LMFDB - Embedded newform 5520.2.be.c.1471.17

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			1471.17
Character	$\chi$	$=$	5520.1471
Dual form			5520.2.be.c.1471.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} -1.00000i q^{5} -0.143131 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} -1.00000i q^{5} -0.143131 q^{7} -1.00000 q^{9} -4.80659 q^{11} +4.78151 q^{13} -1.00000 q^{15} +4.86743i q^{17} -0.379340 q^{19} +0.143131i q^{21} +(-1.15365 - 4.65501i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +8.39933 q^{29} +2.55983i q^{31} +4.80659i q^{33} +0.143131i q^{35} -5.56282i q^{37} -4.78151i q^{39} -5.17202 q^{41} -1.03505 q^{43} +1.00000i q^{45} -8.93486i q^{47} -6.97951 q^{49} +4.86743 q^{51} -13.3413i q^{53} +4.80659i q^{55} +0.379340i q^{57} -6.57340i q^{59} +7.49837i q^{61} +0.143131 q^{63} -4.78151i q^{65} -4.96888 q^{67} +(-4.65501 + 1.15365i) q^{69} +6.77385i q^{71} -8.04529 q^{73} +1.00000i q^{75} +0.687974 q^{77} +1.53132 q^{79} +1.00000 q^{81} -14.3973 q^{83} +4.86743 q^{85} -8.39933i q^{87} -12.6001i q^{89} -0.684383 q^{91} +2.55983 q^{93} +0.379340i q^{95} +5.67311i q^{97} +4.80659 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	−0.143131			−0.0540986			−0.0270493	−	0.999634i	$-0.508611\pi$
$7$	−0.143131			−0.0540986			−0.0270493	+	0.999634i	$0.508611\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	−4.80659			−1.44924			−0.724621	−	0.689148i	$-0.757985\pi$
$11$	−4.80659			−1.44924			−0.724621	+	0.689148i	$0.757985\pi$
$12$		0			0
$13$	4.78151			1.32615			0.663075	−	0.748553i	$-0.269251\pi$
$13$	4.78151			1.32615			0.663075	+	0.748553i	$0.269251\pi$
$14$		0			0
$15$	−1.00000			−0.258199
$16$		0			0
$17$			4.86743i			1.18053i	0.807211	+	0.590263i	$0.200976\pi$
$17$			4.86743i			1.18053i	−0.807211	+	0.590263i	$0.799024\pi$
$18$		0			0
$19$	−0.379340			−0.0870266			−0.0435133	−	0.999053i	$-0.513855\pi$
$19$	−0.379340			−0.0870266			−0.0435133	+	0.999053i	$0.513855\pi$
$20$		0			0
$21$			0.143131i			0.0312338i
$22$		0			0
$23$	−1.15365	−	4.65501i	−0.240552	−	0.970636i
$23$	−1.15365	−	4.65501i	−0.240552	−	0.970636i
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$			1.00000i			0.192450i
$28$		0			0
$29$	8.39933			1.55972			0.779858	−	0.625956i	$-0.215291\pi$
$29$	8.39933			1.55972			0.779858	+	0.625956i	$0.215291\pi$
$30$		0			0
$31$			2.55983i			0.459759i	0.973219	+	0.229879i	$0.0738332\pi$
$31$			2.55983i			0.459759i	−0.973219	+	0.229879i	$0.926167\pi$
$32$		0			0
$33$			4.80659i			0.836720i
$34$		0			0
$35$			0.143131i			0.0241936i
$36$		0			0
$37$		−	5.56282i		−	0.914521i	−0.889333	−	0.457261i	$-0.848831\pi$
