LMFDB - Embedded newform 6048.2.j.d.5615.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6048,2,Mod(5615,6048)]

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([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("6048.5615");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6048 = 2^{5} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6048.j (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:	$48$
Twist minimal:	no (minimal twist has level 1512)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5615.20
Character	$\chi$	$=$	6048.5615
Dual form			6048.2.j.d.5615.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.863507 q^{5} -1.00000i q^{7} +O(q^{10})$	$q-0.863507 q^{5} -1.00000i q^{7} +2.62979i q^{11} +1.01906i q^{13} -2.34382i q^{17} +4.09641 q^{19} +6.23450 q^{23} -4.25435 q^{25} +1.57590 q^{29} +6.29822i q^{31} +0.863507i q^{35} +5.18085i q^{37} -10.1388i q^{41} +0.496156 q^{43} -5.24394 q^{47} -1.00000 q^{49} -10.5477 q^{53} -2.27084i q^{55} -4.40654i q^{59} +9.59653i q^{61} -0.879966i q^{65} +12.6840 q^{67} -2.11023 q^{71} -8.27702 q^{73} +2.62979 q^{77} -9.44227i q^{79} +6.62600i q^{83} +2.02391i q^{85} -10.2551i q^{89} +1.01906 q^{91} -3.53728 q^{95} +15.1206 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) $ 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

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
$5$	−0.863507	−0.386172	−0.193086	−	0.981182i	$-0.561850\pi$
$5$	−0.863507	−0.386172	−0.193086	+	0.981182i	$0.561850\pi$
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.62979i	0.792911i	0.918054	+	0.396455i	$0.129760\pi$
$11$	2.62979i	0.792911i	−0.918054	+	0.396455i	$0.870240\pi$
$12$	0	0
$13$	1.01906i	0.282636i	0.989964	+	0.141318i	$0.0451341\pi$
$13$	1.01906i	0.282636i	−0.989964	+	0.141318i	$0.954866\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.34382i	− 0.568460i	−0.958756	−	0.284230i	$-0.908262\pi$
$17$	− 2.34382i	− 0.568460i	0.958756	−	0.284230i	$-0.0917379\pi$
$18$	0	0
$19$	4.09641	0.939780	0.469890	−	0.882725i	$-0.344294\pi$
$19$	4.09641	0.939780	0.469890	+	0.882725i	$0.344294\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.23450	1.29998	0.649992	−	0.759941i	$-0.274772\pi$
$23$	6.23450	1.29998	0.649992	+	0.759941i	$0.274772\pi$
$24$	0	0
$25$	−4.25435	−0.850871
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.57590	0.292638	0.146319	−	0.989237i	$-0.453257\pi$
$29$	1.57590	0.292638	0.146319	+	0.989237i	$0.453257\pi$
$30$	0	0
$31$	6.29822i	1.13119i	0.824682	+	0.565597i	$0.191354\pi$
$31$	6.29822i	1.13119i	−0.824682	+	0.565597i	$0.808646\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.863507i	0.145959i
$36$	0	0
$37$	5.18085i	0.851727i	0.904787	+	0.425863i	$0.140030\pi$
$37$	5.18085i	0.851727i	−0.904787	+	0.425863i	$0.859970\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 10.1388i	− 1.58342i	−0.610899	−	0.791708i	$-0.709192\pi$
$41$	− 10.1388i	− 1.58342i	0.610899	−	0.791708i	$-0.290808\pi$
$42$	0	0
$43$	0.496156	0.0756630	0.0378315	−	0.999284i	$-0.487955\pi$
$43$	0.496156	0.0756630	0.0378315	+	0.999284i	$0.487955\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.24394	−0.764907	−0.382453	−	0.923975i	$-0.624921\pi$
$47$	−5.24394	−0.764907	−0.382453	+	0.923975i	$0.624921\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.5477	−1.44884	−0.724419	−	0.689360i	$-0.757892\pi$
$53$	−10.5477	−1.44884	−0.724419	+	0.689360i	$0.757892\pi$
$54$	0	0
$55$	− 2.27084i	− 0.306200i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.40654i	− 0.573683i	−0.957978	−	0.286841i	$-0.907395\pi$
$59$	− 4.40654i	− 0.573683i	0.957978	−	0.286841i	$-0.0926053\pi$
$60$	0	0
$61$	9.59653i	1.22871i	0.789030	+	0.614355i	$0.210584\pi$
$61$	9.59653i	1.22871i	−0.789030	+	0.614355i	$0.789416\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.879966i	− 0.109146i
$66$	0	0
$67$	12.6840	1.54959	0.774796	−	0.632212i	$-0.217853\pi$
$67$	12.6840	1.54959	0.774796	+	0.632212i	$0.217853\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−2.11023	−0.250438	−0.125219	−	0.992129i	$-0.539963\pi$
$71$	−2.11023	−0.250438	−0.125219	+	0.992129i	$0.539963\pi$
$72$	0	0
$73$	−8.27702	−0.968752	−0.484376	−	0.874860i	$-0.660953\pi$
$73$	−8.27702	−0.968752	−0.484376	+	0.874860i	$0.660953\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.62979	0.299692
$78$	0	0
$79$	− 9.44227i	− 1.06234i	−0.847266	−	0.531169i	$-0.821753\pi$
$79$	− 9.44227i	− 1.06234i	0.847266	−	0.531169i	$-0.178247\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.62600i	0.727298i	0.931536	+	0.363649i	$0.118469\pi$
