LMFDB - Embedded newform 2023.4.a.v.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2023,4,Mod(1,2023)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2023, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2023.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 2023 = 7 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2023.a (trivial)

Newform invariants

comment:select newform

sage:traces = [56,8,24,240,80,68] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$119.360863942$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	no (minimal twist has level 119)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	2023.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.88913 q^{2} -5.33007 q^{3} +15.9036 q^{4} -4.56956 q^{5} +26.0594 q^{6} +7.00000 q^{7} -38.6415 q^{8} +1.40966 q^{9} +22.3412 q^{10} +12.2239 q^{11} -84.7671 q^{12} -2.40598 q^{13} -34.2239 q^{14} +24.3561 q^{15} +61.6949 q^{16} -6.89203 q^{18} +81.2334 q^{19} -72.6723 q^{20} -37.3105 q^{21} -59.7642 q^{22} +67.0308 q^{23} +205.962 q^{24} -104.119 q^{25} +11.7631 q^{26} +136.398 q^{27} +111.325 q^{28} +54.9426 q^{29} -119.080 q^{30} -206.317 q^{31} +7.49819 q^{32} -65.1543 q^{33} -31.9869 q^{35} +22.4187 q^{36} -83.7840 q^{37} -397.160 q^{38} +12.8240 q^{39} +176.575 q^{40} -30.4104 q^{41} +182.416 q^{42} +256.536 q^{43} +194.404 q^{44} -6.44154 q^{45} -327.722 q^{46} +20.2645 q^{47} -328.838 q^{48} +49.0000 q^{49} +509.052 q^{50} -38.2636 q^{52} +17.1405 q^{53} -666.869 q^{54} -55.8579 q^{55} -270.491 q^{56} -432.980 q^{57} -268.621 q^{58} +693.221 q^{59} +387.349 q^{60} +278.007 q^{61} +1008.71 q^{62} +9.86765 q^{63} -530.219 q^{64} +10.9943 q^{65} +318.548 q^{66} +893.564 q^{67} -357.279 q^{69} +156.388 q^{70} +180.449 q^{71} -54.4716 q^{72} -273.854 q^{73} +409.631 q^{74} +554.962 q^{75} +1291.90 q^{76} +85.5673 q^{77} -62.6983 q^{78} +1068.40 q^{79} -281.919 q^{80} -765.074 q^{81} +148.680 q^{82} -984.805 q^{83} -593.370 q^{84} -1254.24 q^{86} -292.848 q^{87} -472.351 q^{88} -633.249 q^{89} +31.4935 q^{90} -16.8418 q^{91} +1066.03 q^{92} +1099.68 q^{93} -99.0756 q^{94} -371.201 q^{95} -39.9659 q^{96} +233.777 q^{97} -239.567 q^{98} +17.2316 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 8 q^{2} + 24 q^{3} + 240 q^{4} + 80 q^{5} + 68 q^{6} + 392 q^{7} + 96 q^{8} + 576 q^{9} + 80 q^{10} + 176 q^{11} + 288 q^{12} - 96 q^{13} + 56 q^{14} + 192 q^{15} + 1088 q^{16} + 216 q^{18} + 48 q^{19}+ \cdots + 6576 q^{99}+O(q^{100}) $ 56 * q + 8 * q^2 + 24 * q^3 + 240 * q^4 + 80 * q^5 + 68 * q^6 + 392 * q^7 + 96 * q^8 + 576 * q^9 + 80 * q^10 + 176 * q^11 + 288 * q^12 - 96 * q^13 + 56 * q^14 + 192 * q^15 + 1088 * q^16 + 216 * q^18 + 48 * q^19 + 1660 * q^20 + 168 * q^21 + 352 * q^22 + 608 * q^23 + 736 * q^24 + 1792 * q^25 + 864 * q^27 + 1680 * q^28 + 928 * q^29 - 100 * q^30 + 496 * q^31 + 436 * q^32 + 384 * q^33 + 560 * q^35 + 2804 * q^36 + 1392 * q^37 + 624 * q^39 - 1020 * q^40 + 2088 * q^41 + 476 * q^42 + 1312 * q^43 + 2112 * q^44 + 4840 * q^45 + 736 * q^46 + 5228 * q^48 + 2744 * q^49 + 2020 * q^50 - 1536 * q^52 + 1272 * q^53 + 2268 * q^54 - 336 * q^55 + 672 * q^56 + 2448 * q^57 + 1376 * q^58 + 2372 * q^60 + 976 * q^61 + 3776 * q^62 + 4032 * q^63 + 5376 * q^64 + 6080 * q^65 + 976 * q^67 + 1248 * q^69 + 560 * q^70 + 3216 * q^71 + 4532 * q^72 + 4088 * q^73 - 2456 * q^74 - 760 * q^75 - 2640 * q^76 + 1232 * q^77 + 3744 * q^78 + 1792 * q^79 + 13920 * q^80 + 4184 * q^81 + 8204 * q^82 - 2832 * q^83 + 2016 * q^84 + 4908 * q^86 + 832 * q^87 + 2424 * q^88 - 2768 * q^89 + 8660 * q^90 - 672 * q^91 + 6784 * q^92 + 784 * q^93 - 928 * q^94 + 4560 * q^95 + 6656 * q^96 + 9464 * q^97 + 392 * q^98 + 6576 * 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 $

$2$	−4.88913	−1.72857	−0.864284	−	0.503005i	$-0.832228\pi$
$2$	−4.88913	−1.72857	−0.864284	+	0.503005i	$0.832228\pi$
$3$	−5.33007	−1.02577	−0.512886	−	0.858456i	$-0.671424\pi$
$3$	−5.33007	−1.02577	−0.512886	+	0.858456i	$0.671424\pi$
$4$	15.9036	1.98795
$5$	−4.56956	−0.408714	−0.204357	−	0.978896i	$-0.565510\pi$
$5$	−4.56956	−0.408714	−0.204357	+	0.978896i	$0.565510\pi$
$6$	26.0594	1.77312
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−38.6415	−1.70773
$9$	1.40966	0.0522098
$10$	22.3412	0.706490
$11$	12.2239	0.335059	0.167529	−	0.985867i	$-0.446421\pi$
$11$	12.2239	0.335059	0.167529	+	0.985867i	$0.446421\pi$
$12$	−84.7671	−2.03918
$13$	−2.40598	−0.0513306	−0.0256653	−	0.999671i	$-0.508170\pi$
$13$	−2.40598	−0.0513306	−0.0256653	+	0.999671i	$0.508170\pi$
$14$	−34.2239	−0.653337
$15$	24.3561	0.419248
$16$	61.6949	0.963983
$17$	0	0
$17$	0	0
$18$	−6.89203	−0.0902481
$19$	81.2334	0.980854	0.490427	−	0.871482i	$-0.336841\pi$
$19$	81.2334	0.980854	0.490427	+	0.871482i	$0.336841\pi$
$20$	−72.6723	−0.812501
$21$	−37.3105	−0.387706
$22$	−59.7642	−0.579172
$23$	67.0308	0.607690	0.303845	−	0.952721i	$-0.401729\pi$
$23$	67.0308	0.607690	0.303845	+	0.952721i	$0.401729\pi$
$24$	205.962	1.75174
$25$	−104.119	−0.832953
$26$	11.7631	0.0887284
$27$	136.398	0.972217
$28$	111.325	0.751373
$29$	54.9426	0.351813	0.175906	−	0.984407i	$-0.443714\pi$
$29$	54.9426	0.351813	0.175906	+	0.984407i	$0.443714\pi$
$30$	−119.080	−0.724698
$31$	−206.317	−1.19534	−0.597671	−	0.801741i	$-0.703907\pi$
$31$	−206.317	−1.19534	−0.597671	+	0.801741i	$0.703907\pi$
$32$	7.49819	0.0414220
$33$	−65.1543	−0.343694
$34$	0	0
$35$	−31.9869	−0.154479
$36$	22.4187	0.103790
$37$	−83.7840	−0.372270	−0.186135	−	0.982524i	$-0.559596\pi$
$37$	−83.7840	−0.372270	−0.186135	+	0.982524i	$0.559596\pi$
$38$	−397.160	−1.69547
$39$	12.8240	0.0526536
$40$	176.575	0.697973
$41$	−30.4104	−0.115837	−0.0579183	−	0.998321i	$-0.518446\pi$
$41$	−30.4104	−0.115837	−0.0579183	+	0.998321i	$0.518446\pi$
$42$	182.416	0.670175
$43$	256.536	0.909800	0.454900	−	0.890542i	$-0.349675\pi$
