LMFDB - Embedded newform 7488.2.j.c.287.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7488,2,Mod(287,7488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7488, 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("7488.287");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7488 = 2^{6} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7488.j (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$59.7919810335$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			287.15
Character	$\chi$	$=$	7488.287
Dual form			7488.2.j.c.287.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.168038 q^{5} +1.54075i q^{7} +O(q^{10})$	$q-0.168038 q^{5} +1.54075i q^{7} -0.387386i q^{11} +1.00000i q^{13} -1.73150i q^{17} +1.34305 q^{19} +1.98742 q^{23} -4.97176 q^{25} +2.54109 q^{29} -10.0648i q^{31} -0.258905i q^{35} -8.54924i q^{37} +7.11088i q^{41} -10.2465 q^{43} -9.93699 q^{47} +4.62609 q^{49} -10.6359 q^{53} +0.0650957i q^{55} +14.1762i q^{59} +9.97725i q^{61} -0.168038i q^{65} +6.33961 q^{67} -13.2268 q^{71} +8.59301 q^{73} +0.596865 q^{77} +0.335354i q^{79} +7.55737i q^{83} +0.290957i q^{85} -16.9444i q^{89} -1.54075 q^{91} -0.225684 q^{95} -4.00299 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q+O(q^{10}) $ 32 * q	$ 32 q - 16 q^{19} + 32 q^{25} - 32 q^{43} - 64 q^{49} + 32 q^{67} - 16 q^{73} + 112 q^{97}+O(q^{100}) $ 32 * q - 16 * q^19 + 32 * q^25 - 32 * q^43 - 64 * q^49 + 32 * q^67 - 16 * q^73 + 112 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7488\mathbb{Z}\right)^\times$.

$n$	$703$	$5761$	$5825$	$6085$
$\chi(n)$	$-1$	$1$	$-1$	$-1$

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
$5$	−0.168038	−0.0751490	−0.0375745	−	0.999294i	$-0.511963\pi$
$5$	−0.168038	−0.0751490	−0.0375745	+	0.999294i	$0.511963\pi$
$6$	0	0
$7$	1.54075i	0.582348i	0.956670	+	0.291174i	$0.0940459\pi$
$7$	1.54075i	0.582348i	−0.956670	+	0.291174i	$0.905954\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 0.387386i	− 0.116801i	−0.998293	−	0.0584007i	$-0.981400\pi$
$11$	− 0.387386i	− 0.116801i	0.998293	−	0.0584007i	$-0.0186001\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1.73150i	− 0.419950i	−0.977707	−	0.209975i	$-0.932662\pi$
$17$	− 1.73150i	− 0.419950i	0.977707	−	0.209975i	$-0.0673382\pi$
$18$	0	0
$19$	1.34305	0.308118	0.154059	−	0.988062i	$-0.450765\pi$
$19$	1.34305	0.308118	0.154059	+	0.988062i	$0.450765\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.98742	0.414406	0.207203	−	0.978298i	$-0.433564\pi$
$23$	1.98742	0.414406	0.207203	+	0.978298i	$0.433564\pi$
$24$	0	0
$25$	−4.97176	−0.994353
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.54109	0.471868	0.235934	−	0.971769i	$-0.424185\pi$
$29$	2.54109	0.471868	0.235934	+	0.971769i	$0.424185\pi$
$30$	0	0
$31$	− 10.0648i	− 1.80770i	−0.427851	−	0.903849i	$-0.640729\pi$
$31$	− 10.0648i	− 1.80770i	0.427851	−	0.903849i	$-0.359271\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 0.258905i	− 0.0437629i
$36$	0	0
$37$	− 8.54924i	− 1.40549i	−0.711444	−	0.702743i	$-0.751958\pi$
$37$	− 8.54924i	− 1.40549i	0.711444	−	0.702743i	$-0.248042\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.11088i	1.11053i	0.831672	+	0.555266i	$0.187384\pi$
$41$	7.11088i	1.11053i	−0.831672	+	0.555266i	$0.812616\pi$
$42$	0	0
$43$	−10.2465	−1.56258	−0.781288	−	0.624170i	$-0.785437\pi$
$43$	−10.2465	−1.56258	−0.781288	+	0.624170i	$0.785437\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.93699	−1.44946	−0.724729	−	0.689034i	$-0.758035\pi$
$47$	−9.93699	−1.44946	−0.724729	+	0.689034i	$0.758035\pi$
$48$	0	0
$49$	4.62609	0.660870
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.6359	−1.46096	−0.730479	−	0.682935i	$-0.760703\pi$
$53$	−10.6359	−1.46096	−0.730479	+	0.682935i	$0.760703\pi$
$54$	0	0
$55$	0.0650957i	0.00877751i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	14.1762i	1.84558i	0.385300	+	0.922791i	$0.374098\pi$
$59$	14.1762i	1.84558i	−0.385300	+	0.922791i	$0.625902\pi$
$60$	0	0
$61$	9.97725i	1.27746i	0.769433	+	0.638728i	$0.220539\pi$
$61$	9.97725i	1.27746i	−0.769433	+	0.638728i	$0.779461\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 0.168038i	− 0.0208426i
$66$	0	0
$67$	6.33961	0.774506	0.387253	−	0.921973i	$-0.373424\pi$
$67$	6.33961	0.774506	0.387253	+	0.921973i	$0.373424\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−13.2268	−1.56973	−0.784865	−	0.619666i	$-0.787268\pi$
$71$	−13.2268	−1.56973	−0.784865	+	0.619666i	$0.787268\pi$
$72$	0	0
$73$	8.59301	1.00574	0.502868	−	0.864363i	$-0.332278\pi$
$73$	8.59301	1.00574	0.502868	+	0.864363i	$0.332278\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.596865	0.0680191
$78$	0	0
