273$	0	0
$274$	10026.7	2.21070
$275$	4437.32	0.973020
$276$	0	0
$277$	−2573.69	−0.558261	−0.279130	−	0.960253i	$-0.590046\pi$
$277$	−2573.69	−0.558261	−0.279130	+	0.960253i	$0.590046\pi$
$278$	−6440.11	−1.38940
$279$	0	0
$280$	3052.72	0.651553
$281$	−3810.02	−0.808850	−0.404425	−	0.914571i	$-0.632528\pi$
$281$	−3810.02	−0.808850	−0.404425	+	0.914571i	$0.632528\pi$
$282$	0	0
$283$	4899.07	1.02904	0.514522	−	0.857477i	$-0.327970\pi$
$283$	4899.07	1.02904	0.514522	+	0.857477i	$0.327970\pi$
$284$	12017.0	2.51085
$285$	0	0
$286$	23749.6	4.91029
$287$	−668.120	−0.137414
$288$	0	0
$289$	191.200	0.0389171
$290$	−22.5872	−0.00457367
$291$	0	0
$292$	−11777.4	−2.36034
$293$	−1293.08	−0.257824	−0.128912	−	0.991656i	$-0.541149\pi$
$293$	−1293.08	−0.257824	−0.128912	+	0.991656i	$0.541149\pi$
$294$	0	0
$295$	−6011.02	−1.18636
$296$	−15222.1	−2.98908
$297$	0	0
$298$	−11750.1	−2.28411
$299$	11227.8	2.17165
$300$	0	0
$301$	−4366.74	−0.836195
$302$	14986.8	2.85561
$303$	0	0
$304$	−7222.39	−1.36261
$305$	−5486.89	−1.03009
$306$	0	0
$307$	6330.79	1.17693	0.588465	−	0.808523i	$-0.299732\pi$
$307$	6330.79	1.17693	0.588465	+	0.808523i	$0.299732\pi$
$308$	9935.96	1.83816
$309$	0	0
$310$	−6820.78	−1.24966
$311$	−3698.88	−0.674419	−0.337210	−	0.941430i	$-0.609483\pi$
$311$	−3698.88	−0.674419	−0.337210	+	0.941430i	$0.609483\pi$
$312$	0	0
$313$	−9334.88	−1.68575	−0.842873	−	0.538112i	$-0.819138\pi$
$313$	−9334.88	−1.68575	−0.842873	+	0.538112i	$0.819138\pi$
$314$	9459.61	1.70012
$315$	0	0
$316$	11079.5	1.97237
$317$	4744.40	0.840607	0.420303	−	0.907384i	$-0.361924\pi$
$317$	4744.40	0.840607	0.420303	+	0.907384i	$0.361924\pi$
$318$	0	0
$319$	−38.7666	−0.00680412
$320$	−2281.16	−0.398502
$321$	0	0
$322$	6917.64	1.19722
$323$	5927.63	1.02112
$324$	0	0
$325$	−5183.30	−0.884669
$326$	17251.1	2.93083
$327$	0	0
$328$	−3235.98	−0.544747
$329$	−1321.78	−0.221495
$330$	0	0
$331$	−10696.2	−1.77619	−0.888095	−	0.459660i	$-0.847971\pi$
$331$	−10696.2	−1.77619	−0.888095	+	0.459660i	$0.847971\pi$
$332$	11316.4	1.87068
$333$	0	0
$334$	833.742	0.136588
$335$	−6763.57	−1.10309
$336$	0	0
$337$	1801.99	0.291278	0.145639	−	0.989338i	$-0.453476\pi$
$337$	1801.99	0.291278	0.145639	+	0.989338i	$0.453476\pi$
$338$	−16773.8	−2.69933
$339$	0	0
$340$	−9005.38	−1.43643
$341$	−11706.6	−1.85908
$342$	0	0
$343$	5532.26	0.870886
$344$	−21149.9	−3.31490
$345$	0	0
$346$	13610.0	2.11468
$347$	10474.5	1.62047	0.810233	−	0.586108i	$-0.199341\pi$
$347$	10474.5	1.62047	0.810233	+	0.586108i	$0.199341\pi$
$348$	0	0
$349$	−5755.70	−0.882795	−0.441398	−	0.897312i	$-0.645517\pi$
$349$	−5755.70	−0.882795	−0.441398	+	0.897312i	$0.645517\pi$
$350$	−3193.51	−0.487715
$351$	0	0
$352$	4986.37	0.755041
$353$	−521.926	−0.0786949	−0.0393475	−	0.999226i	$-0.512528\pi$
$353$	−521.926	−0.0786949	−0.0393475	+	0.999226i	$0.512528\pi$
$354$	0	0
$355$	5288.01	0.790587
$356$	−9613.84	−1.43127
$357$	0	0
$358$	5276.45	0.778964
$359$	4857.32	0.714093	0.357046	−	0.934087i	$-0.383784\pi$
$359$	4857.32	0.714093	0.357046	+	0.934087i	$0.383784\pi$
$360$	0	0
$361$	24.9052	0.00363102
$362$	4783.17	0.694469
$363$	0	0
$364$	−11606.3	−1.67126
$365$	−5182.54	−0.743196
$366$	0	0
$367$	6077.51	0.864423	0.432212	−	0.901772i	$-0.357733\pi$
$367$	6077.51	0.864423	0.432212	+	0.901772i	$0.357733\pi$
$368$	13111.3	1.85727
$369$	0	0
$370$	−12702.7	−1.78482
$371$	−3090.63	−0.432500
$372$	0	0
$373$	−12103.6	−1.68016	−0.840078	−	0.542465i	$-0.817491\pi$
$373$	−12103.6	−1.68016	−0.840078	+	0.542465i	$0.817491\pi$
$374$	−22761.8	−3.14702
