LMFDB - Embedded newform 624.4.a.i.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [624,4,Mod(1,624)] 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("624.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(624, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,6,0,-20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$36.8171918436$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	624.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} +6.00000 q^{5} -20.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} +13.0000 q^{13} +18.0000 q^{15} -30.0000 q^{17} +16.0000 q^{19} -60.0000 q^{21} +72.0000 q^{23} -89.0000 q^{25} +27.0000 q^{27} -282.000 q^{29} -164.000 q^{31} -72.0000 q^{33} -120.000 q^{35} +110.000 q^{37} +39.0000 q^{39} -126.000 q^{41} -164.000 q^{43} +54.0000 q^{45} +204.000 q^{47} +57.0000 q^{49} -90.0000 q^{51} -738.000 q^{53} -144.000 q^{55} +48.0000 q^{57} -120.000 q^{59} +614.000 q^{61} -180.000 q^{63} +78.0000 q^{65} -848.000 q^{67} +216.000 q^{69} -132.000 q^{71} +218.000 q^{73} -267.000 q^{75} +480.000 q^{77} +1096.00 q^{79} +81.0000 q^{81} -552.000 q^{83} -180.000 q^{85} -846.000 q^{87} +210.000 q^{89} -260.000 q^{91} -492.000 q^{93} +96.0000 q^{95} -1726.00 q^{97} -216.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	−20.0000	−1.07990	−0.539949	−	0.841698i	$-0.681557\pi$
$7$	−20.0000	−1.07990	−0.539949	+	0.841698i	$0.681557\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−24.0000	−0.657843	−0.328921	−	0.944357i	$-0.606685\pi$
$11$	−24.0000	−0.657843	−0.328921	+	0.944357i	$0.606685\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	18.0000	0.309839
$16$	0	0
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	0	0
$19$	16.0000	0.193192	0.0965961	−	0.995324i	$-0.469204\pi$
$19$	16.0000	0.193192	0.0965961	+	0.995324i	$0.469204\pi$
$20$	0	0
$21$	−60.0000	−0.623480
$22$	0	0
$23$	72.0000	0.652741	0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000	0.652741	0.326370	+	0.945242i	$0.394174\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−282.000	−1.80573	−0.902864	−	0.429927i	$-0.858539\pi$
$29$	−282.000	−1.80573	−0.902864	+	0.429927i	$0.858539\pi$
$30$	0	0
$31$	−164.000	−0.950170	−0.475085	−	0.879940i	$-0.657583\pi$
$31$	−164.000	−0.950170	−0.475085	+	0.879940i	$0.657583\pi$
$32$	0	0
$33$	−72.0000	−0.379806
$34$	0	0
$35$	−120.000	−0.579534
$36$	0	0
$37$	110.000	0.488754	0.244377	−	0.969680i	$-0.421417\pi$
$37$	110.000	0.488754	0.244377	+	0.969680i	$0.421417\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	−126.000	−0.479949	−0.239974	−	0.970779i	$-0.577139\pi$
$41$	−126.000	−0.479949	−0.239974	+	0.970779i	$0.577139\pi$
$42$	0	0
$43$	−164.000	−0.581622	−0.290811	−	0.956780i	$-0.593925\pi$
$43$	−164.000	−0.581622	−0.290811	+	0.956780i	$0.593925\pi$
$44$	0	0
$45$	54.0000	0.178885
$46$	0	0
$47$	204.000	0.633116	0.316558	−	0.948573i	$-0.397473\pi$
$47$	204.000	0.633116	0.316558	+	0.948573i	$0.397473\pi$
$48$	0	0
$49$	57.0000	0.166181
$50$	0	0
$51$	−90.0000	−0.247108
$52$	0	0
$53$	−738.000	−1.91268	−0.956341	−	0.292255i	$-0.905595\pi$
$53$	−738.000	−1.91268	−0.956341	+	0.292255i	$0.905595\pi$
$54$	0	0
$55$	−144.000	−0.353036
$56$	0	0
$57$	48.0000	0.111540
$58$	0	0
$59$	−120.000	−0.264791	−0.132396	−	0.991197i	$-0.542267\pi$
$59$	−120.000	−0.264791	−0.132396	+	0.991197i	$0.542267\pi$
$60$	0	0
$61$	614.000	1.28876	0.644382	−	0.764703i	$-0.277115\pi$
$61$	614.000	1.28876	0.644382	+	0.764703i	$0.277115\pi$
$62$	0	0
$63$	−180.000	−0.359966
$64$	0	0
$65$	78.0000	0.148842
$66$	0	0
$67$	−848.000	−1.54626	−0.773132	−	0.634245i	$-0.781311\pi$
$67$	−848.000	−1.54626	−0.773132	+	0.634245i	$0.781311\pi$
$68$	0	0
$69$	216.000	0.376860
$70$	0	0
$71$	−132.000	−0.220641	−0.110321	−	0.993896i	$-0.535188\pi$
$71$	−132.000	−0.220641	−0.110321	+	0.993896i	$0.535188\pi$
$72$	0	0
$73$	218.000	0.349520	0.174760	−	0.984611i	$-0.444085\pi$
$73$	218.000	0.349520	0.174760	+	0.984611i	$0.444085\pi$
$74$	0	0
$75$	−267.000	−0.411073
$76$	0	0
$77$	480.000	0.710404
$78$	0	0
$79$	1096.00	1.56088	0.780441	−	0.625230i	$-0.214995\pi$
$79$	1096.00	1.56088	0.780441	+	0.625230i	$0.214995\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−552.000	−0.729998	−0.364999	−	0.931008i	$-0.618931\pi$
$83$	−552.000	−0.729998	−0.364999	+	0.931008i	$0.618931\pi$
$84$	0	0
$85$	−180.000	−0.229691
$86$	0	0
$87$	−846.000	−1.04254
