LMFDB - Embedded newform 8280.2.p.a.1241.9

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			1241.9
Character	$\chi$	$=$	8280.1241
Dual form			8280.2.p.a.1241.40

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.11398i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.11398i q^{7} +4.47827 q^{11} +3.14793 q^{13} +0.883019 q^{17} -5.10241i q^{19} +(2.92445 - 3.80100i) q^{23} +1.00000 q^{25} -6.05327i q^{29} +4.03536 q^{31} +3.11398i q^{35} -1.93092i q^{37} -7.43849i q^{41} -7.84218i q^{43} +10.5518i q^{47} -2.69687 q^{49} -0.0758038 q^{53} -4.47827 q^{55} +6.25983i q^{59} -10.8293i q^{61} -3.14793 q^{65} +16.2489i q^{67} +12.8342i q^{71} -7.24089 q^{73} -13.9452i q^{77} +3.66883i q^{79} -8.57498 q^{83} -0.883019 q^{85} +10.7476 q^{89} -9.80259i q^{91} +5.10241i q^{95} +7.27568i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$		−	3.11398i		−	1.17697i	−0.808507	−	0.588487i	$-0.799724\pi$
$7$		−	3.11398i		−	1.17697i	0.808507	−	0.588487i	$-0.200276\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	4.47827			1.35025			0.675124	−	0.737704i	$-0.264090\pi$
$11$	4.47827			1.35025			0.675124	+	0.737704i	$0.264090\pi$
$12$		0			0
$13$	3.14793			0.873079			0.436540	−	0.899685i	$-0.356204\pi$
$13$	3.14793			0.873079			0.436540	+	0.899685i	$0.356204\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	0.883019			0.214164			0.107082	−	0.994250i	$-0.465849\pi$
$17$	0.883019			0.214164			0.107082	+	0.994250i	$0.465849\pi$
$18$		0			0
$19$		−	5.10241i		−	1.17057i	−0.810827	−	0.585286i	$-0.800982\pi$
$19$		−	5.10241i		−	1.17057i	0.810827	−	0.585286i	$-0.199018\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.92445	−	3.80100i	0.609790	−	0.792563i
$23$	2.92445	−	3.80100i	0.609790	−	0.792563i
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.05327i		−	1.12406i	−0.827116	−	0.562032i	$-0.810020\pi$
$29$		−	6.05327i		−	1.12406i	0.827116	−	0.562032i	$-0.189980\pi$
$30$		0			0
$31$	4.03536			0.724772			0.362386	−	0.932028i	$-0.381962\pi$
$31$	4.03536			0.724772			0.362386	+	0.932028i	$0.381962\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			3.11398i			0.526359i
$36$		0			0
$37$		−	1.93092i		−	0.317442i	−0.987323	−	0.158721i	$-0.949263\pi$
$37$		−	1.93092i		−	0.317442i	0.987323	−	0.158721i	$-0.0507370\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	7.43849i		−	1.16170i	−0.814012	−	0.580849i	$-0.802721\pi$
$41$		−	7.43849i		−	1.16170i	0.814012	−	0.580849i	$-0.197279\pi$
$42$		0			0
$43$		−	7.84218i		−	1.19592i	−0.801526	−	0.597960i	$-0.795978\pi$
$43$		−	7.84218i		−	1.19592i	0.801526	−	0.597960i	$-0.204022\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			10.5518i			1.53913i	0.638566	+	0.769567i	$0.279528\pi$
$47$			10.5518i			1.53913i	−0.638566	+	0.769567i	$0.720472\pi$
$48$		0			0
$49$	−2.69687			−0.385267
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.0758038			−0.0104125			−0.00520623	−	0.999986i	$-0.501657\pi$
$53$	−0.0758038			−0.0104125			−0.00520623	+	0.999986i	$0.501657\pi$
$54$		0			0
$55$	−4.47827			−0.603849
$56$		0			0
$57$		0			0
$58$		0			0
$59$			6.25983i			0.814961i	0.913214	+	0.407480i	$0.133593\pi$
$59$			6.25983i			0.814961i	−0.913214	+	0.407480i	$0.866407\pi$
$60$		0			0
$61$		−	10.8293i		−	1.38656i	−0.720670	−	0.693278i	$-0.756166\pi$
$61$		−	10.8293i		−	1.38656i	0.720670	−	0.693278i	$-0.243834\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−3.14793			−0.390453
$66$		0			0
$67$			16.2489i			1.98511i	0.121784	+	0.992557i	$0.461139\pi$
$67$			16.2489i			1.98511i	−0.121784	+	0.992557i	$0.538861\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.8342i			1.52314i	0.648083	+	0.761569i	$0.275571\pi$
$71$			12.8342i			1.52314i	−0.648083	+	0.761569i	$0.724429\pi$
$72$		0			0
$73$	−7.24089			−0.847482			−0.423741	−	0.905783i	$-0.639283\pi$
$73$	−7.24089			−0.847482			−0.423741	+	0.905783i	$0.639283\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	13.9452i		−	1.58921i
$78$		0			0
$79$			3.66883i			0.412775i	0.978470	+	0.206388i	$0.0661708\pi$
$79$			3.66883i			0.412775i	−0.978470	+	0.206388i	$0.933829\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−8.57498			−0.941227			−0.470613	−	0.882340i	$-0.655967\pi$
$83$	−8.57498			−0.941227			−0.470613	+	0.882340i	$0.655967\pi$
$84$		0			0
$85$	−0.883019			−0.0957769
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.7476			1.13925			0.569624	−	0.821905i	$-0.307089\pi$
$89$	10.7476			1.13925			0.569624	+	0.821905i	$0.307089\pi$
$90$		0			0
$91$		−	9.80259i		−	1.02759i
$92$		0			0
$93$		0			0
$94$		0			0
$95$			5.10241i			0.523496i
$96$		0			0
$97$			7.27568i			0.738733i	0.929284	+	0.369367i	$0.120425\pi$
$97$			7.27568i			0.738733i	−0.929284	+	0.369367i	$0.879575\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			12.2057i			1.21451i	0.794507	+	0.607255i	$0.207729\pi$