$375$	0	0
$376$	−6401.91	−0.878068
$377$	45.2838	0.00618630
$378$	0	0
$379$	7849.48	1.06385	0.531927	−	0.846790i	$-0.321468\pi$
$379$	7849.48	1.06385	0.531927	+	0.846790i	$0.321468\pi$
$380$	−10458.2	−1.41182
$381$	0	0
$382$	−11413.3	−1.52867
$383$	6360.67	0.848603	0.424301	−	0.905521i	$-0.360520\pi$
$383$	6360.67	0.848603	0.424301	+	0.905521i	$0.360520\pi$
$384$	0	0
$385$	4372.24	0.578779
$386$	7209.67	0.950680
$387$	0	0
$388$	−6621.97	−0.866442
$389$	−919.055	−0.119789	−0.0598945	−	0.998205i	$-0.519076\pi$
$389$	−919.055	−0.119789	−0.0598945	+	0.998205i	$0.519076\pi$
$390$	0	0
$391$	−10760.8	−1.39181
$392$	11512.1	1.48329
$393$	0	0
$394$	16815.2	2.15010
$395$	4875.43	0.621037
$396$	0	0
$397$	−6857.32	−0.866899	−0.433449	−	0.901178i	$-0.642704\pi$
$397$	−6857.32	−0.866899	−0.433449	+	0.901178i	$0.642704\pi$
$398$	2968.21	0.373827
$399$	0	0
$400$	−6052.80	−0.756599
$401$	−7342.59	−0.914392	−0.457196	−	0.889366i	$-0.651146\pi$
$401$	−7342.59	−0.914392	−0.457196	+	0.889366i	$0.651146\pi$
$402$	0	0
$403$	13674.6	1.69028
$404$	18307.6	2.25455
$405$	0	0
$406$	27.9001	0.00341048
$407$	−21801.8	−2.65522
$408$	0	0
$409$	5190.29	0.627489	0.313745	−	0.949507i	$-0.398416\pi$
$409$	5190.29	0.627489	0.313745	+	0.949507i	$0.398416\pi$
$410$	−2700.39	−0.325275
$411$	0	0
$412$	−4386.00	−0.524472
$413$	7424.92	0.884640
$414$	0	0
$415$	4979.68	0.589019
$416$	−5824.65	−0.686483
$417$	0	0
$418$	−26433.9	−3.09312
$419$	−13190.9	−1.53799	−0.768997	−	0.639253i	$-0.779244\pi$
$419$	−13190.9	−1.53799	−0.768997	+	0.639253i	$0.779244\pi$
$420$	0	0
$421$	−12623.5	−1.46135	−0.730677	−	0.682724i	$-0.760795\pi$
$421$	−12623.5	−1.46135	−0.730677	+	0.682724i	$0.760795\pi$
$422$	−8828.32	−1.01838
$423$	0	0
$424$	−14969.2	−1.71455
$425$	4967.71	0.566987
$426$	0	0
$427$	6777.50	0.768118
$428$	17276.9	1.95120
$429$	0	0
$430$	−17649.4	−1.97937
$431$	3904.92	0.436412	0.218206	−	0.975903i	$-0.429980\pi$
$431$	3904.92	0.436412	0.218206	+	0.975903i	$0.429980\pi$
$432$	0	0
$433$	33.0127	0.00366395	0.00183197	−	0.999998i	$-0.499417\pi$
$433$	33.0127	0.00366395	0.00183197	+	0.999998i	$0.499417\pi$
$434$	8425.15	0.931844
$435$	0	0
$436$	−22838.6	−2.50865
$437$	−12496.8	−1.36797
$438$	0	0
$439$	−5372.93	−0.584137	−0.292068	−	0.956397i	$-0.594343\pi$
$439$	−5372.93	−0.584137	−0.292068	+	0.956397i	$0.594343\pi$
$440$	21176.5	2.29444
$441$	0	0
$442$	26588.4	2.86127
$443$	5557.29	0.596015	0.298008	−	0.954563i	$-0.403678\pi$
$443$	5557.29	0.596015	0.298008	+	0.954563i	$0.403678\pi$
$444$	0	0
$445$	−4230.49	−0.450662
$446$	−13195.9	−1.40100
$447$	0	0
$448$	2817.73	0.297155
$449$	−7997.64	−0.840606	−0.420303	−	0.907384i	$-0.638076\pi$
$449$	−7997.64	−0.840606	−0.420303	+	0.907384i	$0.638076\pi$
$450$	0	0
$451$	−4634.71	−0.483902
$452$	−8926.81	−0.928942
$453$	0	0
$454$	20921.7	2.16278
$455$	−5107.28	−0.526226
$456$	0	0
$457$	−15583.5	−1.59511	−0.797553	−	0.603249i	$-0.793872\pi$
$457$	−15583.5	−1.59511	−0.797553	+	0.603249i	$0.793872\pi$
$458$	−32652.7	−3.33135
$459$	0	0
$460$	18985.5	1.92435
$461$	1966.16	0.198640	0.0993199	−	0.995056i	$-0.468333\pi$
$461$	1966.16	0.198640	0.0993199	+	0.995056i	$0.468333\pi$
$462$	0	0
$463$	3402.00	0.341478	0.170739	−	0.985316i	$-0.445385\pi$
$463$	3402.00	0.341478	0.170739	+	0.985316i	$0.445385\pi$
$464$	52.8802	0.00529074
$465$	0	0
$466$	10144.1	1.00841
$467$	−9435.63	−0.934966	−0.467483	−	0.884002i	$-0.654839\pi$
$467$	−9435.63	−0.934966	−0.467483	+	0.884002i	$0.654839\pi$
$468$	0	0
$469$	8354.49	0.822547
$470$	−5342.33	−0.524305
$471$	0	0