$43$	256.536	0.909800	0.454900	+	0.890542i	$0.349675\pi$
$44$	194.404	0.666079
$45$	−6.44154	−0.0213389
$46$	−327.722	−1.05043
$47$	20.2645	0.0628910	0.0314455	−	0.999505i	$-0.489989\pi$
$47$	20.2645	0.0628910	0.0314455	+	0.999505i	$0.489989\pi$
$48$	−328.838	−0.988827
$49$	49.0000	0.142857
$50$	509.052	1.43982
$51$	0	0
$52$	−38.2636	−0.102042
$53$	17.1405	0.0444231	0.0222116	−	0.999753i	$-0.492929\pi$
$53$	17.1405	0.0444231	0.0222116	+	0.999753i	$0.492929\pi$
$54$	−666.869	−1.68054
$55$	−55.8579	−0.136943
$56$	−270.491	−0.645462
$57$	−432.980	−1.00613
$58$	−268.621	−0.608132
$59$	693.221	1.52966	0.764828	−	0.644235i	$-0.222824\pi$
$59$	693.221	1.52966	0.764828	+	0.644235i	$0.222824\pi$
$60$	387.349	0.833442
$61$	278.007	0.583526	0.291763	−	0.956491i	$-0.405758\pi$
$61$	278.007	0.583526	0.291763	+	0.956491i	$0.405758\pi$
$62$	1008.71	2.06623
$63$	9.86765	0.0197334
$64$	−530.219	−1.03558
$65$	10.9943	0.0209795
$66$	318.548	0.594099
$67$	893.564	1.62935	0.814674	−	0.579920i	$-0.196916\pi$
$67$	893.564	1.62935	0.814674	+	0.579920i	$0.196916\pi$
$68$	0	0
$69$	−357.279	−0.623352
$70$	156.388	0.267028
$71$	180.449	0.301625	0.150813	−	0.988562i	$-0.451811\pi$
$71$	180.449	0.301625	0.150813	+	0.988562i	$0.451811\pi$
$72$	−54.4716	−0.0891602
$73$	−273.854	−0.439071	−0.219535	−	0.975605i	$-0.570454\pi$
$73$	−273.854	−0.439071	−0.219535	+	0.975605i	$0.570454\pi$
$74$	409.631	0.643494
$75$	554.962	0.854420
$76$	1291.90	1.94988
$77$	85.5673	0.126640
$78$	−62.6983	−0.0910152
$79$	1068.40	1.52158	0.760789	−	0.648999i	$-0.224812\pi$
$79$	1068.40	1.52158	0.760789	+	0.648999i	$0.224812\pi$
$80$	−281.919	−0.393993
$81$	−765.074	−1.04948
$82$	148.680	0.200231
$83$	−984.805	−1.30237	−0.651183	−	0.758921i	$-0.725727\pi$
$83$	−984.805	−1.30237	−0.651183	+	0.758921i	$0.725727\pi$
$84$	−593.370	−0.770738
$85$	0	0
$86$	−1254.24	−1.57265
$87$	−292.848	−0.360880
$88$	−472.351	−0.572190
$89$	−633.249	−0.754205	−0.377102	−	0.926172i	$-0.623080\pi$
$89$	−633.249	−0.754205	−0.377102	+	0.926172i	$0.623080\pi$
$90$	31.4935	0.0368857
$91$	−16.8418	−0.0194012
$92$	1066.03	1.20806
$93$	1099.68	1.22615
$94$	−99.0756	−0.108711
$95$	−371.201	−0.400889
$96$	−39.9659	−0.0424896
$97$	233.777	0.244706	0.122353	−	0.992487i	$-0.460956\pi$
$97$	233.777	0.244706	0.122353	+	0.992487i	$0.460956\pi$
$98$	−239.567	−0.246938
$99$	17.2316	0.0174933
$100$	−1655.87	−1.65587
$101$	1155.47	1.13835	0.569177	−	0.822215i	$-0.307262\pi$
$101$	1155.47	1.13835	0.569177	+	0.822215i	$0.307262\pi$
$102$	0	0
$103$	−1159.97	−1.10966	−0.554830	−	0.831964i	$-0.687217\pi$
$103$	−1159.97	−1.10966	−0.554830	+	0.831964i	$0.687217\pi$
$104$	92.9707	0.0876589
$105$	170.493	0.158461
$106$	−83.8019	−0.0767883
$107$	1695.72	1.53207	0.766033	−	0.642801i	$-0.222228\pi$
$107$	1695.72	1.53207	0.766033	+	0.642801i	$0.222228\pi$
$108$	2169.22	1.93272
$109$	−502.254	−0.441350	−0.220675	−	0.975347i	$-0.570826\pi$
$109$	−502.254	−0.441350	−0.220675	+	0.975347i	$0.570826\pi$
$110$	273.096	0.236716
$111$	446.575	0.381865
$112$	431.864	0.364351
$113$	−1346.76	−1.12117	−0.560586	−	0.828096i	$-0.689424\pi$
$113$	−1346.76	−1.12117	−0.560586	+	0.828096i	$0.689424\pi$
$114$	2116.89	1.73917
$115$	−306.301	−0.248372
$116$	873.783	0.699385
$117$	−3.39162	−0.00267996
$118$	−3389.25	−2.64411
$119$	0	0
$120$	−941.157	−0.715962
$121$	−1181.58	−0.887736
$122$	−1359.21	−1.00866
$123$	162.089	0.118822
$124$	−3281.18	−2.37628
$125$	1046.97	0.749153
$126$	−48.2442	−0.0341106
$127$	−711.450	−0.497094	−0.248547	−	0.968620i	$-0.579953\pi$
$127$	−711.450	−0.497094	−0.248547	+	0.968620i	$0.579953\pi$
$128$	2532.32	1.74865
$129$	−1367.36	−0.933248
$130$	−53.7523	−0.0362646
$131$	1541.75	1.02827	0.514133	−	0.857710i	$-0.328114\pi$
$131$	1541.75	1.02827	0.514133	+	0.857710i	$0.328114\pi$
$132$	−1036.19	−0.683245
$133$	568.634	0.370728
$134$	−4368.75	−2.81644
$135$	−623.280	−0.397359
$136$	0	0
$137$	−2269.14	−1.41508	−0.707538	−	0.706675i	$-0.750194\pi$
$137$	−2269.14	−1.41508	−0.707538	+	0.706675i	$0.750194\pi$
$138$	1746.78	1.07751
$139$	−366.787	−0.223816	−0.111908	−	0.993719i	$-0.535696\pi$
$139$	−366.787	−0.223816	−0.111908	+	0.993719i	$0.535696\pi$
$140$	−508.706	−0.307097
$141$	−108.011	−0.0645119
$142$	−882.240	−0.521380
$143$	−29.4104	−0.0171988
$144$	86.9690	0.0503293
$145$	−251.063	−0.143791
$146$	1338.91	0.758963
$147$	−261.174	−0.146539
$148$	−1332.46	−0.740053
$149$	356.802	0.196177	0.0980883	−	0.995178i	$-0.468727\pi$
$149$	356.802	0.196177	0.0980883	+	0.995178i	$0.468727\pi$
$150$	−2713.28	−1.47692
$151$	−3265.92	−1.76011	−0.880056	−	0.474870i	$-0.842495\pi$
$151$	−3265.92	−1.76011	−0.880056	+	0.474870i	$0.842495\pi$
$152$	−3138.98	−1.67503
$153$	0	0
$154$	−418.350	−0.218906
$155$	942.778	0.488553
$156$	203.948	0.104672
$157$	−1666.46	−0.847123	−0.423561	−	0.905867i	$-0.639220\pi$
$157$	−1666.46	−0.847123	−0.423561	+	0.905867i	$0.639220\pi$
$158$	−5223.56	−2.63015
$159$	−91.3599	−0.0455680
$160$	−34.2634	−0.0169298
$161$	469.215	0.229685
$162$	3740.54	1.81410
$163$	3735.02	1.79478	0.897390	−	0.441239i	$-0.145461\pi$
$163$	3735.02	1.79478	0.897390	+	0.441239i	$0.145461\pi$
$164$	−483.633	−0.230277
$165$	297.727	0.140473
$166$	4814.84	2.25123
$167$	540.608	0.250500	0.125250	−	0.992125i	$-0.460027\pi$
$167$	540.608	0.250500	0.125250	+	0.992125i	$0.460027\pi$
$168$	1441.74	0.662097
$169$	−2191.21	−0.997365
$170$	0	0
$171$	114.512	0.0512101
$172$	4079.84	1.80863
$173$	−1248.71	−0.548773	−0.274387	−	0.961619i	$-0.588475\pi$
$173$	−1248.71	−0.548773	−0.274387	+	0.961619i	$0.588475\pi$
$174$	1431.77	0.623806
$175$	−728.834	−0.314827
$176$	754.152	0.322991
$177$	−3694.92	−1.56908
$178$	3096.03	1.30369
$179$	−3106.56	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$179$	−3106.56	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$180$	−102.444	−0.0424205
$181$	−627.925	−0.257863	−0.128932	−	0.991653i	$-0.541155\pi$