$88$	0	0
$89$	210.000	0.250112	0.125056	−	0.992150i	$-0.460089\pi$
$89$	210.000	0.250112	0.125056	+	0.992150i	$0.460089\pi$
$90$	0	0
$91$	−260.000	−0.299510
$92$	0	0
$93$	−492.000	−0.548581
$94$	0	0
$95$	96.0000	0.103678
$96$	0	0
$97$	−1726.00	−1.80669	−0.903344	−	0.428917i	$-0.858895\pi$
$97$	−1726.00	−1.80669	−0.903344	+	0.428917i	$0.858895\pi$
$98$	0	0
$99$	−216.000	−0.219281
$100$	0	0
$101$	798.000	0.786178	0.393089	−	0.919500i	$-0.371406\pi$
$101$	798.000	0.786178	0.393089	+	0.919500i	$0.371406\pi$
$102$	0	0
$103$	520.000	0.497448	0.248724	−	0.968574i	$-0.419989\pi$
$103$	520.000	0.497448	0.248724	+	0.968574i	$0.419989\pi$
$104$	0	0
$105$	−360.000	−0.334594
$106$	0	0
$107$	−12.0000	−0.0108419	−0.00542095	−	0.999985i	$-0.501726\pi$
$107$	−12.0000	−0.0108419	−0.00542095	+	0.999985i	$0.501726\pi$
$108$	0	0
$109$	−1834.00	−1.61161	−0.805804	−	0.592182i	$-0.798267\pi$
$109$	−1834.00	−1.61161	−0.805804	+	0.592182i	$0.798267\pi$
$110$	0	0
$111$	330.000	0.282182
$112$	0	0
$113$	−366.000	−0.304694	−0.152347	−	0.988327i	$-0.548683\pi$
$113$	−366.000	−0.304694	−0.152347	+	0.988327i	$0.548683\pi$
$114$	0	0
$115$	432.000	0.350297
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	600.000	0.462201
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	−378.000	−0.277098
$124$	0	0
$125$	−1284.00	−0.918756
$126$	0	0
$127$	−2144.00	−1.49803	−0.749013	−	0.662556i	$-0.769472\pi$
$127$	−2144.00	−1.49803	−0.749013	+	0.662556i	$0.769472\pi$
$128$	0	0
$129$	−492.000	−0.335800
$130$	0	0
$131$	2748.00	1.83278	0.916389	−	0.400289i	$-0.131090\pi$
$131$	2748.00	1.83278	0.916389	+	0.400289i	$0.131090\pi$
$132$	0	0
$133$	−320.000	−0.208628
$134$	0	0
$135$	162.000	0.103280
$136$	0	0
$137$	2754.00	1.71745	0.858723	−	0.512440i	$-0.171258\pi$
$137$	2754.00	1.71745	0.858723	+	0.512440i	$0.171258\pi$
$138$	0	0
$139$	−2252.00	−1.37419	−0.687094	−	0.726568i	$-0.741114\pi$
$139$	−2252.00	−1.37419	−0.687094	+	0.726568i	$0.741114\pi$
$140$	0	0
$141$	612.000	0.365530
$142$	0	0
$143$	−312.000	−0.182453
$144$	0	0
$145$	−1692.00	−0.969055
$146$	0	0
$147$	171.000	0.0959445
$148$	0	0
$149$	−1770.00	−0.973182	−0.486591	−	0.873630i	$-0.661760\pi$
$149$	−1770.00	−0.973182	−0.486591	+	0.873630i	$0.661760\pi$
$150$	0	0
$151$	988.000	0.532466	0.266233	−	0.963909i	$-0.414221\pi$
$151$	988.000	0.532466	0.266233	+	0.963909i	$0.414221\pi$
$152$	0	0
$153$	−270.000	−0.142668
$154$	0	0
$155$	−984.000	−0.509915
$156$	0	0
$157$	326.000	0.165717	0.0828587	−	0.996561i	$-0.473595\pi$
$157$	326.000	0.165717	0.0828587	+	0.996561i	$0.473595\pi$
$158$	0	0
$159$	−2214.00	−1.10429
$160$	0	0
$161$	−1440.00	−0.704894
$162$	0	0
$163$	−1496.00	−0.718870	−0.359435	−	0.933170i	$-0.617031\pi$
$163$	−1496.00	−0.718870	−0.359435	+	0.933170i	$0.617031\pi$
$164$	0	0
$165$	−432.000	−0.203825
$166$	0	0
$167$	−1116.00	−0.517118	−0.258559	−	0.965995i	$-0.583248\pi$
$167$	−1116.00	−0.517118	−0.258559	+	0.965995i	$0.583248\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	144.000	0.0643974
$172$	0	0
$173$	4374.00	1.92225	0.961124	−	0.276116i	$-0.0890472\pi$
$173$	4374.00	1.92225	0.961124	+	0.276116i	$0.0890472\pi$
$174$	0	0
$175$	1780.00	0.768888
$176$	0	0
$177$	−360.000	−0.152877
$178$	0	0
$179$	−12.0000	−0.00501074	−0.00250537	−	0.999997i	$-0.500797\pi$
$179$	−12.0000	−0.00501074	−0.00250537	+	0.999997i	$0.500797\pi$
$180$	0	0
$181$	4718.00	1.93749	0.968746	−	0.248053i	$-0.0797909\pi$
$181$	4718.00	1.93749	0.968746	+	0.248053i	$0.0797909\pi$
$182$	0	0
$183$	1842.00	0.744069
$184$	0	0
$185$	660.000	0.262293
$186$	0	0
$187$	720.000	0.281559
$188$	0	0
$189$	−540.000	−0.207827
$190$	0	0
$191$	1368.00	0.518246	0.259123	−	0.965844i	$-0.416566\pi$
$191$	1368.00	0.518246	0.259123	+	0.965844i	$0.416566\pi$
$192$	0	0
$193$	−3310.00	−1.23450	−0.617251	−	0.786766i	$-0.711754\pi$
$193$	−3310.00	−1.23450	−0.617251	+	0.786766i	$0.711754\pi$
$194$	0	0
$195$	234.000	0.0859338
$196$	0	0
$197$	3126.00	1.13055	0.565275	−	0.824903i	$-0.308770\pi$
$197$	3126.00	1.13055	0.565275	+	0.824903i	$0.308770\pi$
$198$	0	0
$199$	−4664.00	−1.66142	−0.830709	−	0.556707i	$-0.812065\pi$
$199$	−4664.00	−1.66142	−0.830709	+	0.556707i	$0.812065\pi$
$200$	0	0
$201$	−2544.00	−0.892736
$202$	0	0
$203$	5640.00	1.95000
$204$	0	0
$205$	−756.000	−0.257567
$206$	0	0
$207$	648.000	0.217580
$208$	0	0
$209$	−384.000	−0.127090
$210$	0	0
