LMFDB - Embedded newform 6048.2.c.g.3025.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(3025,6048)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6048, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6048.3025");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.c (of order $2$, degree $1$, not 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:	$48.2935231425$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3025.2
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.g.3025.23

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.66698i q^{5} -1.00000 q^{7} +O(q^{10})$	$q-3.66698i q^{5} -1.00000 q^{7} +4.45794i q^{11} +1.51546i q^{13} -3.45516 q^{17} -2.29933i q^{19} +8.76326 q^{23} -8.44677 q^{25} +1.62037i q^{29} +9.26498 q^{31} +3.66698i q^{35} -5.69381i q^{37} -6.86926 q^{41} -0.880042i q^{43} -10.5955 q^{47} +1.00000 q^{49} -11.5036i q^{53} +16.3472 q^{55} -2.99426i q^{59} -8.51202i q^{61} +5.55718 q^{65} -10.6629i q^{67} -10.6172 q^{71} +7.85523 q^{73} -4.45794i q^{77} -4.57730 q^{79} +2.60500i q^{83} +12.6700i q^{85} -2.14381 q^{89} -1.51546i q^{91} -8.43162 q^{95} -13.8317 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 24 q^{7}+O(q^{10}) $ 24 * q - 24 * q^7	$ 24 q - 24 q^{7} - 24 q^{25} + 16 q^{31} + 24 q^{49} + 8 q^{55} - 32 q^{79}+O(q^{100}) $ 24 * q - 24 * q^7 - 24 * q^25 + 16 * q^31 + 24 * q^49 + 8 * q^55 - 32 * q^79

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times$.

$n$	$2593$	$3781$	$3809$	$4159$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$5$	− 3.66698i	− 1.63993i	−0.572417	−	0.819963i	$-0.693994\pi$
$5$	− 3.66698i	− 1.63993i	0.572417	−	0.819963i	$-0.306006\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.45794i	1.34412i	0.740498	+	0.672059i	$0.234590\pi$
$11$	4.45794i	1.34412i	−0.740498	+	0.672059i	$0.765410\pi$
$12$	0	0
$13$	1.51546i	0.420314i	0.977668	+	0.210157i	$0.0673975\pi$
$13$	1.51546i	0.420314i	−0.977668	+	0.210157i	$0.932602\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.45516	−0.838000	−0.419000	−	0.907986i	$-0.637619\pi$
$17$	−3.45516	−0.838000	−0.419000	+	0.907986i	$0.637619\pi$
$18$	0	0
$19$	− 2.29933i	− 0.527503i	−0.964591	−	0.263752i	$-0.915040\pi$
$19$	− 2.29933i	− 0.527503i	0.964591	−	0.263752i	$-0.0849600\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.76326	1.82727	0.913633	−	0.406539i	$-0.133265\pi$
$23$	8.76326	1.82727	0.913633	+	0.406539i	$0.133265\pi$
$24$	0	0
$25$	−8.44677	−1.68935
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.62037i	0.300895i	0.988618	+	0.150448i	$0.0480715\pi$
$29$	1.62037i	0.300895i	−0.988618	+	0.150448i	$0.951928\pi$
$30$	0	0
$31$	9.26498	1.66404	0.832020	−	0.554746i	$-0.187184\pi$
$31$	9.26498	1.66404	0.832020	+	0.554746i	$0.187184\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.66698i	0.619833i
$36$	0	0
$37$	− 5.69381i	− 0.936056i	−0.883714	−	0.468028i	$-0.844965\pi$
$37$	− 5.69381i	− 0.936056i	0.883714	−	0.468028i	$-0.155035\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.86926	−1.07280	−0.536399	−	0.843965i	$-0.680216\pi$
$41$	−6.86926	−1.07280	−0.536399	+	0.843965i	$0.680216\pi$
$42$	0	0
$43$	− 0.880042i	− 0.134205i	−0.997746	−	0.0671026i	$-0.978625\pi$
$43$	− 0.880042i	− 0.134205i	0.997746	−	0.0671026i	$-0.0213755\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10.5955	−1.54551	−0.772753	−	0.634707i	$-0.781121\pi$
$47$	−10.5955	−1.54551	−0.772753	+	0.634707i	$0.781121\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 11.5036i	− 1.58014i	−0.613014	−	0.790072i	$-0.710043\pi$
$53$	− 11.5036i	− 1.58014i	0.613014	−	0.790072i	$-0.289957\pi$
$54$	0	0
$55$	16.3472	2.20425
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.99426i	− 0.389819i	−0.980821	−	0.194909i	$-0.937559\pi$
$59$	− 2.99426i	− 0.389819i	0.980821	−	0.194909i	$-0.0624413\pi$
$60$	0	0
$61$	− 8.51202i	− 1.08985i	−0.838484	−	0.544926i	$-0.816558\pi$
$61$	− 8.51202i	− 1.08985i	0.838484	−	0.544926i	$-0.183442\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.55718	0.689284
$66$	0	0
$67$	− 10.6629i	− 1.30268i	−0.758786	−	0.651341i	$-0.774207\pi$
$67$	− 10.6629i	− 1.30268i	0.758786	−	0.651341i	$-0.225793\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.6172	−1.26003	−0.630015	−	0.776583i	$-0.716951\pi$
$71$	−10.6172	−1.26003	−0.630015	+	0.776583i	$0.716951\pi$
$72$	0	0
$73$	7.85523	0.919385	0.459692	−	0.888078i	$-0.347960\pi$
$73$	7.85523	0.919385	0.459692	+	0.888078i	$0.347960\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4.45794i	− 0.508029i
$78$	0	0
$79$	−4.57730	−0.514986	−0.257493	−	0.966280i	$-0.582896\pi$