$83$	6.62600i	0.727298i	−0.931536	+	0.363649i	$0.881531\pi$
$84$	0	0
$85$	2.02391i	0.219524i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 10.2551i	− 1.08704i	−0.839395	−	0.543522i	$-0.817091\pi$
$89$	− 10.2551i	− 1.08704i	0.839395	−	0.543522i	$-0.182909\pi$
$90$	0	0
$91$	1.01906	0.106827
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−3.53728	−0.362917
$96$	0	0
$97$	15.1206	1.53526	0.767631	−	0.640892i	$-0.221435\pi$
$97$	15.1206	1.53526	0.767631	+	0.640892i	$0.221435\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	7.58948	0.755181	0.377591	−	0.925973i	$-0.376753\pi$
$101$	7.58948	0.755181	0.377591	+	0.925973i	$0.376753\pi$
$102$	0	0
$103$	7.33678i	0.722914i	0.932389	+	0.361457i	$0.117721\pi$
$103$	7.33678i	0.722914i	−0.932389	+	0.361457i	$0.882279\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.1057i	1.84702i	0.383578	+	0.923508i	$0.374692\pi$
$107$	19.1057i	1.84702i	−0.383578	+	0.923508i	$0.625308\pi$
$108$	0	0
$109$	5.46494i	0.523446i	0.965143	+	0.261723i	$0.0842907\pi$
$109$	5.46494i	0.523446i	−0.965143	+	0.261723i	$0.915709\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	5.25502i	0.494351i	0.968971	+	0.247175i	$0.0795024\pi$
$113$	5.25502i	0.494351i	−0.968971	+	0.247175i	$0.920498\pi$
$114$	0	0
$115$	−5.38354	−0.502017
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.34382	−0.214858
$120$	0	0
$121$	4.08422	0.371293
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.99120	0.714755
$126$	0	0
$127$	8.49723i	0.754007i	0.926212	+	0.377004i	$0.123046\pi$
$127$	8.49723i	0.754007i	−0.926212	+	0.377004i	$0.876954\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.91940i	0.517180i	0.965987	+	0.258590i	$0.0832579\pi$
$131$	5.91940i	0.517180i	−0.965987	+	0.258590i	$0.916742\pi$
$132$	0	0
$133$	− 4.09641i	− 0.355204i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 0.115119i	− 0.00983532i	−0.999988	−	0.00491766i	$-0.998435\pi$
$137$	− 0.115119i	− 0.00983532i	0.999988	−	0.00491766i	$-0.00156535\pi$
$138$	0	0
$139$	9.05579	0.768102	0.384051	−	0.923312i	$-0.374529\pi$
$139$	9.05579	0.768102	0.384051	+	0.923312i	$0.374529\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−2.67991	−0.224105
$144$	0	0
$145$	−1.36081	−0.113009
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.56092	0.619415	0.309707	−	0.950832i	$-0.399769\pi$
$149$	7.56092	0.619415	0.309707	+	0.950832i	$0.399769\pi$
$150$	0	0
$151$	11.0274i	0.897401i	0.893682	+	0.448700i	$0.148113\pi$
$151$	11.0274i	0.897401i	−0.893682	+	0.448700i	$0.851887\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 5.43856i	− 0.436836i
$156$	0	0
$157$	6.90136i	0.550788i	0.961331	+	0.275394i	$0.0888083\pi$
$157$	6.90136i	0.550788i	−0.961331	+	0.275394i	$0.911192\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 6.23450i	− 0.491347i
$162$	0	0
$163$	14.0095	1.09731	0.548653	−	0.836050i	$-0.315141\pi$
$163$	14.0095	1.09731	0.548653	+	0.836050i	$0.315141\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−5.97129	−0.462072	−0.231036	−	0.972945i	$-0.574212\pi$
$167$	−5.97129	−0.462072	−0.231036	+	0.972945i	$0.574212\pi$
$168$	0	0
$169$	11.9615	0.920117
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3.37661	−0.256719	−0.128359	−	0.991728i	$-0.540971\pi$
$173$	−3.37661	−0.256719	−0.128359	+	0.991728i	$0.540971\pi$
$174$	0	0
$175$	4.25435i	0.321599i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.3901i	0.851335i	0.904880	+	0.425668i	$0.139961\pi$
$179$	11.3901i	0.851335i	−0.904880	+	0.425668i	$0.860039\pi$
$180$	0	0
$181$	11.4514i	0.851175i	0.904917	+	0.425588i	$0.139933\pi$
$181$	11.4514i	0.851175i	−0.904917	+	0.425588i	$0.860067\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 4.47371i	− 0.328913i
$186$	0	0
$187$	6.16375	0.450738
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−12.1088	−0.876160	−0.438080	−	0.898936i	$-0.644341\pi$
$191$	−12.1088	−0.876160	−0.438080	+	0.898936i	$0.644341\pi$
$192$	0	0
$193$	1.69048	0.121683	0.0608416	−	0.998147i	$-0.480622\pi$
$193$	1.69048	0.121683	0.0608416	+	0.998147i	$0.480622\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.38828	−0.526393	−0.263197	−	0.964742i	$-0.584777\pi$
$197$	−7.38828	−0.526393	−0.263197	+	0.964742i	$0.584777\pi$
$198$	0	0
$199$	21.1249i	1.49750i	0.662851	+	0.748751i	$0.269346\pi$
$199$	21.1249i	1.49750i	−0.662851	+	0.748751i	$0.730654\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.57590i	− 0.110607i
$204$	0	0
$205$	8.75494i	0.611472i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	10.7727i	0.745162i
$210$	0	0
$211$	−5.02364	−0.345841	−0.172921	−	0.984936i	$-0.555320\pi$