$472$	35961.9	3.50695
$473$	−30291.8	−2.94465
$474$	0	0
$475$	5769.13	0.557275
$476$	11123.6	1.07111
$477$	0	0
$478$	5471.11	0.523521
$479$	−10140.0	−0.967240	−0.483620	−	0.875278i	$-0.660678\pi$
$479$	−10140.0	−0.967240	−0.483620	+	0.875278i	$0.660678\pi$
$480$	0	0
$481$	25467.0	2.41413
$482$	−14136.2	−1.33586
$483$	0	0
$484$	46398.4	4.35747
$485$	−2913.94	−0.272815
$486$	0	0
$487$	−17708.9	−1.64778	−0.823890	−	0.566750i	$-0.808201\pi$
$487$	−17708.9	−1.64778	−0.823890	+	0.566750i	$0.808201\pi$
$488$	32826.2	3.04502
$489$	0	0
$490$	9606.72	0.885689
$491$	−6778.26	−0.623012	−0.311506	−	0.950244i	$-0.600833\pi$
$491$	−6778.26	−0.623012	−0.311506	+	0.950244i	$0.600833\pi$
$492$	0	0
$493$	−43.4004	−0.00396482
$494$	30877.8	2.81226
$495$	0	0
$496$	15968.6	1.44558
$497$	−6531.84	−0.589523
$498$	0	0
$499$	−6641.16	−0.595790	−0.297895	−	0.954599i	$-0.596285\pi$
$499$	−6641.16	−0.595790	−0.297895	+	0.954599i	$0.596285\pi$
$500$	−24520.7	−2.19319
$501$	0	0
$502$	16552.7	1.47168
$503$	11033.2	0.978027	0.489013	−	0.872276i	$-0.337357\pi$
$503$	11033.2	0.978027	0.489013	+	0.872276i	$0.337357\pi$
$504$	0	0
$505$	8056.12	0.709886
$506$	47987.3	4.21600
$507$	0	0
$508$	27289.6	2.38343
$509$	−866.633	−0.0754673	−0.0377336	−	0.999288i	$-0.512014\pi$
$509$	−866.633	−0.0754673	−0.0377336	+	0.999288i	$0.512014\pi$
$510$	0	0
$511$	6401.57	0.554185
$512$	24227.0	2.09120
$513$	0	0
$514$	29925.4	2.56800
$515$	−1930.02	−0.165140
$516$	0	0
$517$	−9169.10	−0.779994
$518$	15690.6	1.33090
$519$	0	0
$520$	−24736.6	−2.08610
$521$	−5227.99	−0.439621	−0.219810	−	0.975543i	$-0.570544\pi$
$521$	−5227.99	−0.439621	−0.219810	+	0.975543i	$0.570544\pi$
$522$	0	0
$523$	−1881.06	−0.157272	−0.0786359	−	0.996903i	$-0.525056\pi$
$523$	−1881.06	−0.157272	−0.0786359	+	0.996903i	$0.525056\pi$
$524$	28980.6	2.41607
$525$	0	0
$526$	35198.3	2.91772
$527$	−13105.9	−1.08330
$528$	0	0
$529$	10519.4	0.864584
$530$	−12491.6	−1.02378
$531$	0	0
$532$	12918.1	1.05277
$533$	5413.87	0.439964
$534$	0	0
$535$	7602.58	0.614371
$536$	40464.2	3.26080
$537$	0	0
$538$	10456.6	0.837950
$539$	16488.1	1.31761
$540$	0	0
$541$	−19793.2	−1.57297	−0.786483	−	0.617612i	$-0.788100\pi$
$541$	−19793.2	−1.57297	−0.786483	+	0.617612i	$0.788100\pi$
$542$	−14579.1	−1.15540
$543$	0	0
$544$	5582.39	0.439969
$545$	−10049.9	−0.789894
$546$	0	0
$547$	8455.16	0.660908	0.330454	−	0.943822i	$-0.392798\pi$
$547$	8455.16	0.660908	0.330454	+	0.943822i	$0.392798\pi$
$548$	−33990.9	−2.64967
$549$	0	0
$550$	−22153.2	−1.71748
$551$	−50.4020	−0.00389691
$552$	0	0
$553$	−6022.22	−0.463094
$554$	12849.1	0.985388
$555$	0	0
$556$	21832.3	1.66528
$557$	3437.89	0.261523	0.130761	−	0.991414i	$-0.458258\pi$
$557$	3437.89	0.261523	0.130761	+	0.991414i	$0.458258\pi$
$558$	0	0
$559$	35384.3	2.67728
$560$	−5964.02	−0.450046
$561$	0	0
$562$	19021.4	1.42770
$563$	−13121.8	−0.982270	−0.491135	−	0.871083i	$-0.663418\pi$
$563$	−13121.8	−0.982270	−0.491135	+	0.871083i	$0.663418\pi$
$564$	0	0
$565$	−3928.17	−0.292495
$566$	−24458.4	−1.81637
$567$	0	0
$568$	−31636.4	−2.33703
$569$	10214.4	0.752566	0.376283	−	0.926505i	$-0.377202\pi$
$569$	10214.4	0.752566	0.376283	+	0.926505i	$0.377202\pi$
$570$	0	0
$571$	−9281.29	−0.680228	−0.340114	−	0.940384i	$-0.610466\pi$
$571$	−9281.29	−0.680228	−0.340114	+	0.940384i	$0.610466\pi$
$572$	−80512.5	−5.88531
$573$	0	0
$574$	3335.57	0.242550
$575$	−10473.1	−0.759580
$576$	0	0
$577$	−5951.01	−0.429365	−0.214683	−	0.976684i	$-0.568872\pi$
$577$	−5951.01	−0.429365	−0.214683	+	0.976684i	$0.568872\pi$