$181$	−627.925	−0.257863	−0.128932	+	0.991653i	$0.541155\pi$
$182$	82.3419	0.0335362
$183$	−1481.80	−0.598566
$184$	−2590.17	−1.03777
$185$	382.856	0.152152
$186$	−5376.50	−2.11948
$187$	0	0
$188$	322.277	0.125024
$189$	954.788	0.367464
$190$	1814.85	0.692963
$191$	4511.12	1.70897	0.854485	−	0.519475i	$-0.173873\pi$
$191$	4511.12	1.70897	0.854485	+	0.519475i	$0.173873\pi$
$192$	2826.10	1.06227
$193$	−4300.90	−1.60407	−0.802036	−	0.597276i	$-0.796250\pi$
$193$	−4300.90	−1.60407	−0.802036	+	0.597276i	$0.796250\pi$
$194$	−1142.97	−0.422990
$195$	−58.6002	−0.0215202
$196$	779.275	0.283992
$197$	4388.44	1.58712	0.793562	−	0.608489i	$-0.208224\pi$
$197$	4388.44	1.58712	0.793562	+	0.608489i	$0.208224\pi$
$198$	−84.2475	−0.0302384
$199$	−5417.18	−1.92972	−0.964859	−	0.262767i	$-0.915365\pi$
$199$	−5417.18	−1.92972	−0.964859	+	0.262767i	$0.915365\pi$
$200$	4023.32	1.42246
$201$	−4762.76	−1.67134
$202$	−5649.25	−1.96772
$203$	384.598	0.132973
$204$	0	0
$205$	138.962	0.0473440
$206$	5671.23	1.91812
$207$	94.4908	0.0317274
$208$	−148.436	−0.0494818
$209$	992.989	0.328644
$210$	−833.560	−0.273910
$211$	−5071.17	−1.65457	−0.827283	−	0.561785i	$-0.810115\pi$
$211$	−5071.17	−1.65457	−0.827283	+	0.561785i	$0.810115\pi$
$212$	272.595	0.0883107
$213$	−961.808	−0.309399
$214$	−8290.57	−2.64828
$215$	−1172.26	−0.371848
$216$	−5270.64	−1.66029
$217$	−1444.22	−0.451797
$218$	2455.58	0.762904
$219$	1459.66	0.450387
$220$	−888.340	−0.272236
$221$	0	0
$222$	−2183.36	−0.660079
$223$	−1615.63	−0.485160	−0.242580	−	0.970131i	$-0.577994\pi$
$223$	−1615.63	−0.485160	−0.242580	+	0.970131i	$0.577994\pi$
$224$	52.4873	0.0156561
$225$	−146.773	−0.0434883
$226$	6584.48	1.93802
$227$	4961.70	1.45075	0.725373	−	0.688356i	$-0.241667\pi$
$227$	4961.70	1.45075	0.725373	+	0.688356i	$0.241667\pi$
$228$	−6885.92	−2.00014
$229$	−72.9223	−0.0210430	−0.0105215	−	0.999945i	$-0.503349\pi$
$229$	−72.9223	−0.0210430	−0.0105215	+	0.999945i	$0.503349\pi$
$230$	1497.55	0.429327
$231$	−456.080	−0.129904
$232$	−2123.06	−0.600802
$233$	3979.55	1.11892	0.559461	−	0.828857i	$-0.311008\pi$
$233$	3979.55	1.11892	0.559461	+	0.828857i	$0.311008\pi$
$234$	16.5821	0.00463249
$235$	−92.5998	−0.0257044
$236$	11024.7	3.04087
$237$	−5694.66	−1.56079
$238$	0	0
$239$	−2395.18	−0.648248	−0.324124	−	0.946015i	$-0.605070\pi$
$239$	−2395.18	−0.648248	−0.324124	+	0.946015i	$0.605070\pi$
$240$	1502.65	0.404147
$241$	−3859.15	−1.03149	−0.515746	−	0.856742i	$-0.672485\pi$
$241$	−3859.15	−1.03149	−0.515746	+	0.856742i	$0.672485\pi$
$242$	5776.88	1.53451
$243$	395.143	0.104315
$244$	4421.30	1.16002
$245$	−223.908	−0.0583877
$246$	−792.476	−0.205392
$247$	−195.446	−0.0503478
$248$	7972.41	2.04132
$249$	5249.08	1.33593
$250$	−5118.79	−1.29496
$251$	−5186.97	−1.30438	−0.652188	−	0.758057i	$-0.726149\pi$
$251$	−5186.97	−1.30438	−0.652188	+	0.758057i	$0.726149\pi$
$252$	156.931	0.0392290
$253$	819.378	0.203612
$254$	3478.37	0.859261
$255$	0	0
$256$	−8139.09	−1.98708
$257$	−1815.85	−0.440738	−0.220369	−	0.975417i	$-0.570726\pi$
$257$	−1815.85	−0.440738	−0.220369	+	0.975417i	$0.570726\pi$
$258$	6685.18	1.61318
$259$	−586.488	−0.140705
$260$	174.848	0.0417062
$261$	77.4505	0.0183681
$262$	−7537.79	−1.77743
$263$	−3070.90	−0.719999	−0.360000	−	0.932952i	$-0.617223\pi$
$263$	−3070.90	−0.719999	−0.360000	+	0.932952i	$0.617223\pi$
$264$	2517.66	0.586937
$265$	−78.3244	−0.0181563
$266$	−2780.12	−0.640828
$267$	3375.26	0.773643
$268$	14210.9	3.23905
$269$	−5399.67	−1.22388	−0.611940	−	0.790904i	$-0.709611\pi$
$269$	−5399.67	−1.22388	−0.611940	+	0.790904i	$0.709611\pi$
$270$	3047.30	0.686862
$271$	4553.25	1.02063	0.510314	−	0.859988i	$-0.329529\pi$
$271$	4553.25	1.02063	0.510314	+	0.859988i	$0.329529\pi$
$272$	0	0
$273$	89.7682	0.0199012
$274$	11094.1	2.44605
$275$	−1272.74	−0.279088
$276$	−5682.01	−1.23919
$277$	3093.34	0.670979	0.335489	−	0.942044i	$-0.391098\pi$
$277$	3093.34	0.670979	0.335489	+	0.942044i	$0.391098\pi$
$278$	1793.27	0.386881
$279$	−290.838	−0.0624086
$280$	1236.02	0.263809
$281$	1762.74	0.374222	0.187111	−	0.982339i	$-0.440088\pi$
$281$	1762.74	0.374222	0.187111	+	0.982339i	$0.440088\pi$
$282$	528.080	0.111513
$283$	−274.316	−0.0576198	−0.0288099	−	0.999585i	$-0.509172\pi$
$283$	−274.316	−0.0576198	−0.0288099	+	0.999585i	$0.509172\pi$
$284$	2869.79	0.599615
$285$	1978.53	0.411221
$286$	143.791	0.0297292
$287$	−212.873	−0.0437821
$288$	10.5699	0.00216263
$289$	0	0
$290$	1227.48	0.248552
$291$	−1246.05	−0.251012
$292$	−4355.25	−0.872849
$293$	4475.67	0.892394	0.446197	−	0.894935i	$-0.352778\pi$
$293$	4475.67	0.892394	0.446197	+	0.894935i	$0.352778\pi$
$294$	1276.91	0.253303
$295$	−3167.72	−0.625192
$296$	3237.54	0.635738
$297$	1667.32	0.325750
$298$	−1744.45	−0.339105
$299$	−161.275	−0.0311931
$300$	8825.88	1.69854
$301$	1795.75	0.343872
$302$	15967.5	3.04247
$303$	−6158.75	−1.16769
$304$	5011.68	0.945526
$305$	−1270.37	−0.238495
$306$	0	0
$307$	7535.94	1.40097	0.700487	−	0.713665i	$-0.252966\pi$
$307$	7535.94	1.40097	0.700487	+	0.713665i	$0.252966\pi$
$308$	1360.83	0.251754
$309$	6182.71	1.13826
$310$	−4609.36	−0.844497
$311$	10419.1	1.89972	0.949859	−	0.312679i	$-0.101226\pi$
$311$	10419.1	1.89972	0.949859	+	0.312679i	$0.101226\pi$
$312$	−495.540	−0.0899181
$313$	2239.40	0.404403	0.202201	−	0.979344i	$-0.435190\pi$
$313$	2239.40	0.404403	0.202201	+	0.979344i	$0.435190\pi$
$314$	8147.55	1.46431
$315$	−45.0908	−0.00806533
$316$	16991.4	3.02481
$317$	−1736.45	−0.307662	−0.153831	−	0.988097i	$-0.549161\pi$
$317$	−1736.45	−0.307662	−0.153831	+	0.988097i	$0.549161\pi$
$318$	446.670	0.0787674
$319$	671.613	0.117878
$320$	2422.87	0.423257
$321$	−9038.29	−1.57155
$322$	−2294.05	−0.397027
$323$	0	0
$324$	−12167.4	−2.08632
$325$	250.508	0.0427560
$326$	−18261.0	−3.10240
$327$	2677.05	0.452725
$328$	1175.10	0.197818
