LMFDB - Embedded newform 920.4.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [920,4,Mod(1,920)] 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("920.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 920 = 2^{3} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	920.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$54.2817572053$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 81x^{4} + 161x^{3} + 1520x^{2} - 3915x + 588 $ x^6 - 2x^5 - 81x^4 + 161x^3 + 1520x^2 - 3915*x + 588
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.38774$ of defining polynomial
Character	$\chi$	$=$	920.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-6.38774 q^{3} -5.00000 q^{5} -5.78952 q^{7} +13.8032 q^{9} -3.04007 q^{11} -41.8142 q^{13} +31.9387 q^{15} +89.8459 q^{17} +102.477 q^{19} +36.9819 q^{21} -23.0000 q^{23} +25.0000 q^{25} +84.2976 q^{27} -77.9931 q^{29} -17.8575 q^{31} +19.4192 q^{33} +28.9476 q^{35} +236.953 q^{37} +267.098 q^{39} -318.152 q^{41} -31.9573 q^{43} -69.0161 q^{45} +239.636 q^{47} -309.482 q^{49} -573.913 q^{51} +498.695 q^{53} +15.2004 q^{55} -654.596 q^{57} +276.837 q^{59} -86.9389 q^{61} -79.9140 q^{63} +209.071 q^{65} +1019.33 q^{67} +146.918 q^{69} +1020.60 q^{71} -104.709 q^{73} -159.694 q^{75} +17.6006 q^{77} -1052.52 q^{79} -911.158 q^{81} -1006.59 q^{83} -449.230 q^{85} +498.200 q^{87} -1098.26 q^{89} +242.084 q^{91} +114.069 q^{93} -512.384 q^{95} -489.424 q^{97} -41.9628 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{3} - 30 q^{5} + 28 q^{7} + 4 q^{9} - 3 q^{11} - 28 q^{13} - 10 q^{15} + 24 q^{17} - 3 q^{19} + 60 q^{21} - 138 q^{23} + 150 q^{25} - 97 q^{27} + 76 q^{29} - 381 q^{31} - 3 q^{33} - 140 q^{35}+ \cdots - 525 q^{99}+O(q^{100}) $ 6 * q + 2 * q^3 - 30 * q^5 + 28 * q^7 + 4 * q^9 - 3 * q^11 - 28 * q^13 - 10 * q^15 + 24 * q^17 - 3 * q^19 + 60 * q^21 - 138 * q^23 + 150 * q^25 - 97 * q^27 + 76 * q^29 - 381 * q^31 - 3 * q^33 - 140 * q^35 + 131 * q^37 + 41 * q^39 - 95 * q^41 - 202 * q^43 - 20 * q^45 - 119 * q^47 - 578 * q^49 - 997 * q^51 + 137 * q^53 + 15 * q^55 - 894 * q^57 - 39 * q^59 - 573 * q^61 - 355 * q^63 + 140 * q^65 - 563 * q^67 - 46 * q^69 - 83 * q^71 - 1799 * q^73 + 50 * q^75 - 1410 * q^77 - 1636 * q^79 - 2006 * q^81 - 1191 * q^83 - 120 * q^85 - 1821 * q^87 - 1370 * q^89 - 1251 * q^91 - 2215 * q^93 + 15 * q^95 - 3021 * q^97 - 525 * 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	0
$2$	0	0
$3$	−6.38774	−1.22932	−0.614661	−	0.788792i	$-0.710707\pi$
$3$	−6.38774	−1.22932	−0.614661	+	0.788792i	$0.710707\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−5.78952	−0.312604	−0.156302	−	0.987709i	$-0.549957\pi$
$7$	−5.78952	−0.312604	−0.156302	+	0.987709i	$0.549957\pi$
$8$	0	0
$9$	13.8032	0.511231
$10$	0	0
$11$	−3.04007	−0.0833288	−0.0416644	−	0.999132i	$-0.513266\pi$
$11$	−3.04007	−0.0833288	−0.0416644	+	0.999132i	$0.513266\pi$
$12$	0	0
$13$	−41.8142	−0.892090	−0.446045	−	0.895011i	$-0.647168\pi$
$13$	−41.8142	−0.892090	−0.446045	+	0.895011i	$0.647168\pi$
$14$	0	0
$15$	31.9387	0.549769
$16$	0	0
$17$	89.8459	1.28181	0.640907	−	0.767618i	$-0.278558\pi$
$17$	89.8459	1.28181	0.640907	+	0.767618i	$0.278558\pi$
$18$	0	0
$19$	102.477	1.23736	0.618679	−	0.785644i	$-0.287668\pi$
$19$	102.477	1.23736	0.618679	+	0.785644i	$0.287668\pi$
$20$	0	0
$21$	36.9819	0.384291
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	84.2976	0.600855
$28$	0	0
$29$	−77.9931	−0.499412	−0.249706	−	0.968322i	$-0.580334\pi$
$29$	−77.9931	−0.499412	−0.249706	+	0.968322i	$0.580334\pi$
$30$	0	0
$31$	−17.8575	−0.103461	−0.0517307	−	0.998661i	$-0.516474\pi$
$31$	−17.8575	−0.103461	−0.0517307	+	0.998661i	$0.516474\pi$
$32$	0	0
$33$	19.4192	0.102438
$34$	0	0
$35$	28.9476	0.139801
$36$	0	0
$37$	236.953	1.05283	0.526417	−	0.850226i	$-0.323535\pi$
$37$	236.953	1.05283	0.526417	+	0.850226i	$0.323535\pi$
$38$	0	0
$39$	267.098	1.09667
$40$	0	0
$41$	−318.152	−1.21188	−0.605939	−	0.795511i	$-0.707202\pi$
$41$	−318.152	−1.21188	−0.605939	+	0.795511i	$0.707202\pi$
$42$	0	0
$43$	−31.9573	−0.113336	−0.0566679	−	0.998393i	$-0.518048\pi$
$43$	−31.9573	−0.113336	−0.0566679	+	0.998393i	$0.518048\pi$
$44$	0	0
$45$	−69.0161	−0.228629
$46$	0	0
$47$	239.636	0.743713	0.371856	−	0.928290i	$-0.378721\pi$
$47$	239.636	0.743713	0.371856	+	0.928290i	$0.378721\pi$
$48$	0	0
$49$	−309.482	−0.902278
$50$	0	0
$51$	−573.913	−1.57576
$52$	0	0
$53$	498.695	1.29247	0.646236	−	0.763137i	$-0.276342\pi$
$53$	498.695	1.29247	0.646236	+	0.763137i	$0.276342\pi$
$54$	0	0
$55$	15.2004	0.0372658
$56$	0	0
$57$	−654.596	−1.52111
$58$	0	0
$59$	276.837	0.610867	0.305434	−	0.952213i	$-0.401199\pi$
$59$	276.837	0.610867	0.305434	+	0.952213i	$0.401199\pi$
$60$	0	0