$79$	0.335354i	0.0377303i	0.999822	+	0.0188651i	$0.00600532\pi$
$79$	0.335354i	0.0377303i	−0.999822	+	0.0188651i	$0.993995\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	7.55737i	0.829530i	0.909929	+	0.414765i	$0.136136\pi$
$83$	7.55737i	0.829530i	−0.909929	+	0.414765i	$0.863864\pi$
$84$	0	0
$85$	0.290957i	0.0315588i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 16.9444i	− 1.79610i	−0.439890	−	0.898052i	$-0.644983\pi$
$89$	− 16.9444i	− 1.79610i	0.439890	−	0.898052i	$-0.355017\pi$
$90$	0	0
$91$	−1.54075	−0.161514
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.225684	−0.0231547
$96$	0	0
$97$	−4.00299	−0.406442	−0.203221	−	0.979133i	$-0.565141\pi$
$97$	−4.00299	−0.406442	−0.203221	+	0.979133i	$0.565141\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.5490	1.04967	0.524833	−	0.851205i	$-0.324128\pi$
$101$	10.5490	1.04967	0.524833	+	0.851205i	$0.324128\pi$
$102$	0	0
$103$	− 6.10277i	− 0.601324i	−0.953731	−	0.300662i	$-0.902792\pi$
$103$	− 6.10277i	− 0.601324i	0.953731	−	0.300662i	$-0.0972075\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.72576i	− 0.456856i	−0.973561	−	0.228428i	$-0.926641\pi$
$107$	− 4.72576i	− 0.456856i	0.973561	−	0.228428i	$-0.0733586\pi$
$108$	0	0
$109$	− 4.00421i	− 0.383534i	−0.981440	−	0.191767i	$-0.938578\pi$
$109$	− 4.00421i	− 0.383534i	0.981440	−	0.191767i	$-0.0614218\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 15.4119i	− 1.44983i	−0.688839	−	0.724915i	$-0.741879\pi$
$113$	− 15.4119i	− 1.44983i	0.688839	−	0.724915i	$-0.258121\pi$
$114$	0	0
$115$	−0.333963	−0.0311422
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.66780	0.244557
$120$	0	0
$121$	10.8499	0.986357
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.67564	0.149874
$126$	0	0
$127$	11.0741i	0.982670i	0.870971	+	0.491335i	$0.163491\pi$
$127$	11.0741i	0.982670i	−0.870971	+	0.491335i	$0.836509\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 17.5050i	− 1.52942i	−0.644377	−	0.764708i	$-0.722883\pi$
$131$	− 17.5050i	− 1.52942i	0.644377	−	0.764708i	$-0.277117\pi$
$132$	0	0
$133$	2.06931i	0.179432i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.52611i	− 0.472127i	−0.971738	−	0.236064i	$-0.924143\pi$
$137$	− 5.52611i	− 0.472127i	0.971738	−	0.236064i	$-0.0758574\pi$
$138$	0	0
$139$	2.99265	0.253833	0.126916	−	0.991913i	$-0.459492\pi$
$139$	2.99265	0.253833	0.126916	+	0.991913i	$0.459492\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	0.387386	0.0323949
$144$	0	0
$145$	−0.427000	−0.0354604
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.6665	−1.36538	−0.682688	−	0.730710i	$-0.739189\pi$
$149$	−16.6665	−1.36538	−0.682688	+	0.730710i	$0.739189\pi$
$150$	0	0
$151$	− 13.6429i	− 1.11024i	−0.831770	−	0.555121i	$-0.812672\pi$
$151$	− 13.6429i	− 1.11024i	0.831770	−	0.555121i	$-0.187328\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.69128i	0.135847i
$156$	0	0
$157$	− 18.0549i	− 1.44094i	−0.693488	−	0.720468i	$-0.743927\pi$
$157$	− 18.0549i	− 1.44094i	0.693488	−	0.720468i	$-0.256073\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.06212i	0.241329i
$162$	0	0
$163$	−10.6538	−0.834469	−0.417235	−	0.908799i	$-0.637001\pi$
$163$	−10.6538	−0.834469	−0.417235	+	0.908799i	$0.637001\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.02708	−0.621154	−0.310577	−	0.950548i	$-0.600522\pi$
$167$	−8.02708	−0.621154	−0.310577	+	0.950548i	$0.600522\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−17.5352	−1.33318	−0.666589	−	0.745426i	$-0.732246\pi$
$173$	−17.5352	−1.33318	−0.666589	+	0.745426i	$0.732246\pi$
$174$	0	0
$175$	− 7.66024i	− 0.579060i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.1882i	0.910987i	0.890239	+	0.455493i	$0.150537\pi$
$179$	12.1882i	0.910987i	−0.890239	+	0.455493i	$0.849463\pi$
$180$	0	0
$181$	− 2.10887i	− 0.156751i	−0.996924	−	0.0783755i	$-0.975027\pi$
$181$	− 2.10887i	− 0.156751i	0.996924	−	0.0783755i	$-0.0249733\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.43660i	0.105621i
$186$	0	0
$187$	−0.670758	−0.0490507
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.26559	−0.453362	−0.226681	−	0.973969i	$-0.572788\pi$
$191$	−6.26559	−0.453362	−0.226681	+	0.973969i	$0.572788\pi$
$192$	0	0
$193$	4.79252	0.344973	0.172487	−	0.985012i	$-0.444820\pi$
$193$	4.79252	0.344973	0.172487	+	0.985012i	$0.444820\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.84671	−0.701549	−0.350775	−	0.936460i	$-0.614082\pi$
$197$	−9.84671	−0.701549	−0.350775	+	0.936460i	$0.614082\pi$
$198$	0	0
$199$	− 3.66461i	− 0.259777i	−0.991529	−	0.129889i	$-0.958538\pi$
$199$	− 3.66461i	− 0.259777i	0.991529	−	0.129889i	$-0.0414620\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.91518i	0.274792i