$101$			12.2057i			1.21451i	−0.794507	+	0.607255i	$0.792271\pi$
$102$		0			0
$103$		−	19.1666i		−	1.88854i	−0.329176	−	0.944269i	$-0.606771\pi$
$103$		−	19.1666i		−	1.88854i	0.329176	−	0.944269i	$-0.393229\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	11.5897			1.12042			0.560210	−	0.828351i	$-0.310721\pi$
$107$	11.5897			1.12042			0.560210	+	0.828351i	$0.310721\pi$
$108$		0			0
$109$		−	2.10433i		−	0.201559i	−0.994909	−	0.100779i	$-0.967866\pi$
$109$		−	2.10433i		−	0.201559i	0.994909	−	0.100779i	$-0.0321336\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.701486			−0.0659903			−0.0329951	−	0.999456i	$-0.510505\pi$
$113$	−0.701486			−0.0659903			−0.0329951	+	0.999456i	$0.510505\pi$
$114$		0			0
$115$	−2.92445	+	3.80100i	−0.272706	+	0.354445i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	2.74970i		−	0.252065i
$120$		0			0
$121$	9.05486			0.823169
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	6.09452			0.540801			0.270401	−	0.962748i	$-0.412844\pi$
$127$	6.09452			0.540801			0.270401	+	0.962748i	$0.412844\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	8.02747i		−	0.701363i	−0.936495	−	0.350682i	$-0.885950\pi$
$131$		−	8.02747i		−	0.701363i	0.936495	−	0.350682i	$-0.114050\pi$
$132$		0			0
$133$	−15.8888			−1.37773
$134$		0			0
$135$		0			0
$136$		0			0
$137$	14.6895			1.25501			0.627504	−	0.778613i	$-0.284076\pi$
$137$	14.6895			1.25501			0.627504	+	0.778613i	$0.284076\pi$
$138$		0			0
$139$	−12.7508			−1.08151			−0.540753	−	0.841181i	$-0.681861\pi$
$139$	−12.7508			−1.08151			−0.540753	+	0.841181i	$0.681861\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	14.0973			1.17887
$144$		0			0
$145$			6.05327i			0.502696i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	16.5738			1.35778			0.678890	−	0.734240i	$-0.262461\pi$
$149$	16.5738			1.35778			0.678890	+	0.734240i	$0.262461\pi$
$150$		0			0
$151$	−22.4742			−1.82892			−0.914462	−	0.404672i	$-0.867386\pi$
$151$	−22.4742			−1.82892			−0.914462	+	0.404672i	$0.867386\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.03536			−0.324128
$156$		0			0
$157$			13.7981i			1.10121i	0.834767	+	0.550604i	$0.185602\pi$
$157$			13.7981i			1.10121i	−0.834767	+	0.550604i	$0.814398\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−11.8362	−	9.10668i	−0.932826	−	0.717707i
$162$		0			0
$163$	15.3556			1.20275			0.601373	−	0.798968i	$-0.294620\pi$
$163$	15.3556			1.20275			0.601373	+	0.798968i	$0.294620\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	15.6063i		−	1.20766i	−0.797115	−	0.603828i	$-0.793641\pi$
$167$		−	15.6063i		−	1.20766i	0.797115	−	0.603828i	$-0.206359\pi$
$168$		0			0
$169$	−3.09053			−0.237733
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	10.4952i		−	0.797939i	−0.916964	−	0.398969i	$-0.869368\pi$
$173$		−	10.4952i		−	0.797939i	0.916964	−	0.398969i	$-0.130632\pi$
$174$		0			0
$175$		−	3.11398i		−	0.235395i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			4.51221i			0.337258i	0.985680	+	0.168629i	$0.0539340\pi$
$179$			4.51221i			0.337258i	−0.985680	+	0.168629i	$0.946066\pi$
$180$		0			0
$181$			6.74041i			0.501010i	0.968115	+	0.250505i	$0.0805967\pi$
$181$			6.74041i			0.501010i	−0.968115	+	0.250505i	$0.919403\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			1.93092i			0.141964i
$186$		0			0
$187$	3.95440			0.289174
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.96194			0.648463			0.324232	−	0.945978i	$-0.394894\pi$
$191$	8.96194			0.648463			0.324232	+	0.945978i	$0.394894\pi$
$192$		0			0
$193$	−6.83269			−0.491828			−0.245914	−	0.969292i	$-0.579088\pi$
$193$	−6.83269			−0.491828			−0.245914	+	0.969292i	$0.579088\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			20.0652i			1.42959i	0.699335	+	0.714794i	$0.253479\pi$
$197$			20.0652i			1.42959i	−0.699335	+	0.714794i	$0.746521\pi$
$198$		0			0
$199$		−	4.10760i		−	0.291180i	−0.989345	−	0.145590i	$-0.953492\pi$
$199$		−	4.10760i		−	0.291180i	0.989345	−	0.145590i	$-0.0465081\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−18.8497			−1.32299
$204$		0			0
$205$			7.43849i			0.519527i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	22.8499i		−	1.58056i
$210$		0			0
$211$	−21.5279			−1.48204			−0.741020	−	0.671483i	$-0.765658\pi$
$211$	−21.5279			−1.48204			−0.741020	+	0.671483i	$0.765658\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			7.84218i			0.534832i
$216$		0			0
$217$		−	12.5660i		−	0.853037i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2.77968			0.186982
$222$		0			0
$223$	−17.9075			−1.19917			−0.599586	−	0.800310i	$-0.704668\pi$
