LMFDB - Embedded newform 2592.3.e.i.161.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,3,Mod(161,2592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("2592.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$N$	$=$	$2592 = 2^{5} \cdot 3^{4}$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	2592.e (of order $2$ , degree $1$ , minimal)

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:	no
Analytic conductor:	$70.6268845222$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.10
Character	$\chi$	$=$	2592.161
Dual form			2592.3.e.i.161.9

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.12450i q^{5} +6.84941 q^{7} -5.24998i q^{11} +20.0322 q^{13} +0.0666728i q^{17} +27.2587 q^{19} +21.4411i q^{23} +20.4865 q^{25} -43.0612i q^{29} -16.1303 q^{31} +14.5516i q^{35} -22.3835 q^{37} +2.96299i q^{41} +29.3416 q^{43} -44.3085i q^{47} -2.08563 q^{49} +12.8946i q^{53} +11.1536 q^{55} +86.6134i q^{59} -12.9054 q^{61} +42.5585i q^{65} -122.838 q^{67} -125.360i q^{71} +90.7813 q^{73} -35.9593i q^{77} +48.5047 q^{79} -116.787i q^{83} -0.141647 q^{85} -116.172i q^{89} +137.209 q^{91} +57.9111i q^{95} +46.5562 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q - 120 q^{25} + 72 q^{49} + 192 q^{61} + 24 q^{73} - 192 q^{85} + 264 q^{97}+O(q^{100})$ 24 * q - 120 * q^25 + 72 * q^49 + 192 * q^61 + 24 * q^73 - 192 * q^85 + 264 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times$ .

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.12450i	0.424900i	0.977172	+	0.212450i	$0.0681443\pi$
$5$	2.12450i	0.424900i	−0.977172	+	0.212450i	$0.931856\pi$
$6$	0	0
$7$	6.84941	0.978487	0.489243	−	0.872147i	$-0.337273\pi$
$7$	6.84941	0.978487	0.489243	+	0.872147i	$0.337273\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 5.24998i	− 0.477271i	−0.971109	−	0.238636i	$-0.923300\pi$
$11$	− 5.24998i	− 0.477271i	0.971109	−	0.238636i	$-0.0767002\pi$
$12$	0	0
$13$	20.0322	1.54094	0.770471	−	0.637475i	$-0.220021\pi$
$13$	20.0322	1.54094	0.770471	+	0.637475i	$0.220021\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.0666728i	0.00392193i	0.999998	+	0.00196097i	$0.000624195\pi$
$17$	0.0666728i	0.00392193i	−0.999998	+	0.00196097i	$0.999376\pi$
$18$	0	0
$19$	27.2587	1.43467	0.717334	−	0.696730i	$-0.245362\pi$
$19$	27.2587	1.43467	0.717334	+	0.696730i	$0.245362\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	21.4411i	0.932220i	0.884727	+	0.466110i	$0.154345\pi$
$23$	21.4411i	0.932220i	−0.884727	+	0.466110i	$0.845655\pi$
$24$	0	0
$25$	20.4865	0.819460
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 43.0612i	− 1.48487i	−0.669918	−	0.742435i	$-0.733671\pi$
$29$	− 43.0612i	− 1.48487i	0.669918	−	0.742435i	$-0.266329\pi$
$30$	0	0
$31$	−16.1303	−0.520333	−0.260167	−	0.965564i	$-0.583777\pi$
$31$	−16.1303	−0.520333	−0.260167	+	0.965564i	$0.583777\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.5516i	0.415759i
$36$	0	0
$37$	−22.3835	−0.604961	−0.302480	−	0.953156i	$-0.597815\pi$
$37$	−22.3835	−0.604961	−0.302480	+	0.953156i	$0.597815\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.96299i	0.0722680i	0.999347	+	0.0361340i	$0.0115043\pi$
$41$	2.96299i	0.0722680i	−0.999347	+	0.0361340i	$0.988496\pi$
$42$	0	0
$43$	29.3416	0.682363	0.341182	−	0.939997i	$-0.389173\pi$
$43$	29.3416	0.682363	0.341182	+	0.939997i	$0.389173\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 44.3085i	− 0.942734i	−0.881937	−	0.471367i	$-0.843761\pi$
$47$	− 44.3085i	− 0.942734i	0.881937	−	0.471367i	$-0.156239\pi$
$48$	0	0
$49$	−2.08563	−0.0425638
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.8946i	0.243294i	0.992573	+	0.121647i	$0.0388176\pi$
$53$	12.8946i	0.243294i	−0.992573	+	0.121647i	$0.961182\pi$
$54$	0	0
$55$	11.1536	0.202793
$56$	0	0
$57$	0	0
$58$	0	0
$59$	86.6134i	1.46802i	0.679137	+	0.734012i	$0.262354\pi$
$59$	86.6134i	1.46802i	−0.679137	+	0.734012i	$0.737646\pi$
$60$	0	0
$61$	−12.9054	−0.211564	−0.105782	−	0.994389i	$-0.533735\pi$
$61$	−12.9054	−0.211564	−0.105782	+	0.994389i	$0.533735\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	42.5585i	0.654747i
$66$	0	0
$67$	−122.838	−1.83341	−0.916704	−	0.399568i	$-0.869160\pi$
$67$	−122.838	−1.83341	−0.916704	+	0.399568i	$0.869160\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 125.360i	− 1.76563i	−0.469717	−	0.882817i	$-0.655644\pi$
$71$	− 125.360i	− 1.76563i	0.469717	−	0.882817i	$-0.344356\pi$
$72$	0	0
$73$	90.7813	1.24358	0.621790	−	0.783184i	$-0.286406\pi$
$73$	90.7813	1.24358	0.621790	+	0.783184i	$0.286406\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 35.9593i	− 0.467004i
