LMFDB - Embedded newform 5520.2.be.c.1471.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1471,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 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("5520.1471");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.be (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:	$44.0774219157$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.12
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} +1.00000i q^{5} -2.13359 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} +1.00000i q^{5} -2.13359 q^{7} -1.00000 q^{9} +4.16538 q^{11} +2.81823 q^{13} -1.00000 q^{15} -1.82147i q^{17} -1.32589 q^{19} -2.13359i q^{21} +(4.63191 - 1.24316i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -4.92560 q^{29} -10.5363i q^{31} +4.16538i q^{33} -2.13359i q^{35} -4.95490i q^{37} +2.81823i q^{39} +0.656543 q^{41} -9.49430 q^{43} -1.00000i q^{45} -7.28458i q^{47} -2.44777 q^{49} +1.82147 q^{51} +7.24831i q^{53} +4.16538i q^{55} -1.32589i q^{57} -8.42939i q^{59} -3.54481i q^{61} +2.13359 q^{63} +2.81823i q^{65} -10.6962 q^{67} +(1.24316 + 4.63191i) q^{69} +0.897920i q^{71} +4.48027 q^{73} -1.00000i q^{75} -8.88724 q^{77} +7.73462 q^{79} +1.00000 q^{81} -3.14065 q^{83} +1.82147 q^{85} -4.92560i q^{87} -5.21158i q^{89} -6.01297 q^{91} +10.5363 q^{93} -1.32589i q^{95} -0.00964558i q^{97} -4.16538 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) $ 32 * q - 8 * q^7 - 32 * q^9	$ 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) $ 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5520\mathbb{Z}\right)^\times$.

$n$	$1201$	$1381$	$1841$	$4417$	$4831$
$\chi(n)$	$-1$	$1$	$1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$	−2.13359			−0.806423			−0.403211	−	0.915107i	$-0.632106\pi$
$7$	−2.13359			−0.806423			−0.403211	+	0.915107i	$0.632106\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	4.16538			1.25591			0.627955	−	0.778249i	$-0.283892\pi$
$11$	4.16538			1.25591			0.627955	+	0.778249i	$0.283892\pi$
$12$		0			0
$13$	2.81823			0.781638			0.390819	−	0.920468i	$-0.372192\pi$
$13$	2.81823			0.781638			0.390819	+	0.920468i	$0.372192\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$		−	1.82147i		−	0.441772i	−0.975300	−	0.220886i	$-0.929105\pi$
$17$		−	1.82147i		−	0.441772i	0.975300	−	0.220886i	$-0.0708948\pi$
$18$		0			0
$19$	−1.32589			−0.304180			−0.152090	−	0.988367i	$-0.548600\pi$
$19$	−1.32589			−0.304180			−0.152090	+	0.988367i	$0.548600\pi$
$20$		0			0
$21$		−	2.13359i		−	0.465589i
$22$		0			0
$23$	4.63191	−	1.24316i	0.965819	−	0.259217i
$23$	4.63191	−	1.24316i	0.965819	−	0.259217i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$	−4.92560			−0.914662			−0.457331	−	0.889297i	$-0.651194\pi$
$29$	−4.92560			−0.914662			−0.457331	+	0.889297i	$0.651194\pi$
$30$		0			0
$31$		−	10.5363i		−	1.89237i	−0.323628	−	0.946184i	$-0.604903\pi$
$31$		−	10.5363i		−	1.89237i	0.323628	−	0.946184i	$-0.395097\pi$
$32$		0			0
$33$			4.16538i			0.725100i
$34$		0			0
$35$		−	2.13359i		−	0.360643i
$36$		0			0
$37$		−	4.95490i		−	0.814580i	−0.913299	−	0.407290i	$-0.866474\pi$
$37$		−	4.95490i		−	0.814580i	0.913299	−	0.407290i	$-0.133526\pi$
$38$		0			0
$39$			2.81823i			0.451279i
$40$		0			0
$41$	0.656543			0.102535			0.0512674	−	0.998685i	$-0.483674\pi$
$41$	0.656543			0.102535			0.0512674	+	0.998685i	$0.483674\pi$
$42$		0			0
$43$	−9.49430			−1.44787			−0.723934	−	0.689870i	$-0.757668\pi$
$43$	−9.49430			−1.44787			−0.723934	+	0.689870i	$0.757668\pi$
$44$		0			0
$45$		−	1.00000i		−	0.149071i
$46$		0			0
$47$		−	7.28458i		−	1.06257i	−0.847195	−	0.531283i	$-0.821710\pi$
$47$		−	7.28458i		−	1.06257i	0.847195	−	0.531283i	$-0.178290\pi$
$48$		0			0
$49$	−2.44777			−0.349682
$50$		0			0
$51$	1.82147			0.255057
$52$		0			0
$53$			7.24831i			0.995631i	0.867283	+	0.497816i	$0.165864\pi$
$53$			7.24831i			0.995631i	−0.867283	+	0.497816i	$0.834136\pi$
$54$		0			0
$55$			4.16538i			0.561660i
$56$		0			0
$57$		−	1.32589i		−	0.175619i
$58$		0			0
$59$		−	8.42939i		−	1.09741i	−0.836015	−	0.548707i	$-0.815120\pi$
$59$		−	8.42939i		−	1.09741i	0.836015	−	0.548707i	$-0.184880\pi$
$60$		0			0
$61$		−	3.54481i		−	0.453866i	−0.973910	−	0.226933i	$-0.927130\pi$
$61$		−	3.54481i		−	0.453866i	0.973910	−	0.226933i	$-0.0728698\pi$
$62$		0			0
$63$	2.13359			0.268808
$64$		0			0
$65$			2.81823i			0.349559i
$66$		0			0
$67$	−10.6962			−1.30674			−0.653372	−	0.757037i	$-0.726646\pi$
$67$	−10.6962			−1.30674			−0.653372	+	0.757037i	$0.726646\pi$
$68$		0			0
$69$	1.24316	+	4.63191i	0.149659	+	0.557616i
$70$		0			0
$71$			0.897920i			0.106564i	0.998580	+	0.0532818i	$0.0169681\pi$
$71$			0.897920i			0.106564i	−0.998580	+	0.0532818i	$0.983032\pi$
$72$		0			0
$73$	4.48027			0.524375			0.262188	−	0.965017i	$-0.415556\pi$
$73$	4.48027			0.524375			0.262188	+	0.965017i	$0.415556\pi$
$74$		0			0
$75$		−	1.00000i		−	0.115470i
$76$		0			0
$77$	−8.88724			−1.01280
$78$		0			0
$79$	7.73462			0.870213			0.435106	−	0.900379i	$-0.356711\pi$
$79$	7.73462			0.870213			0.435106	+	0.900379i	$0.356711\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−3.14065			−0.344731			−0.172365	−	0.985033i	$-0.555141\pi$
$83$	−3.14065			−0.344731			−0.172365	+	0.985033i	$0.555141\pi$