$204$	0	0
$205$	− 1.19490i	− 0.0834554i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 0.520281i	− 0.0359886i
$210$	0	0
$211$	−14.6358	−1.00757	−0.503784	−	0.863830i	$-0.668059\pi$
$211$	−14.6358	−1.00757	−0.503784	+	0.863830i	$0.668059\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	1.72180	0.117426
$216$	0	0
$217$	15.5074	1.05271
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.73150	0.116473
$222$	0	0
$223$	− 8.51497i	− 0.570205i	−0.958497	−	0.285102i	$-0.907972\pi$
$223$	− 8.51497i	− 0.570205i	0.958497	−	0.285102i	$-0.0920276\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.176590i	0.0117207i	0.999983	+	0.00586033i	$0.00186541\pi$
$227$	0.176590i	0.0117207i	−0.999983	+	0.00586033i	$0.998135\pi$
$228$	0	0
$229$	7.60979i	0.502869i	0.967874	+	0.251435i	$0.0809023\pi$
$229$	7.60979i	0.502869i	−0.967874	+	0.251435i	$0.919098\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 7.81623i	− 0.512059i	−0.966669	−	0.256029i	$-0.917586\pi$
$233$	− 7.81623i	− 0.512059i	0.966669	−	0.256029i	$-0.0824144\pi$
$234$	0	0
$235$	1.66979	0.108925
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−10.0070	−0.647298	−0.323649	−	0.946177i	$-0.604910\pi$
$239$	−10.0070	−0.647298	−0.323649	+	0.946177i	$0.604910\pi$
$240$	0	0
$241$	13.2194	0.851534	0.425767	−	0.904833i	$-0.360004\pi$
$241$	13.2194	0.851534	0.425767	+	0.904833i	$0.360004\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.777360	−0.0496637
$246$	0	0
$247$	1.34305i	0.0854565i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	29.3019i	1.84952i	0.380555	+	0.924758i	$0.375733\pi$
$251$	29.3019i	1.84952i	−0.380555	+	0.924758i	$0.624267\pi$
$252$	0	0
$253$	− 0.769900i	− 0.0484032i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 20.6672i	− 1.28918i	−0.764527	−	0.644591i	$-0.777028\pi$
$257$	− 20.6672i	− 1.28918i	0.764527	−	0.644591i	$-0.222972\pi$
$258$	0	0
$259$	13.1722	0.818483
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.6948	−0.967785	−0.483893	−	0.875127i	$-0.660777\pi$
$263$	−15.6948	−0.967785	−0.483893	+	0.875127i	$0.660777\pi$
$264$	0	0
$265$	1.78724	0.109789
$266$	0	0
$267$	0	0
$268$	0	0
$269$	12.8133	0.781241	0.390621	−	0.920552i	$-0.372260\pi$
$269$	12.8133	0.781241	0.390621	+	0.920552i	$0.372260\pi$
$270$	0	0
$271$	− 15.0137i	− 0.912020i	−0.889975	−	0.456010i	$-0.849278\pi$
$271$	− 15.0137i	− 0.912020i	0.889975	−	0.456010i	$-0.150722\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.92599i	0.116142i
$276$	0	0
$277$	− 12.5022i	− 0.751182i	−0.926786	−	0.375591i	$-0.877440\pi$
$277$	− 12.5022i	− 0.751182i	0.926786	−	0.375591i	$-0.122560\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 25.3786i	− 1.51396i	−0.653438	−	0.756980i	$-0.726674\pi$
$281$	− 25.3786i	− 1.51396i	0.653438	−	0.756980i	$-0.273326\pi$
$282$	0	0
$283$	−18.5267	−1.10130	−0.550648	−	0.834737i	$-0.685619\pi$
$283$	−18.5267	−1.10130	−0.550648	+	0.834737i	$0.685619\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−10.9561	−0.646717
$288$	0	0
$289$	14.0019	0.823642
$290$	0	0
$291$	0	0
$292$	0	0
$293$	5.13373	0.299916	0.149958	−	0.988692i	$-0.452086\pi$
$293$	5.13373	0.299916	0.149958	+	0.988692i	$0.452086\pi$
$294$	0	0
$295$	− 2.38214i	− 0.138694i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.98742i	0.114936i
$300$	0	0
$301$	− 15.7873i	− 0.909964i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.67656i	− 0.0959995i
$306$	0	0
$307$	7.63239	0.435603	0.217802	−	0.975993i	$-0.430111\pi$
$307$	7.63239	0.435603	0.217802	+	0.975993i	$0.430111\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.9949	−1.53074	−0.765371	−	0.643590i	$-0.777444\pi$
$311$	−26.9949	−1.53074	−0.765371	+	0.643590i	$0.777444\pi$
$312$	0	0
$313$	−5.62323	−0.317844	−0.158922	−	0.987291i	$-0.550802\pi$
$313$	−5.62323	−0.317844	−0.158922	+	0.987291i	$0.550802\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−28.6844	−1.61108	−0.805538	−	0.592544i	$-0.798124\pi$
$317$	−28.6844	−1.61108	−0.805538	+	0.592544i	$0.798124\pi$
$318$	0	0
$319$	− 0.984383i	− 0.0551149i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.32549i	− 0.129394i
$324$	0	0
$325$	− 4.97176i	− 0.275784i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 15.3104i	− 0.844090i
$330$	0	0
$331$	−0.898205	−0.0493698	−0.0246849	−	0.999695i	$-0.507858\pi$
$331$	−0.898205	−0.0493698	−0.0246849	+	0.999695i	$0.507858\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.06530	−0.0582033
$336$	0	0
$337$	25.6793	1.39884	0.699420	−	0.714711i	$-0.253442\pi$
$337$	25.6793	1.39884	0.699420	+	0.714711i	$0.253442\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−3.89898	−0.211142
$342$	0	0