$223$	−17.9075			−1.19917			−0.599586	+	0.800310i	$0.704668\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	20.7605			1.37792			0.688962	−	0.724797i	$-0.258066\pi$
$227$	20.7605			1.37792			0.688962	+	0.724797i	$0.258066\pi$
$228$		0			0
$229$		−	10.4598i		−	0.691202i	−0.938382	−	0.345601i	$-0.887675\pi$
$229$		−	10.4598i		−	0.691202i	0.938382	−	0.345601i	$-0.112325\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5.63036i		−	0.368857i	−0.982846	−	0.184429i	$-0.940957\pi$
$233$		−	5.63036i		−	0.368857i	0.982846	−	0.184429i	$-0.0590434\pi$
$234$		0			0
$235$		−	10.5518i		−	0.688321i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			5.16577i			0.334146i	0.985945	+	0.167073i	$0.0534316\pi$
$239$			5.16577i			0.334146i	−0.985945	+	0.167073i	$0.946568\pi$
$240$		0			0
$241$		−	14.8074i		−	0.953826i	−0.878950	−	0.476913i	$-0.841756\pi$
$241$		−	14.8074i		−	0.953826i	0.878950	−	0.476913i	$-0.158244\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.69687			0.172296
$246$		0			0
$247$		−	16.0620i		−	1.02200i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.5873			−0.983862			−0.491931	−	0.870634i	$-0.663709\pi$
$251$	−15.5873			−0.983862			−0.491931	+	0.870634i	$0.663709\pi$
$252$		0			0
$253$	13.0965	−	17.0219i	0.823368	−	1.07016i
$254$		0			0
$255$		0			0
$256$		0			0
$257$			17.0842i			1.06568i	0.846215	+	0.532842i	$0.178876\pi$
$257$			17.0842i			1.06568i	−0.846215	+	0.532842i	$0.821124\pi$
$258$		0			0
$259$	−6.01286			−0.373621
$260$		0			0
$261$		0			0
$262$		0			0
$263$	14.4715			0.892349			0.446174	−	0.894946i	$-0.352786\pi$
$263$	14.4715			0.892349			0.446174	+	0.894946i	$0.352786\pi$
$264$		0			0
$265$	0.0758038			0.00465659
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	31.2935i		−	1.90800i	−0.299805	−	0.954001i	$-0.596922\pi$
$269$		−	31.2935i		−	1.90800i	0.299805	−	0.954001i	$-0.403078\pi$
$270$		0			0
$271$	18.7146			1.13683			0.568416	−	0.822741i	$-0.307556\pi$
$271$	18.7146			1.13683			0.568416	+	0.822741i	$0.307556\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	4.47827			0.270050
$276$		0			0
$277$	23.5938			1.41761			0.708806	−	0.705404i	$-0.249234\pi$
$277$	23.5938			1.41761			0.708806	+	0.705404i	$0.249234\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−29.2398			−1.74430			−0.872150	−	0.489238i	$-0.837275\pi$
$281$	−29.2398			−1.74430			−0.872150	+	0.489238i	$0.837275\pi$
$282$		0			0
$283$			29.4742i			1.75206i	0.482256	+	0.876030i	$0.339818\pi$
$283$			29.4742i			1.75206i	−0.482256	+	0.876030i	$0.660182\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−23.1633			−1.36729
$288$		0			0
$289$	−16.2203			−0.954134
$290$		0			0
$291$		0			0
$292$		0			0
$293$	10.2625			0.599542			0.299771	−	0.954011i	$-0.403090\pi$
$293$	10.2625			0.599542			0.299771	+	0.954011i	$0.403090\pi$
$294$		0			0
$295$		−	6.25983i		−	0.364462i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	9.20597	−	11.9653i	0.532395	−	0.691970i
$300$		0			0
$301$	−24.4204			−1.40757
$302$		0			0
$303$		0			0
$304$		0			0
$305$			10.8293i			0.620087i
$306$		0			0
$307$	−21.1070			−1.20464			−0.602319	−	0.798255i	$-0.705757\pi$
$307$	−21.1070			−1.20464			−0.602319	+	0.798255i	$0.705757\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	32.2688i		−	1.82980i	−0.403683	−	0.914899i	$-0.632270\pi$
$311$		−	32.2688i		−	1.82980i	0.403683	−	0.914899i	$-0.367730\pi$
$312$		0			0
$313$		−	5.35928i		−	0.302925i	−0.988463	−	0.151462i	$-0.951602\pi$
$313$		−	5.35928i		−	0.302925i	0.988463	−	0.151462i	$-0.0483982\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4.15552i			0.233397i	0.993167	+	0.116699i	$0.0372312\pi$
$317$			4.15552i			0.233397i	−0.993167	+	0.116699i	$0.962769\pi$
$318$		0			0
$319$		−	27.1081i		−	1.51776i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	4.50553i		−	0.250694i
$324$		0			0
$325$	3.14793			0.174616
$326$		0			0
$327$		0			0
$328$		0			0
$329$	32.8580			1.81152
$330$		0			0
$331$	−28.9052			−1.58878			−0.794388	−	0.607411i	$-0.792208\pi$
$331$	−28.9052			−1.58878			−0.794388	+	0.607411i	$0.792208\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	16.2489i		−	0.887770i
$336$		0			0
$337$			10.6804i			0.581796i	0.956754	+	0.290898i	$0.0939541\pi$
$337$			10.6804i			0.581796i	−0.956754	+	0.290898i	$0.906046\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	18.0714			0.978622
$342$		0			0
$343$		−	13.3999i		−	0.723525i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.98373i		−	0.428589i	−0.976769	−	0.214295i	$-0.931255\pi$
$347$		−	7.98373i		−	0.428589i	0.976769	−	0.214295i	$-0.0687453\pi$
$348$		0			0
$349$	−6.02293			−0.322400			−0.161200	−	0.986922i	$-0.551536\pi$
$349$	−6.02293			−0.322400			−0.161200	+	0.986922i	$0.551536\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	13.9415i		−	0.742029i	−0.928627	−	0.371015i	$-0.879010\pi$