$84$		0			0
$85$	1.82147			0.197566
$86$		0			0
$87$		−	4.92560i		−	0.528080i
$88$		0			0
$89$		−	5.21158i		−	0.552427i	−0.961096	−	0.276213i	$-0.910920\pi$
$89$		−	5.21158i		−	0.552427i	0.961096	−	0.276213i	$-0.0890796\pi$
$90$		0			0
$91$	−6.01297			−0.630331
$92$		0			0
$93$	10.5363			1.09256
$94$		0			0
$95$		−	1.32589i		−	0.136034i
$96$		0			0
$97$		−	0.00964558i		−	0.000979360i	−1.00000		0.000489680i	$-0.999844\pi$
$97$		−	0.00964558i		−	0.000979360i	1.00000		0.000489680i	$-0.000155870\pi$
$98$		0			0
$99$	−4.16538			−0.418637
$100$		0			0
$101$	−6.02561			−0.599571			−0.299785	−	0.954007i	$-0.596915\pi$
$101$	−6.02561			−0.599571			−0.299785	+	0.954007i	$0.596915\pi$
$102$		0			0
$103$	6.10648			0.601690			0.300845	−	0.953673i	$-0.402731\pi$
$103$	6.10648			0.601690			0.300845	+	0.953673i	$0.402731\pi$
$104$		0			0
$105$	2.13359			0.208218
$106$		0			0
$107$	3.50759			0.339091			0.169546	−	0.985522i	$-0.445770\pi$
$107$	3.50759			0.339091			0.169546	+	0.985522i	$0.445770\pi$
$108$		0			0
$109$			9.73697i			0.932632i	0.884618	+	0.466316i	$0.154419\pi$
$109$			9.73697i			0.932632i	−0.884618	+	0.466316i	$0.845581\pi$
$110$		0			0
$111$	4.95490			0.470298
$112$		0			0
$113$			1.94423i			0.182898i	0.995810	+	0.0914490i	$0.0291499\pi$
$113$			1.94423i			0.182898i	−0.995810	+	0.0914490i	$0.970850\pi$
$114$		0			0
$115$	1.24316	+	4.63191i	0.115925	+	0.431927i
$116$		0			0
$117$	−2.81823			−0.260546
$118$		0			0
$119$			3.88628i			0.356255i
$120$		0			0
$121$	6.35043			0.577312
$122$		0			0
$123$			0.656543i			0.0591985i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$		−	15.3785i		−	1.36462i	−0.731061	−	0.682312i	$-0.760974\pi$
$127$		−	15.3785i		−	1.36462i	0.731061	−	0.682312i	$-0.239026\pi$
$128$		0			0
$129$		−	9.49430i		−	0.835927i
$130$		0			0
$131$			1.02037i			0.0891498i	0.999006	+	0.0445749i	$0.0141933\pi$
$131$			1.02037i			0.0891498i	−0.999006	+	0.0445749i	$0.985807\pi$
$132$		0			0
$133$	2.82891			0.245298
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$		−	2.73571i		−	0.233727i	−0.993148	−	0.116864i	$-0.962716\pi$
$137$		−	2.73571i		−	0.233727i	0.993148	−	0.116864i	$-0.0372840\pi$
$138$		0			0
$139$		−	20.8691i		−	1.77009i	−0.465504	−	0.885046i	$-0.654127\pi$
$139$		−	20.8691i		−	1.77009i	0.465504	−	0.885046i	$-0.345873\pi$
$140$		0			0
$141$	7.28458			0.613472
$142$		0			0
$143$	11.7390			0.981667
$144$		0			0
$145$		−	4.92560i		−	0.409049i
$146$		0			0
$147$		−	2.44777i		−	0.201889i
$148$		0			0
$149$		−	5.40737i		−	0.442989i	−0.975162	−	0.221494i	$-0.928907\pi$
$149$		−	5.40737i		−	0.442989i	0.975162	−	0.221494i	$-0.0710934\pi$
$150$		0			0
$151$		−	2.34996i		−	0.191237i	−0.995418	−	0.0956185i	$-0.969517\pi$
$151$		−	2.34996i		−	0.191237i	0.995418	−	0.0956185i	$-0.0304829\pi$
$152$		0			0
$153$			1.82147i			0.147257i
$154$		0			0
$155$	10.5363			0.846293
$156$		0			0
$157$			7.86091i			0.627369i	0.949527	+	0.313685i	$0.101563\pi$
$157$			7.86091i			0.627369i	−0.949527	+	0.313685i	$0.898437\pi$
$158$		0			0
$159$	−7.24831			−0.574828
$160$		0			0
$161$	−9.88261	+	2.65240i	−0.778859	+	0.209039i
$162$		0			0
$163$		−	0.585998i		−	0.0458989i	−0.999737	−	0.0229494i	$-0.992694\pi$
$163$		−	0.585998i		−	0.0458989i	0.999737	−	0.0229494i	$-0.00730568\pi$
$164$		0			0
$165$	−4.16538			−0.324275
$166$		0			0
$167$		−	3.15561i		−	0.244189i	−0.992518	−	0.122094i	$-0.961039\pi$
$167$		−	3.15561i		−	0.244189i	0.992518	−	0.122094i	$-0.0389611\pi$
$168$		0			0
$169$	−5.05755			−0.389043
$170$		0			0
$171$	1.32589			0.101393
$172$		0			0
$173$	1.97459			0.150126			0.0750628	−	0.997179i	$-0.476084\pi$
$173$	1.97459			0.150126			0.0750628	+	0.997179i	$0.476084\pi$
$174$		0			0
$175$	2.13359			0.161285
$176$		0			0
$177$	8.42939			0.633592
$178$		0			0
$179$		−	25.9565i		−	1.94008i	−0.242943	−	0.970041i	$-0.578113\pi$
$179$		−	25.9565i		−	1.94008i	0.242943	−	0.970041i	$-0.421887\pi$
$180$		0			0
$181$			15.5579i			1.15641i	0.815892	+	0.578204i	$0.196246\pi$
$181$			15.5579i			1.15641i	−0.815892	+	0.578204i	$0.803754\pi$
$182$		0			0
$183$	3.54481			0.262040
$184$		0			0
$185$	4.95490			0.364291
$186$		0			0
$187$		−	7.58713i		−	0.554826i
$188$		0			0
$189$			2.13359i			0.155196i
$190$		0			0
$191$	15.2728			1.10510			0.552552	−	0.833479i	$-0.313654\pi$
$191$	15.2728			1.10510			0.552552	+	0.833479i	$0.313654\pi$
$192$		0			0
$193$	7.77339			0.559541			0.279771	−	0.960067i	$-0.409742\pi$
$193$	7.77339			0.559541			0.279771	+	0.960067i	$0.409742\pi$
$194$		0			0
$195$	−2.81823			−0.201818
$196$		0			0
$197$	14.3268			1.02074			0.510369	−	0.859955i	$-0.329509\pi$
$197$	14.3268			1.02074			0.510369	+	0.859955i	$0.329509\pi$
$198$		0			0
$199$	16.9685			1.20287			0.601433	−	0.798923i	$-0.294597\pi$
$199$	16.9685			1.20287			0.601433	+	0.798923i	$0.294597\pi$
$200$		0			0
$201$		−	10.6962i		−	0.754450i
$202$		0			0
$203$	10.5092			0.737604
$204$		0			0
$205$			0.656543i			0.0458550i
$206$		0			0
$207$	−4.63191	+	1.24316i	−0.321940	+	0.0864058i
$208$		0			0
$209$	−5.52285			−0.382023
$210$		0			0
$211$		−	10.6786i		−	0.735142i	−0.929995	−	0.367571i	$-0.880189\pi$
$211$		−	10.6786i		−	0.735142i	0.929995	−	0.367571i	$-0.119811\pi$