$79$	−4.57730	−0.514986	−0.257493	+	0.966280i	$0.582896\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.60500i	0.285936i	0.989727	+	0.142968i	$0.0456645\pi$
$83$	2.60500i	0.285936i	−0.989727	+	0.142968i	$0.954335\pi$
$84$	0	0
$85$	12.6700i	1.37426i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.14381	−0.227244	−0.113622	−	0.993524i	$-0.536245\pi$
$89$	−2.14381	−0.227244	−0.113622	+	0.993524i	$0.536245\pi$
$90$	0	0
$91$	− 1.51546i	− 0.158864i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.43162	−0.865066
$96$	0	0
$97$	−13.8317	−1.40440	−0.702199	−	0.711981i	$-0.747798\pi$
$97$	−13.8317	−1.40440	−0.702199	+	0.711981i	$0.747798\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.2333i	− 1.01826i	−0.860691	−	0.509128i	$-0.829968\pi$
$101$	− 10.2333i	− 1.01826i	0.860691	−	0.509128i	$-0.170032\pi$
$102$	0	0
$103$	12.7223	1.25357	0.626784	−	0.779193i	$-0.284371\pi$
$103$	12.7223	1.25357	0.626784	+	0.779193i	$0.284371\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.40575i	0.619268i	0.950856	+	0.309634i	$0.100206\pi$
$107$	6.40575i	0.619268i	−0.950856	+	0.309634i	$0.899794\pi$
$108$	0	0
$109$	− 3.38839i	− 0.324549i	−0.986746	−	0.162274i	$-0.948117\pi$
$109$	− 3.38839i	− 0.324549i	0.986746	−	0.162274i	$-0.0518829\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−1.40668	−0.132329	−0.0661644	−	0.997809i	$-0.521076\pi$
$113$	−1.40668	−0.132329	−0.0661644	+	0.997809i	$0.521076\pi$
$114$	0	0
$115$	− 32.1347i	− 2.99658i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.45516	0.316734
$120$	0	0
$121$	−8.87319	−0.806653
$122$	0	0
$123$	0	0
$124$	0	0
$125$	12.6393i	1.13049i
$126$	0	0
$127$	−10.4597	−0.928152	−0.464076	−	0.885795i	$-0.653614\pi$
$127$	−10.4597	−0.928152	−0.464076	+	0.885795i	$0.653614\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 12.6638i	− 1.10644i	−0.833034	−	0.553221i	$-0.813398\pi$
$131$	− 12.6638i	− 1.10644i	0.833034	−	0.553221i	$-0.186602\pi$
$132$	0	0
$133$	2.29933i	0.199378i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.87026	−0.586966	−0.293483	−	0.955964i	$-0.594814\pi$
$137$	−6.87026	−0.586966	−0.293483	+	0.955964i	$0.594814\pi$
$138$	0	0
$139$	13.4362i	1.13964i	0.821768	+	0.569822i	$0.192988\pi$
$139$	13.4362i	1.13964i	−0.821768	+	0.569822i	$0.807012\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.75584	−0.564952
$144$	0	0
$145$	5.94188	0.493446
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 6.35294i	− 0.520453i	−0.965548	−	0.260227i	$-0.916203\pi$
$149$	− 6.35294i	− 0.520453i	0.965548	−	0.260227i	$-0.0837973\pi$
$150$	0	0
$151$	−19.5918	−1.59436	−0.797179	−	0.603743i	$-0.793675\pi$
$151$	−19.5918	−1.59436	−0.797179	+	0.603743i	$0.793675\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 33.9745i	− 2.72890i
$156$	0	0
$157$	− 22.6978i	− 1.81148i	−0.423834	−	0.905740i	$-0.639316\pi$
$157$	− 22.6978i	− 1.81148i	0.423834	−	0.905740i	$-0.360684\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.76326	−0.690642
$162$	0	0
$163$	3.60107i	0.282058i	0.990005	+	0.141029i	$0.0450410\pi$
$163$	3.60107i	0.282058i	−0.990005	+	0.141029i	$0.954959\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.46351	0.422779	0.211390	−	0.977402i	$-0.432201\pi$
$167$	5.46351	0.422779	0.211390	+	0.977402i	$0.432201\pi$
$168$	0	0
$169$	10.7034	0.823336
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.71944i	− 0.130727i	−0.997862	−	0.0653634i	$-0.979179\pi$
$173$	− 1.71944i	− 0.130727i	0.997862	−	0.0653634i	$-0.0208207\pi$
$174$	0	0
$175$	8.44677	0.638516
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.9956i	0.821846i	0.911670	+	0.410923i	$0.134794\pi$
$179$	10.9956i	0.821846i	−0.911670	+	0.410923i	$0.865206\pi$
$180$	0	0
$181$	26.6227i	1.97885i	0.145040	+	0.989426i	$0.453669\pi$
$181$	26.6227i	1.97885i	−0.145040	+	0.989426i	$0.546331\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−20.8791	−1.53506
$186$	0	0
$187$	− 15.4029i	− 1.12637i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−22.1581	−1.60330	−0.801652	−	0.597791i	$-0.796045\pi$
$191$	−22.1581	−1.60330	−0.801652	+	0.597791i	$0.796045\pi$
$192$	0	0
$193$	11.5599	0.832098	0.416049	−	0.909342i	$-0.363414\pi$
$193$	11.5599	0.832098	0.416049	+	0.909342i	$0.363414\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.1740i	0.938611i	0.883036	+	0.469305i	$0.155496\pi$
$197$	13.1740i	0.938611i	−0.883036	+	0.469305i	$0.844504\pi$
$198$	0	0
$199$	12.1012	0.857828	0.428914	−	0.903345i	$-0.358896\pi$
$199$	12.1012	0.857828	0.428914	+	0.903345i	$0.358896\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.62037i	− 0.113728i
$204$	0	0
