LMFDB - Embedded newform 2592.3.e.i.161.6

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			161.6
Character	$\chi$	$=$	2592.161
Dual form			2592.3.e.i.161.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.25050i q^{5} -7.49710 q^{7} +O(q^{10})$	$q+6.25050i q^{5} -7.49710 q^{7} +7.13244i q^{11} +1.77760 q^{13} -14.7791i q^{17} -19.9460 q^{19} -24.1005i q^{23} -14.0688 q^{25} -46.3357i q^{29} -28.3094 q^{31} -46.8606i q^{35} +63.0770 q^{37} -32.3530i q^{41} +77.6353 q^{43} +44.4235i q^{47} +7.20647 q^{49} -42.2846i q^{53} -44.5813 q^{55} +108.417i q^{59} +50.7716 q^{61} +11.1109i q^{65} -113.897 q^{67} -85.2129i q^{71} -94.5357 q^{73} -53.4726i q^{77} +71.2117 q^{79} +109.585i q^{83} +92.3771 q^{85} +29.6936i q^{89} -13.3269 q^{91} -124.673i q^{95} +124.959 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * 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

Display 100 coefficients 10 coefficients

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
$5$	6.25050i	1.25010i	0.780585	+	0.625050i	$0.214921\pi$
$5$	6.25050i	1.25010i	−0.780585	+	0.625050i	$0.785079\pi$
$6$	0	0
$7$	−7.49710	−1.07101	−0.535507	−	0.844531i	$-0.679879\pi$
$7$	−7.49710	−1.07101	−0.535507	+	0.844531i	$0.679879\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7.13244i	0.648404i	0.945988	+	0.324202i	$0.105096\pi$
$11$	7.13244i	0.648404i	−0.945988	+	0.324202i	$0.894904\pi$
$12$	0	0
$13$	1.77760	0.136739	0.0683694	−	0.997660i	$-0.478220\pi$
$13$	1.77760	0.136739	0.0683694	+	0.997660i	$0.478220\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 14.7791i	− 0.869362i	−0.900585	−	0.434681i	$-0.856861\pi$
$17$	− 14.7791i	− 0.869362i	0.900585	−	0.434681i	$-0.143139\pi$
$18$	0	0
$19$	−19.9460	−1.04979	−0.524896	−	0.851167i	$-0.675896\pi$
$19$	−19.9460	−1.04979	−0.524896	+	0.851167i	$0.675896\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 24.1005i	− 1.04785i	−0.851765	−	0.523923i	$-0.824468\pi$
$23$	− 24.1005i	− 1.04785i	0.851765	−	0.523923i	$-0.175532\pi$
$24$	0	0
$25$	−14.0688	−0.562750
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 46.3357i	− 1.59778i	−0.601475	−	0.798891i	$-0.705420\pi$
$29$	− 46.3357i	− 1.59778i	0.601475	−	0.798891i	$-0.294580\pi$
$30$	0	0
$31$	−28.3094	−0.913206	−0.456603	−	0.889671i	$-0.650934\pi$
$31$	−28.3094	−0.913206	−0.456603	+	0.889671i	$0.650934\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 46.8606i	− 1.33887i
$36$	0	0
$37$	63.0770	1.70478	0.852391	−	0.522904i	$-0.175151\pi$
$37$	63.0770	1.70478	0.852391	+	0.522904i	$0.175151\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 32.3530i	− 0.789097i	−0.918875	−	0.394549i	$-0.870901\pi$
$41$	− 32.3530i	− 0.789097i	0.918875	−	0.394549i	$-0.129099\pi$
$42$	0	0
$43$	77.6353	1.80547	0.902736	−	0.430195i	$-0.141555\pi$
$43$	77.6353	1.80547	0.902736	+	0.430195i	$0.141555\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	44.4235i	0.945182i	0.881282	+	0.472591i	$0.156681\pi$
$47$	44.4235i	0.945182i	−0.881282	+	0.472591i	$0.843319\pi$
$48$	0	0
$49$	7.20647	0.147071
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 42.2846i	− 0.797823i	−0.916990	−	0.398911i	$-0.869388\pi$
$53$	− 42.2846i	− 0.797823i	0.916990	−	0.398911i	$-0.130612\pi$
$54$	0	0
$55$	−44.5813	−0.810570
$56$	0	0
$57$	0	0
$58$	0	0
$59$	108.417i	1.83757i	0.394756	+	0.918786i	$0.370829\pi$
$59$	108.417i	1.83757i	−0.394756	+	0.918786i	$0.629171\pi$
$60$	0	0
$61$	50.7716	0.832322	0.416161	−	0.909291i	$-0.363375\pi$
$61$	50.7716	0.832322	0.416161	+	0.909291i	$0.363375\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	11.1109i	0.170937i
$66$	0	0
$67$	−113.897	−1.69995	−0.849977	−	0.526820i	$-0.823384\pi$
$67$	−113.897	−1.69995	−0.849977	+	0.526820i	$0.823384\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 85.2129i	− 1.20018i	−0.799932	−	0.600091i	$-0.795131\pi$
$71$	− 85.2129i	− 1.20018i	0.799932	−	0.600091i	$-0.204869\pi$
$72$	0	0
$73$	−94.5357	−1.29501	−0.647505	−	0.762061i	$-0.724188\pi$
$73$	−94.5357	−1.29501	−0.647505	+	0.762061i	$0.724188\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 53.4726i	− 0.694449i
$78$	0	0
$79$	71.2117	0.901414	0.450707	−	0.892672i	$-0.351172\pi$
$79$	71.2117	0.901414	0.450707	+	0.892672i	$0.351172\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	109.585i	1.32031i	0.751131	+	0.660153i	$0.229509\pi$
$83$	109.585i	1.32031i	−0.751131	+	0.660153i	$0.770491\pi$
$84$	0	0
$85$	92.3771	1.08679
$86$	0	0
$87$	0	0
$88$	0	0
$89$	29.6936i	0.333635i	0.985988	+	0.166818i	$0.0533491\pi$
$89$	29.6936i	0.333635i	−0.985988	+	0.166818i	$0.946651\pi$
$90$	0	0
$91$	−13.3269	−0.146449