$211$	−5.02364	−0.345841	−0.172921	+	0.984936i	$0.555320\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.428434	−0.0292190
$216$	0	0
$217$	6.29822	0.427551
$218$	0	0
$219$	0	0
$220$	0	0
$221$	2.38849	0.160668
$222$	0	0
$223$	6.37320i	0.426781i	0.976967	+	0.213390i	$0.0684507\pi$
$223$	6.37320i	0.426781i	−0.976967	+	0.213390i	$0.931549\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 7.87985i	− 0.523004i	−0.965203	−	0.261502i	$-0.915782\pi$
$227$	− 7.87985i	− 0.523004i	0.965203	−	0.261502i	$-0.0842177\pi$
$228$	0	0
$229$	− 21.7103i	− 1.43466i	−0.696736	−	0.717328i	$-0.745365\pi$
$229$	− 21.7103i	− 1.43466i	0.696736	−	0.717328i	$-0.254635\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.68586i	− 0.175956i	−0.996122	−	0.0879782i	$-0.971959\pi$
$233$	− 2.68586i	− 0.175956i	0.996122	−	0.0879782i	$-0.0280406\pi$
$234$	0	0
$235$	4.52818	0.295386
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3.96640	0.256565	0.128283	−	0.991738i	$-0.459054\pi$
$239$	3.96640	0.256565	0.128283	+	0.991738i	$0.459054\pi$
$240$	0	0
$241$	17.4310	1.12283	0.561414	−	0.827535i	$-0.310258\pi$
$241$	17.4310	1.12283	0.561414	+	0.827535i	$0.310258\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.863507	0.0551675
$246$	0	0
$247$	4.17449i	0.265616i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	19.8413i	1.25237i	0.779673	+	0.626187i	$0.215385\pi$
$251$	19.8413i	1.25237i	−0.779673	+	0.626187i	$0.784615\pi$
$252$	0	0
$253$	16.3954i	1.03077i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 23.6701i	− 1.47650i	−0.674525	−	0.738252i	$-0.735652\pi$
$257$	− 23.6701i	− 1.47650i	0.674525	−	0.738252i	$-0.264348\pi$
$258$	0	0
$259$	5.18085	0.321923
$260$	0	0
$261$	0	0
$262$	0	0
$263$	30.1325	1.85805	0.929025	−	0.370017i	$-0.120648\pi$
$263$	30.1325	1.85805	0.929025	+	0.370017i	$0.120648\pi$
$264$	0	0
$265$	9.10801	0.559501
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.73011	−0.227429	−0.113714	−	0.993513i	$-0.536275\pi$
$269$	−3.73011	−0.227429	−0.113714	+	0.993513i	$0.536275\pi$
$270$	0	0
$271$	− 7.25451i	− 0.440680i	−0.975423	−	0.220340i	$-0.929283\pi$
$271$	− 7.25451i	− 0.440680i	0.975423	−	0.220340i	$-0.0707167\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 11.1880i	− 0.674665i
$276$	0	0
$277$	− 3.00503i	− 0.180555i	−0.995917	−	0.0902774i	$-0.971225\pi$
$277$	− 3.00503i	− 0.180555i	0.995917	−	0.0902774i	$-0.0287754\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7.67039i	0.457577i	0.973476	+	0.228789i	$0.0734764\pi$
$281$	7.67039i	0.457577i	−0.973476	+	0.228789i	$0.926524\pi$
$282$	0	0
$283$	17.3124	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$283$	17.3124	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.1388	−0.598475
$288$	0	0
$289$	11.5065	0.676853
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.9400	1.34017	0.670084	−	0.742286i	$-0.266258\pi$
$293$	22.9400	1.34017	0.670084	+	0.742286i	$0.266258\pi$
$294$	0	0
$295$	3.80508i	0.221540i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	6.35333i	0.367423i
$300$	0	0
$301$	− 0.496156i	− 0.0285979i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 8.28668i	− 0.474494i
$306$	0	0
$307$	−1.46235	−0.0834608	−0.0417304	−	0.999129i	$-0.513287\pi$
$307$	−1.46235	−0.0834608	−0.0417304	+	0.999129i	$0.513287\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	5.83176	0.330689	0.165344	−	0.986236i	$-0.447126\pi$
$311$	5.83176	0.330689	0.165344	+	0.986236i	$0.447126\pi$
$312$	0	0
$313$	30.4020	1.71842	0.859211	−	0.511621i	$-0.170955\pi$
$313$	30.4020	1.71842	0.859211	+	0.511621i	$0.170955\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.85186	0.497170	0.248585	−	0.968610i	$-0.420035\pi$
$317$	8.85186	0.497170	0.248585	+	0.968610i	$0.420035\pi$
$318$	0	0
$319$	4.14429i	0.232036i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 9.60124i	− 0.534228i
$324$	0	0
$325$	− 4.33544i	− 0.240487i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.24394i	0.289108i
$330$	0	0
$331$	−27.0721	−1.48802	−0.744010	−	0.668169i	$-0.767078\pi$
$331$	−27.0721	−1.48802	−0.744010	+	0.668169i	$0.767078\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.9527	−0.598409
$336$	0	0
$337$	−21.8651	−1.19107	−0.595533	−	0.803331i	$-0.703059\pi$
$337$	−21.8651	−1.19107	−0.595533	+	0.803331i	$0.703059\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.5630	−0.896936
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 16.3033i	− 0.875206i	−0.899168	−	0.437603i	$-0.855828\pi$
$347$	− 16.3033i	− 0.875206i	0.899168	−	0.437603i	$-0.144172\pi$
$348$	0	0