$205$	25.1895i	1.75931i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.2503	0.709027
$210$	0	0
$211$	21.7501i	1.49734i	0.662944	+	0.748669i	$0.269307\pi$
$211$	21.7501i	1.49734i	−0.662944	+	0.748669i	$0.730693\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.22710	−0.220086
$216$	0	0
$217$	−9.26498	−0.628948
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5.23617i	− 0.352223i
$222$	0	0
$223$	3.00739	0.201390	0.100695	−	0.994917i	$-0.467893\pi$
$223$	3.00739	0.201390	0.100695	+	0.994917i	$0.467893\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.1508i	0.740102i	0.929011	+	0.370051i	$0.120660\pi$
$227$	11.1508i	0.740102i	−0.929011	+	0.370051i	$0.879340\pi$
$228$	0	0
$229$	2.46317i	0.162771i	0.996683	+	0.0813853i	$0.0259344\pi$
$229$	2.46317i	0.162771i	−0.996683	+	0.0813853i	$0.974066\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−26.1143	−1.71080	−0.855401	−	0.517966i	$-0.826689\pi$
$233$	−26.1143	−1.71080	−0.855401	+	0.517966i	$0.826689\pi$
$234$	0	0
$235$	38.8534i	2.53451i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−24.4496	−1.58151	−0.790757	−	0.612130i	$-0.790313\pi$
$239$	−24.4496	−1.58151	−0.790757	+	0.612130i	$0.790313\pi$
$240$	0	0
$241$	−9.01348	−0.580609	−0.290305	−	0.956934i	$-0.593757\pi$
$241$	−9.01348	−0.580609	−0.290305	+	0.956934i	$0.593757\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 3.66698i	− 0.234275i
$246$	0	0
$247$	3.48456	0.221717
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 24.7649i	− 1.56315i	−0.623814	−	0.781573i	$-0.714418\pi$
$251$	− 24.7649i	− 1.56315i	0.623814	−	0.781573i	$-0.285582\pi$
$252$	0	0
$253$	39.0661i	2.45606i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	17.1618	1.07052	0.535262	−	0.844686i	$-0.320213\pi$
$257$	17.1618	1.07052	0.535262	+	0.844686i	$0.320213\pi$
$258$	0	0
$259$	5.69381i	0.353796i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.02861	−0.248415	−0.124207	−	0.992256i	$-0.539639\pi$
$263$	−4.02861	−0.248415	−0.124207	+	0.992256i	$0.539639\pi$
$264$	0	0
$265$	−42.1836	−2.59132
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 24.4193i	− 1.48887i	−0.667693	−	0.744436i	$-0.732718\pi$
$269$	− 24.4193i	− 1.48887i	0.667693	−	0.744436i	$-0.267282\pi$
$270$	0	0
$271$	1.21510	0.0738120	0.0369060	−	0.999319i	$-0.488250\pi$
$271$	1.21510	0.0738120	0.0369060	+	0.999319i	$0.488250\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 37.6552i	− 2.27069i
$276$	0	0
$277$	11.1239i	0.668371i	0.942507	+	0.334185i	$0.108461\pi$
$277$	11.1239i	0.668371i	−0.942507	+	0.334185i	$0.891539\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−29.4678	−1.75790	−0.878952	−	0.476910i	$-0.841757\pi$
$281$	−29.4678	−1.75790	−0.878952	+	0.476910i	$0.841757\pi$
$282$	0	0
$283$	− 13.0312i	− 0.774624i	−0.921949	−	0.387312i	$-0.873404\pi$
$283$	− 13.0312i	− 0.774624i	0.921949	−	0.387312i	$-0.126596\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	6.86926	0.405479
$288$	0	0
$289$	−5.06186	−0.297756
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.86503i	0.284218i	0.989851	+	0.142109i	$0.0453883\pi$
$293$	4.86503i	0.284218i	−0.989851	+	0.142109i	$0.954612\pi$
$294$	0	0
$295$	−10.9799	−0.639274
$296$	0	0
$297$	0	0
$298$	0	0
$299$	13.2804i	0.768026i
$300$	0	0
$301$	0.880042i	0.0507248i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−31.2134	−1.78728
$306$	0	0
$307$	− 9.07611i	− 0.518001i	−0.965877	−	0.259000i	$-0.916607\pi$
$307$	− 9.07611i	− 0.518001i	0.965877	−	0.259000i	$-0.0833931\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−1.21020	−0.0686240	−0.0343120	−	0.999411i	$-0.510924\pi$
$311$	−1.21020	−0.0686240	−0.0343120	+	0.999411i	$0.510924\pi$
$312$	0	0
$313$	−3.15697	−0.178443	−0.0892214	−	0.996012i	$-0.528438\pi$
$313$	−3.15697	−0.178443	−0.0892214	+	0.996012i	$0.528438\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	7.08197i	0.397763i	0.980024	+	0.198882i	$0.0637309\pi$
$317$	7.08197i	0.397763i	−0.980024	+	0.198882i	$0.936269\pi$
$318$	0	0
$319$	−7.22351	−0.404439
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.94457i	0.442048i
$324$	0	0
$325$	− 12.8008i	− 0.710060i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	10.5955	0.584146
$330$	0	0
$331$	7.34765i	0.403863i	0.979400	+	0.201932i	$0.0647219\pi$
$331$	7.34765i	0.403863i	−0.979400	+	0.201932i	$0.935278\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−39.1007	−2.13630
$336$	0	0
$337$	−26.9873	−1.47009	−0.735046	−	0.678017i	$-0.762840\pi$
$337$	−26.9873	−1.47009	−0.735046	+	0.678017i	$0.762840\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	41.3027i	2.23667i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	9.06776i	0.486783i	0.969928	+	0.243391i	$0.0782599\pi$
