LMFDB - Embedded newform 6048.2.c.g.3025.17

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			3025.17
Character	$\chi$	$=$	6048.3025
Dual form			6048.2.c.g.3025.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.58470i q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.58470i q^{5} -1.00000 q^{7} +0.790533i q^{11} -0.494366i q^{13} +4.46962 q^{17} +7.55705i q^{19} -0.839644 q^{23} +2.48872 q^{25} +2.76437i q^{29} +0.568664 q^{31} -1.58470i q^{35} -0.343650i q^{37} -5.93014 q^{41} +3.16166i q^{43} -4.80964 q^{47} +1.00000 q^{49} -4.36792i q^{53} -1.25276 q^{55} +4.50798i q^{59} -5.40103i q^{61} +0.783423 q^{65} -7.57396i q^{67} -9.52886 q^{71} +14.1344 q^{73} -0.790533i q^{77} +3.71341 q^{79} -7.30905i q^{83} +7.08301i q^{85} +11.5305 q^{89} +0.494366i q^{91} -11.9757 q^{95} +1.75839 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.58470i	0.708700i	0.935113	+	0.354350i	$0.115298\pi$
$5$	1.58470i	0.708700i	−0.935113	+	0.354350i	$0.884702\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.790533i	0.238355i	0.992873	+	0.119177i	$0.0380257\pi$
$11$	0.790533i	0.238355i	−0.992873	+	0.119177i	$0.961974\pi$
$12$	0	0
$13$	− 0.494366i	− 0.137112i	−0.997647	−	0.0685562i	$-0.978161\pi$
$13$	− 0.494366i	− 0.137112i	0.997647	−	0.0685562i	$-0.0218392\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.46962	1.08404	0.542021	−	0.840365i	$-0.317660\pi$
$17$	4.46962	1.08404	0.542021	+	0.840365i	$0.317660\pi$
$18$	0	0
$19$	7.55705i	1.73371i	0.498564	+	0.866853i	$0.333861\pi$
$19$	7.55705i	1.73371i	−0.498564	+	0.866853i	$0.666139\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.839644	−0.175078	−0.0875390	−	0.996161i	$-0.527900\pi$
$23$	−0.839644	−0.175078	−0.0875390	+	0.996161i	$0.527900\pi$
$24$	0	0
$25$	2.48872	0.497744
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.76437i	0.513331i	0.966500	+	0.256665i	$0.0826238\pi$
$29$	2.76437i	0.513331i	−0.966500	+	0.256665i	$0.917376\pi$
$30$	0	0
$31$	0.568664	0.102135	0.0510675	−	0.998695i	$-0.483738\pi$
$31$	0.568664	0.102135	0.0510675	+	0.998695i	$0.483738\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.58470i	− 0.267864i
$36$	0	0
$37$	− 0.343650i	− 0.0564957i	−0.999601	−	0.0282479i	$-0.991007\pi$
$37$	− 0.343650i	− 0.0564957i	0.999601	−	0.0282479i	$-0.00899277\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.93014	−0.926132	−0.463066	−	0.886324i	$-0.653251\pi$
$41$	−5.93014	−0.926132	−0.463066	+	0.886324i	$0.653251\pi$
$42$	0	0
$43$	3.16166i	0.482149i	0.970507	+	0.241074i	$0.0774998\pi$
$43$	3.16166i	0.482149i	−0.970507	+	0.241074i	$0.922500\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.80964	−0.701558	−0.350779	−	0.936458i	$-0.614083\pi$
$47$	−4.80964	−0.701558	−0.350779	+	0.936458i	$0.614083\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 4.36792i	− 0.599980i	−0.953942	−	0.299990i	$-0.903017\pi$
$53$	− 4.36792i	− 0.599980i	0.953942	−	0.299990i	$-0.0969833\pi$
$54$	0	0
$55$	−1.25276	−0.168922
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.50798i	0.586889i	0.955976	+	0.293445i	$0.0948016\pi$
$59$	4.50798i	0.586889i	−0.955976	+	0.293445i	$0.905198\pi$
$60$	0	0
$61$	− 5.40103i	− 0.691531i	−0.938321	−	0.345766i	$-0.887619\pi$
$61$	− 5.40103i	− 0.691531i	0.938321	−	0.345766i	$-0.112381\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.783423	0.0971716
$66$	0	0
$67$	− 7.57396i	− 0.925307i	−0.886539	−	0.462653i	$-0.846897\pi$
$67$	− 7.57396i	− 0.925307i	0.886539	−	0.462653i	$-0.153103\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.52886	−1.13087	−0.565434	−	0.824794i	$-0.691291\pi$
$71$	−9.52886	−1.13087	−0.565434	+	0.824794i	$0.691291\pi$
$72$	0	0
$73$	14.1344	1.65431	0.827155	−	0.561975i	$-0.189958\pi$
$73$	14.1344	1.65431	0.827155	+	0.561975i	$0.189958\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 0.790533i	− 0.0900896i
$78$	0	0
$79$	3.71341	0.417791	0.208896	−	0.977938i	$-0.433013\pi$
$79$	3.71341	0.417791	0.208896	+	0.977938i	$0.433013\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 7.30905i	− 0.802273i	−0.916018	−	0.401136i	$-0.868615\pi$
$83$	− 7.30905i	− 0.802273i	0.916018	−	0.401136i	$-0.131385\pi$
$84$	0	0
$85$	7.08301i	0.768260i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	11.5305	1.22223	0.611114	−	0.791543i	$-0.290722\pi$
$89$	11.5305	1.22223	0.611114	+	0.791543i	$0.290722\pi$
$90$	0	0
$91$	0.494366i	0.0518236i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.9757	−1.22868
$96$	0	0
$97$	1.75839	0.178538	0.0892689	−	0.996008i	$-0.471547\pi$