$349$	10.3171i	0.552260i	0.961120	+	0.276130i	$0.0890521\pi$
$349$	10.3171i	0.552260i	−0.961120	+	0.276130i	$0.910948\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.62441i	0.0864586i	0.999065	+	0.0432293i	$0.0137646\pi$
$353$	1.62441i	0.0864586i	−0.999065	+	0.0432293i	$0.986235\pi$
$354$	0	0
$355$	1.82220	0.0967121
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.5784	1.40275	0.701376	−	0.712791i	$-0.252569\pi$
$359$	26.5784	1.40275	0.701376	+	0.712791i	$0.252569\pi$
$360$	0	0
$361$	−2.21945	−0.116813
$362$	0	0
$363$	0	0
$364$	0	0
$365$	7.14727	0.374105
$366$	0	0
$367$	14.5506i	0.759534i	0.925082	+	0.379767i	$0.123996\pi$
$367$	14.5506i	0.759534i	−0.925082	+	0.379767i	$0.876004\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	10.5477i	0.547609i
$372$	0	0
$373$	38.2468i	1.98034i	0.139856	+	0.990172i	$0.455336\pi$
$373$	38.2468i	1.98034i	−0.139856	+	0.990172i	$0.544664\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.60594i	0.0827102i
$378$	0	0
$379$	−28.9922	−1.48923	−0.744614	−	0.667495i	$-0.767367\pi$
$379$	−28.9922	−1.48923	−0.744614	+	0.667495i	$0.767367\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	33.7127	1.72264	0.861320	−	0.508063i	$-0.169638\pi$
$383$	33.7127	1.72264	0.861320	+	0.508063i	$0.169638\pi$
$384$	0	0
$385$	−2.27084	−0.115733
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−32.7096	−1.65844	−0.829221	−	0.558922i	$-0.811215\pi$
$389$	−32.7096	−1.65844	−0.829221	+	0.558922i	$0.811215\pi$
$390$	0	0
$391$	− 14.6126i	− 0.738988i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.15347i	0.410245i
$396$	0	0
$397$	− 16.5435i	− 0.830292i	−0.909755	−	0.415146i	$-0.863730\pi$
$397$	− 16.5435i	− 0.830292i	0.909755	−	0.415146i	$-0.136270\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 0.431420i	− 0.0215441i	−0.999942	−	0.0107720i	$-0.996571\pi$
$401$	− 0.431420i	− 0.0215441i	0.999942	−	0.0107720i	$-0.00342892\pi$
$402$	0	0
$403$	−6.41827	−0.319717
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.6245	−0.675343
$408$	0	0
$409$	−7.04622	−0.348413	−0.174207	−	0.984709i	$-0.555736\pi$
$409$	−7.04622	−0.348413	−0.174207	+	0.984709i	$0.555736\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.40654	−0.216832
$414$	0	0
$415$	− 5.72160i	− 0.280862i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 6.00287i	− 0.293259i	−0.989191	−	0.146630i	$-0.953157\pi$
$419$	− 6.00287i	− 0.293259i	0.989191	−	0.146630i	$-0.0468426\pi$
$420$	0	0
$421$	30.8156i	1.50186i	0.660381	+	0.750931i	$0.270395\pi$
$421$	30.8156i	1.50186i	−0.660381	+	0.750931i	$0.729605\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	9.97145i	0.483686i
$426$	0	0
$427$	9.59653	0.464409
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.16376	0.200561	0.100281	−	0.994959i	$-0.468026\pi$
$431$	4.16376	0.200561	0.100281	+	0.994959i	$0.468026\pi$
$432$	0	0
$433$	18.1624	0.872827	0.436414	−	0.899746i	$-0.356248\pi$
$433$	18.1624	0.872827	0.436414	+	0.899746i	$0.356248\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	25.5390	1.22170
$438$	0	0
$439$	20.8931i	0.997173i	0.866840	+	0.498586i	$0.166147\pi$
$439$	20.8931i	0.997173i	−0.866840	+	0.498586i	$0.833853\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.40982i	0.114494i	0.998360	+	0.0572469i	$0.0182322\pi$
$443$	2.40982i	0.114494i	−0.998360	+	0.0572469i	$0.981768\pi$
$444$	0	0
$445$	8.85539i	0.419786i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	37.5452i	1.77187i	0.463810	+	0.885935i	$0.346482\pi$
$449$	37.5452i	1.77187i	−0.463810	+	0.885935i	$0.653518\pi$
$450$	0	0
$451$	26.6629	1.25551
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.879966	−0.0412534
$456$	0	0
$457$	41.1990	1.92721	0.963603	−	0.267336i	$-0.0861433\pi$
$457$	41.1990	1.92721	0.963603	+	0.267336i	$0.0861433\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−22.9421	−1.06852	−0.534260	−	0.845320i	$-0.679410\pi$
$461$	−22.9421	−1.06852	−0.534260	+	0.845320i	$0.679410\pi$
$462$	0	0
$463$	28.5381i	1.32628i	0.748497	+	0.663138i	$0.230776\pi$
$463$	28.5381i	1.32628i	−0.748497	+	0.663138i	$0.769224\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	3.49880i	0.161905i	0.996718	+	0.0809525i	$0.0257962\pi$
$467$	3.49880i	0.161905i	−0.996718	+	0.0809525i	$0.974204\pi$
$468$	0	0
$469$	− 12.6840i	− 0.585691i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.30478i	0.0599940i
$474$	0	0
$475$	−17.4276	−0.799632
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−31.3664	−1.43317	−0.716583	−	0.697502i	$-0.754295\pi$
$479$	−31.3664	−1.43317	−0.716583	+	0.697502i	$0.754295\pi$