$353$		−	13.9415i		−	0.742029i	0.928627	−	0.371015i	$-0.120990\pi$
$354$		0			0
$355$		−	12.8342i		−	0.681168i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−26.6994			−1.40914			−0.704570	−	0.709634i	$-0.748860\pi$
$359$	−26.6994			−1.40914			−0.704570	+	0.709634i	$0.748860\pi$
$360$		0			0
$361$	−7.03457			−0.370241
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.24089			0.379005
$366$		0			0
$367$		−	1.67067i		−	0.0872085i	−0.999049	−	0.0436042i	$-0.986116\pi$
$367$		−	1.67067i		−	0.0872085i	0.999049	−	0.0436042i	$-0.0138841\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			0.236052i			0.0122552i
$372$		0			0
$373$		−	22.2414i		−	1.15161i	−0.817586	−	0.575807i	$-0.804688\pi$
$373$		−	22.2414i		−	1.15161i	0.817586	−	0.575807i	$-0.195312\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	19.0553i		−	0.981396i
$378$		0			0
$379$		−	3.40624i		−	0.174967i	−0.996166	−	0.0874835i	$-0.972118\pi$
$379$		−	3.40624i		−	0.174967i	0.996166	−	0.0874835i	$-0.0278825\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−16.9640			−0.866819			−0.433410	−	0.901197i	$-0.642690\pi$
$383$	−16.9640			−0.866819			−0.433410	+	0.901197i	$0.642690\pi$
$384$		0			0
$385$			13.9452i			0.710714i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	8.77591			0.444957			0.222478	−	0.974938i	$-0.428585\pi$
$389$	8.77591			0.444957			0.222478	+	0.974938i	$0.428585\pi$
$390$		0			0
$391$	2.58235	−	3.35636i	0.130595	−	0.169738i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	3.66883i		−	0.184599i
$396$		0			0
$397$	−26.8283			−1.34647			−0.673237	−	0.739426i	$-0.735097\pi$
$397$	−26.8283			−1.34647			−0.673237	+	0.739426i	$0.735097\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.67443			0.333305			0.166653	−	0.986016i	$-0.446704\pi$
$401$	6.67443			0.333305			0.166653	+	0.986016i	$0.446704\pi$
$402$		0			0
$403$	12.7030			0.632783
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	8.64719i		−	0.428625i
$408$		0			0
$409$	−9.54089			−0.471767			−0.235883	−	0.971781i	$-0.575798\pi$
$409$	−9.54089			−0.471767			−0.235883	+	0.971781i	$0.575798\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	19.4930			0.959187
$414$		0			0
$415$	8.57498			0.420929
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−20.8453			−1.01836			−0.509179	−	0.860661i	$-0.670051\pi$
$419$	−20.8453			−1.01836			−0.509179	+	0.860661i	$0.670051\pi$
$420$		0			0
$421$		−	24.3811i		−	1.18826i	−0.804369	−	0.594130i	$-0.797496\pi$
$421$		−	24.3811i		−	1.18826i	0.804369	−	0.594130i	$-0.202504\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	0.883019			0.0428327
$426$		0			0
$427$	−33.7224			−1.63194
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−20.1910			−0.972568			−0.486284	−	0.873801i	$-0.661648\pi$
$431$	−20.1910			−0.972568			−0.486284	+	0.873801i	$0.661648\pi$
$432$		0			0
$433$			1.45488i			0.0699169i	0.999389	+	0.0349585i	$0.0111299\pi$
$433$			1.45488i			0.0699169i	−0.999389	+	0.0349585i	$0.988870\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−19.3942	−	14.9217i	−0.927753	−	0.713804i
$438$		0			0
$439$	41.6141			1.98613			0.993065	−	0.117565i	$-0.0375089\pi$
$439$	41.6141			1.98613			0.993065	+	0.117565i	$0.0375089\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	8.87180i		−	0.421512i	−0.977539	−	0.210756i	$-0.932407\pi$
$443$		−	8.87180i		−	0.421512i	0.977539	−	0.210756i	$-0.0675925\pi$
$444$		0			0
$445$	−10.7476			−0.509487
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.79633i		−	0.273546i	−0.990602	−	0.136773i	$-0.956327\pi$
$449$		−	5.79633i		−	0.273546i	0.990602	−	0.136773i	$-0.0436730\pi$
$450$		0			0
$451$		−	33.3115i		−	1.56858i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			9.80259i			0.459553i
$456$		0			0
$457$		−	0.893221i		−	0.0417831i	−0.999782	−	0.0208916i	$-0.993350\pi$
$457$		−	0.893221i		−	0.0417831i	0.999782	−	0.0208916i	$-0.00665047\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			8.70685i			0.405518i	0.979229	+	0.202759i	$0.0649909\pi$
$461$			8.70685i			0.405518i	−0.979229	+	0.202759i	$0.935009\pi$
$462$		0			0
$463$	2.04160			0.0948813			0.0474407	−	0.998874i	$-0.484893\pi$
$463$	2.04160			0.0948813			0.0474407	+	0.998874i	$0.484893\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18.3288			0.848157			0.424079	−	0.905625i	$-0.360598\pi$
$467$	18.3288			0.848157			0.424079	+	0.905625i	$0.360598\pi$
$468$		0			0
$469$	50.5986			2.33643
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	35.1193i		−	1.61479i
$474$		0			0
$475$		−	5.10241i		−	0.234115i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−3.12689			−0.142871			−0.0714356	−	0.997445i	$-0.522758\pi$
$479$	−3.12689			−0.142871			−0.0714356	+	0.997445i	$0.522758\pi$
$480$		0			0