$212$		0			0
$213$	−0.897920			−0.0615245
$214$		0			0
$215$		−	9.49430i		−	0.647506i
$216$		0			0
$217$			22.4801i			1.52605i
$218$		0			0
$219$			4.48027i			0.302748i
$220$		0			0
$221$		−	5.13333i		−	0.345305i
$222$		0			0
$223$			5.49270i			0.367818i	0.982943	+	0.183909i	$0.0588752\pi$
$223$			5.49270i			0.367818i	−0.982943	+	0.183909i	$0.941125\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−3.53360			−0.234534			−0.117267	−	0.993100i	$-0.537413\pi$
$227$	−3.53360			−0.234534			−0.117267	+	0.993100i	$0.537413\pi$
$228$		0			0
$229$		−	6.15959i		−	0.407037i	−0.979071	−	0.203518i	$-0.934762\pi$
$229$		−	6.15959i		−	0.407037i	0.979071	−	0.203518i	$-0.0652377\pi$
$230$		0			0
$231$		−	8.88724i		−	0.584738i
$232$		0			0
$233$	17.9823			1.17806			0.589030	−	0.808111i	$-0.299510\pi$
$233$	17.9823			1.17806			0.589030	+	0.808111i	$0.299510\pi$
$234$		0			0
$235$	7.28458			0.475194
$236$		0			0
$237$			7.73462i			0.502418i
$238$		0			0
$239$		−	16.4389i		−	1.06335i	−0.846950	−	0.531673i	$-0.821564\pi$
$239$		−	16.4389i		−	1.06335i	0.846950	−	0.531673i	$-0.178436\pi$
$240$		0			0
$241$		−	15.9706i		−	1.02875i	−0.857564	−	0.514377i	$-0.828023\pi$
$241$		−	15.9706i		−	1.02875i	0.857564	−	0.514377i	$-0.171977\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$		−	2.44777i		−	0.156383i
$246$		0			0
$247$	−3.73667			−0.237759
$248$		0			0
$249$		−	3.14065i		−	0.199030i
$250$		0			0
$251$	−25.9346			−1.63698			−0.818488	−	0.574523i	$-0.805187\pi$
$251$	−25.9346			−1.63698			−0.818488	+	0.574523i	$0.805187\pi$
$252$		0			0
$253$	19.2937	−	5.17825i	1.21298	−	0.325554i
$254$		0			0
$255$			1.82147i			0.114065i
$256$		0			0
$257$	−9.54194			−0.595210			−0.297605	−	0.954689i	$-0.596188\pi$
$257$	−9.54194			−0.595210			−0.297605	+	0.954689i	$0.596188\pi$
$258$		0			0
$259$			10.5717i			0.656896i
$260$		0			0
$261$	4.92560			0.304887
$262$		0			0
$263$	10.6982			0.659681			0.329840	−	0.944037i	$-0.393005\pi$
$263$	10.6982			0.659681			0.329840	+	0.944037i	$0.393005\pi$
$264$		0			0
$265$	−7.24831			−0.445260
$266$		0			0
$267$	5.21158			0.318944
$268$		0			0
$269$	5.41224			0.329990			0.164995	−	0.986294i	$-0.447239\pi$
$269$	5.41224			0.329990			0.164995	+	0.986294i	$0.447239\pi$
$270$		0			0
$271$		−	1.28927i		−	0.0783179i	−0.999233	−	0.0391589i	$-0.987532\pi$
$271$		−	1.28927i		−	0.0783179i	0.999233	−	0.0391589i	$-0.0124679\pi$
$272$		0			0
$273$		−	6.01297i		−	0.363922i
$274$		0			0
$275$	−4.16538			−0.251182
$276$		0			0
$277$	11.7340			0.705027			0.352513	−	0.935807i	$-0.385327\pi$
$277$	11.7340			0.705027			0.352513	+	0.935807i	$0.385327\pi$
$278$		0			0
$279$			10.5363i			0.630790i
$280$		0			0
$281$		−	13.4017i		−	0.799480i	−0.916628	−	0.399740i	$-0.869100\pi$
$281$		−	13.4017i		−	0.799480i	0.916628	−	0.399740i	$-0.130900\pi$
$282$		0			0
$283$	15.1691			0.901707			0.450853	−	0.892598i	$-0.351120\pi$
$283$	15.1691			0.901707			0.450853	+	0.892598i	$0.351120\pi$
$284$		0			0
$285$	1.32589			0.0785390
$286$		0			0
$287$	−1.40080			−0.0826865
$288$		0			0
$289$	13.6822			0.804838
$290$		0			0
$291$	0.00964558			0.000565434
$292$		0			0
$293$		−	0.465132i		−	0.0271733i	−0.999908	−	0.0135867i	$-0.995675\pi$
$293$		−	0.465132i		−	0.0271733i	0.999908	−	0.0135867i	$-0.00432490\pi$
$294$		0			0
$295$	8.42939			0.490778
$296$		0			0
$297$		−	4.16538i		−	0.241700i
$298$		0			0
$299$	13.0538	−	3.50352i	0.754921	−	0.202614i
$300$		0			0
$301$	20.2570			1.16759
$302$		0			0
$303$		−	6.02561i		−	0.346162i
$304$		0			0
$305$	3.54481			0.202975
$306$		0			0
$307$			22.1194i			1.26242i	0.775613	+	0.631209i	$0.217441\pi$
$307$			22.1194i			1.26242i	−0.775613	+	0.631209i	$0.782559\pi$
$308$		0			0
$309$			6.10648i			0.347386i
$310$		0			0
$311$		−	23.0765i		−	1.30855i	−0.756257	−	0.654275i	$-0.772974\pi$
$311$		−	23.0765i		−	1.30855i	0.756257	−	0.654275i	$-0.227026\pi$
$312$		0			0
$313$			13.5444i			0.765576i	0.923836	+	0.382788i	$0.125036\pi$
$313$			13.5444i			0.765576i	−0.923836	+	0.382788i	$0.874964\pi$
$314$		0			0
$315$			2.13359i			0.120214i
$316$		0			0
$317$	24.2326			1.36104			0.680520	−	0.732730i	$-0.261754\pi$
$317$	24.2326			1.36104			0.680520	+	0.732730i	$0.261754\pi$
$318$		0			0
$319$	−20.5170			−1.14873
$320$		0			0
$321$			3.50759i			0.195775i
$322$		0			0
$323$			2.41507i			0.134378i
$324$		0			0
$325$	−2.81823			−0.156328
$326$		0			0
$327$	−9.73697			−0.538455
$328$		0			0
$329$			15.5423i			0.856877i
$330$		0			0
$331$			24.5464i			1.34919i	0.738186	+	0.674597i	$0.235683\pi$
$331$			24.5464i			1.34919i	−0.738186	+	0.674597i	$0.764317\pi$
$332$		0			0
$333$			4.95490i			0.271527i
$334$		0			0
$335$		−	10.6962i		−	0.584394i
$336$		0			0
$337$		−	3.21466i		−	0.175113i	−0.996160	−	0.0875567i	$-0.972094\pi$
$337$		−	3.21466i		−	0.175113i	0.996160	−	0.0875567i	$-0.0279059\pi$
$338$		0			0
$339$	−1.94423			−0.105596
$340$		0			0
$341$		−	43.8876i		−	2.37665i
$342$		0			0
$343$	20.1577			1.08841
$344$		0			0
$345$	−4.63191	+	1.24316i	−0.249373	+	0.0669296i
$346$		0			0
$347$			2.76560i			0.148465i	0.997241	+	0.0742327i	$0.0236508\pi$
$347$			2.76560i			0.148465i	−0.997241	+	0.0742327i	$0.976349\pi$
$348$		0			0
$349$	3.14049			0.168107			0.0840533	−	0.996461i	$-0.473213\pi$