$211$	556.000	0.181406	0.0907029	−	0.995878i	$-0.471089\pi$
$211$	556.000	0.181406	0.0907029	+	0.995878i	$0.471089\pi$
$212$	0	0
$213$	−396.000	−0.127387
$214$	0	0
$215$	−984.000	−0.312131
$216$	0	0
$217$	3280.00	1.02609
$218$	0	0
$219$	654.000	0.201796
$220$	0	0
$221$	−390.000	−0.118707
$222$	0	0
$223$	268.000	0.0804781	0.0402390	−	0.999190i	$-0.487188\pi$
$223$	268.000	0.0804781	0.0402390	+	0.999190i	$0.487188\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	0	0
$227$	−1800.00	−0.526300	−0.263150	−	0.964755i	$-0.584761\pi$
$227$	−1800.00	−0.526300	−0.263150	+	0.964755i	$0.584761\pi$
$228$	0	0
$229$	2990.00	0.862816	0.431408	−	0.902157i	$-0.358017\pi$
$229$	2990.00	0.862816	0.431408	+	0.902157i	$0.358017\pi$
$230$	0	0
$231$	1440.00	0.410152
$232$	0	0
$233$	2826.00	0.794581	0.397291	−	0.917693i	$-0.369951\pi$
$233$	2826.00	0.794581	0.397291	+	0.917693i	$0.369951\pi$
$234$	0	0
$235$	1224.00	0.339766
$236$	0	0
$237$	3288.00	0.901175
$238$	0	0
$239$	1812.00	0.490412	0.245206	−	0.969471i	$-0.421144\pi$
$239$	1812.00	0.490412	0.245206	+	0.969471i	$0.421144\pi$
$240$	0	0
$241$	−1582.00	−0.422845	−0.211422	−	0.977395i	$-0.567810\pi$
$241$	−1582.00	−0.422845	−0.211422	+	0.977395i	$0.567810\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	342.000	0.0891820
$246$	0	0
$247$	208.000	0.0535819
$248$	0	0
$249$	−1656.00	−0.421465
$250$	0	0
$251$	−2148.00	−0.540162	−0.270081	−	0.962838i	$-0.587050\pi$
$251$	−2148.00	−0.540162	−0.270081	+	0.962838i	$0.587050\pi$
$252$	0	0
$253$	−1728.00	−0.429401
$254$	0	0
$255$	−540.000	−0.132612
$256$	0	0
$257$	−558.000	−0.135436	−0.0677181	−	0.997704i	$-0.521572\pi$
$257$	−558.000	−0.135436	−0.0677181	+	0.997704i	$0.521572\pi$
$258$	0	0
$259$	−2200.00	−0.527804
$260$	0	0
$261$	−2538.00	−0.601909
$262$	0	0
$263$	−2112.00	−0.495177	−0.247588	−	0.968865i	$-0.579638\pi$
$263$	−2112.00	−0.495177	−0.247588	+	0.968865i	$0.579638\pi$
$264$	0	0
$265$	−4428.00	−1.02645
$266$	0	0
$267$	630.000	0.144402
$268$	0	0
$269$	5046.00	1.14372	0.571859	−	0.820352i	$-0.306223\pi$
$269$	5046.00	1.14372	0.571859	+	0.820352i	$0.306223\pi$
$270$	0	0
$271$	3796.00	0.850888	0.425444	−	0.904985i	$-0.360118\pi$
$271$	3796.00	0.850888	0.425444	+	0.904985i	$0.360118\pi$
$272$	0	0
$273$	−780.000	−0.172922
$274$	0	0
$275$	2136.00	0.468384
$276$	0	0
$277$	5582.00	1.21079	0.605397	−	0.795924i	$-0.293014\pi$
$277$	5582.00	1.21079	0.605397	+	0.795924i	$0.293014\pi$
$278$	0	0
$279$	−1476.00	−0.316723
$280$	0	0
$281$	−1950.00	−0.413976	−0.206988	−	0.978343i	$-0.566366\pi$
$281$	−1950.00	−0.413976	−0.206988	+	0.978343i	$0.566366\pi$
$282$	0	0
$283$	4732.00	0.993951	0.496976	−	0.867765i	$-0.334444\pi$
$283$	4732.00	0.993951	0.496976	+	0.867765i	$0.334444\pi$
$284$	0	0
$285$	288.000	0.0598584
$286$	0	0
$287$	2520.00	0.518296
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	−5178.00	−1.04309
$292$	0	0
$293$	4998.00	0.996540	0.498270	−	0.867022i	$-0.333969\pi$
$293$	4998.00	0.996540	0.498270	+	0.867022i	$0.333969\pi$
$294$	0	0
$295$	−720.000	−0.142102
$296$	0	0
$297$	−648.000	−0.126602
$298$	0	0
$299$	936.000	0.181038
$300$	0	0
$301$	3280.00	0.628093
$302$	0	0
$303$	2394.00	0.453900
$304$	0	0
$305$	3684.00	0.691624
$306$	0	0
$307$	−6824.00	−1.26862	−0.634310	−	0.773079i	$-0.718716\pi$
$307$	−6824.00	−1.26862	−0.634310	+	0.773079i	$0.718716\pi$
$308$	0	0
$309$	1560.00	0.287202
$310$	0	0
$311$	8760.00	1.59722	0.798608	−	0.601852i	$-0.205570\pi$
$311$	8760.00	1.59722	0.798608	+	0.601852i	$0.205570\pi$
$312$	0	0
$313$	3962.00	0.715481	0.357740	−	0.933821i	$-0.383547\pi$
$313$	3962.00	0.715481	0.357740	+	0.933821i	$0.383547\pi$
$314$	0	0
$315$	−1080.00	−0.193178
$316$	0	0
$317$	7086.00	1.25549	0.627744	−	0.778420i	$-0.283979\pi$
$317$	7086.00	1.25549	0.627744	+	0.778420i	$0.283979\pi$
$318$	0	0
$319$	6768.00	1.18788
$320$	0	0
$321$	−36.0000	−0.00625958
$322$	0	0
$323$	−480.000	−0.0826870
$324$	0	0
$325$	−1157.00	−0.197473
$326$	0	0
$327$	−5502.00	−0.930463
$328$	0	0
$329$	−4080.00	−0.683701
$330$	0	0
$331$	9016.00	1.49717	0.748586	−	0.663037i	$-0.230733\pi$
$331$	9016.00	1.49717	0.748586	+	0.663037i	$0.230733\pi$
$332$	0	0
$333$	990.000	0.162918
$334$	0	0
$335$	−5088.00	−0.829812
$336$	0	0
$337$	2306.00	0.372747	0.186374	−	0.982479i	$-0.440327\pi$
$337$	2306.00	0.372747	0.186374	+	0.982479i	$0.440327\pi$
$338$	0	0
$339$	−1098.00	−0.175915
$340$	0	0
$341$	3936.00	0.625063