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 124.673i	− 1.31234i
$96$	0	0
$97$	124.959	1.28823	0.644116	−	0.764928i	$-0.277225\pi$
$97$	124.959	1.28823	0.644116	+	0.764928i	$0.277225\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	39.3740i	0.389841i	0.980819	+	0.194921i	$0.0624450\pi$
$101$	39.3740i	0.389841i	−0.980819	+	0.194921i	$0.937555\pi$
$102$	0	0
$103$	149.446	1.45093	0.725466	−	0.688258i	$-0.241624\pi$
$103$	149.446	1.45093	0.725466	+	0.688258i	$0.241624\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 0.420211i	− 0.00392721i	−0.999998	−	0.00196360i	$-0.999375\pi$
$107$	− 0.420211i	− 0.00392721i	0.999998	−	0.00196360i	$-0.000625035\pi$
$108$	0	0
$109$	64.4616	0.591391	0.295695	−	0.955282i	$-0.404449\pi$
$109$	64.4616	0.591391	0.295695	+	0.955282i	$0.404449\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	25.9482i	0.229630i	0.993387	+	0.114815i	$0.0366275\pi$
$113$	25.9482i	0.229630i	−0.993387	+	0.114815i	$0.963373\pi$
$114$	0	0
$115$	150.640	1.30991
$116$	0	0
$117$	0	0
$118$	0	0
$119$	110.801i	0.931098i
$120$	0	0
$121$	70.1283	0.579573
$122$	0	0
$123$	0	0
$124$	0	0
$125$	68.3258i	0.546606i
$126$	0	0
$127$	−12.0916	−0.0952097	−0.0476049	−	0.998866i	$-0.515159\pi$
$127$	−12.0916	−0.0952097	−0.0476049	+	0.998866i	$0.515159\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 52.4832i	− 0.400635i	−0.979731	−	0.200318i	$-0.935803\pi$
$131$	− 52.4832i	− 0.400635i	0.979731	−	0.200318i	$-0.0641974\pi$
$132$	0	0
$133$	149.537	1.12434
$134$	0	0
$135$	0	0
$136$	0	0
$137$	139.978i	1.02173i	0.859660	+	0.510867i	$0.170676\pi$
$137$	139.978i	1.02173i	−0.859660	+	0.510867i	$0.829324\pi$
$138$	0	0
$139$	−39.9312	−0.287275	−0.143637	−	0.989630i	$-0.545880\pi$
$139$	−39.9312	−0.287275	−0.143637	+	0.989630i	$0.545880\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12.6787i	0.0886619i
$144$	0	0
$145$	289.621	1.99739
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 3.76693i	− 0.0252814i	−0.999920	−	0.0126407i	$-0.995976\pi$
$149$	− 3.76693i	− 0.0252814i	0.999920	−	0.0126407i	$-0.00402377\pi$
$150$	0	0
$151$	−149.759	−0.991781	−0.495891	−	0.868385i	$-0.665158\pi$
$151$	−149.759	−0.991781	−0.495891	+	0.868385i	$0.665158\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 176.948i	− 1.14160i
$156$	0	0
$157$	117.901	0.750964	0.375482	−	0.926830i	$-0.377477\pi$
$157$	117.901	0.750964	0.375482	+	0.926830i	$0.377477\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	180.684i	1.12226i
$162$	0	0
$163$	111.960	0.686874	0.343437	−	0.939176i	$-0.388409\pi$
$163$	111.960	0.686874	0.343437	+	0.939176i	$0.388409\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.54357i	0.00924291i	0.999989	+	0.00462146i	$0.00147106\pi$
$167$	1.54357i	0.00924291i	−0.999989	+	0.00462146i	$0.998529\pi$
$168$	0	0
$169$	−165.840	−0.981303
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 53.1833i	− 0.307418i	−0.988116	−	0.153709i	$-0.950878\pi$
$173$	− 53.1833i	− 0.307418i	0.988116	−	0.153709i	$-0.0491219\pi$
$174$	0	0
$175$	105.475	0.602713
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 239.444i	− 1.33768i	−0.743407	−	0.668839i	$-0.766792\pi$
$179$	− 239.444i	− 1.33768i	0.743407	−	0.668839i	$-0.233208\pi$
$180$	0	0
$181$	61.1557	0.337877	0.168938	−	0.985627i	$-0.445966\pi$
$181$	61.1557	0.337877	0.168938	+	0.985627i	$0.445966\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	394.263i	2.13115i
$186$	0	0
$187$	105.411	0.563697
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 33.3781i	− 0.174755i	−0.996175	−	0.0873773i	$-0.972151\pi$
$191$	− 33.3781i	− 0.174755i	0.996175	−	0.0873773i	$-0.0278486\pi$
$192$	0	0
$193$	−42.5592	−0.220514	−0.110257	−	0.993903i	$-0.535167\pi$
$193$	−42.5592	−0.220514	−0.110257	+	0.993903i	$0.535167\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 203.372i	− 1.03235i	−0.856484	−	0.516173i	$-0.827356\pi$
$197$	− 203.372i	− 1.03235i	0.856484	−	0.516173i	$-0.172644\pi$
$198$	0	0
$199$	128.160	0.644022	0.322011	−	0.946736i	$-0.395641\pi$
$199$	128.160	0.644022	0.322011	+	0.946736i	$0.395641\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	347.383i	1.71125i
$204$	0	0
$205$	202.222	0.986451
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 142.264i	− 0.680689i
$210$	0	0
$211$	−194.535	−0.921967	−0.460983	−	0.887409i	$-0.652503\pi$
$211$	−194.535	−0.921967	−0.460983	+	0.887409i	$0.652503\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	485.259i	2.25702i
$216$	0	0
$217$	212.238	0.978057
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 26.2715i	− 0.118875i
$222$	0	0