$347$	9.06776i	0.486783i	−0.969928	+	0.243391i	$0.921740\pi$
$348$	0	0
$349$	13.6784i	0.732191i	0.930577	+	0.366095i	$0.119306\pi$
$349$	13.6784i	0.732191i	−0.930577	+	0.366095i	$0.880694\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−0.949067	−0.0505138	−0.0252569	−	0.999681i	$-0.508040\pi$
$353$	−0.949067	−0.0505138	−0.0252569	+	0.999681i	$0.508040\pi$
$354$	0	0
$355$	38.9331i	2.06636i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	15.0508	0.794351	0.397175	−	0.917743i	$-0.369991\pi$
$359$	15.0508	0.794351	0.397175	+	0.917743i	$0.369991\pi$
$360$	0	0
$361$	13.7131	0.721740
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 28.8050i	− 1.50772i
$366$	0	0
$367$	3.70674	0.193490	0.0967451	−	0.995309i	$-0.469157\pi$
$367$	3.70674	0.193490	0.0967451	+	0.995309i	$0.469157\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	11.5036i	0.597239i
$372$	0	0
$373$	− 31.3688i	− 1.62421i	−0.583508	−	0.812107i	$-0.698320\pi$
$373$	− 31.3688i	− 1.62421i	0.583508	−	0.812107i	$-0.301680\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.45562	−0.126471
$378$	0	0
$379$	− 17.6560i	− 0.906930i	−0.891274	−	0.453465i	$-0.850188\pi$
$379$	− 17.6560i	− 0.906930i	0.891274	−	0.453465i	$-0.149812\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	17.7974	0.909402	0.454701	−	0.890644i	$-0.349746\pi$
$383$	17.7974	0.909402	0.454701	+	0.890644i	$0.349746\pi$
$384$	0	0
$385$	−16.3472	−0.833129
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 25.1886i	− 1.27711i	−0.769574	−	0.638557i	$-0.779532\pi$
$389$	− 25.1886i	− 1.27711i	0.769574	−	0.638557i	$-0.220468\pi$
$390$	0	0
$391$	−30.2785	−1.53125
$392$	0	0
$393$	0	0
$394$	0	0
$395$	16.7849i	0.844539i
$396$	0	0
$397$	− 34.3710i	− 1.72503i	−0.506032	−	0.862515i	$-0.668888\pi$
$397$	− 34.3710i	− 1.72503i	0.506032	−	0.862515i	$-0.331112\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.65560	0.482178	0.241089	−	0.970503i	$-0.422495\pi$
$401$	9.65560	0.482178	0.241089	+	0.970503i	$0.422495\pi$
$402$	0	0
$403$	14.0408i	0.699420i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	25.3826	1.25817
$408$	0	0
$409$	28.4161	1.40509	0.702543	−	0.711642i	$-0.252048\pi$
$409$	28.4161	1.40509	0.702543	+	0.711642i	$0.252048\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.99426i	0.147338i
$414$	0	0
$415$	9.55248	0.468913
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 11.5381i	− 0.563673i	−0.959462	−	0.281837i	$-0.909056\pi$
$419$	− 11.5381i	− 0.563673i	0.959462	−	0.281837i	$-0.0909436\pi$
$420$	0	0
$421$	− 6.84814i	− 0.333758i	−0.985977	−	0.166879i	$-0.946631\pi$
$421$	− 6.84814i	− 0.333758i	0.985977	−	0.166879i	$-0.0533689\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.1850	1.41568
$426$	0	0
$427$	8.51202i	0.411925i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−10.4500	−0.503357	−0.251679	−	0.967811i	$-0.580983\pi$
$431$	−10.4500	−0.503357	−0.251679	+	0.967811i	$0.580983\pi$
$432$	0	0
$433$	18.1818	0.873763	0.436882	−	0.899519i	$-0.356083\pi$
$433$	18.1818	0.873763	0.436882	+	0.899519i	$0.356083\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.1497i	− 0.963889i
$438$	0	0
$439$	−7.91836	−0.377923	−0.188961	−	0.981985i	$-0.560512\pi$
$439$	−7.91836	−0.377923	−0.188961	+	0.981985i	$0.560512\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 18.2916i	− 0.869062i	−0.900657	−	0.434531i	$-0.856914\pi$
$443$	− 18.2916i	− 0.869062i	0.900657	−	0.434531i	$-0.143086\pi$
$444$	0	0
$445$	7.86132i	0.372662i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−15.2172	−0.718144	−0.359072	−	0.933310i	$-0.616907\pi$
$449$	−15.2172	−0.718144	−0.359072	+	0.933310i	$0.616907\pi$
$450$	0	0
$451$	− 30.6227i	− 1.44197i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.55718	−0.260525
$456$	0	0
$457$	38.6658	1.80871	0.904354	−	0.426783i	$-0.140353\pi$
$457$	38.6658	1.80871	0.904354	+	0.426783i	$0.140353\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.18888i	0.241670i	0.992673	+	0.120835i	$0.0385572\pi$
$461$	5.18888i	0.241670i	−0.992673	+	0.120835i	$0.961443\pi$
$462$	0	0
$463$	3.81583	0.177337	0.0886683	−	0.996061i	$-0.471739\pi$
$463$	3.81583	0.177337	0.0886683	+	0.996061i	$0.471739\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 7.29246i	− 0.337455i	−0.985663	−	0.168727i	$-0.946034\pi$
$467$	− 7.29246i	− 0.337455i	0.985663	−	0.168727i	$-0.0539657\pi$
$468$	0	0
$469$	10.6629i	0.492367i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.92317	0.180388
$474$	0	0
$475$	19.4220i	0.891140i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.2695	−0.743371	−0.371686	−	0.928359i	$-0.621220\pi$