$78$	0	0
$79$	48.5047	0.613983	0.306992	−	0.951712i	$-0.400678\pi$
$79$	48.5047	0.613983	0.306992	+	0.951712i	$0.400678\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 116.787i	− 1.40707i	−0.710661	−	0.703534i	$-0.751604\pi$
$83$	− 116.787i	− 1.40707i	0.710661	−	0.703534i	$-0.248396\pi$
$84$	0	0
$85$	−0.141647	−0.00166643
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 116.172i	− 1.30531i	−0.757656	−	0.652654i	$-0.773655\pi$
$89$	− 116.172i	− 1.30531i	0.757656	−	0.652654i	$-0.226345\pi$
$90$	0	0
$91$	137.209	1.50779
$92$	0	0
$93$	0	0
$94$	0	0
$95$	57.9111i	0.609591i
$96$	0	0
$97$	46.5562	0.479961	0.239980	−	0.970778i	$-0.422859\pi$
$97$	46.5562	0.479961	0.239980	+	0.970778i	$0.422859\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	135.120i	1.33782i	0.743343	+	0.668911i	$0.233239\pi$
$101$	135.120i	1.33782i	−0.743343	+	0.668911i	$0.766761\pi$
$102$	0	0
$103$	−15.8330	−0.153718	−0.0768592	−	0.997042i	$-0.524489\pi$
$103$	−15.8330	−0.153718	−0.0768592	+	0.997042i	$0.524489\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	191.129i	1.78625i	0.449810	+	0.893124i	$0.351492\pi$
$107$	191.129i	1.78625i	−0.449810	+	0.893124i	$0.648508\pi$
$108$	0	0
$109$	144.818	1.32861	0.664304	−	0.747463i	$-0.268728\pi$
$109$	144.818	1.32861	0.664304	+	0.747463i	$0.268728\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 96.4026i	− 0.853120i	−0.904459	−	0.426560i	$-0.859725\pi$
$113$	− 96.4026i	− 0.853120i	0.904459	−	0.426560i	$-0.140275\pi$
$114$	0	0
$115$	−45.5516	−0.396101
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.456669i	0.00383756i
$120$	0	0
$121$	93.4377	0.772212
$122$	0	0
$123$	0	0
$124$	0	0
$125$	96.6361i	0.773089i
$126$	0	0
$127$	−127.727	−1.00572	−0.502861	−	0.864367i	$-0.667719\pi$
$127$	−127.727	−1.00572	−0.502861	+	0.864367i	$0.667719\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 32.6907i	− 0.249548i	−0.992185	−	0.124774i	$-0.960179\pi$
$131$	− 32.6907i	− 0.249548i	0.992185	−	0.124774i	$-0.0398205\pi$
$132$	0	0
$133$	186.706	1.40380
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 184.427i	− 1.34618i	−0.739561	−	0.673089i	$-0.764967\pi$
$137$	− 184.427i	− 1.34618i	0.739561	−	0.673089i	$-0.235033\pi$
$138$	0	0
$139$	−238.705	−1.71730	−0.858652	−	0.512558i	$-0.828698\pi$
$139$	−238.705	−1.71730	−0.858652	+	0.512558i	$0.828698\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 105.169i	− 0.735447i
$144$	0	0
$145$	91.4837	0.630922
$146$	0	0
$147$	0	0
$148$	0	0
$149$	256.660i	1.72255i	0.508141	+	0.861274i	$0.330333\pi$
$149$	256.660i	1.72255i	−0.508141	+	0.861274i	$0.669667\pi$
$150$	0	0
$151$	−83.2741	−0.551484	−0.275742	−	0.961232i	$-0.588924\pi$
$151$	−83.2741	−0.551484	−0.275742	+	0.961232i	$0.588924\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 34.2689i	− 0.221090i
$156$	0	0
$157$	159.564	1.01633	0.508166	−	0.861259i	$-0.330324\pi$
$157$	159.564	1.01633	0.508166	+	0.861259i	$0.330324\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	146.859i	0.912165i
$162$	0	0
$163$	18.4140	0.112970	0.0564848	−	0.998403i	$-0.482011\pi$
$163$	18.4140	0.112970	0.0564848	+	0.998403i	$0.482011\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	202.147i	1.21046i	0.796051	+	0.605229i	$0.206919\pi$
$167$	202.147i	1.21046i	−0.796051	+	0.605229i	$0.793081\pi$
$168$	0	0
$169$	232.291	1.37450
$170$	0	0
$171$	0	0
$172$	0	0
$173$	100.965i	0.583613i	0.956477	+	0.291807i	$0.0942564\pi$
$173$	100.965i	0.583613i	−0.956477	+	0.291807i	$0.905744\pi$
$174$	0	0
$175$	140.320	0.801830
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 45.8305i	− 0.256036i	−0.991772	−	0.128018i	$-0.959138\pi$
$179$	− 45.8305i	− 0.256036i	0.991772	−	0.128018i	$-0.0408616\pi$
$180$	0	0
$181$	55.7708	0.308126	0.154063	−	0.988061i	$-0.450764\pi$
$181$	55.7708	0.308126	0.154063	+	0.988061i	$0.450764\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 47.5539i	− 0.257048i
$186$	0	0
$187$	0.350031	0.00187182
$188$	0	0
$189$	0	0
$190$	0	0
$191$	316.396i	1.65652i	0.560341	+	0.828262i	$0.310670\pi$
$191$	316.396i	1.65652i	−0.560341	+	0.828262i	$0.689330\pi$
$192$	0	0
$193$	375.293	1.94452	0.972262	−	0.233895i	$-0.0751474\pi$
$193$	375.293	1.94452	0.972262	+	0.233895i	$0.0751474\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 46.6367i	− 0.236734i	−0.992970	−	0.118367i	$-0.962234\pi$
$197$	− 46.6367i	− 0.236734i	0.992970	−	0.118367i	$-0.0377660\pi$
$198$	0	0
$199$	−192.041	−0.965033	−0.482516	−	0.875887i	$-0.660277\pi$
$199$	−192.041	−0.965033	−0.482516	+	0.875887i	$0.660277\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 294.944i	− 1.45293i
