LMFDB - Embedded newform 8280.2.p.a.1241.37

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1241,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8280.1241");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.p (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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1241.37
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.72943i q^{7} +O(q^{10})$	$q-1.00000 q^{5} +2.72943i q^{7} -6.18590 q^{11} -1.56797 q^{13} -7.34000 q^{17} -2.41299i q^{19} +(-4.64138 - 1.20731i) q^{23} +1.00000 q^{25} +7.31830i q^{29} +5.07406 q^{31} -2.72943i q^{35} +8.91623i q^{37} +1.74173i q^{41} +11.9685i q^{43} -0.0866043i q^{47} -0.449787 q^{49} -2.22559 q^{53} +6.18590 q^{55} +0.102805i q^{59} -10.4353i q^{61} +1.56797 q^{65} -8.46561i q^{67} +12.4723i q^{71} -11.2668 q^{73} -16.8840i q^{77} -1.00635i q^{79} +4.30642 q^{83} +7.34000 q^{85} -3.93965 q^{89} -4.27966i q^{91} +2.41299i q^{95} -14.9674i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 48 q^{5}+O(q^{10}) $ 48 * q - 48 * q^5	$ 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) $ 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8280\mathbb{Z}\right)^\times$.

$n$	$1657$	$2071$	$3961$	$4141$	$4601$
$\chi(n)$	$1$	$1$	$-1$	$1$	$-1$

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
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$			2.72943i			1.03163i	0.856701	+	0.515814i	$0.172510\pi$
$7$			2.72943i			1.03163i	−0.856701	+	0.515814i	$0.827490\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−6.18590			−1.86512			−0.932559	−	0.361016i	$-0.882430\pi$
$11$	−6.18590			−1.86512			−0.932559	+	0.361016i	$0.882430\pi$
$12$		0			0
$13$	−1.56797			−0.434876			−0.217438	−	0.976074i	$-0.569770\pi$
$13$	−1.56797			−0.434876			−0.217438	+	0.976074i	$0.569770\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.34000			−1.78021			−0.890105	−	0.455755i	$-0.849369\pi$
$17$	−7.34000			−1.78021			−0.890105	+	0.455755i	$0.849369\pi$
$18$		0			0
$19$		−	2.41299i		−	0.553579i	−0.960931	−	0.276789i	$-0.910730\pi$
$19$		−	2.41299i		−	0.553579i	0.960931	−	0.276789i	$-0.0892704\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.64138	−	1.20731i	−0.967794	−	0.251742i
$23$	−4.64138	−	1.20731i	−0.967794	−	0.251742i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$			7.31830i			1.35897i	0.733687	+	0.679487i	$0.237798\pi$
$29$			7.31830i			1.35897i	−0.733687	+	0.679487i	$0.762202\pi$
$30$		0			0
$31$	5.07406			0.911329			0.455664	−	0.890152i	$-0.349402\pi$
$31$	5.07406			0.911329			0.455664	+	0.890152i	$0.349402\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.72943i		−	0.461358i
$36$		0			0
$37$			8.91623i			1.46582i	0.680326	+	0.732910i	$0.261838\pi$
$37$			8.91623i			1.46582i	−0.680326	+	0.732910i	$0.738162\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.74173i			0.272012i	0.990708	+	0.136006i	$0.0434266\pi$
$41$			1.74173i			0.272012i	−0.990708	+	0.136006i	$0.956573\pi$
$42$		0			0
$43$			11.9685i			1.82517i	0.408884	+	0.912587i	$0.365918\pi$
$43$			11.9685i			1.82517i	−0.408884	+	0.912587i	$0.634082\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	0.0866043i		−	0.0126325i	−0.999980	−	0.00631627i	$-0.997989\pi$
$47$		−	0.0866043i		−	0.0126325i	0.999980	−	0.00631627i	$-0.00201054\pi$
$48$		0			0
$49$	−0.449787			−0.0642553
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−2.22559			−0.305709			−0.152854	−	0.988249i	$-0.548847\pi$
$53$	−2.22559			−0.305709			−0.152854	+	0.988249i	$0.548847\pi$
$54$		0			0
$55$	6.18590			0.834107
$56$		0			0
$57$		0			0
$58$		0			0
$59$			0.102805i			0.0133841i	0.999978	+	0.00669203i	$0.00213016\pi$
$59$			0.102805i			0.0133841i	−0.999978	+	0.00669203i	$0.997870\pi$
$60$		0			0
$61$		−	10.4353i		−	1.33610i	−0.744115	−	0.668051i	$-0.767129\pi$
$61$		−	10.4353i		−	1.33610i	0.744115	−	0.668051i	$-0.232871\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.56797			0.194483
$66$		0			0
$67$		−	8.46561i		−	1.03424i	−0.855913	−	0.517120i	$-0.827004\pi$
$67$		−	8.46561i		−	1.03424i	0.855913	−	0.517120i	$-0.172996\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.4723i			1.48019i	0.672505	+	0.740093i	$0.265219\pi$
$71$			12.4723i			1.48019i	−0.672505	+	0.740093i	$0.734781\pi$
$72$		0			0
$73$	−11.2668			−1.31868			−0.659339	−	0.751846i	$-0.729164\pi$
$73$	−11.2668			−1.31868			−0.659339	+	0.751846i	$0.729164\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	16.8840i		−	1.92411i
$78$		0			0
$79$		−	1.00635i		−	0.113223i	−0.998396	−	0.0566113i	$-0.981970\pi$
$79$		−	1.00635i		−	0.113223i	0.998396	−	0.0566113i	$-0.0180296\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.30642			0.472692			0.236346	−	0.971669i	$-0.424050\pi$
$83$	4.30642			0.472692			0.236346	+	0.971669i	$0.424050\pi$
$84$		0			0
$85$	7.34000			0.796134