$349$	3.14049			0.168107			0.0840533	+	0.996461i	$0.473213\pi$
$350$		0			0
$351$		−	2.81823i		−	0.150426i
$352$		0			0
$353$	−31.9728			−1.70174			−0.850871	−	0.525375i	$-0.823925\pi$
$353$	−31.9728			−1.70174			−0.850871	+	0.525375i	$0.823925\pi$
$354$		0			0
$355$	−0.897920			−0.0476567
$356$		0			0
$357$	−3.88628			−0.205684
$358$		0			0
$359$	−8.48175			−0.447650			−0.223825	−	0.974629i	$-0.571854\pi$
$359$	−8.48175			−0.447650			−0.223825	+	0.974629i	$0.571854\pi$
$360$		0			0
$361$	−17.2420			−0.907474
$362$		0			0
$363$			6.35043i			0.333311i
$364$		0			0
$365$			4.48027i			0.234508i
$366$		0			0
$367$	−5.00433			−0.261224			−0.130612	−	0.991434i	$-0.541694\pi$
$367$	−5.00433			−0.261224			−0.130612	+	0.991434i	$0.541694\pi$
$368$		0			0
$369$	−0.656543			−0.0341783
$370$		0			0
$371$		−	15.4649i		−	0.802900i
$372$		0			0
$373$			13.6544i			0.706998i	0.935435	+	0.353499i	$0.115008\pi$
$373$			13.6544i			0.706998i	−0.935435	+	0.353499i	$0.884992\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−13.8815			−0.714934
$378$		0			0
$379$	0.658572			0.0338285			0.0169143	−	0.999857i	$-0.494616\pi$
$379$	0.658572			0.0338285			0.0169143	+	0.999857i	$0.494616\pi$
$380$		0			0
$381$	15.3785			0.787866
$382$		0			0
$383$	−13.6020			−0.695028			−0.347514	−	0.937675i	$-0.612974\pi$
$383$	−13.6020			−0.695028			−0.347514	+	0.937675i	$0.612974\pi$
$384$		0			0
$385$		−	8.88724i		−	0.452936i
$386$		0			0
$387$	9.49430			0.482622
$388$		0			0
$389$			32.7804i			1.66203i	0.556247	+	0.831017i	$0.312241\pi$
$389$			32.7804i			1.66203i	−0.556247	+	0.831017i	$0.687759\pi$
$390$		0			0
$391$	−2.26438	−	8.43688i	−0.114515	−	0.426672i
$392$		0			0
$393$	−1.02037			−0.0514707
$394$		0			0
$395$			7.73462i			0.389171i
$396$		0			0
$397$	7.70037			0.386470			0.193235	−	0.981152i	$-0.438102\pi$
$397$	7.70037			0.386470			0.193235	+	0.981152i	$0.438102\pi$
$398$		0			0
$399$			2.82891i			0.141623i
$400$		0			0
$401$		−	0.572120i		−	0.0285703i	−0.999898	−	0.0142851i	$-0.995453\pi$
$401$		−	0.572120i		−	0.0285703i	0.999898	−	0.0142851i	$-0.00454726\pi$
$402$		0			0
$403$		−	29.6937i		−	1.47915i
$404$		0			0
$405$			1.00000i			0.0496904i
$406$		0			0
$407$		−	20.6390i		−	1.02304i
$408$		0			0
$409$	2.81116			0.139003			0.0695014	−	0.997582i	$-0.477859\pi$
$409$	2.81116			0.139003			0.0695014	+	0.997582i	$0.477859\pi$
$410$		0			0
$411$	2.73571			0.134942
$412$		0			0
$413$			17.9849i			0.884980i
$414$		0			0
$415$		−	3.14065i		−	0.154168i
$416$		0			0
$417$	20.8691			1.02196
$418$		0			0
$419$	19.4496			0.950174			0.475087	−	0.879939i	$-0.342417\pi$
$419$	19.4496			0.950174			0.475087	+	0.879939i	$0.342417\pi$
$420$		0			0
$421$			10.1604i			0.495186i	0.968864	+	0.247593i	$0.0796395\pi$
$421$			10.1604i			0.495186i	−0.968864	+	0.247593i	$0.920360\pi$
$422$		0			0
$423$			7.28458i			0.354188i
$424$		0			0
$425$			1.82147i			0.0883544i
$426$		0			0
$427$			7.56318i			0.366008i
$428$		0			0
$429$			11.7390i			0.566766i
$430$		0			0
$431$	−25.8128			−1.24336			−0.621679	−	0.783272i	$-0.713549\pi$
$431$	−25.8128			−1.24336			−0.621679	+	0.783272i	$0.713549\pi$
$432$		0			0
$433$			37.3789i			1.79631i	0.439676	+	0.898157i	$0.355093\pi$
$433$			37.3789i			1.79631i	−0.439676	+	0.898157i	$0.644907\pi$
$434$		0			0
$435$	4.92560			0.236165
$436$		0			0
$437$	−6.14140	+	1.64830i	−0.293783	+	0.0788488i
$438$		0			0
$439$			5.86390i			0.279869i	0.990161	+	0.139934i	$0.0446892\pi$
$439$			5.86390i			0.279869i	−0.990161	+	0.139934i	$0.955311\pi$
$440$		0			0
$441$	2.44777			0.116561
$442$		0			0
$443$			12.8409i			0.610091i	0.952338	+	0.305046i	$0.0986718\pi$
$443$			12.8409i			0.610091i	−0.952338	+	0.305046i	$0.901328\pi$
$444$		0			0
$445$	5.21158			0.247053
$446$		0			0
$447$	5.40737			0.255760
$448$		0			0
$449$	−22.9627			−1.08368			−0.541838	−	0.840483i	$-0.682271\pi$
$449$	−22.9627			−1.08368			−0.541838	+	0.840483i	$0.682271\pi$
$450$		0			0
$451$	2.73476			0.128775
$452$		0			0
$453$	2.34996			0.110411
$454$		0			0
$455$		−	6.01297i		−	0.281892i
$456$		0			0
$457$			26.7913i			1.25324i	0.779323	+	0.626622i	$0.215563\pi$
$457$			26.7913i			1.25324i	−0.779323	+	0.626622i	$0.784437\pi$
$458$		0			0
$459$	−1.82147			−0.0850190
$460$		0			0
$461$	21.6671			1.00914			0.504569	−	0.863371i	$-0.331651\pi$
$461$	21.6671			1.00914			0.504569	+	0.863371i	$0.331651\pi$
$462$		0			0
$463$		−	35.5825i		−	1.65366i	−0.562454	−	0.826829i	$-0.690143\pi$
$463$		−	35.5825i		−	1.65366i	0.562454	−	0.826829i	$-0.309857\pi$
$464$		0			0
$465$			10.5363i			0.488608i
$466$		0			0
$467$	−18.6086			−0.861104			−0.430552	−	0.902566i	$-0.641681\pi$
$467$	−18.6086			−0.861104			−0.430552	+	0.902566i	$0.641681\pi$
$468$		0			0
$469$	22.8213			1.05379
$470$		0			0
$471$	−7.86091			−0.362212
$472$		0			0
$473$	−39.5474			−1.81839
$474$		0			0
$475$	1.32589			0.0608361
$476$		0			0
$477$		−	7.24831i		−	0.331877i
$478$		0			0
$479$	−15.9689			−0.729637			−0.364818	−	0.931079i	$-0.618869\pi$
$479$	−15.9689			−0.729637			−0.364818	+	0.931079i	$0.618869\pi$
$480$		0			0
$481$		−	13.9641i		−	0.636706i
$482$		0			0
$483$	−2.65240	−	9.88261i	−0.120689	−	0.449674i
$484$		0			0
$485$	0.00964558			0.000437983