$37$		−	5.56282i		−	0.914521i	0.889333	−	0.457261i	$-0.151169\pi$
$38$		0			0
$39$		−	4.78151i		−	0.765654i
$40$		0			0
$41$	−5.17202			−0.807734			−0.403867	−	0.914818i	$-0.632334\pi$
$41$	−5.17202			−0.807734			−0.403867	+	0.914818i	$0.632334\pi$
$42$		0			0
$43$	−1.03505			−0.157843			−0.0789215	−	0.996881i	$-0.525148\pi$
$43$	−1.03505			−0.157843			−0.0789215	+	0.996881i	$0.525148\pi$
$44$		0			0
$45$			1.00000i			0.149071i
$46$		0			0
$47$		−	8.93486i		−	1.30328i	−0.758527	−	0.651642i	$-0.774081\pi$
$47$		−	8.93486i		−	1.30328i	0.758527	−	0.651642i	$-0.225919\pi$
$48$		0			0
$49$	−6.97951			−0.997073
$50$		0			0
$51$	4.86743			0.681577
$52$		0			0
$53$		−	13.3413i		−	1.83256i	−0.400534	−	0.916282i	$-0.631175\pi$
$53$		−	13.3413i		−	1.83256i	0.400534	−	0.916282i	$-0.368825\pi$
$54$		0			0
$55$			4.80659i			0.648121i
$56$		0			0
$57$			0.379340i			0.0502448i
$58$		0			0
$59$		−	6.57340i		−	0.855783i	−0.903830	−	0.427892i	$-0.859257\pi$
$59$		−	6.57340i		−	0.855783i	0.903830	−	0.427892i	$-0.140743\pi$
$60$		0			0
$61$			7.49837i			0.960068i	0.877250	+	0.480034i	$0.159376\pi$
$61$			7.49837i			0.960068i	−0.877250	+	0.480034i	$0.840624\pi$
$62$		0			0
$63$	0.143131			0.0180329
$64$		0			0
$65$		−	4.78151i		−	0.593073i
$66$		0			0
$67$	−4.96888			−0.607046			−0.303523	−	0.952824i	$-0.598163\pi$
$67$	−4.96888			−0.607046			−0.303523	+	0.952824i	$0.598163\pi$
$68$		0			0
$69$	−4.65501	+	1.15365i	−0.560397	+	0.138883i
$70$		0			0
$71$			6.77385i			0.803908i	0.915660	+	0.401954i	$0.131669\pi$
$71$			6.77385i			0.803908i	−0.915660	+	0.401954i	$0.868331\pi$
$72$		0			0
$73$	−8.04529			−0.941631			−0.470815	−	0.882232i	$-0.656040\pi$
$73$	−8.04529			−0.941631			−0.470815	+	0.882232i	$0.656040\pi$
$74$		0			0
$75$			1.00000i			0.115470i
$76$		0			0
$77$	0.687974			0.0784019
$78$		0			0
$79$	1.53132			0.172287			0.0861433	−	0.996283i	$-0.472546\pi$
$79$	1.53132			0.172287			0.0861433	+	0.996283i	$0.472546\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	−14.3973			−1.58031			−0.790157	−	0.612905i	$-0.790001\pi$
$83$	−14.3973			−1.58031			−0.790157	+	0.612905i	$0.790001\pi$
$84$		0			0
$85$	4.86743			0.527947
$86$		0			0
$87$		−	8.39933i		−	0.900503i
$88$		0			0
$89$		−	12.6001i		−	1.33561i	−0.744337	−	0.667804i	$-0.767234\pi$
$89$		−	12.6001i		−	1.33561i	0.744337	−	0.667804i	$-0.232766\pi$
$90$		0			0
$91$	−0.684383			−0.0717429
$92$		0			0
$93$	2.55983			0.265442
$94$		0			0
$95$			0.379340i			0.0389195i
$96$		0			0
$97$			5.67311i			0.576017i	0.957628	+	0.288009i	$0.0929932\pi$
$97$			5.67311i			0.576017i	−0.957628	+	0.288009i	$0.907007\pi$