$329$	141.851	0.0237706
$330$	−1455.62	−0.242816
$331$	−3587.98	−0.595811	−0.297906	−	0.954595i	$-0.596288\pi$
$331$	−3587.98	−0.595811	−0.297906	+	0.954595i	$0.596288\pi$
$332$	−15661.9	−2.58903
$333$	−118.107	−0.0194362
$334$	−2643.10	−0.433006
$335$	−4083.20	−0.665937
$336$	−2301.87	−0.373741
$337$	5117.37	0.827183	0.413592	−	0.910463i	$-0.364274\pi$
$337$	5117.37	0.827183	0.413592	+	0.910463i	$0.364274\pi$
$338$	10713.1	1.72401
$339$	7178.33	1.15007
$340$	0	0
$341$	−2522.00	−0.400510
$342$	−559.863	−0.0885202
$343$	343.000	0.0539949
$344$	−9912.96	−1.55369
$345$	1632.61	0.254773
$346$	6105.11	0.948592
$347$	7787.39	1.20475	0.602376	−	0.798212i	$-0.294221\pi$
$347$	7787.39	1.20475	0.602376	+	0.798212i	$0.294221\pi$
$348$	−4657.32	−0.717410
$349$	−6896.66	−1.05779	−0.528897	−	0.848686i	$-0.677394\pi$
$349$	−6896.66	−1.05779	−0.528897	+	0.848686i	$0.677394\pi$
$350$	3563.36	0.544199
$351$	−328.171	−0.0499045
$352$	91.6571	0.0138788
$353$	3774.78	0.569154	0.284577	−	0.958653i	$-0.408147\pi$
$353$	3774.78	0.569154	0.284577	+	0.958653i	$0.408147\pi$
$354$	18064.9	2.71226
$355$	−824.574	−0.123278
$356$	−10070.9	−1.49932
$357$	0	0
$358$	15188.4	2.24226
$359$	−6489.78	−0.954088	−0.477044	−	0.878879i	$-0.658292\pi$
$359$	−6489.78	−0.954088	−0.477044	+	0.878879i	$0.658292\pi$
$360$	248.911	0.0364410
$361$	−260.136	−0.0379263
$362$	3070.00	0.445734
$363$	6297.89	0.910615
$364$	−267.845	−0.0385684
$365$	1251.39	0.179454
$366$	7244.69	1.03466
$367$	12082.8	1.71858	0.859288	−	0.511492i	$-0.170907\pi$
$367$	12082.8	1.71858	0.859288	+	0.511492i	$0.170907\pi$
$368$	4135.46	0.585803
$369$	−42.8684	−0.00604780
$370$	−1871.83	−0.263005
$371$	119.983	0.0167904
$372$	17488.9	2.43752
$373$	4796.64	0.665846	0.332923	−	0.942954i	$-0.391965\pi$
$373$	4796.64	0.665846	0.332923	+	0.942954i	$0.391965\pi$
$374$	0	0
$375$	−5580.44	−0.768461
$376$	−783.051	−0.107401
$377$	−132.191	−0.0180588
$378$	−4668.08	−0.635186
$379$	3459.63	0.468890	0.234445	−	0.972129i	$-0.424673\pi$
$379$	3459.63	0.468890	0.234445	+	0.972129i	$0.424673\pi$
$380$	−5903.42	−0.796945
$381$	3792.08	0.509906
$382$	−22055.5	−2.95407
$383$	14128.4	1.88492	0.942462	−	0.334315i	$-0.108505\pi$
$383$	14128.4	1.88492	0.942462	+	0.334315i	$0.108505\pi$
$384$	−13497.5	−1.79372
$385$	−391.005	−0.0517597
$386$	21027.7	2.77275
$387$	361.630	0.0475005
$388$	3717.89	0.486462
$389$	−13545.2	−1.76547	−0.882737	−	0.469868i	$-0.844301\pi$
$389$	−13545.2	−1.76547	−0.882737	+	0.469868i	$0.844301\pi$
$390$	286.504	0.0371992
$391$	0	0
$392$	−1893.44	−0.243962
$393$	−8217.62	−1.05477
$394$	−21455.7	−2.74345
$395$	−4882.13	−0.621890
$396$	274.044	0.0347758
$397$	−12475.6	−1.57716	−0.788582	−	0.614929i	$-0.789185\pi$
$397$	−12475.6	−1.57716	−0.788582	+	0.614929i	$0.789185\pi$
$398$	26485.3	3.33565
$399$	−3030.86	−0.380282
$400$	−6423.62	−0.802952
$401$	2649.19	0.329911	0.164955	−	0.986301i	$-0.447252\pi$
$401$	2649.19	0.329911	0.164955	+	0.986301i	$0.447252\pi$
$402$	23285.7	2.88902
$403$	496.394	0.0613577
$404$	18376.1	2.26298
$405$	3496.05	0.428939
$406$	−1880.35	−0.229852
$407$	−1024.17	−0.124732
$408$	0	0
$409$	9345.41	1.12983	0.564916	−	0.825149i	$-0.308909\pi$
$409$	9345.41	1.12983	0.564916	+	0.825149i	$0.308909\pi$
$410$	−679.403	−0.0818374
$411$	12094.7	1.45155
$412$	−18447.6	−2.20594
$413$	4852.55	0.578155
$414$	−461.978	−0.0548429
$415$	4500.13	0.532295
$416$	−18.0405	−0.00212622
$417$	1955.00	0.229585
$418$	−4854.85	−0.568083
$419$	13135.2	1.53150	0.765750	−	0.643138i	$-0.222368\pi$
$419$	13135.2	1.53150	0.765750	+	0.643138i	$0.222368\pi$
$420$	2711.44	0.315011
$421$	−4924.90	−0.570131	−0.285065	−	0.958508i	$-0.592015\pi$
$421$	−4924.90	−0.570131	−0.285065	+	0.958508i	$0.592015\pi$
$422$	24793.6	2.86003
$423$	28.5661	0.00328353
$424$	−662.334	−0.0758627
$425$	0	0
$426$	4702.40	0.534817
$427$	1946.05	0.220552
$428$	26967.9	3.04566
$429$	156.760	0.0176420
$430$	5731.32	0.642764
$431$	−6365.31	−0.711384	−0.355692	−	0.934603i	$-0.615755\pi$
$431$	−6365.31	−0.711384	−0.355692	+	0.934603i	$0.615755\pi$
$432$	8415.08	0.937201
$433$	1601.53	0.177747	0.0888735	−	0.996043i	$-0.471673\pi$
$433$	1601.53	0.177747	0.0888735	+	0.996043i	$0.471673\pi$
$434$	7060.97	0.780962
$435$	1338.19	0.147497
$436$	−7987.62	−0.877380
$437$	5445.14	0.596055
$438$	−7136.47	−0.778524
$439$	−983.541	−0.106929	−0.0534645	−	0.998570i	$-0.517026\pi$
$439$	−983.541	−0.106929	−0.0534645	+	0.998570i	$0.517026\pi$
$440$	2158.43	0.233862
$441$	69.0735	0.00745854
$442$	0	0
$443$	−153.603	−0.0164738	−0.00823688	−	0.999966i	$-0.502622\pi$
$443$	−153.603	−0.0164738	−0.00823688	+	0.999966i	$0.502622\pi$
$444$	7102.13	0.759127
$445$	2893.67	0.308254
$446$	7899.03	0.838632
$447$	−1901.78	−0.201233
$448$	−3711.53	−0.391414
$449$	11814.8	1.24182	0.620909	−	0.783883i	$-0.286764\pi$
$449$	11814.8	1.24182	0.620909	+	0.783883i	$0.286764\pi$
$450$	717.592	0.0751724
$451$	−371.734	−0.0388121
$452$	−21418.3	−2.22883
$453$	17407.6	1.80548
$454$	−24258.4	−2.50771
$455$	76.9598	0.00792952
$456$	16731.0	1.71820
$457$	13741.4	1.40655	0.703277	−	0.710916i	$-0.251720\pi$
$457$	13741.4	1.40655	0.703277	+	0.710916i	$0.251720\pi$
$458$	356.527	0.0363742
$459$	0	0
$460$	−4871.28	−0.493749
$461$	−5182.54	−0.523590	−0.261795	−	0.965123i	$-0.584314\pi$
$461$	−5182.54	−0.523590	−0.261795	+	0.965123i	$0.584314\pi$
$462$	2229.83	0.224548
$463$	5240.52	0.526021	0.263011	−	0.964793i	$-0.415285\pi$
$463$	5240.52	0.526021	0.263011	+	0.964793i	$0.415285\pi$
$464$	3389.67	0.339142
$465$	−5025.07	−0.501145
$466$	−19456.5	−1.93413
$467$	−4002.51	−0.396604	−0.198302	−	0.980141i	$-0.563543\pi$
$467$	−4002.51	−0.396604	−0.198302	+	0.980141i	$0.563543\pi$
$468$	−53.9388	−0.00532761
$469$	6254.95	0.615835
$470$	452.732	0.0444319
$471$	8882.37	0.868955
$472$	−26787.1	−2.61224
$473$	3135.88	0.304837
$474$	27841.9	2.69794