$204$	0	0
$205$	−6.29487	−0.0307067
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 143.108i	− 0.684725i
$210$	0	0
$211$	315.180	1.49374	0.746872	−	0.664968i	$-0.231555\pi$
$211$	315.180	1.49374	0.746872	+	0.664968i	$0.231555\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	62.3363i	0.289936i
$216$	0	0
$217$	−110.483	−0.509139
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.33561i	0.00604347i
$222$	0	0
$223$	184.372	0.826779	0.413390	−	0.910554i	$-0.364345\pi$
$223$	184.372	0.826779	0.413390	+	0.910554i	$0.364345\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 132.566i	− 0.583992i	−0.956420	−	0.291996i	$-0.905681\pi$
$227$	− 132.566i	− 0.583992i	0.956420	−	0.291996i	$-0.0943193\pi$
$228$	0	0
$229$	−24.0705	−0.105111	−0.0525556	−	0.998618i	$-0.516737\pi$
$229$	−24.0705	−0.105111	−0.0525556	+	0.998618i	$0.516737\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 135.751i	− 0.582621i	−0.956629	−	0.291311i	$-0.905909\pi$
$233$	− 135.751i	− 0.582621i	0.956629	−	0.291311i	$-0.0940913\pi$
$234$	0	0
$235$	94.1335	0.400568
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 429.285i	− 1.79617i	−0.439821	−	0.898085i	$-0.644958\pi$
$239$	− 429.285i	− 1.79617i	0.439821	−	0.898085i	$-0.355042\pi$
$240$	0	0
$241$	−182.403	−0.756858	−0.378429	−	0.925630i	$-0.623536\pi$
$241$	−182.403	−0.756858	−0.378429	+	0.925630i	$0.623536\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 4.43092i	− 0.0180854i
$246$	0	0
$247$	546.053	2.21074
$248$	0	0
$249$	0	0
$250$	0	0
$251$	69.0483i	0.275093i	0.990495	+	0.137546i	$0.0439216\pi$
$251$	69.0483i	0.275093i	−0.990495	+	0.137546i	$0.956078\pi$
$252$	0	0
$253$	112.565	0.444922
$254$	0	0
$255$	0	0
$256$	0	0
$257$	106.339i	0.413770i	0.978365	+	0.206885i	$0.0663326\pi$
$257$	106.339i	0.413770i	−0.978365	+	0.206885i	$0.933667\pi$
$258$	0	0
$259$	−153.314	−0.591946
$260$	0	0
$261$	0	0
$262$	0	0
$263$	242.212i	0.920960i	0.887670	+	0.460480i	$0.152323\pi$
$263$	242.212i	0.920960i	−0.887670	+	0.460480i	$0.847677\pi$
$264$	0	0
$265$	−27.3946	−0.103376
$266$	0	0
$267$	0	0
$268$	0	0
$269$	143.438i	0.533228i	0.963803	+	0.266614i	$0.0859049\pi$
$269$	143.438i	0.533228i	−0.963803	+	0.266614i	$0.914095\pi$
$270$	0	0
$271$	59.8857	0.220980	0.110490	−	0.993877i	$-0.464758\pi$
$271$	59.8857	0.220980	0.110490	+	0.993877i	$0.464758\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 107.554i	− 0.391105i
$276$	0	0
$277$	314.916	1.13688	0.568441	−	0.822724i	$-0.307547\pi$
$277$	314.916	1.13688	0.568441	+	0.822724i	$0.307547\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 40.6153i	− 0.144539i	−0.997385	−	0.0722693i	$-0.976976\pi$
$281$	− 40.6153i	− 0.144539i	0.997385	−	0.0722693i	$-0.0230241\pi$
$282$	0	0
$283$	101.897	0.360059	0.180030	−	0.983661i	$-0.442381\pi$
$283$	101.897	0.360059	0.180030	+	0.983661i	$0.442381\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.2947i	0.0707133i
$288$	0	0
$289$	288.996	0.999985
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 134.113i	− 0.457724i	−0.973459	−	0.228862i	$-0.926500\pi$
$293$	− 134.113i	− 0.457724i	0.973459	−	0.228862i	$-0.0735004\pi$
$294$	0	0
$295$	−184.010	−0.623764
$296$	0	0
$297$	0	0
$298$	0	0
$299$	429.513i	1.43650i
$300$	0	0
$301$	200.973	0.667683
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 27.4176i	− 0.0898937i
$306$	0	0
$307$	−129.799	−0.422799	−0.211400	−	0.977400i	$-0.567802\pi$
$307$	−129.799	−0.422799	−0.211400	+	0.977400i	$0.567802\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	202.089i	0.649803i	0.945748	+	0.324902i	$0.105331\pi$
$311$	202.089i	0.649803i	−0.945748	+	0.324902i	$0.894669\pi$
$312$	0	0
$313$	337.004	1.07669	0.538346	−	0.842724i	$-0.319050\pi$
$313$	337.004	1.07669	0.538346	+	0.842724i	$0.319050\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	310.337i	0.978980i	0.872009	+	0.489490i	$0.162817\pi$
$317$	310.337i	0.978980i	−0.872009	+	0.489490i	$0.837183\pi$
$318$	0	0
$319$	−226.071	−0.708686
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1.81741i	0.00562667i
$324$	0	0
$325$	410.390	1.26274
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 303.487i	− 0.922452i
$330$	0	0
$331$	164.459	0.496854	0.248427	−	0.968651i	$-0.420086\pi$
$331$	164.459	0.496854	0.248427	+	0.968651i	$0.420086\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 260.970i	− 0.779015i
$336$	0	0
$337$	−189.077	−0.561060	−0.280530	−	0.959845i	$-0.590510\pi$
$337$	−189.077	−0.561060	−0.280530	+	0.959845i	$0.590510\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	84.6840i	0.248340i