$97$	1.75839	0.178538	0.0892689	+	0.996008i	$0.471547\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.5623i	1.74752i	0.486362	+	0.873758i	$0.338324\pi$
$101$	17.5623i	1.74752i	−0.486362	+	0.873758i	$0.661676\pi$
$102$	0	0
$103$	−8.30641	−0.818454	−0.409227	−	0.912432i	$-0.634202\pi$
$103$	−8.30641	−0.818454	−0.409227	+	0.912432i	$0.634202\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.7750i	1.81505i	0.419996	+	0.907526i	$0.362032\pi$
$107$	18.7750i	1.81505i	−0.419996	+	0.907526i	$0.637968\pi$
$108$	0	0
$109$	0.165074i	0.0158113i	0.999969	+	0.00790563i	$0.00251646\pi$
$109$	0.165074i	0.0158113i	−0.999969	+	0.00790563i	$0.997484\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.8486	−1.02055	−0.510275	−	0.860012i	$-0.670456\pi$
$113$	−10.8486	−1.02055	−0.510275	+	0.860012i	$0.670456\pi$
$114$	0	0
$115$	− 1.33059i	− 0.124078i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.46962	−0.409729
$120$	0	0
$121$	10.3751	0.943187
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.8674i	1.06145i
$126$	0	0
$127$	−21.0320	−1.86629	−0.933145	−	0.359499i	$-0.882948\pi$
$127$	−21.0320	−1.86629	−0.933145	+	0.359499i	$0.882948\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 5.17978i	− 0.452560i	−0.974062	−	0.226280i	$-0.927344\pi$
$131$	− 5.17978i	− 0.452560i	0.974062	−	0.226280i	$-0.0726564\pi$
$132$	0	0
$133$	− 7.55705i	− 0.655279i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.19905	−0.700492	−0.350246	−	0.936658i	$-0.613902\pi$
$137$	−8.19905	−0.700492	−0.350246	+	0.936658i	$0.613902\pi$
$138$	0	0
$139$	3.89763i	0.330593i	0.986244	+	0.165296i	$0.0528581\pi$
$139$	3.89763i	0.330593i	−0.986244	+	0.165296i	$0.947142\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.390813	0.0326814
$144$	0	0
$145$	−4.38070	−0.363798
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.71034i	0.303963i	0.988383	+	0.151981i	$0.0485654\pi$
$149$	3.71034i	0.303963i	−0.988383	+	0.151981i	$0.951435\pi$
$150$	0	0
$151$	4.08172	0.332165	0.166083	−	0.986112i	$-0.446888\pi$
$151$	4.08172	0.332165	0.166083	+	0.986112i	$0.446888\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.901163i	0.0723832i
$156$	0	0
$157$	9.64250i	0.769555i	0.923009	+	0.384778i	$0.125722\pi$
$157$	9.64250i	0.769555i	−0.923009	+	0.384778i	$0.874278\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.839644	0.0661732
$162$	0	0
$163$	6.79301i	0.532069i	0.963963	+	0.266035i	$0.0857136\pi$
$163$	6.79301i	0.532069i	−0.963963	+	0.266035i	$0.914286\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.35614	−0.414471	−0.207235	−	0.978291i	$-0.566447\pi$
$167$	−5.35614	−0.414471	−0.207235	+	0.978291i	$0.566447\pi$
$168$	0	0
$169$	12.7556	0.981200
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.3862i	1.16979i	0.811110	+	0.584894i	$0.198864\pi$
$173$	15.3862i	1.16979i	−0.811110	+	0.584894i	$0.801136\pi$
$174$	0	0
$175$	−2.48872	−0.188129
$176$	0	0
$177$	0	0
$178$	0	0
$179$	19.5360i	1.46019i	0.683347	+	0.730094i	$0.260524\pi$
$179$	19.5360i	1.46019i	−0.683347	+	0.730094i	$0.739476\pi$
$180$	0	0
$181$	− 1.07045i	− 0.0795657i	−0.999208	−	0.0397828i	$-0.987333\pi$
$181$	− 1.07045i	− 0.0795657i	0.999208	−	0.0397828i	$-0.0126666\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0.544583	0.0400386
$186$	0	0
$187$	3.53338i	0.258386i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.7019	1.28086	0.640431	−	0.768016i	$-0.278756\pi$
$191$	17.7019	1.28086	0.640431	+	0.768016i	$0.278756\pi$
$192$	0	0
$193$	−10.5405	−0.758719	−0.379359	−	0.925249i	$-0.623856\pi$
$193$	−10.5405	−0.758719	−0.379359	+	0.925249i	$0.623856\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.1573i	1.07991i	0.841692	+	0.539957i	$0.181560\pi$
$197$	15.1573i	1.07991i	−0.841692	+	0.539957i	$0.818440\pi$
$198$	0	0
$199$	−19.8834	−1.40950	−0.704750	−	0.709456i	$-0.748941\pi$
$199$	−19.8834	−1.40950	−0.704750	+	0.709456i	$0.748941\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 2.76437i	− 0.194021i
$204$	0	0
$205$	− 9.39751i	− 0.656350i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.97410	−0.413237
$210$	0	0
$211$	− 26.0319i	− 1.79211i	−0.443942	−	0.896055i	$-0.646420\pi$
$211$	− 26.0319i	− 1.79211i	0.443942	−	0.896055i	$-0.353580\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.01029	−0.341699
$216$	0	0
$217$	−0.568664	−0.0386034
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.20963i	− 0.148635i
$222$	0	0
$223$	−21.1231	−1.41451	−0.707255	−	0.706959i	$-0.750067\pi$
