LMFDB - Embedded newform 2592.2.c.c.2591.23

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2592,2,Mod(2591,2592)]

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([1, 0, 1]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2592.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.c (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:	$20.6972242039$
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			2591.23
Character	$\chi$	$=$	2592.2591
Dual form			2592.2.c.c.2591.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.93668i q^{5} -1.11004i q^{7} +O(q^{10})$	$q+3.93668i q^{5} -1.11004i q^{7} -3.26603 q^{11} -0.249825 q^{13} +5.86838i q^{17} -2.19827i q^{19} -5.58641 q^{23} -10.4974 q^{25} -2.72014i q^{29} -10.3576i q^{31} +4.36988 q^{35} -0.333076 q^{37} -6.10144i q^{41} +9.82629i q^{43} -9.41480 q^{47} +5.76781 q^{49} +4.75157i q^{53} -12.8573i q^{55} +6.53047 q^{59} -2.14671 q^{61} -0.983480i q^{65} +0.578699i q^{67} -3.26444 q^{71} -12.6045 q^{73} +3.62543i q^{77} -8.85728i q^{79} +4.68113 q^{83} -23.1019 q^{85} -4.75157i q^{89} +0.277316i q^{91} +8.65387 q^{95} -1.83273 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q - 24 q^{25} - 24 q^{49} + 24 q^{73} - 24 q^{97}+O(q^{100}) $ 24 * q - 24 * q^25 - 24 * q^49 + 24 * q^73 - 24 * 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
$5$	3.93668i	1.76053i	0.474478	+	0.880267i	$0.342637\pi$
$5$	3.93668i	1.76053i	−0.474478	+	0.880267i	$0.657363\pi$
$6$	0	0
$7$	− 1.11004i	− 0.419557i	−0.977749	−	0.209778i	$-0.932726\pi$
$7$	− 1.11004i	− 0.419557i	0.977749	−	0.209778i	$-0.0672742\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.26603	−0.984744	−0.492372	−	0.870385i	$-0.663870\pi$
$11$	−3.26603	−0.984744	−0.492372	+	0.870385i	$0.663870\pi$
$12$	0	0
$13$	−0.249825	−0.0692890	−0.0346445	−	0.999400i	$-0.511030\pi$
$13$	−0.249825	−0.0692890	−0.0346445	+	0.999400i	$0.511030\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.86838i	1.42329i	0.702538	+	0.711646i	$0.252050\pi$
$17$	5.86838i	1.42329i	−0.702538	+	0.711646i	$0.747950\pi$
$18$	0	0
$19$	− 2.19827i	− 0.504317i	−0.967686	−	0.252159i	$-0.918860\pi$
$19$	− 2.19827i	− 0.504317i	0.967686	−	0.252159i	$-0.0811405\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.58641	−1.16485	−0.582423	−	0.812886i	$-0.697895\pi$
$23$	−5.58641	−1.16485	−0.582423	+	0.812886i	$0.697895\pi$
$24$	0	0
$25$	−10.4974	−2.09948
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 2.72014i	− 0.505118i	−0.967581	−	0.252559i	$-0.918728\pi$
$29$	− 2.72014i	− 0.505118i	0.967581	−	0.252559i	$-0.0812722\pi$
$30$	0	0
$31$	− 10.3576i	− 1.86029i	−0.367197	−	0.930143i	$-0.619683\pi$
$31$	− 10.3576i	− 1.86029i	0.367197	−	0.930143i	$-0.380317\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.36988	0.738644
$36$	0	0
$37$	−0.333076	−0.0547574	−0.0273787	−	0.999625i	$-0.508716\pi$
$37$	−0.333076	−0.0547574	−0.0273787	+	0.999625i	$0.508716\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 6.10144i	− 0.952884i	−0.879206	−	0.476442i	$-0.841926\pi$
$41$	− 6.10144i	− 0.952884i	0.879206	−	0.476442i	$-0.158074\pi$
$42$	0	0
$43$	9.82629i	1.49850i	0.662290	+	0.749248i	$0.269585\pi$
$43$	9.82629i	1.49850i	−0.662290	+	0.749248i	$0.730415\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.41480	−1.37329	−0.686645	−	0.726993i	$-0.740917\pi$
$47$	−9.41480	−1.37329	−0.686645	+	0.726993i	$0.740917\pi$
$48$	0	0
$49$	5.76781	0.823972
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.75157i	0.652678i	0.945253	+	0.326339i	$0.105815\pi$
$53$	4.75157i	0.652678i	−0.945253	+	0.326339i	$0.894185\pi$
$54$	0	0
$55$	− 12.8573i	− 1.73368i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.53047	0.850195	0.425097	−	0.905148i	$-0.360240\pi$
$59$	6.53047	0.850195	0.425097	+	0.905148i	$0.360240\pi$
$60$	0	0
$61$	−2.14671	−0.274858	−0.137429	−	0.990512i	$-0.543884\pi$
$61$	−2.14671	−0.274858	−0.137429	+	0.990512i	$0.543884\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.983480i	− 0.121986i
$66$	0	0
$67$	0.578699i	0.0706993i	0.999375	+	0.0353497i	$0.0112545\pi$
$67$	0.578699i	0.0706993i	−0.999375	+	0.0353497i	$0.988746\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−3.26444	−0.387418	−0.193709	−	0.981059i	$-0.562052\pi$
$71$	−3.26444	−0.387418	−0.193709	+	0.981059i	$0.562052\pi$
$72$	0	0
$73$	−12.6045	−1.47525	−0.737623	−	0.675213i	$-0.764052\pi$
$73$	−12.6045	−1.47525	−0.737623	+	0.675213i	$0.764052\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.62543i	0.413156i
$78$	0	0
$79$	− 8.85728i	− 0.996522i	−0.867027	−	0.498261i	$-0.833972\pi$
$79$	− 8.85728i	− 0.996522i	0.867027	−	0.498261i	$-0.166028\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.68113	0.513820	0.256910	−	0.966435i	$-0.417296\pi$