$61$	−86.9389	−0.182482	−0.0912409	−	0.995829i	$-0.529083\pi$
$61$	−86.9389	−0.182482	−0.0912409	+	0.995829i	$0.529083\pi$
$62$	0	0
$63$	−79.9140	−0.159813
$64$	0	0
$65$	209.071	0.398955
$66$	0	0
$67$	1019.33	1.85866	0.929332	−	0.369246i	$-0.120384\pi$
$67$	1019.33	1.85866	0.929332	+	0.369246i	$0.120384\pi$
$68$	0	0
$69$	146.918	0.256331
$70$	0	0
$71$	1020.60	1.70596	0.852978	−	0.521947i	$-0.174794\pi$
$71$	1020.60	1.70596	0.852978	+	0.521947i	$0.174794\pi$
$72$	0	0
$73$	−104.709	−0.167880	−0.0839398	−	0.996471i	$-0.526750\pi$
$73$	−104.709	−0.167880	−0.0839398	+	0.996471i	$0.526750\pi$
$74$	0	0
$75$	−159.694	−0.245864
$76$	0	0
$77$	17.6006	0.0260490
$78$	0	0
$79$	−1052.52	−1.49896	−0.749482	−	0.662025i	$-0.769697\pi$
$79$	−1052.52	−1.49896	−0.749482	+	0.662025i	$0.769697\pi$
$80$	0	0
$81$	−911.158	−1.24987
$82$	0	0
$83$	−1006.59	−1.33118	−0.665589	−	0.746318i	$-0.731820\pi$
$83$	−1006.59	−1.33118	−0.665589	+	0.746318i	$0.731820\pi$
$84$	0	0
$85$	−449.230	−0.573245
$86$	0	0
$87$	498.200	0.613938
$88$	0	0
$89$	−1098.26	−1.30804	−0.654022	−	0.756476i	$-0.726919\pi$
$89$	−1098.26	−1.30804	−0.654022	+	0.756476i	$0.726919\pi$
$90$	0	0
$91$	242.084	0.278871
$92$	0	0
$93$	114.069	0.127187
$94$	0	0
$95$	−512.384	−0.553364
$96$	0	0
$97$	−489.424	−0.512304	−0.256152	−	0.966637i	$-0.582455\pi$
$97$	−489.424	−0.512304	−0.256152	+	0.966637i	$0.582455\pi$
$98$	0	0
$99$	−41.9628	−0.0426002
$100$	0	0
$101$	−434.531	−0.428094	−0.214047	−	0.976823i	$-0.568665\pi$
$101$	−434.531	−0.428094	−0.214047	+	0.976823i	$0.568665\pi$
$102$	0	0
$103$	409.330	0.391577	0.195789	−	0.980646i	$-0.437273\pi$
$103$	409.330	0.391577	0.195789	+	0.980646i	$0.437273\pi$
$104$	0	0
$105$	−184.910	−0.171860
$106$	0	0
$107$	−1944.48	−1.75683	−0.878413	−	0.477902i	$-0.841397\pi$
$107$	−1944.48	−1.75683	−0.878413	+	0.477902i	$0.841397\pi$
$108$	0	0
$109$	322.594	0.283476	0.141738	−	0.989904i	$-0.454731\pi$
$109$	322.594	0.283476	0.141738	+	0.989904i	$0.454731\pi$
$110$	0	0
$111$	−1513.60	−1.29427
$112$	0	0
$113$	−2353.00	−1.95886	−0.979432	−	0.201777i	$-0.935328\pi$
$113$	−2353.00	−1.95886	−0.979432	+	0.201777i	$0.935328\pi$
$114$	0	0
$115$	115.000	0.0932505
$116$	0	0
$117$	−577.171	−0.456064
$118$	0	0
$119$	−520.164	−0.400701
$120$	0	0
$121$	−1321.76	−0.993056
$122$	0	0
$123$	2032.27	1.48979
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−133.612	−0.0933557	−0.0466779	−	0.998910i	$-0.514863\pi$
$127$	−133.612	−0.0933557	−0.0466779	+	0.998910i	$0.514863\pi$
$128$	0	0
$129$	204.135	0.139326
$130$	0	0
$131$	−775.474	−0.517202	−0.258601	−	0.965984i	$-0.583261\pi$
$131$	−775.474	−0.517202	−0.258601	+	0.965984i	$0.583261\pi$
$132$	0	0
$133$	−593.292	−0.386804
$134$	0	0
$135$	−421.488	−0.268710
$136$	0	0
$137$	987.824	0.616026	0.308013	−	0.951382i	$-0.400336\pi$
$137$	987.824	0.616026	0.308013	+	0.951382i	$0.400336\pi$
$138$	0	0
$139$	2051.26	1.25170	0.625848	−	0.779945i	$-0.284753\pi$
$139$	2051.26	1.25170	0.625848	+	0.779945i	$0.284753\pi$
$140$	0	0
$141$	−1530.73	−0.914262
$142$	0	0
$143$	127.118	0.0743368
$144$	0	0
$145$	389.966	0.223344
$146$	0	0
$147$	1976.89	1.10919
$148$	0	0
$149$	−2554.07	−1.40428	−0.702141	−	0.712038i	$-0.747772\pi$
$149$	−2554.07	−1.40428	−0.702141	+	0.712038i	$0.747772\pi$
$150$	0	0
$151$	−1312.77	−0.707495	−0.353748	−	0.935341i	$-0.615093\pi$
$151$	−1312.77	−0.707495	−0.353748	+	0.935341i	$0.615093\pi$
$152$	0	0
$153$	1240.16	0.655303
$154$	0	0
$155$	89.2875	0.0462693
$156$	0	0
$157$	−3430.46	−1.74382	−0.871912	−	0.489662i	$-0.837120\pi$
$157$	−3430.46	−1.74382	−0.871912	+	0.489662i	$0.837120\pi$
$158$	0	0
$159$	−3185.54	−1.58886
$160$	0	0
$161$	133.159	0.0651825
$162$	0	0
$163$	190.995	0.0917785	0.0458892	−	0.998947i	$-0.485388\pi$
$163$	190.995	0.0917785	0.0458892	+	0.998947i	$0.485388\pi$
$164$	0	0
$165$	−97.0960	−0.0458116
$166$	0	0
$167$	621.559	0.288010	0.144005	−	0.989577i	$-0.454002\pi$
$167$	621.559	0.288010	0.144005	+	0.989577i	$0.454002\pi$
$168$	0	0
$169$	−448.574	−0.204176
$170$	0	0
$171$	1414.51	0.632576
$172$	0	0
$173$	3722.24	1.63582	0.817910	−	0.575346i	$-0.195133\pi$
$173$	3722.24	1.63582	0.817910	+	0.575346i	$0.195133\pi$
$174$	0	0
$175$	−144.738	−0.0625209
$176$	0	0
$177$	−1768.37	−0.750952
$178$	0	0
$179$	170.146	0.0710464	0.0355232	−	0.999369i	$-0.488690\pi$
$179$	170.146	0.0710464	0.0355232	+	0.999369i	$0.488690\pi$
$180$	0	0
$181$	−80.7800	−0.0331731	−0.0165865	−	0.999862i	$-0.505280\pi$
$181$	−80.7800	−0.0331731	−0.0165865	+	0.999862i	$0.505280\pi$
$182$	0	0
$183$	555.343	0.224329
$184$	0	0
$185$	−1184.77	−0.470842
$186$	0	0
$187$	−273.138	−0.106812
$188$	0	0
$189$	−488.042	−0.187830
$190$	0	0
$191$	−1105.26	−0.418712	−0.209356	−	0.977839i	$-0.567137\pi$
$191$	−1105.26	−0.418712	−0.209356	+	0.977839i	$0.567137\pi$