$480$	0	0
$481$	−5.27960	−0.240729
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.0567	−0.592876
$486$	0	0
$487$	39.7445i	1.80100i	0.434861	+	0.900498i	$0.356798\pi$
$487$	39.7445i	1.80100i	−0.434861	+	0.900498i	$0.643202\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 6.77799i	− 0.305886i	−0.988235	−	0.152943i	$-0.951125\pi$
$491$	− 6.77799i	− 0.305886i	0.988235	−	0.152943i	$-0.0488751\pi$
$492$	0	0
$493$	− 3.69364i	− 0.166353i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.11023i	0.0946566i
$498$	0	0
$499$	8.19185	0.366718	0.183359	−	0.983046i	$-0.441303\pi$
$499$	8.19185	0.366718	0.183359	+	0.983046i	$0.441303\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.5183	−1.67286	−0.836429	−	0.548075i	$-0.815361\pi$
$503$	−37.5183	−1.67286	−0.836429	+	0.548075i	$0.815361\pi$
$504$	0	0
$505$	−6.55357	−0.291630
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.84007	0.303181	0.151590	−	0.988443i	$-0.451561\pi$
$509$	6.84007	0.303181	0.151590	+	0.988443i	$0.451561\pi$
$510$	0	0
$511$	8.27702i	0.366154i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 6.33536i	− 0.279169i
$516$	0	0
$517$	− 13.7904i	− 0.606503i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.2860i	0.888746i	0.895842	+	0.444373i	$0.146574\pi$
$521$	20.2860i	0.888746i	−0.895842	+	0.444373i	$0.853426\pi$
$522$	0	0
$523$	36.4847	1.59537	0.797683	−	0.603077i	$-0.206059\pi$
$523$	36.4847	1.59537	0.797683	+	0.603077i	$0.206059\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14.7619	0.643038
$528$	0	0
$529$	15.8690	0.689956
$530$	0	0
$531$	0	0
$532$	0	0
$533$	10.3321	0.447531
$534$	0	0
$535$	− 16.4979i	− 0.713267i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.62979i	− 0.113273i
$540$	0	0
$541$	− 16.1841i	− 0.695810i	−0.937530	−	0.347905i	$-0.886893\pi$
$541$	− 16.1841i	− 0.695810i	0.937530	−	0.347905i	$-0.113107\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 4.71901i	− 0.202140i
$546$	0	0
$547$	−7.35520	−0.314486	−0.157243	−	0.987560i	$-0.550261\pi$
$547$	−7.35520	−0.314486	−0.157243	+	0.987560i	$0.550261\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.45555	0.275015
$552$	0	0
$553$	−9.44227	−0.401526
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.2146	−1.53446	−0.767231	−	0.641371i	$-0.778366\pi$
$557$	−36.2146	−1.53446	−0.767231	+	0.641371i	$0.778366\pi$
$558$	0	0
$559$	0.505613i	0.0213851i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 39.2015i	− 1.65215i	−0.563562	−	0.826074i	$-0.690570\pi$
$563$	− 39.2015i	− 1.65215i	0.563562	−	0.826074i	$-0.309430\pi$
$564$	0	0
$565$	− 4.53775i	− 0.190905i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 29.8748i	− 1.25242i	−0.779655	−	0.626209i	$-0.784606\pi$
$569$	− 29.8748i	− 1.25242i	0.779655	−	0.626209i	$-0.215394\pi$
$570$	0	0
$571$	−4.67642	−0.195702	−0.0978511	−	0.995201i	$-0.531197\pi$
$571$	−4.67642	−0.195702	−0.0978511	+	0.995201i	$0.531197\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−26.5238	−1.10612
$576$	0	0
$577$	−39.3888	−1.63978	−0.819889	−	0.572522i	$-0.805965\pi$
$577$	−39.3888	−1.63978	−0.819889	+	0.572522i	$0.805965\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.62600	0.274893
$582$	0	0
$583$	− 27.7382i	− 1.14880i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 41.1486i	− 1.69838i	−0.528085	−	0.849191i	$-0.677090\pi$
$587$	− 41.1486i	− 1.69838i	0.528085	−	0.849191i	$-0.322910\pi$
$588$	0	0
$589$	25.8001i	1.06307i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.7274i	1.26182i	0.775854	+	0.630912i	$0.217319\pi$
$593$	30.7274i	1.26182i	−0.775854	+	0.630912i	$0.782681\pi$
$594$	0	0
$595$	2.02391	0.0829721
$596$	0	0
$597$	0	0
$598$	0	0
$599$	44.7307	1.82765	0.913824	−	0.406110i	$-0.133115\pi$
$599$	44.7307	1.82765	0.913824	+	0.406110i	$0.133115\pi$
$600$	0	0
$601$	−20.8116	−0.848923	−0.424461	−	0.905446i	$-0.639537\pi$
$601$	−20.8116	−0.848923	−0.424461	+	0.905446i	$0.639537\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−3.52675	−0.143383
$606$	0	0
$607$	21.1941i	0.860243i	0.902771	+	0.430121i	$0.141529\pi$
$607$	21.1941i	0.860243i	−0.902771	+	0.430121i	$0.858471\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5.34389i	− 0.216191i
$612$	0	0
$613$	− 3.95265i	− 0.159646i	−0.996809	−	0.0798229i	$-0.974565\pi$
$613$	− 3.95265i	− 0.159646i	0.996809	−	0.0798229i	$-0.0254355\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 17.1751i	− 0.691445i	−0.938337	−	0.345722i	$-0.887634\pi$
$617$	− 17.1751i	− 0.691445i	0.938337	−	0.345722i	$-0.112366\pi$
$618$	0	0
$619$	−22.4666	−0.903007	−0.451504	−	0.892269i	$-0.649112\pi$