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−3.93965			−0.417602			−0.208801	−	0.977958i	$-0.566956\pi$
$89$	−3.93965			−0.417602			−0.208801	+	0.977958i	$0.566956\pi$
$90$		0			0
$91$		−	4.27966i		−	0.448630i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			2.41299i			0.247568i
$96$		0			0
$97$		−	14.9674i		−	1.51971i	−0.650095	−	0.759853i	$-0.725271\pi$
$97$		−	14.9674i		−	1.51971i	0.650095	−	0.759853i	$-0.274729\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	12.1022i		−	1.20421i	−0.798415	−	0.602107i	$-0.794328\pi$
$101$		−	12.1022i		−	1.20421i	0.798415	−	0.602107i	$-0.205672\pi$
$102$		0			0
$103$		−	9.74573i		−	0.960276i	−0.877193	−	0.480138i	$-0.840587\pi$
$103$		−	9.74573i		−	0.960276i	0.877193	−	0.480138i	$-0.159413\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−10.0549			−0.972048			−0.486024	−	0.873945i	$-0.661553\pi$
$107$	−10.0549			−0.972048			−0.486024	+	0.873945i	$0.661553\pi$
$108$		0			0
$109$		−	15.9444i		−	1.52719i	−0.645694	−	0.763596i	$-0.723432\pi$
$109$		−	15.9444i		−	1.52719i	0.645694	−	0.763596i	$-0.276568\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.4113			0.979410			0.489705	−	0.871888i	$-0.337104\pi$
$113$	10.4113			0.979410			0.489705	+	0.871888i	$0.337104\pi$
$114$		0			0
$115$	4.64138	+	1.20731i	0.432811	+	0.112582i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	20.0340i		−	1.83651i
$120$		0			0
$121$	27.2654			2.47867
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−3.05363			−0.270966			−0.135483	−	0.990780i	$-0.543259\pi$
$127$	−3.05363			−0.270966			−0.135483	+	0.990780i	$0.543259\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			7.35861i			0.642925i	0.946922	+	0.321462i	$0.104174\pi$
$131$			7.35861i			0.642925i	−0.946922	+	0.321462i	$0.895826\pi$
$132$		0			0
$133$	6.58609			0.571087
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.68321			−0.656421			−0.328211	−	0.944605i	$-0.606446\pi$
$137$	−7.68321			−0.656421			−0.328211	+	0.944605i	$0.606446\pi$
$138$		0			0
$139$	−17.6844			−1.49997			−0.749986	−	0.661453i	$-0.769940\pi$
$139$	−17.6844			−1.49997			−0.749986	+	0.661453i	$0.769940\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	9.69930			0.811096
$144$		0			0
$145$		−	7.31830i		−	0.607752i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	24.2027			1.98276			0.991380	−	0.131014i	$-0.0418234\pi$
$149$	24.2027			1.98276			0.991380	+	0.131014i	$0.0418234\pi$
$150$		0			0
$151$	10.3560			0.842759			0.421379	−	0.906885i	$-0.361546\pi$
$151$	10.3560			0.842759			0.421379	+	0.906885i	$0.361546\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.07406			−0.407559
$156$		0			0
$157$			5.29703i			0.422749i	0.977405	+	0.211375i	$0.0677940\pi$
$157$			5.29703i			0.422749i	−0.977405	+	0.211375i	$0.932206\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	3.29527	−	12.6683i	0.259704	−	0.998403i
$162$		0			0
$163$	−4.95215			−0.387882			−0.193941	−	0.981013i	$-0.562127\pi$
$163$	−4.95215			−0.387882			−0.193941	+	0.981013i	$0.562127\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			8.20101i			0.634613i	0.948323	+	0.317307i	$0.102778\pi$
$167$			8.20101i			0.634613i	−0.948323	+	0.317307i	$0.897222\pi$
$168$		0			0
$169$	−10.5415			−0.810883
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	18.3258i		−	1.39328i	−0.717420	−	0.696641i	$-0.754677\pi$
$173$		−	18.3258i		−	1.39328i	0.717420	−	0.696641i	$-0.245323\pi$
$174$		0			0
$175$			2.72943i			0.206325i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	6.21672i		−	0.464659i	−0.972637	−	0.232330i	$-0.925365\pi$
$179$		−	6.21672i		−	0.464659i	0.972637	−	0.232330i	$-0.0746348\pi$
$180$		0			0
$181$			10.4391i			0.775932i	0.921674	+	0.387966i	$0.126822\pi$
$181$			10.4391i			0.775932i	−0.921674	+	0.387966i	$0.873178\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		−	8.91623i		−	0.655534i
$186$		0			0
$187$	45.4045			3.32030
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.7744			1.64790			0.823949	−	0.566664i	$-0.191766\pi$
$191$	22.7744			1.64790			0.823949	+	0.566664i	$0.191766\pi$
$192$		0			0
$193$	−16.6166			−1.19609			−0.598046	−	0.801462i	$-0.704056\pi$
$193$	−16.6166			−1.19609			−0.598046	+	0.801462i	$0.704056\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	18.7600i		−	1.33660i	−0.743894	−	0.668298i	$-0.767023\pi$
$197$		−	18.7600i		−	1.33660i	0.743894	−	0.668298i	$-0.232977\pi$
$198$		0			0
$199$			22.2158i			1.57483i	0.616420	+	0.787417i	$0.288582\pi$
$199$			22.2158i			1.57483i	−0.616420	+	0.787417i	$0.711418\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−19.9748			−1.40196
$204$		0			0
$205$		−	1.74173i		−	0.121647i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			14.9265i			1.03249i
$210$		0			0
$211$	24.8053			1.70766			0.853832	−	0.520549i	$-0.174273\pi$
$211$	24.8053			1.70766			0.853832	+	0.520549i	$0.174273\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	11.9685i		−	0.816242i