$475$	−8457.95	−0.817005
$476$	0	0
$477$	24.1623	0.00231932
$478$	11710.3	1.12054
$479$	11572.2	1.10386	0.551929	−	0.833891i	$-0.313892\pi$
$479$	11572.2	1.10386	0.551929	+	0.833891i	$0.313892\pi$
$480$	182.626	0.0173661
$481$	201.582	0.0191089
$482$	18867.9	1.78300
$483$	−2500.95	−0.235605
$484$	−18791.3	−1.76477
$485$	−1068.26	−0.100015
$486$	−1931.91	−0.180315
$487$	14012.6	1.30384	0.651920	−	0.758288i	$-0.273964\pi$
$487$	14012.6	1.30384	0.651920	+	0.758288i	$0.273964\pi$
$488$	−10742.6	−0.996506
$489$	−19907.9	−1.84104
$490$	1094.72	0.100927
$491$	5225.13	0.480259	0.240129	−	0.970741i	$-0.422810\pi$
$491$	5225.13	0.480259	0.240129	+	0.970741i	$0.422810\pi$
$492$	2577.80	0.236212
$493$	0	0
$494$	955.559	0.0870296
$495$	−78.7408	−0.00714977
$496$	−12728.7	−1.15229
$497$	1263.15	0.114004
$498$	−25663.4	−2.30925
$499$	−3252.73	−0.291808	−0.145904	−	0.989299i	$-0.546609\pi$
$499$	−3252.73	−0.291808	−0.145904	+	0.989299i	$0.546609\pi$
$500$	16650.6	1.48928
$501$	−2881.48	−0.256956
$502$	25359.7	2.25470
$503$	3033.67	0.268916	0.134458	−	0.990919i	$-0.457071\pi$
$503$	3033.67	0.268916	0.134458	+	0.990919i	$0.457071\pi$
$504$	−381.301	−0.0336994
$505$	−5280.00	−0.465261
$506$	−4006.04	−0.351957
$507$	11679.3	1.02307
$508$	−11314.6	−0.988196
$509$	−1536.89	−0.133834	−0.0669171	−	0.997759i	$-0.521316\pi$
$509$	−1536.89	−0.133834	−0.0669171	+	0.997759i	$0.521316\pi$
$510$	0	0
$511$	−1916.98	−0.165953
$512$	19534.5	1.68615
$513$	11080.1	0.953603
$514$	8877.92	0.761845
$515$	5300.54	0.453534
$516$	−21745.8	−1.85525
$517$	247.711	0.0210722
$518$	2867.41	0.243218
$519$	6655.72	0.562917
$520$	−424.835	−0.0358274
$521$	2453.24	0.206293	0.103146	−	0.994666i	$-0.467109\pi$
$521$	2453.24	0.206293	0.103146	+	0.994666i	$0.467109\pi$
$522$	−378.665	−0.0317505
$523$	15144.2	1.26617	0.633087	−	0.774080i	$-0.281787\pi$
$523$	15144.2	1.26617	0.633087	+	0.774080i	$0.281787\pi$
$524$	24519.3	2.04414
$525$	3884.74	0.322941
$526$	15014.0	1.24457
$527$	0	0
$528$	−4019.69	−0.331315
$529$	−7673.88	−0.630712
$530$	382.938	0.0313845
$531$	977.209	0.0798630
$532$	9043.30	0.736987
$533$	73.1667	0.00594597
$534$	−16502.1	−1.33729
$535$	−7748.68	−0.626177
$536$	−34528.7	−2.78249
$537$	16558.2	1.33061
$538$	26399.7	2.11556
$539$	598.971	0.0478655
$540$	−9912.38	−0.789928
$541$	−3932.38	−0.312507	−0.156253	−	0.987717i	$-0.549942\pi$
$541$	−3932.38	−0.312507	−0.156253	+	0.987717i	$0.549942\pi$
$542$	−22261.4	−1.76422
$543$	3346.88	0.264509
$544$	0	0
$545$	2295.08	0.180386
$546$	−438.888	−0.0344005
$547$	8651.68	0.676269	0.338135	−	0.941098i	$-0.390204\pi$
$547$	8651.68	0.676269	0.338135	+	0.941098i	$0.390204\pi$
$548$	−36087.4	−2.81309
$549$	391.896	0.0304658
$550$	6222.60	0.482423
$551$	4463.17	0.345077
$552$	13805.8	1.06452
$553$	7478.82	0.575103
$554$	−15123.8	−1.15983
$555$	−2040.65	−0.156073
$556$	−5833.22	−0.444935
$557$	1534.71	0.116746	0.0583732	−	0.998295i	$-0.481409\pi$
$557$	1534.71	0.116746	0.0583732	+	0.998295i	$0.481409\pi$
$558$	1421.94	0.107877
$559$	−617.220	−0.0467006
$560$	−1973.43	−0.148915
$561$	0	0
$562$	−8618.27	−0.646868
$563$	−25084.0	−1.87774	−0.938868	−	0.344277i	$-0.888124\pi$
$563$	−25084.0	−1.87774	−0.938868	+	0.344277i	$0.888124\pi$
$564$	−1717.76	−0.128246
$565$	6154.10	0.458239
$566$	1341.17	0.0995997
$567$	−5355.52	−0.396668
$568$	−6972.84	−0.515095
$569$	−3239.35	−0.238665	−0.119333	−	0.992854i	$-0.538075\pi$
$569$	−3239.35	−0.238665	−0.119333	+	0.992854i	$0.538075\pi$
$570$	−9673.27	−0.710822
$571$	−2403.43	−0.176148	−0.0880739	−	0.996114i	$-0.528071\pi$
$571$	−2403.43	−0.176148	−0.0880739	+	0.996114i	$0.528071\pi$
$572$	−467.731	−0.0341902
$573$	−24044.6	−1.75302
$574$	1040.76	0.0756804
$575$	−6979.18	−0.506178
$576$	−747.430	−0.0540676
$577$	−15163.7	−1.09406	−0.547029	−	0.837114i	$-0.684241\pi$
$577$	−15163.7	−1.09406	−0.547029	+	0.837114i	$0.684241\pi$
$578$	0	0
$579$	22924.1	1.64541
$580$	−3992.80	−0.285848
$581$	−6893.64	−0.492248
$582$	6092.09	0.433892
$583$	209.523	0.0148844
$584$	10582.1	0.749815
$585$	15.4982	0.00109534
$586$	−21882.1	−1.54256
$587$	−1892.33	−0.133058	−0.0665289	−	0.997784i	$-0.521192\pi$
$587$	−1892.33	−0.133058	−0.0665289	+	0.997784i	$0.521192\pi$
$588$	−4153.59	−0.291312
$589$	−16759.8	−1.17246
$590$	15487.4	1.08069
$591$	−23390.7	−1.62803
$592$	−5169.04	−0.358862
$593$	−10086.6	−0.698494	−0.349247	−	0.937031i	$-0.613563\pi$
$593$	−10086.6	−0.698494	−0.349247	+	0.937031i	$0.613563\pi$
$594$	−8151.74	−0.563081
$595$	0	0
$596$	5674.42	0.389989
$597$	28874.0	1.97945
$598$	788.492	0.0539194
$599$	−7544.15	−0.514600	−0.257300	−	0.966332i	$-0.582833\pi$
$599$	−7544.15	−0.514600	−0.257300	+	0.966332i	$0.582833\pi$
$600$	−21444.6	−1.45912
$601$	10158.8	0.689495	0.344748	−	0.938695i	$-0.387964\pi$
$601$	10158.8	0.689495	0.344748	+	0.938695i	$0.387964\pi$
$602$	−8779.67	−0.594406
$603$	1259.63	0.0850678
$604$	−51939.8	−3.49901
$605$	5399.28	0.362830
$606$	30110.9	2.01843
$607$	−7954.41	−0.531894	−0.265947	−	0.963988i	$-0.585685\pi$
$607$	−7954.41	−0.531894	−0.265947	+	0.963988i	$0.585685\pi$
$608$	609.103	0.0406289
$609$	−2049.93	−0.136400
$610$	6210.99	0.412255
$611$	−48.7559	−0.00322823
$612$	0	0
$613$	20246.8	1.33403	0.667016	−	0.745043i	$-0.267571\pi$
$613$	20246.8	1.33403	0.667016	+	0.745043i	$0.267571\pi$
$614$	−36844.2	−2.42168
$615$	−740.678	−0.0485642
$616$	−3306.45	−0.216268
$617$	9349.12	0.610019	0.305009	−	0.952349i	$-0.401340\pi$
$617$	9349.12	0.610019	0.305009	+	0.952349i	$0.401340\pi$
$618$	−30228.1	−1.96756
$619$	−12426.7	−0.806902	−0.403451	−	0.915001i	$-0.632189\pi$
$619$	−12426.7	−0.806902	−0.403451	+	0.915001i	$0.632189\pi$
$620$	14993.5	0.971217
$621$	9142.88	0.590807
$622$	−50940.2	−3.28379
$623$	−4432.74	−0.285063
$624$	791.177	0.0507571
$625$	8230.68	0.526763
$626$	−10948.7	−0.699038
$627$	−5292.70	−0.337114
$628$	−26502.7	−1.68403