$342$	0	0
$343$	5720.00	0.900440
$344$	0	0
$345$	1296.00	0.202244
$346$	0	0
$347$	11076.0	1.71352	0.856759	−	0.515717i	$-0.172474\pi$
$347$	11076.0	1.71352	0.856759	+	0.515717i	$0.172474\pi$
$348$	0	0
$349$	2342.00	0.359210	0.179605	−	0.983739i	$-0.442518\pi$
$349$	2342.00	0.359210	0.179605	+	0.983739i	$0.442518\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	4650.00	0.701118	0.350559	−	0.936541i	$-0.385992\pi$
$353$	4650.00	0.701118	0.350559	+	0.936541i	$0.385992\pi$
$354$	0	0
$355$	−792.000	−0.118408
$356$	0	0
$357$	1800.00	0.266852
$358$	0	0
$359$	11268.0	1.65655	0.828276	−	0.560320i	$-0.189322\pi$
$359$	11268.0	1.65655	0.828276	+	0.560320i	$0.189322\pi$
$360$	0	0
$361$	−6603.00	−0.962677
$362$	0	0
$363$	−2265.00	−0.327498
$364$	0	0
$365$	1308.00	0.187572
$366$	0	0
$367$	7288.00	1.03660	0.518298	−	0.855200i	$-0.326566\pi$
$367$	7288.00	1.03660	0.518298	+	0.855200i	$0.326566\pi$
$368$	0	0
$369$	−1134.00	−0.159983
$370$	0	0
$371$	14760.0	2.06550
$372$	0	0
$373$	−9970.00	−1.38399	−0.691993	−	0.721904i	$-0.743267\pi$
$373$	−9970.00	−1.38399	−0.691993	+	0.721904i	$0.743267\pi$
$374$	0	0
$375$	−3852.00	−0.530444
$376$	0	0
$377$	−3666.00	−0.500819
$378$	0	0
$379$	−13448.0	−1.82263	−0.911316	−	0.411708i	$-0.864932\pi$
$379$	−13448.0	−1.82263	−0.911316	+	0.411708i	$0.864932\pi$
$380$	0	0
$381$	−6432.00	−0.864885
$382$	0	0
$383$	−11820.0	−1.57696	−0.788478	−	0.615064i	$-0.789130\pi$
$383$	−11820.0	−1.57696	−0.788478	+	0.615064i	$0.789130\pi$
$384$	0	0
$385$	2880.00	0.381243
$386$	0	0
$387$	−1476.00	−0.193874
$388$	0	0
$389$	174.000	0.0226790	0.0113395	−	0.999936i	$-0.496390\pi$
$389$	174.000	0.0226790	0.0113395	+	0.999936i	$0.496390\pi$
$390$	0	0
$391$	−2160.00	−0.279376
$392$	0	0
$393$	8244.00	1.05815
$394$	0	0
$395$	6576.00	0.837657
$396$	0	0
$397$	−2986.00	−0.377489	−0.188744	−	0.982026i	$-0.560442\pi$
$397$	−2986.00	−0.377489	−0.188744	+	0.982026i	$0.560442\pi$
$398$	0	0
$399$	−960.000	−0.120451
$400$	0	0
$401$	−10566.0	−1.31581	−0.657906	−	0.753100i	$-0.728558\pi$
$401$	−10566.0	−1.31581	−0.657906	+	0.753100i	$0.728558\pi$
$402$	0	0
$403$	−2132.00	−0.263530
$404$	0	0
$405$	486.000	0.0596285
$406$	0	0
$407$	−2640.00	−0.321523
$408$	0	0
$409$	−7270.00	−0.878920	−0.439460	−	0.898262i	$-0.644830\pi$
$409$	−7270.00	−0.878920	−0.439460	+	0.898262i	$0.644830\pi$
$410$	0	0
$411$	8262.00	0.991568
$412$	0	0
$413$	2400.00	0.285947
$414$	0	0
$415$	−3312.00	−0.391758
$416$	0	0
$417$	−6756.00	−0.793388
$418$	0	0
$419$	7308.00	0.852074	0.426037	−	0.904706i	$-0.359909\pi$
$419$	7308.00	0.852074	0.426037	+	0.904706i	$0.359909\pi$
$420$	0	0
$421$	−5938.00	−0.687412	−0.343706	−	0.939077i	$-0.611682\pi$
$421$	−5938.00	−0.687412	−0.343706	+	0.939077i	$0.611682\pi$
$422$	0	0
$423$	1836.00	0.211039
$424$	0	0
$425$	2670.00	0.304739
$426$	0	0
$427$	−12280.0	−1.39174
$428$	0	0
$429$	−936.000	−0.105339
$430$	0	0
$431$	−11532.0	−1.28881	−0.644405	−	0.764685i	$-0.722895\pi$
$431$	−11532.0	−1.28881	−0.644405	+	0.764685i	$0.722895\pi$
$432$	0	0
$433$	−718.000	−0.0796879	−0.0398440	−	0.999206i	$-0.512686\pi$
$433$	−718.000	−0.0796879	−0.0398440	+	0.999206i	$0.512686\pi$
$434$	0	0
$435$	−5076.00	−0.559484
$436$	0	0
$437$	1152.00	0.126104
$438$	0	0
$439$	−8984.00	−0.976726	−0.488363	−	0.872640i	$-0.662406\pi$
$439$	−8984.00	−0.976726	−0.488363	+	0.872640i	$0.662406\pi$
$440$	0	0
$441$	513.000	0.0553936
$442$	0	0
$443$	−2604.00	−0.279277	−0.139639	−	0.990203i	$-0.544594\pi$
$443$	−2604.00	−0.279277	−0.139639	+	0.990203i	$0.544594\pi$
$444$	0	0
$445$	1260.00	0.134224
$446$	0	0
$447$	−5310.00	−0.561867
$448$	0	0
$449$	−13206.0	−1.38804	−0.694020	−	0.719956i	$-0.744162\pi$
$449$	−13206.0	−1.38804	−0.694020	+	0.719956i	$0.744162\pi$
$450$	0	0
$451$	3024.00	0.315731
$452$	0	0
$453$	2964.00	0.307419
$454$	0	0
$455$	−1560.00	−0.160734
$456$	0	0
$457$	8426.00	0.862476	0.431238	−	0.902238i	$-0.358077\pi$
$457$	8426.00	0.862476	0.431238	+	0.902238i	$0.358077\pi$
$458$	0	0
$459$	−810.000	−0.0823694
$460$	0	0
$461$	16686.0	1.68578	0.842890	−	0.538086i	$-0.180852\pi$
$461$	16686.0	1.68578	0.842890	+	0.538086i	$0.180852\pi$
$462$	0	0
$463$	−15932.0	−1.59919	−0.799593	−	0.600543i	$-0.794951\pi$
$463$	−15932.0	−1.59919	−0.799593	+	0.600543i	$0.794951\pi$
$464$	0	0
$465$	−2952.00	−0.294399
$466$	0	0
$467$	−18540.0	−1.83711	−0.918553	−	0.395297i	$-0.870642\pi$