$342$	0	0
$343$	−349.906	−1.02013
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 431.072i	− 1.24228i	−0.783699	−	0.621141i	$-0.786669\pi$
$347$	− 431.072i	− 1.24228i	0.783699	−	0.621141i	$-0.213331\pi$
$348$	0	0
$349$	366.650	1.05057	0.525287	−	0.850925i	$-0.323958\pi$
$349$	366.650	1.05057	0.525287	+	0.850925i	$0.323958\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	58.5238i	0.165790i	0.996558	+	0.0828949i	$0.0264166\pi$
$353$	58.5238i	0.165790i	−0.996558	+	0.0828949i	$0.973583\pi$
$354$	0	0
$355$	266.328	0.750218
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 48.0665i	− 0.133890i	−0.997757	−	0.0669449i	$-0.978675\pi$
$359$	− 48.0665i	− 0.133890i	0.997757	−	0.0669449i	$-0.0213252\pi$
$360$	0	0
$361$	382.036	1.05827
$362$	0	0
$363$	0	0
$364$	0	0
$365$	192.865i	0.528397i
$366$	0	0
$367$	−114.347	−0.311573	−0.155787	−	0.987791i	$-0.549791\pi$
$367$	−114.347	−0.311573	−0.155787	+	0.987791i	$0.549791\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	88.3203i	0.238060i
$372$	0	0
$373$	345.956	0.927496	0.463748	−	0.885967i	$-0.346504\pi$
$373$	345.956	0.927496	0.463748	+	0.885967i	$0.346504\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 862.613i	− 2.28810i
$378$	0	0
$379$	−323.425	−0.853364	−0.426682	−	0.904402i	$-0.640318\pi$
$379$	−323.425	−0.853364	−0.426682	+	0.904402i	$0.640318\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 519.463i	− 1.35630i	−0.734923	−	0.678151i	$-0.762782\pi$
$383$	− 519.463i	− 1.35630i	0.734923	−	0.678151i	$-0.237218\pi$
$384$	0	0
$385$	76.3955	0.198430
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 171.077i	− 0.439787i	−0.975524	−	0.219893i	$-0.929429\pi$
$389$	− 171.077i	− 0.439787i	0.975524	−	0.219893i	$-0.0705710\pi$
$390$	0	0
$391$	−1.42954	−0.00365610
$392$	0	0
$393$	0	0
$394$	0	0
$395$	103.048i	0.260882i
$396$	0	0
$397$	159.942	0.402878	0.201439	−	0.979501i	$-0.435438\pi$
$397$	159.942	0.402878	0.201439	+	0.979501i	$0.435438\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 655.884i	− 1.63562i	−0.575487	−	0.817811i	$-0.695187\pi$
$401$	− 655.884i	− 1.63562i	0.575487	−	0.817811i	$-0.304813\pi$
$402$	0	0
$403$	−323.127	−0.801803
$404$	0	0
$405$	0	0
$406$	0	0
$407$	117.513i	0.288730i
$408$	0	0
$409$	66.3179	0.162146	0.0810732	−	0.996708i	$-0.474165\pi$
$409$	66.3179	0.162146	0.0810732	+	0.996708i	$0.474165\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	593.250i	1.43644i
$414$	0	0
$415$	248.114	0.597864
$416$	0	0
$417$	0	0
$418$	0	0
$419$	744.076i	1.77584i	0.460001	+	0.887918i	$0.347849\pi$
$419$	744.076i	1.77584i	−0.460001	+	0.887918i	$0.652151\pi$
$420$	0	0
$421$	−681.117	−1.61785	−0.808927	−	0.587909i	$-0.799951\pi$
$421$	−681.117	−1.61785	−0.808927	+	0.587909i	$0.799951\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.36589i	0.00321386i
$426$	0	0
$427$	−88.3945	−0.207013
$428$	0	0
$429$	0	0
$430$	0	0
$431$	251.031i	0.582439i	0.956656	+	0.291220i	$0.0940610\pi$
$431$	251.031i	0.582439i	−0.956656	+	0.291220i	$0.905939\pi$
$432$	0	0
$433$	−87.8113	−0.202797	−0.101399	−	0.994846i	$-0.532332\pi$
$433$	−87.8113	−0.202797	−0.101399	+	0.994846i	$0.532332\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	584.455i	1.33743i
$438$	0	0
$439$	−393.268	−0.895827	−0.447913	−	0.894077i	$-0.647833\pi$
$439$	−393.268	−0.895827	−0.447913	+	0.894077i	$0.647833\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 49.6560i	− 0.112090i	−0.998428	−	0.0560452i	$-0.982151\pi$
$443$	− 49.6560i	− 0.112090i	0.998428	−	0.0560452i	$-0.0178491\pi$
$444$	0	0
$445$	246.809	0.554626
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 276.997i	− 0.616919i	−0.951237	−	0.308459i	$-0.900187\pi$
$449$	− 276.997i	− 0.616919i	0.951237	−	0.308459i	$-0.0998134\pi$
$450$	0	0
$451$	15.5556	0.0344914
$452$	0	0
$453$	0	0
$454$	0	0
$455$	291.501i	0.640661i
$456$	0	0
$457$	−7.13408	−0.0156107	−0.00780534	−	0.999970i	$-0.502485\pi$
$457$	−7.13408	−0.0156107	−0.00780534	+	0.999970i	$0.502485\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	344.475i	0.747235i	0.927583	+	0.373617i	$0.121883\pi$
$461$	344.475i	0.747235i	−0.927583	+	0.373617i	$0.878117\pi$
$462$	0	0
$463$	−618.903	−1.33672	−0.668362	−	0.743836i	$-0.733004\pi$
$463$	−618.903	−1.33672	−0.668362	+	0.743836i	$0.733004\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 682.172i	− 1.46075i	−0.683044	−	0.730377i	$-0.739344\pi$
$467$	− 682.172i	− 1.46075i	0.683044	−	0.730377i	$-0.260656\pi$
$468$	0	0
$469$	−841.369	−1.79396
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 154.043i	− 0.325672i