$486$		0			0
$487$			15.6968i			0.711288i	0.934621	+	0.355644i	$0.115738\pi$
$487$			15.6968i			0.711288i	−0.934621	+	0.355644i	$0.884262\pi$
$488$		0			0
$489$	0.585998			0.0264997
$490$		0			0
$491$			32.2500i			1.45542i	0.685884	+	0.727710i	$0.259416\pi$
$491$			32.2500i			1.45542i	−0.685884	+	0.727710i	$0.740584\pi$
$492$		0			0
$493$			8.97185i			0.404072i
$494$		0			0
$495$		−	4.16538i		−	0.187220i
$496$		0			0
$497$		−	1.91580i		−	0.0859353i
$498$		0			0
$499$		−	11.7721i		−	0.526993i	−0.964660	−	0.263496i	$-0.915124\pi$
$499$		−	11.7721i		−	0.526993i	0.964660	−	0.263496i	$-0.0848757\pi$
$500$		0			0
$501$	3.15561			0.140982
$502$		0			0
$503$	7.65715			0.341415			0.170708	−	0.985322i	$-0.445395\pi$
$503$	7.65715			0.341415			0.170708	+	0.985322i	$0.445395\pi$
$504$		0			0
$505$		−	6.02561i		−	0.268136i
$506$		0			0
$507$		−	5.05755i		−	0.224614i
$508$		0			0
$509$	1.45059			0.0642961			0.0321481	−	0.999483i	$-0.489765\pi$
$509$	1.45059			0.0642961			0.0321481	+	0.999483i	$0.489765\pi$
$510$		0			0
$511$	−9.55907			−0.422868
$512$		0			0
$513$			1.32589i			0.0585395i
$514$		0			0
$515$			6.10648i			0.269084i
$516$		0			0
$517$		−	30.3431i		−	1.33449i
$518$		0			0
$519$			1.97459i			0.0866751i
$520$		0			0
$521$		−	6.69276i		−	0.293215i	−0.989195	−	0.146607i	$-0.953165\pi$
$521$		−	6.69276i		−	0.293215i	0.989195	−	0.146607i	$-0.0468354\pi$
$522$		0			0
$523$	29.7009			1.29873			0.649365	−	0.760477i	$-0.275035\pi$
$523$	29.7009			1.29873			0.649365	+	0.760477i	$0.275035\pi$
$524$		0			0
$525$			2.13359i			0.0931177i
$526$		0			0
$527$	−19.1915			−0.835995
$528$		0			0
$529$	19.9091	−	11.5164i	0.865613	−	0.500714i
$530$		0			0
$531$			8.42939i			0.365805i
$532$		0			0
$533$	1.85029			0.0801451
$534$		0			0
$535$			3.50759i			0.151646i
$536$		0			0
$537$	25.9565			1.12011
$538$		0			0
$539$	−10.1959			−0.439169
$540$		0			0
$541$	43.6607			1.87712			0.938560	−	0.345117i	$-0.112161\pi$
$541$	43.6607			1.87712			0.938560	+	0.345117i	$0.112161\pi$
$542$		0			0
$543$	−15.5579			−0.667652
$544$		0			0
$545$	−9.73697			−0.417086
$546$		0			0
$547$		−	11.1381i		−	0.476231i	−0.971237	−	0.238115i	$-0.923470\pi$
$547$		−	11.1381i		−	0.476231i	0.971237	−	0.238115i	$-0.0765296\pi$
$548$		0			0
$549$			3.54481i			0.151289i
$550$		0			0
$551$	6.53082			0.278222
$552$		0			0
$553$	−16.5025			−0.701760
$554$		0			0
$555$			4.95490i			0.210324i
$556$		0			0
$557$			24.6097i			1.04275i	0.853328	+	0.521374i	$0.174580\pi$
$557$			24.6097i			1.04275i	−0.853328	+	0.521374i	$0.825420\pi$
$558$		0			0
$559$	−26.7572			−1.13171
$560$		0			0
$561$	7.58713			0.320329
$562$		0			0
$563$	−1.64103			−0.0691613			−0.0345807	−	0.999402i	$-0.511010\pi$
$563$	−1.64103			−0.0691613			−0.0345807	+	0.999402i	$0.511010\pi$
$564$		0			0
$565$	−1.94423			−0.0817945
$566$		0			0
$567$	−2.13359			−0.0896026
$568$		0			0
$569$		−	39.1957i		−	1.64317i	−0.570087	−	0.821585i	$-0.693090\pi$
$569$		−	39.1957i		−	1.64317i	0.570087	−	0.821585i	$-0.306910\pi$
$570$		0			0
$571$	−43.5236			−1.82141			−0.910704	−	0.413060i	$-0.864460\pi$
$571$	−43.5236			−1.82141			−0.910704	+	0.413060i	$0.864460\pi$
$572$		0			0
$573$			15.2728i			0.638032i
$574$		0			0
$575$	−4.63191	+	1.24316i	−0.193164	+	0.0518435i
$576$		0			0
$577$	−36.8449			−1.53387			−0.766936	−	0.641724i	$-0.778220\pi$
$577$	−36.8449			−1.53387			−0.766936	+	0.641724i	$0.778220\pi$
$578$		0			0
$579$			7.77339i			0.323051i
$580$		0			0
$581$	6.70086			0.277999
$582$		0			0
$583$			30.1920i			1.25042i
$584$		0			0
$585$		−	2.81823i		−	0.116520i
$586$		0			0
$587$		−	22.7311i		−	0.938214i	−0.883141	−	0.469107i	$-0.844576\pi$
$587$		−	22.7311i		−	0.938214i	0.883141	−	0.469107i	$-0.155424\pi$
$588$		0			0
$589$			13.9699i			0.575621i
$590$		0			0
$591$			14.3268i			0.589324i
$592$		0			0
$593$	−20.3848			−0.837103			−0.418551	−	0.908193i	$-0.637462\pi$
$593$	−20.3848			−0.837103			−0.418551	+	0.908193i	$0.637462\pi$
$594$		0			0
$595$	−3.88628			−0.159322
$596$		0			0
$597$			16.9685i			0.694475i
$598$		0			0
$599$		−	40.9408i		−	1.67279i	−0.548124	−	0.836397i	$-0.684658\pi$
$599$		−	40.9408i		−	1.67279i	0.548124	−	0.836397i	$-0.315342\pi$
$600$		0			0
$601$	37.7717			1.54074			0.770369	−	0.637598i	$-0.220072\pi$
$601$	37.7717			1.54074			0.770369	+	0.637598i	$0.220072\pi$
$602$		0			0
$603$	10.6962			0.435582
$604$		0			0
$605$			6.35043i			0.258182i
$606$		0			0
$607$			25.2408i			1.02449i	0.858839	+	0.512246i	$0.171186\pi$
$607$			25.2408i			1.02449i	−0.858839	+	0.512246i	$0.828814\pi$
$608$		0			0
$609$			10.5092i			0.425856i
$610$		0			0
$611$		−	20.5297i		−	0.830541i
$612$		0			0
$613$			0.132012i			0.00533192i	0.999996	+	0.00266596i	$0.000848603\pi$
$613$			0.132012i			0.00533192i	−0.999996	+	0.00266596i	$0.999151\pi$
$614$		0			0
$615$	−0.656543			−0.0264744
$616$		0			0
$617$			8.92309i			0.359230i	0.983737	+	0.179615i	$0.0574852\pi$
$617$			8.92309i			0.359230i	−0.983737	+	0.179615i	$0.942515\pi$
$618$		0			0
$619$	30.3910			1.22152			0.610758	−	0.791817i	$-0.290865\pi$
$619$	30.3910			1.22152			0.610758	+	0.791817i	$0.290865\pi$
$620$		0			0
$621$	−1.24316	−	4.63191i	−0.0498864	−	0.185872i
$622$		0			0
