LMFDB - Embedded newform 2240.4.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2240,4,Mod(1,2240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2240, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2240.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2240 = 2^{6} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2240.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$132.164278413$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	2240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} +5.00000 q^{5} -7.00000 q^{7} +37.0000 q^{9} -68.0000 q^{11} -34.0000 q^{13} +40.0000 q^{15} +74.0000 q^{17} +128.000 q^{19} -56.0000 q^{21} -80.0000 q^{23} +25.0000 q^{25} +80.0000 q^{27} -286.000 q^{29} -24.0000 q^{31} -544.000 q^{33} -35.0000 q^{35} -294.000 q^{37} -272.000 q^{39} +66.0000 q^{41} +124.000 q^{43} +185.000 q^{45} +312.000 q^{47} +49.0000 q^{49} +592.000 q^{51} +34.0000 q^{53} -340.000 q^{55} +1024.00 q^{57} -168.000 q^{59} -170.000 q^{61} -259.000 q^{63} -170.000 q^{65} -564.000 q^{67} -640.000 q^{69} +616.000 q^{71} +250.000 q^{73} +200.000 q^{75} +476.000 q^{77} -944.000 q^{79} -359.000 q^{81} -672.000 q^{83} +370.000 q^{85} -2288.00 q^{87} -1430.00 q^{89} +238.000 q^{91} -192.000 q^{93} +640.000 q^{95} -1270.00 q^{97} -2516.00 q^{99} +O(q^{100})$

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
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−68.0000	−1.86389	−0.931944	−	0.362602i	$-0.881889\pi$
$11$	−68.0000	−1.86389	−0.931944	+	0.362602i	$0.881889\pi$
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	0	0
$15$	40.0000	0.688530
$16$	0	0
$17$	74.0000	1.05574	0.527872	−	0.849324i	$-0.322990\pi$
$17$	74.0000	1.05574	0.527872	+	0.849324i	$0.322990\pi$
$18$	0	0
$19$	128.000	1.54554	0.772769	−	0.634688i	$-0.218871\pi$
$19$	128.000	1.54554	0.772769	+	0.634688i	$0.218871\pi$
$20$	0	0
$21$	−56.0000	−0.581914
$22$	0	0
$23$	−80.0000	−0.725268	−0.362634	−	0.931932i	$-0.618122\pi$
$23$	−80.0000	−0.725268	−0.362634	+	0.931932i	$0.618122\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	−286.000	−1.83134	−0.915670	−	0.401931i	$-0.868339\pi$
$29$	−286.000	−1.83134	−0.915670	+	0.401931i	$0.868339\pi$
$30$	0	0
$31$	−24.0000	−0.139049	−0.0695246	−	0.997580i	$-0.522148\pi$
$31$	−24.0000	−0.139049	−0.0695246	+	0.997580i	$0.522148\pi$
$32$	0	0
$33$	−544.000	−2.86964
$34$	0	0
$35$	−35.0000	−0.169031
$36$	0	0
$37$	−294.000	−1.30631	−0.653153	−	0.757226i	$-0.726554\pi$
$37$	−294.000	−1.30631	−0.653153	+	0.757226i	$0.726554\pi$
$38$	0	0
$39$	−272.000	−1.11679
$40$	0	0
$41$	66.0000	0.251402	0.125701	−	0.992068i	$-0.459882\pi$
$41$	66.0000	0.251402	0.125701	+	0.992068i	$0.459882\pi$
$42$	0	0
$43$	124.000	0.439763	0.219882	−	0.975527i	$-0.429433\pi$
$43$	124.000	0.439763	0.219882	+	0.975527i	$0.429433\pi$
$44$	0	0
$45$	185.000	0.612848
$46$	0	0
$47$	312.000	0.968295	0.484148	−	0.874986i	$-0.339130\pi$
$47$	312.000	0.968295	0.484148	+	0.874986i	$0.339130\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	592.000	1.62542
$52$	0	0
$53$	34.0000	0.0881181	0.0440590	−	0.999029i	$-0.485971\pi$
$53$	34.0000	0.0881181	0.0440590	+	0.999029i	$0.485971\pi$
$54$	0	0
$55$	−340.000	−0.833556
$56$	0	0
$57$	1024.00	2.37951
$58$	0	0
$59$	−168.000	−0.370707	−0.185354	−	0.982672i	$-0.559343\pi$
$59$	−168.000	−0.370707	−0.185354	+	0.982672i	$0.559343\pi$
$60$	0	0
$61$	−170.000	−0.356824	−0.178412	−	0.983956i	$-0.557096\pi$
$61$	−170.000	−0.356824	−0.178412	+	0.983956i	$0.557096\pi$
$62$	0	0
$63$	−259.000	−0.517951
$64$	0	0
$65$	−170.000	−0.324399
$66$	0	0
$67$	−564.000	−1.02841	−0.514206	−	0.857667i	$-0.671913\pi$
$67$	−564.000	−1.02841	−0.514206	+	0.857667i	$0.671913\pi$
$68$	0	0
$69$	−640.000	−1.11662
$70$	0	0
$71$	616.000	1.02966	0.514829	−	0.857293i	$-0.327855\pi$
$71$	616.000	1.02966	0.514829	+	0.857293i	$0.327855\pi$
$72$	0	0
$73$	250.000	0.400826	0.200413	−	0.979712i	$-0.435772\pi$
$73$	250.000	0.400826	0.200413	+	0.979712i	$0.435772\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	476.000	0.704484
$78$	0	0
$79$	−944.000	−1.34441	−0.672204	−	0.740366i	$-0.734652\pi$
$79$	−944.000	−1.34441	−0.672204	+	0.740366i	$0.734652\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	−672.000	−0.888694	−0.444347	−	0.895855i	$-0.646564\pi$
$83$	−672.000	−0.888694	−0.444347	+	0.895855i	$0.646564\pi$
$84$	0	0
$85$	370.000	0.472143
$86$	0	0
$87$	−2288.00	−2.81953