$223$	−21.1231	−1.41451	−0.707255	+	0.706959i	$0.750067\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.07343i	0.602224i	0.953589	+	0.301112i	$0.0973579\pi$
$227$	9.07343i	0.602224i	−0.953589	+	0.301112i	$0.902642\pi$
$228$	0	0
$229$	15.8950i	1.05037i	0.850987	+	0.525187i	$0.176005\pi$
$229$	15.8950i	1.05037i	−0.850987	+	0.525187i	$0.823995\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−25.2633	−1.65505	−0.827527	−	0.561426i	$-0.810253\pi$
$233$	−25.2633	−1.65505	−0.827527	+	0.561426i	$0.810253\pi$
$234$	0	0
$235$	− 7.62185i	− 0.497194i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−3.13797	−0.202979	−0.101489	−	0.994837i	$-0.532361\pi$
$239$	−3.13797	−0.202979	−0.101489	+	0.994837i	$0.532361\pi$
$240$	0	0
$241$	7.05736	0.454604	0.227302	−	0.973824i	$-0.427009\pi$
$241$	7.05736	0.454604	0.227302	+	0.973824i	$0.427009\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.58470i	0.101243i
$246$	0	0
$247$	3.73595	0.237713
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.8195i	0.809158i	0.914503	+	0.404579i	$0.132582\pi$
$251$	12.8195i	0.809158i	−0.914503	+	0.404579i	$0.867418\pi$
$252$	0	0
$253$	− 0.663767i	− 0.0417307i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.1680	−0.696639	−0.348319	−	0.937376i	$-0.613248\pi$
$257$	−11.1680	−0.696639	−0.348319	+	0.937376i	$0.613248\pi$
$258$	0	0
$259$	0.343650i	0.0213534i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.1277	−0.932817	−0.466408	−	0.884570i	$-0.654452\pi$
$263$	−15.1277	−0.932817	−0.466408	+	0.884570i	$0.654452\pi$
$264$	0	0
$265$	6.92185	0.425206
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.70567i	0.286910i	0.989657	+	0.143455i	$0.0458212\pi$
$269$	4.70567i	0.286910i	−0.989657	+	0.143455i	$0.954179\pi$
$270$	0	0
$271$	19.6194	1.19179	0.595897	−	0.803061i	$-0.296797\pi$
$271$	19.6194	1.19179	0.595897	+	0.803061i	$0.296797\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.96742i	0.118640i
$276$	0	0
$277$	− 10.0661i	− 0.604811i	−0.953179	−	0.302406i	$-0.902210\pi$
$277$	− 10.0661i	− 0.604811i	0.953179	−	0.302406i	$-0.0977897\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−28.6669	−1.71013	−0.855063	−	0.518525i	$-0.826481\pi$
$281$	−28.6669	−1.71013	−0.855063	+	0.518525i	$0.826481\pi$
$282$	0	0
$283$	8.69308i	0.516750i	0.966045	+	0.258375i	$0.0831870\pi$
$283$	8.69308i	0.516750i	−0.966045	+	0.258375i	$0.916813\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.93014	0.350045
$288$	0	0
$289$	2.97746	0.175145
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 23.9254i	− 1.39774i	−0.715250	−	0.698868i	$-0.753687\pi$
$293$	− 23.9254i	− 1.39774i	0.715250	−	0.698868i	$-0.246313\pi$
$294$	0	0
$295$	−7.14381	−0.415928
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.415091i	0.0240054i
$300$	0	0
$301$	− 3.16166i	− 0.182235i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	8.55903	0.490089
$306$	0	0
$307$	7.88720i	0.450146i	0.974342	+	0.225073i	$0.0722621\pi$
$307$	7.88720i	0.450146i	−0.974342	+	0.225073i	$0.927738\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−34.7041	−1.96789	−0.983944	−	0.178480i	$-0.942882\pi$
$311$	−34.7041	−1.96789	−0.983944	+	0.178480i	$0.942882\pi$
$312$	0	0
$313$	−11.2387	−0.635249	−0.317624	−	0.948217i	$-0.602885\pi$
$313$	−11.2387	−0.635249	−0.317624	+	0.948217i	$0.602885\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 30.2902i	− 1.70127i	−0.525761	−	0.850633i	$-0.676219\pi$
$317$	− 30.2902i	− 1.70127i	0.525761	−	0.850633i	$-0.323781\pi$
$318$	0	0
$319$	−2.18533	−0.122355
$320$	0	0
$321$	0	0
$322$	0	0
$323$	33.7771i	1.87941i
$324$	0	0
$325$	− 1.23034i	− 0.0682469i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.80964	0.265164
$330$	0	0
$331$	20.2855i	1.11499i	0.830180	+	0.557495i	$0.188237\pi$
$331$	20.2855i	1.11499i	−0.830180	+	0.557495i	$0.811763\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.0025	0.655765
$336$	0	0
$337$	2.73774	0.149134	0.0745671	−	0.997216i	$-0.476242\pi$
$337$	2.73774	0.149134	0.0745671	+	0.997216i	$0.476242\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0.449548i	0.0243444i
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.91592i	0.317583i	0.987312	+	0.158792i	$0.0507598\pi$
$347$	5.91592i	0.317583i	−0.987312	+	0.158792i	$0.949240\pi$
$348$	0	0
$349$	− 26.5968i	− 1.42370i	−0.702333	−	0.711848i	$-0.747858\pi$
$349$	− 26.5968i	− 1.42370i	0.702333	−	0.711848i	$-0.252142\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−21.8800	−1.16455	−0.582277	−	0.812990i	$-0.697838\pi$