$479$	−16.2695	−0.743371	−0.371686	+	0.928359i	$0.621220\pi$
$480$	0	0
$481$	8.62876	0.393438
$482$	0	0
$483$	0	0
$484$	0	0
$485$	50.7207i	2.30311i
$486$	0	0
$487$	−7.16667	−0.324753	−0.162376	−	0.986729i	$-0.551916\pi$
$487$	−7.16667	−0.324753	−0.162376	+	0.986729i	$0.551916\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 17.8247i	− 0.804418i	−0.915548	−	0.402209i	$-0.868243\pi$
$491$	− 17.8247i	− 0.804418i	0.915548	−	0.402209i	$-0.131757\pi$
$492$	0	0
$493$	− 5.59864i	− 0.252150i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	10.6172	0.476247
$498$	0	0
$499$	− 37.6121i	− 1.68375i	−0.539673	−	0.841875i	$-0.681452\pi$
$499$	− 37.6121i	− 1.68375i	0.539673	−	0.841875i	$-0.318548\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.3707	0.908286	0.454143	−	0.890929i	$-0.349945\pi$
$503$	20.3707	0.908286	0.454143	+	0.890929i	$0.349945\pi$
$504$	0	0
$505$	−37.5255	−1.66986
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 26.6481i	− 1.18116i	−0.806980	−	0.590578i	$-0.798900\pi$
$509$	− 26.6481i	− 1.18116i	0.806980	−	0.590578i	$-0.201100\pi$
$510$	0	0
$511$	−7.85523	−0.347495
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 46.6526i	− 2.05576i
$516$	0	0
$517$	− 47.2339i	− 2.07734i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−13.2614	−0.580992	−0.290496	−	0.956876i	$-0.593820\pi$
$521$	−13.2614	−0.580992	−0.290496	+	0.956876i	$0.593820\pi$
$522$	0	0
$523$	− 21.3786i	− 0.934821i	−0.884040	−	0.467411i	$-0.845187\pi$
$523$	− 21.3786i	− 0.934821i	0.884040	−	0.467411i	$-0.154813\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.0120	−1.39447
$528$	0	0
$529$	53.7948	2.33890
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 10.4101i	− 0.450912i
$534$	0	0
$535$	23.4898	1.01555
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.45794i	0.192017i
$540$	0	0
$541$	− 31.4969i	− 1.35416i	−0.735910	−	0.677080i	$-0.763245\pi$
$541$	− 31.4969i	− 1.35416i	0.735910	−	0.677080i	$-0.236755\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−12.4252	−0.532235
$546$	0	0
$547$	− 1.38840i	− 0.0593637i	−0.999559	−	0.0296819i	$-0.990551\pi$
$547$	− 1.38840i	− 0.0593637i	0.999559	−	0.0296819i	$-0.00944942\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.72578	0.158723
$552$	0	0
$553$	4.57730	0.194647
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.8246i	0.458654i	0.973349	+	0.229327i	$0.0736525\pi$
$557$	10.8246i	0.458654i	−0.973349	+	0.229327i	$0.926347\pi$
$558$	0	0
$559$	1.33367	0.0564083
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 7.56473i	− 0.318816i	−0.987213	−	0.159408i	$-0.949042\pi$
$563$	− 7.56473i	− 0.318816i	0.987213	−	0.159408i	$-0.0509585\pi$
$564$	0	0
$565$	5.15826i	0.217009i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−21.9063	−0.918360	−0.459180	−	0.888343i	$-0.651857\pi$
$569$	−21.9063	−0.918360	−0.459180	+	0.888343i	$0.651857\pi$
$570$	0	0
$571$	43.6061i	1.82486i	0.409235	+	0.912429i	$0.365796\pi$
$571$	43.6061i	1.82486i	−0.409235	+	0.912429i	$0.634204\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−74.0213	−3.08690
$576$	0	0
$577$	12.6116	0.525028	0.262514	−	0.964928i	$-0.415448\pi$
$577$	12.6116	0.525028	0.262514	+	0.964928i	$0.415448\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 2.60500i	− 0.108073i
$582$	0	0
$583$	51.2824	2.12390
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 10.5118i	− 0.433866i	−0.976186	−	0.216933i	$-0.930395\pi$
$587$	− 10.5118i	− 0.433866i	0.976186	−	0.216933i	$-0.0696054\pi$
$588$	0	0
$589$	− 21.3033i	− 0.877787i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−23.9765	−0.984598	−0.492299	−	0.870426i	$-0.663843\pi$
$593$	−23.9765	−0.984598	−0.492299	+	0.870426i	$0.663843\pi$
$594$	0	0
$595$	− 12.6700i	− 0.519420i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.2302	1.31689	0.658446	−	0.752628i	$-0.271214\pi$
$599$	32.2302	1.31689	0.658446	+	0.752628i	$0.271214\pi$
$600$	0	0
$601$	43.3405	1.76790	0.883949	−	0.467584i	$-0.154875\pi$
$601$	43.3405	1.76790	0.883949	+	0.467584i	$0.154875\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	32.5378i	1.32285i
$606$	0	0
$607$	13.2372	0.537283	0.268642	−	0.963240i	$-0.413425\pi$
$607$	13.2372	0.537283	0.268642	+	0.963240i	$0.413425\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 16.0570i	− 0.649598i
$612$	0	0
$613$	− 6.23612i	− 0.251874i	−0.992038	−	0.125937i	$-0.959806\pi$
$613$	− 6.23612i	− 0.251874i	0.992038	−	0.125937i	$-0.0401938\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7.66713	0.308667	0.154334	−	0.988019i	$-0.450677\pi$