$223$	−190.291	−0.853322	−0.426661	−	0.904412i	$-0.640310\pi$
$223$	−190.291	−0.853322	−0.426661	+	0.904412i	$0.640310\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 370.578i	− 1.63250i	−0.577696	−	0.816252i	$-0.696048\pi$
$227$	− 370.578i	− 1.63250i	0.577696	−	0.816252i	$-0.303952\pi$
$228$	0	0
$229$	407.370	1.77891	0.889455	−	0.457023i	$-0.151084\pi$
$229$	407.370	1.77891	0.889455	+	0.457023i	$0.151084\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 21.5267i	− 0.0923895i	−0.998932	−	0.0461947i	$-0.985291\pi$
$233$	− 21.5267i	− 0.0923895i	0.998932	−	0.0461947i	$-0.0147095\pi$
$234$	0	0
$235$	−277.669	−1.18157
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 198.631i	− 0.831092i	−0.909572	−	0.415546i	$-0.863590\pi$
$239$	− 198.631i	− 0.831092i	0.909572	−	0.415546i	$-0.136410\pi$
$240$	0	0
$241$	277.881	1.15303	0.576516	−	0.817086i	$-0.304412\pi$
$241$	277.881	1.15303	0.576516	+	0.817086i	$0.304412\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	45.0441i	0.183853i
$246$	0	0
$247$	−35.4561	−0.143547
$248$	0	0
$249$	0	0
$250$	0	0
$251$	128.325i	0.511253i	0.966776	+	0.255627i	$0.0822817\pi$
$251$	128.325i	0.511253i	−0.966776	+	0.255627i	$0.917718\pi$
$252$	0	0
$253$	171.895	0.679428
$254$	0	0
$255$	0	0
$256$	0	0
$257$	72.0053i	0.280176i	0.990139	+	0.140088i	$0.0447386\pi$
$257$	72.0053i	0.280176i	−0.990139	+	0.140088i	$0.955261\pi$
$258$	0	0
$259$	−472.894	−1.82585
$260$	0	0
$261$	0	0
$262$	0	0
$263$	131.385i	0.499563i	0.968302	+	0.249782i	$0.0803588\pi$
$263$	131.385i	0.499563i	−0.968302	+	0.249782i	$0.919641\pi$
$264$	0	0
$265$	264.300	0.997358
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 38.8284i	− 0.144344i	−0.997392	−	0.0721718i	$-0.977007\pi$
$269$	− 38.8284i	− 0.144344i	0.997392	−	0.0721718i	$-0.0229930\pi$
$270$	0	0
$271$	−0.802525	−0.00296135	−0.00148067	−	0.999999i	$-0.500471\pi$
$271$	−0.802525	−0.00296135	−0.00148067	+	0.999999i	$0.500471\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 100.345i	− 0.364889i
$276$	0	0
$277$	−122.386	−0.441828	−0.220914	−	0.975293i	$-0.570904\pi$
$277$	−122.386	−0.441828	−0.220914	+	0.975293i	$0.570904\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	293.454i	1.04432i	0.852848	+	0.522160i	$0.174874\pi$
$281$	293.454i	1.04432i	−0.852848	+	0.522160i	$0.825126\pi$
$282$	0	0
$283$	181.238	0.640418	0.320209	−	0.947347i	$-0.396247\pi$
$283$	181.238	0.640418	0.320209	+	0.947347i	$0.396247\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	242.554i	0.845134i
$288$	0	0
$289$	70.5768	0.244210
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 335.493i	− 1.14503i	−0.819896	−	0.572513i	$-0.805969\pi$
$293$	− 335.493i	− 1.14503i	0.819896	−	0.572513i	$-0.194031\pi$
$294$	0	0
$295$	−677.659	−2.29715
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 42.8411i	− 0.143281i
$300$	0	0
$301$	−582.039	−1.93369
$302$	0	0
$303$	0	0
$304$	0	0
$305$	317.348i	1.04049i
$306$	0	0
$307$	289.210	0.942053	0.471027	−	0.882119i	$-0.343884\pi$
$307$	289.210	0.942053	0.471027	+	0.882119i	$0.343884\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 27.3649i	− 0.0879900i	−0.999032	−	0.0439950i	$-0.985991\pi$
$311$	− 27.3649i	− 0.0879900i	0.999032	−	0.0439950i	$-0.0140086\pi$
$312$	0	0
$313$	587.143	1.87586	0.937928	−	0.346830i	$-0.112742\pi$
$313$	587.143	1.87586	0.937928	+	0.346830i	$0.112742\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 376.525i	− 1.18778i	−0.804547	−	0.593889i	$-0.797592\pi$
$317$	− 376.525i	− 1.18778i	0.804547	−	0.593889i	$-0.202408\pi$
$318$	0	0
$319$	330.487	1.03601
$320$	0	0
$321$	0	0
$322$	0	0
$323$	294.785i	0.912648i
$324$	0	0
$325$	−25.0087	−0.0769497
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 333.048i	− 1.01230i
$330$	0	0
$331$	−2.96775	−0.00896602	−0.00448301	−	0.999990i	$-0.501427\pi$
$331$	−2.96775	−0.00896602	−0.00448301	+	0.999990i	$0.501427\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 711.912i	− 2.12511i
$336$	0	0
$337$	658.470	1.95392	0.976958	−	0.213432i	$-0.0684643\pi$
$337$	658.470	1.95392	0.976958	+	0.213432i	$0.0684643\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 201.915i	− 0.592126i
$342$	0	0
$343$	313.330	0.913499
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 295.545i	− 0.851715i	−0.904790	−	0.425857i	$-0.859973\pi$
$347$	− 295.545i	− 0.851715i	0.904790	−	0.425857i	$-0.140027\pi$
$348$	0	0
$349$	−212.758	−0.609622	−0.304811	−	0.952413i	$-0.598593\pi$
$349$	−212.758	−0.609622	−0.304811	+	0.952413i	$0.598593\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	143.514i	0.406555i	0.979121	+	0.203278i	$0.0651594\pi$