$216$		0			0
$217$			13.8493i			0.940152i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	11.5089			0.774172
$222$		0			0
$223$	25.9505			1.73777			0.868887	−	0.495010i	$-0.164836\pi$
$223$	25.9505			1.73777			0.868887	+	0.495010i	$0.164836\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	11.0258			0.731808			0.365904	−	0.930653i	$-0.380760\pi$
$227$	11.0258			0.731808			0.365904	+	0.930653i	$0.380760\pi$
$228$		0			0
$229$			17.6864i			1.16875i	0.811483	+	0.584376i	$0.198661\pi$
$229$			17.6864i			1.16875i	−0.811483	+	0.584376i	$0.801339\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			21.3444i			1.39832i	0.714965	+	0.699160i	$0.246443\pi$
$233$			21.3444i			1.39832i	−0.714965	+	0.699160i	$0.753557\pi$
$234$		0			0
$235$			0.0866043i			0.00564944i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			4.47318i			0.289346i	0.989480	+	0.144673i	$0.0462130\pi$
$239$			4.47318i			0.289346i	−0.989480	+	0.144673i	$0.953787\pi$
$240$		0			0
$241$		−	5.95063i		−	0.383314i	−0.981462	−	0.191657i	$-0.938614\pi$
$241$		−	5.95063i		−	0.383314i	0.981462	−	0.191657i	$-0.0613861\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.449787			0.0287358
$246$		0			0
$247$			3.78350i			0.240738i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−25.0983			−1.58419			−0.792094	−	0.610399i	$-0.791009\pi$
$251$	−25.0983			−1.58419			−0.792094	+	0.610399i	$0.791009\pi$
$252$		0			0
$253$	28.7111	+	7.46830i	1.80505	+	0.469528i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	6.12015i		−	0.381764i	−0.981613	−	0.190882i	$-0.938865\pi$
$257$		−	6.12015i		−	0.381764i	0.981613	−	0.190882i	$-0.0611348\pi$
$258$		0			0
$259$	−24.3362			−1.51218
$260$		0			0
$261$		0			0
$262$		0			0
$263$	16.5265			1.01907			0.509534	−	0.860451i	$-0.329818\pi$
$263$	16.5265			1.01907			0.509534	+	0.860451i	$0.329818\pi$
$264$		0			0
$265$	2.22559			0.136717
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	16.7090i		−	1.01876i	−0.860541	−	0.509382i	$-0.829874\pi$
$269$		−	16.7090i		−	1.01876i	0.860541	−	0.509382i	$-0.170126\pi$
$270$		0			0
$271$	−13.5676			−0.824173			−0.412086	−	0.911145i	$-0.635200\pi$
$271$	−13.5676			−0.824173			−0.412086	+	0.911145i	$0.635200\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−6.18590			−0.373024
$276$		0			0
$277$	−14.8253			−0.890768			−0.445384	−	0.895340i	$-0.646933\pi$
$277$	−14.8253			−0.890768			−0.445384	+	0.895340i	$0.646933\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.77942			−0.225461			−0.112731	−	0.993626i	$-0.535960\pi$
$281$	−3.77942			−0.225461			−0.112731	+	0.993626i	$0.535960\pi$
$282$		0			0
$283$		−	20.5953i		−	1.22427i	−0.790755	−	0.612133i	$-0.790312\pi$
$283$		−	20.5953i		−	1.22427i	0.790755	−	0.612133i	$-0.209688\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.75392			−0.280615
$288$		0			0
$289$	36.8756			2.16915
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−30.0552			−1.75585			−0.877923	−	0.478803i	$-0.841071\pi$
$293$	−30.0552			−1.75585			−0.877923	+	0.478803i	$0.841071\pi$
$294$		0			0
$295$		−	0.102805i		−	0.00598554i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	7.27754	+	1.89303i	0.420871	+	0.109477i
$300$		0			0
$301$	−32.6671			−1.88290
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.4353i			0.597523i
$306$		0			0
$307$	34.0696			1.94445			0.972227	−	0.234041i	$-0.0751949\pi$
$307$	34.0696			1.94445			0.972227	+	0.234041i	$0.0751949\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$			9.51737i			0.539681i	0.962905	+	0.269840i	$0.0869709\pi$
$311$			9.51737i			0.539681i	−0.962905	+	0.269840i	$0.913029\pi$
$312$		0			0
$313$		−	7.53633i		−	0.425979i	−0.977055	−	0.212989i	$-0.931680\pi$
$313$		−	7.53633i		−	0.425979i	0.977055	−	0.212989i	$-0.0683199\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	14.3174i		−	0.804147i	−0.915607	−	0.402074i	$-0.868290\pi$
$317$		−	14.3174i		−	0.804147i	0.915607	−	0.402074i	$-0.131710\pi$
$318$		0			0
$319$		−	45.2703i		−	2.53465i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			17.7114i			0.985486i
$324$		0			0
$325$	−1.56797			−0.0869753
$326$		0			0
$327$		0			0
$328$		0			0
$329$	0.236380			0.0130321
$330$		0			0
$331$	−0.171170			−0.00940834			−0.00470417	−	0.999989i	$-0.501497\pi$
$331$	−0.171170			−0.00940834			−0.00470417	+	0.999989i	$0.501497\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$			8.46561i			0.462526i
$336$		0			0
$337$			16.9754i			0.924710i	0.886695	+	0.462355i	$0.152995\pi$
$337$			16.9754i			0.924710i	−0.886695	+	0.462355i	$0.847005\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−31.3876			−1.69974
$342$		0			0
$343$			17.8783i			0.965340i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	4.13762i		−	0.222119i	−0.993814	−	0.111059i	$-0.964576\pi$