$629$	0	0
$630$	220.455	0.0139415
$631$	−480.557	−0.0303180	−0.0151590	−	0.999885i	$-0.504825\pi$
$631$	−480.557	−0.0303180	−0.0151590	+	0.999885i	$0.504825\pi$
$632$	−41284.7	−2.59845
$633$	27029.7	1.69721
$634$	8489.74	0.531815
$635$	3251.01	0.203169
$636$	−1452.95	−0.0905867
$637$	−117.893	−0.00733295
$638$	−3283.60	−0.203760
$639$	254.373	0.0157478
$640$	−11571.6	−0.714699
$641$	7237.13	0.445943	0.222971	−	0.974825i	$-0.428424\pi$
$641$	7237.13	0.445943	0.222971	+	0.974825i	$0.428424\pi$
$642$	44189.4	2.71653
$643$	−24889.7	−1.52652	−0.763262	−	0.646089i	$-0.776403\pi$
$643$	−24889.7	−1.52652	−0.763262	+	0.646089i	$0.776403\pi$
$644$	7462.20	0.456602
$645$	6248.22	0.381432
$646$	0	0
$647$	1948.66	0.118408	0.0592039	−	0.998246i	$-0.481144\pi$
$647$	1948.66	0.118408	0.0592039	+	0.998246i	$0.481144\pi$
$648$	29563.6	1.79224
$649$	8473.87	0.512525
$650$	−1224.77	−0.0739066
$651$	7697.79	0.463441
$652$	59400.1	3.56792
$653$	17239.8	1.03315	0.516574	−	0.856243i	$-0.327207\pi$
$653$	17239.8	1.03315	0.516574	+	0.856243i	$0.327207\pi$
$654$	−13088.4	−0.782566
$655$	−7045.10	−0.420267
$656$	−1876.16	−0.111664
$657$	−386.042	−0.0229238
$658$	−693.529	−0.0410890
$659$	31017.3	1.83348	0.916739	−	0.399486i	$-0.130811\pi$
$659$	31017.3	1.83348	0.916739	+	0.399486i	$0.130811\pi$
$660$	4734.91	0.279252
$661$	−26768.9	−1.57517	−0.787587	−	0.616203i	$-0.788670\pi$
$661$	−26768.9	−1.57517	−0.787587	+	0.616203i	$0.788670\pi$
$662$	17542.1	1.02990
$663$	0	0
$664$	38054.4	2.22409
$665$	−2598.41	−0.151522
$666$	577.441	0.0335967
$667$	3682.84	0.213793
$668$	8597.59	0.497980
$669$	8611.44	0.497664
$670$	19963.3	1.15112
$671$	3398.33	0.195516
$672$	−279.761	−0.0160596
$673$	23372.0	1.33867	0.669334	−	0.742961i	$-0.266579\pi$
$673$	23372.0	1.33867	0.669334	+	0.742961i	$0.266579\pi$
$674$	−25019.5	−1.42984
$675$	−14201.7	−0.809811
$676$	−34848.1	−1.98271
$677$	−1225.58	−0.0695761	−0.0347880	−	0.999395i	$-0.511076\pi$
$677$	−1225.58	−0.0695761	−0.0347880	+	0.999395i	$0.511076\pi$
$678$	−35095.8	−1.98797
$679$	1636.44	0.0924901
$680$	0	0
$681$	−26446.2	−1.48814
$682$	12330.4	0.692309
$683$	8494.76	0.475904	0.237952	−	0.971277i	$-0.423524\pi$
$683$	8494.76	0.475904	0.237952	+	0.971277i	$0.423524\pi$
$684$	1821.15	0.101803
$685$	10369.0	0.578361
$686$	−1676.97	−0.0933339
$687$	388.681	0.0215853
$688$	15827.0	0.877032
$689$	−41.2396	−0.00228027
$690$	−7982.02	−0.440392
$691$	−26061.7	−1.43478	−0.717390	−	0.696671i	$-0.754664\pi$
$691$	−26061.7	−1.43478	−0.717390	+	0.696671i	$0.754664\pi$
$692$	−19859.0	−1.09093
$693$	120.621	0.00661186
$694$	−38073.6	−2.08250
$695$	1676.05	0.0914768
$696$	11316.1	0.616286
$697$	0	0
$698$	33718.7	1.82847
$699$	−21211.3	−1.14776
$700$	−11591.1	−0.625858
$701$	13770.4	0.741942	0.370971	−	0.928644i	$-0.379025\pi$
$701$	13770.4	0.741942	0.370971	+	0.928644i	$0.379025\pi$
$702$	1604.47	0.0862633
$703$	−6806.06	−0.365143
$704$	−6481.34	−0.346981
$705$	493.563	0.0263669
$706$	−18455.4	−0.983822
$707$	8088.30	0.430257
$708$	−58762.4	−3.11924
$709$	17007.7	0.900902	0.450451	−	0.892801i	$-0.351263\pi$
$709$	17007.7	0.900902	0.450451	+	0.892801i	$0.351263\pi$
$710$	4031.45	0.213095
$711$	1506.09	0.0794413
$712$	24469.7	1.28798
$713$	−13829.6	−0.726398
$714$	0	0
$715$	134.393	0.00702938
$716$	−49405.4	−2.57872
$717$	12766.5	0.664955
$718$	31729.4	1.64921
$719$	−27197.5	−1.41070	−0.705351	−	0.708858i	$-0.749211\pi$
$719$	−27197.5	−1.41070	−0.705351	+	0.708858i	$0.749211\pi$
$720$	−397.410	−0.0205703
$721$	−8119.77	−0.419412
$722$	1271.84	0.0655581
$723$	20569.5	1.05808
$724$	−9986.24	−0.512618
$725$	−5720.57	−0.293044
$726$	−30791.2	−1.57406
$727$	12605.2	0.643054	0.321527	−	0.946900i	$-0.395804\pi$
$727$	12605.2	0.643054	0.321527	+	0.946900i	$0.395804\pi$
$728$	650.795	0.0331319
$729$	18550.9	0.942481
$730$	−6118.21	−0.310199
$731$	0	0
$732$	−23565.8	−1.18992
$733$	−8015.42	−0.403897	−0.201948	−	0.979396i	$-0.564727\pi$
$733$	−8015.42	−0.403897	−0.201948	+	0.979396i	$0.564727\pi$
$734$	−59074.4	−2.97067
$735$	1193.45	0.0598925
$736$	502.609	0.0251718
$737$	10922.8	0.545927
$738$	209.589	0.0104540
$739$	35097.9	1.74709	0.873544	−	0.486746i	$-0.161816\pi$
$739$	35097.9	1.74709	0.873544	+	0.486746i	$0.161816\pi$
$740$	6088.78	0.302470
$741$	1041.74	0.0516454
$742$	−586.614	−0.0290233
$743$	−1506.29	−0.0743749	−0.0371875	−	0.999308i	$-0.511840\pi$
$743$	−1506.29	−0.0743749	−0.0371875	+	0.999308i	$0.511840\pi$
$744$	−42493.5	−2.09393
$745$	−1630.43	−0.0801801
$746$	−23451.4	−1.15096
$747$	−1388.24	−0.0679963
$748$	0	0
$749$	11870.0	0.579067
$750$	27283.5	1.32834
$751$	8011.24	0.389260	0.194630	−	0.980877i	$-0.437649\pi$
$751$	8011.24	0.389260	0.194630	+	0.980877i	$0.437649\pi$
$752$	1250.21	0.0606258
$753$	27646.9	1.33799
$754$	646.296	0.0312158
$755$	14923.8	0.719382
$756$	15184.5	0.730498
$757$	6688.33	0.321125	0.160562	−	0.987026i	$-0.448669\pi$
$757$	6688.33	0.321125	0.160562	+	0.987026i	$0.448669\pi$
$758$	−16914.6	−0.810508
$759$	−4367.34	−0.208860
$760$	14343.8	0.684610
$761$	3293.24	0.156872	0.0784362	−	0.996919i	$-0.475007\pi$
$761$	3293.24	0.156872	0.0784362	+	0.996919i	$0.475007\pi$
$762$	−18540.0	−0.881406
$763$	−3515.78	−0.166815
$764$	71742.9	3.39734
$765$	0	0
$766$	−69075.4	−3.25822
$767$	−1667.87	−0.0785182
$768$	43381.9	2.03830
$769$	27195.0	1.27526	0.637631	−	0.770342i	$-0.279914\pi$
$769$	27195.0	1.27526	0.637631	+	0.770342i	$0.279914\pi$
$770$	1911.67	0.0894701
$771$	9678.61	0.452097
$772$	−68399.7	−3.18881
$773$	2088.57	0.0971805	0.0485902	−	0.998819i	$-0.484527\pi$
$773$	2088.57	0.0971805	0.0485902	+	0.998819i	$0.484527\pi$
$774$	−1768.05	−0.0821078
$775$	21481.5	0.995664
$776$	−9033.50	−0.417892
$777$	3126.02	0.144331
$778$	66224.2	3.05174
$779$	−2470.34	−0.113619
$780$	−931.952	−0.0427811
$781$	2205.80	0.101062
$782$	0	0
$783$	7494.07	0.342039