$353$	143.514i	0.406555i	−0.979121	+	0.203278i	$0.934841\pi$
$354$	0	0
$355$	532.623	1.50035
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 270.973i	− 0.754799i	−0.926051	−	0.377400i	$-0.876818\pi$
$359$	− 270.973i	− 0.754799i	0.926051	−	0.377400i	$-0.123182\pi$
$360$	0	0
$361$	36.8443	0.102062
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 590.896i	− 1.61889i
$366$	0	0
$367$	401.711	1.09458	0.547290	−	0.836943i	$-0.315660\pi$
$367$	401.711	1.09458	0.547290	+	0.836943i	$0.315660\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	317.012i	0.854479i
$372$	0	0
$373$	293.781	0.787616	0.393808	−	0.919193i	$-0.371157\pi$
$373$	293.781	0.787616	0.393808	+	0.919193i	$0.371157\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 82.3665i	− 0.218479i
$378$	0	0
$379$	280.802	0.740903	0.370452	−	0.928852i	$-0.379203\pi$
$379$	280.802	0.740903	0.370452	+	0.928852i	$0.379203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	631.235i	1.64813i	0.566493	+	0.824066i	$0.308300\pi$
$383$	631.235i	1.64813i	−0.566493	+	0.824066i	$0.691700\pi$
$384$	0	0
$385$	334.231	0.868131
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 436.534i	− 1.12219i	−0.827750	−	0.561097i	$-0.810379\pi$
$389$	− 436.534i	− 1.12219i	0.827750	−	0.561097i	$-0.189621\pi$
$390$	0	0
$391$	−356.184	−0.910958
$392$	0	0
$393$	0	0
$394$	0	0
$395$	445.109i	1.12686i
$396$	0	0
$397$	−405.382	−1.02111	−0.510557	−	0.859844i	$-0.670561\pi$
$397$	−405.382	−1.02111	−0.510557	+	0.859844i	$0.670561\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 77.3182i	− 0.192814i	−0.995342	−	0.0964068i	$-0.969265\pi$
$401$	− 77.3182i	− 0.192814i	0.995342	−	0.0964068i	$-0.0307350\pi$
$402$	0	0
$403$	−50.3229	−0.124871
$404$	0	0
$405$	0	0
$406$	0	0
$407$	449.893i	1.10539i
$408$	0	0
$409$	−170.661	−0.417265	−0.208632	−	0.977994i	$-0.566901\pi$
$409$	−170.661	−0.417265	−0.208632	+	0.977994i	$0.566901\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 812.811i	− 1.96807i
$414$	0	0
$415$	−684.964	−1.65051
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 142.639i	− 0.340427i	−0.985407	−	0.170214i	$-0.945554\pi$
$419$	− 142.639i	− 0.340427i	0.985407	−	0.170214i	$-0.0544458\pi$
$420$	0	0
$421$	615.638	1.46232	0.731162	−	0.682204i	$-0.238979\pi$
$421$	615.638	1.46232	0.731162	+	0.682204i	$0.238979\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	207.924i	0.489233i
$426$	0	0
$427$	−380.640	−0.891428
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 405.961i	− 0.941905i	−0.882159	−	0.470953i	$-0.843910\pi$
$431$	− 405.961i	− 0.941905i	0.882159	−	0.470953i	$-0.156090\pi$
$432$	0	0
$433$	50.1302	0.115774	0.0578870	−	0.998323i	$-0.481564\pi$
$433$	50.1302	0.115774	0.0578870	+	0.998323i	$0.481564\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	480.709i	1.10002i
$438$	0	0
$439$	727.104	1.65627	0.828136	−	0.560527i	$-0.189401\pi$
$439$	727.104	1.65627	0.828136	+	0.560527i	$0.189401\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 374.206i	− 0.844709i	−0.906431	−	0.422355i	$-0.861204\pi$
$443$	− 374.206i	− 0.844709i	0.906431	−	0.422355i	$-0.138796\pi$
$444$	0	0
$445$	−185.600	−0.417078
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 225.368i	− 0.501934i	−0.967996	−	0.250967i	$-0.919251\pi$
$449$	− 225.368i	− 0.501934i	0.967996	−	0.250967i	$-0.0807486\pi$
$450$	0	0
$451$	230.756	0.511654
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 83.2996i	− 0.183076i
$456$	0	0
$457$	−780.525	−1.70793	−0.853966	−	0.520328i	$-0.825810\pi$
$457$	−780.525	−1.70793	−0.853966	+	0.520328i	$0.825810\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 547.426i	− 1.18748i	−0.804659	−	0.593738i	$-0.797652\pi$
$461$	− 547.426i	− 1.18748i	0.804659	−	0.593738i	$-0.202348\pi$
$462$	0	0
$463$	−349.327	−0.754486	−0.377243	−	0.926114i	$-0.623128\pi$
$463$	−349.327	−0.754486	−0.377243	+	0.926114i	$0.623128\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 499.358i	− 1.06929i	−0.845077	−	0.534645i	$-0.820445\pi$
$467$	− 499.358i	− 1.06929i	0.845077	−	0.534645i	$-0.179555\pi$
$468$	0	0
$469$	853.896	1.82067
$470$	0	0
$471$	0	0
$472$	0	0
$473$	553.729i	1.17067i
$474$	0	0
$475$	280.616	0.590770
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 660.072i	− 1.37802i	−0.724751	−	0.689011i	$-0.758045\pi$
$479$	− 660.072i	− 1.37802i	0.724751	−	0.689011i	$-0.241955\pi$
$480$	0	0
$481$	112.126	0.233110
$482$	0	0
$483$	0	0
$484$	0	0
$485$	781.053i	1.61042i
$486$	0	0
$487$	217.861	0.447352	0.223676	−	0.974664i	$-0.428194\pi$
$487$	217.861	0.447352	0.223676	+	0.974664i	$0.428194\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 153.243i	− 0.312103i	−0.987749	−	0.156052i	$-0.950123\pi$