$353$	−21.8800	−1.16455	−0.582277	+	0.812990i	$0.697838\pi$
$354$	0	0
$355$	− 15.1004i	− 0.801446i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	18.4834	0.975519	0.487759	−	0.872978i	$-0.337814\pi$
$359$	18.4834	0.975519	0.487759	+	0.872978i	$0.337814\pi$
$360$	0	0
$361$	−38.1090	−2.00573
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22.3989i	1.17241i
$366$	0	0
$367$	32.8686	1.71573	0.857863	−	0.513878i	$-0.171792\pi$
$367$	32.8686	1.71573	0.857863	+	0.513878i	$0.171792\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	4.36792i	0.226771i
$372$	0	0
$373$	4.11202i	0.212912i	0.994317	+	0.106456i	$0.0339504\pi$
$373$	4.11202i	0.212912i	−0.994317	+	0.106456i	$0.966050\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.36661	0.0703840
$378$	0	0
$379$	− 10.6063i	− 0.544811i	−0.962183	−	0.272406i	$-0.912181\pi$
$379$	− 10.6063i	− 0.544811i	0.962183	−	0.272406i	$-0.0878192\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0.193380	0.00988125	0.00494063	−	0.999988i	$-0.498427\pi$
$383$	0.193380	0.00988125	0.00494063	+	0.999988i	$0.498427\pi$
$384$	0	0
$385$	1.25276	0.0638466
$386$	0	0
$387$	0	0
$388$	0	0
$389$	30.2102i	1.53172i	0.643008	+	0.765859i	$0.277686\pi$
$389$	30.2102i	1.53172i	−0.643008	+	0.765859i	$0.722314\pi$
$390$	0	0
$391$	−3.75289	−0.189792
$392$	0	0
$393$	0	0
$394$	0	0
$395$	5.88465i	0.296089i
$396$	0	0
$397$	− 19.1773i	− 0.962480i	−0.876589	−	0.481240i	$-0.840186\pi$
$397$	− 19.1773i	− 0.962480i	0.876589	−	0.481240i	$-0.159814\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−33.6813	−1.68196	−0.840981	−	0.541065i	$-0.818021\pi$
$401$	−33.6813	−1.68196	−0.840981	+	0.541065i	$0.818021\pi$
$402$	0	0
$403$	− 0.281128i	− 0.0140040i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.271667	0.0134660
$408$	0	0
$409$	15.0414	0.743751	0.371876	−	0.928283i	$-0.378715\pi$
$409$	15.0414	0.743751	0.371876	+	0.928283i	$0.378715\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 4.50798i	− 0.221823i
$414$	0	0
$415$	11.5827	0.568571
$416$	0	0
$417$	0	0
$418$	0	0
$419$	15.8239i	0.773048i	0.922279	+	0.386524i	$0.126324\pi$
$419$	15.8239i	0.773048i	−0.922279	+	0.386524i	$0.873676\pi$
$420$	0	0
$421$	− 2.62118i	− 0.127748i	−0.997958	−	0.0638742i	$-0.979654\pi$
$421$	− 2.62118i	− 0.127748i	0.997958	−	0.0638742i	$-0.0203457\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	11.1236	0.539575
$426$	0	0
$427$	5.40103i	0.261374i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.5464	1.03785	0.518927	−	0.854819i	$-0.326332\pi$
$431$	21.5464	1.03785	0.518927	+	0.854819i	$0.326332\pi$
$432$	0	0
$433$	12.4258	0.597145	0.298572	−	0.954387i	$-0.403490\pi$
$433$	12.4258	0.597145	0.298572	+	0.954387i	$0.403490\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 6.34523i	− 0.303534i
$438$	0	0
$439$	24.2735	1.15851	0.579256	−	0.815146i	$-0.303343\pi$
$439$	24.2735	1.15851	0.579256	+	0.815146i	$0.303343\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 0.931584i	− 0.0442609i	−0.999755	−	0.0221304i	$-0.992955\pi$
$443$	− 0.931584i	− 0.0442609i	0.999755	−	0.0221304i	$-0.00704492\pi$
$444$	0	0
$445$	18.2724i	0.866193i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	20.8693	0.984883	0.492441	−	0.870346i	$-0.336105\pi$
$449$	20.8693	0.984883	0.492441	+	0.870346i	$0.336105\pi$
$450$	0	0
$451$	− 4.68797i	− 0.220748i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.783423	−0.0367274
$456$	0	0
$457$	−31.3346	−1.46577	−0.732885	−	0.680353i	$-0.761827\pi$
$457$	−31.3346	−1.46577	−0.732885	+	0.680353i	$0.761827\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.90910i	0.0889156i	0.999011	+	0.0444578i	$0.0141560\pi$
$461$	1.90910i	0.0889156i	−0.999011	+	0.0444578i	$0.985844\pi$
$462$	0	0
$463$	−18.6081	−0.864793	−0.432397	−	0.901683i	$-0.642332\pi$
$463$	−18.6081	−0.864793	−0.432397	+	0.901683i	$0.642332\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 17.7972i	− 0.823554i	−0.911285	−	0.411777i	$-0.864908\pi$
$467$	− 17.7972i	− 0.823554i	0.911285	−	0.411777i	$-0.135092\pi$
$468$	0	0
$469$	7.57396i	0.349733i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.49940	−0.114922
$474$	0	0
$475$	18.8074i	0.862941i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.6737	−0.990294	−0.495147	−	0.868809i	$-0.664886\pi$
$479$	−21.6737	−0.990294	−0.495147	+	0.868809i	$0.664886\pi$
$480$	0	0
$481$	−0.169889	−0.00774627
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.78653i	0.126530i
$486$	0	0
$487$	38.6259	1.75030	0.875152	−	0.483848i	$-0.160761\pi$