$343$	17.9129i	0.967205i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.42479i	− 0.398584i	−0.979940	−	0.199292i	$-0.936136\pi$
$347$	− 7.42479i	− 0.398584i	0.979940	−	0.199292i	$-0.0638642\pi$
$348$	0	0
$349$	5.26314i	0.281730i	0.990029	+	0.140865i	$0.0449883\pi$
$349$	5.26314i	0.281730i	−0.990029	+	0.140865i	$0.955012\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 30.2397i	− 1.60950i	−0.593614	−	0.804750i	$-0.702300\pi$
$353$	− 30.2397i	− 1.60950i	0.593614	−	0.804750i	$-0.297700\pi$
$354$	0	0
$355$	2.22260	0.117964
$356$	0	0
$357$	0	0
$358$	0	0
$359$	23.2725	1.22827	0.614137	−	0.789200i	$-0.289504\pi$
$359$	23.2725	1.22827	0.614137	+	0.789200i	$0.289504\pi$
$360$	0	0
$361$	−17.1962	−0.905063
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1.44395	−0.0755800
$366$	0	0
$367$	− 13.0111i	− 0.679172i	−0.940575	−	0.339586i	$-0.889713\pi$
$367$	− 13.0111i	− 0.679172i	0.940575	−	0.339586i	$-0.110287\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 16.3873i	− 0.850787i
$372$	0	0
$373$	− 28.2067i	− 1.46049i	−0.683187	−	0.730243i	$-0.739407\pi$
$373$	− 28.2067i	− 1.46049i	0.683187	−	0.730243i	$-0.260593\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.54109i	0.130873i
$378$	0	0
$379$	−26.2426	−1.34799	−0.673997	−	0.738734i	$-0.735424\pi$
$379$	−26.2426	−1.34799	−0.673997	+	0.738734i	$0.735424\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.0509	0.564676	0.282338	−	0.959315i	$-0.408890\pi$
$383$	11.0509	0.564676	0.282338	+	0.959315i	$0.408890\pi$
$384$	0	0
$385$	−0.100296	−0.00511157
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−20.9802	−1.06374	−0.531870	−	0.846826i	$-0.678511\pi$
$389$	−20.9802	−1.06374	−0.531870	+	0.846826i	$0.678511\pi$
$390$	0	0
$391$	− 3.44121i	− 0.174030i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 0.0563523i	− 0.00283539i
$396$	0	0
$397$	23.4126i	1.17505i	0.809207	+	0.587523i	$0.199897\pi$
$397$	23.4126i	1.17505i	−0.809207	+	0.587523i	$0.800103\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	27.3086i	1.36373i	0.731480	+	0.681863i	$0.238830\pi$
$401$	27.3086i	1.36373i	−0.731480	+	0.681863i	$0.761170\pi$
$402$	0	0
$403$	10.0648	0.501365
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3.31186	−0.164163
$408$	0	0
$409$	23.9012	1.18184	0.590920	−	0.806731i	$-0.298765\pi$
$409$	23.9012	1.18184	0.590920	+	0.806731i	$0.298765\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−21.8419	−1.07477
$414$	0	0
$415$	− 1.26993i	− 0.0623383i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.3885i	0.800628i	0.916378	+	0.400314i	$0.131099\pi$
$419$	16.3885i	0.800628i	−0.916378	+	0.400314i	$0.868901\pi$
$420$	0	0
$421$	− 9.49832i	− 0.462920i	−0.972844	−	0.231460i	$-0.925650\pi$
$421$	− 9.49832i	− 0.462920i	0.972844	−	0.231460i	$-0.0743502\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.60859i	0.417578i
$426$	0	0
$427$	−15.3724	−0.743925
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.8086	−1.05048	−0.525241	−	0.850954i	$-0.676025\pi$
$431$	−21.8086	−1.05048	−0.525241	+	0.850954i	$0.676025\pi$
$432$	0	0
$433$	−33.3622	−1.60329	−0.801643	−	0.597803i	$-0.796041\pi$
$433$	−33.3622	−1.60329	−0.801643	+	0.597803i	$0.796041\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.66921	0.127686
$438$	0	0
$439$	− 26.1723i	− 1.24914i	−0.780971	−	0.624568i	$-0.785275\pi$
$439$	− 26.1723i	− 1.24914i	0.780971	−	0.624568i	$-0.214725\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.6892i	1.83818i	0.394049	+	0.919089i	$0.371074\pi$
$443$	38.6892i	1.83818i	−0.394049	+	0.919089i	$0.628926\pi$
$444$	0	0
$445$	2.84731i	0.134975i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 18.2761i	− 0.862500i	−0.902232	−	0.431250i	$-0.858073\pi$
$449$	− 18.2761i	− 0.862500i	0.902232	−	0.431250i	$-0.141927\pi$
$450$	0	0
$451$	2.75466	0.129712
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0.258905	0.0121376
$456$	0	0
$457$	23.1311	1.08203	0.541014	−	0.841013i	$-0.318040\pi$
$457$	23.1311	1.08203	0.541014	+	0.841013i	$0.318040\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	35.8813	1.67116	0.835580	−	0.549368i	$-0.185132\pi$
$461$	35.8813	1.67116	0.835580	+	0.549368i	$0.185132\pi$
$462$	0	0
$463$	6.66683i	0.309834i	0.987928	+	0.154917i	$0.0495110\pi$
$463$	6.66683i	0.309834i	−0.987928	+	0.154917i	$0.950489\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 16.1819i	− 0.748809i	−0.927266	−	0.374404i	$-0.877847\pi$
$467$	− 16.1819i	− 0.748809i	0.927266	−	0.374404i	$-0.122153\pi$
$468$	0	0
$469$	9.76774i	0.451033i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	3.96936i	0.182511i
$474$	0	0
$475$	−6.67735	−0.306378