$491$	− 153.243i	− 0.312103i	0.987749	−	0.156052i	$-0.0498766\pi$
$492$	0	0
$493$	−684.802	−1.38905
$494$	0	0
$495$	0	0
$496$	0	0
$497$	638.849i	1.28541i
$498$	0	0
$499$	307.386	0.616005	0.308002	−	0.951386i	$-0.400340\pi$
$499$	307.386	0.616005	0.308002	+	0.951386i	$0.400340\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 797.972i	− 1.58643i	−0.608945	−	0.793213i	$-0.708407\pi$
$503$	− 797.972i	− 1.58643i	0.608945	−	0.793213i	$-0.291593\pi$
$504$	0	0
$505$	−246.107	−0.487341
$506$	0	0
$507$	0	0
$508$	0	0
$509$	300.690i	0.590747i	0.955382	+	0.295373i	$0.0954441\pi$
$509$	300.690i	0.590747i	−0.955382	+	0.295373i	$0.904556\pi$
$510$	0	0
$511$	708.744	1.38697
$512$	0	0
$513$	0	0
$514$	0	0
$515$	934.112i	1.81381i
$516$	0	0
$517$	−316.848	−0.612859
$518$	0	0
$519$	0	0
$520$	0	0
$521$	928.990i	1.78309i	0.452932	+	0.891545i	$0.350378\pi$
$521$	928.990i	1.78309i	−0.452932	+	0.891545i	$0.649622\pi$
$522$	0	0
$523$	−519.086	−0.992516	−0.496258	−	0.868175i	$-0.665293\pi$
$523$	−519.086	−0.992516	−0.496258	+	0.868175i	$0.665293\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	418.389i	0.793906i
$528$	0	0
$529$	−51.8328	−0.0979826
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 57.5108i	− 0.107900i
$534$	0	0
$535$	2.62653	0.00490940
$536$	0	0
$537$	0	0
$538$	0	0
$539$	51.3997i	0.0953613i
$540$	0	0
$541$	−109.959	−0.203251	−0.101625	−	0.994823i	$-0.532404\pi$
$541$	−109.959	−0.203251	−0.101625	+	0.994823i	$0.532404\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	402.917i	0.739298i
$546$	0	0
$547$	−978.239	−1.78837	−0.894185	−	0.447697i	$-0.852244\pi$
$547$	−978.239	−1.78837	−0.894185	+	0.447697i	$0.852244\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	924.213i	1.67734i
$552$	0	0
$553$	−533.881	−0.965427
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 322.002i	− 0.578100i	−0.957314	−	0.289050i	$-0.906661\pi$
$557$	− 322.002i	− 0.578100i	0.957314	−	0.289050i	$-0.0933394\pi$
$558$	0	0
$559$	138.005	0.246878
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 201.614i	− 0.358106i	−0.983839	−	0.179053i	$-0.942697\pi$
$563$	− 201.614i	− 0.358106i	0.983839	−	0.179053i	$-0.0573034\pi$
$564$	0	0
$565$	−162.189	−0.287060
$566$	0	0
$567$	0	0
$568$	0	0
$569$	142.935i	0.251204i	0.992081	+	0.125602i	$0.0400863\pi$
$569$	142.935i	0.251204i	−0.992081	+	0.125602i	$0.959914\pi$
$570$	0	0
$571$	91.4410	0.160142	0.0800709	−	0.996789i	$-0.474485\pi$
$571$	91.4410	0.160142	0.0800709	+	0.996789i	$0.474485\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	339.064i	0.589676i
$576$	0	0
$577$	404.739	0.701455	0.350727	−	0.936478i	$-0.385934\pi$
$577$	404.739	0.701455	0.350727	+	0.936478i	$0.385934\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 821.572i	− 1.41407i
$582$	0	0
$583$	301.592	0.517311
$584$	0	0
$585$	0	0
$586$	0	0
$587$	811.575i	1.38258i	0.722577	+	0.691291i	$0.242958\pi$
$587$	811.575i	1.38258i	−0.722577	+	0.691291i	$0.757042\pi$
$588$	0	0
$589$	564.660	0.958676
$590$	0	0
$591$	0	0
$592$	0	0
$593$	95.3026i	0.160713i	0.996766	+	0.0803563i	$0.0256058\pi$
$593$	95.3026i	0.160713i	−0.996766	+	0.0803563i	$0.974394\pi$
$594$	0	0
$595$	−692.560	−1.16397
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 440.787i	− 0.735871i	−0.929851	−	0.367936i	$-0.880065\pi$
$599$	− 440.787i	− 0.735871i	0.929851	−	0.367936i	$-0.119935\pi$
$600$	0	0
$601$	−180.963	−0.301103	−0.150551	−	0.988602i	$-0.548105\pi$
$601$	−180.963	−0.301103	−0.150551	+	0.988602i	$0.548105\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	438.337i	0.724524i
$606$	0	0
$607$	−555.993	−0.915969	−0.457984	−	0.888960i	$-0.651428\pi$
$607$	−555.993	−0.915969	−0.457984	+	0.888960i	$0.651428\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	78.9674i	0.129243i
$612$	0	0
$613$	−335.352	−0.547067	−0.273534	−	0.961862i	$-0.588192\pi$
$613$	−335.352	−0.547067	−0.273534	+	0.961862i	$0.588192\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1178.54i	1.91011i	0.296433	+	0.955053i	$0.404203\pi$
$617$	1178.54i	1.91011i	−0.296433	+	0.955053i	$0.595797\pi$
$618$	0	0
$619$	898.584	1.45167	0.725835	−	0.687868i	$-0.241453\pi$
$619$	898.584	1.45167	0.725835	+	0.687868i	$0.241453\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 222.615i	− 0.357328i
$624$	0	0
$625$	−778.789	−1.24606
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 932.224i	− 1.48207i
$630$	0	0
$631$	431.017	0.683070	0.341535	−	0.939869i	$-0.389053\pi$
$631$	431.017	0.683070	0.341535	+	0.939869i	$0.389053\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 75.5788i	− 0.119022i