$578$	−954.559	−0.0686927
$579$	0	0
$580$	76.5717	0.00548184
$581$	−6150.99	−0.439218
$582$	0	0
$583$	−21439.5	−1.52304
$584$	31005.4	2.19694
$585$	0	0
$586$	6455.66	0.455087
$587$	14230.1	1.00058	0.500288	−	0.865859i	$-0.333227\pi$
$587$	14230.1	1.00058	0.500288	+	0.865859i	$0.333227\pi$
$588$	0	0
$589$	−15220.2	−1.06475
$590$	30009.8	2.09404
$591$	0	0
$592$	29739.1	2.06464
$593$	−13136.3	−0.909687	−0.454843	−	0.890571i	$-0.650305\pi$
$593$	−13136.3	−0.909687	−0.454843	+	0.890571i	$0.650305\pi$
$594$	0	0
$595$	4894.86	0.337260
$596$	39833.4	2.73765
$597$	0	0
$598$	−56054.6	−3.83318
$599$	4416.99	0.301291	0.150646	−	0.988588i	$-0.451865\pi$
$599$	4416.99	0.301291	0.150646	+	0.988588i	$0.451865\pi$
$600$	0	0
$601$	5334.69	0.362074	0.181037	−	0.983476i	$-0.442055\pi$
$601$	5334.69	0.362074	0.181037	+	0.983476i	$0.442055\pi$
$602$	21800.8	1.47597
$603$	0	0
$604$	−50806.0	−3.42263
$605$	20417.2	1.37203
$606$	0	0
$607$	4241.78	0.283639	0.141819	−	0.989893i	$-0.454705\pi$
$607$	4241.78	0.283639	0.141819	+	0.989893i	$0.454705\pi$
$608$	6482.97	0.432433
$609$	0	0
$610$	27393.1	1.81822
$611$	10710.6	0.709170
$612$	0	0
$613$	22776.3	1.50069	0.750346	−	0.661045i	$-0.229887\pi$
$613$	22776.3	1.50069	0.750346	+	0.661045i	$0.229887\pi$
$614$	−31606.3	−2.07740
$615$	0	0
$616$	−26157.6	−1.71091
$617$	771.714	0.0503534	0.0251767	−	0.999683i	$-0.491985\pi$
$617$	771.714	0.0503534	0.0251767	+	0.999683i	$0.491985\pi$
$618$	0	0
$619$	4715.99	0.306222	0.153111	−	0.988209i	$-0.451071\pi$
$619$	4715.99	0.306222	0.153111	+	0.988209i	$0.451071\pi$
$620$	23122.8	1.49780
$621$	0	0
$622$	18466.6	1.19042
$623$	5225.58	0.336049
$624$	0	0
$625$	−2098.46	−0.134301
$626$	46604.1	2.97552
$627$	0	0
$628$	−32068.6	−2.03770
$629$	−24407.8	−1.54722
$630$	0	0
$631$	−8300.60	−0.523679	−0.261840	−	0.965111i	$-0.584329\pi$
$631$	−8300.60	−0.523679	−0.261840	+	0.965111i	$0.584329\pi$
$632$	−29168.1	−1.83583
$633$	0	0
$634$	−23686.3	−1.48376
$635$	12008.6	0.750467
$636$	0	0
$637$	−19260.0	−1.19797
$638$	193.541	0.0120100
$639$	0	0
$640$	16044.1	0.990934
$641$	−4118.22	−0.253760	−0.126880	−	0.991918i	$-0.540496\pi$
$641$	−4118.22	−0.253760	−0.126880	+	0.991918i	$0.540496\pi$
$642$	0	0
$643$	−4255.68	−0.261007	−0.130504	−	0.991448i	$-0.541659\pi$
$643$	−4255.68	−0.261007	−0.130504	+	0.991448i	$0.541659\pi$
$644$	−23451.2	−1.43495
$645$	0	0
$646$	−29593.5	−1.80239
$647$	5729.55	0.348148	0.174074	−	0.984733i	$-0.444307\pi$
$647$	5729.55	0.348148	0.174074	+	0.984733i	$0.444307\pi$
$648$	0	0
$649$	51506.2	3.11525
$650$	25877.4	1.56153
$651$	0	0
$652$	−58482.1	−3.51279
$653$	16215.9	0.971789	0.485895	−	0.874017i	$-0.338494\pi$
$653$	16215.9	0.971789	0.485895	+	0.874017i	$0.338494\pi$
$654$	0	0
$655$	12752.7	0.760745
$656$	6322.05	0.376272
$657$	0	0
$658$	6598.94	0.390963
$659$	−16498.1	−0.975230	−0.487615	−	0.873059i	$-0.662133\pi$
$659$	−16498.1	−0.975230	−0.487615	+	0.873059i	$0.662133\pi$
$660$	0	0
$661$	8032.25	0.472645	0.236323	−	0.971675i	$-0.424058\pi$
$661$	8032.25	0.472645	0.236323	+	0.971675i	$0.424058\pi$
$662$	53400.7	3.13516
$663$	0	0
$664$	−29791.7	−1.74118
$665$	5684.52	0.331483
$666$	0	0
$667$	91.4982	0.00531158
$668$	−2826.43	−0.163709
$669$	0	0
$670$	33766.9	1.94706
$671$	47015.1	2.70492
$672$	0	0
$673$	4041.56	0.231487	0.115743	−	0.993279i	$-0.463075\pi$
$673$	4041.56	0.231487	0.115743	+	0.993279i	$0.463075\pi$
$674$	−8996.38	−0.514136
$675$	0	0
$676$	56864.1	3.23533
$677$	2659.19	0.150962	0.0754808	−	0.997147i	$-0.475951\pi$
$677$	2659.19	0.150962	0.0754808	+	0.997147i	$0.475951\pi$