$784$	3023.05	0.137712
$785$	7615.01	0.346231
$786$	40177.0	1.82324
$787$	3292.28	0.149120	0.0745598	−	0.997217i	$-0.476245\pi$
$787$	3292.28	0.149120	0.0745598	+	0.997217i	$0.476245\pi$
$788$	69791.9	3.15512
$789$	16368.1	0.738556
$790$	23869.4	1.07498
$791$	−9427.32	−0.423764
$792$	−665.856	−0.0298739
$793$	−668.878	−0.0299528
$794$	60995.0	2.72624
$795$	417.475	0.0186243
$796$	−86152.5	−3.83618
$797$	−8037.18	−0.357204	−0.178602	−	0.983921i	$-0.557157\pi$
$797$	−8037.18	−0.357204	−0.178602	+	0.983921i	$0.557157\pi$
$798$	14818.3	0.657344
$799$	0	0
$800$	−780.705	−0.0345026
$801$	−892.668	−0.0393769
$802$	−12952.2	−0.570273
$803$	−3347.56	−0.147114
$804$	−75744.9	−3.32253
$805$	−2144.11	−0.0938756
$806$	−2426.93	−0.106061
$807$	28780.6	1.25542
$808$	−44649.2	−1.94400
$809$	−1871.47	−0.0813317	−0.0406658	−	0.999173i	$-0.512948\pi$
$809$	−1871.47	−0.0813317	−0.0406658	+	0.999173i	$0.512948\pi$
$810$	−17092.6	−0.741450
$811$	−12499.0	−0.541181	−0.270591	−	0.962695i	$-0.587219\pi$
$811$	−12499.0	−0.541181	−0.270591	+	0.962695i	$0.587219\pi$
$812$	6116.48	0.264343
$813$	−24269.1	−1.04693
$814$	5007.29	0.215608
$815$	−17067.4	−0.733551
$816$	0	0
$817$	20839.3	0.892381
$818$	−45690.9	−1.95299
$819$	−23.7413	−0.00101293
$820$	2209.99	0.0941174
$821$	38596.2	1.64070	0.820351	−	0.571860i	$-0.193778\pi$
$821$	38596.2	1.64070	0.820351	+	0.571860i	$0.193778\pi$
$822$	−59132.3	−2.50910
$823$	−30277.9	−1.28241	−0.641204	−	0.767370i	$-0.721565\pi$
$823$	−30277.9	−1.28241	−0.641204	+	0.767370i	$0.721565\pi$
$824$	44822.9	1.89500
$825$	6783.81	0.286281
$826$	−23724.7	−0.999381
$827$	29636.7	1.24615	0.623076	−	0.782161i	$-0.285883\pi$
$827$	29636.7	1.24615	0.623076	+	0.782161i	$0.285883\pi$
$828$	1502.74	0.0630723
$829$	−21176.6	−0.887207	−0.443603	−	0.896223i	$-0.646300\pi$
$829$	−21176.6	−0.887207	−0.443603	+	0.896223i	$0.646300\pi$
$830$	−22001.7	−0.920108
$831$	−16487.7	−0.688272
$832$	1275.69	0.0531571
$833$	0	0
$834$	−9558.25	−0.396852
$835$	−2470.34	−0.102383
$836$	15792.1	0.653326
$837$	−28141.3	−1.16213
$838$	−64219.9	−2.64730
$839$	1007.38	0.0414526	0.0207263	−	0.999785i	$-0.493402\pi$
$839$	1007.38	0.0414526	0.0207263	+	0.999785i	$0.493402\pi$
$840$	−6588.10	−0.270608
$841$	−21370.3	−0.876228
$842$	24078.5	0.985509
$843$	−9395.54	−0.383867
$844$	−80649.6	−3.28919
$845$	10012.9	0.407637
$846$	−139.663	−0.00567580
$847$	−8271.03	−0.335533
$848$	1057.48	0.0428231
$849$	1462.13	0.0591048
$850$	0	0
$851$	−5616.11	−0.226225
$852$	−15296.2	−0.615069
$853$	46656.5	1.87279	0.936393	−	0.350952i	$-0.114142\pi$
$853$	46656.5	1.87279	0.936393	+	0.350952i	$0.114142\pi$
$854$	−9514.47	−0.381240
$855$	−523.268	−0.0209303
$856$	−65525.1	−2.61636
$857$	−30696.9	−1.22356	−0.611778	−	0.791030i	$-0.709545\pi$
$857$	−30696.9	−1.22356	−0.611778	+	0.791030i	$0.709545\pi$
$858$	−766.418	−0.0304954
$859$	18499.8	0.734815	0.367408	−	0.930060i	$-0.380245\pi$
$859$	18499.8	0.734815	0.367408	+	0.930060i	$0.380245\pi$
$860$	−18643.1	−0.739214
$861$	1134.63	0.0449105
$862$	31120.8	1.22967
$863$	−36103.3	−1.42407	−0.712035	−	0.702144i	$-0.752226\pi$
$863$	−36103.3	−1.42407	−0.712035	+	0.702144i	$0.752226\pi$
$864$	1022.74	0.0402712
$865$	5706.07	0.224291
$866$	−7830.07	−0.307248
$867$	0	0
$868$	−22968.2	−0.898148
$869$	13060.1	0.509818
$870$	−6542.56	−0.254958
$871$	−2149.90	−0.0836354
$872$	19407.9	0.753707
$873$	329.547	0.0127760
$874$	−26622.0	−1.03032
$875$	7328.82	0.283153
$876$	23213.8	0.895344
$877$	7764.50	0.298961	0.149480	−	0.988765i	$-0.452240\pi$
$877$	7764.50	0.298961	0.149480	+	0.988765i	$0.452240\pi$
$878$	4808.66	0.184834
$879$	−23855.6	−0.915394
$880$	−3446.15	−0.132011
$881$	10059.7	0.384700	0.192350	−	0.981326i	$-0.438389\pi$
$881$	10059.7	0.384700	0.192350	+	0.981326i	$0.438389\pi$
$882$	−337.709	−0.0128926
$883$	−37488.9	−1.42877	−0.714383	−	0.699755i	$-0.753293\pi$
$883$	−37488.9	−1.42877	−0.714383	+	0.699755i	$0.753293\pi$
$884$	0	0
$885$	16884.1	0.641304
$886$	750.983	0.0284760
$887$	9284.73	0.351466	0.175733	−	0.984438i	$-0.443770\pi$
$887$	9284.73	0.351466	0.175733	+	0.984438i	$0.443770\pi$
$888$	−17256.3	−0.652122
$889$	−4980.15	−0.187884
$890$	−14147.5	−0.532838
$891$	−9352.19	−0.351639
$892$	−25694.3	−0.964472
$893$	1646.15	0.0616869
$894$	9298.03	0.347844
$895$	14195.6	0.530176
$896$	17726.2	0.660929
$897$	859.605	0.0319971
$898$	−57764.2	−2.14657
$899$	−11335.6	−0.420537
$900$	−2334.21	−0.0864523
$901$	0	0
$902$	1817.45	0.0670893
$903$	−9571.50	−0.352735
$904$	52040.9	1.91466
$905$	2869.34	0.105392
$906$	−85108.0	−3.12089
$907$	18911.5	0.692334	0.346167	−	0.938173i	$-0.387483\pi$
$907$	18911.5	0.692334	0.346167	+	0.938173i	$0.387483\pi$
$908$	78908.7	2.88401
$909$	1628.83	0.0594332
$910$	−376.266	−0.0137067
$911$	37184.7	1.35234	0.676171	−	0.736745i	$-0.263638\pi$
$911$	37184.7	1.35234	0.676171	+	0.736745i	$0.263638\pi$
$912$	−26712.6	−0.969895
$913$	−12038.2	−0.436369
$914$	−67183.4	−2.43132
$915$	6771.16	0.244642
$916$	−1159.73	−0.0418323
$917$	10792.2	0.388648
$918$	0	0
$919$	−10489.0	−0.376495	−0.188247	−	0.982122i	$-0.560281\pi$
$919$	−10489.0	−0.376495	−0.188247	+	0.982122i	$0.560281\pi$
$920$	11835.9	0.424152
$921$	−40167.1	−1.43708
$922$	25338.1	0.905061
$923$	−434.157	−0.0154826
$924$	−7253.30	−0.258242
$925$	8723.52	0.310084
$926$	−25621.6	−0.909263
$927$	−1635.16	−0.0579351
$928$	411.970	0.0145728
$929$	−47367.3	−1.67284	−0.836421	−	0.548088i	$-0.815356\pi$
$929$	−47367.3	−1.67284	−0.836421	+	0.548088i	$0.815356\pi$
$930$	24568.2	0.866262
$931$	3980.44	0.140122
$932$	63289.0	2.22436
$933$	−55534.5	−1.94868
$934$	19568.8	0.685558
$935$	0	0
$936$	131.057	0.00457665
$937$	30379.3	1.05918	0.529588	−	0.848255i	$-0.322347\pi$
$937$	30379.3	1.05918	0.529588	+	0.848255i	$0.322347\pi$
$938$	−30581.2	−1.06451
$939$	−11936.1	−0.414826
$940$	−1472.67	−0.0510990