$88$	0	0
$89$	−1430.00	−1.70314	−0.851571	−	0.524239i	$-0.824350\pi$
$89$	−1430.00	−1.70314	−0.851571	+	0.524239i	$0.824350\pi$
$90$	0	0
$91$	238.000	0.274167
$92$	0	0
$93$	−192.000	−0.214080
$94$	0	0
$95$	640.000	0.691185
$96$	0	0
$97$	−1270.00	−1.32937	−0.664685	−	0.747123i	$-0.731434\pi$
$97$	−1270.00	−1.32937	−0.664685	+	0.747123i	$0.731434\pi$
$98$	0	0
$99$	−2516.00	−2.55422
$100$	0	0
$101$	−930.000	−0.916222	−0.458111	−	0.888895i	$-0.651474\pi$
$101$	−930.000	−0.916222	−0.458111	+	0.888895i	$0.651474\pi$
$102$	0	0
$103$	160.000	0.153061	0.0765304	−	0.997067i	$-0.475616\pi$
$103$	160.000	0.153061	0.0765304	+	0.997067i	$0.475616\pi$
$104$	0	0
$105$	−280.000	−0.260240
$106$	0	0
$107$	−1404.00	−1.26850	−0.634251	−	0.773127i	$-0.718692\pi$
$107$	−1404.00	−1.26850	−0.634251	+	0.773127i	$0.718692\pi$
$108$	0	0
$109$	274.000	0.240775	0.120387	−	0.992727i	$-0.461586\pi$
$109$	274.000	0.240775	0.120387	+	0.992727i	$0.461586\pi$
$110$	0	0
$111$	−2352.00	−2.01119
$112$	0	0
$113$	−798.000	−0.664332	−0.332166	−	0.943221i	$-0.607779\pi$
$113$	−798.000	−0.664332	−0.332166	+	0.943221i	$0.607779\pi$
$114$	0	0
$115$	−400.000	−0.324349
$116$	0	0
$117$	−1258.00	−0.994035
$118$	0	0
$119$	−518.000	−0.399033
$120$	0	0
$121$	3293.00	2.47408
$122$	0	0
$123$	528.000	0.387058
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−904.000	−0.631630	−0.315815	−	0.948821i	$-0.602278\pi$
$127$	−904.000	−0.631630	−0.315815	+	0.948821i	$0.602278\pi$
$128$	0	0
$129$	992.000	0.677060
$130$	0	0
$131$	2080.00	1.38726	0.693628	−	0.720334i	$-0.256011\pi$
$131$	2080.00	1.38726	0.693628	+	0.720334i	$0.256011\pi$
$132$	0	0
$133$	−896.000	−0.584158
$134$	0	0
$135$	400.000	0.255011
$136$	0	0
$137$	2218.00	1.38319	0.691593	−	0.722287i	$-0.256909\pi$
$137$	2218.00	1.38319	0.691593	+	0.722287i	$0.256909\pi$
$138$	0	0
$139$	−2432.00	−1.48403	−0.742013	−	0.670386i	$-0.766129\pi$
$139$	−2432.00	−1.48403	−0.742013	+	0.670386i	$0.766129\pi$
$140$	0	0
$141$	2496.00	1.49079
$142$	0	0
$143$	2312.00	1.35202
$144$	0	0
$145$	−1430.00	−0.819000
$146$	0	0
$147$	392.000	0.219943
$148$	0	0
$149$	−1598.00	−0.878612	−0.439306	−	0.898337i	$-0.644776\pi$
$149$	−1598.00	−0.878612	−0.439306	+	0.898337i	$0.644776\pi$
$150$	0	0
$151$	−2672.00	−1.44003	−0.720014	−	0.693959i	$-0.755865\pi$
$151$	−2672.00	−1.44003	−0.720014	+	0.693959i	$0.755865\pi$
$152$	0	0
$153$	2738.00	1.44676
$154$	0	0
$155$	−120.000	−0.0621847
$156$	0	0
$157$	−1170.00	−0.594753	−0.297376	−	0.954760i	$-0.596112\pi$
$157$	−1170.00	−0.594753	−0.297376	+	0.954760i	$0.596112\pi$
$158$	0	0
$159$	272.000	0.135667
$160$	0	0
$161$	560.000	0.274125
$162$	0	0
$163$	−1580.00	−0.759234	−0.379617	−	0.925144i	$-0.623944\pi$
$163$	−1580.00	−0.759234	−0.379617	+	0.925144i	$0.623944\pi$
$164$	0	0
$165$	−2720.00	−1.28334
$166$	0	0
$167$	−704.000	−0.326211	−0.163105	−	0.986609i	$-0.552151\pi$
$167$	−704.000	−0.326211	−0.163105	+	0.986609i	$0.552151\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	4736.00	2.11796
$172$	0	0
$173$	−18.0000	−0.00791049	−0.00395524	−	0.999992i	$-0.501259\pi$
$173$	−18.0000	−0.00791049	−0.00395524	+	0.999992i	$0.501259\pi$
$174$	0	0
$175$	−175.000	−0.0755929
$176$	0	0
$177$	−1344.00	−0.570741
$178$	0	0
$179$	4268.00	1.78215	0.891076	−	0.453854i	$-0.149951\pi$
$179$	4268.00	1.78215	0.891076	+	0.453854i	$0.149951\pi$
$180$	0	0
$181$	−282.000	−0.115806	−0.0579030	−	0.998322i	$-0.518441\pi$
$181$	−282.000	−0.115806	−0.0579030	+	0.998322i	$0.518441\pi$
$182$	0	0
$183$	−1360.00	−0.549367
$184$	0	0
$185$	−1470.00	−0.584198
$186$	0	0
$187$	−5032.00	−1.96779
$188$	0	0
$189$	−560.000	−0.215524
$190$	0	0
$191$	1424.00	0.539461	0.269730	−	0.962936i	$-0.413065\pi$
$191$	1424.00	0.539461	0.269730	+	0.962936i	$0.413065\pi$
$192$	0	0
$193$	−1870.00	−0.697438	−0.348719	−	0.937227i	$-0.613383\pi$
$193$	−1870.00	−0.697438	−0.348719	+	0.937227i	$0.613383\pi$
$194$	0	0
$195$	−1360.00	−0.499444
$196$	0	0
$197$	3618.00	1.30849	0.654243	−	0.756284i	$-0.272987\pi$
$197$	3618.00	1.30849	0.654243	+	0.756284i	$0.272987\pi$
$198$	0	0
$199$	2736.00	0.974623	0.487311	−	0.873228i	$-0.337978\pi$
$199$	2736.00	0.974623	0.487311	+	0.873228i	$0.337978\pi$
$200$	0	0
$201$	−4512.00	−1.58334
$202$	0	0
$203$	2002.00	0.692182
$204$	0	0
$205$	330.000	0.112430
$206$	0	0
$207$	−2960.00	−0.993885
$208$	0	0
$209$	−8704.00	−2.88071
$210$	0	0
$211$	348.000	0.113542	0.0567709	−	0.998387i	$-0.481920\pi$