$474$	0	0
$475$	558.435	1.17565
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 138.757i	− 0.289681i	−0.989455	−	0.144841i	$-0.953733\pi$
$479$	− 138.757i	− 0.289681i	0.989455	−	0.144841i	$-0.0462670\pi$
$480$	0	0
$481$	−448.393	−0.932209
$482$	0	0
$483$	0	0
$484$	0	0
$485$	98.9087i	0.203935i
$486$	0	0
$487$	630.925	1.29553	0.647766	−	0.761839i	$-0.275703\pi$
$487$	630.925	1.29553	0.647766	+	0.761839i	$0.275703\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	405.911i	0.826703i	0.910572	+	0.413351i	$0.135642\pi$
$491$	405.911i	0.826703i	−0.910572	+	0.413351i	$0.864358\pi$
$492$	0	0
$493$	2.87101	0.00582356
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 858.642i	− 1.72765i
$498$	0	0
$499$	104.177	0.208772	0.104386	−	0.994537i	$-0.466712\pi$
$499$	104.177	0.208772	0.104386	+	0.994537i	$0.466712\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	275.389i	0.547493i	0.961802	+	0.273746i	$0.0882629\pi$
$503$	275.389i	0.547493i	−0.961802	+	0.273746i	$0.911737\pi$
$504$	0	0
$505$	−287.063	−0.568441
$506$	0	0
$507$	0	0
$508$	0	0
$509$	763.424i	1.49985i	0.661523	+	0.749925i	$0.269911\pi$
$509$	763.424i	1.49985i	−0.661523	+	0.749925i	$0.730089\pi$
$510$	0	0
$511$	621.798	1.21683
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 33.6372i	− 0.0653150i
$516$	0	0
$517$	−232.619	−0.449940
$518$	0	0
$519$	0	0
$520$	0	0
$521$	727.642i	1.39663i	0.715793	+	0.698313i	$0.246065\pi$
$521$	727.642i	1.39663i	−0.715793	+	0.698313i	$0.753935\pi$
$522$	0	0
$523$	−585.050	−1.11864	−0.559321	−	0.828951i	$-0.688938\pi$
$523$	−585.050	−1.11864	−0.559321	+	0.828951i	$0.688938\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 1.07545i	− 0.00204071i
$528$	0	0
$529$	69.2810	0.130966
$530$	0	0
$531$	0	0
$532$	0	0
$533$	59.3553i	0.111361i
$534$	0	0
$535$	−406.053	−0.758978
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.9495i	0.0203145i
$540$	0	0
$541$	−359.365	−0.664260	−0.332130	−	0.943234i	$-0.607767\pi$
$541$	−359.365	−0.664260	−0.332130	+	0.943234i	$0.607767\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	307.667i	0.564526i
$546$	0	0
$547$	−97.9640	−0.179093	−0.0895467	−	0.995983i	$-0.528542\pi$
$547$	−97.9640	−0.179093	−0.0895467	+	0.995983i	$0.528542\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1173.79i	− 2.13029i
$552$	0	0
$553$	332.228	0.600775
$554$	0	0
$555$	0	0
$556$	0	0
$557$	550.731i	0.988745i	0.869250	+	0.494373i	$0.164602\pi$
$557$	550.731i	0.988745i	−0.869250	+	0.494373i	$0.835398\pi$
$558$	0	0
$559$	587.779	1.05148
$560$	0	0
$561$	0	0
$562$	0	0
$563$	536.763i	0.953398i	0.879067	+	0.476699i	$0.158167\pi$
$563$	536.763i	0.953398i	−0.879067	+	0.476699i	$0.841833\pi$
$564$	0	0
$565$	204.807	0.362491
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 551.945i	− 0.970025i	−0.874507	−	0.485013i	$-0.838815\pi$
$569$	− 551.945i	− 0.970025i	0.874507	−	0.485013i	$-0.161185\pi$
$570$	0	0
$571$	−819.521	−1.43524	−0.717619	−	0.696436i	$-0.754768\pi$
$571$	−819.521	−1.43524	−0.717619	+	0.696436i	$0.754768\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	439.252i	0.763917i
$576$	0	0
$577$	−828.718	−1.43625	−0.718126	−	0.695913i	$-0.755000\pi$
$577$	−828.718	−1.43625	−0.718126	+	0.695913i	$0.755000\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 799.920i	− 1.37680i
$582$	0	0
$583$	67.6964	0.116117
$584$	0	0
$585$	0	0
$586$	0	0
$587$	76.6443i	0.130570i	0.997867	+	0.0652848i	$0.0207956\pi$
$587$	76.6443i	0.130570i	−0.997867	+	0.0652848i	$0.979204\pi$
$588$	0	0
$589$	−439.691	−0.746505
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 943.303i	− 1.59073i	−0.606130	−	0.795366i	$-0.707279\pi$
$593$	− 943.303i	− 1.59073i	0.606130	−	0.795366i	$-0.292721\pi$
$594$	0	0
$595$	−0.970195	−0.00163058
$596$	0	0
$597$	0	0
$598$	0	0
$599$	329.462i	0.550021i	0.961441	+	0.275010i	$0.0886813\pi$
$599$	329.462i	0.550021i	−0.961441	+	0.275010i	$0.911319\pi$
$600$	0	0
$601$	−762.581	−1.26885	−0.634427	−	0.772983i	$-0.718764\pi$
$601$	−762.581	−1.26885	−0.634427	+	0.772983i	$0.718764\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	198.508i	0.328113i
$606$	0	0
$607$	−959.375	−1.58052	−0.790259	−	0.612773i	$-0.790054\pi$
$607$	−959.375	−1.58052	−0.790259	+	0.612773i	$0.790054\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 887.599i	− 1.45270i
$612$	0	0
$613$	573.016	0.934773	0.467386	−	0.884053i	$-0.345196\pi$
$613$	573.016	0.934773	0.467386	+	0.884053i	$0.345196\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 112.234i	− 0.181903i	−0.995855	−	0.0909515i	$-0.971009\pi$