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−17.5308	−0.801001	−0.400500	−	0.916297i	$-0.631164\pi$
$479$	−17.5308	−0.801001	−0.400500	+	0.916297i	$0.631164\pi$
$480$	0	0
$481$	8.54924	0.389812
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.672655	0.0305437
$486$	0	0
$487$	− 18.2821i	− 0.828441i	−0.910177	−	0.414221i	$-0.864054\pi$
$487$	− 18.2821i	− 0.828441i	0.910177	−	0.414221i	$-0.135946\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	37.6339i	1.69839i	0.528077	+	0.849196i	$0.322913\pi$
$491$	37.6339i	1.69839i	−0.528077	+	0.849196i	$0.677087\pi$
$492$	0	0
$493$	− 4.39988i	− 0.198161i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 20.3792i	− 0.914130i
$498$	0	0
$499$	−1.16797	−0.0522855	−0.0261428	−	0.999658i	$-0.508322\pi$
$499$	−1.16797	−0.0522855	−0.0261428	+	0.999658i	$0.508322\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−37.3048	−1.66334	−0.831669	−	0.555271i	$-0.812614\pi$
$503$	−37.3048	−1.66334	−0.831669	+	0.555271i	$0.812614\pi$
$504$	0	0
$505$	−1.77264	−0.0788814
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.1555	0.627433	0.313716	−	0.949517i	$-0.398426\pi$
$509$	14.1555	0.627433	0.313716	+	0.949517i	$0.398426\pi$
$510$	0	0
$511$	13.2397i	0.585689i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.02550i	0.0451888i
$516$	0	0
$517$	3.84945i	0.169299i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 0.0724408i	− 0.00317369i	−0.999999	−	0.00158684i	$-0.999495\pi$
$521$	− 0.0724408i	− 0.00317369i	0.999999	−	0.00158684i	$-0.000505109\pi$
$522$	0	0
$523$	32.2889	1.41190	0.705948	−	0.708264i	$-0.250521\pi$
$523$	32.2889	1.41190	0.705948	+	0.708264i	$0.250521\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.4272	−0.759142
$528$	0	0
$529$	−19.0502	−0.828268
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−7.11088	−0.308006
$534$	0	0
$535$	0.794108i	0.0343323i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.79209i	− 0.0771906i
$540$	0	0
$541$	0.290719i	0.0124990i	0.999980	+	0.00624949i	$0.00198929\pi$
$541$	0.290719i	0.0124990i	−0.999980	+	0.00624949i	$0.998011\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0.672861i	0.0288222i
$546$	0	0
$547$	−10.9505	−0.468209	−0.234104	−	0.972211i	$-0.575216\pi$
$547$	−10.9505	−0.468209	−0.234104	+	0.972211i	$0.575216\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.41282	0.145391
$552$	0	0
$553$	−0.516696	−0.0219722
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.1469	0.726536	0.363268	−	0.931685i	$-0.381661\pi$
$557$	17.1469	0.726536	0.363268	+	0.931685i	$0.381661\pi$
$558$	0	0
$559$	− 10.2465i	− 0.433381i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 28.7604i	− 1.21211i	−0.795423	−	0.606054i	$-0.792751\pi$
$563$	− 28.7604i	− 1.21211i	0.795423	−	0.606054i	$-0.207249\pi$
$564$	0	0
$565$	2.58979i	0.108953i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	14.2206i	0.596160i	0.954541	+	0.298080i	$0.0963463\pi$
$569$	14.2206i	0.596160i	−0.954541	+	0.298080i	$0.903654\pi$
$570$	0	0
$571$	−15.5649	−0.651371	−0.325686	−	0.945478i	$-0.605595\pi$
$571$	−15.5649	−0.651371	−0.325686	+	0.945478i	$0.605595\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.88099	−0.412066
$576$	0	0
$577$	−15.4037	−0.641266	−0.320633	−	0.947203i	$-0.603896\pi$
$577$	−15.4037	−0.641266	−0.320633	+	0.947203i	$0.603896\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−11.6440	−0.483075
$582$	0	0
$583$	4.12022i	0.170642i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	29.7720i	1.22882i	0.788986	+	0.614412i	$0.210607\pi$
$587$	29.7720i	1.22882i	−0.788986	+	0.614412i	$0.789393\pi$
$588$	0	0
$589$	− 13.5176i	− 0.556984i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.60830i	0.353501i	0.984256	+	0.176750i	$0.0565585\pi$
$593$	8.60830i	0.353501i	−0.984256	+	0.176750i	$0.943442\pi$
$594$	0	0
$595$	−0.448293	−0.0183782
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−20.7285	−0.846942	−0.423471	−	0.905910i	$-0.639189\pi$
$599$	−20.7285	−0.846942	−0.423471	+	0.905910i	$0.639189\pi$
$600$	0	0
$601$	−13.2440	−0.540233	−0.270117	−	0.962828i	$-0.587062\pi$
$601$	−13.2440	−0.540233	−0.270117	+	0.962828i	$0.587062\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−1.82320	−0.0741237
$606$	0	0
$607$	− 38.6883i	− 1.57031i	−0.619301	−	0.785154i	$-0.712584\pi$
$607$	− 38.6883i	− 1.57031i	0.619301	−	0.785154i	$-0.287416\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 9.93699i	− 0.402007i
$612$	0	0
$613$	17.6639i	0.713439i	0.934212	+	0.356719i	$0.116105\pi$
$613$	17.6639i	0.713439i	−0.934212	+	0.356719i	$0.883895\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 41.1818i	− 1.65792i	−0.559311	−	0.828958i	$-0.688934\pi$