$83$	4.68113	0.513820	0.256910	+	0.966435i	$0.417296\pi$
$84$	0	0
$85$	−23.1019	−2.50576
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 4.75157i	− 0.503665i	−0.967771	−	0.251833i	$-0.918967\pi$
$89$	− 4.75157i	− 0.503665i	0.967771	−	0.251833i	$-0.0810333\pi$
$90$	0	0
$91$	0.277316i	0.0290707i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.65387	0.887868
$96$	0	0
$97$	−1.83273	−0.186085	−0.0930426	−	0.995662i	$-0.529659\pi$
$97$	−1.83273	−0.186085	−0.0930426	+	0.995662i	$0.529659\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.59353i	− 0.954592i	−0.878743	−	0.477296i	$-0.841617\pi$
$101$	− 9.59353i	− 0.954592i	0.878743	−	0.477296i	$-0.158383\pi$
$102$	0	0
$103$	0.390307i	0.0384581i	0.999815	+	0.0192291i	$0.00612118\pi$
$103$	0.390307i	0.0384581i	−0.999815	+	0.0192291i	$0.993879\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.67656	0.742121	0.371060	−	0.928609i	$-0.378994\pi$
$107$	7.67656	0.742121	0.371060	+	0.928609i	$0.378994\pi$
$108$	0	0
$109$	−4.72961	−0.453015	−0.226507	−	0.974009i	$-0.572731\pi$
$109$	−4.72961	−0.453015	−0.226507	+	0.974009i	$0.572731\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 7.32905i	− 0.689459i	−0.938702	−	0.344730i	$-0.887971\pi$
$113$	− 7.32905i	− 0.689459i	0.938702	−	0.344730i	$-0.112029\pi$
$114$	0	0
$115$	− 21.9919i	− 2.05075i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.51416	0.597152
$120$	0	0
$121$	−0.333076	−0.0302797
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 21.6415i	− 1.93568i
$126$	0	0
$127$	− 0.116971i	− 0.0103795i	−0.999987	−	0.00518973i	$-0.998348\pi$
$127$	− 0.116971i	− 0.0103795i	0.999987	−	0.00518973i	$-0.00165195\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.91774	0.342295	0.171147	−	0.985245i	$-0.445253\pi$
$131$	3.91774	0.342295	0.171147	+	0.985245i	$0.445253\pi$
$132$	0	0
$133$	−2.44017	−0.211590
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.42877i	0.634683i	0.948311	+	0.317341i	$0.102790\pi$
$137$	7.42877i	0.634683i	−0.948311	+	0.317341i	$0.897210\pi$
$138$	0	0
$139$	− 14.9426i	− 1.26741i	−0.773574	−	0.633706i	$-0.781533\pi$
$139$	− 14.9426i	− 1.26741i	0.773574	−	0.633706i	$-0.218467\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.815935	0.0682319
$144$	0	0
$145$	10.7083	0.889278
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.70406i	0.139602i	0.997561	+	0.0698010i	$0.0222364\pi$
$149$	1.70406i	0.139602i	−0.997561	+	0.0698010i	$0.977764\pi$
$150$	0	0
$151$	6.01468i	0.489468i	0.969590	+	0.244734i	$0.0787006\pi$
$151$	6.01468i	0.489468i	−0.969590	+	0.244734i	$0.921299\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	40.7746	3.27510
$156$	0	0
$157$	5.85727	0.467461	0.233731	−	0.972301i	$-0.424907\pi$
$157$	5.85727	0.467461	0.233731	+	0.972301i	$0.424907\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.20115i	0.488719i
$162$	0	0
$163$	4.27956i	0.335201i	0.985855	+	0.167601i	$0.0536019\pi$
$163$	4.27956i	0.335201i	−0.985855	+	0.167601i	$0.946398\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6295	−1.44159	−0.720795	−	0.693148i	$-0.756223\pi$
$167$	−18.6295	−1.44159	−0.720795	+	0.693148i	$0.756223\pi$
$168$	0	0
$169$	−12.9376	−0.995199
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.81618i	0.670282i	0.942168	+	0.335141i	$0.108784\pi$
$173$	8.81618i	0.670282i	−0.942168	+	0.335141i	$0.891216\pi$
$174$	0	0
$175$	11.6526i	0.880852i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.21764	0.464728	0.232364	−	0.972629i	$-0.425354\pi$
$179$	6.21764	0.464728	0.232364	+	0.972629i	$0.425354\pi$
$180$	0	0
$181$	−20.6618	−1.53578	−0.767888	−	0.640584i	$-0.778692\pi$
$181$	−20.6618	−1.53578	−0.767888	+	0.640584i	$0.778692\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1.31121i	− 0.0964023i
$186$	0	0
$187$	− 19.1663i	− 1.40158i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.7443	−0.994504	−0.497252	−	0.867606i	$-0.665658\pi$
$191$	−13.7443	−0.994504	−0.497252	+	0.867606i	$0.665658\pi$
$192$	0	0
$193$	−17.1621	−1.23536	−0.617678	−	0.786431i	$-0.711926\pi$
$193$	−17.1621	−1.23536	−0.617678	+	0.786431i	$0.711926\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 5.65685i	− 0.403034i	−0.979485	−	0.201517i	$-0.935413\pi$
$197$	− 5.65685i	− 0.403034i	0.979485	−	0.201517i	$-0.0645872\pi$
$198$	0	0
$199$	10.8015i	0.765700i	0.923811	+	0.382850i	$0.125057\pi$
$199$	10.8015i	0.765700i	−0.923811	+	0.382850i	$0.874943\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.01948	−0.211926
$204$	0	0
$205$	24.0194	1.67759
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.17960i	0.496623i
$210$	0	0
$211$	− 7.44312i	− 0.512405i	−0.966623	−	0.256203i	$-0.917529\pi$