$467$	−18540.0	−1.83711	−0.918553	+	0.395297i	$0.870642\pi$
$468$	0	0
$469$	16960.0	1.66981
$470$	0	0
$471$	978.000	0.0956770
$472$	0	0
$473$	3936.00	0.382616
$474$	0	0
$475$	−1424.00	−0.137553
$476$	0	0
$477$	−6642.00	−0.637560
$478$	0	0
$479$	−6180.00	−0.589502	−0.294751	−	0.955574i	$-0.595237\pi$
$479$	−6180.00	−0.589502	−0.294751	+	0.955574i	$0.595237\pi$
$480$	0	0
$481$	1430.00	0.135556
$482$	0	0
$483$	−4320.00	−0.406971
$484$	0	0
$485$	−10356.0	−0.969571
$486$	0	0
$487$	−11756.0	−1.09387	−0.546936	−	0.837175i	$-0.684206\pi$
$487$	−11756.0	−1.09387	−0.546936	+	0.837175i	$0.684206\pi$
$488$	0	0
$489$	−4488.00	−0.415040
$490$	0	0
$491$	−1908.00	−0.175370	−0.0876852	−	0.996148i	$-0.527947\pi$
$491$	−1908.00	−0.175370	−0.0876852	+	0.996148i	$0.527947\pi$
$492$	0	0
$493$	8460.00	0.772858
$494$	0	0
$495$	−1296.00	−0.117679
$496$	0	0
$497$	2640.00	0.238270
$498$	0	0
$499$	8944.00	0.802382	0.401191	−	0.915995i	$-0.368596\pi$
$499$	8944.00	0.802382	0.401191	+	0.915995i	$0.368596\pi$
$500$	0	0
$501$	−3348.00	−0.298558
$502$	0	0
$503$	6528.00	0.578666	0.289333	−	0.957228i	$-0.406566\pi$
$503$	6528.00	0.578666	0.289333	+	0.957228i	$0.406566\pi$
$504$	0	0
$505$	4788.00	0.421907
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−12114.0	−1.05490	−0.527450	−	0.849586i	$-0.676852\pi$
$509$	−12114.0	−1.05490	−0.527450	+	0.849586i	$0.676852\pi$
$510$	0	0
$511$	−4360.00	−0.377446
$512$	0	0
$513$	432.000	0.0371799
$514$	0	0
$515$	3120.00	0.266958
$516$	0	0
$517$	−4896.00	−0.416491
$518$	0	0
$519$	13122.0	1.10981
$520$	0	0
$521$	−14310.0	−1.20333	−0.601663	−	0.798750i	$-0.705495\pi$
$521$	−14310.0	−1.20333	−0.601663	+	0.798750i	$0.705495\pi$
$522$	0	0
$523$	18340.0	1.53337	0.766685	−	0.642024i	$-0.221905\pi$
$523$	18340.0	1.53337	0.766685	+	0.642024i	$0.221905\pi$
$524$	0	0
$525$	5340.00	0.443918
$526$	0	0
$527$	4920.00	0.406677
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	−1080.00	−0.0882637
$532$	0	0
$533$	−1638.00	−0.133114
$534$	0	0
$535$	−72.0000	−0.00581838
$536$	0	0
$537$	−36.0000	−0.00289295
$538$	0	0
$539$	−1368.00	−0.109321
$540$	0	0
$541$	9254.00	0.735417	0.367708	−	0.929941i	$-0.380142\pi$
$541$	9254.00	0.735417	0.367708	+	0.929941i	$0.380142\pi$
$542$	0	0
$543$	14154.0	1.11861
$544$	0	0
$545$	−11004.0	−0.864880
$546$	0	0
$547$	−17444.0	−1.36353	−0.681766	−	0.731571i	$-0.738788\pi$
$547$	−17444.0	−1.36353	−0.681766	+	0.731571i	$0.738788\pi$
$548$	0	0
$549$	5526.00	0.429588
$550$	0	0
$551$	−4512.00	−0.348852
$552$	0	0
$553$	−21920.0	−1.68559
$554$	0	0
$555$	1980.00	0.151435
$556$	0	0
$557$	−3714.00	−0.282526	−0.141263	−	0.989972i	$-0.545116\pi$
$557$	−3714.00	−0.282526	−0.141263	+	0.989972i	$0.545116\pi$
$558$	0	0
$559$	−2132.00	−0.161313
$560$	0	0
$561$	2160.00	0.162558
$562$	0	0
$563$	13812.0	1.03394	0.516968	−	0.856004i	$-0.327060\pi$
$563$	13812.0	1.03394	0.516968	+	0.856004i	$0.327060\pi$
$564$	0	0
$565$	−2196.00	−0.163516
$566$	0	0
$567$	−1620.00	−0.119989
$568$	0	0
$569$	−15942.0	−1.17456	−0.587279	−	0.809385i	$-0.699801\pi$
$569$	−15942.0	−1.17456	−0.587279	+	0.809385i	$0.699801\pi$
$570$	0	0
$571$	−1604.00	−0.117557	−0.0587787	−	0.998271i	$-0.518721\pi$
$571$	−1604.00	−0.117557	−0.0587787	+	0.998271i	$0.518721\pi$
$572$	0	0
$573$	4104.00	0.299210
$574$	0	0
$575$	−6408.00	−0.464751
$576$	0	0
$577$	−10654.0	−0.768686	−0.384343	−	0.923190i	$-0.625572\pi$
$577$	−10654.0	−0.768686	−0.384343	+	0.923190i	$0.625572\pi$
$578$	0	0
$579$	−9930.00	−0.712740
$580$	0	0
$581$	11040.0	0.788324
$582$	0	0
$583$	17712.0	1.25824
$584$	0	0
$585$	702.000	0.0496139
$586$	0	0
$587$	9984.00	0.702017	0.351008	−	0.936372i	$-0.385839\pi$
$587$	9984.00	0.702017	0.351008	+	0.936372i	$0.385839\pi$
$588$	0	0
$589$	−2624.00	−0.183565
$590$	0	0
$591$	9378.00	0.652723
$592$	0	0
$593$	12618.0	0.873793	0.436896	−	0.899512i	$-0.356078\pi$
$593$	12618.0	0.873793	0.436896	+	0.899512i	$0.356078\pi$
$594$	0	0
$595$	3600.00	0.248043
$596$	0	0
$597$	−13992.0	−0.959220
$598$	0	0
$599$	−11184.0	−0.762881	−0.381441	−	0.924393i	$-0.624572\pi$
$599$	−11184.0	−0.762881	−0.381441	+	0.924393i	$0.624572\pi$
$600$	0	0
$601$	2810.00	0.190719	0.0953596	−	0.995443i	$-0.469600\pi$
$601$	2810.00	0.190719	0.0953596	+	0.995443i	$0.469600\pi$
$602$	0	0
$603$	−7632.00	−0.515421
$604$	0	0
$605$	−4530.00	−0.304414
$606$	0	0
$607$	−1064.00	−0.0711473	−0.0355737	−	0.999367i	$-0.511326\pi$