$617$	7.66713	0.308667	0.154334	+	0.988019i	$0.450677\pi$
$618$	0	0
$619$	− 27.9974i	− 1.12531i	−0.826692	−	0.562655i	$-0.809780\pi$
$619$	− 27.9974i	− 1.12531i	0.826692	−	0.562655i	$-0.190220\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	2.14381	0.0858900
$624$	0	0
$625$	4.11412	0.164565
$626$	0	0
$627$	0	0
$628$	0	0
$629$	19.6730i	0.784415i
$630$	0	0
$631$	−18.2806	−0.727739	−0.363869	−	0.931450i	$-0.618545\pi$
$631$	−18.2806	−0.727739	−0.363869	+	0.931450i	$0.618545\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	38.3557i	1.52210i
$636$	0	0
$637$	1.51546i	0.0600449i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	5.99361	0.236733	0.118367	−	0.992970i	$-0.462234\pi$
$641$	5.99361	0.236733	0.118367	+	0.992970i	$0.462234\pi$
$642$	0	0
$643$	− 29.0547i	− 1.14581i	−0.819623	−	0.572903i	$-0.805817\pi$
$643$	− 29.0547i	− 1.14581i	0.819623	−	0.572903i	$-0.194183\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−26.7288	−1.05082	−0.525408	−	0.850850i	$-0.676087\pi$
$647$	−26.7288	−1.05082	−0.525408	+	0.850850i	$0.676087\pi$
$648$	0	0
$649$	13.3482	0.523963
$650$	0	0
$651$	0	0
$652$	0	0
$653$	41.0011i	1.60450i	0.596991	+	0.802248i	$0.296363\pi$
$653$	41.0011i	1.60450i	−0.596991	+	0.802248i	$0.703637\pi$
$654$	0	0
$655$	−46.4380	−1.81448
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.91818i	0.0747219i	0.999302	+	0.0373609i	$0.0118951\pi$
$659$	1.91818i	0.0747219i	−0.999302	+	0.0373609i	$0.988105\pi$
$660$	0	0
$661$	10.9347i	0.425312i	0.977127	+	0.212656i	$0.0682114\pi$
$661$	10.9347i	0.425312i	−0.977127	+	0.212656i	$0.931789\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.43162	0.326964
$666$	0	0
$667$	14.1997i	0.549816i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	37.9460	1.46489
$672$	0	0
$673$	10.0071	0.385745	0.192872	−	0.981224i	$-0.438220\pi$
$673$	10.0071	0.385745	0.192872	+	0.981224i	$0.438220\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 35.4441i	− 1.36223i	−0.732178	−	0.681113i	$-0.761496\pi$
$677$	− 35.4441i	− 1.36223i	0.732178	−	0.681113i	$-0.238504\pi$
$678$	0	0
$679$	13.8317	0.530812
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.8397i	1.33310i	0.745459	+	0.666551i	$0.232230\pi$
$683$	34.8397i	1.33310i	−0.745459	+	0.666551i	$0.767770\pi$
$684$	0	0
$685$	25.1931i	0.962580i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	17.4333	0.664157
$690$	0	0
$691$	− 16.2381i	− 0.617727i	−0.951106	−	0.308864i	$-0.900051\pi$
$691$	− 16.2381i	− 0.617727i	0.951106	−	0.308864i	$-0.0999486\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	49.2704	1.86893
$696$	0	0
$697$	23.7344	0.899004
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 28.8081i	− 1.08807i	−0.839064	−	0.544033i	$-0.816897\pi$
$701$	− 28.8081i	− 1.08807i	0.839064	−	0.544033i	$-0.183103\pi$
$702$	0	0
$703$	−13.0920	−0.493773
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.2333i	0.384864i
$708$	0	0
$709$	− 18.1039i	− 0.679906i	−0.940442	−	0.339953i	$-0.889589\pi$
$709$	− 18.1039i	− 0.679906i	0.940442	−	0.339953i	$-0.110411\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	81.1915	3.04064
$714$	0	0
$715$	24.7736i	0.926479i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	12.7731	0.476357	0.238179	−	0.971221i	$-0.423450\pi$
$719$	12.7731	0.476357	0.238179	+	0.971221i	$0.423450\pi$
$720$	0	0
$721$	−12.7223	−0.473804
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 13.6869i	− 0.508319i
$726$	0	0
$727$	10.1654	0.377012	0.188506	−	0.982072i	$-0.439635\pi$
$727$	10.1654	0.377012	0.188506	+	0.982072i	$0.439635\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.04069i	0.112464i
$732$	0	0
$733$	− 26.1041i	− 0.964178i	−0.876122	−	0.482089i	$-0.839878\pi$
$733$	− 26.1041i	− 0.964178i	0.876122	−	0.482089i	$-0.160122\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	47.5345	1.75096
$738$	0	0
$739$	− 9.90517i	− 0.364368i	−0.983264	−	0.182184i	$-0.941683\pi$
$739$	− 9.90517i	− 0.364368i	0.983264	−	0.182184i	$-0.0583166\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−33.5353	−1.23029	−0.615146	−	0.788413i	$-0.710903\pi$
$743$	−33.5353	−1.23029	−0.615146	+	0.788413i	$0.710903\pi$
$744$	0	0
$745$	−23.2961	−0.853504
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 6.40575i	− 0.234061i
$750$	0	0
$751$	−53.8436	−1.96478	−0.982391	−	0.186839i	$-0.940176\pi$
$751$	−53.8436	−1.96478	−0.982391	+	0.186839i	$0.940176\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	71.8428i	2.61463i
$756$	0	0
$757$	1.18921i	0.0432225i	0.999766	+	0.0216113i	$0.00687962\pi$