$192$	0	0
$193$	−5280.24	−1.96933	−0.984663	−	0.174464i	$-0.944181\pi$
$193$	−5280.24	−1.96933	−0.984663	+	0.174464i	$0.944181\pi$
$194$	0	0
$195$	−1335.49	−0.490444
$196$	0	0
$197$	1592.44	0.575922	0.287961	−	0.957642i	$-0.407023\pi$
$197$	1592.44	0.575922	0.287961	+	0.957642i	$0.407023\pi$
$198$	0	0
$199$	3585.69	1.27730	0.638651	−	0.769497i	$-0.279493\pi$
$199$	3585.69	1.27730	0.638651	+	0.769497i	$0.279493\pi$
$200$	0	0
$201$	−6511.19	−2.28489
$202$	0	0
$203$	451.542	0.156119
$204$	0	0
$205$	1590.76	0.541968
$206$	0	0
$207$	−317.474	−0.106599
$208$	0	0
$209$	−311.537	−0.103108
$210$	0	0
$211$	140.259	0.0457623	0.0228812	−	0.999738i	$-0.492716\pi$
$211$	140.259	0.0457623	0.0228812	+	0.999738i	$0.492716\pi$
$212$	0	0
$213$	−6519.33	−2.09717
$214$	0	0
$215$	159.786	0.0506853
$216$	0	0
$217$	103.386	0.0323425
$218$	0	0
$219$	668.851	0.206378
$220$	0	0
$221$	−3756.83	−1.14349
$222$	0	0
$223$	−457.275	−0.137316	−0.0686579	−	0.997640i	$-0.521872\pi$
$223$	−457.275	−0.137316	−0.0686579	+	0.997640i	$0.521872\pi$
$224$	0	0
$225$	345.081	0.102246
$226$	0	0
$227$	−207.632	−0.0607093	−0.0303546	−	0.999539i	$-0.509664\pi$
$227$	−207.632	−0.0607093	−0.0303546	+	0.999539i	$0.509664\pi$
$228$	0	0
$229$	−2372.26	−0.684555	−0.342278	−	0.939599i	$-0.611198\pi$
$229$	−2372.26	−0.684555	−0.342278	+	0.939599i	$0.611198\pi$
$230$	0	0
$231$	−112.428	−0.0320225
$232$	0	0
$233$	6027.61	1.69477	0.847385	−	0.530978i	$-0.178175\pi$
$233$	6027.61	1.69477	0.847385	+	0.530978i	$0.178175\pi$
$234$	0	0
$235$	−1198.18	−0.332599
$236$	0	0
$237$	6723.25	1.84271
$238$	0	0
$239$	4764.03	1.28937	0.644686	−	0.764448i	$-0.276988\pi$
$239$	4764.03	1.28937	0.644686	+	0.764448i	$0.276988\pi$
$240$	0	0
$241$	−195.211	−0.0521771	−0.0260885	−	0.999660i	$-0.508305\pi$
$241$	−195.211	−0.0521771	−0.0260885	+	0.999660i	$0.508305\pi$
$242$	0	0
$243$	3544.21	0.935642
$244$	0	0
$245$	1547.41	0.403511
$246$	0	0
$247$	−4284.99	−1.10384
$248$	0	0
$249$	6429.85	1.63645
$250$	0	0
$251$	−5890.81	−1.48137	−0.740686	−	0.671851i	$-0.765499\pi$
$251$	−5890.81	−1.48137	−0.740686	+	0.671851i	$0.765499\pi$
$252$	0	0
$253$	69.9217	0.0173753
$254$	0	0
$255$	2869.56	0.704702
$256$	0	0
$257$	4390.17	1.06557	0.532784	−	0.846251i	$-0.321146\pi$
$257$	4390.17	1.06557	0.532784	+	0.846251i	$0.321146\pi$
$258$	0	0
$259$	−1371.84	−0.329121
$260$	0	0
$261$	−1076.56	−0.255315
$262$	0	0
$263$	1403.84	0.329142	0.164571	−	0.986365i	$-0.447376\pi$
$263$	1403.84	0.329142	0.164571	+	0.986365i	$0.447376\pi$
$264$	0	0
$265$	−2493.48	−0.578011
$266$	0	0
$267$	7015.43	1.60801
$268$	0	0
$269$	2725.26	0.617704	0.308852	−	0.951110i	$-0.400055\pi$
$269$	2725.26	0.617704	0.308852	+	0.951110i	$0.400055\pi$
$270$	0	0
$271$	230.366	0.0516375	0.0258188	−	0.999667i	$-0.491781\pi$
$271$	230.366	0.0516375	0.0258188	+	0.999667i	$0.491781\pi$
$272$	0	0
$273$	−1546.37	−0.342822
$274$	0	0
$275$	−76.0019	−0.0166658
$276$	0	0
$277$	5516.86	1.19666	0.598332	−	0.801248i	$-0.295830\pi$
$277$	5516.86	1.19666	0.598332	+	0.801248i	$0.295830\pi$
$278$	0	0
$279$	−246.491	−0.0528926
$280$	0	0
$281$	−5384.04	−1.14301	−0.571504	−	0.820600i	$-0.693640\pi$
$281$	−5384.04	−1.14301	−0.571504	+	0.820600i	$0.693640\pi$
$282$	0	0
$283$	2449.05	0.514420	0.257210	−	0.966355i	$-0.417197\pi$
$283$	2449.05	0.514420	0.257210	+	0.966355i	$0.417197\pi$
$284$	0	0
$285$	3272.98	0.680262
$286$	0	0
$287$	1841.94	0.378838
$288$	0	0
$289$	3159.29	0.643047
$290$	0	0
$291$	3126.31	0.629786
$292$	0	0
$293$	5257.98	1.04838	0.524189	−	0.851602i	$-0.324369\pi$
$293$	5257.98	1.04838	0.524189	+	0.851602i	$0.324369\pi$
$294$	0	0
$295$	−1384.19	−0.273188
$296$	0	0
$297$	−256.271	−0.0500685
$298$	0	0
$299$	961.726	0.186014
$300$	0	0
$301$	185.017	0.0354293
$302$	0	0
$303$	2775.67	0.526265
$304$	0	0
$305$	434.695	0.0816083
$306$	0	0
$307$	−5730.30	−1.06530	−0.532648	−	0.846337i	$-0.678803\pi$
$307$	−5730.30	−1.06530	−0.532648	+	0.846337i	$0.678803\pi$
$308$	0	0
$309$	−2614.69	−0.481374
$310$	0	0
$311$	−6887.88	−1.25587	−0.627936	−	0.778265i	$-0.716100\pi$
$311$	−6887.88	−1.25587	−0.627936	+	0.778265i	$0.716100\pi$
$312$	0	0
$313$	−4574.19	−0.826033	−0.413017	−	0.910723i	$-0.635525\pi$
$313$	−4574.19	−0.826033	−0.413017	+	0.910723i	$0.635525\pi$
$314$	0	0
$315$	399.570	0.0714705
$316$	0	0
$317$	−6481.61	−1.14840	−0.574201	−	0.818714i	$-0.694687\pi$
$317$	−6481.61	−1.14840	−0.574201	+	0.818714i	$0.694687\pi$
$318$	0	0
$319$	237.105	0.0416154
$320$	0	0
$321$	12420.9	2.15970
$322$	0	0
$323$	9207.13	1.58606
$324$	0	0
$325$	−1045.35	−0.178418
$326$	0	0
$327$	−2060.65	−0.348483
$328$	0	0
$329$	−1387.38	−0.232488
$330$	0	0
$331$	−5626.40	−0.934305	−0.467153	−	0.884177i	$-0.654720\pi$