$211$	348.000	0.113542	0.0567709	+	0.998387i	$0.481920\pi$
$212$	0	0
$213$	4928.00	1.58526
$214$	0	0
$215$	620.000	0.196668
$216$	0	0
$217$	168.000	0.0525557
$218$	0	0
$219$	2000.00	0.617112
$220$	0	0
$221$	−2516.00	−0.765812
$222$	0	0
$223$	5888.00	1.76811	0.884057	−	0.467378i	$-0.154801\pi$
$223$	5888.00	1.76811	0.884057	+	0.467378i	$0.154801\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	4304.00	1.25844	0.629221	−	0.777226i	$-0.283374\pi$
$227$	4304.00	1.25844	0.629221	+	0.777226i	$0.283374\pi$
$228$	0	0
$229$	−3674.00	−1.06020	−0.530098	−	0.847937i	$-0.677845\pi$
$229$	−3674.00	−1.06020	−0.530098	+	0.847937i	$0.677845\pi$
$230$	0	0
$231$	3808.00	1.08462
$232$	0	0
$233$	−838.000	−0.235619	−0.117809	−	0.993036i	$-0.537587\pi$
$233$	−838.000	−0.235619	−0.117809	+	0.993036i	$0.537587\pi$
$234$	0	0
$235$	1560.00	0.433035
$236$	0	0
$237$	−7552.00	−2.06985
$238$	0	0
$239$	5832.00	1.57841	0.789207	−	0.614128i	$-0.210492\pi$
$239$	5832.00	1.57841	0.789207	+	0.614128i	$0.210492\pi$
$240$	0	0
$241$	1690.00	0.451711	0.225856	−	0.974161i	$-0.427482\pi$
$241$	1690.00	0.451711	0.225856	+	0.974161i	$0.427482\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	245.000	0.0638877
$246$	0	0
$247$	−4352.00	−1.12110
$248$	0	0
$249$	−5376.00	−1.36823
$250$	0	0
$251$	−5760.00	−1.44848	−0.724239	−	0.689549i	$-0.757809\pi$
$251$	−5760.00	−1.44848	−0.724239	+	0.689549i	$0.757809\pi$
$252$	0	0
$253$	5440.00	1.35182
$254$	0	0
$255$	2960.00	0.726911
$256$	0	0
$257$	−350.000	−0.0849510	−0.0424755	−	0.999098i	$-0.513524\pi$
$257$	−350.000	−0.0849510	−0.0424755	+	0.999098i	$0.513524\pi$
$258$	0	0
$259$	2058.00	0.493737
$260$	0	0
$261$	−10582.0	−2.50961
$262$	0	0
$263$	−2968.00	−0.695873	−0.347937	−	0.937518i	$-0.613118\pi$
$263$	−2968.00	−0.695873	−0.347937	+	0.937518i	$0.613118\pi$
$264$	0	0
$265$	170.000	0.0394076
$266$	0	0
$267$	−11440.0	−2.62216
$268$	0	0
$269$	4494.00	1.01860	0.509301	−	0.860588i	$-0.329904\pi$
$269$	4494.00	1.01860	0.509301	+	0.860588i	$0.329904\pi$
$270$	0	0
$271$	1312.00	0.294090	0.147045	−	0.989130i	$-0.453024\pi$
$271$	1312.00	0.294090	0.147045	+	0.989130i	$0.453024\pi$
$272$	0	0
$273$	1904.00	0.422107
$274$	0	0
$275$	−1700.00	−0.372778
$276$	0	0
$277$	5234.00	1.13531	0.567654	−	0.823267i	$-0.307851\pi$
$277$	5234.00	1.13531	0.567654	+	0.823267i	$0.307851\pi$
$278$	0	0
$279$	−888.000	−0.190549
$280$	0	0
$281$	−5718.00	−1.21390	−0.606952	−	0.794738i	$-0.707608\pi$
$281$	−5718.00	−1.21390	−0.606952	+	0.794738i	$0.707608\pi$
$282$	0	0
$283$	3752.00	0.788103	0.394052	−	0.919088i	$-0.371073\pi$
$283$	3752.00	0.788103	0.394052	+	0.919088i	$0.371073\pi$
$284$	0	0
$285$	5120.00	1.06415
$286$	0	0
$287$	−462.000	−0.0950209
$288$	0	0
$289$	563.000	0.114594
$290$	0	0
$291$	−10160.0	−2.04670
$292$	0	0
$293$	−7698.00	−1.53489	−0.767444	−	0.641116i	$-0.778472\pi$
$293$	−7698.00	−1.53489	−0.767444	+	0.641116i	$0.778472\pi$
$294$	0	0
$295$	−840.000	−0.165785
$296$	0	0
$297$	−5440.00	−1.06283
$298$	0	0
$299$	2720.00	0.526093
$300$	0	0
$301$	−868.000	−0.166215
$302$	0	0
$303$	−7440.00	−1.41062
$304$	0	0
$305$	−850.000	−0.159577
$306$	0	0
$307$	1456.00	0.270679	0.135339	−	0.990799i	$-0.456788\pi$
$307$	1456.00	0.270679	0.135339	+	0.990799i	$0.456788\pi$
$308$	0	0
$309$	1280.00	0.235653
$310$	0	0
$311$	9592.00	1.74891	0.874457	−	0.485103i	$-0.161218\pi$
$311$	9592.00	1.74891	0.874457	+	0.485103i	$0.161218\pi$
$312$	0	0
$313$	3570.00	0.644691	0.322346	−	0.946622i	$-0.395529\pi$
$313$	3570.00	0.644691	0.322346	+	0.946622i	$0.395529\pi$
$314$	0	0
$315$	−1295.00	−0.231635
$316$	0	0
$317$	3866.00	0.684972	0.342486	−	0.939523i	$-0.388731\pi$
$317$	3866.00	0.684972	0.342486	+	0.939523i	$0.388731\pi$
$318$	0	0
$319$	19448.0	3.41341
$320$	0	0
$321$	−11232.0	−1.95299
$322$	0	0
$323$	9472.00	1.63169
$324$	0	0
$325$	−850.000	−0.145075
$326$	0	0
$327$	2192.00	0.370697
$328$	0	0
$329$	−2184.00	−0.365981
$330$	0	0
$331$	2940.00	0.488209	0.244104	−	0.969749i	$-0.421506\pi$
$331$	2940.00	0.488209	0.244104	+	0.969749i	$0.421506\pi$
$332$	0	0
$333$	−10878.0	−1.79012
$334$	0	0
$335$	−2820.00	−0.459920
$336$	0	0
$337$	−414.000	−0.0669199	−0.0334600	−	0.999440i	$-0.510653\pi$
$337$	−414.000	−0.0669199	−0.0334600	+	0.999440i	$0.510653\pi$
$338$	0	0
$339$	−6384.00	−1.02281
$340$	0	0
$341$	1632.00	0.259172
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−3200.00	−0.499369
$346$	0	0