$98$		0			0
$99$	4.80659			0.483081
$100$		0			0
$101$	−17.3244			−1.72384			−0.861921	−	0.507043i	$-0.830739\pi$
$101$	−17.3244			−1.72384			−0.861921	+	0.507043i	$0.830739\pi$
$102$		0			0
$103$	−12.2942			−1.21138			−0.605691	−	0.795700i	$-0.707103\pi$
$103$	−12.2942			−1.21138			−0.605691	+	0.795700i	$0.707103\pi$
$104$		0			0
$105$	0.143131			0.0139682
$106$		0			0
$107$	−1.02774			−0.0993557			−0.0496779	−	0.998765i	$-0.515819\pi$
$107$	−1.02774			−0.0993557			−0.0496779	+	0.998765i	$0.515819\pi$
$108$		0			0
$109$			16.9971i			1.62802i	0.580849	+	0.814011i	$0.302721\pi$
$109$			16.9971i			1.62802i	−0.580849	+	0.814011i	$0.697279\pi$
$110$		0			0
$111$	−5.56282			−0.527999
$112$		0			0
$113$			0.637571i			0.0599776i	0.999550	+	0.0299888i	$0.00954716\pi$
$113$			0.637571i			0.0599776i	−0.999550	+	0.0299888i	$0.990453\pi$
$114$		0			0
$115$	−4.65501	+	1.15365i	−0.434082	+	0.107578i
$116$		0			0
$117$	−4.78151			−0.442050
$118$		0			0
$119$		−	0.696682i		−	0.0638648i
$120$		0			0
$121$	12.1033			1.10030
$122$		0			0
$123$			5.17202i			0.466345i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	3.74112i		−	0.331970i	−0.986128	−	0.165985i	$-0.946920\pi$
$127$		−	3.74112i		−	0.331970i	0.986128	−	0.165985i	$-0.0530804\pi$
$128$		0			0
$129$			1.03505i			0.0911308i
$130$		0			0
$131$			11.9550i			1.04451i	0.852790	+	0.522255i	$0.174909\pi$
$131$			11.9550i			1.04451i	−0.852790	+	0.522255i	$0.825091\pi$
$132$		0			0
$133$	0.0542955			0.00470802
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$			8.16863i			0.697893i	0.937143	+	0.348947i	$0.113461\pi$
$137$			8.16863i			0.697893i	−0.937143	+	0.348947i	$0.886539\pi$
$138$		0			0
$139$			0.870322i			0.0738198i	0.999319	+	0.0369099i	$0.0117514\pi$
$139$			0.870322i			0.0738198i	−0.999319	+	0.0369099i	$0.988249\pi$
$140$		0			0
$141$	−8.93486			−0.752451
$142$		0			0
$143$	−22.9827			−1.92191
$144$		0			0
$145$		−	8.39933i		−	0.697526i
$146$		0			0
$147$			6.97951i			0.575661i
$148$		0			0
$149$		−	11.4050i		−	0.934337i	−0.884168	−	0.467168i	$-0.845274\pi$
$149$		−	11.4050i		−	0.934337i	0.884168	−	0.467168i	$-0.154726\pi$
$150$		0			0
$151$		−	5.66516i		−	0.461024i	−0.973069	−	0.230512i	$-0.925960\pi$
$151$		−	5.66516i		−	0.461024i	0.973069	−	0.230512i	$-0.0740401\pi$
$152$		0			0
$153$		−	4.86743i		−	0.393509i
$154$		0			0
$155$	2.55983			0.205610
$156$		0			0
$157$			5.94964i			0.474833i	0.971408	+	0.237417i	$0.0763006\pi$
$157$			5.94964i			0.474833i	−0.971408	+	0.237417i	$0.923699\pi$
$158$		0			0
$159$	−13.3413			−1.05803
$160$		0			0
$161$	0.165123	+	0.666278i	0.0130135	+	0.0525100i