$211$	− 7.44312i	− 0.512405i	0.966623	−	0.256203i	$-0.0824715\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−38.6829	−2.63815
$216$	0	0
$217$	−11.4974	−0.780495
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.46607i	− 0.0986185i
$222$	0	0
$223$	− 21.4387i	− 1.43564i	−0.696227	−	0.717822i	$-0.745139\pi$
$223$	− 21.4387i	− 1.43564i	0.696227	−	0.717822i	$-0.254861\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−26.7914	−1.77821	−0.889104	−	0.457706i	$-0.848671\pi$
$227$	−26.7914	−1.77821	−0.889104	+	0.457706i	$0.848671\pi$
$228$	0	0
$229$	−0.0832514	−0.00550141	−0.00275070	−	0.999996i	$-0.500876\pi$
$229$	−0.0832514	−0.00550141	−0.00275070	+	0.999996i	$0.500876\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 9.29262i	− 0.608780i	−0.952548	−	0.304390i	$-0.901547\pi$
$233$	− 9.29262i	− 0.608780i	0.952548	−	0.304390i	$-0.0984526\pi$
$234$	0	0
$235$	− 37.0630i	− 2.41773i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.0896	−0.976062	−0.488031	−	0.872826i	$-0.662285\pi$
$239$	−15.0896	−0.976062	−0.488031	+	0.872826i	$0.662285\pi$
$240$	0	0
$241$	24.0385	1.54845	0.774227	−	0.632908i	$-0.218139\pi$
$241$	24.0385	1.54845	0.774227	+	0.632908i	$0.218139\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	22.7060i	1.45063i
$246$	0	0
$247$	0.549182i	0.0349436i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−18.4712	−1.16589	−0.582947	−	0.812510i	$-0.698101\pi$
$251$	−18.4712	−1.16589	−0.582947	+	0.812510i	$0.698101\pi$
$252$	0	0
$253$	18.2454	1.14708
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.1494i	1.19451i	0.802052	+	0.597254i	$0.203742\pi$
$257$	19.1494i	1.19451i	−0.802052	+	0.597254i	$0.796258\pi$
$258$	0	0
$259$	0.369729i	0.0229738i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.79860	−0.234232	−0.117116	−	0.993118i	$-0.537365\pi$
$263$	−3.79860	−0.234232	−0.117116	+	0.993118i	$0.537365\pi$
$264$	0	0
$265$	−18.7054	−1.14906
$266$	0	0
$267$	0	0
$268$	0	0
$269$	22.0593i	1.34498i	0.740106	+	0.672490i	$0.234775\pi$
$269$	22.0593i	1.34498i	−0.740106	+	0.672490i	$0.765225\pi$
$270$	0	0
$271$	16.7153i	1.01538i	0.861540	+	0.507690i	$0.169501\pi$
$271$	16.7153i	1.01538i	−0.861540	+	0.507690i	$0.830499\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	34.2848	2.06745
$276$	0	0
$277$	−18.5150	−1.11246	−0.556231	−	0.831028i	$-0.687753\pi$
$277$	−18.5150	−1.11246	−0.556231	+	0.831028i	$0.687753\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 5.07930i	− 0.303006i	−0.988457	−	0.151503i	$-0.951589\pi$
$281$	− 5.07930i	− 0.303006i	0.988457	−	0.151503i	$-0.0484113\pi$
$282$	0	0
$283$	22.8672i	1.35931i	0.733531	+	0.679656i	$0.237871\pi$
$283$	22.8672i	1.35931i	−0.733531	+	0.679656i	$0.762129\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.77285	−0.399789
$288$	0	0
$289$	−17.4379	−1.02576
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 0.185071i	− 0.0108120i	−0.999985	−	0.00540599i	$-0.998279\pi$
$293$	− 0.185071i	− 0.0108120i	0.999985	−	0.00540599i	$-0.00172079\pi$
$294$	0	0
$295$	25.7083i	1.49680i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.39562	0.0807111
$300$	0	0
$301$	10.9076	0.628704
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.45090i	− 0.483897i
$306$	0	0
$307$	18.7966i	1.07278i	0.843971	+	0.536388i	$0.180212\pi$
$307$	18.7966i	1.07278i	−0.843971	+	0.536388i	$0.819788\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	7.30417	0.414181	0.207091	−	0.978322i	$-0.433600\pi$
$311$	7.30417	0.414181	0.207091	+	0.978322i	$0.433600\pi$
$312$	0	0
$313$	22.4945	1.27146	0.635732	−	0.771910i	$-0.280698\pi$
$313$	22.4945	1.27146	0.635732	+	0.771910i	$0.280698\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 10.3352i	− 0.580481i	−0.956954	−	0.290240i	$-0.906265\pi$
$317$	− 10.3352i	− 0.580481i	0.956954	−	0.290240i	$-0.0937353\pi$
$318$	0	0
$319$	8.88406i	0.497412i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.9003	0.717791
$324$	0	0
$325$	2.62252	0.145471
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.4508i	0.576173i
$330$	0	0
$331$	28.4915i	1.56604i	0.621999	+	0.783018i	$0.286321\pi$
$331$	28.4915i	1.56604i	−0.621999	+	0.783018i	$0.713679\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.27815	−0.124469
$336$	0	0
$337$	23.4357	1.27662	0.638312	−	0.769778i	$-0.279633\pi$
$337$	23.4357	1.27662	0.638312	+	0.769778i	$0.279633\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	33.8283i	1.83191i
$342$	0	0
$343$	− 14.1728i	− 0.765260i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.9063	−1.17599	−0.587995	−	0.808865i	$-0.700083\pi$
$347$	−21.9063	−1.17599	−0.587995	+	0.808865i	$0.700083\pi$
$348$	0	0
$349$	−28.8521	−1.54442	−0.772209	−	0.635369i	$-0.780848\pi$
$349$	−28.8521	−1.54442	−0.772209	+	0.635369i	$0.780848\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8.37085i	0.445535i	0.974872	+	0.222768i	$0.0715092\pi$