$347$	5068.00	0.784048	0.392024	−	0.919955i	$-0.371775\pi$
$347$	5068.00	0.784048	0.392024	+	0.919955i	$0.371775\pi$
$348$	0	0
$349$	−2882.00	−0.442034	−0.221017	−	0.975270i	$-0.570938\pi$
$349$	−2882.00	−0.442034	−0.221017	+	0.975270i	$0.570938\pi$
$350$	0	0
$351$	−2720.00	−0.413626
$352$	0	0
$353$	−6350.00	−0.957440	−0.478720	−	0.877968i	$-0.658899\pi$
$353$	−6350.00	−0.957440	−0.478720	+	0.877968i	$0.658899\pi$
$354$	0	0
$355$	3080.00	0.460477
$356$	0	0
$357$	−4144.00	−0.614352
$358$	0	0
$359$	4656.00	0.684497	0.342248	−	0.939610i	$-0.388812\pi$
$359$	4656.00	0.684497	0.342248	+	0.939610i	$0.388812\pi$
$360$	0	0
$361$	9525.00	1.38869
$362$	0	0
$363$	26344.0	3.80909
$364$	0	0
$365$	1250.00	0.179255
$366$	0	0
$367$	−8336.00	−1.18566	−0.592828	−	0.805329i	$-0.701989\pi$
$367$	−8336.00	−1.18566	−0.592828	+	0.805329i	$0.701989\pi$
$368$	0	0
$369$	2442.00	0.344513
$370$	0	0
$371$	−238.000	−0.0333055
$372$	0	0
$373$	−4862.00	−0.674919	−0.337460	−	0.941340i	$-0.609568\pi$
$373$	−4862.00	−0.674919	−0.337460	+	0.941340i	$0.609568\pi$
$374$	0	0
$375$	1000.00	0.137706
$376$	0	0
$377$	9724.00	1.32841
$378$	0	0
$379$	3932.00	0.532911	0.266456	−	0.963847i	$-0.414147\pi$
$379$	3932.00	0.532911	0.266456	+	0.963847i	$0.414147\pi$
$380$	0	0
$381$	−7232.00	−0.972458
$382$	0	0
$383$	−8120.00	−1.08332	−0.541661	−	0.840597i	$-0.682204\pi$
$383$	−8120.00	−1.08332	−0.541661	+	0.840597i	$0.682204\pi$
$384$	0	0
$385$	2380.00	0.315055
$386$	0	0
$387$	4588.00	0.602639
$388$	0	0
$389$	4346.00	0.566455	0.283227	−	0.959053i	$-0.408595\pi$
$389$	4346.00	0.566455	0.283227	+	0.959053i	$0.408595\pi$
$390$	0	0
$391$	−5920.00	−0.765696
$392$	0	0
$393$	16640.0	2.13582
$394$	0	0
$395$	−4720.00	−0.601238
$396$	0	0
$397$	4494.00	0.568129	0.284065	−	0.958805i	$-0.408317\pi$
$397$	4494.00	0.568129	0.284065	+	0.958805i	$0.408317\pi$
$398$	0	0
$399$	−7168.00	−0.899371
$400$	0	0
$401$	2578.00	0.321045	0.160523	−	0.987032i	$-0.448682\pi$
$401$	2578.00	0.321045	0.160523	+	0.987032i	$0.448682\pi$
$402$	0	0
$403$	816.000	0.100863
$404$	0	0
$405$	−1795.00	−0.220233
$406$	0	0
$407$	19992.0	2.43481
$408$	0	0
$409$	3730.00	0.450945	0.225473	−	0.974249i	$-0.427607\pi$
$409$	3730.00	0.450945	0.225473	+	0.974249i	$0.427607\pi$
$410$	0	0
$411$	17744.0	2.12955
$412$	0	0
$413$	1176.00	0.140114
$414$	0	0
$415$	−3360.00	−0.397436
$416$	0	0
$417$	−19456.0	−2.28481
$418$	0	0
$419$	2872.00	0.334860	0.167430	−	0.985884i	$-0.446453\pi$
$419$	2872.00	0.334860	0.167430	+	0.985884i	$0.446453\pi$
$420$	0	0
$421$	2210.00	0.255840	0.127920	−	0.991784i	$-0.459170\pi$
$421$	2210.00	0.255840	0.127920	+	0.991784i	$0.459170\pi$
$422$	0	0
$423$	11544.0	1.32692
$424$	0	0
$425$	1850.00	0.211149
$426$	0	0
$427$	1190.00	0.134867
$428$	0	0
$429$	18496.0	2.08157
$430$	0	0
$431$	−456.000	−0.0509623	−0.0254811	−	0.999675i	$-0.508112\pi$
$431$	−456.000	−0.0509623	−0.0254811	+	0.999675i	$0.508112\pi$
$432$	0	0
$433$	−5318.00	−0.590223	−0.295112	−	0.955463i	$-0.595357\pi$
$433$	−5318.00	−0.590223	−0.295112	+	0.955463i	$0.595357\pi$
$434$	0	0
$435$	−11440.0	−1.26093
$436$	0	0
$437$	−10240.0	−1.12093
$438$	0	0
$439$	16264.0	1.76820	0.884098	−	0.467301i	$-0.154774\pi$
$439$	16264.0	1.76820	0.884098	+	0.467301i	$0.154774\pi$
$440$	0	0
$441$	1813.00	0.195767
$442$	0	0
$443$	−4812.00	−0.516084	−0.258042	−	0.966134i	$-0.583077\pi$
$443$	−4812.00	−0.516084	−0.258042	+	0.966134i	$0.583077\pi$
$444$	0	0
$445$	−7150.00	−0.761669
$446$	0	0
$447$	−12784.0	−1.35271
$448$	0	0
$449$	−2590.00	−0.272226	−0.136113	−	0.990693i	$-0.543461\pi$
$449$	−2590.00	−0.272226	−0.136113	+	0.990693i	$0.543461\pi$
$450$	0	0
$451$	−4488.00	−0.468585
$452$	0	0
$453$	−21376.0	−2.21707
$454$	0	0
$455$	1190.00	0.122611
$456$	0	0
$457$	−14294.0	−1.46312	−0.731559	−	0.681778i	$-0.761207\pi$
$457$	−14294.0	−1.46312	−0.731559	+	0.681778i	$0.761207\pi$
$458$	0	0
$459$	5920.00	0.602009
$460$	0	0
$461$	−8258.00	−0.834302	−0.417151	−	0.908837i	$-0.636971\pi$
$461$	−8258.00	−0.834302	−0.417151	+	0.908837i	$0.636971\pi$
$462$	0	0
$463$	−7344.00	−0.737159	−0.368580	−	0.929596i	$-0.620156\pi$
$463$	−7344.00	−0.737159	−0.368580	+	0.929596i	$0.620156\pi$
$464$	0	0
$465$	−960.000	−0.0957396
$466$	0	0
$467$	−6288.00	−0.623071	−0.311535	−	0.950235i	$-0.600843\pi$
$467$	−6288.00	−0.623071	−0.311535	+	0.950235i	$0.600843\pi$
$468$	0	0
$469$	3948.00	0.388703
$470$	0	0
$471$	−9360.00	−0.915682
$472$	0	0
$473$	−8432.00	−0.819670