$941$	−41609.5	−1.44148	−0.720740	−	0.693206i	$-0.756198\pi$
$941$	−41609.5	−1.44148	−0.720740	+	0.693206i	$0.756198\pi$
$942$	−43427.0	−1.50205
$943$	−2038.43	−0.0703928
$944$	42768.2	1.47456
$945$	−4362.96	−0.150188
$946$	−15331.7	−0.526931
$947$	29893.9	1.02579	0.512893	−	0.858452i	$-0.328574\pi$
$947$	29893.9	1.02579	0.512893	+	0.858452i	$0.328574\pi$
$948$	−90565.4	−3.10277
$949$	658.886	0.0225378
$950$	41352.0	1.41225
$951$	9255.42	0.315592
$952$	0	0
$953$	22595.5	0.768039	0.384020	−	0.923325i	$-0.374540\pi$
$953$	22595.5	0.768039	0.384020	+	0.923325i	$0.374540\pi$
$954$	−118.133	−0.00400910
$955$	−20613.9	−0.698480
$956$	−38091.9	−1.28868
$957$	−3579.74	−0.120916
$958$	−56578.0	−1.90809
$959$	−15884.0	−0.534848
$960$	−12914.1	−0.434166
$961$	12775.7	0.428845
$962$	−985.562	−0.0330310
$963$	2390.39	0.0799888
$964$	−61374.2	−2.05055
$965$	19653.2	0.655607
$966$	12227.5	0.407259
$967$	−10872.7	−0.361574	−0.180787	−	0.983522i	$-0.557864\pi$
$967$	−10872.7	−0.361574	−0.180787	+	0.983522i	$0.557864\pi$
$968$	45657.9	1.51601
$969$	0	0
$970$	5222.85	0.172882
$971$	−11270.8	−0.372501	−0.186251	−	0.982502i	$-0.559634\pi$
$971$	−11270.8	−0.372501	−0.186251	+	0.982502i	$0.559634\pi$
$972$	6284.19	0.207372
$973$	−2567.51	−0.0845946
$974$	−68509.2	−2.25377
$975$	−1335.23	−0.0438579
$976$	17151.6	0.562509
$977$	40400.9	1.32297	0.661484	−	0.749959i	$-0.269927\pi$
$977$	40400.9	1.32297	0.661484	+	0.749959i	$0.269927\pi$
$978$	97332.3	3.18235
$979$	−7740.77	−0.252703
$980$	−3560.94	−0.116072
$981$	−708.009	−0.0230428
$982$	−25546.3	−0.830159
$983$	11094.4	0.359975	0.179987	−	0.983669i	$-0.442394\pi$
$983$	11094.4	0.359975	0.179987	+	0.983669i	$0.442394\pi$
$984$	−6263.39	−0.202916
$985$	−20053.3	−0.648680
$986$	0	0
$987$	−756.078	−0.0243832
$988$	−3108.28	−0.100089
$989$	17195.8	0.552877
$990$	384.974	0.0123589
$991$	14523.4	0.465542	0.232771	−	0.972532i	$-0.425221\pi$
$991$	14523.4	0.465542	0.232771	+	0.972532i	$0.425221\pi$
$992$	−1547.00	−0.0495135
$993$	19124.2	0.611167
$994$	−6175.68	−0.197063
$995$	24754.2	0.788703
$996$	83479.1	2.65576
$997$	−20326.8	−0.645694	−0.322847	−	0.946451i	$-0.604640\pi$
$997$	−20326.8	−0.645694	−0.322847	+	0.946451i	$0.604640\pi$
$998$	15903.0	0.504411
$999$	−11428.0	−0.361928

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2023.4.a.v.1.5		56
17.5	odd	16		119.4.k.a.8.26	✓	112
17.7	odd	16		119.4.k.a.15.26	yes	112
17.16	even	2		2023.4.a.u.1.5		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
119.4.k.a.8.26	✓	112	17.5	odd	16
119.4.k.a.15.26	yes	112	17.7	odd	16
2023.4.a.u.1.5		56	17.16	even	2
2023.4.a.v.1.5		56	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2023 = 7 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2023.a (trivial)

\(f(q)\)	\(=\)	\(q-4.88913 q^{2} -5.33007 q^{3} +15.9036 q^{4} -4.56956 q^{5} +26.0594 q^{6} +7.00000 q^{7} -38.6415 q^{8} +1.40966 q^{9} +22.3412 q^{10} +12.2239 q^{11} -84.7671 q^{12} -2.40598 q^{13} -34.2239 q^{14} +24.3561 q^{15} +61.6949 q^{16} -6.89203 q^{18} +81.2334 q^{19} -72.6723 q^{20} -37.3105 q^{21} -59.7642 q^{22} +67.0308 q^{23} +205.962 q^{24} -104.119 q^{25} +11.7631 q^{26} +136.398 q^{27} +111.325 q^{28} +54.9426 q^{29} -119.080 q^{30} -206.317 q^{31} +7.49819 q^{32} -65.1543 q^{33} -31.9869 q^{35} +22.4187 q^{36} -83.7840 q^{37} -397.160 q^{38} +12.8240 q^{39} +176.575 q^{40} -30.4104 q^{41} +182.416 q^{42} +256.536 q^{43} +194.404 q^{44} -6.44154 q^{45} -327.722 q^{46} +20.2645 q^{47} -328.838 q^{48} +49.0000 q^{49} +509.052 q^{50} -38.2636 q^{52} +17.1405 q^{53} -666.869 q^{54} -55.8579 q^{55} -270.491 q^{56} -432.980 q^{57} -268.621 q^{58} +693.221 q^{59} +387.349 q^{60} +278.007 q^{61} +1008.71 q^{62} +9.86765 q^{63} -530.219 q^{64} +10.9943 q^{65} +318.548 q^{66} +893.564 q^{67} -357.279 q^{69} +156.388 q^{70} +180.449 q^{71} -54.4716 q^{72} -273.854 q^{73} +409.631 q^{74} +554.962 q^{75} +1291.90 q^{76} +85.5673 q^{77} -62.6983 q^{78} +1068.40 q^{79} -281.919 q^{80} -765.074 q^{81} +148.680 q^{82} -984.805 q^{83} -593.370 q^{84} -1254.24 q^{86} -292.848 q^{87} -472.351 q^{88} -633.249 q^{89} +31.4935 q^{90} -16.8418 q^{91} +1066.03 q^{92} +1099.68 q^{93} -99.0756 q^{94} -371.201 q^{95} -39.9659 q^{96} +233.777 q^{97} -239.567 q^{98} +17.2316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 8 q^{2} + 24 q^{3} + 240 q^{4} + 80 q^{5} + 68 q^{6} + 392 q^{7} + 96 q^{8} + 576 q^{9} + 80 q^{10} + 176 q^{11} + 288 q^{12} - 96 q^{13} + 56 q^{14} + 192 q^{15} + 1088 q^{16} + 216 q^{18} + 48 q^{19}+ \cdots + 6576 q^{99}+O(q^{100}) \) 56 * q + 8 * q^2 + 24 * q^3 + 240 * q^4 + 80 * q^5 + 68 * q^6 + 392 * q^7 + 96 * q^8 + 576 * q^9 + 80 * q^10 + 176 * q^11 + 288 * q^12 - 96 * q^13 + 56 * q^14 + 192 * q^15 + 1088 * q^16 + 216 * q^18 + 48 * q^19 + 1660 * q^20 + 168 * q^21 + 352 * q^22 + 608 * q^23 + 736 * q^24 + 1792 * q^25 + 864 * q^27 + 1680 * q^28 + 928 * q^29 - 100 * q^30 + 496 * q^31 + 436 * q^32 + 384 * q^33 + 560 * q^35 + 2804 * q^36 + 1392 * q^37 + 624 * q^39 - 1020 * q^40 + 2088 * q^41 + 476 * q^42 + 1312 * q^43 + 2112 * q^44 + 4840 * q^45 + 736 * q^46 + 5228 * q^48 + 2744 * q^49 + 2020 * q^50 - 1536 * q^52 + 1272 * q^53 + 2268 * q^54 - 336 * q^55 + 672 * q^56 + 2448 * q^57 + 1376 * q^58 + 2372 * q^60 + 976 * q^61 + 3776 * q^62 + 4032 * q^63 + 5376 * q^64 + 6080 * q^65 + 976 * q^67 + 1248 * q^69 + 560 * q^70 + 3216 * q^71 + 4532 * q^72 + 4088 * q^73 - 2456 * q^74 - 760 * q^75 - 2640 * q^76 + 1232 * q^77 + 3744 * q^78 + 1792 * q^79 + 13920 * q^80 + 4184 * q^81 + 8204 * q^82 - 2832 * q^83 + 2016 * q^84 + 4908 * q^86 + 832 * q^87 + 2424 * q^88 - 2768 * q^89 + 8660 * q^90 - 672 * q^91 + 6784 * q^92 + 784 * q^93 - 928 * q^94 + 4560 * q^95 + 6656 * q^96 + 9464 * q^97 + 392 * q^98 + 6576 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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