$481$		−	6.07842i		−	0.277152i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	7.27568i		−	0.330371i
$486$		0			0
$487$	−18.3594			−0.831944			−0.415972	−	0.909378i	$-0.636558\pi$
$487$	−18.3594			−0.831944			−0.415972	+	0.909378i	$0.636558\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			35.6832i			1.61036i	0.593031	+	0.805179i	$0.297931\pi$
$491$			35.6832i			1.61036i	−0.593031	+	0.805179i	$0.702069\pi$
$492$		0			0
$493$		−	5.34515i		−	0.240734i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	39.9654			1.79269
$498$		0			0
$499$	−8.80282			−0.394068			−0.197034	−	0.980397i	$-0.563131\pi$
$499$	−8.80282			−0.394068			−0.197034	+	0.980397i	$0.563131\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	31.1071			1.38700			0.693498	−	0.720459i	$-0.256069\pi$
$503$	31.1071			1.38700			0.693498	+	0.720459i	$0.256069\pi$
$504$		0			0
$505$		−	12.2057i		−	0.543145i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	22.1731i		−	0.982804i	−0.870933	−	0.491402i	$-0.836485\pi$
$509$		−	22.1731i		−	0.982804i	0.870933	−	0.491402i	$-0.163515\pi$
$510$		0			0
$511$			22.5480i			0.997464i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			19.1666i			0.844580i
$516$		0			0
$517$			47.2536i			2.07821i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	24.5183			1.07417			0.537084	−	0.843529i	$-0.319526\pi$
$521$	24.5183			1.07417			0.537084	+	0.843529i	$0.319526\pi$
$522$		0			0
$523$			20.0475i			0.876617i	0.898825	+	0.438308i	$0.144422\pi$
$523$			20.0475i			0.876617i	−0.898825	+	0.438308i	$0.855578\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3.56330			0.155220
$528$		0			0
$529$	−5.89518	−	22.2317i	−0.256312	−	0.966594i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	23.4159i		−	1.01425i
$534$		0			0
$535$	−11.5897			−0.501067
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0773			−0.520205
$540$		0			0
$541$	41.3267			1.77677			0.888386	−	0.459097i	$-0.151827\pi$
$541$	41.3267			1.77677			0.888386	+	0.459097i	$0.151827\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			2.10433i			0.0901397i
$546$		0			0
$547$	−33.1246			−1.41631			−0.708154	−	0.706058i	$-0.750472\pi$
$547$	−33.1246			−1.41631			−0.708154	+	0.706058i	$0.750472\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−30.8862			−1.31580
$552$		0			0
$553$	11.4247			0.485826
$554$		0			0
$555$		0			0
$556$		0			0
$557$	38.3960			1.62689			0.813446	−	0.581640i	$-0.197589\pi$
$557$	38.3960			1.62689			0.813446	+	0.581640i	$0.197589\pi$
$558$		0			0
$559$		−	24.6866i		−	1.04413i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−0.320847			−0.0135221			−0.00676105	−	0.999977i	$-0.502152\pi$
$563$	−0.320847			−0.0135221			−0.00676105	+	0.999977i	$0.502152\pi$
$564$		0			0
$565$	0.701486			0.0295117
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−28.3121			−1.18691			−0.593453	−	0.804869i	$-0.702236\pi$
$569$	−28.3121			−1.18691			−0.593453	+	0.804869i	$0.702236\pi$
$570$		0			0
$571$		−	8.22703i		−	0.344291i	−0.985072	−	0.172145i	$-0.944930\pi$
$571$		−	8.22703i		−	0.344291i	0.985072	−	0.172145i	$-0.0550699\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.92445	−	3.80100i	0.121958	−	0.158513i
$576$		0			0
$577$	23.9294			0.996193			0.498097	−	0.867122i	$-0.334033\pi$
$577$	23.9294			0.996193			0.498097	+	0.867122i	$0.334033\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$			26.7023i			1.10780i
$582$		0			0
$583$	−0.339470			−0.0140594
$584$		0			0
$585$		0			0
$586$		0			0
$587$			7.66746i			0.316470i	0.987401	+	0.158235i	$0.0505804\pi$
$587$			7.66746i			0.316470i	−0.987401	+	0.158235i	$0.949420\pi$
$588$		0			0
$589$		−	20.5901i		−	0.848398i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.7517i			1.87880i	0.342830	+	0.939398i	$0.388615\pi$
$593$			45.7517i			1.87880i	−0.342830	+	0.939398i	$0.611385\pi$
$594$		0			0
$595$			2.74970i			0.112727i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			2.84552i			0.116265i	0.998309	+	0.0581324i	$0.0185145\pi$
$599$			2.84552i			0.116265i	−0.998309	+	0.0581324i	$0.981485\pi$
$600$		0			0
$601$	20.4884			0.835739			0.417870	−	0.908507i	$-0.362777\pi$
$601$	20.4884			0.835739			0.417870	+	0.908507i	$0.362777\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9.05486			−0.368132
$606$		0			0
$607$	26.9550			1.09407			0.547035	−	0.837110i	$-0.315757\pi$
$607$	26.9550			1.09407			0.547035	+	0.837110i	$0.315757\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			33.2162i			1.34379i
$612$		0			0
$613$			19.4529i			0.785694i	0.919604	+	0.392847i	$0.128510\pi$
$613$			19.4529i			0.785694i	−0.919604	+	0.392847i	$0.871490\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−13.6152			−0.548126			−0.274063	−	0.961712i	$-0.588368\pi$