$474$	0	0
$475$	3200.00	0.309108
$476$	0	0
$477$	1258.00	0.120754
$478$	0	0
$479$	392.000	0.0373924	0.0186962	−	0.999825i	$-0.494048\pi$
$479$	392.000	0.0373924	0.0186962	+	0.999825i	$0.494048\pi$
$480$	0	0
$481$	9996.00	0.947564
$482$	0	0
$483$	4480.00	0.422044
$484$	0	0
$485$	−6350.00	−0.594513
$486$	0	0
$487$	−512.000	−0.0476405	−0.0238203	−	0.999716i	$-0.507583\pi$
$487$	−512.000	−0.0476405	−0.0238203	+	0.999716i	$0.507583\pi$
$488$	0	0
$489$	−12640.0	−1.16892
$490$	0	0
$491$	−13308.0	−1.22318	−0.611590	−	0.791175i	$-0.709470\pi$
$491$	−13308.0	−1.22318	−0.611590	+	0.791175i	$0.709470\pi$
$492$	0	0
$493$	−21164.0	−1.93343
$494$	0	0
$495$	−12580.0	−1.14228
$496$	0	0
$497$	−4312.00	−0.389174
$498$	0	0
$499$	6252.00	0.560878	0.280439	−	0.959872i	$-0.409520\pi$
$499$	6252.00	0.560878	0.280439	+	0.959872i	$0.409520\pi$
$500$	0	0
$501$	−5632.00	−0.502234
$502$	0	0
$503$	12568.0	1.11407	0.557037	−	0.830488i	$-0.311938\pi$
$503$	12568.0	1.11407	0.557037	+	0.830488i	$0.311938\pi$
$504$	0	0
$505$	−4650.00	−0.409747
$506$	0	0
$507$	−8328.00	−0.729506
$508$	0	0
$509$	5502.00	0.479120	0.239560	−	0.970882i	$-0.422997\pi$
$509$	5502.00	0.479120	0.239560	+	0.970882i	$0.422997\pi$
$510$	0	0
$511$	−1750.00	−0.151498
$512$	0	0
$513$	10240.0	0.881300
$514$	0	0
$515$	800.000	0.0684509
$516$	0	0
$517$	−21216.0	−1.80479
$518$	0	0
$519$	−144.000	−0.0121790
$520$	0	0
$521$	−19086.0	−1.60494	−0.802469	−	0.596694i	$-0.796481\pi$
$521$	−19086.0	−1.60494	−0.802469	+	0.596694i	$0.796481\pi$
$522$	0	0
$523$	16184.0	1.35311	0.676555	−	0.736392i	$-0.263472\pi$
$523$	16184.0	1.35311	0.676555	+	0.736392i	$0.263472\pi$
$524$	0	0
$525$	−1400.00	−0.116383
$526$	0	0
$527$	−1776.00	−0.146800
$528$	0	0
$529$	−5767.00	−0.473987
$530$	0	0
$531$	−6216.00	−0.508006
$532$	0	0
$533$	−2244.00	−0.182361
$534$	0	0
$535$	−7020.00	−0.567292
$536$	0	0
$537$	34144.0	2.74380
$538$	0	0
$539$	−3332.00	−0.266270
$540$	0	0
$541$	16042.0	1.27486	0.637430	−	0.770508i	$-0.279997\pi$
$541$	16042.0	1.27486	0.637430	+	0.770508i	$0.279997\pi$
$542$	0	0
$543$	−2256.00	−0.178295
$544$	0	0
$545$	1370.00	0.107678
$546$	0	0
$547$	6180.00	0.483067	0.241534	−	0.970392i	$-0.422350\pi$
$547$	6180.00	0.483067	0.241534	+	0.970392i	$0.422350\pi$
$548$	0	0
$549$	−6290.00	−0.488981
$550$	0	0
$551$	−36608.0	−2.83041
$552$	0	0
$553$	6608.00	0.508139
$554$	0	0
$555$	−11760.0	−0.899431
$556$	0	0
$557$	−9078.00	−0.690569	−0.345285	−	0.938498i	$-0.612218\pi$
$557$	−9078.00	−0.690569	−0.345285	+	0.938498i	$0.612218\pi$
$558$	0	0
$559$	−4216.00	−0.318994
$560$	0	0
$561$	−40256.0	−3.02961
$562$	0	0
$563$	−9464.00	−0.708455	−0.354227	−	0.935159i	$-0.615256\pi$
$563$	−9464.00	−0.708455	−0.354227	+	0.935159i	$0.615256\pi$
$564$	0	0
$565$	−3990.00	−0.297098
$566$	0	0
$567$	2513.00	0.186131
$568$	0	0
$569$	15722.0	1.15835	0.579174	−	0.815204i	$-0.303375\pi$
$569$	15722.0	1.15835	0.579174	+	0.815204i	$0.303375\pi$
$570$	0	0
$571$	17084.0	1.25209	0.626045	−	0.779787i	$-0.284673\pi$
$571$	17084.0	1.25209	0.626045	+	0.779787i	$0.284673\pi$
$572$	0	0
$573$	11392.0	0.830554
$574$	0	0
$575$	−2000.00	−0.145054
$576$	0	0
$577$	−16526.0	−1.19235	−0.596175	−	0.802854i	$-0.703314\pi$
$577$	−16526.0	−1.19235	−0.596175	+	0.802854i	$0.703314\pi$
$578$	0	0
$579$	−14960.0	−1.07378
$580$	0	0
$581$	4704.00	0.335895
$582$	0	0
$583$	−2312.00	−0.164242
$584$	0	0
$585$	−6290.00	−0.444546
$586$	0	0
$587$	−2832.00	−0.199130	−0.0995649	−	0.995031i	$-0.531745\pi$
$587$	−2832.00	−0.199130	−0.0995649	+	0.995031i	$0.531745\pi$
$588$	0	0
$589$	−3072.00	−0.214906
$590$	0	0
$591$	28944.0	2.01455
$592$	0	0
$593$	9666.00	0.669368	0.334684	−	0.942330i	$-0.391370\pi$
$593$	9666.00	0.669368	0.334684	+	0.942330i	$0.391370\pi$
$594$	0	0
$595$	−2590.00	−0.178453
$596$	0	0
$597$	21888.0	1.50053
$598$	0	0
$599$	4536.00	0.309409	0.154704	−	0.987961i	$-0.450557\pi$
$599$	4536.00	0.309409	0.154704	+	0.987961i	$0.450557\pi$
$600$	0	0
$601$	−15542.0	−1.05486	−0.527430	−	0.849598i	$-0.676844\pi$
$601$	−15542.0	−1.05486	−0.527430	+	0.849598i	$0.676844\pi$
$602$	0	0
$603$	−20868.0	−1.40930
$604$	0	0
$605$	16465.0	1.10644
$606$	0	0
$607$	−23648.0	−1.58129	−0.790645	−	0.612275i	$-0.790254\pi$
$607$	−23648.0	−1.58129	−0.790645	+	0.612275i	$0.790254\pi$
$608$	0	0
$609$	16016.0	1.06568
$610$	0	0
$611$	−10608.0	−0.702379
$612$	0	0
$613$	906.000	0.0596949	0.0298475	−	0.999554i	$-0.490498\pi$