$623$			11.1194i			0.445489i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	5.52285i		−	0.220561i
$628$		0			0
$629$	−9.02520			−0.359858
$630$		0			0
$631$	−3.75251			−0.149385			−0.0746924	−	0.997207i	$-0.523798\pi$
$631$	−3.75251			−0.149385			−0.0746924	+	0.997207i	$0.523798\pi$
$632$		0			0
$633$	10.6786			0.424435
$634$		0			0
$635$	15.3785			0.610279
$636$		0			0
$637$	−6.89840			−0.273325
$638$		0			0
$639$		−	0.897920i		−	0.0355212i
$640$		0			0
$641$		−	4.00796i		−	0.158305i	−0.996863	−	0.0791524i	$-0.974779\pi$
$641$		−	4.00796i		−	0.158305i	0.996863	−	0.0791524i	$-0.0252214\pi$
$642$		0			0
$643$	43.5751			1.71843			0.859217	−	0.511612i	$-0.170952\pi$
$643$	43.5751			1.71843			0.859217	+	0.511612i	$0.170952\pi$
$644$		0			0
$645$	9.49430			0.373838
$646$		0			0
$647$		−	39.2034i		−	1.54124i	−0.637293	−	0.770621i	$-0.719946\pi$
$647$		−	39.2034i		−	1.54124i	0.637293	−	0.770621i	$-0.280054\pi$
$648$		0			0
$649$		−	35.1117i		−	1.37825i
$650$		0			0
$651$	−22.4801			−0.881065
$652$		0			0
$653$	−40.2961			−1.57691			−0.788455	−	0.615093i	$-0.789119\pi$
$653$	−40.2961			−1.57691			−0.788455	+	0.615093i	$0.789119\pi$
$654$		0			0
$655$	−1.02037			−0.0398690
$656$		0			0
$657$	−4.48027			−0.174792
$658$		0			0
$659$	10.1481			0.395315			0.197658	−	0.980271i	$-0.436667\pi$
$659$	10.1481			0.395315			0.197658	+	0.980271i	$0.436667\pi$
$660$		0			0
$661$		−	8.80651i		−	0.342534i	−0.985225	−	0.171267i	$-0.945214\pi$
$661$		−	8.80651i		−	0.342534i	0.985225	−	0.171267i	$-0.0547860\pi$
$662$		0			0
$663$	5.13333			0.199362
$664$		0			0
$665$			2.82891i			0.109701i
$666$		0			0
$667$	−22.8149	+	6.12333i	−0.883398	+	0.237096i
$668$		0			0
$669$	−5.49270			−0.212360
$670$		0			0
$671$		−	14.7655i		−	0.570015i
$672$		0			0
$673$	15.2734			0.588748			0.294374	−	0.955690i	$-0.404889\pi$
$673$	15.2734			0.588748			0.294374	+	0.955690i	$0.404889\pi$
$674$		0			0
$675$			1.00000i			0.0384900i
$676$		0			0
$677$		−	37.8079i		−	1.45307i	−0.687127	−	0.726537i	$-0.741129\pi$
$677$		−	37.8079i		−	1.45307i	0.687127	−	0.726537i	$-0.258871\pi$
$678$		0			0
$679$			0.0205798i			0.000789779i
$680$		0			0
$681$		−	3.53360i		−	0.135408i
$682$		0			0
$683$		−	3.32230i		−	0.127124i	−0.997978	−	0.0635622i	$-0.979754\pi$
$683$		−	3.32230i		−	0.127124i	0.997978	−	0.0635622i	$-0.0202461\pi$
$684$		0			0
$685$	2.73571			0.104526
$686$		0			0
$687$	6.15959			0.235003
$688$		0			0
$689$			20.4274i			0.778223i
$690$		0			0
$691$		−	8.07215i		−	0.307079i	−0.988143	−	0.153540i	$-0.950933\pi$
$691$		−	8.07215i		−	0.307079i	0.988143	−	0.153540i	$-0.0490672\pi$
$692$		0			0
$693$	8.88724			0.337598
$694$		0			0
$695$	20.8691			0.791609
$696$		0			0
$697$		−	1.19588i		−	0.0452970i
$698$		0			0
$699$			17.9823i			0.680153i
$700$		0			0
$701$		−	45.3917i		−	1.71442i	−0.514965	−	0.857211i	$-0.672195\pi$
$701$		−	45.3917i		−	1.71442i	0.514965	−	0.857211i	$-0.327805\pi$
$702$		0			0
$703$			6.56965i			0.247779i
$704$		0			0
$705$			7.28458i			0.274353i
$706$		0			0
$707$	12.8562			0.483508
$708$		0			0
$709$		−	24.3181i		−	0.913285i	−0.889650	−	0.456643i	$-0.849052\pi$
$709$		−	24.3181i		−	0.913285i	0.889650	−	0.456643i	$-0.150948\pi$
$710$		0			0
$711$	−7.73462			−0.290071
$712$		0			0
$713$	−13.0983	−	48.8030i	−0.490535	−	1.82769i
$714$		0			0
$715$			11.7390i			0.439015i
$716$		0			0
$717$	16.4389			0.613923
$718$		0			0
$719$			13.0606i			0.487079i	0.969891	+	0.243539i	$0.0783085\pi$
$719$			13.0606i			0.487079i	−0.969891	+	0.243539i	$0.921691\pi$
$720$		0			0
$721$	−13.0288			−0.485216
$722$		0			0
$723$	15.9706			0.593951
$724$		0			0
$725$	4.92560			0.182932
$726$		0			0
$727$	44.1977			1.63920			0.819600	−	0.572935i	$-0.194195\pi$
$727$	44.1977			1.63920			0.819600	+	0.572935i	$0.194195\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$			17.2936i			0.639627i
$732$		0			0
$733$			6.33078i			0.233833i	0.993142	+	0.116916i	$0.0373010\pi$
$733$			6.33078i			0.233833i	−0.993142	+	0.116916i	$0.962699\pi$
$734$		0			0
$735$	2.44777			0.0902875
$736$		0			0
$737$	−44.5537			−1.64115
$738$		0			0
$739$			18.0828i			0.665186i	0.943070	+	0.332593i	$0.107924\pi$
$739$			18.0828i			0.665186i	−0.943070	+	0.332593i	$0.892076\pi$
$740$		0			0
$741$		−	3.73667i		−	0.137270i
$742$		0			0
$743$	23.5187			0.862818			0.431409	−	0.902156i	$-0.358017\pi$
$743$	23.5187			0.862818			0.431409	+	0.902156i	$0.358017\pi$
$744$		0			0
$745$	5.40737			0.198111
$746$		0			0
$747$	3.14065			0.114910
$748$		0			0
$749$	−7.48377			−0.273451
$750$		0			0
$751$	−2.98033			−0.108754			−0.0543769	−	0.998520i	$-0.517317\pi$
$751$	−2.98033			−0.108754			−0.0543769	+	0.998520i	$0.517317\pi$
$752$		0			0
$753$		−	25.9346i		−	0.945109i
$754$		0			0
$755$	2.34996			0.0855238
$756$		0			0
$757$		−	16.0118i		−	0.581958i	−0.956729	−	0.290979i	$-0.906019\pi$
$757$		−	16.0118i		−	0.581958i	0.956729	−	0.290979i	$-0.0939810\pi$
$758$		0			0
$759$	5.17825	+	19.2937i	0.187959	+	0.700316i
$760$		0			0
$761$	1.15577			0.0418968			0.0209484	−	0.999781i	$-0.493331\pi$
$761$	1.15577			0.0418968			0.0209484	+	0.999781i	$0.493331\pi$
$762$		0			0
$763$		−	20.7747i		−	0.752096i
$764$		0			0
$765$	−1.82147			−0.0658554
$766$		0			0
$767$		−	23.7560i		−	0.857780i