$617$	− 41.1818i	− 1.65792i	0.559311	−	0.828958i	$-0.311066\pi$
$618$	0	0
$619$	−38.9030	−1.56364	−0.781822	−	0.623502i	$-0.785709\pi$
$619$	−38.9030	−1.56364	−0.781822	+	0.623502i	$0.785709\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	26.1071	1.04596
$624$	0	0
$625$	24.5772	0.983090
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−14.8030	−0.590233
$630$	0	0
$631$	− 19.2707i	− 0.767156i	−0.923509	−	0.383578i	$-0.874692\pi$
$631$	− 19.2707i	− 0.767156i	0.923509	−	0.383578i	$-0.125308\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.86088i	− 0.0738466i
$636$	0	0
$637$	4.62609i	0.183292i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	36.6399i	1.44719i	0.690225	+	0.723595i	$0.257512\pi$
$641$	36.6399i	1.44719i	−0.690225	+	0.723595i	$0.742488\pi$
$642$	0	0
$643$	13.1724	0.519471	0.259735	−	0.965680i	$-0.416365\pi$
$643$	13.1724	0.519471	0.259735	+	0.965680i	$0.416365\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.16396	−0.242330	−0.121165	−	0.992632i	$-0.538663\pi$
$647$	−6.16396	−0.242330	−0.121165	+	0.992632i	$0.538663\pi$
$648$	0	0
$649$	5.49166	0.215567
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20.0192	−0.783410	−0.391705	−	0.920091i	$-0.628115\pi$
$653$	−20.0192	−0.783410	−0.391705	+	0.920091i	$0.628115\pi$
$654$	0	0
$655$	2.94150i	0.114934i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 40.8666i	− 1.59194i	−0.605337	−	0.795969i	$-0.706962\pi$
$659$	− 40.8666i	− 1.59194i	0.605337	−	0.795969i	$-0.293038\pi$
$660$	0	0
$661$	27.1667i	1.05666i	0.849038	+	0.528332i	$0.177182\pi$
$661$	27.1667i	1.05666i	−0.849038	+	0.528332i	$0.822818\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 0.347723i	− 0.0134841i
$666$	0	0
$667$	5.05021	0.195545
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.86505	0.149209
$672$	0	0
$673$	43.4105	1.67335	0.836676	−	0.547698i	$-0.184496\pi$
$673$	43.4105	1.67335	0.836676	+	0.547698i	$0.184496\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9.71499	0.373378	0.186689	−	0.982419i	$-0.440224\pi$
$677$	9.71499	0.373378	0.186689	+	0.982419i	$0.440224\pi$
$678$	0	0
$679$	− 6.16760i	− 0.236691i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 10.6441i	− 0.407285i	−0.979045	−	0.203642i	$-0.934722\pi$
$683$	− 10.6441i	− 0.407285i	0.979045	−	0.203642i	$-0.0652780\pi$
$684$	0	0
$685$	0.928597i	0.0354799i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 10.6359i	− 0.405197i
$690$	0	0
$691$	3.16807	0.120519	0.0602596	−	0.998183i	$-0.480807\pi$
$691$	3.16807	0.120519	0.0602596	+	0.998183i	$0.480807\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.502879	−0.0190753
$696$	0	0
$697$	12.3125	0.466368
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−35.0927	−1.32543	−0.662717	−	0.748870i	$-0.730597\pi$
$701$	−35.0927	−1.32543	−0.662717	+	0.748870i	$0.730597\pi$
$702$	0	0
$703$	− 11.4821i	− 0.433055i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.2534i	0.611272i
$708$	0	0
$709$	− 18.7683i	− 0.704859i	−0.935838	−	0.352430i	$-0.885356\pi$
$709$	− 18.7683i	− 0.704859i	0.935838	−	0.352430i	$-0.114644\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 20.0031i	− 0.749121i
$714$	0	0
$715$	−0.0650957	−0.00243444
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.3173	0.384771	0.192385	−	0.981319i	$-0.438378\pi$
$719$	10.3173	0.384771	0.192385	+	0.981319i	$0.438378\pi$
$720$	0	0
$721$	9.40283	0.350180
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.6337	−0.469203
$726$	0	0
$727$	− 1.10742i	− 0.0410721i	−0.999789	−	0.0205361i	$-0.993463\pi$
$727$	− 1.10742i	− 0.0410721i	0.999789	−	0.0205361i	$-0.00653729\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.7418i	0.656203i
$732$	0	0
$733$	− 46.4201i	− 1.71457i	−0.514845	−	0.857283i	$-0.672151\pi$
$733$	− 46.4201i	− 1.71457i	0.514845	−	0.857283i	$-0.327849\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.45588i	− 0.0904634i
$738$	0	0
$739$	20.4446	0.752069	0.376034	−	0.926606i	$-0.377287\pi$
$739$	20.4446	0.752069	0.376034	+	0.926606i	$0.377287\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	8.04710	0.295220	0.147610	−	0.989046i	$-0.452842\pi$
$743$	8.04710	0.295220	0.147610	+	0.989046i	$0.452842\pi$
$744$	0	0
$745$	2.80062	0.102607
$746$	0	0
$747$	0	0
$748$	0	0
$749$	7.28121	0.266050
$750$	0	0
$751$	22.0224i	0.803610i	0.915725	+	0.401805i	$0.131617\pi$
$751$	22.0224i	0.803610i	−0.915725	+	0.401805i	$0.868383\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2.29252i	0.0834335i
$756$	0	0
$757$	− 27.8553i	− 1.01242i	−0.862411	−	0.506208i	$-0.831047\pi$
$757$	− 27.8553i	− 1.01242i	0.862411	−	0.506208i	$-0.168953\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.17276i	0.115012i	0.998345	+	0.0575061i	$0.0183149\pi$