$613$	906.000	0.0596949	0.0298475	+	0.999554i	$0.490498\pi$
$614$	0	0
$615$	2640.00	0.173098
$616$	0	0
$617$	26666.0	1.73992	0.869962	−	0.493119i	$-0.164143\pi$
$617$	26666.0	1.73992	0.869962	+	0.493119i	$0.164143\pi$
$618$	0	0
$619$	3352.00	0.217655	0.108827	−	0.994061i	$-0.465290\pi$
$619$	3352.00	0.217655	0.108827	+	0.994061i	$0.465290\pi$
$620$	0	0
$621$	−6400.00	−0.413564
$622$	0	0
$623$	10010.0	0.643727
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−69632.0	−4.43514
$628$	0	0
$629$	−21756.0	−1.37912
$630$	0	0
$631$	2792.00	0.176145	0.0880727	−	0.996114i	$-0.471929\pi$
$631$	2792.00	0.176145	0.0880727	+	0.996114i	$0.471929\pi$
$632$	0	0
$633$	2784.00	0.174809
$634$	0	0
$635$	−4520.00	−0.282474
$636$	0	0
$637$	−1666.00	−0.103625
$638$	0	0
$639$	22792.0	1.41101
$640$	0	0
$641$	19330.0	1.19109	0.595545	−	0.803322i	$-0.296936\pi$
$641$	19330.0	1.19109	0.595545	+	0.803322i	$0.296936\pi$
$642$	0	0
$643$	15160.0	0.929785	0.464893	−	0.885367i	$-0.346093\pi$
$643$	15160.0	0.929785	0.464893	+	0.885367i	$0.346093\pi$
$644$	0	0
$645$	4960.00	0.302790
$646$	0	0
$647$	−23472.0	−1.42624	−0.713122	−	0.701040i	$-0.752719\pi$
$647$	−23472.0	−1.42624	−0.713122	+	0.701040i	$0.752719\pi$
$648$	0	0
$649$	11424.0	0.690957
$650$	0	0
$651$	1344.00	0.0809148
$652$	0	0
$653$	−542.000	−0.0324810	−0.0162405	−	0.999868i	$-0.505170\pi$
$653$	−542.000	−0.0324810	−0.0162405	+	0.999868i	$0.505170\pi$
$654$	0	0
$655$	10400.0	0.620399
$656$	0	0
$657$	9250.00	0.549280
$658$	0	0
$659$	19764.0	1.16828	0.584140	−	0.811653i	$-0.301432\pi$
$659$	19764.0	1.16828	0.584140	+	0.811653i	$0.301432\pi$
$660$	0	0
$661$	−4130.00	−0.243023	−0.121512	−	0.992590i	$-0.538774\pi$
$661$	−4130.00	−0.243023	−0.121512	+	0.992590i	$0.538774\pi$
$662$	0	0
$663$	−20128.0	−1.17904
$664$	0	0
$665$	−4480.00	−0.261244
$666$	0	0
$667$	22880.0	1.32821
$668$	0	0
$669$	47104.0	2.72219
$670$	0	0
$671$	11560.0	0.665080
$672$	0	0
$673$	−24366.0	−1.39560	−0.697801	−	0.716292i	$-0.745838\pi$
$673$	−24366.0	−1.39560	−0.697801	+	0.716292i	$0.745838\pi$
$674$	0	0
$675$	2000.00	0.114044
$676$	0	0
$677$	16790.0	0.953164	0.476582	−	0.879130i	$-0.341876\pi$
$677$	16790.0	0.953164	0.476582	+	0.879130i	$0.341876\pi$
$678$	0	0
$679$	8890.00	0.502455
$680$	0	0
$681$	34432.0	1.93750
$682$	0	0
$683$	7764.00	0.434965	0.217483	−	0.976064i	$-0.430215\pi$
$683$	7764.00	0.434965	0.217483	+	0.976064i	$0.430215\pi$
$684$	0	0
$685$	11090.0	0.618580
$686$	0	0
$687$	−29392.0	−1.63228
$688$	0	0
$689$	−1156.00	−0.0639189
$690$	0	0
$691$	2064.00	0.113630	0.0568149	−	0.998385i	$-0.481905\pi$
$691$	2064.00	0.113630	0.0568149	+	0.998385i	$0.481905\pi$
$692$	0	0
$693$	17612.0	0.965403
$694$	0	0
$695$	−12160.0	−0.663676
$696$	0	0
$697$	4884.00	0.265416
$698$	0	0
$699$	−6704.00	−0.362759
$700$	0	0
$701$	−1230.00	−0.0662717	−0.0331358	−	0.999451i	$-0.510549\pi$
$701$	−1230.00	−0.0662717	−0.0331358	+	0.999451i	$0.510549\pi$
$702$	0	0
$703$	−37632.0	−2.01894
$704$	0	0
$705$	12480.0	0.666701
$706$	0	0
$707$	6510.00	0.346300
$708$	0	0
$709$	−5158.00	−0.273220	−0.136610	−	0.990625i	$-0.543621\pi$
$709$	−5158.00	−0.273220	−0.136610	+	0.990625i	$0.543621\pi$
$710$	0	0
$711$	−34928.0	−1.84234
$712$	0	0
$713$	1920.00	0.100848
$714$	0	0
$715$	11560.0	0.604643
$716$	0	0
$717$	46656.0	2.43013
$718$	0	0
$719$	−23800.0	−1.23448	−0.617239	−	0.786775i	$-0.711749\pi$
$719$	−23800.0	−1.23448	−0.617239	+	0.786775i	$0.711749\pi$
$720$	0	0
$721$	−1120.00	−0.0578516
$722$	0	0
$723$	13520.0	0.695455
$724$	0	0
$725$	−7150.00	−0.366268
$726$	0	0
$727$	−12832.0	−0.654625	−0.327313	−	0.944916i	$-0.606143\pi$
$727$	−12832.0	−0.654625	−0.327313	+	0.944916i	$0.606143\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	9176.00	0.464277
$732$	0	0
$733$	−15370.0	−0.774494	−0.387247	−	0.921976i	$-0.626574\pi$
$733$	−15370.0	−0.774494	−0.387247	+	0.921976i	$0.626574\pi$
$734$	0	0
$735$	1960.00	0.0983615
$736$	0	0
$737$	38352.0	1.91684
$738$	0	0
$739$	−16940.0	−0.843231	−0.421616	−	0.906775i	$-0.638537\pi$
$739$	−16940.0	−0.843231	−0.421616	+	0.906775i	$0.638537\pi$
$740$	0	0
$741$	−34816.0	−1.72604
$742$	0	0
$743$	39792.0	1.96477	0.982387	−	0.186858i	$-0.0598305\pi$
$743$	39792.0	1.96477	0.982387	+	0.186858i	$0.0598305\pi$
$744$	0	0
$745$	−7990.00	−0.392927
$746$	0	0
$747$	−24864.0	−1.21784
$748$	0	0
$749$	9828.00	0.479449
$750$	0	0
$751$	−776.000	−0.0377052	−0.0188526	−	0.999822i	$-0.506001\pi$