$353$	8.37085i	0.445535i	−0.974872	+	0.222768i	$0.928491\pi$
$354$	0	0
$355$	− 12.8511i	− 0.682063i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	35.2621	1.86106	0.930532	−	0.366210i	$-0.119345\pi$
$359$	35.2621	1.86106	0.930532	+	0.366210i	$0.119345\pi$
$360$	0	0
$361$	14.1676	0.745664
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 49.6199i	− 2.59722i
$366$	0	0
$367$	6.43097i	0.335694i	0.985813	+	0.167847i	$0.0536814\pi$
$367$	6.43097i	0.335694i	−0.985813	+	0.167847i	$0.946319\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.27445	0.273836
$372$	0	0
$373$	19.6848	1.01924	0.509621	−	0.860399i	$-0.329786\pi$
$373$	19.6848	1.01924	0.509621	+	0.860399i	$0.329786\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.679560i	0.0349991i
$378$	0	0
$379$	− 1.50534i	− 0.0773240i	−0.999252	−	0.0386620i	$-0.987690\pi$
$379$	− 1.50534i	− 0.0773240i	0.999252	−	0.0386620i	$-0.0123096\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−28.1326	−1.43751	−0.718754	−	0.695264i	$-0.755287\pi$
$383$	−28.1326	−1.43751	−0.718754	+	0.695264i	$0.755287\pi$
$384$	0	0
$385$	−14.2721	−0.727375
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.0277i	0.812638i	0.913731	+	0.406319i	$0.133188\pi$
$389$	16.0277i	0.812638i	−0.913731	+	0.406319i	$0.866812\pi$
$390$	0	0
$391$	− 32.7832i	− 1.65792i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	34.8683	1.75441
$396$	0	0
$397$	13.7725	0.691224	0.345612	−	0.938377i	$-0.387671\pi$
$397$	13.7725	0.691224	0.345612	+	0.938377i	$0.387671\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.4803i	1.82174i	0.412693	+	0.910870i	$0.364588\pi$
$401$	36.4803i	1.82174i	−0.412693	+	0.910870i	$0.635412\pi$
$402$	0	0
$403$	2.58760i	0.128897i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.08784	0.0539220
$408$	0	0
$409$	−2.83350	−0.140107	−0.0700537	−	0.997543i	$-0.522317\pi$
$409$	−2.83350	−0.140107	−0.0700537	+	0.997543i	$0.522317\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7.24910i	− 0.356705i
$414$	0	0
$415$	18.4281i	0.904599i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	13.2943	0.649471	0.324736	−	0.945805i	$-0.394725\pi$
$419$	13.2943	0.649471	0.324736	+	0.945805i	$0.394725\pi$
$420$	0	0
$421$	−19.8052	−0.965246	−0.482623	−	0.875828i	$-0.660316\pi$
$421$	−19.8052	−0.965246	−0.482623	+	0.875828i	$0.660316\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 61.6029i	− 2.98818i
$426$	0	0
$427$	2.38294i	0.115319i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.7886	−0.567837	−0.283919	−	0.958848i	$-0.591635\pi$
$431$	−11.7886	−0.567837	−0.283919	+	0.958848i	$0.591635\pi$
$432$	0	0
$433$	33.5993	1.61468	0.807341	−	0.590086i	$-0.200906\pi$
$433$	33.5993	1.61468	0.807341	+	0.590086i	$0.200906\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	12.2804i	0.587452i
$438$	0	0
$439$	4.47070i	0.213375i	0.994293	+	0.106687i	$0.0340244\pi$
$439$	4.47070i	0.213375i	−0.994293	+	0.106687i	$0.965976\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.06840	0.0982726	0.0491363	−	0.998792i	$-0.484353\pi$
$443$	2.06840	0.0982726	0.0491363	+	0.998792i	$0.484353\pi$
$444$	0	0
$445$	18.7054	0.886721
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.297420i	0.0140361i	0.999975	+	0.00701807i	$0.00223394\pi$
$449$	0.297420i	0.0140361i	−0.999975	+	0.00701807i	$0.997766\pi$
$450$	0	0
$451$	19.9274i	0.938347i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.09170	−0.0511799
$456$	0	0
$457$	2.16978	0.101498	0.0507491	−	0.998711i	$-0.483839\pi$
$457$	2.16978	0.101498	0.0507491	+	0.998711i	$0.483839\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 41.5468i	− 1.93503i	−0.252819	−	0.967514i	$-0.581358\pi$
$461$	− 41.5468i	− 1.93503i	0.252819	−	0.967514i	$-0.418642\pi$
$462$	0	0
$463$	19.4303i	0.903001i	0.892271	+	0.451501i	$0.149111\pi$
$463$	19.4303i	0.903001i	−0.892271	+	0.451501i	$0.850889\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.5573	0.673630	0.336815	−	0.941571i	$-0.390650\pi$
$467$	14.5573	0.673630	0.336815	+	0.941571i	$0.390650\pi$
$468$	0	0
$469$	0.642380	0.0296624
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 32.0929i	− 1.47563i
$474$	0	0
$475$	23.0761i	1.05881i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−26.3301	−1.20305	−0.601526	−	0.798854i	$-0.705440\pi$
$479$	−26.3301	−1.20305	−0.601526	+	0.798854i	$0.705440\pi$
$480$	0	0
$481$	0.0832108	0.00379409
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.21485i	− 0.327609i
$486$	0	0
$487$	− 21.6173i	− 0.979574i	−0.871842	−	0.489787i	$-0.837074\pi$
$487$	− 21.6173i	− 0.979574i	0.871842	−	0.489787i	$-0.162926\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.3252	0.781873	0.390937	−	0.920418i	$-0.372151\pi$