$617$	−13.6152			−0.548126			−0.274063	+	0.961712i	$0.588368\pi$
$618$		0			0
$619$		−	39.0869i		−	1.57103i	−0.618839	−	0.785517i	$-0.712397\pi$
$619$		−	39.0869i		−	1.57103i	0.618839	−	0.785517i	$-0.287603\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	33.4679i		−	1.34086i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	1.70504i		−	0.0679845i
$630$		0			0
$631$		−	37.1522i		−	1.47901i	−0.673153	−	0.739503i	$-0.735061\pi$
$631$		−	37.1522i		−	1.47901i	0.673153	−	0.739503i	$-0.264939\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−6.09452			−0.241854
$636$		0			0
$637$	−8.48955			−0.336368
$638$		0			0
$639$		0			0
$640$		0			0
$641$	15.9575			0.630285			0.315142	−	0.949044i	$-0.397948\pi$
$641$	15.9575			0.630285			0.315142	+	0.949044i	$0.397948\pi$
$642$		0			0
$643$			3.44670i			0.135924i	0.997688	+	0.0679622i	$0.0216497\pi$
$643$			3.44670i			0.135924i	−0.997688	+	0.0679622i	$0.978350\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			35.9502i			1.41335i	0.707538	+	0.706675i	$0.249806\pi$
$647$			35.9502i			1.41335i	−0.707538	+	0.706675i	$0.750194\pi$
$648$		0			0
$649$			28.0332i			1.10040i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			50.0031i			1.95677i	0.206782	+	0.978387i	$0.433701\pi$
$653$			50.0031i			1.95677i	−0.206782	+	0.978387i	$0.566299\pi$
$654$		0			0
$655$			8.02747i			0.313659i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	3.00204			0.116943			0.0584714	−	0.998289i	$-0.481377\pi$
$659$	3.00204			0.116943			0.0584714	+	0.998289i	$0.481377\pi$
$660$		0			0
$661$		−	10.3208i		−	0.401434i	−0.979649	−	0.200717i	$-0.935673\pi$
$661$		−	10.3208i		−	0.401434i	0.979649	−	0.200717i	$-0.0643272\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	15.8888			0.616141
$666$		0			0
$667$	−23.0085	−	17.7025i	−0.890891	−	0.685442i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	48.4967i		−	1.87219i
$672$		0			0
$673$	15.3153			0.590362			0.295181	−	0.955441i	$-0.404620\pi$
$673$	15.3153			0.590362			0.295181	+	0.955441i	$0.404620\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	40.7480			1.56607			0.783036	−	0.621977i	$-0.213670\pi$
$677$	40.7480			1.56607			0.783036	+	0.621977i	$0.213670\pi$
$678$		0			0
$679$	22.6563			0.869469
$680$		0			0
$681$		0			0
$682$		0			0
$683$			8.22473i			0.314711i	0.987542	+	0.157355i	$0.0502968\pi$
$683$			8.22473i			0.314711i	−0.987542	+	0.157355i	$0.949703\pi$
$684$		0			0
$685$	−14.6895			−0.561257
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.238625			−0.00909090
$690$		0			0
$691$	12.1677			0.462880			0.231440	−	0.972849i	$-0.425656\pi$
$691$	12.1677			0.462880			0.231440	+	0.972849i	$0.425656\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	12.7508			0.483664
$696$		0			0
$697$		−	6.56833i		−	0.248793i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	29.3960			1.11027			0.555136	−	0.831760i	$-0.312666\pi$
$701$	29.3960			1.11027			0.555136	+	0.831760i	$0.312666\pi$
$702$		0			0
$703$	−9.85236			−0.371589
$704$		0			0
$705$		0			0
$706$		0			0
$707$	38.0082			1.42944
$708$		0			0
$709$			24.3197i			0.913345i	0.889635	+	0.456672i	$0.150959\pi$
$709$			24.3197i			0.913345i	−0.889635	+	0.456672i	$0.849041\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	11.8012	−	15.3384i	0.441959	−	0.574428i
$714$		0			0
$715$	−14.0973			−0.527208
$716$		0			0
$717$		0			0
$718$		0			0
$719$			49.4040i			1.84246i	0.389019	+	0.921230i	$0.372814\pi$
$719$			49.4040i			1.84246i	−0.389019	+	0.921230i	$0.627186\pi$
$720$		0			0
$721$	−59.6843			−2.22276
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	6.05327i		−	0.224813i
$726$		0			0
$727$		−	19.9760i		−	0.740869i	−0.928859	−	0.370434i	$-0.879209\pi$
$727$		−	19.9760i		−	0.740869i	0.928859	−	0.370434i	$-0.120791\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	6.92479i		−	0.256123i
$732$		0			0
$733$			23.8227i			0.879913i	0.898019	+	0.439957i	$0.145006\pi$
$733$			23.8227i			0.879913i	−0.898019	+	0.439957i	$0.854994\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			72.7667i			2.68039i
$738$		0			0
$739$	40.2757			1.48157			0.740784	−	0.671744i	$-0.234454\pi$
$739$	40.2757			1.48157			0.740784	+	0.671744i	$0.234454\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−31.0024			−1.13737			−0.568684	−	0.822556i	$-0.692547\pi$
$743$	−31.0024			−1.13737			−0.568684	+	0.822556i	$0.692547\pi$
$744$		0			0
$745$	−16.5738			−0.607218
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	36.0901i		−	1.31870i
$750$		0			0
$751$		−	21.1572i		−	0.772037i	−0.922491	−	0.386019i	$-0.873850\pi$
$751$		−	21.1572i		−	0.772037i	0.922491	−	0.386019i	$-0.126150\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	22.4742			0.817920
$756$		0			0