$347$		−	4.13762i		−	0.222119i	0.993814	−	0.111059i	$-0.0354244\pi$
$348$		0			0
$349$	5.48433			0.293569			0.146785	−	0.989168i	$-0.453108\pi$
$349$	5.48433			0.293569			0.146785	+	0.989168i	$0.453108\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	30.8604i		−	1.64253i	−0.570545	−	0.821267i	$-0.693268\pi$
$353$		−	30.8604i		−	1.64253i	0.570545	−	0.821267i	$-0.306732\pi$
$354$		0			0
$355$		−	12.4723i		−	0.661959i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	16.0049			0.844707			0.422353	−	0.906431i	$-0.361204\pi$
$359$	16.0049			0.844707			0.422353	+	0.906431i	$0.361204\pi$
$360$		0			0
$361$	13.1775			0.693551
$362$		0			0
$363$		0			0
$364$		0			0
$365$	11.2668			0.589731
$366$		0			0
$367$		−	12.5743i		−	0.656374i	−0.944613	−	0.328187i	$-0.893562\pi$
$367$		−	12.5743i		−	0.656374i	0.944613	−	0.328187i	$-0.106438\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	6.07460i		−	0.315377i
$372$		0			0
$373$			10.7448i			0.556345i	0.960531	+	0.278173i	$0.0897287\pi$
$373$			10.7448i			0.556345i	−0.960531	+	0.278173i	$0.910271\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	11.4749i		−	0.590986i
$378$		0			0
$379$			4.11204i			0.211221i	0.994408	+	0.105611i	$0.0336797\pi$
$379$			4.11204i			0.211221i	−0.994408	+	0.105611i	$0.966320\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4.62303			0.236226			0.118113	−	0.993000i	$-0.462316\pi$
$383$	4.62303			0.236226			0.118113	+	0.993000i	$0.462316\pi$
$384$		0			0
$385$			16.8840i			0.860487i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.17964			0.262618			0.131309	−	0.991341i	$-0.458082\pi$
$389$	5.17964			0.262618			0.131309	+	0.991341i	$0.458082\pi$
$390$		0			0
$391$	34.0677	+	8.86166i	1.72288	+	0.448153i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			1.00635i			0.0506347i
$396$		0			0
$397$	−20.4023			−1.02396			−0.511980	−	0.858997i	$-0.671088\pi$
$397$	−20.4023			−1.02396			−0.511980	+	0.858997i	$0.671088\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	24.7937			1.23814			0.619069	−	0.785336i	$-0.287510\pi$
$401$	24.7937			1.23814			0.619069	+	0.785336i	$0.287510\pi$
$402$		0			0
$403$	−7.95597			−0.396315
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	55.1549i		−	2.73393i
$408$		0			0
$409$	10.9243			0.540174			0.270087	−	0.962836i	$-0.412948\pi$
$409$	10.9243			0.540174			0.270087	+	0.962836i	$0.412948\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.280599			−0.0138074
$414$		0			0
$415$	−4.30642			−0.211394
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−31.0588			−1.51732			−0.758661	−	0.651486i	$-0.774146\pi$
$419$	−31.0588			−1.51732			−0.758661	+	0.651486i	$0.774146\pi$
$420$		0			0
$421$			35.6421i			1.73709i	0.495610	+	0.868545i	$0.334944\pi$
$421$			35.6421i			1.73709i	−0.495610	+	0.868545i	$0.665056\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.34000			−0.356042
$426$		0			0
$427$	28.4824			1.37836
$428$		0			0
$429$		0			0
$430$		0			0
$431$	30.5109			1.46966			0.734830	−	0.678251i	$-0.237262\pi$
$431$	30.5109			1.46966			0.734830	+	0.678251i	$0.237262\pi$
$432$		0			0
$433$		−	31.4222i		−	1.51006i	−0.655693	−	0.755028i	$-0.727623\pi$
$433$		−	31.4222i		−	1.51006i	0.655693	−	0.755028i	$-0.272377\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.91323	+	11.1996i	−0.139359	+	0.535750i
$438$		0			0
$439$	−18.5846			−0.886993			−0.443497	−	0.896276i	$-0.646262\pi$
$439$	−18.5846			−0.886993			−0.443497	+	0.896276i	$0.646262\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			22.6267i			1.07503i	0.843255	+	0.537513i	$0.180636\pi$
$443$			22.6267i			1.07503i	−0.843255	+	0.537513i	$0.819364\pi$
$444$		0			0
$445$	3.93965			0.186757
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	18.5270i		−	0.874344i	−0.899378	−	0.437172i	$-0.855980\pi$
$449$		−	18.5270i		−	0.874344i	0.899378	−	0.437172i	$-0.144020\pi$
$450$		0			0
$451$		−	10.7741i		−	0.507335i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			4.27966i			0.200634i
$456$		0			0
$457$			18.8954i			0.883890i	0.897042	+	0.441945i	$0.145711\pi$
$457$			18.8954i			0.883890i	−0.897042	+	0.441945i	$0.854289\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	15.0747i		−	0.702098i	−0.936357	−	0.351049i	$-0.885825\pi$
$461$		−	15.0747i		−	0.702098i	0.936357	−	0.351049i	$-0.114175\pi$
$462$		0			0
$463$	12.1878			0.566413			0.283207	−	0.959059i	$-0.408602\pi$
$463$	12.1878			0.566413			0.283207	+	0.959059i	$0.408602\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−31.2902			−1.44794			−0.723968	−	0.689833i	$-0.757684\pi$
$467$	−31.2902			−1.44794			−0.723968	+	0.689833i	$0.757684\pi$
$468$		0			0
$469$	23.1063			1.06695
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	74.0357i		−	3.40416i
$474$		0			0
$475$		−	2.41299i		−	0.110716i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	7.04788			0.322026			0.161013	−	0.986952i	$-0.448524\pi$