$487$	38.6259	1.75030	0.875152	+	0.483848i	$0.160761\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.9010i	0.807863i	0.914789	+	0.403931i	$0.132357\pi$
$491$	17.9010i	0.807863i	−0.914789	+	0.403931i	$0.867643\pi$
$492$	0	0
$493$	12.3557i	0.556472i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	9.52886	0.427428
$498$	0	0
$499$	− 24.3201i	− 1.08872i	−0.838853	−	0.544359i	$-0.816773\pi$
$499$	− 24.3201i	− 1.08872i	0.838853	−	0.544359i	$-0.183227\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	36.3076	1.61888	0.809438	−	0.587205i	$-0.199772\pi$
$503$	36.3076	1.61888	0.809438	+	0.587205i	$0.199772\pi$
$504$	0	0
$505$	−27.8310	−1.23846
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.12375i	− 0.138458i	−0.997601	−	0.0692290i	$-0.977946\pi$
$509$	− 3.12375i	− 0.138458i	0.997601	−	0.0692290i	$-0.0220539\pi$
$510$	0	0
$511$	−14.1344	−0.625270
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 13.1632i	− 0.580039i
$516$	0	0
$517$	− 3.80218i	− 0.167220i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.68044	0.0736213	0.0368107	−	0.999322i	$-0.488280\pi$
$521$	1.68044	0.0736213	0.0368107	+	0.999322i	$0.488280\pi$
$522$	0	0
$523$	26.5738i	1.16199i	0.813906	+	0.580996i	$0.197337\pi$
$523$	26.5738i	1.16199i	−0.813906	+	0.580996i	$0.802663\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.54171	0.110719
$528$	0	0
$529$	−22.2950	−0.969348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.93166i	0.126984i
$534$	0	0
$535$	−29.7529	−1.28633
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.790533i	0.0340507i
$540$	0	0
$541$	26.0022i	1.11792i	0.829194	+	0.558962i	$0.188800\pi$
$541$	26.0022i	1.11792i	−0.829194	+	0.558962i	$0.811200\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−0.261594	−0.0112054
$546$	0	0
$547$	13.4108i	0.573406i	0.958019	+	0.286703i	$0.0925594\pi$
$547$	13.4108i	0.573406i	−0.958019	+	0.286703i	$0.907441\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−20.8905	−0.889964
$552$	0	0
$553$	−3.71341	−0.157910
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.9728i	1.65133i	0.564161	+	0.825665i	$0.309200\pi$
$557$	38.9728i	1.65133i	−0.564161	+	0.825665i	$0.690800\pi$
$558$	0	0
$559$	1.56302	0.0661086
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 15.8415i	− 0.667642i	−0.942637	−	0.333821i	$-0.891662\pi$
$563$	− 15.8415i	− 0.667642i	0.942637	−	0.333821i	$-0.108338\pi$
$564$	0	0
$565$	− 17.1918i	− 0.723264i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.4790	0.732758	0.366379	−	0.930466i	$-0.380597\pi$
$569$	17.4790	0.732758	0.366379	+	0.930466i	$0.380597\pi$
$570$	0	0
$571$	11.4195i	0.477893i	0.971033	+	0.238946i	$0.0768020\pi$
$571$	11.4195i	0.477893i	−0.971033	+	0.238946i	$0.923198\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.08964	−0.0871439
$576$	0	0
$577$	27.4108	1.14113	0.570564	−	0.821253i	$-0.306725\pi$
$577$	27.4108	1.14113	0.570564	+	0.821253i	$0.306725\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.30905i	0.303231i
$582$	0	0
$583$	3.45299	0.143008
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 23.8334i	− 0.983709i	−0.870677	−	0.491855i	$-0.836319\pi$
$587$	− 23.8334i	− 0.983709i	0.870677	−	0.491855i	$-0.163681\pi$
$588$	0	0
$589$	4.29742i	0.177072i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.0780	1.15302	0.576512	−	0.817088i	$-0.304413\pi$
$593$	28.0780	1.15302	0.576512	+	0.817088i	$0.304413\pi$
$594$	0	0
$595$	− 7.08301i	− 0.290375i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−21.6780	−0.885738	−0.442869	−	0.896586i	$-0.646039\pi$
$599$	−21.6780	−0.885738	−0.442869	+	0.896586i	$0.646039\pi$
$600$	0	0
$601$	5.83369	0.237961	0.118981	−	0.992897i	$-0.462037\pi$
$601$	5.83369	0.237961	0.118981	+	0.992897i	$0.462037\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.4414i	0.668437i
$606$	0	0
$607$	29.5358	1.19882	0.599410	−	0.800442i	$-0.295402\pi$
$607$	29.5358	1.19882	0.599410	+	0.800442i	$0.295402\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	2.37772i	0.0961923i
$612$	0	0
$613$	19.5561i	0.789861i	0.918711	+	0.394931i	$0.129231\pi$
$613$	19.5561i	0.789861i	−0.918711	+	0.394931i	$0.870769\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−33.8435	−1.36249	−0.681244	−	0.732056i	$-0.738561\pi$
$617$	−33.8435	−1.36249	−0.681244	+	0.732056i	$0.738561\pi$
$618$	0	0
$619$	20.6581i	0.830320i	0.909748	+	0.415160i	$0.136274\pi$
$619$	20.6581i	0.830320i	−0.909748	+	0.415160i	$0.863726\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11.5305	−0.461959