$331$	−5626.40	−0.934305	−0.467153	+	0.884177i	$0.654720\pi$
$332$	0	0
$333$	3270.72	0.538241
$334$	0	0
$335$	−5096.63	−0.831220
$336$	0	0
$337$	1655.51	0.267601	0.133800	−	0.991008i	$-0.457282\pi$
$337$	1655.51	0.267601	0.133800	+	0.991008i	$0.457282\pi$
$338$	0	0
$339$	15030.3	2.40807
$340$	0	0
$341$	54.2881	0.00862131
$342$	0	0
$343$	3777.55	0.594661
$344$	0	0
$345$	−734.590	−0.114635
$346$	0	0
$347$	3969.00	0.614026	0.307013	−	0.951705i	$-0.400670\pi$
$347$	3969.00	0.614026	0.307013	+	0.951705i	$0.400670\pi$
$348$	0	0
$349$	−2786.18	−0.427337	−0.213669	−	0.976906i	$-0.568541\pi$
$349$	−2786.18	−0.427337	−0.213669	+	0.976906i	$0.568541\pi$
$350$	0	0
$351$	−3524.83	−0.536016
$352$	0	0
$353$	−635.463	−0.0958138	−0.0479069	−	0.998852i	$-0.515255\pi$
$353$	−635.463	−0.0958138	−0.0479069	+	0.998852i	$0.515255\pi$
$354$	0	0
$355$	−5103.00	−0.762927
$356$	0	0
$357$	3322.68	0.492590
$358$	0	0
$359$	10070.3	1.48047	0.740237	−	0.672346i	$-0.234713\pi$
$359$	10070.3	1.48047	0.740237	+	0.672346i	$0.234713\pi$
$360$	0	0
$361$	3642.51	0.531056
$362$	0	0
$363$	8443.05	1.22079
$364$	0	0
$365$	523.543	0.0750780
$366$	0	0
$367$	−10473.3	−1.48964	−0.744822	−	0.667263i	$-0.767466\pi$
$367$	−10473.3	−1.48964	−0.744822	+	0.667263i	$0.767466\pi$
$368$	0	0
$369$	−4391.52	−0.619549
$370$	0	0
$371$	−2887.20	−0.404033
$372$	0	0
$373$	5752.25	0.798499	0.399250	−	0.916842i	$-0.369271\pi$
$373$	5752.25	0.798499	0.399250	+	0.916842i	$0.369271\pi$
$374$	0	0
$375$	798.468	0.109954
$376$	0	0
$377$	3261.22	0.445521
$378$	0	0
$379$	−105.211	−0.0142595	−0.00712974	−	0.999975i	$-0.502269\pi$
$379$	−105.211	−0.0142595	−0.00712974	+	0.999975i	$0.502269\pi$
$380$	0	0
$381$	853.481	0.114764
$382$	0	0
$383$	−7728.47	−1.03109	−0.515544	−	0.856863i	$-0.672410\pi$
$383$	−7728.47	−1.03109	−0.515544	+	0.856863i	$0.672410\pi$
$384$	0	0
$385$	−88.0028	−0.0116494
$386$	0	0
$387$	−441.113	−0.0579407
$388$	0	0
$389$	−10791.3	−1.40653	−0.703266	−	0.710927i	$-0.748276\pi$
$389$	−10791.3	−1.40653	−0.703266	+	0.710927i	$0.748276\pi$
$390$	0	0
$391$	−2066.46	−0.267277
$392$	0	0
$393$	4953.52	0.635807
$394$	0	0
$395$	5262.62	0.670357
$396$	0	0
$397$	12131.5	1.53366	0.766831	−	0.641849i	$-0.221832\pi$
$397$	12131.5	1.53366	0.766831	+	0.641849i	$0.221832\pi$
$398$	0	0
$399$	3789.79	0.475506
$400$	0	0
$401$	−11439.2	−1.42455	−0.712275	−	0.701901i	$-0.752335\pi$
$401$	−11439.2	−1.42455	−0.712275	+	0.701901i	$0.752335\pi$
$402$	0	0
$403$	746.697	0.0922968
$404$	0	0
$405$	4555.79	0.558961
$406$	0	0
$407$	−720.356	−0.0877315
$408$	0	0
$409$	−6923.37	−0.837013	−0.418507	−	0.908214i	$-0.637446\pi$
$409$	−6923.37	−0.837013	−0.418507	+	0.908214i	$0.637446\pi$
$410$	0	0
$411$	−6309.96	−0.757293
$412$	0	0
$413$	−1602.75	−0.190960
$414$	0	0
$415$	5032.96	0.595321
$416$	0	0
$417$	−13102.9	−1.53874
$418$	0	0
$419$	11554.9	1.34724	0.673622	−	0.739076i	$-0.264738\pi$
$419$	11554.9	1.34724	0.673622	+	0.739076i	$0.264738\pi$
$420$	0	0
$421$	−3922.75	−0.454117	−0.227058	−	0.973881i	$-0.572911\pi$
$421$	−3922.75	−0.454117	−0.227058	+	0.973881i	$0.572911\pi$
$422$	0	0
$423$	3307.75	0.380209
$424$	0	0
$425$	2246.15	0.256363
$426$	0	0
$427$	503.334	0.0570446
$428$	0	0
$429$	−811.998	−0.0913838
$430$	0	0
$431$	7190.68	0.803626	0.401813	−	0.915722i	$-0.368380\pi$
$431$	7190.68	0.803626	0.401813	+	0.915722i	$0.368380\pi$
$432$	0	0
$433$	2292.30	0.254413	0.127206	−	0.991876i	$-0.459399\pi$
$433$	2292.30	0.254413	0.127206	+	0.991876i	$0.459399\pi$
$434$	0	0
$435$	−2491.00	−0.274562
$436$	0	0
$437$	−2356.97	−0.258007
$438$	0	0
$439$	−11453.9	−1.24525	−0.622627	−	0.782519i	$-0.713935\pi$
$439$	−11453.9	−1.24525	−0.622627	+	0.782519i	$0.713935\pi$
$440$	0	0
$441$	−4271.84	−0.461272
$442$	0	0
$443$	15928.7	1.70835	0.854173	−	0.519990i	$-0.174064\pi$
$443$	15928.7	1.70835	0.854173	+	0.519990i	$0.174064\pi$
$444$	0	0
$445$	5491.32	0.584975
$446$	0	0
$447$	16314.8	1.72631
$448$	0	0
$449$	−13852.4	−1.45598	−0.727988	−	0.685590i	$-0.759544\pi$
$449$	−13852.4	−1.45598	−0.727988	+	0.685590i	$0.759544\pi$
$450$	0	0
$451$	967.205	0.100984
$452$	0	0
$453$	8385.64	0.869739
$454$	0	0
$455$	−1210.42	−0.124715
$456$	0	0
$457$	1993.34	0.204036	0.102018	−	0.994783i	$-0.467470\pi$
$457$	1993.34	0.204036	0.102018	+	0.994783i	$0.467470\pi$
$458$	0	0
$459$	7573.79	0.770184
$460$	0	0
$461$	−2772.98	−0.280153	−0.140076	−	0.990141i	$-0.544735\pi$
$461$	−2772.98	−0.280153	−0.140076	+	0.990141i	$0.544735\pi$
$462$	0	0
$463$	−13282.8	−1.33327	−0.666635	−	0.745385i	$-0.732266\pi$
$463$	−13282.8	−1.33327	−0.666635	+	0.745385i	$0.732266\pi$
$464$	0	0
$465$	−570.345	−0.0568799
$466$	0	0
$467$	7196.43	0.713086	0.356543	−	0.934279i	$-0.383955\pi$
$467$	7196.43	0.713086	0.356543	+	0.934279i	$0.383955\pi$