$479$	7.04788			0.322026			0.161013	+	0.986952i	$0.448524\pi$
$480$		0			0
$481$		−	13.9804i		−	0.637450i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			14.9674i			0.679633i
$486$		0			0
$487$	19.3153			0.875260			0.437630	−	0.899155i	$-0.355818\pi$
$487$	19.3153			0.875260			0.437630	+	0.899155i	$0.355818\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			22.3584i			1.00902i	0.863405	+	0.504511i	$0.168327\pi$
$491$			22.3584i			1.00902i	−0.863405	+	0.504511i	$0.831673\pi$
$492$		0			0
$493$		−	53.7163i		−	2.41926i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−34.0422			−1.52700
$498$		0			0
$499$	3.51343			0.157283			0.0786413	−	0.996903i	$-0.474942\pi$
$499$	3.51343			0.157283			0.0786413	+	0.996903i	$0.474942\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	16.8774			0.752527			0.376264	−	0.926513i	$-0.377209\pi$
$503$	16.8774			0.752527			0.376264	+	0.926513i	$0.377209\pi$
$504$		0			0
$505$			12.1022i			0.538541i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			8.62970i			0.382505i	0.981541	+	0.191252i	$0.0612549\pi$
$509$			8.62970i			0.382505i	−0.981541	+	0.191252i	$0.938745\pi$
$510$		0			0
$511$		−	30.7519i		−	1.36038i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			9.74573i			0.429448i
$516$		0			0
$517$			0.535725i			0.0235612i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6.09279			0.266930			0.133465	−	0.991054i	$-0.457390\pi$
$521$	6.09279			0.266930			0.133465	+	0.991054i	$0.457390\pi$
$522$		0			0
$523$		−	18.5658i		−	0.811826i	−0.913912	−	0.405913i	$-0.866954\pi$
$523$		−	18.5658i		−	0.811826i	0.913912	−	0.405913i	$-0.133046\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−37.2436			−1.62236
$528$		0			0
$529$	20.0848	+	11.2072i	0.873252	+	0.487268i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	2.73097i		−	0.118292i
$534$		0			0
$535$	10.0549			0.434713
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.78234			0.119844
$540$		0			0
$541$	−19.1292			−0.822427			−0.411213	−	0.911539i	$-0.634895\pi$
$541$	−19.1292			−0.822427			−0.411213	+	0.911539i	$0.634895\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			15.9444i			0.682981i
$546$		0			0
$547$	−16.3021			−0.697028			−0.348514	−	0.937304i	$-0.613314\pi$
$547$	−16.3021			−0.697028			−0.348514	+	0.937304i	$0.613314\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	17.6590			0.752299
$552$		0			0
$553$	2.74675			0.116804
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−33.7685			−1.43082			−0.715409	−	0.698706i	$-0.753760\pi$
$557$	−33.7685			−1.43082			−0.715409	+	0.698706i	$0.753760\pi$
$558$		0			0
$559$		−	18.7662i		−	0.793725i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	22.4345			0.945501			0.472750	−	0.881196i	$-0.343261\pi$
$563$	22.4345			0.945501			0.472750	+	0.881196i	$0.343261\pi$
$564$		0			0
$565$	−10.4113			−0.438005
$566$		0			0
$567$		0			0
$568$		0			0
$569$	27.0265			1.13301			0.566505	−	0.824059i	$-0.308295\pi$
$569$	27.0265			1.13301			0.566505	+	0.824059i	$0.308295\pi$
$570$		0			0
$571$		−	10.7398i		−	0.449449i	−0.974422	−	0.224724i	$-0.927852\pi$
$571$		−	10.7398i		−	0.449449i	0.974422	−	0.224724i	$-0.0721482\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−4.64138	−	1.20731i	−0.193559	−	0.0503483i
$576$		0			0
$577$	−8.54243			−0.355626			−0.177813	−	0.984064i	$-0.556902\pi$
$577$	−8.54243			−0.355626			−0.177813	+	0.984064i	$0.556902\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			11.7541i			0.487642i
$582$		0			0
$583$	13.7673			0.570183
$584$		0			0
$585$		0			0
$586$		0			0
$587$			29.1545i			1.20334i	0.798746	+	0.601668i	$0.205497\pi$
$587$			29.1545i			1.20334i	−0.798746	+	0.601668i	$0.794503\pi$
$588$		0			0
$589$		−	12.2437i		−	0.504492i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			3.40263i			0.139729i	0.997556	+	0.0698645i	$0.0222567\pi$
$593$			3.40263i			0.139729i	−0.997556	+	0.0698645i	$0.977743\pi$
$594$		0			0
$595$			20.0340i			0.821314i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	4.55038i		−	0.185924i	−0.995670	−	0.0929618i	$-0.970367\pi$
$599$		−	4.55038i		−	0.185924i	0.995670	−	0.0929618i	$-0.0296334\pi$
$600$		0			0
$601$	−4.01391			−0.163731			−0.0818655	−	0.996643i	$-0.526088\pi$
$601$	−4.01391			−0.163731			−0.0818655	+	0.996643i	$0.526088\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−27.2654			−1.10849
$606$		0			0
$607$	−24.0608			−0.976599			−0.488299	−	0.872676i	$-0.662383\pi$
$607$	−24.0608			−0.976599			−0.488299	+	0.872676i	$0.662383\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			0.135793i			0.00549359i
$612$		0			0
$613$			24.2044i			0.977606i	0.872394	+	0.488803i	$0.162566\pi$
$613$			24.2044i			0.977606i	−0.872394	+	0.488803i	$0.837434\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	44.3709			1.78631			0.893153	−	0.449753i	$-0.148488\pi$