$624$	0	0
$625$	−6.36268	−0.254507
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.53598i	− 0.0612437i
$630$	0	0
$631$	42.3648	1.68651	0.843257	−	0.537510i	$-0.180635\pi$
$631$	42.3648	1.68651	0.843257	+	0.537510i	$0.180635\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 33.3295i	− 1.32264i
$636$	0	0
$637$	− 0.494366i	− 0.0195875i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11.6219	−0.459039	−0.229519	−	0.973304i	$-0.573715\pi$
$641$	−11.6219	−0.459039	−0.229519	+	0.973304i	$0.573715\pi$
$642$	0	0
$643$	− 40.4455i	− 1.59502i	−0.603309	−	0.797508i	$-0.706151\pi$
$643$	− 40.4455i	− 1.59502i	0.603309	−	0.797508i	$-0.293849\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−29.9913	−1.17908	−0.589540	−	0.807739i	$-0.700691\pi$
$647$	−29.9913	−1.17908	−0.589540	+	0.807739i	$0.700691\pi$
$648$	0	0
$649$	−3.56371	−0.139888
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 14.9263i	− 0.584110i	−0.956402	−	0.292055i	$-0.905661\pi$
$653$	− 14.9263i	− 0.584110i	0.956402	−	0.292055i	$-0.0943390\pi$
$654$	0	0
$655$	8.20842	0.320729
$656$	0	0
$657$	0	0
$658$	0	0
$659$	27.3581i	1.06572i	0.846204	+	0.532859i	$0.178883\pi$
$659$	27.3581i	1.06572i	−0.846204	+	0.532859i	$0.821117\pi$
$660$	0	0
$661$	14.3560i	0.558385i	0.960235	+	0.279192i	$0.0900667\pi$
$661$	14.3560i	0.558385i	−0.960235	+	0.279192i	$0.909933\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.9757	0.464397
$666$	0	0
$667$	− 2.32109i	− 0.0898729i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.26970	0.164830
$672$	0	0
$673$	20.5132	0.790727	0.395364	−	0.918525i	$-0.370619\pi$
$673$	20.5132	0.790727	0.395364	+	0.918525i	$0.370619\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 43.5656i	− 1.67436i	−0.546926	−	0.837181i	$-0.684202\pi$
$677$	− 43.5656i	− 1.67436i	0.546926	−	0.837181i	$-0.315798\pi$
$678$	0	0
$679$	−1.75839	−0.0674810
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 16.9585i	− 0.648898i	−0.945903	−	0.324449i	$-0.894821\pi$
$683$	− 16.9585i	− 0.648898i	0.945903	−	0.324449i	$-0.105179\pi$
$684$	0	0
$685$	− 12.9931i	− 0.496439i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2.15935	−0.0822647
$690$	0	0
$691$	− 27.7324i	− 1.05499i	−0.849558	−	0.527495i	$-0.823131\pi$
$691$	− 27.7324i	− 1.05499i	0.849558	−	0.527495i	$-0.176869\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.17659	−0.234291
$696$	0	0
$697$	−26.5054	−1.00397
$698$	0	0
$699$	0	0
$700$	0	0
$701$	29.5427i	1.11581i	0.829905	+	0.557905i	$0.188395\pi$
$701$	29.5427i	1.11581i	−0.829905	+	0.557905i	$0.811605\pi$
$702$	0	0
$703$	2.59698	0.0979470
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.5623i	− 0.660499i
$708$	0	0
$709$	45.9348i	1.72512i	0.505955	+	0.862560i	$0.331140\pi$
$709$	45.9348i	1.72512i	−0.505955	+	0.862560i	$0.668860\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.477476	−0.0178816
$714$	0	0
$715$	0.619322i	0.0231613i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	16.3756	0.610708	0.305354	−	0.952239i	$-0.401225\pi$
$719$	16.3756	0.610708	0.305354	+	0.952239i	$0.401225\pi$
$720$	0	0
$721$	8.30641	0.309347
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.87974i	0.255507i
$726$	0	0
$727$	−3.43696	−0.127470	−0.0637348	−	0.997967i	$-0.520301\pi$
$727$	−3.43696	−0.127470	−0.0637348	+	0.997967i	$0.520301\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.1314i	0.522669i
$732$	0	0
$733$	35.3171i	1.30447i	0.758019	+	0.652233i	$0.226167\pi$
$733$	35.3171i	1.30447i	−0.758019	+	0.652233i	$0.773833\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.98747	0.220551
$738$	0	0
$739$	− 41.3212i	− 1.52003i	−0.649908	−	0.760013i	$-0.725193\pi$
$739$	− 41.3212i	− 1.52003i	0.649908	−	0.760013i	$-0.274807\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−22.9110	−0.840524	−0.420262	−	0.907403i	$-0.638062\pi$
$743$	−22.9110	−0.840524	−0.420262	+	0.907403i	$0.638062\pi$
$744$	0	0
$745$	−5.87978	−0.215418
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 18.7750i	− 0.686025i
$750$	0	0
$751$	48.2764	1.76163	0.880815	−	0.473461i	$-0.156995\pi$
$751$	48.2764	1.76163	0.880815	+	0.473461i	$0.156995\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.46830i	0.235406i
$756$	0	0
$757$	− 26.9389i	− 0.979112i	−0.871972	−	0.489556i	$-0.837159\pi$
$757$	− 26.9389i	− 0.979112i	0.871972	−	0.489556i	$-0.162841\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.98366	0.108158	0.0540789	−	0.998537i	$-0.482778\pi$