$468$	0	0
$469$	−5901.40	−0.581027
$470$	0	0
$471$	21912.9	2.14372
$472$	0	0
$473$	97.1524	0.00944413
$474$	0	0
$475$	2561.92	0.247472
$476$	0	0
$477$	6883.60	0.660752
$478$	0	0
$479$	−6023.00	−0.574525	−0.287263	−	0.957852i	$-0.592745\pi$
$479$	−6023.00	−0.574525	−0.287263	+	0.957852i	$0.592745\pi$
$480$	0	0
$481$	−9908.01	−0.939223
$482$	0	0
$483$	−850.584	−0.0801303
$484$	0	0
$485$	2447.12	0.229109
$486$	0	0
$487$	−16386.8	−1.52476	−0.762380	−	0.647130i	$-0.775969\pi$
$487$	−16386.8	−1.52476	−0.762380	+	0.647130i	$0.775969\pi$
$488$	0	0
$489$	−1220.03	−0.112825
$490$	0	0
$491$	−1991.00	−0.182999	−0.0914993	−	0.995805i	$-0.529166\pi$
$491$	−1991.00	−0.182999	−0.0914993	+	0.995805i	$0.529166\pi$
$492$	0	0
$493$	−7007.37	−0.640154
$494$	0	0
$495$	209.814	0.0190514
$496$	0	0
$497$	−5908.78	−0.533290
$498$	0	0
$499$	3012.35	0.270244	0.135122	−	0.990829i	$-0.456857\pi$
$499$	3012.35	0.270244	0.135122	+	0.990829i	$0.456857\pi$
$500$	0	0
$501$	−3970.35	−0.354057
$502$	0	0
$503$	13147.4	1.16543	0.582716	−	0.812676i	$-0.301990\pi$
$503$	13147.4	1.16543	0.582716	+	0.812676i	$0.301990\pi$
$504$	0	0
$505$	2172.66	0.191449
$506$	0	0
$507$	2865.37	0.250997
$508$	0	0
$509$	−1885.12	−0.164158	−0.0820790	−	0.996626i	$-0.526156\pi$
$509$	−1885.12	−0.164158	−0.0820790	+	0.996626i	$0.526156\pi$
$510$	0	0
$511$	606.212	0.0524799
$512$	0	0
$513$	8638.55	0.743472
$514$	0	0
$515$	−2046.65	−0.175119
$516$	0	0
$517$	−728.511	−0.0619727
$518$	0	0
$519$	−23776.7	−2.01095
$520$	0	0
$521$	8179.75	0.687833	0.343917	−	0.939000i	$-0.388246\pi$
$521$	8179.75	0.687833	0.343917	+	0.939000i	$0.388246\pi$
$522$	0	0
$523$	−2525.47	−0.211149	−0.105575	−	0.994411i	$-0.533668\pi$
$523$	−2525.47	−0.211149	−0.105575	+	0.994411i	$0.533668\pi$
$524$	0	0
$525$	924.548	0.0768583
$526$	0	0
$527$	−1604.42	−0.132618
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	3821.25	0.312294
$532$	0	0
$533$	13303.3	1.08110
$534$	0	0
$535$	9722.42	0.785677
$536$	0	0
$537$	−1086.85	−0.0873389
$538$	0	0
$539$	940.847	0.0751858
$540$	0	0
$541$	10062.6	0.799680	0.399840	−	0.916585i	$-0.369066\pi$
$541$	10062.6	0.799680	0.399840	+	0.916585i	$0.369066\pi$
$542$	0	0
$543$	516.002	0.0407804
$544$	0	0
$545$	−1612.97	−0.126774
$546$	0	0
$547$	−4559.71	−0.356415	−0.178208	−	0.983993i	$-0.557030\pi$
$547$	−4559.71	−0.356415	−0.178208	+	0.983993i	$0.557030\pi$
$548$	0	0
$549$	−1200.04	−0.0932903
$550$	0	0
$551$	−7992.49	−0.617952
$552$	0	0
$553$	6093.60	0.468583
$554$	0	0
$555$	7567.98	0.578816
$556$	0	0
$557$	−17074.5	−1.29887	−0.649433	−	0.760419i	$-0.724994\pi$
$557$	−17074.5	−1.29887	−0.649433	+	0.760419i	$0.724994\pi$
$558$	0	0
$559$	1336.27	0.101106
$560$	0	0
$561$	1744.74	0.131306
$562$	0	0
$563$	−2166.16	−0.162154	−0.0810772	−	0.996708i	$-0.525836\pi$
$563$	−2166.16	−0.162154	−0.0810772	+	0.996708i	$0.525836\pi$
$564$	0	0
$565$	11765.0	0.876030
$566$	0	0
$567$	5275.16	0.390716
$568$	0	0
$569$	−8376.91	−0.617185	−0.308593	−	0.951194i	$-0.599858\pi$
$569$	−8376.91	−0.617185	−0.308593	+	0.951194i	$0.599858\pi$
$570$	0	0
$571$	5555.57	0.407169	0.203585	−	0.979057i	$-0.434741\pi$
$571$	5555.57	0.407169	0.203585	+	0.979057i	$0.434741\pi$
$572$	0	0
$573$	7060.13	0.514732
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−16122.8	−1.16326	−0.581630	−	0.813454i	$-0.697585\pi$
$577$	−16122.8	−1.16326	−0.581630	+	0.813454i	$0.697585\pi$
$578$	0	0
$579$	33728.8	2.42094
$580$	0	0
$581$	5827.68	0.416132
$582$	0	0
$583$	−1516.07	−0.107700
$584$	0	0
$585$	2885.85	0.203958
$586$	0	0
$587$	−26083.7	−1.83405	−0.917027	−	0.398826i	$-0.869418\pi$
$587$	−26083.7	−1.83405	−0.917027	+	0.398826i	$0.869418\pi$
$588$	0	0
$589$	−1829.98	−0.128019
$590$	0	0
$591$	−10172.1	−0.707993
$592$	0	0
$593$	−12843.5	−0.889410	−0.444705	−	0.895677i	$-0.646691\pi$
$593$	−12843.5	−0.889410	−0.444705	+	0.895677i	$0.646691\pi$
$594$	0	0
$595$	2600.82	0.179199
$596$	0	0
$597$	−22904.5	−1.57021
$598$	0	0
$599$	3857.91	0.263155	0.131578	−	0.991306i	$-0.457996\pi$
$599$	3857.91	0.263155	0.131578	+	0.991306i	$0.457996\pi$
$600$	0	0
$601$	−1568.28	−0.106442	−0.0532210	−	0.998583i	$-0.516949\pi$
$601$	−1568.28	−0.106442	−0.0532210	+	0.998583i	$0.516949\pi$
$602$	0	0
$603$	14070.0	0.950206
$604$	0	0
$605$	6608.79	0.444108
$606$	0	0
$607$	−13332.7	−0.891528	−0.445764	−	0.895150i	$-0.647068\pi$
$607$	−13332.7	−0.891528	−0.445764	+	0.895150i	$0.647068\pi$
$608$	0	0
$609$	−2884.34	−0.191920
$610$	0	0
$611$	−10020.2	−0.663459
$612$	0	0
$613$	17285.0	1.13889	0.569443	−	0.822031i	$-0.307159\pi$
$613$	17285.0	1.13889	0.569443	+	0.822031i	$0.307159\pi$
$614$	0	0
$615$	−10161.4	−0.666253
$616$	0	0
$617$	17342.5	1.13157	0.565787	−	0.824551i	$-0.308572\pi$
$617$	17342.5	1.13157	0.565787	+	0.824551i	$0.308572\pi$