$491$	17.3252	0.781873	0.390937	+	0.920418i	$0.372151\pi$
$492$	0	0
$493$	15.9629	0.718931
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.62367i	0.162544i
$498$	0	0
$499$	24.7661i	1.10868i	0.832289	+	0.554342i	$0.187030\pi$
$499$	24.7661i	1.10868i	−0.832289	+	0.554342i	$0.812970\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11.4045	−0.508500	−0.254250	−	0.967139i	$-0.581829\pi$
$503$	−11.4045	−0.508500	−0.254250	+	0.967139i	$0.581829\pi$
$504$	0	0
$505$	37.7666	1.68059
$506$	0	0
$507$	0	0
$508$	0	0
$509$	23.4511i	1.03945i	0.854333	+	0.519726i	$0.173966\pi$
$509$	23.4511i	1.03945i	−0.854333	+	0.519726i	$0.826034\pi$
$510$	0	0
$511$	13.9915i	0.618950i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.53651	−0.0677069
$516$	0	0
$517$	30.7490	1.35234
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 12.9913i	− 0.569160i	−0.958652	−	0.284580i	$-0.908146\pi$
$521$	− 12.9913i	− 0.569160i	0.958652	−	0.284580i	$-0.0918541\pi$
$522$	0	0
$523$	18.8040i	0.822243i	0.911581	+	0.411121i	$0.134863\pi$
$523$	18.8040i	0.822243i	−0.911581	+	0.411121i	$0.865137\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.7826	2.64773
$528$	0	0
$529$	8.20798	0.356869
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.52429i	0.0660244i
$534$	0	0
$535$	30.2201i	1.30653i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.8378	−0.811401
$540$	0	0
$541$	−9.13908	−0.392920	−0.196460	−	0.980512i	$-0.562945\pi$
$541$	−9.13908	−0.392920	−0.196460	+	0.980512i	$0.562945\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 18.6189i	− 0.797548i
$546$	0	0
$547$	17.7995i	0.761052i	0.924770	+	0.380526i	$0.124257\pi$
$547$	17.7995i	0.761052i	−0.924770	+	0.380526i	$0.875743\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−5.97960	−0.254740
$552$	0	0
$553$	−9.83196	−0.418098
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 19.4205i	− 0.822872i	−0.911439	−	0.411436i	$-0.865027\pi$
$557$	− 19.4205i	− 0.822872i	0.911439	−	0.411436i	$-0.134973\pi$
$558$	0	0
$559$	− 2.45485i	− 0.103829i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−30.4155	−1.28186	−0.640930	−	0.767599i	$-0.721451\pi$
$563$	−30.4155	−1.28186	−0.640930	+	0.767599i	$0.721451\pi$
$564$	0	0
$565$	28.8521	1.21382
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 9.68207i	− 0.405893i	−0.979190	−	0.202947i	$-0.934948\pi$
$569$	− 9.68207i	− 0.405893i	0.979190	−	0.202947i	$-0.0650518\pi$
$570$	0	0
$571$	− 5.75585i	− 0.240875i	−0.992721	−	0.120437i	$-0.961570\pi$
$571$	− 5.75585i	− 0.240875i	0.992721	−	0.120437i	$-0.0384297\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	58.6429	2.44558
$576$	0	0
$577$	−19.7436	−0.821936	−0.410968	−	0.911650i	$-0.634809\pi$
$577$	−19.7436	−0.821936	−0.410968	+	0.911650i	$0.634809\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 5.19625i	− 0.215577i
$582$	0	0
$583$	− 15.5188i	− 0.642721i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	27.2029	1.12278	0.561392	−	0.827550i	$-0.310266\pi$
$587$	27.2029	1.12278	0.561392	+	0.827550i	$0.310266\pi$
$588$	0	0
$589$	−22.7689	−0.938174
$590$	0	0
$591$	0	0
$592$	0	0
$593$	34.4946i	1.41652i	0.705950	+	0.708261i	$0.250520\pi$
$593$	34.4946i	1.41652i	−0.705950	+	0.708261i	$0.749480\pi$
$594$	0	0
$595$	25.6441i	1.05131i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.19418	−0.253087	−0.126544	−	0.991961i	$-0.540388\pi$
$599$	−6.19418	−0.253087	−0.126544	+	0.991961i	$0.540388\pi$
$600$	0	0
$601$	25.7688	1.05113	0.525565	−	0.850753i	$-0.323854\pi$
$601$	25.7688	1.05113	0.525565	+	0.850753i	$0.323854\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 1.31121i	− 0.0533084i
$606$	0	0
$607$	− 42.9827i	− 1.74461i	−0.488958	−	0.872307i	$-0.662623\pi$
$607$	− 42.9827i	− 1.74461i	0.488958	−	0.872307i	$-0.337377\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.35205	0.0951539
$612$	0	0
$613$	33.0975	1.33679	0.668397	−	0.743805i	$-0.266981\pi$
$613$	33.0975	1.33679	0.668397	+	0.743805i	$0.266981\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.65222i	− 0.187291i	−0.995606	−	0.0936456i	$-0.970148\pi$
$617$	− 4.65222i	− 0.187291i	0.995606	−	0.0936456i	$-0.0298521\pi$
$618$	0	0
$619$	7.51405i	0.302015i	0.988533	+	0.151008i	$0.0482518\pi$
$619$	7.51405i	0.302015i	−0.988533	+	0.151008i	$0.951748\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5.27445	−0.211316
$624$	0	0
$625$	32.7087	1.30835
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.95462i	− 0.0779358i
$630$	0	0
$631$	− 41.0242i	− 1.63315i	−0.577241	−	0.816574i	$-0.695871\pi$
$631$	− 41.0242i	− 1.63315i	0.577241	−	0.816574i	$-0.304129\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.460475	0.0182734
$636$	0	0
$637$	−1.44094	−0.0570922