$678$	0	0
$679$	3599.36	0.203432
$680$	23707.8	1.33699
$681$	0	0
$682$	58444.8	3.28148
$683$	3156.08	0.176814	0.0884070	−	0.996084i	$-0.471822\pi$
$683$	3156.08	0.176814	0.0884070	+	0.996084i	$0.471822\pi$
$684$	0	0
$685$	−14957.4	−0.834298
$686$	−27619.6	−1.53720
$687$	0	0
$688$	41320.1	2.28970
$689$	25043.8	1.38475
$690$	0	0
$691$	−13262.6	−0.730152	−0.365076	−	0.930978i	$-0.618957\pi$
$691$	−13262.6	−0.730152	−0.365076	+	0.930978i	$0.618957\pi$
$692$	−46138.6	−2.53458
$693$	0	0
$694$	−52293.7	−2.86029
$695$	9607.14	0.524345
$696$	0	0
$697$	−5188.70	−0.281974
$698$	28735.2	1.55823
$699$	0	0
$700$	10826.2	0.584558
$701$	−29738.7	−1.60230	−0.801151	−	0.598462i	$-0.795779\pi$
$701$	−29738.7	−1.60230	−0.801151	+	0.598462i	$0.795779\pi$
$702$	0	0
$703$	−28345.3	−1.52072
$704$	19546.4	1.04643
$705$	0	0
$706$	2605.70	0.138905
$707$	−9951.06	−0.529347
$708$	0	0
$709$	−18519.8	−0.980998	−0.490499	−	0.871442i	$-0.663185\pi$
$709$	−18519.8	−0.980998	−0.490499	+	0.871442i	$0.663185\pi$
$710$	−26400.2	−1.39547
$711$	0	0
$712$	25309.6	1.33219
$713$	27630.3	1.45128
$714$	0	0
$715$	−35428.9	−1.85310
$716$	−17887.5	−0.933639
$717$	0	0
$718$	−24250.0	−1.26045
$719$	22202.2	1.15160	0.575801	−	0.817590i	$-0.304690\pi$
$719$	22202.2	1.15160	0.575801	+	0.817590i	$0.304690\pi$
$720$	0	0
$721$	2384.00	0.123141
$722$	−124.338	−0.00640913
$723$	0	0
$724$	−16215.2	−0.832366
$725$	−42.2399	−0.00216379
$726$	0	0
$727$	9160.95	0.467347	0.233673	−	0.972315i	$-0.424925\pi$
$727$	9160.95	0.467347	0.233673	+	0.972315i	$0.424925\pi$
$728$	30555.1	1.55556
$729$	0	0
$730$	25873.7	1.31182
$731$	−33912.6	−1.71587
$732$	0	0
$733$	−3111.09	−0.156768	−0.0783838	−	0.996923i	$-0.524976\pi$
$733$	−3111.09	−0.156768	−0.0783838	+	0.996923i	$0.524976\pi$
$734$	−30341.8	−1.52580
$735$	0	0
$736$	−11769.0	−0.589417
$737$	57954.6	2.89659
$738$	0	0
$739$	31241.6	1.55513	0.777565	−	0.628803i	$-0.216455\pi$
$739$	31241.6	1.55513	0.777565	+	0.628803i	$0.216455\pi$
$740$	43062.8	2.13922
$741$	0	0
$742$	15429.9	0.763408
$743$	5547.21	0.273900	0.136950	−	0.990578i	$-0.456270\pi$
$743$	5547.21	0.273900	0.136950	+	0.990578i	$0.456270\pi$
$744$	0	0
$745$	17528.4	0.861999
$746$	60426.6	2.96565
$747$	0	0
$748$	77163.8	3.77191
$749$	−9390.85	−0.458123
$750$	0	0
$751$	−35957.6	−1.74715	−0.873577	−	0.486686i	$-0.838206\pi$
$751$	−35957.6	−1.74715	−0.873577	+	0.486686i	$0.838206\pi$
$752$	12507.3	0.606506
$753$	0	0
$754$	−226.078	−0.0109195
$755$	−22356.8	−1.07768
$756$	0	0
$757$	17735.8	0.851543	0.425771	−	0.904831i	$-0.360003\pi$
$757$	17735.8	0.851543	0.425771	+	0.904831i	$0.360003\pi$
$758$	−39188.3	−1.87781
$759$	0	0
$760$	27532.4	1.31409
$761$	4821.03	0.229648	0.114824	−	0.993386i	$-0.463370\pi$
$761$	4821.03	0.229648	0.114824	+	0.993386i	$0.463370\pi$
$762$	0	0
$763$	12413.9	0.589007
$764$	38691.5	1.83221
$765$	0	0
$766$	−31755.4	−1.49787
$767$	−60165.2	−2.83238
$768$	0	0
$769$	14356.6	0.673226	0.336613	−	0.941643i	$-0.390719\pi$
$769$	14356.6	0.673226	0.336613	+	0.941643i	$0.390719\pi$
$770$	−21828.3	−1.02161
$771$	0	0
$772$	−24441.2	−1.13945
$773$	12595.3	0.586056	0.293028	−	0.956104i	$-0.405337\pi$
$773$	12595.3	0.586056	0.293028	+	0.956104i	$0.405337\pi$
$774$	0	0
$775$	−12755.4	−0.591211
$776$	17433.1	0.806461
$777$	0	0
$778$	4588.35	0.211440
$779$	−6025.76	−0.277144
$780$	0	0
$781$	−45311.0	−2.07600
$782$	53723.2	2.45670
$783$	0	0
$784$	−22490.9	−1.02455
$785$	−14111.5	−0.641608
$786$	0	0
$787$	−13898.2	−0.629500	−0.314750	−	0.949175i	$-0.601921\pi$