$618$	0	0
$619$	10636.7	0.690670	0.345335	−	0.938479i	$-0.387765\pi$
$619$	10636.7	0.690670	0.345335	+	0.938479i	$0.387765\pi$
$620$	0	0
$621$	−1938.84	−0.125287
$622$	0	0
$623$	6358.42	0.408900
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1990.02	0.126752
$628$	0	0
$629$	21289.3	1.34954
$630$	0	0
$631$	−19106.3	−1.20540	−0.602702	−	0.797966i	$-0.705909\pi$
$631$	−19106.3	−1.20540	−0.602702	+	0.797966i	$0.705909\pi$
$632$	0	0
$633$	−895.940	−0.0562566
$634$	0	0
$635$	668.062	0.0417500
$636$	0	0
$637$	12940.7	0.804914
$638$	0	0
$639$	14087.6	0.872137
$640$	0	0
$641$	12081.8	0.744466	0.372233	−	0.928139i	$-0.378592\pi$
$641$	12081.8	0.744466	0.372233	+	0.928139i	$0.378592\pi$
$642$	0	0
$643$	−1075.78	−0.0659794	−0.0329897	−	0.999456i	$-0.510503\pi$
$643$	−1075.78	−0.0659794	−0.0329897	+	0.999456i	$0.510503\pi$
$644$	0	0
$645$	−1020.67	−0.0623085
$646$	0	0
$647$	1762.84	0.107117	0.0535584	−	0.998565i	$-0.482944\pi$
$647$	1762.84	0.107117	0.0535584	+	0.998565i	$0.482944\pi$
$648$	0	0
$649$	−841.607	−0.0509028
$650$	0	0
$651$	−660.405	−0.0397593
$652$	0	0
$653$	−22927.7	−1.37401	−0.687007	−	0.726651i	$-0.741076\pi$
$653$	−22927.7	−1.37401	−0.687007	+	0.726651i	$0.741076\pi$
$654$	0	0
$655$	3877.37	0.231300
$656$	0	0
$657$	−1445.32	−0.0858252
$658$	0	0
$659$	−21744.5	−1.28535	−0.642674	−	0.766140i	$-0.722175\pi$
$659$	−21744.5	−1.28535	−0.642674	+	0.766140i	$0.722175\pi$
$660$	0	0
$661$	12575.4	0.739982	0.369991	−	0.929035i	$-0.379361\pi$
$661$	12575.4	0.739982	0.369991	+	0.929035i	$0.379361\pi$
$662$	0	0
$663$	23997.7	1.40572
$664$	0	0
$665$	2966.46	0.172984
$666$	0	0
$667$	1793.84	0.104135
$668$	0	0
$669$	2920.96	0.168805
$670$	0	0
$671$	264.301	0.0152060
$672$	0	0
$673$	−8350.98	−0.478316	−0.239158	−	0.970981i	$-0.576871\pi$
$673$	−8350.98	−0.478316	−0.239158	+	0.970981i	$0.576871\pi$
$674$	0	0
$675$	2107.44	0.120171
$676$	0	0
$677$	12815.0	0.727507	0.363754	−	0.931495i	$-0.381495\pi$
$677$	12815.0	0.727507	0.363754	+	0.931495i	$0.381495\pi$
$678$	0	0
$679$	2833.53	0.160149
$680$	0	0
$681$	1326.30	0.0746312
$682$	0	0
$683$	20368.0	1.14108	0.570542	−	0.821268i	$-0.306733\pi$
$683$	20368.0	1.14108	0.570542	+	0.821268i	$0.306733\pi$
$684$	0	0
$685$	−4939.12	−0.275495
$686$	0	0
$687$	15153.4	0.841538
$688$	0	0
$689$	−20852.5	−1.15300
$690$	0	0
$691$	−1766.94	−0.0972759	−0.0486379	−	0.998816i	$-0.515488\pi$
$691$	−1766.94	−0.0972759	−0.0486379	+	0.998816i	$0.515488\pi$
$692$	0	0
$693$	242.945	0.0133170
$694$	0	0
$695$	−10256.3	−0.559775
$696$	0	0
$697$	−28584.6	−1.55340
$698$	0	0
$699$	−38502.8	−2.08342
$700$	0	0
$701$	7068.17	0.380829	0.190414	−	0.981704i	$-0.439017\pi$
$701$	7068.17	0.380829	0.190414	+	0.981704i	$0.439017\pi$
$702$	0	0
$703$	24282.2	1.30273
$704$	0	0
$705$	7653.66	0.408870
$706$	0	0
$707$	2515.73	0.133824
$708$	0	0
$709$	−5015.87	−0.265691	−0.132845	−	0.991137i	$-0.542411\pi$
$709$	−5015.87	−0.265691	−0.132845	+	0.991137i	$0.542411\pi$
$710$	0	0
$711$	−14528.2	−0.766316
$712$	0	0
$713$	410.723	0.0215732
$714$	0	0
$715$	−635.591	−0.0332444
$716$	0	0
$717$	−30431.4	−1.58505
$718$	0	0
$719$	15104.0	0.783426	0.391713	−	0.920088i	$-0.371883\pi$
$719$	15104.0	0.783426	0.391713	+	0.920088i	$0.371883\pi$
$720$	0	0
$721$	−2369.82	−0.122409
$722$	0	0
$723$	1246.96	0.0641424
$724$	0	0
$725$	−1949.83	−0.0998825
$726$	0	0
$727$	−22043.5	−1.12455	−0.562275	−	0.826950i	$-0.690074\pi$
$727$	−22043.5	−1.12455	−0.562275	+	0.826950i	$0.690074\pi$
$728$	0	0
$729$	1961.79	0.0996694
$730$	0	0
$731$	−2871.23	−0.145275
$732$	0	0
$733$	−17431.2	−0.878355	−0.439178	−	0.898400i	$-0.644730\pi$
$733$	−17431.2	−0.878355	−0.439178	+	0.898400i	$0.644730\pi$
$734$	0	0
$735$	−9884.44	−0.496045
$736$	0	0
$737$	−3098.83	−0.154880
$738$	0	0
$739$	−39019.7	−1.94230	−0.971152	−	0.238463i	$-0.923356\pi$
$739$	−39019.7	−1.94230	−0.971152	+	0.238463i	$0.923356\pi$
$740$	0	0
$741$	27371.4	1.35697
$742$	0	0
$743$	2408.03	0.118899	0.0594495	−	0.998231i	$-0.481065\pi$
$743$	2408.03	0.118899	0.0594495	+	0.998231i	$0.481065\pi$
$744$	0	0
$745$	12770.4	0.628014
$746$	0	0
$747$	−13894.2	−0.680539
$748$	0	0
$749$	11257.6	0.549192
$750$	0	0
$751$	−13056.5	−0.634406	−0.317203	−	0.948358i	$-0.602744\pi$
$751$	−13056.5	−0.634406	−0.317203	+	0.948358i	$0.602744\pi$
$752$	0	0
$753$	37628.9	1.82108
$754$	0	0
$755$	6563.85	0.316401
$756$	0	0
$757$	−16457.2	−0.790156	−0.395078	−	0.918648i	$-0.629282\pi$
$757$	−16457.2	−0.790156	−0.395078	+	0.918648i	$0.629282\pi$
$758$	0	0
$759$	−446.642	−0.0213598
$760$	0	0
$761$	20147.2	0.959705	0.479853	−	0.877349i	$-0.340690\pi$
$761$	20147.2	0.959705	0.479853	+	0.877349i	$0.340690\pi$
$762$	0	0
$763$	−1867.66	−0.0886160
$764$	0	0
$765$	−6200.82	−0.293060