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.3598i	0.962153i	0.876679	+	0.481076i	$0.159754\pi$
$641$	24.3598i	0.962153i	−0.876679	+	0.481076i	$0.840246\pi$
$642$	0	0
$643$	28.0687i	1.10692i	0.832875	+	0.553461i	$0.186693\pi$
$643$	28.0687i	1.10692i	−0.832875	+	0.553461i	$0.813307\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−18.3596	−0.721791	−0.360895	−	0.932606i	$-0.617529\pi$
$647$	−18.3596	−0.721791	−0.360895	+	0.932606i	$0.617529\pi$
$648$	0	0
$649$	−21.3287	−0.837224
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 41.2174i	− 1.61296i	−0.591260	−	0.806481i	$-0.701369\pi$
$653$	− 41.2174i	− 1.61296i	0.591260	−	0.806481i	$-0.298631\pi$
$654$	0	0
$655$	15.4229i	0.602622i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−13.5439	−0.527595	−0.263797	−	0.964578i	$-0.584975\pi$
$659$	−13.5439	−0.527595	−0.263797	+	0.964578i	$0.584975\pi$
$660$	0	0
$661$	25.9743	1.01028	0.505141	−	0.863037i	$-0.331440\pi$
$661$	25.9743	1.01028	0.505141	+	0.863037i	$0.331440\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 9.60616i	− 0.372511i
$666$	0	0
$667$	15.1958i	0.588385i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	7.01121	0.270665
$672$	0	0
$673$	−14.4104	−0.555482	−0.277741	−	0.960656i	$-0.589586\pi$
$673$	−14.4104	−0.555482	−0.277741	+	0.960656i	$0.589586\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	33.0759i	1.27121i	0.772015	+	0.635604i	$0.219249\pi$
$677$	33.0759i	1.27121i	−0.772015	+	0.635604i	$0.780751\pi$
$678$	0	0
$679$	2.03440i	0.0780733i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−31.1501	−1.19193	−0.595963	−	0.803012i	$-0.703229\pi$
$683$	−31.1501	−1.19193	−0.595963	+	0.803012i	$0.703229\pi$
$684$	0	0
$685$	−29.2447	−1.11738
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.18706i	− 0.0452234i
$690$	0	0
$691$	13.8114i	0.525412i	0.964876	+	0.262706i	$0.0846150\pi$
$691$	13.8114i	0.525412i	−0.964876	+	0.262706i	$0.915385\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	58.8240	2.23132
$696$	0	0
$697$	35.8056	1.35623
$698$	0	0
$699$	0	0
$700$	0	0
$701$	30.8782i	1.16625i	0.812381	+	0.583127i	$0.198171\pi$
$701$	30.8782i	1.16625i	−0.812381	+	0.583127i	$0.801829\pi$
$702$	0	0
$703$	0.732191i	0.0276151i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−10.6492	−0.400505
$708$	0	0
$709$	30.6137	1.14972	0.574861	−	0.818251i	$-0.305056\pi$
$709$	30.6137	1.14972	0.574861	+	0.818251i	$0.305056\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	57.8620i	2.16695i
$714$	0	0
$715$	3.21207i	0.120125i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−34.2682	−1.27799	−0.638995	−	0.769211i	$-0.720649\pi$
$719$	−34.2682	−1.27799	−0.638995	+	0.769211i	$0.720649\pi$
$720$	0	0
$721$	0.433258	0.0161354
$722$	0	0
$723$	0	0
$724$	0	0
$725$	28.5545i	1.06049i
$726$	0	0
$727$	8.31257i	0.308296i	0.988048	+	0.154148i	$0.0492633\pi$
$727$	8.31257i	0.308296i	−0.988048	+	0.154148i	$0.950737\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−57.6645	−2.13280
$732$	0	0
$733$	−35.3844	−1.30695	−0.653477	−	0.756946i	$-0.726690\pi$
$733$	−35.3844	−1.30695	−0.653477	+	0.756946i	$0.726690\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.89005i	− 0.0696207i
$738$	0	0
$739$	− 30.3812i	− 1.11759i	−0.829305	−	0.558795i	$-0.811264\pi$
$739$	− 30.3812i	− 1.11759i	0.829305	−	0.558795i	$-0.188736\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−34.5459	−1.26737	−0.633684	−	0.773592i	$-0.718458\pi$
$743$	−34.5459	−1.26737	−0.633684	+	0.773592i	$0.718458\pi$
$744$	0	0
$745$	−6.70833	−0.245774
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 8.52131i	− 0.311362i
$750$	0	0
$751$	10.0994i	0.368532i	0.982876	+	0.184266i	$0.0589908\pi$
$751$	10.0994i	0.368532i	−0.982876	+	0.184266i	$0.941009\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−23.6779	−0.861725
$756$	0	0
$757$	38.7446	1.40820	0.704099	−	0.710102i	$-0.251351\pi$
$757$	38.7446	1.40820	0.704099	+	0.710102i	$0.251351\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	35.4511i	1.28510i	0.766243	+	0.642551i	$0.222124\pi$
$761$	35.4511i	1.28510i	−0.766243	+	0.642551i	$0.777876\pi$
$762$	0	0
$763$	5.25007i	0.190065i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.63147	−0.0589091
$768$	0	0
$769$	−35.8736	−1.29364	−0.646818	−	0.762644i	$-0.723901\pi$
$769$	−35.8736	−1.29364	−0.646818	+	0.762644i	$0.723901\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	0.214276i	0.00770697i	0.999993	+	0.00385349i	$0.00122661\pi$
$773$	0.214276i	0.00770697i	−0.999993	+	0.00385349i	$0.998773\pi$
$774$	0	0
$775$	108.728i	3.90564i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−13.4126	−0.480556
$780$	0	0
$781$	10.6618	0.381508
$782$	0	0
$783$	0	0
$784$	0	0
$785$	23.0582i	0.822981i