$607$	−1064.00	−0.0711473	−0.0355737	+	0.999367i	$0.511326\pi$
$608$	0	0
$609$	16920.0	1.12583
$610$	0	0
$611$	2652.00	0.175595
$612$	0	0
$613$	−20914.0	−1.37799	−0.688996	−	0.724766i	$-0.741948\pi$
$613$	−20914.0	−1.37799	−0.688996	+	0.724766i	$0.741948\pi$
$614$	0	0
$615$	−2268.00	−0.148707
$616$	0	0
$617$	9714.00	0.633826	0.316913	−	0.948455i	$-0.397354\pi$
$617$	9714.00	0.633826	0.316913	+	0.948455i	$0.397354\pi$
$618$	0	0
$619$	14848.0	0.964122	0.482061	−	0.876138i	$-0.339888\pi$
$619$	14848.0	0.964122	0.482061	+	0.876138i	$0.339888\pi$
$620$	0	0
$621$	1944.00	0.125620
$622$	0	0
$623$	−4200.00	−0.270095
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	−1152.00	−0.0733755
$628$	0	0
$629$	−3300.00	−0.209189
$630$	0	0
$631$	−19172.0	−1.20955	−0.604774	−	0.796397i	$-0.706737\pi$
$631$	−19172.0	−1.20955	−0.604774	+	0.796397i	$0.706737\pi$
$632$	0	0
$633$	1668.00	0.104735
$634$	0	0
$635$	−12864.0	−0.803925
$636$	0	0
$637$	741.000	0.0460902
$638$	0	0
$639$	−1188.00	−0.0735470
$640$	0	0
$641$	−11502.0	−0.708739	−0.354369	−	0.935105i	$-0.615304\pi$
$641$	−11502.0	−0.708739	−0.354369	+	0.935105i	$0.615304\pi$
$642$	0	0
$643$	15568.0	0.954809	0.477404	−	0.878684i	$-0.341578\pi$
$643$	15568.0	0.954809	0.477404	+	0.878684i	$0.341578\pi$
$644$	0	0
$645$	−2952.00	−0.180209
$646$	0	0
$647$	−1128.00	−0.0685414	−0.0342707	−	0.999413i	$-0.510911\pi$
$647$	−1128.00	−0.0685414	−0.0342707	+	0.999413i	$0.510911\pi$
$648$	0	0
$649$	2880.00	0.174191
$650$	0	0
$651$	9840.00	0.592412
$652$	0	0
$653$	8118.00	0.486496	0.243248	−	0.969964i	$-0.421787\pi$
$653$	8118.00	0.486496	0.243248	+	0.969964i	$0.421787\pi$
$654$	0	0
$655$	16488.0	0.983572
$656$	0	0
$657$	1962.00	0.116507
$658$	0	0
$659$	−13572.0	−0.802261	−0.401131	−	0.916021i	$-0.631383\pi$
$659$	−13572.0	−0.802261	−0.401131	+	0.916021i	$0.631383\pi$
$660$	0	0
$661$	−13138.0	−0.773085	−0.386542	−	0.922272i	$-0.626331\pi$
$661$	−13138.0	−0.773085	−0.386542	+	0.922272i	$0.626331\pi$
$662$	0	0
$663$	−1170.00	−0.0685355
$664$	0	0
$665$	−1920.00	−0.111962
$666$	0	0
$667$	−20304.0	−1.17867
$668$	0	0
$669$	804.000	0.0464640
$670$	0	0
$671$	−14736.0	−0.847805
$672$	0	0
$673$	−718.000	−0.0411246	−0.0205623	−	0.999789i	$-0.506546\pi$
$673$	−718.000	−0.0411246	−0.0205623	+	0.999789i	$0.506546\pi$
$674$	0	0
$675$	−2403.00	−0.137024
$676$	0	0
$677$	−2994.00	−0.169969	−0.0849843	−	0.996382i	$-0.527084\pi$
$677$	−2994.00	−0.169969	−0.0849843	+	0.996382i	$0.527084\pi$
$678$	0	0
$679$	34520.0	1.95104
$680$	0	0
$681$	−5400.00	−0.303860
$682$	0	0
$683$	−27384.0	−1.53414	−0.767071	−	0.641562i	$-0.778287\pi$
$683$	−27384.0	−1.53414	−0.767071	+	0.641562i	$0.778287\pi$
$684$	0	0
$685$	16524.0	0.921678
$686$	0	0
$687$	8970.00	0.498147
$688$	0	0
$689$	−9594.00	−0.530482
$690$	0	0
$691$	−27632.0	−1.52123	−0.760616	−	0.649202i	$-0.775103\pi$
$691$	−27632.0	−1.52123	−0.760616	+	0.649202i	$0.775103\pi$
$692$	0	0
$693$	4320.00	0.236801
$694$	0	0
$695$	−13512.0	−0.737467
$696$	0	0
$697$	3780.00	0.205420
$698$	0	0
$699$	8478.00	0.458752
$700$	0	0
$701$	19062.0	1.02705	0.513525	−	0.858075i	$-0.328339\pi$
$701$	19062.0	1.02705	0.513525	+	0.858075i	$0.328339\pi$
$702$	0	0
$703$	1760.00	0.0944234
$704$	0	0
$705$	3672.00	0.196164
$706$	0	0
$707$	−15960.0	−0.848992
$708$	0	0
$709$	3854.00	0.204147	0.102073	−	0.994777i	$-0.467452\pi$
$709$	3854.00	0.204147	0.102073	+	0.994777i	$0.467452\pi$
$710$	0	0
$711$	9864.00	0.520294
$712$	0	0
$713$	−11808.0	−0.620215
$714$	0	0
$715$	−1872.00	−0.0979144
$716$	0	0
$717$	5436.00	0.283140
$718$	0	0
$719$	−20976.0	−1.08800	−0.544001	−	0.839085i	$-0.683091\pi$
$719$	−20976.0	−1.08800	−0.544001	+	0.839085i	$0.683091\pi$
$720$	0	0
$721$	−10400.0	−0.537193
$722$	0	0
$723$	−4746.00	−0.244130
$724$	0	0
$725$	25098.0	1.28568
$726$	0	0
$727$	29464.0	1.50311	0.751554	−	0.659672i	$-0.229305\pi$
$727$	29464.0	1.50311	0.751554	+	0.659672i	$0.229305\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	4920.00	0.248937
$732$	0	0
$733$	−2698.00	−0.135952	−0.0679761	−	0.997687i	$-0.521654\pi$
$733$	−2698.00	−0.135952	−0.0679761	+	0.997687i	$0.521654\pi$
$734$	0	0
$735$	1026.00	0.0514892
$736$	0	0
$737$	20352.0	1.01720
$738$	0	0
$739$	−632.000	−0.0314594	−0.0157297	−	0.999876i	$-0.505007\pi$
$739$	−632.000	−0.0314594	−0.0157297	+	0.999876i	$0.505007\pi$
$740$	0	0
$741$	624.000	0.0309355
$742$	0	0
$743$	20844.0	1.02920	0.514598	−	0.857432i	$-0.327941\pi$