$617$	44.3709			1.78631			0.893153	+	0.449753i	$0.148488\pi$
$618$		0			0
$619$		−	33.0294i		−	1.32756i	−0.747926	−	0.663782i	$-0.768950\pi$
$619$		−	33.0294i		−	1.32756i	0.747926	−	0.663782i	$-0.231050\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	10.7530i		−	0.430810i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	65.4451i		−	2.60947i
$630$		0			0
$631$			2.46903i			0.0982905i	0.998792	+	0.0491452i	$0.0156497\pi$
$631$			2.46903i			0.0982905i	−0.998792	+	0.0491452i	$0.984350\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	3.05363			0.121180
$636$		0			0
$637$	0.705252			0.0279431
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−34.7810			−1.37377			−0.686883	−	0.726768i	$-0.741021\pi$
$641$	−34.7810			−1.37377			−0.686883	+	0.726768i	$0.741021\pi$
$642$		0			0
$643$			34.0505i			1.34282i	0.741087	+	0.671409i	$0.234311\pi$
$643$			34.0505i			1.34282i	−0.741087	+	0.671409i	$0.765689\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			8.32463i			0.327275i	0.986521	+	0.163637i	$0.0523227\pi$
$647$			8.32463i			0.327275i	−0.986521	+	0.163637i	$0.947677\pi$
$648$		0			0
$649$		−	0.635941i		−	0.0249629i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	8.54152i		−	0.334256i	−0.985935	−	0.167128i	$-0.946551\pi$
$653$		−	8.54152i		−	0.334256i	0.985935	−	0.167128i	$-0.0534493\pi$
$654$		0			0
$655$		−	7.35861i		−	0.287525i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.56783			0.138983			0.0694914	−	0.997583i	$-0.477862\pi$
$659$	3.56783			0.138983			0.0694914	+	0.997583i	$0.477862\pi$
$660$		0			0
$661$		−	0.357224i		−	0.0138944i	−0.999976	−	0.00694721i	$-0.997789\pi$
$661$		−	0.357224i		−	0.0138944i	0.999976	−	0.00694721i	$-0.00221138\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.58609			−0.255398
$666$		0			0
$667$	8.83547	−	33.9670i	0.342111	−	1.31521i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			64.5517i			2.49199i
$672$		0			0
$673$	11.4213			0.440260			0.220130	−	0.975471i	$-0.429352\pi$
$673$	11.4213			0.440260			0.220130	+	0.975471i	$0.429352\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−29.2218			−1.12308			−0.561542	−	0.827449i	$-0.689792\pi$
$677$	−29.2218			−1.12308			−0.561542	+	0.827449i	$0.689792\pi$
$678$		0			0
$679$	40.8524			1.56777
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	0.467688i		−	0.0178956i	−0.999960	−	0.00894780i	$-0.997152\pi$
$683$		−	0.467688i		−	0.0178956i	0.999960	−	0.00894780i	$-0.00284821\pi$
$684$		0			0
$685$	7.68321			0.293561
$686$		0			0
$687$		0			0
$688$		0			0
$689$	3.48966			0.132945
$690$		0			0
$691$	−19.9850			−0.760265			−0.380133	−	0.924932i	$-0.624122\pi$
$691$	−19.9850			−0.760265			−0.380133	+	0.924932i	$0.624122\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	17.6844			0.670808
$696$		0			0
$697$		−	12.7843i		−	0.484239i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−13.4329			−0.507354			−0.253677	−	0.967289i	$-0.581640\pi$
$701$	−13.4329			−0.507354			−0.253677	+	0.967289i	$0.581640\pi$
$702$		0			0
$703$	21.5148			0.811446
$704$		0			0
$705$		0			0
$706$		0			0
$707$	33.0321			1.24230
$708$		0			0
$709$		−	29.9396i		−	1.12440i	−0.827000	−	0.562202i	$-0.809954\pi$
$709$		−	29.9396i		−	1.12440i	0.827000	−	0.562202i	$-0.190046\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−23.5506	−	6.12597i	−0.881979	−	0.229419i
$714$		0			0
$715$	−9.69930			−0.362733
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	29.0706i		−	1.08415i	−0.840330	−	0.542075i	$-0.817639\pi$
$719$		−	29.0706i		−	1.08415i	0.840330	−	0.542075i	$-0.182361\pi$
$720$		0			0
$721$	26.6003			0.990647
$722$		0			0
$723$		0			0
$724$		0			0
$725$			7.31830i			0.271795i
$726$		0			0
$727$			28.5122i			1.05746i	0.848791	+	0.528729i	$0.177331\pi$
$727$			28.5122i			1.05746i	−0.848791	+	0.528729i	$0.822669\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	87.8485i		−	3.24919i
$732$		0			0
$733$		−	21.0285i		−	0.776705i	−0.921511	−	0.388353i	$-0.873044\pi$
$733$		−	21.0285i		−	0.776705i	0.921511	−	0.388353i	$-0.126956\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			52.3674i			1.92898i
$738$		0			0
$739$	24.6546			0.906932			0.453466	−	0.891273i	$-0.350187\pi$
$739$	24.6546			0.906932			0.453466	+	0.891273i	$0.350187\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	11.5216			0.422685			0.211342	−	0.977412i	$-0.432216\pi$
$743$	11.5216			0.422685			0.211342	+	0.977412i	$0.432216\pi$
$744$		0			0
$745$	−24.2027			−0.886718
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	27.4443i		−	1.00279i
$750$		0			0
$751$			17.5858i			0.641715i	0.947127	+	0.320858i	$0.103971\pi$
$751$			17.5858i			0.641715i	−0.947127	+	0.320858i	$0.896029\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−10.3560			−0.376893
$756$		0			0
$757$		−	4.47669i		−	0.162708i	−0.996685	−	0.0813540i	$-0.974076\pi$