$162$		0			0
$163$			12.0905i			0.946997i	0.880795	+	0.473499i	$0.157009\pi$
$163$			12.0905i			0.946997i	−0.880795	+	0.473499i	$0.842991\pi$
$164$		0			0
$165$	4.80659			0.374193
$166$		0			0
$167$		−	14.1105i		−	1.09191i	−0.837816	−	0.545953i	$-0.816168\pi$
$167$		−	14.1105i		−	1.09191i	0.837816	−	0.545953i	$-0.183832\pi$
$168$		0			0
$169$	9.86279			0.758676
$170$		0			0
$171$	0.379340			0.0290089
$172$		0			0
$173$	3.68070			0.279838			0.139919	−	0.990163i	$-0.455316\pi$
$173$	3.68070			0.279838			0.139919	+	0.990163i	$0.455316\pi$
$174$		0			0
$175$	0.143131			0.0108197
$176$		0			0
$177$	−6.57340			−0.494087
$178$		0			0
$179$			9.27393i			0.693166i	0.938019	+	0.346583i	$0.112658\pi$
$179$			9.27393i			0.693166i	−0.938019	+	0.346583i	$0.887342\pi$
$180$		0			0
$181$			5.10162i			0.379200i	0.981861	+	0.189600i	$0.0607191\pi$
$181$			5.10162i			0.379200i	−0.981861	+	0.189600i	$0.939281\pi$
$182$		0			0
$183$	7.49837			0.554295
$184$		0			0
$185$	−5.56282			−0.408986
$186$		0			0
$187$		−	23.3958i		−	1.71087i
$188$		0			0
$189$		−	0.143131i		−	0.0104113i
$190$		0			0
$191$	−20.1633			−1.45897			−0.729484	−	0.683997i	$-0.760240\pi$
$191$	−20.1633			−1.45897			−0.729484	+	0.683997i	$0.760240\pi$
$192$		0			0
$193$	1.06748			0.0768392			0.0384196	−	0.999262i	$-0.487768\pi$
$193$	1.06748			0.0768392			0.0384196	+	0.999262i	$0.487768\pi$
$194$		0			0
$195$	−4.78151			−0.342411
$196$		0			0
$197$	4.31277			0.307272			0.153636	−	0.988128i	$-0.450902\pi$
$197$	4.31277			0.307272			0.153636	+	0.988128i	$0.450902\pi$
$198$		0			0
$199$	−9.35032			−0.662827			−0.331413	−	0.943486i	$-0.607525\pi$
$199$	−9.35032			−0.662827			−0.331413	+	0.943486i	$0.607525\pi$
$200$		0			0
$201$			4.96888i			0.350478i
$202$		0			0
$203$	−1.20221			−0.0843784
$204$		0			0
$205$			5.17202i			0.361230i
$206$		0			0
$207$	1.15365	+	4.65501i	0.0801840	+	0.323545i
$208$		0			0
$209$	1.82333			0.126123
$210$		0			0
$211$		−	19.8715i		−	1.36801i	−0.729478	−	0.684004i	$-0.760237\pi$
$211$		−	19.8715i		−	1.36801i	0.729478	−	0.684004i	$-0.239763\pi$
$212$		0			0
$213$	6.77385			0.464136
$214$		0			0
$215$			1.03505i			0.0705896i
$216$		0			0
$217$		−	0.366392i		−	0.0248723i
$218$		0			0
$219$			8.04529i			0.543651i
$220$		0			0
$221$			23.2737i			1.56556i
$222$		0			0
$223$			13.4715i			0.902116i	0.892495	+	0.451058i	$0.148953\pi$
$223$			13.4715i			0.902116i	−0.892495	+	0.451058i	$0.851047\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−3.56731			−0.236771			−0.118385	−	0.992968i	$-0.537772\pi$
$227$	−3.56731			−0.236771			−0.118385	+	0.992968i	$0.537772\pi$
$228$		0			0