$619$	−22.4666	−0.903007	−0.451504	+	0.892269i	$0.649112\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.2551	−0.410864
$624$	0	0
$625$	14.3713	0.574852
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.1430	0.484173
$630$	0	0
$631$	− 19.1896i	− 0.763926i	−0.924178	−	0.381963i	$-0.875248\pi$
$631$	− 19.1896i	− 0.763926i	0.924178	−	0.381963i	$-0.124752\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.33742i	− 0.291177i
$636$	0	0
$637$	− 1.01906i	− 0.0403766i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	15.7596i	0.622467i	0.950333	+	0.311234i	$0.100742\pi$
$641$	15.7596i	0.622467i	−0.950333	+	0.311234i	$0.899258\pi$
$642$	0	0
$643$	47.6538	1.87928	0.939642	−	0.342160i	$-0.111158\pi$
$643$	47.6538	1.87928	0.939642	+	0.342160i	$0.111158\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.45095	0.0963567	0.0481783	−	0.998839i	$-0.484658\pi$
$647$	2.45095	0.0963567	0.0481783	+	0.998839i	$0.484658\pi$
$648$	0	0
$649$	11.5883	0.454879
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−26.8786	−1.05184	−0.525920	−	0.850534i	$-0.676279\pi$
$653$	−26.8786	−1.05184	−0.525920	+	0.850534i	$0.676279\pi$
$654$	0	0
$655$	− 5.11144i	− 0.199721i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	38.9485i	1.51722i	0.651545	+	0.758610i	$0.274121\pi$
$659$	38.9485i	1.51722i	−0.651545	+	0.758610i	$0.725879\pi$
$660$	0	0
$661$	− 5.36184i	− 0.208552i	−0.994548	−	0.104276i	$-0.966748\pi$
$661$	− 5.36184i	− 0.208552i	0.994548	−	0.104276i	$-0.0332525\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.53728i	0.137170i
$666$	0	0
$667$	9.82498	0.380425
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−25.2368	−0.974258
$672$	0	0
$673$	−36.5666	−1.40954	−0.704770	−	0.709436i	$-0.748950\pi$
$673$	−36.5666	−1.40954	−0.704770	+	0.709436i	$0.748950\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.9463	0.728166	0.364083	−	0.931367i	$-0.381382\pi$
$677$	18.9463	0.728166	0.364083	+	0.931367i	$0.381382\pi$
$678$	0	0
$679$	− 15.1206i	− 0.580275i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 15.1090i	− 0.578130i	−0.957309	−	0.289065i	$-0.906656\pi$
$683$	− 15.1090i	− 0.578130i	0.957309	−	0.289065i	$-0.0933444\pi$
$684$	0	0
$685$	0.0994065i	0.00379813i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 10.7487i	− 0.409494i
$690$	0	0
$691$	12.9717	0.493467	0.246733	−	0.969083i	$-0.420643\pi$
$691$	12.9717	0.493467	0.246733	+	0.969083i	$0.420643\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7.81974	−0.296620
$696$	0	0
$697$	−23.7636	−0.900109
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−9.31510	−0.351826	−0.175913	−	0.984406i	$-0.556288\pi$
$701$	−9.31510	−0.351826	−0.175913	+	0.984406i	$0.556288\pi$
$702$	0	0
$703$	21.2229i	0.800436i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 7.58948i	− 0.285432i
$708$	0	0
$709$	− 19.1984i	− 0.721012i	−0.932757	−	0.360506i	$-0.882604\pi$
$709$	− 19.1984i	− 0.721012i	0.932757	−	0.360506i	$-0.117396\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.2663i	1.47053i
$714$	0	0
$715$	2.31412	0.0865433
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−2.53286	−0.0944596	−0.0472298	−	0.998884i	$-0.515039\pi$
$719$	−2.53286	−0.0944596	−0.0472298	+	0.998884i	$0.515039\pi$
$720$	0	0
$721$	7.33678	0.273236
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−6.70446	−0.248997
$726$	0	0
$727$	− 38.2306i	− 1.41790i	−0.705261	−	0.708948i	$-0.749170\pi$
$727$	− 38.2306i	− 1.41790i	0.705261	−	0.708948i	$-0.250830\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 1.16290i	− 0.0430114i
$732$	0	0
$733$	− 20.4492i	− 0.755307i	−0.925947	−	0.377654i	$-0.876731\pi$
$733$	− 20.4492i	− 0.755307i	0.925947	−	0.377654i	$-0.123269\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	33.3561i	1.22869i
$738$	0	0
$739$	−43.3758	−1.59560	−0.797802	−	0.602920i	$-0.794004\pi$
$739$	−43.3758	−1.59560	−0.797802	+	0.602920i	$0.794004\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	28.3974	1.04180	0.520900	−	0.853618i	$-0.325596\pi$
$743$	28.3974	1.04180	0.520900	+	0.853618i	$0.325596\pi$
$744$	0	0
$745$	−6.52891	−0.239201
$746$	0	0
$747$	0	0
$748$	0	0
$749$	19.1057	0.698107
$750$	0	0
$751$	− 48.3659i	− 1.76490i	−0.470409	−	0.882448i	$-0.655894\pi$
$751$	− 48.3659i	− 1.76490i	0.470409	−	0.882448i	$-0.344106\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 9.52228i	− 0.346551i
$756$	0	0
$757$	38.1581i	1.38688i	0.720515	+	0.693439i	$0.243905\pi$
$757$	38.1581i	1.38688i	−0.720515	+	0.693439i	$0.756095\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	17.9144i	0.649396i	0.945818	+	0.324698i	$0.105263\pi$