$766$	0	0
$767$	−11575.7	−0.544949
$768$	0	0
$769$	17051.3	0.799591	0.399795	−	0.916604i	$-0.369081\pi$
$769$	17051.3	0.799591	0.399795	+	0.916604i	$0.369081\pi$
$770$	0	0
$771$	−28043.2	−1.30993
$772$	0	0
$773$	36283.3	1.68825	0.844125	−	0.536146i	$-0.180120\pi$
$773$	36283.3	1.68825	0.844125	+	0.536146i	$0.180120\pi$
$774$	0	0
$775$	−446.438	−0.0206923
$776$	0	0
$777$	8762.99	0.404595
$778$	0	0
$779$	−32603.2	−1.49953
$780$	0	0
$781$	−3102.70	−0.142155
$782$	0	0
$783$	−6574.63	−0.300074
$784$	0	0
$785$	17152.3	0.779862
$786$	0	0
$787$	−17235.7	−0.780670	−0.390335	−	0.920673i	$-0.627641\pi$
$787$	−17235.7	−0.780670	−0.390335	+	0.920673i	$0.627641\pi$
$788$	0	0
$789$	−8967.35	−0.404621
$790$	0	0
$791$	13622.7	0.612349
$792$	0	0
$793$	3635.28	0.162790
$794$	0	0
$795$	15927.7	0.710562
$796$	0	0
$797$	26857.3	1.19365	0.596823	−	0.802373i	$-0.296430\pi$
$797$	26857.3	1.19365	0.596823	+	0.802373i	$0.296430\pi$
$798$	0	0
$799$	21530.3	0.953302
$800$	0	0
$801$	−15159.6	−0.668712
$802$	0	0
$803$	318.322	0.0139892
$804$	0	0
$805$	−665.794	−0.0291505
$806$	0	0
$807$	−17408.3	−0.759356
$808$	0	0
$809$	−3922.75	−0.170478	−0.0852389	−	0.996361i	$-0.527165\pi$
$809$	−3922.75	−0.170478	−0.0852389	+	0.996361i	$0.527165\pi$
$810$	0	0
$811$	−32637.4	−1.41314	−0.706568	−	0.707645i	$-0.749758\pi$
$811$	−32637.4	−1.41314	−0.706568	+	0.707645i	$0.749758\pi$
$812$	0	0
$813$	−1471.52	−0.0634791
$814$	0	0
$815$	−954.976	−0.0410446
$816$	0	0
$817$	−3274.88	−0.140237
$818$	0	0
$819$	3341.54	0.142568
$820$	0	0
$821$	6375.47	0.271017	0.135509	−	0.990776i	$-0.456733\pi$
$821$	6375.47	0.271017	0.135509	+	0.990776i	$0.456733\pi$
$822$	0	0
$823$	−43501.6	−1.84249	−0.921246	−	0.388981i	$-0.872827\pi$
$823$	−43501.6	−1.84249	−0.921246	+	0.388981i	$0.872827\pi$
$824$	0	0
$825$	485.480	0.0204876
$826$	0	0
$827$	−40508.7	−1.70330	−0.851648	−	0.524114i	$-0.824397\pi$
$827$	−40508.7	−1.70330	−0.851648	+	0.524114i	$0.824397\pi$
$828$	0	0
$829$	26594.0	1.11417	0.557085	−	0.830456i	$-0.311920\pi$
$829$	26594.0	1.11417	0.557085	+	0.830456i	$0.311920\pi$
$830$	0	0
$831$	−35240.3	−1.47108
$832$	0	0
$833$	−27805.7	−1.15655
$834$	0	0
$835$	−3107.79	−0.128802
$836$	0	0
$837$	−1505.34	−0.0621652
$838$	0	0
$839$	8885.84	0.365642	0.182821	−	0.983146i	$-0.441477\pi$
$839$	8885.84	0.365642	0.182821	+	0.983146i	$0.441477\pi$
$840$	0	0
$841$	−18306.1	−0.750587
$842$	0	0
$843$	34391.9	1.40512
$844$	0	0
$845$	2242.87	0.0913101
$846$	0	0
$847$	7652.34	0.310434
$848$	0	0
$849$	−15643.9	−0.632388
$850$	0	0
$851$	−5449.93	−0.219531
$852$	0	0
$853$	−812.221	−0.0326025	−0.0163012	−	0.999867i	$-0.505189\pi$
$853$	−812.221	−0.0326025	−0.0163012	+	0.999867i	$0.505189\pi$
$854$	0	0
$855$	−7072.56	−0.282896
$856$	0	0
$857$	8502.24	0.338893	0.169446	−	0.985539i	$-0.445802\pi$
$857$	8502.24	0.338893	0.169446	+	0.985539i	$0.445802\pi$
$858$	0	0
$859$	−37323.1	−1.48248	−0.741239	−	0.671241i	$-0.765761\pi$
$859$	−37323.1	−1.48248	−0.741239	+	0.671241i	$0.765761\pi$
$860$	0	0
$861$	−11765.9	−0.465714
$862$	0	0
$863$	−12734.0	−0.502282	−0.251141	−	0.967951i	$-0.580806\pi$
$863$	−12734.0	−0.502282	−0.251141	+	0.967951i	$0.580806\pi$
$864$	0	0
$865$	−18611.2	−0.731561
$866$	0	0
$867$	−20180.7	−0.790512
$868$	0	0
$869$	3199.75	0.124907
$870$	0	0
$871$	−42622.3	−1.65810
$872$	0	0
$873$	−6755.63	−0.261906
$874$	0	0
$875$	723.689	0.0279602
$876$	0	0
$877$	−41366.0	−1.59274	−0.796368	−	0.604812i	$-0.793248\pi$
$877$	−41366.0	−1.59274	−0.796368	+	0.604812i	$0.793248\pi$
$878$	0	0
$879$	−33586.6	−1.28879
$880$	0	0
$881$	−7609.60	−0.291003	−0.145502	−	0.989358i	$-0.546480\pi$
$881$	−7609.60	−0.291003	−0.145502	+	0.989358i	$0.546480\pi$
$882$	0	0
$883$	15972.5	0.608739	0.304369	−	0.952554i	$-0.401554\pi$
$883$	15972.5	0.608739	0.304369	+	0.952554i	$0.401554\pi$
$884$	0	0
$885$	8841.83	0.335836
$886$	0	0
$887$	39795.1	1.50641	0.753207	−	0.657784i	$-0.228506\pi$
$887$	39795.1	1.50641	0.753207	+	0.657784i	$0.228506\pi$
$888$	0	0
$889$	773.551	0.0291834
$890$	0	0
$891$	2769.99	0.104151
$892$	0	0
$893$	24557.2	0.920239
$894$	0	0
$895$	−850.730	−0.0317729
$896$	0	0
$897$	−6143.26	−0.228670
$898$	0	0
$899$	1392.76	0.0516699
$900$	0	0
$901$	44805.7	1.65671
$902$	0	0
$903$	−1181.84	−0.0435539
$904$	0	0
$905$	403.900	0.0148355
$906$	0	0
$907$	−37348.3	−1.36729	−0.683644	−	0.729815i	$-0.739606\pi$
$907$	−37348.3	−1.36729	−0.683644	+	0.729815i	$0.739606\pi$
$908$	0	0
$909$	−5997.93	−0.218855
$910$	0	0
$911$	13218.4	0.480729	0.240365	−	0.970683i	$-0.422733\pi$
$911$	13218.4	0.480729	0.240365	+	0.970683i	$0.422733\pi$
$912$	0	0
$913$	3060.11	0.110926
$914$	0	0
$915$	−2776.72	−0.100323
$916$	0	0
$917$	4489.62	0.161680
$918$	0	0
$919$	−40709.0	−1.46122	−0.730612	−	0.682792i	$-0.760765\pi$