$743$	20844.0	1.02920	0.514598	+	0.857432i	$0.327941\pi$
$744$	0	0
$745$	−10620.0	−0.522264
$746$	0	0
$747$	−4968.00	−0.243333
$748$	0	0
$749$	240.000	0.0117082
$750$	0	0
$751$	−272.000	−0.0132163	−0.00660814	−	0.999978i	$-0.502103\pi$
$751$	−272.000	−0.0132163	−0.00660814	+	0.999978i	$0.502103\pi$
$752$	0	0
$753$	−6444.00	−0.311862
$754$	0	0
$755$	5928.00	0.285751
$756$	0	0
$757$	37550.0	1.80288	0.901439	−	0.432907i	$-0.142512\pi$
$757$	37550.0	1.80288	0.901439	+	0.432907i	$0.142512\pi$
$758$	0	0
$759$	−5184.00	−0.247915
$760$	0	0
$761$	33330.0	1.58766	0.793832	−	0.608138i	$-0.208083\pi$
$761$	33330.0	1.58766	0.793832	+	0.608138i	$0.208083\pi$
$762$	0	0
$763$	36680.0	1.74037
$764$	0	0
$765$	−1620.00	−0.0765637
$766$	0	0
$767$	−1560.00	−0.0734398
$768$	0	0
$769$	−15406.0	−0.722438	−0.361219	−	0.932481i	$-0.617639\pi$
$769$	−15406.0	−0.722438	−0.361219	+	0.932481i	$0.617639\pi$
$770$	0	0
$771$	−1674.00	−0.0781941
$772$	0	0
$773$	−29514.0	−1.37328	−0.686640	−	0.726998i	$-0.740915\pi$
$773$	−29514.0	−1.37328	−0.686640	+	0.726998i	$0.740915\pi$
$774$	0	0
$775$	14596.0	0.676521
$776$	0	0
$777$	−6600.00	−0.304728
$778$	0	0
$779$	−2016.00	−0.0927223
$780$	0	0
$781$	3168.00	0.145147
$782$	0	0
$783$	−7614.00	−0.347512
$784$	0	0
$785$	1956.00	0.0889333
$786$	0	0
$787$	−33176.0	−1.50266	−0.751332	−	0.659924i	$-0.770588\pi$
$787$	−33176.0	−1.50266	−0.751332	+	0.659924i	$0.770588\pi$
$788$	0	0
$789$	−6336.00	−0.285890
$790$	0	0
$791$	7320.00	0.329038
$792$	0	0
$793$	7982.00	0.357439
$794$	0	0
$795$	−13284.0	−0.592623
$796$	0	0
$797$	−16746.0	−0.744258	−0.372129	−	0.928181i	$-0.621372\pi$
$797$	−16746.0	−0.744258	−0.372129	+	0.928181i	$0.621372\pi$
$798$	0	0
$799$	−6120.00	−0.270976
$800$	0	0
$801$	1890.00	0.0833706
$802$	0	0
$803$	−5232.00	−0.229929
$804$	0	0
$805$	−8640.00	−0.378286
$806$	0	0
$807$	15138.0	0.660326
$808$	0	0
$809$	−15846.0	−0.688647	−0.344324	−	0.938851i	$-0.611892\pi$
$809$	−15846.0	−0.688647	−0.344324	+	0.938851i	$0.611892\pi$
$810$	0	0
$811$	−22952.0	−0.993778	−0.496889	−	0.867814i	$-0.665524\pi$
$811$	−22952.0	−0.993778	−0.496889	+	0.867814i	$0.665524\pi$
$812$	0	0
$813$	11388.0	0.491260
$814$	0	0
$815$	−8976.00	−0.385786
$816$	0	0
$817$	−2624.00	−0.112365
$818$	0	0
$819$	−2340.00	−0.0998367
$820$	0	0
$821$	−37146.0	−1.57906	−0.789528	−	0.613715i	$-0.789674\pi$
$821$	−37146.0	−1.57906	−0.789528	+	0.613715i	$0.789674\pi$
$822$	0	0
$823$	9592.00	0.406265	0.203133	−	0.979151i	$-0.434888\pi$
$823$	9592.00	0.406265	0.203133	+	0.979151i	$0.434888\pi$
$824$	0	0
$825$	6408.00	0.270422
$826$	0	0
$827$	39960.0	1.68022	0.840112	−	0.542413i	$-0.182489\pi$
$827$	39960.0	1.68022	0.840112	+	0.542413i	$0.182489\pi$
$828$	0	0
$829$	−3706.00	−0.155265	−0.0776325	−	0.996982i	$-0.524736\pi$
$829$	−3706.00	−0.155265	−0.0776325	+	0.996982i	$0.524736\pi$
$830$	0	0
$831$	16746.0	0.699052
$832$	0	0
$833$	−1710.00	−0.0711260
$834$	0	0
$835$	−6696.00	−0.277515
$836$	0	0
$837$	−4428.00	−0.182860
$838$	0	0
$839$	−9756.00	−0.401448	−0.200724	−	0.979648i	$-0.564329\pi$
$839$	−9756.00	−0.401448	−0.200724	+	0.979648i	$0.564329\pi$
$840$	0	0
$841$	55135.0	2.26065
$842$	0	0
$843$	−5850.00	−0.239009
$844$	0	0
$845$	1014.00	0.0412813
$846$	0	0
$847$	15100.0	0.612565
$848$	0	0
$849$	14196.0	0.573858
$850$	0	0
$851$	7920.00	0.319029
$852$	0	0
$853$	11342.0	0.455267	0.227633	−	0.973747i	$-0.426901\pi$
$853$	11342.0	0.455267	0.227633	+	0.973747i	$0.426901\pi$
$854$	0	0
$855$	864.000	0.0345593
$856$	0	0
$857$	−16134.0	−0.643089	−0.321544	−	0.946895i	$-0.604202\pi$
$857$	−16134.0	−0.643089	−0.321544	+	0.946895i	$0.604202\pi$
$858$	0	0
$859$	20932.0	0.831421	0.415710	−	0.909497i	$-0.363533\pi$
$859$	20932.0	0.831421	0.415710	+	0.909497i	$0.363533\pi$
$860$	0	0
$861$	7560.00	0.299238
$862$	0	0
$863$	−10044.0	−0.396178	−0.198089	−	0.980184i	$-0.563474\pi$
$863$	−10044.0	−0.396178	−0.198089	+	0.980184i	$0.563474\pi$
$864$	0	0
$865$	26244.0	1.03159
$866$	0	0
$867$	−12039.0	−0.471587
$868$	0	0
$869$	−26304.0	−1.02681
$870$	0	0
$871$	−11024.0	−0.428856
$872$	0	0
$873$	−15534.0	−0.602229
$874$	0	0
$875$	25680.0	0.992163
$876$	0	0
$877$	−26314.0	−1.01318	−0.506591	−	0.862186i	$-0.669095\pi$
$877$	−26314.0	−1.01318	−0.506591	+	0.862186i	$0.669095\pi$
$878$	0	0
$879$	14994.0	0.575353
$880$	0	0
$881$	37506.0	1.43429	0.717145	−	0.696924i	$-0.245449\pi$