$751$	−776.000	−0.0377052	−0.0188526	+	0.999822i	$0.506001\pi$
$752$	0	0
$753$	−46080.0	−2.23008
$754$	0	0
$755$	−13360.0	−0.644000
$756$	0	0
$757$	17882.0	0.858563	0.429282	−	0.903171i	$-0.358767\pi$
$757$	17882.0	0.858563	0.429282	+	0.903171i	$0.358767\pi$
$758$	0	0
$759$	43520.0	2.08126
$760$	0	0
$761$	6946.00	0.330870	0.165435	−	0.986221i	$-0.447097\pi$
$761$	6946.00	0.330870	0.165435	+	0.986221i	$0.447097\pi$
$762$	0	0
$763$	−1918.00	−0.0910043
$764$	0	0
$765$	13690.0	0.647010
$766$	0	0
$767$	5712.00	0.268903
$768$	0	0
$769$	35002.0	1.64136	0.820679	−	0.571389i	$-0.193595\pi$
$769$	35002.0	1.64136	0.820679	+	0.571389i	$0.193595\pi$
$770$	0	0
$771$	−2800.00	−0.130791
$772$	0	0
$773$	14414.0	0.670680	0.335340	−	0.942097i	$-0.391149\pi$
$773$	14414.0	0.670680	0.335340	+	0.942097i	$0.391149\pi$
$774$	0	0
$775$	−600.000	−0.0278099
$776$	0	0
$777$	16464.0	0.760158
$778$	0	0
$779$	8448.00	0.388551
$780$	0	0
$781$	−41888.0	−1.91917
$782$	0	0
$783$	−22880.0	−1.04427
$784$	0	0
$785$	−5850.00	−0.265981
$786$	0	0
$787$	7312.00	0.331188	0.165594	−	0.986194i	$-0.447046\pi$
$787$	7312.00	0.331188	0.165594	+	0.986194i	$0.447046\pi$
$788$	0	0
$789$	−23744.0	−1.07137
$790$	0	0
$791$	5586.00	0.251094
$792$	0	0
$793$	5780.00	0.258832
$794$	0	0
$795$	1360.00	0.0606720
$796$	0	0
$797$	−4290.00	−0.190664	−0.0953322	−	0.995446i	$-0.530391\pi$
$797$	−4290.00	−0.190664	−0.0953322	+	0.995446i	$0.530391\pi$
$798$	0	0
$799$	23088.0	1.02227
$800$	0	0
$801$	−52910.0	−2.33394
$802$	0	0
$803$	−17000.0	−0.747095
$804$	0	0
$805$	2800.00	0.122593
$806$	0	0
$807$	35952.0	1.56824
$808$	0	0
$809$	−28246.0	−1.22754	−0.613768	−	0.789487i	$-0.710347\pi$
$809$	−28246.0	−1.22754	−0.613768	+	0.789487i	$0.710347\pi$
$810$	0	0
$811$	15656.0	0.677875	0.338937	−	0.940809i	$-0.389933\pi$
$811$	15656.0	0.677875	0.338937	+	0.940809i	$0.389933\pi$
$812$	0	0
$813$	10496.0	0.452781
$814$	0	0
$815$	−7900.00	−0.339540
$816$	0	0
$817$	15872.0	0.679671
$818$	0	0
$819$	8806.00	0.375710
$820$	0	0
$821$	27506.0	1.16926	0.584632	−	0.811298i	$-0.301239\pi$
$821$	27506.0	1.16926	0.584632	+	0.811298i	$0.301239\pi$
$822$	0	0
$823$	36552.0	1.54814	0.774072	−	0.633097i	$-0.218217\pi$
$823$	36552.0	1.54814	0.774072	+	0.633097i	$0.218217\pi$
$824$	0	0
$825$	−13600.0	−0.573929
$826$	0	0
$827$	−18892.0	−0.794364	−0.397182	−	0.917740i	$-0.630012\pi$
$827$	−18892.0	−0.794364	−0.397182	+	0.917740i	$0.630012\pi$
$828$	0	0
$829$	19446.0	0.814701	0.407351	−	0.913272i	$-0.366453\pi$
$829$	19446.0	0.814701	0.407351	+	0.913272i	$0.366453\pi$
$830$	0	0
$831$	41872.0	1.74792
$832$	0	0
$833$	3626.00	0.150820
$834$	0	0
$835$	−3520.00	−0.145886
$836$	0	0
$837$	−1920.00	−0.0792890
$838$	0	0
$839$	12416.0	0.510903	0.255452	−	0.966822i	$-0.417776\pi$
$839$	12416.0	0.510903	0.255452	+	0.966822i	$0.417776\pi$
$840$	0	0
$841$	57407.0	2.35381
$842$	0	0
$843$	−45744.0	−1.86893
$844$	0	0
$845$	−5205.00	−0.211902
$846$	0	0
$847$	−23051.0	−0.935114
$848$	0	0
$849$	30016.0	1.21336
$850$	0	0
$851$	23520.0	0.947421
$852$	0	0
$853$	−19210.0	−0.771088	−0.385544	−	0.922689i	$-0.625986\pi$
$853$	−19210.0	−0.771088	−0.385544	+	0.922689i	$0.625986\pi$
$854$	0	0
$855$	23680.0	0.947180
$856$	0	0
$857$	530.000	0.0211254	0.0105627	−	0.999944i	$-0.496638\pi$
$857$	530.000	0.0211254	0.0105627	+	0.999944i	$0.496638\pi$
$858$	0	0
$859$	2624.00	0.104226	0.0521128	−	0.998641i	$-0.483404\pi$
$859$	2624.00	0.104226	0.0521128	+	0.998641i	$0.483404\pi$
$860$	0	0
$861$	−3696.00	−0.146294
$862$	0	0
$863$	37136.0	1.46480	0.732401	−	0.680874i	$-0.238400\pi$
$863$	37136.0	1.46480	0.732401	+	0.680874i	$0.238400\pi$
$864$	0	0
$865$	−90.0000	−0.00353768
$866$	0	0
$867$	4504.00	0.176429
$868$	0	0
$869$	64192.0	2.50583
$870$	0	0
$871$	19176.0	0.745986
$872$	0	0
$873$	−46990.0	−1.82173
$874$	0	0
$875$	−875.000	−0.0338062
$876$	0	0
$877$	−44606.0	−1.71749	−0.858744	−	0.512404i	$-0.828755\pi$
$877$	−44606.0	−1.71749	−0.858744	+	0.512404i	$0.828755\pi$
$878$	0	0
$879$	−61584.0	−2.36311
$880$	0	0
$881$	−41806.0	−1.59873	−0.799364	−	0.600847i	$-0.794830\pi$
$881$	−41806.0	−1.59873	−0.799364	+	0.600847i	$0.794830\pi$
$882$	0	0
$883$	−8324.00	−0.317242	−0.158621	−	0.987340i	$-0.550705\pi$
$883$	−8324.00	−0.317242	−0.158621	+	0.987340i	$0.550705\pi$
$884$	0	0
$885$	−6720.00	−0.255243
$886$	0	0
$887$	−6592.00	−0.249535	−0.124768	−	0.992186i	$-0.539819\pi$
$887$	−6592.00	−0.249535	−0.124768	+	0.992186i	$0.539819\pi$