$761$	2.98366	0.108158	0.0540789	+	0.998537i	$0.482778\pi$
$762$	0	0
$763$	− 0.165074i	− 0.00597609i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.22859	0.0804698
$768$	0	0
$769$	1.13907	0.0410760	0.0205380	−	0.999789i	$-0.493462\pi$
$769$	1.13907	0.0410760	0.0205380	+	0.999789i	$0.493462\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	8.40051i	0.302145i	0.988523	+	0.151073i	$0.0482728\pi$
$773$	8.40051i	0.302145i	−0.988523	+	0.151073i	$0.951727\pi$
$774$	0	0
$775$	1.41525	0.0508371
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 44.8143i	− 1.60564i
$780$	0	0
$781$	− 7.53288i	− 0.269548i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−15.2805	−0.545384
$786$	0	0
$787$	13.6837i	0.487771i	0.969804	+	0.243886i	$0.0784222\pi$
$787$	13.6837i	0.487771i	−0.969804	+	0.243886i	$0.921578\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	10.8486	0.385731
$792$	0	0
$793$	−2.67009	−0.0948175
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 43.1786i	− 1.52946i	−0.644349	−	0.764731i	$-0.722872\pi$
$797$	− 43.1786i	− 1.52946i	0.644349	−	0.764731i	$-0.277128\pi$
$798$	0	0
$799$	−21.4972	−0.760518
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.1737i	0.394312i
$804$	0	0
$805$	1.33059i	0.0468970i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	24.5341	0.862573	0.431287	−	0.902215i	$-0.358060\pi$
$809$	24.5341	0.862573	0.431287	+	0.902215i	$0.358060\pi$
$810$	0	0
$811$	0.566589i	0.0198956i	0.999951	+	0.00994781i	$0.00316654\pi$
$811$	0.566589i	0.0198956i	−0.999951	+	0.00994781i	$0.996833\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.7649	−0.377078
$816$	0	0
$817$	−23.8928	−0.835904
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.22205i	0.0426499i	0.999773	+	0.0213250i	$0.00678846\pi$
$821$	1.22205i	0.0426499i	−0.999773	+	0.0213250i	$0.993212\pi$
$822$	0	0
$823$	46.7218	1.62862	0.814309	−	0.580431i	$-0.197116\pi$
$823$	46.7218	1.62862	0.814309	+	0.580431i	$0.197116\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.2221i	1.22479i	0.790551	+	0.612396i	$0.209794\pi$
$827$	35.2221i	1.22479i	−0.790551	+	0.612396i	$0.790206\pi$
$828$	0	0
$829$	− 1.09826i	− 0.0381443i	−0.999818	−	0.0190721i	$-0.993929\pi$
$829$	− 1.09826i	− 0.0381443i	0.999818	−	0.0190721i	$-0.00607122\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.46962	0.154863
$834$	0	0
$835$	− 8.48789i	− 0.293735i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.78246	−0.268680	−0.134340	−	0.990935i	$-0.542891\pi$
$839$	−7.78246	−0.268680	−0.134340	+	0.990935i	$0.542891\pi$
$840$	0	0
$841$	21.3583	0.736492
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20.2138i	0.695377i
$846$	0	0
$847$	−10.3751	−0.356491
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0.288544i	0.00989116i
$852$	0	0
$853$	35.3178i	1.20926i	0.796507	+	0.604630i	$0.206679\pi$
$853$	35.3178i	1.20926i	−0.796507	+	0.604630i	$0.793321\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	48.1812	1.64584	0.822919	−	0.568159i	$-0.192344\pi$
$857$	48.1812	1.64584	0.822919	+	0.568159i	$0.192344\pi$
$858$	0	0
$859$	− 23.2897i	− 0.794633i	−0.917682	−	0.397317i	$-0.869941\pi$
$859$	− 23.2897i	− 0.794633i	0.917682	−	0.397317i	$-0.130059\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	22.2727	0.758173	0.379086	−	0.925361i	$-0.376238\pi$
$863$	22.2727	0.758173	0.379086	+	0.925361i	$0.376238\pi$
$864$	0	0
$865$	−24.3825	−0.829029
$866$	0	0
$867$	0	0
$868$	0	0
$869$	2.93557i	0.0995825i
$870$	0	0
$871$	−3.74431	−0.126871
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 11.8674i	− 0.401191i
$876$	0	0
$877$	− 52.5939i	− 1.77597i	−0.459872	−	0.887985i	$-0.652105\pi$
$877$	− 52.5939i	− 1.77597i	0.459872	−	0.887985i	$-0.347895\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	25.9486	0.874229	0.437115	−	0.899406i	$-0.356000\pi$
$881$	25.9486	0.874229	0.437115	+	0.899406i	$0.356000\pi$
$882$	0	0
$883$	6.37782i	0.214631i	0.994225	+	0.107315i	$0.0342255\pi$
$883$	6.37782i	0.214631i	−0.994225	+	0.107315i	$0.965775\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.60567	0.255373	0.127687	−	0.991815i	$-0.459245\pi$
$887$	7.60567	0.255373	0.127687	+	0.991815i	$0.459245\pi$
$888$	0	0
$889$	21.0320	0.705392
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 36.3467i	− 1.21629i
$894$	0	0
$895$	−30.9587	−1.03484
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.57200i	0.0524291i
$900$	0	0
$901$	− 19.5229i	− 0.650403i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.69634	0.0563882
$906$	0	0