$757$		−	6.67631i		−	0.242655i	−0.992613	−	0.121327i	$-0.961285\pi$
$757$		−	6.67631i		−	0.242655i	0.992613	−	0.121327i	$-0.0387151\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			30.0783i			1.09034i	0.838326	+	0.545170i	$0.183535\pi$
$761$			30.0783i			1.09034i	−0.838326	+	0.545170i	$0.816465\pi$
$762$		0			0
$763$	−6.55285			−0.237229
$764$		0			0
$765$		0			0
$766$		0			0
$767$			19.7055i			0.711525i
$768$		0			0
$769$			27.0065i			0.973879i	0.873436	+	0.486940i	$0.161887\pi$
$769$			27.0065i			0.973879i	−0.873436	+	0.486940i	$0.838113\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	2.38625			0.0858273			0.0429137	−	0.999079i	$-0.486336\pi$
$773$	2.38625			0.0858273			0.0429137	+	0.999079i	$0.486336\pi$
$774$		0			0
$775$	4.03536			0.144954
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−37.9542			−1.35985
$780$		0			0
$781$			57.4749i			2.05661i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	13.7981i		−	0.492475i
$786$		0			0
$787$		−	34.7898i		−	1.24012i	−0.784553	−	0.620061i	$-0.787108\pi$
$787$		−	34.7898i		−	1.24012i	0.784553	−	0.620061i	$-0.212892\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			2.18441i			0.0776688i
$792$		0			0
$793$		−	34.0900i		−	1.21057i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.08953			0.109437			0.0547183	−	0.998502i	$-0.482574\pi$
$797$	3.08953			0.109437			0.0547183	+	0.998502i	$0.482574\pi$
$798$		0			0
$799$			9.31742i			0.329626i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−32.4266			−1.14431
$804$		0			0
$805$	11.8362	+	9.10668i	0.417172	+	0.320968i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	23.0248i		−	0.809509i	−0.914425	−	0.404754i	$-0.867357\pi$
$809$		−	23.0248i		−	0.809509i	0.914425	−	0.404754i	$-0.132643\pi$
$810$		0			0
$811$	−10.8349			−0.380464			−0.190232	−	0.981739i	$-0.560924\pi$
$811$	−10.8349			−0.380464			−0.190232	+	0.981739i	$0.560924\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.3556			−0.537885
$816$		0			0
$817$	−40.0140			−1.39991
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	49.1847i		−	1.71656i	−0.513184	−	0.858279i	$-0.671534\pi$
$821$		−	49.1847i		−	1.71656i	0.513184	−	0.858279i	$-0.328466\pi$
$822$		0			0
$823$	−16.1018			−0.561272			−0.280636	−	0.959814i	$-0.590545\pi$
$823$	−16.1018			−0.561272			−0.280636	+	0.959814i	$0.590545\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	37.8362			1.31569			0.657847	−	0.753151i	$-0.271467\pi$
$827$	37.8362			1.31569			0.657847	+	0.753151i	$0.271467\pi$
$828$		0			0
$829$	15.8212			0.549492			0.274746	−	0.961517i	$-0.411406\pi$
$829$	15.8212			0.549492			0.274746	+	0.961517i	$0.411406\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−2.38139			−0.0825101
$834$		0			0
$835$			15.6063i			0.540080i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−34.6495			−1.19623			−0.598117	−	0.801409i	$-0.704084\pi$
$839$	−34.6495			−1.19623			−0.598117	+	0.801409i	$0.704084\pi$
$840$		0			0
$841$	−7.64203			−0.263518
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3.09053			0.106317
$846$		0			0
$847$		−	28.1966i		−	0.968848i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.33944	−	5.64689i	−0.251593	−	0.193573i
$852$		0			0
$853$	32.6051			1.11638			0.558189	−	0.829714i	$-0.311496\pi$
$853$	32.6051			1.11638			0.558189	+	0.829714i	$0.311496\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			8.88951i			0.303660i	0.988407	+	0.151830i	$0.0485166\pi$
$857$			8.88951i			0.303660i	−0.988407	+	0.151830i	$0.951483\pi$
$858$		0			0
$859$	−16.8947			−0.576440			−0.288220	−	0.957564i	$-0.593063\pi$
$859$	−16.8947			−0.576440			−0.288220	+	0.957564i	$0.593063\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			8.95441i			0.304812i	0.988318	+	0.152406i	$0.0487021\pi$
$863$			8.95441i			0.304812i	−0.988318	+	0.152406i	$0.951298\pi$
$864$		0			0
$865$			10.4952i			0.356849i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			16.4300i			0.557349i
$870$		0			0
$871$			51.1503i			1.73316i
$872$		0			0
$873$		0			0
$874$		0			0
$875$			3.11398i			0.105272i
$876$		0			0
$877$	−12.5212			−0.422812			−0.211406	−	0.977398i	$-0.567804\pi$
$877$	−12.5212			−0.422812			−0.211406	+	0.977398i	$0.567804\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.49581			0.0503951			0.0251976	−	0.999682i	$-0.491979\pi$
$881$	1.49581			0.0503951			0.0251976	+	0.999682i	$0.491979\pi$
$882$		0			0
$883$	−30.1463			−1.01450			−0.507252	−	0.861798i	$-0.669339\pi$
$883$	−30.1463			−1.01450			−0.507252	+	0.861798i	$0.669339\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			34.1209i			1.14567i	0.819671	+	0.572834i	$0.194156\pi$
$887$			34.1209i			1.14567i	−0.819671	+	0.572834i	$0.805844\pi$
$888$		0			0
$889$		−	18.9782i		−	0.636509i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	53.8394			1.80167