$919$	−40709.0	−1.46122	−0.730612	+	0.682792i	$0.760765\pi$
$920$	0	0
$921$	36603.7	1.30959
$922$	0	0
$923$	−42675.5	−1.52187
$924$	0	0
$925$	5923.83	0.210567
$926$	0	0
$927$	5650.07	0.200186
$928$	0	0
$929$	−34275.6	−1.21049	−0.605246	−	0.796039i	$-0.706925\pi$
$929$	−34275.6	−1.21049	−0.605246	+	0.796039i	$0.706925\pi$
$930$	0	0
$931$	−31714.7	−1.11644
$932$	0	0
$933$	43998.0	1.54387
$934$	0	0
$935$	1365.69	0.0477678
$936$	0	0
$937$	−9681.42	−0.337544	−0.168772	−	0.985655i	$-0.553980\pi$
$937$	−9681.42	−0.337544	−0.168772	+	0.985655i	$0.553980\pi$
$938$	0	0
$939$	29218.7	1.01546
$940$	0	0
$941$	−29631.7	−1.02653	−0.513266	−	0.858230i	$-0.671564\pi$
$941$	−29631.7	−1.02653	−0.513266	+	0.858230i	$0.671564\pi$
$942$	0	0
$943$	7317.49	0.252694
$944$	0	0
$945$	2440.21	0.0840000
$946$	0	0
$947$	−13770.7	−0.472531	−0.236265	−	0.971689i	$-0.575923\pi$
$947$	−13770.7	−0.472531	−0.236265	+	0.971689i	$0.575923\pi$
$948$	0	0
$949$	4378.30	0.149764
$950$	0	0
$951$	41402.9	1.41176
$952$	0	0
$953$	−42244.4	−1.43592	−0.717959	−	0.696085i	$-0.754924\pi$
$953$	−42244.4	−1.43592	−0.717959	+	0.696085i	$0.754924\pi$
$954$	0	0
$955$	5526.31	0.187254
$956$	0	0
$957$	−1514.56	−0.0511587
$958$	0	0
$959$	−5719.02	−0.192572
$960$	0	0
$961$	−29472.1	−0.989296
$962$	0	0
$963$	−26840.2	−0.898144
$964$	0	0
$965$	26401.2	0.880710
$966$	0	0
$967$	−8247.06	−0.274258	−0.137129	−	0.990553i	$-0.543788\pi$
$967$	−8247.06	−0.274258	−0.137129	+	0.990553i	$0.543788\pi$
$968$	0	0
$969$	−58812.8	−1.94978
$970$	0	0
$971$	−58514.6	−1.93390	−0.966952	−	0.254957i	$-0.917939\pi$
$971$	−58514.6	−1.93390	−0.966952	+	0.254957i	$0.917939\pi$
$972$	0	0
$973$	−11875.8	−0.391286
$974$	0	0
$975$	6677.45	0.219333
$976$	0	0
$977$	−16995.2	−0.556526	−0.278263	−	0.960505i	$-0.589759\pi$
$977$	−16995.2	−0.556526	−0.278263	+	0.960505i	$0.589759\pi$
$978$	0	0
$979$	3338.81	0.108998
$980$	0	0
$981$	4452.84	0.144922
$982$	0	0
$983$	−34851.6	−1.13082	−0.565409	−	0.824811i	$-0.691282\pi$
$983$	−34851.6	−1.13082	−0.565409	+	0.824811i	$0.691282\pi$
$984$	0	0
$985$	−7962.20	−0.257560
$986$	0	0
$987$	8862.20	0.285802
$988$	0	0
$989$	735.017	0.0236321
$990$	0	0
$991$	−6949.64	−0.222768	−0.111384	−	0.993777i	$-0.535528\pi$
$991$	−6949.64	−0.222768	−0.111384	+	0.993777i	$0.535528\pi$
$992$	0	0
$993$	35940.0	1.14856
$994$	0	0
$995$	−17928.5	−0.571227
$996$	0	0
$997$	20280.1	0.644210	0.322105	−	0.946704i	$-0.395610\pi$
$997$	20280.1	0.644210	0.322105	+	0.946704i	$0.395610\pi$
$998$	0	0
$999$	19974.6	0.632600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	920.4.a.b.1.2	✓	6
4.3	odd	2		1840.4.a.s.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.4.a.b.1.2	✓	6	1.1	even	1	trivial
1840.4.a.s.1.5		6	4.3	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:	\( N \)	\(=\)	\( 920 = 2^{3} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	920.a (trivial)

\(f(q)\)	\(=\)	\(q-6.38774 q^{3} -5.00000 q^{5} -5.78952 q^{7} +13.8032 q^{9} -3.04007 q^{11} -41.8142 q^{13} +31.9387 q^{15} +89.8459 q^{17} +102.477 q^{19} +36.9819 q^{21} -23.0000 q^{23} +25.0000 q^{25} +84.2976 q^{27} -77.9931 q^{29} -17.8575 q^{31} +19.4192 q^{33} +28.9476 q^{35} +236.953 q^{37} +267.098 q^{39} -318.152 q^{41} -31.9573 q^{43} -69.0161 q^{45} +239.636 q^{47} -309.482 q^{49} -573.913 q^{51} +498.695 q^{53} +15.2004 q^{55} -654.596 q^{57} +276.837 q^{59} -86.9389 q^{61} -79.9140 q^{63} +209.071 q^{65} +1019.33 q^{67} +146.918 q^{69} +1020.60 q^{71} -104.709 q^{73} -159.694 q^{75} +17.6006 q^{77} -1052.52 q^{79} -911.158 q^{81} -1006.59 q^{83} -449.230 q^{85} +498.200 q^{87} -1098.26 q^{89} +242.084 q^{91} +114.069 q^{93} -512.384 q^{95} -489.424 q^{97} -41.9628 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{3} - 30 q^{5} + 28 q^{7} + 4 q^{9} - 3 q^{11} - 28 q^{13} - 10 q^{15} + 24 q^{17} - 3 q^{19} + 60 q^{21} - 138 q^{23} + 150 q^{25} - 97 q^{27} + 76 q^{29} - 381 q^{31} - 3 q^{33} - 140 q^{35}+ \cdots - 525 q^{99}+O(q^{100}) \) 6 * q + 2 * q^3 - 30 * q^5 + 28 * q^7 + 4 * q^9 - 3 * q^11 - 28 * q^13 - 10 * q^15 + 24 * q^17 - 3 * q^19 + 60 * q^21 - 138 * q^23 + 150 * q^25 - 97 * q^27 + 76 * q^29 - 381 * q^31 - 3 * q^33 - 140 * q^35 + 131 * q^37 + 41 * q^39 - 95 * q^41 - 202 * q^43 - 20 * q^45 - 119 * q^47 - 578 * q^49 - 997 * q^51 + 137 * q^53 + 15 * q^55 - 894 * q^57 - 39 * q^59 - 573 * q^61 - 355 * q^63 + 140 * q^65 - 563 * q^67 - 46 * q^69 - 83 * q^71 - 1799 * q^73 + 50 * q^75 - 1410 * q^77 - 1636 * q^79 - 2006 * q^81 - 1191 * q^83 - 120 * q^85 - 1821 * q^87 - 1370 * q^89 - 1251 * q^91 - 2215 * q^93 + 15 * q^95 - 3021 * q^97 - 525 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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