$768$		0			0
$769$		−	34.8520i		−	1.25679i	−0.777893	−	0.628397i	$-0.783711\pi$
$769$		−	34.8520i		−	1.25679i	0.777893	−	0.628397i	$-0.216289\pi$
$770$		0			0
$771$		−	9.54194i		−	0.343645i
$772$		0			0
$773$		−	23.6988i		−	0.852387i	−0.904632	−	0.426194i	$-0.859854\pi$
$773$		−	23.6988i		−	0.852387i	0.904632	−	0.426194i	$-0.140146\pi$
$774$		0			0
$775$			10.5363i			0.378474i
$776$		0			0
$777$	−10.5717			−0.379259
$778$		0			0
$779$	−0.870505			−0.0311891
$780$		0			0
$781$			3.74018i			0.133834i
$782$		0			0
$783$			4.92560i			0.176027i
$784$		0			0
$785$	−7.86091			−0.280568
$786$		0			0
$787$	−33.5648			−1.19646			−0.598229	−	0.801326i	$-0.704128\pi$
$787$	−33.5648			−1.19646			−0.598229	+	0.801326i	$0.704128\pi$
$788$		0			0
$789$			10.6982i			0.380867i
$790$		0			0
$791$		−	4.14821i		−	0.147493i
$792$		0			0
$793$		−	9.99009i		−	0.354759i
$794$		0			0
$795$		−	7.24831i		−	0.257071i
$796$		0			0
$797$			6.33001i			0.224221i	0.993696	+	0.112110i	$0.0357610\pi$
$797$			6.33001i			0.224221i	−0.993696	+	0.112110i	$0.964239\pi$
$798$		0			0
$799$	−13.2687			−0.469411
$800$		0			0
$801$			5.21158i			0.184142i
$802$		0			0
$803$	18.6620			0.658569
$804$		0			0
$805$	−2.65240	−	9.88261i	−0.0934850	−	0.348316i
$806$		0			0
$807$			5.41224i			0.190520i
$808$		0			0
$809$	13.2593			0.466172			0.233086	−	0.972456i	$-0.425118\pi$
$809$	13.2593			0.466172			0.233086	+	0.972456i	$0.425118\pi$
$810$		0			0
$811$			22.3485i			0.784763i	0.919803	+	0.392381i	$0.128349\pi$
$811$			22.3485i			0.784763i	−0.919803	+	0.392381i	$0.871651\pi$
$812$		0			0
$813$	1.28927			0.0452168
$814$		0			0
$815$	0.585998			0.0205266
$816$		0			0
$817$	12.5884			0.440413
$818$		0			0
$819$	6.01297			0.210110
$820$		0			0
$821$	−34.4325			−1.20170			−0.600851	−	0.799361i	$-0.705172\pi$
$821$	−34.4325			−1.20170			−0.600851	+	0.799361i	$0.705172\pi$
$822$		0			0
$823$			47.9688i			1.67209i	0.548663	+	0.836044i	$0.315137\pi$
$823$			47.9688i			1.67209i	−0.548663	+	0.836044i	$0.684863\pi$
$824$		0			0
$825$		−	4.16538i		−	0.145020i
$826$		0			0
$827$	1.74922			0.0608264			0.0304132	−	0.999537i	$-0.490318\pi$
$827$	1.74922			0.0608264			0.0304132	+	0.999537i	$0.490318\pi$
$828$		0			0
$829$	51.7581			1.79763			0.898817	−	0.438324i	$-0.144428\pi$
$829$	51.7581			1.79763			0.898817	+	0.438324i	$0.144428\pi$
$830$		0			0
$831$			11.7340i			0.407047i
$832$		0			0
$833$			4.45855i			0.154480i
$834$		0			0
$835$	3.15561			0.109205
$836$		0			0
$837$	−10.5363			−0.364187
$838$		0			0
$839$	−39.4538			−1.36210			−0.681048	−	0.732239i	$-0.738476\pi$
$839$	−39.4538			−1.36210			−0.681048	+	0.732239i	$0.738476\pi$
$840$		0			0
$841$	−4.73842			−0.163394
$842$		0			0
$843$	13.4017			0.461580
$844$		0			0
$845$		−	5.05755i		−	0.173985i
$846$		0			0
$847$	−13.5492			−0.465557
$848$		0			0
$849$			15.1691i			0.520601i
$850$		0			0
$851$	−6.15974	−	22.9506i	−0.211153	−	0.786737i
$852$		0			0
$853$	−46.7294			−1.59999			−0.799993	−	0.600010i	$-0.795163\pi$
$853$	−46.7294			−1.59999			−0.799993	+	0.600010i	$0.795163\pi$
$854$		0			0
$855$			1.32589i			0.0453445i
$856$		0			0
$857$	−12.7629			−0.435974			−0.217987	−	0.975952i	$-0.569949\pi$
$857$	−12.7629			−0.435974			−0.217987	+	0.975952i	$0.569949\pi$
$858$		0			0
$859$			8.08644i			0.275906i	0.990439	+	0.137953i	$0.0440523\pi$
$859$			8.08644i			0.275906i	−0.990439	+	0.137953i	$0.955948\pi$
$860$		0			0
$861$		−	1.40080i		−	0.0477391i
$862$		0			0
$863$		−	28.9619i		−	0.985873i	−0.870065	−	0.492937i	$-0.835923\pi$
$863$		−	28.9619i		−	0.985873i	0.870065	−	0.492937i	$-0.164077\pi$
$864$		0			0
$865$			1.97459i			0.0671382i
$866$		0			0
$867$			13.6822i			0.464673i
$868$		0			0
$869$	32.2177			1.09291
$870$		0			0
$871$	−30.1443			−1.02140
$872$		0			0
$873$			0.00964558i			0.000326453i
$874$		0			0
$875$			2.13359i			0.0721287i
$876$		0			0
$877$	−1.42668			−0.0481756			−0.0240878	−	0.999710i	$-0.507668\pi$
$877$	−1.42668			−0.0481756			−0.0240878	+	0.999710i	$0.507668\pi$
$878$		0			0
$879$	0.465132			0.0156885
$880$		0			0
$881$			11.8097i			0.397879i	0.980012	+	0.198940i	$0.0637498\pi$
$881$			11.8097i			0.397879i	−0.980012	+	0.198940i	$0.936250\pi$
$882$		0			0
$883$			18.1971i			0.612380i	0.951970	+	0.306190i	$0.0990543\pi$
$883$			18.1971i			0.612380i	−0.951970	+	0.306190i	$0.900946\pi$
$884$		0			0
$885$			8.42939i			0.283351i
$886$		0			0
$887$			44.1807i			1.48344i	0.670709	+	0.741721i	$0.265990\pi$
$887$			44.1807i			1.48344i	−0.670709	+	0.741721i	$0.734010\pi$
$888$		0			0
$889$			32.8116i			1.10046i
$890$		0			0
$891$	4.16538			0.139546
$892$		0			0
$893$			9.65856i			0.323211i
$894$		0			0
$895$	25.9565			0.867631
$896$		0			0
$897$	3.50352	+	13.0538i	0.116979	+	0.435854i
$898$		0			0
$899$			51.8975i			1.73088i
$900$		0			0
$901$	13.2026			0.439842
$902$		0			0
$903$			20.2570i			0.674110i
$904$		0			0
$905$	−15.5579			−0.517161
$906$		0			0
$907$	−37.3038			−1.23865			−0.619326	−	0.785134i	$-0.712594\pi$
$907$	−37.3038			−1.23865			−0.619326	+	0.785134i	$0.712594\pi$
$908$		0			0
$909$	6.02561			0.199857
$910$		0			0
$911$	23.4621			0.777334			0.388667	−	0.921378i	$-0.372936\pi$
$911$	23.4621			0.777334			0.388667	+	0.921378i	$0.372936\pi$