$761$	17.9144i	0.649396i	−0.945818	+	0.324698i	$0.894737\pi$
$762$	0	0
$763$	5.46494	0.197844
$764$	0	0
$765$	0	0
$766$	0	0
$767$	4.49053	0.162144
$768$	0	0
$769$	11.7804	0.424811	0.212405	−	0.977182i	$-0.431870\pi$
$769$	11.7804	0.424811	0.212405	+	0.977182i	$0.431870\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38.7391	1.39335	0.696675	−	0.717387i	$-0.254662\pi$
$773$	38.7391	1.39335	0.696675	+	0.717387i	$0.254662\pi$
$774$	0	0
$775$	− 26.7949i	− 0.962500i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 41.5327i	− 1.48806i
$780$	0	0
$781$	− 5.54945i	− 0.198575i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 5.95937i	− 0.212699i
$786$	0	0
$787$	−30.0068	−1.06963	−0.534814	−	0.844970i	$-0.679618\pi$
$787$	−30.0068	−1.06963	−0.534814	+	0.844970i	$0.679618\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.25502	0.186847
$792$	0	0
$793$	−9.77945	−0.347278
$794$	0	0
$795$	0	0
$796$	0	0
$797$	46.9406	1.66272	0.831361	−	0.555733i	$-0.187562\pi$
$797$	46.9406	1.66272	0.831361	+	0.555733i	$0.187562\pi$
$798$	0	0
$799$	12.2909i	0.434819i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 21.7668i	− 0.768134i
$804$	0	0
$805$	5.38354i	0.189745i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 0.613418i	− 0.0215666i	−0.999942	−	0.0107833i	$-0.996567\pi$
$809$	− 0.613418i	− 0.0215666i	0.999942	−	0.0107833i	$-0.00343250\pi$
$810$	0	0
$811$	7.41924	0.260525	0.130262	−	0.991480i	$-0.458418\pi$
$811$	7.41924	0.260525	0.130262	+	0.991480i	$0.458418\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.0973	−0.423749
$816$	0	0
$817$	2.03246	0.0711066
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.6799	−1.31504	−0.657519	−	0.753438i	$-0.728394\pi$
$821$	−37.6799	−1.31504	−0.657519	+	0.753438i	$0.728394\pi$
$822$	0	0
$823$	− 11.9871i	− 0.417844i	−0.977932	−	0.208922i	$-0.933005\pi$
$823$	− 11.9871i	− 0.417844i	0.977932	−	0.208922i	$-0.0669955\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.1978i	− 1.25872i	−0.777113	−	0.629361i	$-0.783317\pi$
$827$	− 36.1978i	− 1.25872i	0.777113	−	0.629361i	$-0.216683\pi$
$828$	0	0
$829$	− 23.3923i	− 0.812449i	−0.913773	−	0.406225i	$-0.866845\pi$
$829$	− 23.3923i	− 0.812449i	0.913773	−	0.406225i	$-0.133155\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.34382i	0.0812086i
$834$	0	0
$835$	5.15625	0.178439
$836$	0	0
$837$	0	0
$838$	0	0
$839$	54.5591	1.88359	0.941794	−	0.336190i	$-0.109139\pi$
$839$	54.5591	1.88359	0.941794	+	0.336190i	$0.109139\pi$
$840$	0	0
$841$	−26.5165	−0.914363
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.3289	−0.355324
$846$	0	0
$847$	− 4.08422i	− 0.140335i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	32.3000i	1.10723i
$852$	0	0
$853$	− 8.05394i	− 0.275762i	−0.990449	−	0.137881i	$-0.955971\pi$
$853$	− 8.05394i	− 0.275762i	0.990449	−	0.137881i	$-0.0440291\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.83566i	0.267661i	0.991004	+	0.133830i	$0.0427278\pi$
$857$	7.83566i	0.267661i	−0.991004	+	0.133830i	$0.957272\pi$
$858$	0	0
$859$	16.4766	0.562173	0.281087	−	0.959682i	$-0.409305\pi$
$859$	16.4766	0.562173	0.281087	+	0.959682i	$0.409305\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−36.1099	−1.22920	−0.614598	−	0.788841i	$-0.710682\pi$
$863$	−36.1099	−1.22920	−0.614598	+	0.788841i	$0.710682\pi$
$864$	0	0
$865$	2.91572	0.0991376
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.8312	0.842339
$870$	0	0
$871$	12.9257i	0.437971i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 7.99120i	− 0.270152i
$876$	0	0
$877$	30.7080i	1.03694i	0.855097	+	0.518468i	$0.173497\pi$
$877$	30.7080i	1.03694i	−0.855097	+	0.518468i	$0.826503\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 8.54573i	− 0.287913i	−0.989584	−	0.143956i	$-0.954017\pi$
$881$	− 8.54573i	− 0.287913i	0.989584	−	0.143956i	$-0.0459825\pi$
$882$	0	0
$883$	−2.67385	−0.0899823	−0.0449912	−	0.998987i	$-0.514326\pi$
$883$	−2.67385	−0.0899823	−0.0449912	+	0.998987i	$0.514326\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.4850	0.855704	0.427852	−	0.903849i	$-0.359270\pi$
$887$	25.4850	0.855704	0.427852	+	0.903849i	$0.359270\pi$
$888$	0	0
$889$	8.49723	0.284988
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−21.4813	−0.718844
$894$	0	0
$895$	− 9.83543i	− 0.328762i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.92539i	0.331030i
$900$	0	0
$901$	24.7219i	0.823606i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 9.88836i	− 0.328700i
$906$	0	0
$907$	9.90655	0.328942	0.164471	−	0.986382i	$-0.447408\pi$