$786$	0	0
$787$	33.5765i	1.19687i	0.801170	+	0.598437i	$0.204211\pi$
$787$	33.5765i	1.19687i	−0.801170	+	0.598437i	$0.795789\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.13556	−0.289267
$792$	0	0
$793$	0.536302	0.0190446
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.9723i	1.27420i	0.770780	+	0.637102i	$0.219867\pi$
$797$	35.9723i	1.27420i	−0.770780	+	0.637102i	$0.780133\pi$
$798$	0	0
$799$	− 55.2497i	− 1.95459i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	41.1667	1.45274
$804$	0	0
$805$	−24.4119	−0.860408
$806$	0	0
$807$	0	0
$808$	0	0
$809$	7.82300i	0.275042i	0.990499	+	0.137521i	$0.0439135\pi$
$809$	7.82300i	0.275042i	−0.990499	+	0.137521i	$0.956087\pi$
$810$	0	0
$811$	40.2969i	1.41502i	0.706705	+	0.707508i	$0.250181\pi$
$811$	40.2969i	1.41502i	−0.706705	+	0.707508i	$0.749819\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−16.8473	−0.590134
$816$	0	0
$817$	21.6008	0.755717
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.5013i	0.401399i	0.979653	+	0.200699i	$0.0643214\pi$
$821$	11.5013i	0.401399i	−0.979653	+	0.200699i	$0.935679\pi$
$822$	0	0
$823$	54.5432i	1.90126i	0.310334	+	0.950628i	$0.399559\pi$
$823$	54.5432i	1.90126i	−0.310334	+	0.950628i	$0.600441\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	2.34342	0.0814887	0.0407443	−	0.999170i	$-0.487027\pi$
$827$	2.34342	0.0814887	0.0407443	+	0.999170i	$0.487027\pi$
$828$	0	0
$829$	15.5991	0.541778	0.270889	−	0.962611i	$-0.412682\pi$
$829$	15.5991	0.541778	0.270889	+	0.962611i	$0.412682\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	33.8477i	1.17275i
$834$	0	0
$835$	− 73.3381i	− 2.53797i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−3.59451	−0.124096	−0.0620481	−	0.998073i	$-0.519763\pi$
$839$	−3.59451	−0.124096	−0.0620481	+	0.998073i	$0.519763\pi$
$840$	0	0
$841$	21.6008	0.744856
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 50.9311i	− 1.75208i
$846$	0	0
$847$	0.369729i	0.0127040i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.86070	0.0637840
$852$	0	0
$853$	38.2990	1.31133	0.655666	−	0.755051i	$-0.272388\pi$
$853$	38.2990	1.31133	0.655666	+	0.755051i	$0.272388\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 31.0461i	− 1.06051i	−0.847837	−	0.530257i	$-0.822095\pi$
$857$	− 31.0461i	− 1.06051i	0.847837	−	0.530257i	$-0.177905\pi$
$858$	0	0
$859$	− 43.2926i	− 1.47712i	−0.674186	−	0.738562i	$-0.735505\pi$
$859$	− 43.2926i	− 1.47712i	0.674186	−	0.738562i	$-0.264495\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−28.1206	−0.957237	−0.478619	−	0.878023i	$-0.658862\pi$
$863$	−28.1206	−0.957237	−0.478619	+	0.878023i	$0.658862\pi$
$864$	0	0
$865$	−34.7064	−1.18005
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28.9281i	0.981319i
$870$	0	0
$871$	− 0.144573i	− 0.00489868i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.0230	−0.812127
$876$	0	0
$877$	−53.5928	−1.80970	−0.904849	−	0.425732i	$-0.860017\pi$
$877$	−53.5928	−1.80970	−0.904849	+	0.425732i	$0.860017\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 26.3409i	− 0.887449i	−0.896163	−	0.443724i	$-0.853657\pi$
$881$	− 26.3409i	− 0.887449i	0.896163	−	0.443724i	$-0.146343\pi$
$882$	0	0
$883$	− 47.8523i	− 1.61036i	−0.593032	−	0.805179i	$-0.702069\pi$
$883$	− 47.8523i	− 1.61036i	0.593032	−	0.805179i	$-0.297931\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	18.5338	0.622304	0.311152	−	0.950360i	$-0.399285\pi$
$887$	18.5338	0.622304	0.311152	+	0.950360i	$0.399285\pi$
$888$	0	0
$889$	−0.129842	−0.00435477
$890$	0	0
$891$	0	0
$892$	0	0
$893$	20.6963i	0.692574i
$894$	0	0
$895$	24.4768i	0.818170i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−28.1743	−0.939664
$900$	0	0
$901$	−27.8840	−0.928952
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 81.3386i	− 2.70379i
$906$	0	0
$907$	− 43.7506i	− 1.45271i	−0.687318	−	0.726357i	$-0.741212\pi$
$907$	− 43.7506i	− 1.45271i	0.687318	−	0.726357i	$-0.258788\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	44.8409	1.48565	0.742823	−	0.669488i	$-0.233487\pi$
$911$	44.8409	1.48565	0.742823	+	0.669488i	$0.233487\pi$
$912$	0	0
$913$	−15.2887	−0.505981
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.34886i	− 0.143612i
$918$	0	0
$919$	− 13.3225i	− 0.439469i	−0.975560	−	0.219734i	$-0.929481\pi$
$919$	− 13.3225i	− 0.439469i	0.975560	−	0.219734i	$-0.0705191\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0.815540	0.0268438
$924$	0	0
$925$	3.49644	0.114962
$926$	0	0
$927$	0	0
$928$	0	0
$929$	19.1507i	0.628314i	0.949371	+	0.314157i	$0.101722\pi$
$929$	19.1507i	0.628314i	−0.949371	+	0.314157i	$0.898278\pi$
$930$	0	0
$931$	− 12.6792i	− 0.415543i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	75.4515	2.46753
$936$	0	0