$787$	−13898.2	−0.629500	−0.314750	+	0.949175i	$0.601921\pi$
$788$	−57004.6	−2.57703
$789$	0	0
$790$	−24340.4	−1.09620
$791$	4852.15	0.218107
$792$	0	0
$793$	−54919.0	−2.45931
$794$	34234.9	1.53017
$795$	0	0
$796$	−10062.4	−0.448056
$797$	−12039.3	−0.535073	−0.267537	−	0.963548i	$-0.586210\pi$
$797$	−12039.3	−0.535073	−0.267537	+	0.963548i	$0.586210\pi$
$798$	0	0
$799$	−10265.1	−0.454509
$800$	5433.12	0.240112
$801$	0	0
$802$	36657.6	1.61400
$803$	44407.3	1.95156
$804$	0	0
$805$	−10319.5	−0.451820
$806$	−68270.2	−2.98352
$807$	0	0
$808$	−48197.1	−2.09847
$809$	−18226.1	−0.792084	−0.396042	−	0.918232i	$-0.629617\pi$
$809$	−18226.1	−0.792084	−0.396042	+	0.918232i	$0.629617\pi$
$810$	0	0
$811$	27538.2	1.19235	0.596176	−	0.802854i	$-0.296686\pi$
$811$	27538.2	1.19235	0.596176	+	0.802854i	$0.296686\pi$
$812$	−94.5827	−0.00408769
$813$	0	0
$814$	108845.	4.68674
$815$	−25734.6	−1.10607
$816$	0	0
$817$	−39383.6	−1.68648
$818$	−25912.3	−1.10758
$819$	0	0
$820$	9154.46	0.389863
$821$	9185.36	0.390465	0.195232	−	0.980757i	$-0.437454\pi$
$821$	9185.36	0.390465	0.195232	+	0.980757i	$0.437454\pi$
$822$	0	0
$823$	15830.5	0.670494	0.335247	−	0.942130i	$-0.391180\pi$
$823$	15830.5	0.670494	0.335247	+	0.942130i	$0.391180\pi$
$824$	11546.7	0.488165
$825$	0	0
$826$	−37068.7	−1.56148
$827$	−37452.5	−1.57479	−0.787396	−	0.616448i	$-0.788571\pi$
$827$	−37452.5	−1.57479	−0.787396	+	0.616448i	$0.788571\pi$
$828$	0	0
$829$	8708.50	0.364848	0.182424	−	0.983220i	$-0.441606\pi$
$829$	8708.50	0.364848	0.182424	+	0.983220i	$0.441606\pi$
$830$	−24860.9	−1.03968
$831$	0	0
$832$	−22832.5	−0.951410
$833$	18459.0	0.767786
$834$	0	0
$835$	−1243.75	−0.0515469
$836$	89612.2	3.70730
$837$	0	0
$838$	65855.3	2.71472
$839$	−243.466	−0.0100183	−0.00500917	−	0.999987i	$-0.501594\pi$
$839$	−243.466	−0.0100183	−0.00500917	+	0.999987i	$0.501594\pi$
$840$	0	0
$841$	−24388.6	−0.999985
$842$	63022.2	2.57944
$843$	0	0
$844$	29928.5	1.22059
$845$	25022.6	1.01870
$846$	0	0
$847$	−25219.7	−1.02309
$848$	29244.9	1.18429
$849$	0	0
$850$	−24801.2	−1.00079
$851$	51457.3	2.07278
$852$	0	0
$853$	−6652.76	−0.267041	−0.133521	−	0.991046i	$-0.542628\pi$
$853$	−6652.76	−0.267041	−0.133521	+	0.991046i	$0.542628\pi$
$854$	−33836.5	−1.35581
$855$	0	0
$856$	−45483.7	−1.81612
$857$	41713.7	1.66268	0.831339	−	0.555766i	$-0.187575\pi$
$857$	41713.7	1.66268	0.831339	+	0.555766i	$0.187575\pi$
$858$	0	0
$859$	7884.86	0.313187	0.156594	−	0.987663i	$-0.449949\pi$
$859$	7884.86	0.313187	0.156594	+	0.987663i	$0.449949\pi$
$860$	59832.3	2.37240
$861$	0	0
$862$	−19495.2	−0.770313
$863$	−38269.7	−1.50952	−0.754760	−	0.656001i	$-0.772247\pi$
$863$	−38269.7	−1.50952	−0.754760	+	0.656001i	$0.772247\pi$
$864$	0	0
$865$	−20302.9	−0.798059
$866$	−164.815	−0.00646725
$867$	0	0
$868$	−28561.7	−1.11688
$869$	−41775.8	−1.63078
$870$	0	0
$871$	−67697.6	−2.63358
$872$	60125.4	2.33498
$873$	0	0
$874$	62390.1	2.41462
$875$	13328.2	0.514942
$876$	0	0
$877$	17712.0	0.681974	0.340987	−	0.940068i	$-0.389239\pi$
$877$	17712.0	0.681974	0.340987	+	0.940068i	$0.389239\pi$
$878$	26824.2	1.03106
$879$	0	0
$880$	−41372.1	−1.58483
$881$	−17285.6	−0.661028	−0.330514	−	0.943801i	$-0.607222\pi$
$881$	−17285.6	−0.661028	−0.330514	+	0.943801i	$0.607222\pi$
$882$	0	0
$883$	−42908.1	−1.63530	−0.817651	−	0.575715i	$-0.804724\pi$
$883$	−42908.1	−1.63530	−0.817651	+	0.575715i	$0.804724\pi$
$884$	−90136.2	−3.42942
$885$	0	0
$886$	−27744.6	−1.05203
$887$	−25656.3	−0.971200	−0.485600	−	0.874181i	$-0.661399\pi$
$887$	−25656.3	−0.971200	−0.485600	+	0.874181i	$0.661399\pi$