$757$		−	4.47669i		−	0.162708i	0.996685	−	0.0813540i	$-0.0259244\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			28.4348i			1.03076i	0.856962	+	0.515380i	$0.172349\pi$
$761$			28.4348i			1.03076i	−0.856962	+	0.515380i	$0.827651\pi$
$762$		0			0
$763$	43.5190			1.57549
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	0.161195i		−	0.00582041i
$768$		0			0
$769$			1.96762i			0.0709542i	0.999370	+	0.0354771i	$0.0112951\pi$
$769$			1.96762i			0.0709542i	−0.999370	+	0.0354771i	$0.988705\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	9.57418			0.344359			0.172180	−	0.985066i	$-0.444919\pi$
$773$	9.57418			0.344359			0.172180	+	0.985066i	$0.444919\pi$
$774$		0			0
$775$	5.07406			0.182266
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4.20277			0.150580
$780$		0			0
$781$		−	77.1522i		−	2.76072i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	5.29703i		−	0.189059i
$786$		0			0
$787$			24.5145i			0.873846i	0.899499	+	0.436923i	$0.143932\pi$
$787$			24.5145i			0.873846i	−0.899499	+	0.436923i	$0.856068\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			28.4168i			1.01039i
$792$		0			0
$793$			16.3622i			0.581040i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−23.7348			−0.840730			−0.420365	−	0.907355i	$-0.638098\pi$
$797$	−23.7348			−0.840730			−0.420365	+	0.907355i	$0.638098\pi$
$798$		0			0
$799$			0.635675i			0.0224886i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	69.6952			2.45949
$804$		0			0
$805$	−3.29527	+	12.6683i	−0.116143	+	0.446500i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			20.3700i			0.716173i	0.933688	+	0.358086i	$0.116571\pi$
$809$			20.3700i			0.716173i	−0.933688	+	0.358086i	$0.883429\pi$
$810$		0			0
$811$	−29.7817			−1.04578			−0.522889	−	0.852401i	$-0.675146\pi$
$811$	−29.7817			−1.04578			−0.522889	+	0.852401i	$0.675146\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	4.95215			0.173466
$816$		0			0
$817$	28.8798			1.01038
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	39.1570i		−	1.36659i	−0.730144	−	0.683294i	$-0.760547\pi$
$821$		−	39.1570i		−	1.36659i	0.730144	−	0.683294i	$-0.239453\pi$
$822$		0			0
$823$	−25.7551			−0.897765			−0.448882	−	0.893591i	$-0.648178\pi$
$823$	−25.7551			−0.897765			−0.448882	+	0.893591i	$0.648178\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−52.0640			−1.81044			−0.905221	−	0.424940i	$-0.860295\pi$
$827$	−52.0640			−1.81044			−0.905221	+	0.424940i	$0.860295\pi$
$828$		0			0
$829$	−39.2168			−1.36205			−0.681027	−	0.732258i	$-0.738466\pi$
$829$	−39.2168			−1.36205			−0.681027	+	0.732258i	$0.738466\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.30143			0.114388
$834$		0			0
$835$		−	8.20101i		−	0.283808i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.94263			0.170639			0.0853193	−	0.996354i	$-0.472809\pi$
$839$	4.94263			0.170639			0.0853193	+	0.996354i	$0.472809\pi$
$840$		0			0
$841$	−24.5575			−0.846812
$842$		0			0
$843$		0			0
$844$		0			0
$845$	10.5415			0.362638
$846$		0			0
$847$			74.4189i			2.55706i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	10.7647	−	41.3836i	0.369008	−	1.41861i
$852$		0			0
$853$	−55.6499			−1.90542			−0.952709	−	0.303885i	$-0.901716\pi$
$853$	−55.6499			−1.90542			−0.952709	+	0.303885i	$0.901716\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	7.55971i		−	0.258235i	−0.991629	−	0.129117i	$-0.958786\pi$
$857$		−	7.55971i		−	0.258235i	0.991629	−	0.129117i	$-0.0412144\pi$
$858$		0			0
$859$	9.60679			0.327779			0.163890	−	0.986479i	$-0.447596\pi$
$859$	9.60679			0.327779			0.163890	+	0.986479i	$0.447596\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			35.6954i			1.21508i	0.794287	+	0.607542i	$0.207844\pi$
$863$			35.6954i			1.21508i	−0.794287	+	0.607542i	$0.792156\pi$
$864$		0			0
$865$			18.3258i			0.623095i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			6.22515i			0.211174i
$870$		0			0
$871$			13.2738i			0.449766i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	2.72943i		−	0.0922716i
$876$		0			0
$877$	−47.6491			−1.60900			−0.804498	−	0.593955i	$-0.797566\pi$
$877$	−47.6491			−1.60900			−0.804498	+	0.593955i	$0.797566\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	9.37496			0.315851			0.157925	−	0.987451i	$-0.449519\pi$
$881$	9.37496			0.315851			0.157925	+	0.987451i	$0.449519\pi$
$882$		0			0
$883$	37.2157			1.25241			0.626203	−	0.779660i	$-0.284608\pi$
$883$	37.2157			1.25241			0.626203	+	0.779660i	$0.284608\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			4.34045i			0.145738i	0.997342	+	0.0728690i	$0.0232155\pi$
$887$			4.34045i			0.145738i	−0.997342	+	0.0728690i	$0.976785\pi$
$888$		0			0
$889$		−	8.33467i		−	0.279536i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−0.208975			−0.00699310
$894$		0			0
$895$			6.21672i			0.207802i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			37.1335i			1.23847i
$900$		0			0