$761$	3.17276i	0.115012i	−0.998345	+	0.0575061i	$0.981685\pi$
$762$	0	0
$763$	6.16949	0.223351
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−14.1762	−0.511872
$768$	0	0
$769$	0.797046	0.0287422	0.0143711	−	0.999897i	$-0.495425\pi$
$769$	0.797046	0.0287422	0.0143711	+	0.999897i	$0.495425\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	42.3500	1.52322	0.761612	−	0.648033i	$-0.224408\pi$
$773$	42.3500	1.52322	0.761612	+	0.648033i	$0.224408\pi$
$774$	0	0
$775$	50.0400i	1.79749i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	9.55030i	0.342175i
$780$	0	0
$781$	5.12388i	0.183347i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	3.03391i	0.108285i
$786$	0	0
$787$	40.7505	1.45260	0.726299	−	0.687379i	$-0.241239\pi$
$787$	40.7505	1.45260	0.726299	+	0.687379i	$0.241239\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	23.7459	0.844306
$792$	0	0
$793$	−9.97725	−0.354303
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−12.4794	−0.442041	−0.221021	−	0.975269i	$-0.570939\pi$
$797$	−12.4794	−0.442041	−0.221021	+	0.975269i	$0.570939\pi$
$798$	0	0
$799$	17.2059i	0.608699i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 3.32882i	− 0.117471i
$804$	0	0
$805$	− 0.514553i	− 0.0181356i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.23836i	0.324803i	0.986725	+	0.162402i	$0.0519240\pi$
$809$	9.23836i	0.324803i	−0.986725	+	0.162402i	$0.948076\pi$
$810$	0	0
$811$	−35.6925	−1.25333	−0.626666	−	0.779288i	$-0.715581\pi$
$811$	−35.6925	−1.25333	−0.626666	+	0.779288i	$0.715581\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.79024	0.0627095
$816$	0	0
$817$	−13.7616	−0.481458
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.2203	1.15940	0.579699	−	0.814831i	$-0.303170\pi$
$821$	33.2203	1.15940	0.579699	+	0.814831i	$0.303170\pi$
$822$	0	0
$823$	− 13.2768i	− 0.462799i	−0.972859	−	0.231399i	$-0.925670\pi$
$823$	− 13.2768i	− 0.462799i	0.972859	−	0.231399i	$-0.0743304\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	51.1345i	1.77812i	0.457789	+	0.889061i	$0.348642\pi$
$827$	51.1345i	1.77812i	−0.457789	+	0.889061i	$0.651358\pi$
$828$	0	0
$829$	− 26.1335i	− 0.907654i	−0.891090	−	0.453827i	$-0.850058\pi$
$829$	− 26.1335i	− 0.907654i	0.891090	−	0.453827i	$-0.149942\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 8.01006i	− 0.277532i
$834$	0	0
$835$	1.34886	0.0466791
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−28.9715	−1.00021	−0.500103	−	0.865966i	$-0.666705\pi$
$839$	−28.9715	−1.00021	−0.500103	+	0.865966i	$0.666705\pi$
$840$	0	0
$841$	−22.5429	−0.777340
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.168038	0.00578069
$846$	0	0
$847$	16.7170i	0.574404i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 16.9909i	− 0.582442i
$852$	0	0
$853$	6.54751i	0.224182i	0.993698	+	0.112091i	$0.0357549\pi$
$853$	6.54751i	0.224182i	−0.993698	+	0.112091i	$0.964245\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	24.2044i	0.826806i	0.910548	+	0.413403i	$0.135660\pi$
$857$	24.2044i	0.826806i	−0.910548	+	0.413403i	$0.864340\pi$
$858$	0	0
$859$	−43.0073	−1.46739	−0.733695	−	0.679479i	$-0.762206\pi$
$859$	−43.0073	−1.46739	−0.733695	+	0.679479i	$0.762206\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	11.0415	0.375857	0.187929	−	0.982183i	$-0.439823\pi$
$863$	11.0415	0.375857	0.187929	+	0.982183i	$0.439823\pi$
$864$	0	0
$865$	2.94659	0.100187
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0.129912	0.00440695
$870$	0	0
$871$	6.33961i	0.214809i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.58174i	0.0872786i
$876$	0	0
$877$	44.0498i	1.48746i	0.668482	+	0.743729i	$0.266944\pi$
$877$	44.0498i	1.48746i	−0.668482	+	0.743729i	$0.733056\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.85121i	0.197132i	0.995131	+	0.0985661i	$0.0314256\pi$
$881$	5.85121i	0.197132i	−0.995131	+	0.0985661i	$0.968574\pi$
$882$	0	0
$883$	16.9433	0.570186	0.285093	−	0.958500i	$-0.407976\pi$
$883$	16.9433	0.570186	0.285093	+	0.958500i	$0.407976\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.8177	1.53841	0.769204	−	0.639003i	$-0.220653\pi$
$887$	45.8177	1.53841	0.769204	+	0.639003i	$0.220653\pi$
$888$	0	0
$889$	−17.0625	−0.572256
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−13.3459	−0.446604
$894$	0	0
$895$	− 2.04808i	− 0.0684597i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 25.5756i	− 0.852996i
$900$	0	0
$901$	18.4161i	0.613529i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.354371i	0.0117797i
$906$	0	0
$907$	−21.8572	−0.725756	−0.362878	−	0.931837i	$-0.618206\pi$