$229$			9.73343i			0.643203i	0.946875	+	0.321602i	$0.104221\pi$
$229$			9.73343i			0.643203i	−0.946875	+	0.321602i	$0.895779\pi$
$230$		0			0
$231$		−	0.687974i		−	0.0452654i
$232$		0			0
$233$	19.7008			1.29065			0.645323	−	0.763910i	$-0.276723\pi$
$233$	19.7008			1.29065			0.645323	+	0.763910i	$0.276723\pi$
$234$		0			0
$235$	−8.93486			−0.582846
$236$		0			0
$237$		−	1.53132i		−	0.0994697i
$238$		0			0
$239$			9.71151i			0.628185i	0.949392	+	0.314093i	$0.101700\pi$
$239$			9.71151i			0.628185i	−0.949392	+	0.314093i	$0.898300\pi$
$240$		0			0
$241$			11.3349i			0.730144i	0.930979	+	0.365072i	$0.118956\pi$
$241$			11.3349i			0.730144i	−0.930979	+	0.365072i	$0.881044\pi$
$242$		0			0
$243$		−	1.00000i		−	0.0641500i
$244$		0			0
$245$			6.97951i			0.445905i
$246$		0			0
$247$	−1.81382			−0.115410
$248$		0			0
$249$			14.3973i			0.912394i
$250$		0			0
$251$	−12.4895			−0.788329			−0.394164	−	0.919040i	$-0.628966\pi$
$251$	−12.4895			−0.788329			−0.394164	+	0.919040i	$0.628966\pi$
$252$		0			0
$253$	5.54511	+	22.3747i	0.348618	+	1.40669i
$254$		0			0
$255$		−	4.86743i		−	0.304810i
$256$		0			0
$257$	−4.82467			−0.300954			−0.150477	−	0.988613i	$-0.548081\pi$
$257$	−4.82467			−0.300954			−0.150477	+	0.988613i	$0.548081\pi$
$258$		0			0
$259$			0.796214i			0.0494743i
$260$		0			0
$261$	−8.39933			−0.519905
$262$		0			0
$263$	−6.75865			−0.416756			−0.208378	−	0.978048i	$-0.566818\pi$
$263$	−6.75865			−0.416756			−0.208378	+	0.978048i	$0.566818\pi$
$264$		0			0
$265$	−13.3413			−0.819547
$266$		0			0
$267$	−12.6001			−0.771114
$268$		0			0
$269$	9.49855			0.579137			0.289568	−	0.957157i	$-0.406488\pi$
$269$	9.49855			0.579137			0.289568	+	0.957157i	$0.406488\pi$
$270$		0			0
$271$		−	17.6288i		−	1.07087i	−0.844576	−	0.535435i	$-0.820148\pi$
$271$		−	17.6288i		−	1.07087i	0.844576	−	0.535435i	$-0.179852\pi$
$272$		0			0
$273$			0.684383i			0.0414208i
$274$		0			0
$275$	4.80659			0.289848
$276$		0			0
$277$	8.47253			0.509065			0.254533	−	0.967064i	$-0.418078\pi$
$277$	8.47253			0.509065			0.254533	+	0.967064i	$0.418078\pi$
$278$		0			0
$279$		−	2.55983i		−	0.153253i
$280$		0			0
$281$		−	10.6546i		−	0.635603i	−0.948157	−	0.317801i	$-0.897055\pi$
$281$		−	10.6546i		−	0.635603i	0.948157	−	0.317801i	$-0.102945\pi$
$282$		0			0
$283$	9.84325			0.585120			0.292560	−	0.956247i	$-0.405493\pi$
$283$	9.84325			0.585120			0.292560	+	0.956247i	$0.405493\pi$
$284$		0			0
$285$	0.379340			0.0224702
$286$		0			0
$287$	0.740278			0.0436973
$288$		0			0
$289$	−6.69189			−0.393641
$290$		0			0
$291$	5.67311			0.332564
$292$		0			0