$901$	16.3358			0.544226
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	10.4391i		−	0.347007i
$906$		0			0
$907$			54.0040i			1.79317i	0.442869	+	0.896586i	$0.353961\pi$
$907$			54.0040i			1.79317i	−0.442869	+	0.896586i	$0.646039\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−34.9003			−1.15630			−0.578149	−	0.815931i	$-0.696225\pi$
$911$	−34.9003			−1.15630			−0.578149	+	0.815931i	$0.696225\pi$
$912$		0			0
$913$	−26.6391			−0.881626
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−20.0848			−0.663259
$918$		0			0
$919$		−	45.4629i		−	1.49968i	−0.661618	−	0.749841i	$-0.730130\pi$
$919$		−	45.4629i		−	1.49968i	0.661618	−	0.749841i	$-0.269870\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	19.5561i		−	0.643698i
$924$		0			0
$925$			8.91623i			0.293164i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	27.0106i		−	0.886189i	−0.896475	−	0.443095i	$-0.853881\pi$
$929$		−	27.0106i		−	0.886189i	0.896475	−	0.443095i	$-0.146119\pi$
$930$		0			0
$931$			1.08533i			0.0355703i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−45.4045			−1.48489
$936$		0			0
$937$			3.53725i			0.115557i	0.998329	+	0.0577785i	$0.0184017\pi$
$937$			3.53725i			0.115557i	−0.998329	+	0.0577785i	$0.981598\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−47.1381			−1.53666			−0.768330	−	0.640054i	$-0.778912\pi$
$941$	−47.1381			−1.53666			−0.768330	+	0.640054i	$0.778912\pi$
$942$		0			0
$943$	2.10281	−	8.08401i	0.0684768	−	0.263252i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			32.7551i			1.06440i	0.846620	+	0.532198i	$0.178634\pi$
$947$			32.7551i			1.06440i	−0.846620	+	0.532198i	$0.821366\pi$
$948$		0			0
$949$	17.6660			0.573462
$950$		0			0
$951$		0			0
$952$		0			0
$953$	16.2297			0.525732			0.262866	−	0.964832i	$-0.415332\pi$
$953$	16.2297			0.525732			0.262866	+	0.964832i	$0.415332\pi$
$954$		0			0
$955$	−22.7744			−0.736963
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	20.9708i		−	0.677182i
$960$		0			0
$961$	−5.25389			−0.169480
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.6166			0.534909
$966$		0			0
$967$	−13.2331			−0.425547			−0.212774	−	0.977102i	$-0.568250\pi$
$967$	−13.2331			−0.425547			−0.212774	+	0.977102i	$0.568250\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	61.6495			1.97843			0.989213	−	0.146487i	$-0.0467965\pi$
$971$	61.6495			1.97843			0.989213	+	0.146487i	$0.0467965\pi$
$972$		0			0
$973$		−	48.2684i		−	1.54741i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	27.3250			0.874204			0.437102	−	0.899412i	$-0.356005\pi$
$977$	27.3250			0.874204			0.437102	+	0.899412i	$0.356005\pi$
$978$		0			0
$979$	24.3703			0.778878
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−12.6774			−0.404346			−0.202173	−	0.979350i	$-0.564800\pi$
$983$	−12.6774			−0.404346			−0.202173	+	0.979350i	$0.564800\pi$
$984$		0			0
$985$			18.7600i			0.597744i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	14.4497	−	55.5502i	0.459472	−	1.76639i
$990$		0			0
$991$	3.34958			0.106403			0.0532015	−	0.998584i	$-0.483057\pi$
$991$	3.34958			0.106403			0.0532015	+	0.998584i	$0.483057\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	22.2158i		−	0.704287i
$996$		0			0
$997$	0.651517			0.0206337			0.0103169	−	0.999947i	$-0.496716\pi$
$997$	0.651517			0.0206337			0.0103169	+	0.999947i	$0.496716\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.p.a.1241.37	yes	48
3.2	odd	2		8280.2.p.b.1241.37	yes	48
23.22	odd	2		8280.2.p.b.1241.12	yes	48
69.68	even	2	inner	8280.2.p.a.1241.12	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.12	✓	48	69.68	even	2	inner
8280.2.p.a.1241.37	yes	48	1.1	even	1	trivial
8280.2.p.b.1241.12	yes	48	23.22	odd	2
8280.2.p.b.1241.37	yes	48	3.2	odd	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.p (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.72943i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +2.72943i q^{7} -6.18590 q^{11} -1.56797 q^{13} -7.34000 q^{17} -2.41299i q^{19} +(-4.64138 - 1.20731i) q^{23} +1.00000 q^{25} +7.31830i q^{29} +5.07406 q^{31} -2.72943i q^{35} +8.91623i q^{37} +1.74173i q^{41} +11.9685i q^{43} -0.0866043i q^{47} -0.449787 q^{49} -2.22559 q^{53} +6.18590 q^{55} +0.102805i q^{59} -10.4353i q^{61} +1.56797 q^{65} -8.46561i q^{67} +12.4723i q^{71} -11.2668 q^{73} -16.8840i q^{77} -1.00635i q^{79} +4.30642 q^{83} +7.34000 q^{85} -3.93965 q^{89} -4.27966i q^{91} +2.41299i q^{95} -14.9674i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 48 * q - 48 * q^5	\( 48 q - 48 q^{5} - 8 q^{11} + 4 q^{23} + 48 q^{25} + 8 q^{31} - 32 q^{49} + 8 q^{55} - 16 q^{73} - 32 q^{83} - 8 q^{89}+O(q^{100}) \) 48 * q - 48 * q^5 - 8 * q^11 + 4 * q^23 + 48 * q^25 + 8 * q^31 - 32 * q^49 + 8 * q^55 - 16 * q^73 - 32 * q^83 - 8 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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