$636$	0	0
$637$	12.8103	0.0201103
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 896.033i	− 1.39787i	−0.715187	−	0.698933i	$-0.753658\pi$
$641$	− 896.033i	− 1.39787i	0.715187	−	0.698933i	$-0.246342\pi$
$642$	0	0
$643$	221.708	0.344802	0.172401	−	0.985027i	$-0.444848\pi$
$643$	221.708	0.344802	0.172401	+	0.985027i	$0.444848\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 536.838i	− 0.829734i	−0.909882	−	0.414867i	$-0.863828\pi$
$647$	− 536.838i	− 0.829734i	0.909882	−	0.414867i	$-0.136172\pi$
$648$	0	0
$649$	−773.276	−1.19149
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 558.614i	− 0.855458i	−0.903907	−	0.427729i	$-0.859314\pi$
$653$	− 558.614i	− 0.855458i	0.903907	−	0.427729i	$-0.140686\pi$
$654$	0	0
$655$	328.046	0.500834
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1077.51i	1.63507i	0.575881	+	0.817534i	$0.304659\pi$
$659$	1077.51i	1.63507i	−0.575881	+	0.817534i	$0.695341\pi$
$660$	0	0
$661$	−295.759	−0.447442	−0.223721	−	0.974653i	$-0.571820\pi$
$661$	−295.759	−0.447442	−0.223721	+	0.974653i	$0.571820\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	934.683i	1.40554i
$666$	0	0
$667$	−1116.71	−1.67423
$668$	0	0
$669$	0	0
$670$	0	0
$671$	362.126i	0.539681i
$672$	0	0
$673$	−567.447	−0.843161	−0.421580	−	0.906791i	$-0.638524\pi$
$673$	−567.447	−0.843161	−0.421580	+	0.906791i	$0.638524\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	939.817i	1.38821i	0.719875	+	0.694104i	$0.244199\pi$
$677$	939.817i	1.38821i	−0.719875	+	0.694104i	$0.755801\pi$
$678$	0	0
$679$	−936.826	−1.37971
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 14.2541i	− 0.0208699i	−0.999946	−	0.0104349i	$-0.996678\pi$
$683$	− 14.2541i	− 0.0208699i	0.999946	−	0.0104349i	$-0.00332161\pi$
$684$	0	0
$685$	−874.930	−1.27727
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 75.1653i	− 0.109093i
$690$	0	0
$691$	166.271	0.240623	0.120312	−	0.992736i	$-0.461611\pi$
$691$	166.271	0.240623	0.120312	+	0.992736i	$0.461611\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 249.590i	− 0.359122i
$696$	0	0
$697$	−478.150	−0.686011
$698$	0	0
$699$	0	0
$700$	0	0
$701$	506.174i	0.722074i	0.932551	+	0.361037i	$0.117577\pi$
$701$	506.174i	0.722074i	−0.932551	+	0.361037i	$0.882423\pi$
$702$	0	0
$703$	−1258.14	−1.78967
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 295.191i	− 0.417526i
$708$	0	0
$709$	519.986	0.733408	0.366704	−	0.930338i	$-0.380486\pi$
$709$	519.986	0.733408	0.366704	+	0.930338i	$0.380486\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	682.270i	0.956900i
$714$	0	0
$715$	−79.2479	−0.110836
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 467.630i	− 0.650390i	−0.945647	−	0.325195i	$-0.894570\pi$
$719$	− 467.630i	− 0.650390i	0.945647	−	0.325195i	$-0.105430\pi$
$720$	0	0
$721$	−1120.41	−1.55397
$722$	0	0
$723$	0	0
$724$	0	0
$725$	651.885i	0.899152i
$726$	0	0
$727$	−524.741	−0.721790	−0.360895	−	0.932607i	$-0.617529\pi$
$727$	−524.741	−0.721790	−0.360895	+	0.932607i	$0.617529\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1147.38i	− 1.56961i
$732$	0	0
$733$	658.695	0.898629	0.449315	−	0.893374i	$-0.351668\pi$
$733$	658.695	0.898629	0.449315	+	0.893374i	$0.351668\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 812.363i	− 1.10226i
$738$	0	0
$739$	1209.27	1.63637	0.818183	−	0.574957i	$-0.194981\pi$
$739$	1209.27	1.63637	0.818183	+	0.574957i	$0.194981\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	392.195i	0.527853i	0.964543	+	0.263926i	$0.0850176\pi$
$743$	392.195i	0.527853i	−0.964543	+	0.263926i	$0.914982\pi$
$744$	0	0
$745$	23.5452	0.0316043
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.15036i	0.00420609i
$750$	0	0
$751$	−1116.99	−1.48733	−0.743666	−	0.668551i	$-0.766915\pi$
$751$	−1116.99	−1.48733	−0.743666	+	0.668551i	$0.766915\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 936.069i	− 1.23983i
$756$	0	0
$757$	−623.892	−0.824164	−0.412082	−	0.911147i	$-0.635198\pi$
$757$	−623.892	−0.824164	−0.412082	+	0.911147i	$0.635198\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 961.237i	− 1.26312i	−0.775326	−	0.631562i	$-0.782414\pi$
$761$	− 961.237i	− 1.26312i	0.775326	−	0.631562i	$-0.217586\pi$
$762$	0	0
$763$	−483.275	−0.633388
$764$	0	0
$765$	0	0
$766$	0	0
$767$	192.722i	0.251267i
$768$	0	0
$769$	219.306	0.285183	0.142592	−	0.989782i	$-0.454456\pi$
$769$	219.306	0.285183	0.142592	+	0.989782i	$0.454456\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	761.394i	0.984986i	0.870316	+	0.492493i	$0.163914\pi$
$773$	761.394i	0.984986i	−0.870316	+	0.492493i	$0.836086\pi$