$617$	− 112.234i	− 0.181903i	0.995855	−	0.0909515i	$-0.0289908\pi$
$618$	0	0
$619$	−45.1174	−0.0728876	−0.0364438	−	0.999336i	$-0.511603\pi$
$619$	−45.1174	−0.0728876	−0.0364438	+	0.999336i	$0.511603\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 795.712i	− 1.27723i
$624$	0	0
$625$	306.859	0.490974
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.49237i	− 0.00237261i
$630$	0	0
$631$	1118.03	1.77185	0.885923	−	0.463832i	$-0.153526\pi$
$631$	1118.03	1.77185	0.885923	+	0.463832i	$0.153526\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 271.355i	− 0.427331i
$636$	0	0
$637$	−41.7798	−0.0655884
$638$	0	0
$639$	0	0
$640$	0	0
$641$	673.551i	1.05078i	0.850861	+	0.525390i	$0.176081\pi$
$641$	673.551i	1.05078i	−0.850861	+	0.525390i	$0.823919\pi$
$642$	0	0
$643$	570.492	0.887235	0.443617	−	0.896216i	$-0.353695\pi$
$643$	570.492	0.887235	0.443617	+	0.896216i	$0.353695\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 376.087i	− 0.581278i	−0.956833	−	0.290639i	$-0.906132\pi$
$647$	− 376.087i	− 0.581278i	0.956833	−	0.290639i	$-0.0938678\pi$
$648$	0	0
$649$	454.719	0.700646
$650$	0	0
$651$	0	0
$652$	0	0
$653$	643.420i	0.985330i	0.870219	+	0.492665i	$0.163977\pi$
$653$	643.420i	0.985330i	−0.870219	+	0.492665i	$0.836023\pi$
$654$	0	0
$655$	69.4515	0.106033
$656$	0	0
$657$	0	0
$658$	0	0
$659$	161.681i	0.245342i	0.992447	+	0.122671i	$0.0391461\pi$
$659$	161.681i	0.245342i	−0.992447	+	0.122671i	$0.960854\pi$
$660$	0	0
$661$	−717.266	−1.08512	−0.542561	−	0.840016i	$-0.682545\pi$
$661$	−717.266	−1.08512	−0.542561	+	0.840016i	$0.682545\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	396.657i	0.596476i
$666$	0	0
$667$	923.278	1.38423
$668$	0	0
$669$	0	0
$670$	0	0
$671$	67.7532i	0.100974i
$672$	0	0
$673$	−164.830	−0.244918	−0.122459	−	0.992474i	$-0.539078\pi$
$673$	−164.830	−0.244918	−0.122459	+	0.992474i	$0.539078\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	811.953i	1.19934i	0.800247	+	0.599670i	$0.204701\pi$
$677$	811.953i	1.19934i	−0.800247	+	0.599670i	$0.795299\pi$
$678$	0	0
$679$	318.882	0.469635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	546.230i	0.799751i	0.916569	+	0.399876i	$0.130947\pi$
$683$	546.230i	0.799751i	−0.916569	+	0.399876i	$0.869053\pi$
$684$	0	0
$685$	391.814	0.571992
$686$	0	0
$687$	0	0
$688$	0	0
$689$	258.308i	0.374902i
$690$	0	0
$691$	23.7822	0.0344171	0.0172085	−	0.999852i	$-0.494522\pi$
$691$	23.7822	0.0344171	0.0172085	+	0.999852i	$0.494522\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 507.130i	− 0.729683i
$696$	0	0
$697$	−0.197551	−0.000283430 0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 959.326i	− 1.36851i	−0.729242	−	0.684255i	$-0.760127\pi$
$701$	− 959.326i	− 1.36851i	0.729242	−	0.684255i	$-0.239873\pi$
$702$	0	0
$703$	−610.146	−0.867917
$704$	0	0
$705$	0	0
$706$	0	0
$707$	925.491i	1.30904i
$708$	0	0
$709$	−436.916	−0.616243	−0.308121	−	0.951347i	$-0.599700\pi$
$709$	−436.916	−0.616243	−0.308121	+	0.951347i	$0.599700\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 345.851i	− 0.485065i
$714$	0	0
$715$	223.432	0.312492
$716$	0	0
$717$	0	0
$718$	0	0
$719$	643.581i	0.895106i	0.894257	+	0.447553i	$0.147704\pi$
$719$	643.581i	0.895106i	−0.894257	+	0.447553i	$0.852296\pi$
$720$	0	0
$721$	−108.447	−0.150411
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 882.174i	− 1.21679i
$726$	0	0
$727$	730.998	1.00550	0.502749	−	0.864432i	$-0.332322\pi$
$727$	730.998	1.00550	0.502749	+	0.864432i	$0.332322\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.95629i	0.00267618i
$732$	0	0
$733$	−942.809	−1.28623	−0.643117	−	0.765768i	$-0.722359\pi$
$733$	−942.809	−1.28623	−0.643117	+	0.765768i	$0.722359\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	644.899i	0.875033i
$738$	0	0
$739$	−354.141	−0.479216	−0.239608	−	0.970870i	$-0.577019\pi$
$739$	−354.141	−0.479216	−0.239608	+	0.970870i	$0.577019\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 125.849i	− 0.169380i	−0.996407	−	0.0846899i	$-0.973010\pi$
$743$	− 125.849i	− 0.169380i	0.996407	−	0.0846899i	$-0.0269900\pi$
$744$	0	0
$745$	−545.274	−0.731911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1309.12i	1.74782i
$750$	0	0
$751$	−551.404	−0.734226	−0.367113	−	0.930176i	$-0.619654\pi$
$751$	−551.404	−0.734226	−0.367113	+	0.930176i	$0.619654\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 176.916i	− 0.234326i
$756$	0	0
$757$	−816.186	−1.07818	−0.539092	−	0.842247i	$-0.681233\pi$
$757$	−816.186	−1.07818	−0.539092	+	0.842247i	$0.681233\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 943.192i	− 1.23941i	−0.784834	−	0.619706i	$-0.787252\pi$