$937$	17.6708	0.577281	0.288640	−	0.957438i	$-0.406797\pi$
$937$	17.6708	0.577281	0.288640	+	0.957438i	$0.406797\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.9173i	0.910080i	0.890471	+	0.455040i	$0.150375\pi$
$941$	27.9173i	0.910080i	−0.890471	+	0.455040i	$0.849625\pi$
$942$	0	0
$943$	34.0851i	1.10996i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−6.30400	−0.204852	−0.102426	−	0.994741i	$-0.532661\pi$
$947$	−6.30400	−0.204852	−0.102426	+	0.994741i	$0.532661\pi$
$948$	0	0
$949$	3.14892	0.102218
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 8.17205i	− 0.264719i	−0.991202	−	0.132359i	$-0.957745\pi$
$953$	− 8.17205i	− 0.264719i	0.991202	−	0.132359i	$-0.0422553\pi$
$954$	0	0
$955$	− 54.1069i	− 1.75086i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.24625	0.266285
$960$	0	0
$961$	−76.2806	−2.46066
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 67.5617i	− 2.17489i
$966$	0	0
$967$	− 26.5619i	− 0.854173i	−0.904211	−	0.427087i	$-0.859540\pi$
$967$	− 26.5619i	− 0.854173i	0.904211	−	0.427087i	$-0.140460\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.2192	−1.48324	−0.741622	−	0.670818i	$-0.765943\pi$
$971$	−46.2192	−1.48324	−0.741622	+	0.670818i	$0.765943\pi$
$972$	0	0
$973$	−16.5869	−0.531751
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 35.9553i	− 1.15031i	−0.818044	−	0.575155i	$-0.804942\pi$
$977$	− 35.9553i	− 1.15031i	0.818044	−	0.575155i	$-0.195058\pi$
$978$	0	0
$979$	15.5188i	0.495981i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.1375	0.706076	0.353038	−	0.935609i	$-0.385149\pi$
$983$	22.1375	0.706076	0.353038	+	0.935609i	$0.385149\pi$
$984$	0	0
$985$	22.2692	0.709556
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 54.8937i	− 1.74552i
$990$	0	0
$991$	49.0851i	1.55924i	0.626254	+	0.779619i	$0.284587\pi$
$991$	49.0851i	1.55924i	−0.626254	+	0.779619i	$0.715413\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−42.5221	−1.34804
$996$	0	0
$997$	19.1896	0.607740	0.303870	−	0.952714i	$-0.401721\pi$
$997$	19.1896	0.607740	0.303870	+	0.952714i	$0.401721\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.c.c.2591.23		24
3.2	odd	2	inner	2592.2.c.c.2591.1		24
4.3	odd	2	inner	2592.2.c.c.2591.24		24
8.3	odd	2		5184.2.c.m.5183.2		24
8.5	even	2		5184.2.c.m.5183.1		24
9.2	odd	6		288.2.s.a.95.8	yes	24
9.4	even	3		288.2.s.a.191.5	yes	24
9.5	odd	6		864.2.s.a.575.2		24
9.7	even	3		864.2.s.a.287.1		24
12.11	even	2	inner	2592.2.c.c.2591.2		24
24.5	odd	2		5184.2.c.m.5183.23		24
24.11	even	2		5184.2.c.m.5183.24		24
36.7	odd	6		864.2.s.a.287.2		24
36.11	even	6		288.2.s.a.95.5	✓	24
36.23	even	6		864.2.s.a.575.1		24
36.31	odd	6		288.2.s.a.191.8	yes	24
72.5	odd	6		1728.2.s.g.575.12		24
72.11	even	6		576.2.s.g.383.8		24
72.13	even	6		576.2.s.g.191.8		24
72.29	odd	6		576.2.s.g.383.5		24
72.43	odd	6		1728.2.s.g.1151.12		24
72.59	even	6		1728.2.s.g.575.11		24
72.61	even	6		1728.2.s.g.1151.11		24
72.67	odd	6		576.2.s.g.191.5		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.2.s.a.95.5	✓	24	36.11	even	6
288.2.s.a.95.8	yes	24	9.2	odd	6
288.2.s.a.191.5	yes	24	9.4	even	3
288.2.s.a.191.8	yes	24	36.31	odd	6
576.2.s.g.191.5		24	72.67	odd	6
576.2.s.g.191.8		24	72.13	even	6
576.2.s.g.383.5		24	72.29	odd	6
576.2.s.g.383.8		24	72.11	even	6
864.2.s.a.287.1		24	9.7	even	3
864.2.s.a.287.2		24	36.7	odd	6
864.2.s.a.575.1		24	36.23	even	6
864.2.s.a.575.2		24	9.5	odd	6
1728.2.s.g.575.11		24	72.59	even	6
1728.2.s.g.575.12		24	72.5	odd	6
1728.2.s.g.1151.11		24	72.61	even	6
1728.2.s.g.1151.12		24	72.43	odd	6
2592.2.c.c.2591.1		24	3.2	odd	2	inner
2592.2.c.c.2591.2		24	12.11	even	2	inner
2592.2.c.c.2591.23		24	1.1	even	1	trivial
2592.2.c.c.2591.24		24	4.3	odd	2	inner
5184.2.c.m.5183.1		24	8.5	even	2
5184.2.c.m.5183.2		24	8.3	odd	2
5184.2.c.m.5183.23		24	24.5	odd	2
5184.2.c.m.5183.24		24	24.11	even	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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.93668i q^{5} -1.11004i q^{7} +O(q^{10})\)	\(q+3.93668i q^{5} -1.11004i q^{7} -3.26603 q^{11} -0.249825 q^{13} +5.86838i q^{17} -2.19827i q^{19} -5.58641 q^{23} -10.4974 q^{25} -2.72014i q^{29} -10.3576i q^{31} +4.36988 q^{35} -0.333076 q^{37} -6.10144i q^{41} +9.82629i q^{43} -9.41480 q^{47} +5.76781 q^{49} +4.75157i q^{53} -12.8573i q^{55} +6.53047 q^{59} -2.14671 q^{61} -0.983480i q^{65} +0.578699i q^{67} -3.26444 q^{71} -12.6045 q^{73} +3.62543i q^{77} -8.85728i q^{79} +4.68113 q^{83} -23.1019 q^{85} -4.75157i q^{89} +0.277316i q^{91} +8.65387 q^{95} -1.83273 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q - 24 q^{25} - 24 q^{49} + 24 q^{73} - 24 q^{97}+O(q^{100}) \) 24 * q - 24 * q^25 - 24 * q^49 + 24 * q^73 - 24 * q^97

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