$888$	0	0
$889$	6328.00	0.238734
$890$	0	0
$891$	24412.0	0.917882
$892$	0	0
$893$	39936.0	1.49654
$894$	0	0
$895$	21340.0	0.797003
$896$	0	0
$897$	21760.0	0.809972
$898$	0	0
$899$	6864.00	0.254647
$900$	0	0
$901$	2516.00	0.0930301
$902$	0	0
$903$	−6944.00	−0.255905
$904$	0	0
$905$	−1410.00	−0.0517900
$906$	0	0
$907$	−43948.0	−1.60890	−0.804448	−	0.594023i	$-0.797539\pi$
$907$	−43948.0	−1.60890	−0.804448	+	0.594023i	$0.797539\pi$
$908$	0	0
$909$	−34410.0	−1.25556
$910$	0	0
$911$	14936.0	0.543196	0.271598	−	0.962411i	$-0.412448\pi$
$911$	14936.0	0.543196	0.271598	+	0.962411i	$0.412448\pi$
$912$	0	0
$913$	45696.0	1.65643
$914$	0	0
$915$	−6800.00	−0.245684
$916$	0	0
$917$	−14560.0	−0.524333
$918$	0	0
$919$	8104.00	0.290888	0.145444	−	0.989366i	$-0.453539\pi$
$919$	8104.00	0.290888	0.145444	+	0.989366i	$0.453539\pi$
$920$	0	0
$921$	11648.0	0.416737
$922$	0	0
$923$	−20944.0	−0.746891
$924$	0	0
$925$	−7350.00	−0.261261
$926$	0	0
$927$	5920.00	0.209750
$928$	0	0
$929$	−43622.0	−1.54057	−0.770286	−	0.637699i	$-0.779887\pi$
$929$	−43622.0	−1.54057	−0.770286	+	0.637699i	$0.779887\pi$
$930$	0	0
$931$	6272.00	0.220791
$932$	0	0
$933$	76736.0	2.69263
$934$	0	0
$935$	−25160.0	−0.880021
$936$	0	0
$937$	−30278.0	−1.05564	−0.527822	−	0.849355i	$-0.676991\pi$
$937$	−30278.0	−1.05564	−0.527822	+	0.849355i	$0.676991\pi$
$938$	0	0
$939$	28560.0	0.992567
$940$	0	0
$941$	−45594.0	−1.57951	−0.789757	−	0.613420i	$-0.789793\pi$
$941$	−45594.0	−1.57951	−0.789757	+	0.613420i	$0.789793\pi$
$942$	0	0
$943$	−5280.00	−0.182333
$944$	0	0
$945$	−2800.00	−0.0963852
$946$	0	0
$947$	−5836.00	−0.200258	−0.100129	−	0.994974i	$-0.531926\pi$
$947$	−5836.00	−0.200258	−0.100129	+	0.994974i	$0.531926\pi$
$948$	0	0
$949$	−8500.00	−0.290750
$950$	0	0
$951$	30928.0	1.05458
$952$	0	0
$953$	7274.00	0.247249	0.123624	−	0.992329i	$-0.460548\pi$
$953$	7274.00	0.247249	0.123624	+	0.992329i	$0.460548\pi$
$954$	0	0
$955$	7120.00	0.241254
$956$	0	0
$957$	155584.	5.25529
$958$	0	0
$959$	−15526.0	−0.522795
$960$	0	0
$961$	−29215.0	−0.980665
$962$	0	0
$963$	−51948.0	−1.73832
$964$	0	0
$965$	−9350.00	−0.311904
$966$	0	0
$967$	−2432.00	−0.0808768	−0.0404384	−	0.999182i	$-0.512875\pi$
$967$	−2432.00	−0.0808768	−0.0404384	+	0.999182i	$0.512875\pi$
$968$	0	0
$969$	75776.0	2.51215
$970$	0	0
$971$	7496.00	0.247743	0.123871	−	0.992298i	$-0.460469\pi$
$971$	7496.00	0.247743	0.123871	+	0.992298i	$0.460469\pi$
$972$	0	0
$973$	17024.0	0.560909
$974$	0	0
$975$	−6800.00	−0.223358
$976$	0	0
$977$	−56574.0	−1.85257	−0.926286	−	0.376822i	$-0.877017\pi$
$977$	−56574.0	−1.85257	−0.926286	+	0.376822i	$0.877017\pi$
$978$	0	0
$979$	97240.0	3.17447
$980$	0	0
$981$	10138.0	0.329950
$982$	0	0
$983$	−25776.0	−0.836345	−0.418172	−	0.908368i	$-0.637329\pi$
$983$	−25776.0	−0.836345	−0.418172	+	0.908368i	$0.637329\pi$
$984$	0	0
$985$	18090.0	0.585173
$986$	0	0
$987$	−17472.0	−0.563465
$988$	0	0
$989$	−9920.00	−0.318946
$990$	0	0
$991$	−60080.0	−1.92584	−0.962918	−	0.269793i	$-0.913045\pi$
$991$	−60080.0	−1.92584	−0.962918	+	0.269793i	$0.913045\pi$
$992$	0	0
$993$	23520.0	0.751646
$994$	0	0
$995$	13680.0	0.435864
$996$	0	0
$997$	−49906.0	−1.58529	−0.792647	−	0.609680i	$-0.791298\pi$
$997$	−49906.0	−1.58529	−0.792647	+	0.609680i	$0.791298\pi$
$998$	0	0
$999$	−23520.0	−0.744885

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2240.4.a.bi.1.1		1
4.3	odd	2		2240.4.a.d.1.1		1
8.3	odd	2		560.4.a.q.1.1		1
8.5	even	2		70.4.a.a.1.1	✓	1
24.5	odd	2		630.4.a.s.1.1		1
40.13	odd	4		350.4.c.n.99.2		2
40.29	even	2		350.4.a.v.1.1		1
40.37	odd	4		350.4.c.n.99.1		2
56.5	odd	6		490.4.e.j.361.1		2
56.13	odd	2		490.4.a.g.1.1		1
56.37	even	6		490.4.e.r.361.1		2
56.45	odd	6		490.4.e.j.471.1		2
56.53	even	6		490.4.e.r.471.1		2
280.69	odd	2		2450.4.a.x.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.4.a.a.1.1	✓	1	8.5	even	2
350.4.a.v.1.1		1	40.29	even	2
350.4.c.n.99.1		2	40.37	odd	4
350.4.c.n.99.2		2	40.13	odd	4
490.4.a.g.1.1		1	56.13	odd	2
490.4.e.j.361.1		2	56.5	odd	6
490.4.e.j.471.1		2	56.45	odd	6
490.4.e.r.361.1		2	56.37	even	6
490.4.e.r.471.1		2	56.53	even	6
560.4.a.q.1.1		1	8.3	odd	2
630.4.a.s.1.1		1	24.5	odd	2
2240.4.a.d.1.1		1	4.3	odd	2
2240.4.a.bi.1.1		1	1.1	even	1	trivial
2450.4.a.x.1.1		1	280.69	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2240.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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