$761$	− 943.192i	− 1.23941i	0.784834	−	0.619706i	$-0.212748\pi$
$762$	0	0
$763$	991.919	1.30002
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1735.06i	2.26214i
$768$	0	0
$769$	−544.577	−0.708163	−0.354081	−	0.935215i	$-0.615206\pi$
$769$	−544.577	−0.708163	−0.354081	+	0.935215i	$0.615206\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	24.6789i	0.0319262i	0.999873	+	0.0159631i	$0.00508143\pi$
$773$	24.6789i	0.0319262i	−0.999873	+	0.0159631i	$0.994919\pi$
$774$	0	0
$775$	−330.454	−0.426392
$776$	0	0
$777$	0	0
$778$	0	0
$779$	80.7671i	0.103681i
$780$	0	0
$781$	−658.138	−0.842686
$782$	0	0
$783$	0	0
$784$	0	0
$785$	338.994i	0.431840i
$786$	0	0
$787$	716.409	0.910304	0.455152	−	0.890414i	$-0.349585\pi$
$787$	716.409	0.910304	0.455152	+	0.890414i	$0.349585\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 660.300i	− 0.834767i
$792$	0	0
$793$	−258.525	−0.326008
$794$	0	0
$795$	0	0
$796$	0	0
$797$	962.391i	1.20752i	0.797167	+	0.603758i	$0.206331\pi$
$797$	962.391i	1.20752i	−0.797167	+	0.603758i	$0.793669\pi$
$798$	0	0
$799$	2.95417	0.00369734
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 476.600i	− 0.593525i
$804$	0	0
$805$	−312.001	−0.387579
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 674.254i	− 0.833441i	−0.909035	−	0.416721i	$-0.863179\pi$
$809$	− 674.254i	− 0.833441i	0.909035	−	0.416721i	$-0.136821\pi$
$810$	0	0
$811$	−36.7884	−0.0453618	−0.0226809	−	0.999743i	$-0.507220\pi$
$811$	−36.7884	−0.0453618	−0.0226809	+	0.999743i	$0.507220\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	39.1207i	0.0480008i
$816$	0	0
$817$	799.814	0.978964
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1515.18i	− 1.84552i	−0.385369	−	0.922762i	$-0.625926\pi$
$821$	− 1515.18i	− 1.84552i	0.385369	−	0.922762i	$-0.374074\pi$
$822$	0	0
$823$	762.423	0.926395	0.463198	−	0.886255i	$-0.346702\pi$
$823$	762.423	0.926395	0.463198	+	0.886255i	$0.346702\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 648.915i	− 0.784662i	−0.919824	−	0.392331i	$-0.871669\pi$
$827$	− 648.915i	− 0.784662i	0.919824	−	0.392331i	$-0.128331\pi$
$828$	0	0
$829$	−958.477	−1.15618	−0.578092	−	0.815971i	$-0.696203\pi$
$829$	−958.477	−1.15618	−0.578092	+	0.815971i	$0.696203\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 0.139055i	0 0.000166932i
$834$	0	0
$835$	−429.461	−0.514324
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 287.975i	− 0.343236i	−0.985164	−	0.171618i	$-0.945100\pi$
$839$	− 287.975i	− 0.343236i	0.985164	−	0.171618i	$-0.0548995\pi$
$840$	0	0
$841$	−1013.27	−1.20484
$842$	0	0
$843$	0	0
$844$	0	0
$845$	493.502i	0.584027i
$846$	0	0
$847$	639.993	0.755599
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 479.927i	− 0.563956i
$852$	0	0
$853$	−666.572	−0.781445	−0.390722	−	0.920509i	$-0.627775\pi$
$853$	−666.572	−0.781445	−0.390722	+	0.920509i	$0.627775\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 731.580i	− 0.853653i	−0.904334	−	0.426826i	$-0.859632\pi$
$857$	− 731.580i	− 0.853653i	0.904334	−	0.426826i	$-0.140368\pi$
$858$	0	0
$859$	−776.313	−0.903740	−0.451870	−	0.892084i	$-0.649243\pi$
$859$	−776.313	−0.903740	−0.451870	+	0.892084i	$0.649243\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	925.254i	1.07214i	0.844175	+	0.536068i	$0.180091\pi$
$863$	925.254i	1.07214i	−0.844175	+	0.536068i	$0.819909\pi$
$864$	0	0
$865$	−214.501	−0.247977
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 254.649i	− 0.293037i
$870$	0	0
$871$	−2460.73	−2.82517
$872$	0	0
$873$	0	0
$874$	0	0
$875$	661.900i	0.756457i
$876$	0	0
$877$	−696.151	−0.793787	−0.396893	−	0.917865i	$-0.629912\pi$
$877$	−696.151	−0.793787	−0.396893	+	0.917865i	$0.629912\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	183.830i	0.208660i	0.994543	+	0.104330i	$0.0332699\pi$
$881$	183.830i	0.208660i	−0.994543	+	0.104330i	$0.966730\pi$
$882$	0	0
$883$	−823.282	−0.932369	−0.466184	−	0.884688i	$-0.654372\pi$
$883$	−823.282	−0.932369	−0.466184	+	0.884688i	$0.654372\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	365.372i	0.411919i	0.978561	+	0.205959i	$0.0660315\pi$
$887$	365.372i	0.411919i	−0.978561	+	0.205959i	$0.933969\pi$
$888$	0	0
$889$	−874.852	−0.984085
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1207.79i	− 1.35251i
$894$	0	0
$895$	97.3670	0.108790
$896$	0	0
$897$	0	0
$898$	0	0
$899$	694.592i	0.772627i
$900$	0	0
$901$	−0.859719	−0.000954183 0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	118.485i	0.130923i
$906$	0	0
$907$	−306.404	−0.337822	−0.168911	−	0.985631i	$-0.554025\pi$