$774$	0	0
$775$	398.278	0.513907
$776$	0	0
$777$	0	0
$778$	0	0
$779$	645.314i	0.828388i
$780$	0	0
$781$	607.776	0.778202
$782$	0	0
$783$	0	0
$784$	0	0
$785$	736.942i	0.938780i
$786$	0	0
$787$	−526.046	−0.668420	−0.334210	−	0.942499i	$-0.608469\pi$
$787$	−526.046	−0.668420	−0.334210	+	0.942499i	$0.608469\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 194.536i	− 0.245937i
$792$	0	0
$793$	90.2518	0.113811
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1420.42i	− 1.78221i	−0.453794	−	0.891107i	$-0.649930\pi$
$797$	− 1420.42i	− 1.78221i	0.453794	−	0.891107i	$-0.350070\pi$
$798$	0	0
$799$	656.542	0.821705
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 674.271i	− 0.839689i
$804$	0	0
$805$	−1129.36	−1.40294
$806$	0	0
$807$	0	0
$808$	0	0
$809$	749.612i	0.926590i	0.886204	+	0.463295i	$0.153333\pi$
$809$	749.612i	0.926590i	−0.886204	+	0.463295i	$0.846667\pi$
$810$	0	0
$811$	238.446	0.294015	0.147008	−	0.989135i	$-0.453036\pi$
$811$	238.446	0.294015	0.147008	+	0.989135i	$0.453036\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	699.809i	0.858661i
$816$	0	0
$817$	−1548.52	−1.89537
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1083.04i	− 1.31917i	−0.751631	−	0.659583i	$-0.770733\pi$
$821$	− 1083.04i	− 1.31917i	0.751631	−	0.659583i	$-0.229267\pi$
$822$	0	0
$823$	1575.76	1.91466	0.957328	−	0.289004i	$-0.0933242\pi$
$823$	1575.76	1.91466	0.957328	+	0.289004i	$0.0933242\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	124.799i	0.150906i	0.997149	+	0.0754529i	$0.0240403\pi$
$827$	124.799i	0.150906i	−0.997149	+	0.0754529i	$0.975960\pi$
$828$	0	0
$829$	426.486	0.514458	0.257229	−	0.966350i	$-0.417190\pi$
$829$	426.486	0.514458	0.257229	+	0.966350i	$0.417190\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 106.506i	− 0.127858i
$834$	0	0
$835$	−9.64806	−0.0115546
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 497.342i	− 0.592779i	−0.955067	−	0.296389i	$-0.904217\pi$
$839$	− 497.342i	− 0.592779i	0.955067	−	0.296389i	$-0.0957826\pi$
$840$	0	0
$841$	−1306.00	−1.55291
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 1036.58i	− 1.22673i
$846$	0	0
$847$	−525.759	−0.620730
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1520.18i	− 1.78635i
$852$	0	0
$853$	−1534.96	−1.79949	−0.899743	−	0.436420i	$-0.856246\pi$
$853$	−1534.96	−1.79949	−0.899743	+	0.436420i	$0.856246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1218.30i	− 1.42158i	−0.703402	−	0.710792i	$-0.748337\pi$
$857$	− 1218.30i	− 1.42158i	0.703402	−	0.710792i	$-0.251663\pi$
$858$	0	0
$859$	579.441	0.674553	0.337276	−	0.941406i	$-0.390494\pi$
$859$	579.441	0.674553	0.337276	+	0.941406i	$0.390494\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1361.09i	1.57716i	0.614934	+	0.788579i	$0.289183\pi$
$863$	1361.09i	1.57716i	−0.614934	+	0.788579i	$0.710817\pi$
$864$	0	0
$865$	332.422	0.384303
$866$	0	0
$867$	0	0
$868$	0	0
$869$	507.913i	0.584480i
$870$	0	0
$871$	−202.464	−0.232450
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 512.245i	− 0.585423i
$876$	0	0
$877$	−301.895	−0.344236	−0.172118	−	0.985076i	$-0.555061\pi$
$877$	−301.895	−0.344236	−0.172118	+	0.985076i	$0.555061\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1170.94i	− 1.32911i	−0.747240	−	0.664554i	$-0.768622\pi$
$881$	− 1170.94i	− 1.32911i	0.747240	−	0.664554i	$-0.231378\pi$
$882$	0	0
$883$	1339.68	1.51719	0.758597	−	0.651560i	$-0.225885\pi$
$883$	1339.68	1.51719	0.758597	+	0.651560i	$0.225885\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	16.6709i	0.0187948i	0.999956	+	0.00939738i	$0.00299132\pi$
$887$	16.6709i	0.0187948i	−0.999956	+	0.00939738i	$0.997009\pi$
$888$	0	0
$889$	90.6522	0.101971
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 886.073i	− 0.992243i
$894$	0	0
$895$	1496.65	1.67223
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1311.74i	1.45911i
$900$	0	0
$901$	−624.930	−0.693596
$902$	0	0
$903$	0	0
$904$	0	0
$905$	382.254i	0.422380i
$906$	0	0
$907$	−451.336	−0.497614	−0.248807	−	0.968553i	$-0.580038\pi$
$907$	−451.336	−0.497614	−0.248807	+	0.968553i	$0.580038\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1435.42i	1.57565i	0.615896	+	0.787827i	$0.288794\pi$
$911$	1435.42i	1.57565i	−0.615896	+	0.787827i	$0.711206\pi$
$912$	0	0
$913$	−781.611	−0.856091
$914$	0	0
$915$	0	0
$916$	0	0
$917$	393.472i	0.429086i
$918$	0	0
$919$	380.633	0.414181	0.207091	−	0.978322i	$-0.433600\pi$
$919$	380.633	0.414181	0.207091	+	0.978322i	$0.433600\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 151.475i	− 0.164111i
$924$	0	0
$925$	−887.414	−0.959367
$926$	0	0