$907$	21.2861i	0.706794i	0.935473	+	0.353397i	$0.114974\pi$
$907$	21.2861i	0.706794i	−0.935473	+	0.353397i	$0.885026\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.0894	−1.59327	−0.796637	−	0.604458i	$-0.793390\pi$
$911$	−48.0894	−1.59327	−0.796637	+	0.604458i	$0.793390\pi$
$912$	0	0
$913$	5.77805	0.191226
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.17978i	0.171052i
$918$	0	0
$919$	−33.6403	−1.10969	−0.554846	−	0.831953i	$-0.687223\pi$
$919$	−33.6403	−1.10969	−0.554846	+	0.831953i	$0.687223\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	4.71074i	0.155056i
$924$	0	0
$925$	− 0.855249i	− 0.0281204i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	35.9109	1.17820	0.589100	−	0.808060i	$-0.299483\pi$
$929$	35.9109	1.17820	0.589100	+	0.808060i	$0.299483\pi$
$930$	0	0
$931$	7.55705i	0.247672i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.59936	−0.183119
$936$	0	0
$937$	6.70862	0.219161	0.109581	−	0.993978i	$-0.465049\pi$
$937$	6.70862	0.219161	0.109581	+	0.993978i	$0.465049\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.81896i	0.189693i	0.995492	+	0.0948464i	$0.0302360\pi$
$941$	5.81896i	0.189693i	−0.995492	+	0.0948464i	$0.969764\pi$
$942$	0	0
$943$	4.97921	0.162145
$944$	0	0
$945$	0	0
$946$	0	0
$947$	22.8143i	0.741365i	0.928760	+	0.370682i	$0.120876\pi$
$947$	22.8143i	0.741365i	−0.928760	+	0.370682i	$0.879124\pi$
$948$	0	0
$949$	− 6.98758i	− 0.226826i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−0.659700	−0.0213698	−0.0106849	−	0.999943i	$-0.503401\pi$
$953$	−0.659700	−0.0213698	−0.0106849	+	0.999943i	$0.503401\pi$
$954$	0	0
$955$	28.0522i	0.907747i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.19905	0.264761
$960$	0	0
$961$	−30.6766	−0.989568
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 16.7035i	− 0.537704i
$966$	0	0
$967$	3.46722	0.111498	0.0557491	−	0.998445i	$-0.482245\pi$
$967$	3.46722	0.111498	0.0557491	+	0.998445i	$0.482245\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.84056i	0.0911580i	0.998961	+	0.0455790i	$0.0145133\pi$
$971$	2.84056i	0.0911580i	−0.998961	+	0.0455790i	$0.985487\pi$
$972$	0	0
$973$	− 3.89763i	− 0.124952i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.3811	0.556071	0.278035	−	0.960571i	$-0.410317\pi$
$977$	17.3811	0.556071	0.278035	+	0.960571i	$0.410317\pi$
$978$	0	0
$979$	9.11522i	0.291324i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	54.8615	1.74981	0.874905	−	0.484294i	$-0.160924\pi$
$983$	54.8615	1.74981	0.874905	+	0.484294i	$0.160924\pi$
$984$	0	0
$985$	−24.0198	−0.765336
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2.65467i	− 0.0844136i
$990$	0	0
$991$	20.9952	0.666933	0.333467	−	0.942762i	$-0.391782\pi$
$991$	20.9952	0.666933	0.333467	+	0.942762i	$0.391782\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 31.5093i	− 0.998913i
$996$	0	0
$997$	− 44.0239i	− 1.39425i	−0.716950	−	0.697125i	$-0.754462\pi$
$997$	− 44.0239i	− 1.39425i	0.716950	−	0.697125i	$-0.245538\pi$
$998$	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.3025.17		24
3.2	odd	2	inner	6048.2.c.g.3025.7		24
4.3	odd	2		1512.2.c.f.757.6	yes	24
8.3	odd	2		1512.2.c.f.757.5	✓	24
8.5	even	2	inner	6048.2.c.g.3025.8		24
12.11	even	2		1512.2.c.f.757.19	yes	24
24.5	odd	2	inner	6048.2.c.g.3025.18		24
24.11	even	2		1512.2.c.f.757.20	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.c.f.757.5	✓	24	8.3	odd	2
1512.2.c.f.757.6	yes	24	4.3	odd	2
1512.2.c.f.757.19	yes	24	12.11	even	2
1512.2.c.f.757.20	yes	24	24.11	even	2
6048.2.c.g.3025.7		24	3.2	odd	2	inner
6048.2.c.g.3025.8		24	8.5	even	2	inner
6048.2.c.g.3025.17		24	1.1	even	1	trivial
6048.2.c.g.3025.18		24	24.5	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)

\(f(q)\)	\(=\)	\(q+1.58470i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.58470i q^{5} -1.00000 q^{7} +0.790533i q^{11} -0.494366i q^{13} +4.46962 q^{17} +7.55705i q^{19} -0.839644 q^{23} +2.48872 q^{25} +2.76437i q^{29} +0.568664 q^{31} -1.58470i q^{35} -0.343650i q^{37} -5.93014 q^{41} +3.16166i q^{43} -4.80964 q^{47} +1.00000 q^{49} -4.36792i q^{53} -1.25276 q^{55} +4.50798i q^{59} -5.40103i q^{61} +0.783423 q^{65} -7.57396i q^{67} -9.52886 q^{71} +14.1344 q^{73} -0.790533i q^{77} +3.71341 q^{79} -7.30905i q^{83} +7.08301i q^{85} +11.5305 q^{89} +0.494366i q^{91} -11.9757 q^{95} +1.75839 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 24 q^{7}+O(q^{10}) \) 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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