$888$	0	0
$889$	−14833.2	−0.559607
$890$	21120.6	0.795465
$891$	0	0
$892$	44734.8	1.67919
$893$	−11921.1	−0.446724
$894$	0	0
$895$	−7871.23	−0.293973
$896$	−19817.9	−0.738918
$897$	0	0
$898$	39928.0	1.48376
$899$	111.438	0.00413421
$900$	0	0
$901$	−24002.2	−0.887491
$902$	23138.6	0.854138
$903$	0	0
$904$	23500.9	0.864634
$905$	−7135.37	−0.262086
$906$	0	0
$907$	4491.13	0.164416	0.0822081	−	0.996615i	$-0.473803\pi$
$907$	4491.13	0.164416	0.0822081	+	0.996615i	$0.473803\pi$
$908$	−70925.5	−2.59223
$909$	0	0
$910$	25497.9	0.928843
$911$	−51196.0	−1.86191	−0.930955	−	0.365134i	$-0.881023\pi$
$911$	−51196.0	−1.86191	−0.930955	+	0.365134i	$0.881023\pi$
$912$	0	0
$913$	−42669.0	−1.54670
$914$	77799.9	2.81553
$915$	0	0
$916$	110694.	3.99284
$917$	−15752.3	−0.567271
$918$	0	0
$919$	8166.93	0.293147	0.146574	−	0.989200i	$-0.453175\pi$
$919$	8166.93	0.293147	0.146574	+	0.989200i	$0.453175\pi$
$920$	−49981.6	−1.79113
$921$	0	0
$922$	−9815.97	−0.350620
$923$	52928.4	1.88750
$924$	0	0
$925$	−23755.1	−0.844393
$926$	−16984.4	−0.602744
$927$	0	0
$928$	−47.4664	−0.00167905
$929$	17007.5	0.600642	0.300321	−	0.953838i	$-0.402906\pi$
$929$	17007.5	0.600642	0.300321	+	0.953838i	$0.402906\pi$
$930$	0	0
$931$	21436.9	0.754634
$932$	−34389.2	−1.20864
$933$	0	0
$934$	47107.1	1.65031
$935$	33955.3	1.18766
$936$	0	0
$937$	18463.6	0.643734	0.321867	−	0.946785i	$-0.395690\pi$
$937$	18463.6	0.643734	0.321867	+	0.946785i	$0.395690\pi$
$938$	−41709.5	−1.45188
$939$	0	0
$940$	18110.8	0.628413
$941$	29962.6	1.03800	0.518998	−	0.854776i	$-0.326305\pi$
$941$	29962.6	1.03800	0.518998	+	0.854776i	$0.326305\pi$
$942$	0	0
$943$	10939.0	0.377755
$944$	−70257.9	−2.42235
$945$	0	0
$946$	151231.	5.19762
$947$	−18376.7	−0.630583	−0.315292	−	0.948995i	$-0.602102\pi$
$947$	−18376.7	−0.630583	−0.315292	+	0.948995i	$0.602102\pi$
$948$	0	0
$949$	−51872.8	−1.77435
$950$	−28802.2	−0.983649
$951$	0	0
$952$	−29284.3	−0.996963
$953$	15184.4	0.516130	0.258065	−	0.966128i	$-0.416915\pi$
$953$	15184.4	0.516130	0.258065	+	0.966128i	$0.416915\pi$
$954$	0	0
$955$	17025.9	0.576906
$956$	−18547.4	−0.627474
$957$	0	0
$958$	50623.6	1.70728
$959$	18475.7	0.622118
$960$	0	0
$961$	3860.55	0.129588
$962$	−127143.	−4.26118
$963$	0	0
$964$	47922.4	1.60112
$965$	−10755.1	−0.358777
$966$	0	0
$967$	40202.6	1.33695	0.668473	−	0.743736i	$-0.266948\pi$
$967$	40202.6	1.33695	0.668473	+	0.743736i	$0.266948\pi$
$968$	−122149.	−4.05582
$969$	0	0
$970$	14547.8	0.481547
$971$	−18805.9	−0.621535	−0.310768	−	0.950486i	$-0.600586\pi$
$971$	−18805.9	−0.621535	−0.310768	+	0.950486i	$0.600586\pi$
$972$	0	0
$973$	−11866.9	−0.390992
$974$	88411.3	2.90850
$975$	0	0
$976$	−64131.7	−2.10329
$977$	43.0888	0.00141098	0.000705492	−	1.00000i	$-0.499775\pi$
$977$	43.0888	0.00141098	0.000705492		1.00000i	$0.499775\pi$
$978$	0	0
$979$	36249.6	1.18339
$980$	−32567.3	−1.06156
$981$	0	0
$982$	33840.3	1.09968
$983$	−21591.5	−0.700572	−0.350286	−	0.936643i	$-0.613916\pi$
$983$	−21591.5	−0.700572	−0.350286	+	0.936643i	$0.613916\pi$
$984$	0	0
$985$	−25084.4	−0.811427
$986$	216.675	0.00699832
$987$	0	0
$988$	−104677.	−3.37068
$989$	71495.8	2.29872
$990$	0	0
$991$	6694.67	0.214595	0.107297	−	0.994227i	$-0.465780\pi$
$991$	6694.67	0.214595	0.107297	+	0.994227i	$0.465780\pi$
$992$	−14333.7	−0.458766
$993$	0	0
$994$	32610.0	1.04057
$995$	−4427.88	−0.141079
$996$	0	0
$997$	17958.2	0.570454	0.285227	−	0.958460i	$-0.407931\pi$
$997$	17958.2	0.570454	0.285227	+	0.958460i	$0.407931\pi$
$998$	33155.8	1.05163