$881$	37506.0	1.43429	0.717145	+	0.696924i	$0.245449\pi$
$882$	0	0
$883$	6388.00	0.243458	0.121729	−	0.992563i	$-0.461156\pi$
$883$	6388.00	0.243458	0.121729	+	0.992563i	$0.461156\pi$
$884$	0	0
$885$	−2160.00	−0.0820425
$886$	0	0
$887$	5472.00	0.207138	0.103569	−	0.994622i	$-0.466974\pi$
$887$	5472.00	0.207138	0.103569	+	0.994622i	$0.466974\pi$
$888$	0	0
$889$	42880.0	1.61772
$890$	0	0
$891$	−1944.00	−0.0730937
$892$	0	0
$893$	3264.00	0.122313
$894$	0	0
$895$	−72.0000	−0.00268904
$896$	0	0
$897$	2808.00	0.104522
$898$	0	0
$899$	46248.0	1.71575
$900$	0	0
$901$	22140.0	0.818635
$902$	0	0
$903$	9840.00	0.362630
$904$	0	0
$905$	28308.0	1.03977
$906$	0	0
$907$	7180.00	0.262853	0.131427	−	0.991326i	$-0.458044\pi$
$907$	7180.00	0.262853	0.131427	+	0.991326i	$0.458044\pi$
$908$	0	0
$909$	7182.00	0.262059
$910$	0	0
$911$	−27624.0	−1.00464	−0.502318	−	0.864683i	$-0.667519\pi$
$911$	−27624.0	−1.00464	−0.502318	+	0.864683i	$0.667519\pi$
$912$	0	0
$913$	13248.0	0.480224
$914$	0	0
$915$	11052.0	0.399309
$916$	0	0
$917$	−54960.0	−1.97921
$918$	0	0
$919$	30256.0	1.08602	0.543011	−	0.839726i	$-0.317284\pi$
$919$	30256.0	1.08602	0.543011	+	0.839726i	$0.317284\pi$
$920$	0	0
$921$	−20472.0	−0.732438
$922$	0	0
$923$	−1716.00	−0.0611948
$924$	0	0
$925$	−9790.00	−0.347993
$926$	0	0
$927$	4680.00	0.165816
$928$	0	0
$929$	−1926.00	−0.0680194	−0.0340097	−	0.999422i	$-0.510828\pi$
$929$	−1926.00	−0.0680194	−0.0340097	+	0.999422i	$0.510828\pi$
$930$	0	0
$931$	912.000	0.0321048
$932$	0	0
$933$	26280.0	0.922153
$934$	0	0
$935$	4320.00	0.151101
$936$	0	0
$937$	3962.00	0.138135	0.0690677	−	0.997612i	$-0.477998\pi$
$937$	3962.00	0.138135	0.0690677	+	0.997612i	$0.477998\pi$
$938$	0	0
$939$	11886.0	0.413083
$940$	0	0
$941$	−1074.00	−0.0372066	−0.0186033	−	0.999827i	$-0.505922\pi$
$941$	−1074.00	−0.0372066	−0.0186033	+	0.999827i	$0.505922\pi$
$942$	0	0
$943$	−9072.00	−0.313282
$944$	0	0
$945$	−3240.00	−0.111531
$946$	0	0
$947$	−4848.00	−0.166356	−0.0831778	−	0.996535i	$-0.526507\pi$
$947$	−4848.00	−0.166356	−0.0831778	+	0.996535i	$0.526507\pi$
$948$	0	0
$949$	2834.00	0.0969394
$950$	0	0
$951$	21258.0	0.724856
$952$	0	0
$953$	762.000	0.0259009	0.0129505	−	0.999916i	$-0.495878\pi$
$953$	762.000	0.0259009	0.0129505	+	0.999916i	$0.495878\pi$
$954$	0	0
$955$	8208.00	0.278120
$956$	0	0
$957$	20304.0	0.685826
$958$	0	0
$959$	−55080.0	−1.85467
$960$	0	0
$961$	−2895.00	−0.0971770
$962$	0	0
$963$	−108.000	−0.00361397
$964$	0	0
$965$	−19860.0	−0.662504
$966$	0	0
$967$	−35804.0	−1.19067	−0.595336	−	0.803477i	$-0.702981\pi$
$967$	−35804.0	−1.19067	−0.595336	+	0.803477i	$0.702981\pi$
$968$	0	0
$969$	−1440.00	−0.0477394
$970$	0	0
$971$	4260.00	0.140793	0.0703964	−	0.997519i	$-0.477574\pi$
$971$	4260.00	0.140793	0.0703964	+	0.997519i	$0.477574\pi$
$972$	0	0
$973$	45040.0	1.48398
$974$	0	0
$975$	−3471.00	−0.114011
$976$	0	0
$977$	−28710.0	−0.940137	−0.470069	−	0.882630i	$-0.655771\pi$
$977$	−28710.0	−0.940137	−0.470069	+	0.882630i	$0.655771\pi$
$978$	0	0
$979$	−5040.00	−0.164534
$980$	0	0
$981$	−16506.0	−0.537203
$982$	0	0
$983$	49524.0	1.60689	0.803444	−	0.595381i	$-0.202999\pi$
$983$	49524.0	1.60689	0.803444	+	0.595381i	$0.202999\pi$
$984$	0	0
$985$	18756.0	0.606717
$986$	0	0
$987$	−12240.0	−0.394735
$988$	0	0
$989$	−11808.0	−0.379649
$990$	0	0
$991$	−44408.0	−1.42348	−0.711739	−	0.702444i	$-0.752092\pi$
$991$	−44408.0	−1.42348	−0.711739	+	0.702444i	$0.752092\pi$
$992$	0	0
$993$	27048.0	0.864393
$994$	0	0
$995$	−27984.0	−0.891610
$996$	0	0
$997$	18398.0	0.584424	0.292212	−	0.956354i	$-0.405609\pi$
$997$	18398.0	0.584424	0.292212	+	0.956354i	$0.405609\pi$
$998$	0	0
$999$	2970.00	0.0940607

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.a.i.1.1		1
3.2	odd	2		1872.4.a.e.1.1		1
4.3	odd	2		78.4.a.e.1.1	✓	1
8.3	odd	2		2496.4.a.k.1.1		1
8.5	even	2		2496.4.a.b.1.1		1
12.11	even	2		234.4.a.b.1.1		1
20.19	odd	2		1950.4.a.c.1.1		1
52.31	even	4		1014.4.b.c.337.1		2
52.47	even	4		1014.4.b.c.337.2		2
52.51	odd	2		1014.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.4.a.e.1.1	✓	1	4.3	odd	2
234.4.a.b.1.1		1	12.11	even	2
624.4.a.i.1.1		1	1.1	even	1	trivial
1014.4.a.b.1.1		1	52.51	odd	2
1014.4.b.c.337.1		2	52.31	even	4
1014.4.b.c.337.2		2	52.47	even	4
1872.4.a.e.1.1		1	3.2	odd	2
1950.4.a.c.1.1		1	20.19	odd	2
2496.4.a.b.1.1		1	8.5	even	2
2496.4.a.k.1.1		1	8.3	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more