$757$	1.18921i	0.0432225i	−0.999766	+	0.0216113i	$0.993120\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−38.0810	−1.38043	−0.690217	−	0.723602i	$-0.742485\pi$
$761$	−38.0810	−1.38043	−0.690217	+	0.723602i	$0.742485\pi$
$762$	0	0
$763$	3.38839i	0.122668i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.53769	0.163846
$768$	0	0
$769$	−38.6053	−1.39214	−0.696071	−	0.717973i	$-0.745070\pi$
$769$	−38.6053	−1.39214	−0.696071	+	0.717973i	$0.745070\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 18.5902i	− 0.668642i	−0.942459	−	0.334321i	$-0.891493\pi$
$773$	− 18.5902i	− 0.668642i	0.942459	−	0.334321i	$-0.108507\pi$
$774$	0	0
$775$	−78.2592	−2.81115
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.7947i	0.565905i
$780$	0	0
$781$	− 47.3308i	− 1.69363i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−83.2324	−2.97069
$786$	0	0
$787$	17.3947i	0.620056i	0.950727	+	0.310028i	$0.100338\pi$
$787$	17.3947i	0.620056i	−0.950727	+	0.310028i	$0.899662\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.40668	0.0500156
$792$	0	0
$793$	12.8997	0.458080
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 15.3683i	− 0.544374i	−0.962244	−	0.272187i	$-0.912253\pi$
$797$	− 15.3683i	− 0.544374i	0.962244	−	0.272187i	$-0.0877470\pi$
$798$	0	0
$799$	36.6090	1.29513
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.0181i	1.23576i
$804$	0	0
$805$	32.1347i	1.13260i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.7888	1.04732	0.523659	−	0.851928i	$-0.324567\pi$
$809$	29.7888	1.04732	0.523659	+	0.851928i	$0.324567\pi$
$810$	0	0
$811$	− 6.50515i	− 0.228427i	−0.993456	−	0.114213i	$-0.963565\pi$
$811$	− 6.50515i	− 0.228427i	0.993456	−	0.114213i	$-0.0364347\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.2051	0.462553
$816$	0	0
$817$	−2.02351	−0.0707937
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.89581i	− 0.135965i	−0.997687	−	0.0679823i	$-0.978344\pi$
$821$	− 3.89581i	− 0.135965i	0.997687	−	0.0679823i	$-0.0216562\pi$
$822$	0	0
$823$	−47.1577	−1.64381	−0.821907	−	0.569621i	$-0.807090\pi$
$823$	−47.1577	−1.64381	−0.821907	+	0.569621i	$0.807090\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	30.9915i	1.07768i	0.842408	+	0.538841i	$0.181138\pi$
$827$	30.9915i	1.07768i	−0.842408	+	0.538841i	$0.818862\pi$
$828$	0	0
$829$	− 34.2090i	− 1.18813i	−0.804417	−	0.594065i	$-0.797522\pi$
$829$	− 34.2090i	− 1.18813i	0.804417	−	0.594065i	$-0.202478\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.45516	−0.119714
$834$	0	0
$835$	− 20.0346i	− 0.693326i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	40.9980	1.41541	0.707704	−	0.706509i	$-0.249731\pi$
$839$	40.9980	1.41541	0.707704	+	0.706509i	$0.249731\pi$
$840$	0	0
$841$	26.3744	0.909462
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 39.2491i	− 1.35021i
$846$	0	0
$847$	8.87319	0.304886
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 49.8963i	− 1.71042i
$852$	0	0
$853$	26.3408i	0.901891i	0.892551	+	0.450946i	$0.148913\pi$
$853$	26.3408i	0.901891i	−0.892551	+	0.450946i	$0.851087\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−56.4187	−1.92723	−0.963613	−	0.267300i	$-0.913868\pi$
$857$	−56.4187	−1.92723	−0.963613	+	0.267300i	$0.913868\pi$
$858$	0	0
$859$	34.5693i	1.17949i	0.807589	+	0.589745i	$0.200772\pi$
$859$	34.5693i	1.17949i	−0.807589	+	0.589745i	$0.799228\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−1.52709	−0.0519826	−0.0259913	−	0.999662i	$-0.508274\pi$
$863$	−1.52709	−0.0519826	−0.0259913	+	0.999662i	$0.508274\pi$
$864$	0	0
$865$	−6.30517	−0.214382
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 20.4053i	− 0.692202i
$870$	0	0
$871$	16.1593	0.547535
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 12.6393i	− 0.427285i
$876$	0	0
$877$	− 20.5638i	− 0.694389i	−0.937793	−	0.347195i	$-0.887134\pi$
$877$	− 20.5638i	− 0.694389i	0.937793	−	0.347195i	$-0.112866\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−16.8073	−0.566252	−0.283126	−	0.959083i	$-0.591371\pi$
$881$	−16.8073	−0.566252	−0.283126	+	0.959083i	$0.591371\pi$
$882$	0	0
$883$	− 13.7849i	− 0.463900i	−0.972728	−	0.231950i	$-0.925489\pi$
$883$	− 13.7849i	− 0.463900i	0.972728	−	0.231950i	$-0.0745105\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.0037	0.839541	0.419771	−	0.907630i	$-0.362111\pi$
$887$	25.0037	0.839541	0.419771	+	0.907630i	$0.362111\pi$
$888$	0	0
$889$	10.4597	0.350808
$890$	0	0
$891$	0	0
$892$	0	0
$893$	24.3625i	0.815260i
$894$	0	0
$895$	40.3205	1.34777
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.0127i	0.500702i
$900$	0	0
$901$	39.7469i	1.32416i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	97.6251	3.24517
$906$	0	0