$907$	−21.8572	−0.725756	−0.362878	+	0.931837i	$0.618206\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.8509	1.51911	0.759553	−	0.650445i	$-0.225418\pi$
$911$	45.8509	1.51911	0.759553	+	0.650445i	$0.225418\pi$
$912$	0	0
$913$	2.92762	0.0968902
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.9708	0.890653
$918$	0	0
$919$	19.8831i	0.655884i	0.944698	+	0.327942i	$0.106355\pi$
$919$	19.8831i	0.655884i	−0.944698	+	0.327942i	$0.893645\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 13.2268i	− 0.435365i
$924$	0	0
$925$	42.5048i	1.39755i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.3512i	0.503657i	0.967772	+	0.251828i	$0.0810318\pi$
$929$	15.3512i	0.503657i	−0.967772	+	0.251828i	$0.918968\pi$
$930$	0	0
$931$	6.21309	0.203626
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0.112713	0.00368611
$936$	0	0
$937$	−5.24357	−0.171300	−0.0856500	−	0.996325i	$-0.527297\pi$
$937$	−5.24357	−0.171300	−0.0856500	+	0.996325i	$0.527297\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−38.0488	−1.24035	−0.620177	−	0.784462i	$-0.712939\pi$
$941$	−38.0488	−1.24035	−0.620177	+	0.784462i	$0.712939\pi$
$942$	0	0
$943$	14.1323i	0.460211i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 34.0971i	− 1.10801i	−0.832515	−	0.554003i	$-0.813100\pi$
$947$	− 34.0971i	− 1.10801i	0.832515	−	0.554003i	$-0.186900\pi$
$948$	0	0
$949$	8.59301i	0.278941i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	17.5843i	0.569610i	0.958585	+	0.284805i	$0.0919289\pi$
$953$	17.5843i	0.569610i	−0.958585	+	0.284805i	$0.908071\pi$
$954$	0	0
$955$	1.05286	0.0340697
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.51434	0.274943
$960$	0	0
$961$	−70.3010	−2.26777
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−0.805327	−0.0259244
$966$	0	0
$967$	− 50.1078i	− 1.61136i	−0.592353	−	0.805679i	$-0.701801\pi$
$967$	− 50.1078i	− 1.61136i	0.592353	−	0.805679i	$-0.298199\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	11.0284i	0.353918i	0.984218	+	0.176959i	$0.0566260\pi$
$971$	11.0284i	0.353918i	−0.984218	+	0.176959i	$0.943374\pi$
$972$	0	0
$973$	4.61092i	0.147819i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.17438i	0.0695647i	0.999395	+	0.0347824i	$0.0110738\pi$
$977$	2.17438i	0.0695647i	−0.999395	+	0.0347824i	$0.988926\pi$
$978$	0	0
$979$	−6.56403	−0.209787
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.0684	−0.544396	−0.272198	−	0.962241i	$-0.587751\pi$
$983$	−17.0684	−0.544396	−0.272198	+	0.962241i	$0.587751\pi$
$984$	0	0
$985$	1.65462	0.0527207
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.3641	−0.647541
$990$	0	0
$991$	33.3918i	1.06073i	0.847770	+	0.530364i	$0.177945\pi$
$991$	33.3918i	1.06073i	−0.847770	+	0.530364i	$0.822055\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0.615794i	0.0195220i
$996$	0	0
$997$	49.6216i	1.57153i	0.618525	+	0.785765i	$0.287731\pi$
$997$	49.6216i	1.57153i	−0.618525	+	0.785765i	$0.712269\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7488.2.j.c.287.15	✓	32
3.2	odd	2	inner	7488.2.j.c.287.17	yes	32
4.3	odd	2		7488.2.j.d.287.15	yes	32
8.3	odd	2	inner	7488.2.j.c.287.18	yes	32
8.5	even	2		7488.2.j.d.287.18	yes	32
12.11	even	2		7488.2.j.d.287.17	yes	32
24.5	odd	2		7488.2.j.d.287.16	yes	32
24.11	even	2	inner	7488.2.j.c.287.16	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7488.2.j.c.287.15	✓	32	1.1	even	1	trivial
7488.2.j.c.287.16	yes	32	24.11	even	2	inner
7488.2.j.c.287.17	yes	32	3.2	odd	2	inner
7488.2.j.c.287.18	yes	32	8.3	odd	2	inner
7488.2.j.d.287.15	yes	32	4.3	odd	2
7488.2.j.d.287.16	yes	32	24.5	odd	2
7488.2.j.d.287.17	yes	32	12.11	even	2
7488.2.j.d.287.18	yes	32	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7488 = 2^{6} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7488.j (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.168038 q^{5} +1.54075i q^{7} +O(q^{10})\)	\(q-0.168038 q^{5} +1.54075i q^{7} -0.387386i q^{11} +1.00000i q^{13} -1.73150i q^{17} +1.34305 q^{19} +1.98742 q^{23} -4.97176 q^{25} +2.54109 q^{29} -10.0648i q^{31} -0.258905i q^{35} -8.54924i q^{37} +7.11088i q^{41} -10.2465 q^{43} -9.93699 q^{47} +4.62609 q^{49} -10.6359 q^{53} +0.0650957i q^{55} +14.1762i q^{59} +9.97725i q^{61} -0.168038i q^{65} +6.33961 q^{67} -13.2268 q^{71} +8.59301 q^{73} +0.596865 q^{77} +0.335354i q^{79} +7.55737i q^{83} +0.290957i q^{85} -16.9444i q^{89} -1.54075 q^{91} -0.225684 q^{95} -4.00299 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q+O(q^{10}) \) 32 * q	\( 32 q - 16 q^{19} + 32 q^{25} - 32 q^{43} - 64 q^{49} + 32 q^{67} - 16 q^{73} + 112 q^{97}+O(q^{100}) \) 32 * q - 16 * q^19 + 32 * q^25 - 32 * q^43 - 64 * q^49 + 32 * q^67 - 16 * q^73 + 112 * q^97

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