$907$	9.90655	0.328942	0.164471	+	0.986382i	$0.447408\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.7839	−1.45062	−0.725312	−	0.688420i	$-0.758304\pi$
$911$	−43.7839	−1.45062	−0.725312	+	0.688420i	$0.758304\pi$
$912$	0	0
$913$	−17.4250	−0.576683
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.91940	0.195476
$918$	0	0
$919$	− 8.90469i	− 0.293739i	−0.989156	−	0.146869i	$-0.953080\pi$
$919$	− 8.90469i	− 0.293739i	0.989156	−	0.146869i	$-0.0469197\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2.15045i	− 0.0707828i
$924$	0	0
$925$	− 22.0412i	− 0.724710i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 27.2093i	− 0.892709i	−0.894856	−	0.446354i	$-0.852722\pi$
$929$	− 27.2093i	− 0.892709i	0.894856	−	0.446354i	$-0.147278\pi$
$930$	0	0
$931$	−4.09641	−0.134254
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5.32245	−0.174063
$936$	0	0
$937$	11.2863	0.368708	0.184354	−	0.982860i	$-0.440981\pi$
$937$	11.2863	0.368708	0.184354	+	0.982860i	$0.440981\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	42.7661	1.39414	0.697068	−	0.717005i	$-0.254488\pi$
$941$	42.7661	1.39414	0.697068	+	0.717005i	$0.254488\pi$
$942$	0	0
$943$	− 63.2104i	− 2.05842i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.63182i	− 0.0530271i	−0.999648	−	0.0265135i	$-0.991559\pi$
$947$	− 1.63182i	− 0.0530271i	0.999648	−	0.0265135i	$-0.00844051\pi$
$948$	0	0
$949$	− 8.43479i	− 0.273805i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 30.4300i	− 0.985723i	−0.870108	−	0.492862i	$-0.835951\pi$
$953$	− 30.4300i	− 0.985723i	0.870108	−	0.492862i	$-0.164049\pi$
$954$	0	0
$955$	10.4560	0.338349
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.115119	−0.00371740
$960$	0	0
$961$	−8.66758	−0.279599
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−1.45974	−0.0469906
$966$	0	0
$967$	55.0276i	1.76957i	0.466003	+	0.884783i	$0.345694\pi$
$967$	55.0276i	1.76957i	−0.466003	+	0.884783i	$0.654306\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.1605i	0.614891i	0.951566	+	0.307445i	$0.0994741\pi$
$971$	19.1605i	0.614891i	−0.951566	+	0.307445i	$0.900526\pi$
$972$	0	0
$973$	− 9.05579i	− 0.290315i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 19.0482i	− 0.609405i	−0.952448	−	0.304703i	$-0.901443\pi$
$977$	− 19.0482i	− 0.609405i	0.952448	−	0.304703i	$-0.0985571\pi$
$978$	0	0
$979$	26.9689	0.861928
$980$	0	0
$981$	0	0
$982$	0	0
$983$	22.5016	0.717689	0.358845	−	0.933397i	$-0.383171\pi$
$983$	22.5016	0.717689	0.358845	+	0.933397i	$0.383171\pi$
$984$	0	0
$985$	6.37984	0.203279
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.09328	0.0983607
$990$	0	0
$991$	− 30.0467i	− 0.954466i	−0.878777	−	0.477233i	$-0.841640\pi$
$991$	− 30.0467i	− 0.954466i	0.878777	−	0.477233i	$-0.158360\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 18.2415i	− 0.578294i
$996$	0	0
$997$	61.6652i	1.95296i	0.215619	+	0.976478i	$0.430823\pi$
$997$	61.6652i	1.95296i	−0.215619	+	0.976478i	$0.569177\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6048.2.j.d.5615.20		48
3.2	odd	2	inner	6048.2.j.d.5615.29		48
4.3	odd	2		1512.2.j.d.323.35	yes	48
8.3	odd	2	inner	6048.2.j.d.5615.30		48
8.5	even	2		1512.2.j.d.323.13	✓	48
12.11	even	2		1512.2.j.d.323.14	yes	48
24.5	odd	2		1512.2.j.d.323.36	yes	48
24.11	even	2	inner	6048.2.j.d.5615.19		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1512.2.j.d.323.13	✓	48	8.5	even	2
1512.2.j.d.323.14	yes	48	12.11	even	2
1512.2.j.d.323.35	yes	48	4.3	odd	2
1512.2.j.d.323.36	yes	48	24.5	odd	2
6048.2.j.d.5615.19		48	24.11	even	2	inner
6048.2.j.d.5615.20		48	1.1	even	1	trivial
6048.2.j.d.5615.29		48	3.2	odd	2	inner
6048.2.j.d.5615.30		48	8.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6048.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.863507 q^{5} -1.00000i q^{7} +O(q^{10})\)	\(q-0.863507 q^{5} -1.00000i q^{7} +2.62979i q^{11} +1.01906i q^{13} -2.34382i q^{17} +4.09641 q^{19} +6.23450 q^{23} -4.25435 q^{25} +1.57590 q^{29} +6.29822i q^{31} +0.863507i q^{35} +5.18085i q^{37} -10.1388i q^{41} +0.496156 q^{43} -5.24394 q^{47} -1.00000 q^{49} -10.5477 q^{53} -2.27084i q^{55} -4.40654i q^{59} +9.59653i q^{61} -0.879966i q^{65} +12.6840 q^{67} -2.11023 q^{71} -8.27702 q^{73} +2.62979 q^{77} -9.44227i q^{79} +6.62600i q^{83} +2.02391i q^{85} -10.2551i q^{89} +1.01906 q^{91} -3.53728 q^{95} +15.1206 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q - 16 q^{19} + 48 q^{25} - 64 q^{43} - 48 q^{49} - 32 q^{67}+O(q^{100}) \) 48 * q - 16 * q^19 + 48 * q^25 - 64 * q^43 - 48 * q^49 - 32 * q^67

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