$907$	23.5827i	0.783052i	0.920167	+	0.391526i	$0.128053\pi$
$907$	23.5827i	0.783052i	−0.920167	+	0.391526i	$0.871947\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	29.1770	0.966677	0.483338	−	0.875434i	$-0.339424\pi$
$911$	29.1770	0.966677	0.483338	+	0.875434i	$0.339424\pi$
$912$	0	0
$913$	−11.6129	−0.384331
$914$	0	0
$915$	0	0
$916$	0	0
$917$	12.6638i	0.418196i
$918$	0	0
$919$	−51.6854	−1.70494	−0.852472	−	0.522772i	$-0.824898\pi$
$919$	−51.6854	−1.70494	−0.852472	+	0.522772i	$0.824898\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 16.0900i	− 0.529609i
$924$	0	0
$925$	48.0943i	1.58133i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20.0819	−0.658865	−0.329432	−	0.944179i	$-0.606857\pi$
$929$	−20.0819	−0.658865	−0.329432	+	0.944179i	$0.606857\pi$
$930$	0	0
$931$	− 2.29933i	− 0.0753576i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−56.4821	−1.84716
$936$	0	0
$937$	−19.1583	−0.625875	−0.312937	−	0.949774i	$-0.601313\pi$
$937$	−19.1583	−0.625875	−0.312937	+	0.949774i	$0.601313\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 54.2409i	− 1.76820i	−0.467296	−	0.884101i	$-0.654772\pi$
$941$	− 54.2409i	− 1.76820i	0.467296	−	0.884101i	$-0.345228\pi$
$942$	0	0
$943$	−60.1971	−1.96029
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.7355i	1.55120i	0.631228	+	0.775598i	$0.282551\pi$
$947$	47.7355i	1.55120i	−0.631228	+	0.775598i	$0.717449\pi$
$948$	0	0
$949$	11.9043i	0.386430i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	11.2075	0.363046	0.181523	−	0.983387i	$-0.441897\pi$
$953$	11.2075	0.363046	0.181523	+	0.983387i	$0.441897\pi$
$954$	0	0
$955$	81.2534i	2.62930i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.87026	0.221852
$960$	0	0
$961$	54.8399	1.76903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 42.3899i	− 1.36458i
$966$	0	0
$967$	22.5979	0.726701	0.363350	−	0.931653i	$-0.381633\pi$
$967$	22.5979	0.726701	0.363350	+	0.931653i	$0.381633\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.5586i	1.49414i	0.664747	+	0.747068i	$0.268539\pi$
$971$	46.5586i	1.49414i	−0.664747	+	0.747068i	$0.731461\pi$
$972$	0	0
$973$	− 13.4362i	− 0.430745i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.5968	−1.39479	−0.697393	−	0.716689i	$-0.745657\pi$
$977$	−43.5968	−1.39479	−0.697393	+	0.716689i	$0.745657\pi$
$978$	0	0
$979$	− 9.55697i	− 0.305442i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.5407	−0.718937	−0.359469	−	0.933157i	$-0.617042\pi$
$983$	−22.5407	−0.718937	−0.359469	+	0.933157i	$0.617042\pi$
$984$	0	0
$985$	48.3090	1.53925
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 7.71204i	− 0.245229i
$990$	0	0
$991$	−24.2444	−0.770148	−0.385074	−	0.922886i	$-0.625824\pi$
$991$	−24.2444	−0.770148	−0.385074	+	0.922886i	$0.625824\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 44.3747i	− 1.40677i
$996$	0	0
$997$	49.3412i	1.56265i	0.624124	+	0.781325i	$0.285456\pi$
$997$	49.3412i	1.56265i	−0.624124	+	0.781325i	$0.714544\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.c.g.3025.2		24
3.2	odd	2	inner	6048.2.c.g.3025.24		24
4.3	odd	2		1512.2.c.f.757.16	yes	24
8.3	odd	2		1512.2.c.f.757.15	yes	24
8.5	even	2	inner	6048.2.c.g.3025.23		24
12.11	even	2		1512.2.c.f.757.9	✓	24
24.5	odd	2	inner	6048.2.c.g.3025.1		24
24.11	even	2		1512.2.c.f.757.10	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.f.757.9	✓	24	12.11	even	2
1512.2.c.f.757.10	yes	24	24.11	even	2
1512.2.c.f.757.15	yes	24	8.3	odd	2
1512.2.c.f.757.16	yes	24	4.3	odd	2
6048.2.c.g.3025.1		24	24.5	odd	2	inner
6048.2.c.g.3025.2		24	1.1	even	1	trivial
6048.2.c.g.3025.23		24	8.5	even	2	inner
6048.2.c.g.3025.24		24	3.2	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:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.66698i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-3.66698i q^{5} -1.00000 q^{7} +4.45794i q^{11} +1.51546i q^{13} -3.45516 q^{17} -2.29933i q^{19} +8.76326 q^{23} -8.44677 q^{25} +1.62037i q^{29} +9.26498 q^{31} +3.66698i q^{35} -5.69381i q^{37} -6.86926 q^{41} -0.880042i q^{43} -10.5955 q^{47} +1.00000 q^{49} -11.5036i q^{53} +16.3472 q^{55} -2.99426i q^{59} -8.51202i q^{61} +5.55718 q^{65} -10.6629i q^{67} -10.6172 q^{71} +7.85523 q^{73} -4.45794i q^{77} -4.57730 q^{79} +2.60500i q^{83} +12.6700i q^{85} -2.14381 q^{89} -1.51546i q^{91} -8.43162 q^{95} -13.8317 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 24 * q - 24 * q^7	\( 24 q - 24 q^{7} - 24 q^{25} + 16 q^{31} + 24 q^{49} + 8 q^{55} - 32 q^{79}+O(q^{100}) \) 24 * q - 24 * q^7 - 24 * q^25 + 16 * q^31 + 24 * q^49 + 8 * q^55 - 32 * q^79

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