$927$	0	0
$928$	0	0
$929$	989.157i	1.06475i	0.846507	+	0.532377i	$0.178701\pi$
$929$	989.157i	1.06475i	−0.846507	+	0.532377i	$0.821299\pi$
$930$	0	0
$931$	−143.741	−0.154394
$932$	0	0
$933$	0	0
$934$	0	0
$935$	658.874i	0.704678i
$936$	0	0
$937$	−549.202	−0.586128	−0.293064	−	0.956093i	$-0.594675\pi$
$937$	−549.202	−0.586128	−0.293064	+	0.956093i	$0.594675\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	468.207i	0.497563i	0.968560	+	0.248781i	$0.0800301\pi$
$941$	468.207i	0.497563i	−0.968560	+	0.248781i	$0.919970\pi$
$942$	0	0
$943$	−779.722	−0.826853
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 301.225i	− 0.318084i	−0.987272	−	0.159042i	$-0.949160\pi$
$947$	− 301.225i	− 0.318084i	0.987272	−	0.159042i	$-0.0508405\pi$
$948$	0	0
$949$	−168.047	−0.177078
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 319.384i	− 0.335135i	−0.985861	−	0.167567i	$-0.946409\pi$
$953$	− 319.384i	− 0.335135i	0.985861	−	0.167567i	$-0.0535912\pi$
$954$	0	0
$955$	208.630	0.218461
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1049.43i	− 1.09429i
$960$	0	0
$961$	−159.578	−0.166054
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 266.016i	− 0.275664i
$966$	0	0
$967$	604.173	0.624791	0.312395	−	0.949952i	$-0.398869\pi$
$967$	604.173	0.624791	0.312395	+	0.949952i	$0.398869\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	465.068i	0.478958i	0.970902	+	0.239479i	$0.0769766\pi$
$971$	465.068i	0.478958i	−0.970902	+	0.239479i	$0.923023\pi$
$972$	0	0
$973$	299.368	0.307675
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 383.951i	− 0.392990i	−0.980505	−	0.196495i	$-0.937044\pi$
$977$	− 383.951i	− 0.392990i	0.980505	−	0.196495i	$-0.0629560\pi$
$978$	0	0
$979$	−211.788	−0.216330
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 676.882i	− 0.688588i	−0.938862	−	0.344294i	$-0.888118\pi$
$983$	− 676.882i	− 0.688588i	0.938862	−	0.344294i	$-0.111882\pi$
$984$	0	0
$985$	1271.18	1.29054
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1871.05i	− 1.89186i
$990$	0	0
$991$	1097.79	1.10776	0.553878	−	0.832598i	$-0.313147\pi$
$991$	1097.79	1.10776	0.553878	+	0.832598i	$0.313147\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	801.067i	0.805092i
$996$	0	0
$997$	181.425	0.181971	0.0909857	−	0.995852i	$-0.470998\pi$
$997$	181.425	0.181971	0.0909857	+	0.995852i	$0.470998\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.3.e.i.161.6		24
3.2	odd	2	inner	2592.3.e.i.161.5		24
4.3	odd	2	inner	2592.3.e.i.161.20		24
9.2	odd	6		288.3.q.b.257.12	yes	24
9.4	even	3		288.3.q.b.65.12	yes	24
9.5	odd	6		864.3.q.a.737.4		24
9.7	even	3		864.3.q.a.449.4		24
12.11	even	2	inner	2592.3.e.i.161.19		24
36.7	odd	6		864.3.q.a.449.3		24
36.11	even	6		288.3.q.b.257.1	yes	24
36.23	even	6		864.3.q.a.737.3		24
36.31	odd	6		288.3.q.b.65.1	✓	24
72.5	odd	6		1728.3.q.k.1601.10		24
72.11	even	6		576.3.q.l.257.12		24
72.13	even	6		576.3.q.l.65.1		24
72.29	odd	6		576.3.q.l.257.1		24
72.43	odd	6		1728.3.q.k.449.9		24
72.59	even	6		1728.3.q.k.1601.9		24
72.61	even	6		1728.3.q.k.449.10		24
72.67	odd	6		576.3.q.l.65.12		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.b.65.1	✓	24	36.31	odd	6
288.3.q.b.65.12	yes	24	9.4	even	3
288.3.q.b.257.1	yes	24	36.11	even	6
288.3.q.b.257.12	yes	24	9.2	odd	6
576.3.q.l.65.1		24	72.13	even	6
576.3.q.l.65.12		24	72.67	odd	6
576.3.q.l.257.1		24	72.29	odd	6
576.3.q.l.257.12		24	72.11	even	6
864.3.q.a.449.3		24	36.7	odd	6
864.3.q.a.449.4		24	9.7	even	3
864.3.q.a.737.3		24	36.23	even	6
864.3.q.a.737.4		24	9.5	odd	6
1728.3.q.k.449.9		24	72.43	odd	6
1728.3.q.k.449.10		24	72.61	even	6
1728.3.q.k.1601.9		24	72.59	even	6
1728.3.q.k.1601.10		24	72.5	odd	6
2592.3.e.i.161.5		24	3.2	odd	2	inner
2592.3.e.i.161.6		24	1.1	even	1	trivial
2592.3.e.i.161.19		24	12.11	even	2	inner
2592.3.e.i.161.20		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+6.25050i q^{5} -7.49710 q^{7} +O(q^{10})\)	\(q+6.25050i q^{5} -7.49710 q^{7} +7.13244i q^{11} +1.77760 q^{13} -14.7791i q^{17} -19.9460 q^{19} -24.1005i q^{23} -14.0688 q^{25} -46.3357i q^{29} -28.3094 q^{31} -46.8606i q^{35} +63.0770 q^{37} -32.3530i q^{41} +77.6353 q^{43} +44.4235i q^{47} +7.20647 q^{49} -42.2846i q^{53} -44.5813 q^{55} +108.417i q^{59} +50.7716 q^{61} +11.1109i q^{65} -113.897 q^{67} -85.2129i q^{71} -94.5357 q^{73} -53.4726i q^{77} +71.2117 q^{79} +109.585i q^{83} +92.3771 q^{85} +29.6936i q^{89} -13.3269 q^{91} -124.673i q^{95} +124.959 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * 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

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