$912$		0			0
$913$	−13.0820			−0.432951
$914$		0			0
$915$			3.54481i			0.117188i
$916$		0			0
$917$		−	2.17705i		−	0.0718924i
$918$		0			0
$919$	−9.06407			−0.298996			−0.149498	−	0.988762i	$-0.547766\pi$
$919$	−9.06407			−0.298996			−0.149498	+	0.988762i	$0.547766\pi$
$920$		0			0
$921$	−22.1194			−0.728858
$922$		0			0
$923$			2.53055i			0.0832941i
$924$		0			0
$925$			4.95490i			0.162916i
$926$		0			0
$927$	−6.10648			−0.200563
$928$		0			0
$929$	−47.0444			−1.54348			−0.771738	−	0.635941i	$-0.780612\pi$
$929$	−47.0444			−1.54348			−0.771738	+	0.635941i	$0.780612\pi$
$930$		0			0
$931$	3.24548			0.106366
$932$		0			0
$933$	23.0765			0.755491
$934$		0			0
$935$	7.58713			0.248126
$936$		0			0
$937$		−	14.6329i		−	0.478035i	−0.971015	−	0.239018i	$-0.923175\pi$
$937$		−	14.6329i		−	0.478035i	0.971015	−	0.239018i	$-0.0768254\pi$
$938$		0			0
$939$	−13.5444			−0.442005
$940$		0			0
$941$		−	5.08380i		−	0.165727i	−0.996561	−	0.0828636i	$-0.973593\pi$
$941$		−	5.08380i		−	0.165727i	0.996561	−	0.0828636i	$-0.0264066\pi$
$942$		0			0
$943$	3.04105	−	0.816190i	0.0990301	−	0.0265788i
$944$		0			0
$945$	−2.13359			−0.0694058
$946$		0			0
$947$			20.1881i			0.656026i	0.944673	+	0.328013i	$0.106379\pi$
$947$			20.1881i			0.656026i	−0.944673	+	0.328013i	$0.893621\pi$
$948$		0			0
$949$	12.6264			0.409872
$950$		0			0
$951$			24.2326i			0.785796i
$952$		0			0
$953$			1.31411i			0.0425683i	0.999773	+	0.0212842i	$0.00677547\pi$
$953$			1.31411i			0.0425683i	−0.999773	+	0.0212842i	$0.993225\pi$
$954$		0			0
$955$			15.2728i			0.494217i
$956$		0			0
$957$		−	20.5170i		−	0.663222i
$958$		0			0
$959$			5.83689i			0.188483i
$960$		0			0
$961$	−80.0129			−2.58106
$962$		0			0
$963$	−3.50759			−0.113030
$964$		0			0
$965$			7.77339i			0.250234i
$966$		0			0
$967$			40.1946i			1.29257i	0.763096	+	0.646286i	$0.223679\pi$
$967$			40.1946i			1.29257i	−0.763096	+	0.646286i	$0.776321\pi$
$968$		0			0
$969$	−2.41507			−0.0775833
$970$		0			0
$971$	−35.6262			−1.14330			−0.571649	−	0.820498i	$-0.693696\pi$
$971$	−35.6262			−1.14330			−0.571649	+	0.820498i	$0.693696\pi$
$972$		0			0
$973$			44.5262i			1.42744i
$974$		0			0
$975$		−	2.81823i		−	0.0902557i
$976$		0			0
$977$			24.2767i			0.776679i	0.921516	+	0.388339i	$0.126951\pi$
$977$			24.2767i			0.776679i	−0.921516	+	0.388339i	$0.873049\pi$
$978$		0			0
$979$		−	21.7082i		−	0.693798i
$980$		0			0
$981$		−	9.73697i		−	0.310877i
$982$		0			0
$983$	51.9923			1.65830			0.829148	−	0.559029i	$-0.188826\pi$
$983$	51.9923			1.65830			0.829148	+	0.559029i	$0.188826\pi$
$984$		0			0
$985$			14.3268i			0.456488i
$986$		0			0
$987$	−15.5423			−0.494718
$988$		0			0
$989$	−43.9767	+	11.8030i	−1.39838	+	0.375312i
$990$		0			0
$991$		−	23.3018i		−	0.740207i	−0.928991	−	0.370103i	$-0.879322\pi$
$991$		−	23.3018i		−	0.740207i	0.928991	−	0.370103i	$-0.120678\pi$
$992$		0			0
$993$	−24.5464			−0.778957
$994$		0			0
$995$			16.9685i			0.537938i
$996$		0			0
$997$	17.2606			0.546649			0.273324	−	0.961922i	$-0.411877\pi$
$997$	17.2606			0.546649			0.273324	+	0.961922i	$0.411877\pi$
$998$		0			0
$999$	−4.95490			−0.156766

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.12	yes	32
4.3	odd	2		5520.2.be.d.1471.11	yes	32
23.22	odd	2		5520.2.be.d.1471.12	yes	32
92.91	even	2	inner	5520.2.be.c.1471.11	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.11	✓	32	92.91	even	2	inner
5520.2.be.c.1471.12	yes	32	1.1	even	1	trivial
5520.2.be.d.1471.11	yes	32	4.3	odd	2
5520.2.be.d.1471.12	yes	32	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:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.be (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.13359 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} +1.00000i q^{5} -2.13359 q^{7} -1.00000 q^{9} +4.16538 q^{11} +2.81823 q^{13} -1.00000 q^{15} -1.82147i q^{17} -1.32589 q^{19} -2.13359i q^{21} +(4.63191 - 1.24316i) q^{23} -1.00000 q^{25} -1.00000i q^{27} -4.92560 q^{29} -10.5363i q^{31} +4.16538i q^{33} -2.13359i q^{35} -4.95490i q^{37} +2.81823i q^{39} +0.656543 q^{41} -9.49430 q^{43} -1.00000i q^{45} -7.28458i q^{47} -2.44777 q^{49} +1.82147 q^{51} +7.24831i q^{53} +4.16538i q^{55} -1.32589i q^{57} -8.42939i q^{59} -3.54481i q^{61} +2.13359 q^{63} +2.81823i q^{65} -10.6962 q^{67} +(1.24316 + 4.63191i) q^{69} +0.897920i q^{71} +4.48027 q^{73} -1.00000i q^{75} -8.88724 q^{77} +7.73462 q^{79} +1.00000 q^{81} -3.14065 q^{83} +1.82147 q^{85} -4.92560i q^{87} -5.21158i q^{89} -6.01297 q^{91} +10.5363 q^{93} -1.32589i q^{95} -0.00964558i q^{97} -4.16538 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 32 * q - 8 * q^7 - 32 * q^9	\( 32 q - 8 q^{7} - 32 q^{9} + 8 q^{11} - 8 q^{13} - 32 q^{15} - 32 q^{25} + 4 q^{29} + 20 q^{41} + 52 q^{49} - 4 q^{51} + 8 q^{63} + 32 q^{67} - 40 q^{73} - 24 q^{77} + 32 q^{79} + 32 q^{81} - 4 q^{85} - 48 q^{91} - 8 q^{99}+O(q^{100}) \) 32 * q - 8 * q^7 - 32 * q^9 + 8 * q^11 - 8 * q^13 - 32 * q^15 - 32 * q^25 + 4 * q^29 + 20 * q^41 + 52 * q^49 - 4 * q^51 + 8 * q^63 + 32 * q^67 - 40 * q^73 - 24 * q^77 + 32 * q^79 + 32 * q^81 - 4 * q^85 - 48 * q^91 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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