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1503.4.a.d.1.3		23
3.2	odd	2		501.4.a.d.1.21	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.4.a.d.1.21	✓	23	3.2	odd	2
1503.4.a.d.1.3		23	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1503 = 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1503.a (trivial)

\(f(q)\)	\(=\)	\(q-4.99247 q^{2} +16.9247 q^{4} +7.44760 q^{5} -9.19941 q^{7} -44.5565 q^{8} +O(q^{10})\)	\(q-4.99247 q^{2} +16.9247 q^{4} +7.44760 q^{5} -9.19941 q^{7} -44.5565 q^{8} -37.1819 q^{10} -63.8158 q^{11} +74.5441 q^{13} +45.9277 q^{14} +87.0489 q^{16} -71.4437 q^{17} -82.9693 q^{19} +126.049 q^{20} +318.598 q^{22} +150.620 q^{23} -69.5333 q^{25} -372.159 q^{26} -155.698 q^{28} +0.607477 q^{29} +183.444 q^{31} -78.1370 q^{32} +356.680 q^{34} -68.5135 q^{35} +341.637 q^{37} +414.222 q^{38} -331.839 q^{40} +72.6264 q^{41} +474.676 q^{43} -1080.07 q^{44} -751.966 q^{46} +143.681 q^{47} -258.371 q^{49} +347.143 q^{50} +1261.64 q^{52} +335.960 q^{53} -475.274 q^{55} +409.893 q^{56} -3.03281 q^{58} -807.109 q^{59} -736.732 q^{61} -915.836 q^{62} -306.295 q^{64} +555.174 q^{65} -908.155 q^{67} -1209.17 q^{68} +342.051 q^{70} +710.028 q^{71} -695.867 q^{73} -1705.61 q^{74} -1404.23 q^{76} +587.067 q^{77} +654.632 q^{79} +648.305 q^{80} -362.585 q^{82} +668.629 q^{83} -532.084 q^{85} -2369.81 q^{86} +2843.41 q^{88} -568.035 q^{89} -685.761 q^{91} +2549.20 q^{92} -717.322 q^{94} -617.922 q^{95} -391.260 q^{97} +1289.91 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 11 q^{2} + 103 q^{4} - 48 q^{5} + 28 q^{7} - 147 q^{8}+O(q^{10}) \) 23 * q - 11 * q^2 + 103 * q^4 - 48 * q^5 + 28 * q^7 - 147 * q^8	\( 23 q - 11 q^{2} + 103 q^{4} - 48 q^{5} + 28 q^{7} - 147 q^{8} + 43 q^{10} - 204 q^{11} + 34 q^{13} - 101 q^{14} + 587 q^{16} - 368 q^{17} + 448 q^{19} - 543 q^{20} + 307 q^{22} - 406 q^{23} + 863 q^{25} - 335 q^{26} + 115 q^{28} - 556 q^{29} + 776 q^{31} - 1402 q^{32} + 672 q^{34} - 730 q^{35} + 270 q^{37} - 238 q^{38} + 694 q^{40} - 816 q^{41} - 84 q^{43} - 2035 q^{44} - 327 q^{46} - 1420 q^{47} + 1131 q^{49} - 430 q^{50} - 828 q^{52} - 1430 q^{53} - 758 q^{55} + 433 q^{56} - 2233 q^{58} - 3110 q^{59} + 278 q^{61} - 2044 q^{62} + 859 q^{64} - 2332 q^{65} + 802 q^{67} - 1384 q^{68} - 4411 q^{70} - 1696 q^{71} - 1048 q^{73} - 225 q^{74} - 2804 q^{76} - 986 q^{77} + 460 q^{79} - 4026 q^{80} - 3401 q^{82} - 1886 q^{83} + 360 q^{85} - 633 q^{86} - 116 q^{88} - 6404 q^{89} + 3304 q^{91} - 5135 q^{92} - 1860 q^{94} - 796 q^{95} + 1040 q^{97} - 5905 q^{98}+O(q^{100}) \) 23 * q - 11 * q^2 + 103 * q^4 - 48 * q^5 + 28 * q^7 - 147 * q^8 + 43 * q^10 - 204 * q^11 + 34 * q^13 - 101 * q^14 + 587 * q^16 - 368 * q^17 + 448 * q^19 - 543 * q^20 + 307 * q^22 - 406 * q^23 + 863 * q^25 - 335 * q^26 + 115 * q^28 - 556 * q^29 + 776 * q^31 - 1402 * q^32 + 672 * q^34 - 730 * q^35 + 270 * q^37 - 238 * q^38 + 694 * q^40 - 816 * q^41 - 84 * q^43 - 2035 * q^44 - 327 * q^46 - 1420 * q^47 + 1131 * q^49 - 430 * q^50 - 828 * q^52 - 1430 * q^53 - 758 * q^55 + 433 * q^56 - 2233 * q^58 - 3110 * q^59 + 278 * q^61 - 2044 * q^62 + 859 * q^64 - 2332 * q^65 + 802 * q^67 - 1384 * q^68 - 4411 * q^70 - 1696 * q^71 - 1048 * q^73 - 225 * q^74 - 2804 * q^76 - 986 * q^77 + 460 * q^79 - 4026 * q^80 - 3401 * q^82 - 1886 * q^83 + 360 * q^85 - 633 * q^86 - 116 * q^88 - 6404 * q^89 + 3304 * q^91 - 5135 * q^92 - 1860 * q^94 - 796 * q^95 + 1040 * q^97 - 5905 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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