$907$	−306.404	−0.337822	−0.168911	+	0.985631i	$0.554025\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	788.139i	0.865136i	0.901601	+	0.432568i	$0.142392\pi$
$911$	788.139i	0.865136i	−0.901601	+	0.432568i	$0.857608\pi$
$912$	0	0
$913$	−613.128	−0.671554
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 223.912i	− 0.244179i
$918$	0	0
$919$	−1319.46	−1.43575	−0.717876	−	0.696171i	$-0.754886\pi$
$919$	−1319.46	−1.43575	−0.717876	+	0.696171i	$0.754886\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2511.24i	− 2.72074i
$924$	0	0
$925$	−458.560	−0.495741
$926$	0	0
$927$	0	0
$928$	0	0
$929$	469.930i	0.505845i	0.967487	+	0.252922i	$0.0813917\pi$
$929$	469.930i	0.505845i	−0.967487	+	0.252922i	$0.918608\pi$
$930$	0	0
$931$	−56.8515	−0.0610649
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.743642i	0 0.000795339i
$936$	0	0
$937$	−1619.12	−1.72798	−0.863991	−	0.503508i	$-0.832042\pi$
$937$	−1619.12	−1.72798	−0.863991	+	0.503508i	$0.832042\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 668.395i	− 0.710302i	−0.934809	−	0.355151i	$-0.884429\pi$
$941$	− 668.395i	− 0.710302i	0.934809	−	0.355151i	$-0.115571\pi$
$942$	0	0
$943$	−63.5296	−0.0673697
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 992.522i	− 1.04807i	−0.851697	−	0.524035i	$-0.824426\pi$
$947$	− 992.522i	− 1.04807i	0.851697	−	0.524035i	$-0.175574\pi$
$948$	0	0
$949$	1818.55	1.91628
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 565.650i	− 0.593546i	−0.954948	−	0.296773i	$-0.904089\pi$
$953$	− 565.650i	− 0.593546i	0.954948	−	0.296773i	$-0.0959105\pi$
$954$	0	0
$955$	−672.184	−0.703858
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1263.21i	− 1.31722i
$960$	0	0
$961$	−700.812	−0.729253
$962$	0	0
$963$	0	0
$964$	0	0
$965$	797.311i	0.826229i
$966$	0	0
$967$	1030.39	1.06555	0.532775	−	0.846257i	$-0.321149\pi$
$967$	1030.39	1.06555	0.532775	+	0.846257i	$0.321149\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 454.101i	− 0.467663i	−0.972277	−	0.233832i	$-0.924874\pi$
$971$	− 454.101i	− 0.467663i	0.972277	−	0.233832i	$-0.0751265\pi$
$972$	0	0
$973$	−1634.99	−1.68036
$974$	0	0
$975$	0	0
$976$	0	0
$977$	846.744i	0.866678i	0.901231	+	0.433339i	$0.142665\pi$
$977$	846.744i	0.866678i	−0.901231	+	0.433339i	$0.857335\pi$
$978$	0	0
$979$	−609.904	−0.622986
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1433.24i	1.45803i	0.684499	+	0.729014i	$0.260021\pi$
$983$	1433.24i	1.45803i	−0.684499	+	0.729014i	$0.739979\pi$
$984$	0	0
$985$	99.0797	0.100588
$986$	0	0
$987$	0	0
$988$	0	0
$989$	629.115i	0.636113i
$990$	0	0
$991$	−955.562	−0.964240	−0.482120	−	0.876105i	$-0.660133\pi$
$991$	−955.562	−0.964240	−0.482120	+	0.876105i	$0.660133\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 407.992i	− 0.410043i
$996$	0	0
$997$	−589.446	−0.591219	−0.295610	−	0.955309i	$-0.595523\pi$
$997$	−589.446	−0.591219	−0.295610	+	0.955309i	$0.595523\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.3.e.i.161.10		24
3.2	odd	2	inner	2592.3.e.i.161.9		24
4.3	odd	2	inner	2592.3.e.i.161.16		24
9.2	odd	6		864.3.q.a.449.7		24
9.4	even	3		864.3.q.a.737.7		24
9.5	odd	6		288.3.q.b.65.7	yes	24
9.7	even	3		288.3.q.b.257.7	yes	24
12.11	even	2	inner	2592.3.e.i.161.15		24
36.7	odd	6		288.3.q.b.257.6	yes	24
36.11	even	6		864.3.q.a.449.8		24
36.23	even	6		288.3.q.b.65.6	✓	24
36.31	odd	6		864.3.q.a.737.8		24
72.5	odd	6		576.3.q.l.65.6		24
72.11	even	6		1728.3.q.k.449.6		24
72.13	even	6		1728.3.q.k.1601.5		24
72.29	odd	6		1728.3.q.k.449.5		24
72.43	odd	6		576.3.q.l.257.7		24
72.59	even	6		576.3.q.l.65.7		24
72.61	even	6		576.3.q.l.257.6		24
72.67	odd	6		1728.3.q.k.1601.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.b.65.6	✓	24	36.23	even	6
288.3.q.b.65.7	yes	24	9.5	odd	6
288.3.q.b.257.6	yes	24	36.7	odd	6
288.3.q.b.257.7	yes	24	9.7	even	3
576.3.q.l.65.6		24	72.5	odd	6
576.3.q.l.65.7		24	72.59	even	6
576.3.q.l.257.6		24	72.61	even	6
576.3.q.l.257.7		24	72.43	odd	6
864.3.q.a.449.7		24	9.2	odd	6
864.3.q.a.449.8		24	36.11	even	6
864.3.q.a.737.7		24	9.4	even	3
864.3.q.a.737.8		24	36.31	odd	6
1728.3.q.k.449.5		24	72.29	odd	6
1728.3.q.k.449.6		24	72.11	even	6
1728.3.q.k.1601.5		24	72.13	even	6
1728.3.q.k.1601.6		24	72.67	odd	6
2592.3.e.i.161.9		24	3.2	odd	2	inner
2592.3.e.i.161.10		24	1.1	even	1	trivial
2592.3.e.i.161.15		24	12.11	even	2	inner
2592.3.e.i.161.16		24	4.3	odd	2	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2592.3.e.i.161.10

Level	$2592$
Weight	$3$
Character	2592.161
Analytic conductor	$70.627$
Analytic rank	$0$
Dimension	$24$
Inner twists	$4$