$894$		0			0
$895$		−	4.51221i		−	0.150826i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	24.4271i		−	0.814690i
$900$		0			0
$901$	−0.0669363			−0.00222997
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	6.74041i		−	0.224059i
$906$		0			0
$907$			16.0295i			0.532250i	0.963939	+	0.266125i	$0.0857434\pi$
$907$			16.0295i			0.532250i	−0.963939	+	0.266125i	$0.914257\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	17.3730			0.575592			0.287796	−	0.957692i	$-0.407078\pi$
$911$	17.3730			0.575592			0.287796	+	0.957692i	$0.407078\pi$
$912$		0			0
$913$	−38.4010			−1.27089
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−24.9974			−0.825486
$918$		0			0
$919$			26.9824i			0.890066i	0.895514	+	0.445033i	$0.146808\pi$
$919$			26.9824i			0.890066i	−0.895514	+	0.445033i	$0.853192\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			40.4012i			1.32982i
$924$		0			0
$925$		−	1.93092i		−	0.0634884i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		−	48.3303i		−	1.58566i	−0.609440	−	0.792832i	$-0.708606\pi$
$929$		−	48.3303i		−	1.58566i	0.609440	−	0.792832i	$-0.291394\pi$
$930$		0			0
$931$			13.7605i			0.450983i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.95440			−0.129323
$936$		0			0
$937$		−	35.9801i		−	1.17542i	−0.809072	−	0.587709i	$-0.800030\pi$
$937$		−	35.9801i		−	1.17542i	0.809072	−	0.587709i	$-0.199970\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	41.7455			1.36087			0.680433	−	0.732811i	$-0.261792\pi$
$941$	41.7455			1.36087			0.680433	+	0.732811i	$0.261792\pi$
$942$		0			0
$943$	−28.2737	−	21.7535i	−0.920718	−	0.708391i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	11.9410i		−	0.388032i	−0.980998	−	0.194016i	$-0.937849\pi$
$947$		−	11.9410i		−	0.388032i	0.980998	−	0.194016i	$-0.0621513\pi$
$948$		0			0
$949$	−22.7938			−0.739919
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−49.2889			−1.59662			−0.798312	−	0.602244i	$-0.794273\pi$
$953$	−49.2889			−1.59662			−0.798312	+	0.602244i	$0.794273\pi$
$954$		0			0
$955$	−8.96194			−0.290002
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	45.7428i		−	1.47711i
$960$		0			0
$961$	−14.7159			−0.474706
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6.83269			0.219952
$966$		0			0
$967$	−56.1448			−1.80549			−0.902747	−	0.430172i	$-0.858453\pi$
$967$	−56.1448			−1.80549			−0.902747	+	0.430172i	$0.858453\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−19.5530			−0.627486			−0.313743	−	0.949508i	$-0.601583\pi$
$971$	−19.5530			−0.627486			−0.313743	+	0.949508i	$0.601583\pi$
$972$		0			0
$973$			39.7056i			1.27290i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	11.1198			0.355754			0.177877	−	0.984053i	$-0.443077\pi$
$977$	11.1198			0.355754			0.177877	+	0.984053i	$0.443077\pi$
$978$		0			0
$979$	48.1308			1.53827
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−13.7058			−0.437146			−0.218573	−	0.975821i	$-0.570140\pi$
$983$	−13.7058			−0.437146			−0.218573	+	0.975821i	$0.570140\pi$
$984$		0			0
$985$		−	20.0652i		−	0.639331i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−29.8081	−	22.9341i	−0.947843	−	0.729260i
$990$		0			0
$991$	18.4668			0.586616			0.293308	−	0.956018i	$-0.405244\pi$
$991$	18.4668			0.586616			0.293308	+	0.956018i	$0.405244\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			4.10760i			0.130220i
$996$		0			0
$997$	45.1282			1.42923			0.714613	−	0.699520i	$-0.246603\pi$
$997$	45.1282			1.42923			0.714613	+	0.699520i	$0.246603\pi$
$998$		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.1241.9	✓	48
3.2	odd	2		8280.2.p.b.1241.9	yes	48
23.22	odd	2		8280.2.p.b.1241.40	yes	48
69.68	even	2	inner	8280.2.p.a.1241.40	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.p.a.1241.9	✓	48	1.1	even	1	trivial
8280.2.p.a.1241.40	yes	48	69.68	even	2	inner
8280.2.p.b.1241.9	yes	48	3.2	odd	2
8280.2.p.b.1241.40	yes	48	23.22	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.11398i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.11398i q^{7} +4.47827 q^{11} +3.14793 q^{13} +0.883019 q^{17} -5.10241i q^{19} +(2.92445 - 3.80100i) q^{23} +1.00000 q^{25} -6.05327i q^{29} +4.03536 q^{31} +3.11398i q^{35} -1.93092i q^{37} -7.43849i q^{41} -7.84218i q^{43} +10.5518i q^{47} -2.69687 q^{49} -0.0758038 q^{53} -4.47827 q^{55} +6.25983i q^{59} -10.8293i q^{61} -3.14793 q^{65} +16.2489i q^{67} +12.8342i q^{71} -7.24089 q^{73} -13.9452i q^{77} +3.66883i q^{79} -8.57498 q^{83} -0.883019 q^{85} +10.7476 q^{89} -9.80259i q^{91} +5.10241i q^{95} +7.27568i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 48 q^{5}+O(q^{10}) \) 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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