$293$			29.3775i			1.71625i	0.513439	+	0.858126i	$0.328371\pi$
$293$			29.3775i			1.71625i	−0.513439	+	0.858126i	$0.671629\pi$
$294$		0			0
$295$	−6.57340			−0.382718
$296$		0			0
$297$		−	4.80659i		−	0.278907i
$298$		0			0
$299$	−5.51617	−	22.2579i	−0.319008	−	1.28721i
$300$		0			0
$301$	0.148148			0.00853909
$302$		0			0
$303$			17.3244i			0.995260i
$304$		0			0
$305$	7.49837			0.429355
$306$		0			0
$307$		−	15.6395i		−	0.892591i	−0.894886	−	0.446296i	$-0.852743\pi$
$307$		−	15.6395i		−	0.892591i	0.894886	−	0.446296i	$-0.147257\pi$
$308$		0			0
$309$			12.2942i			0.699391i
$310$		0			0
$311$		−	16.7393i		−	0.949201i	−0.880201	−	0.474600i	$-0.842593\pi$
$311$		−	16.7393i		−	0.949201i	0.880201	−	0.474600i	$-0.157407\pi$
$312$		0			0
$313$		−	17.0663i		−	0.964646i	−0.875993	−	0.482323i	$-0.839793\pi$
$313$		−	17.0663i		−	0.964646i	0.875993	−	0.482323i	$-0.160207\pi$
$314$		0			0
$315$		−	0.143131i		−	0.00806454i
$316$		0			0
$317$	−29.6407			−1.66479			−0.832394	−	0.554184i	$-0.813030\pi$
$317$	−29.6407			−1.66479			−0.832394	+	0.554184i	$0.813030\pi$
$318$		0			0
$319$	−40.3721			−2.26041
$320$		0			0
$321$			1.02774i			0.0573630i
$322$		0			0
$323$		−	1.84641i		−	0.102737i
$324$		0			0
$325$	−4.78151			−0.265230
$326$		0			0
$327$	16.9971			0.939939
$328$		0			0
$329$			1.27886i			0.0705058i
$330$		0			0
$331$			4.03408i			0.221733i	0.993835	+	0.110866i	$0.0353626\pi$
$331$			4.03408i			0.221733i	−0.993835	+	0.110866i	$0.964637\pi$
$332$		0			0
$333$			5.56282i			0.304840i
$334$		0			0
$335$			4.96888i			0.271479i
$336$		0			0
$337$		−	1.54962i		−	0.0844132i	−0.999109	−	0.0422066i	$-0.986561\pi$
$337$		−	1.54962i		−	0.0844132i	0.999109	−	0.0422066i	$-0.0134388\pi$
$338$		0			0
$339$	0.637571			0.0346281
$340$		0			0
$341$		−	12.3040i		−	0.666302i
$342$		0			0
$343$	2.00091			0.108039
$344$		0			0
$345$	1.15365	+	4.65501i	0.0621103	+	0.250617i
$346$		0			0
$347$			9.87984i			0.530377i	0.964197	+	0.265189i	$0.0854343\pi$
$347$			9.87984i			0.530377i	−0.964197	+	0.265189i	$0.914566\pi$
$348$		0			0
$349$	6.41716			0.343503			0.171751	−	0.985140i	$-0.445057\pi$
$349$	6.41716			0.343503			0.171751	+	0.985140i	$0.445057\pi$
$350$		0			0
$351$			4.78151i			0.255218i
$352$		0			0
$353$	−6.23647			−0.331934			−0.165967	−	0.986131i	$-0.553074\pi$
$353$	−6.23647			−0.331934			−0.165967	+	0.986131i	$0.553074\pi$
$354$		0			0
$355$	6.77385			0.359519
$356$		0			0
$357$	−0.696682			−0.0368723
$358$		0			0
$359$	−4.19870			−0.221599			−0.110799	−	0.993843i	$-0.535341\pi$
$359$	−4.19870			−0.221599			−0.110799	+	0.993843i	$0.535341\pi$
$360$		0			0
$361$	−18.8561			−0.992426
$362$		0			0
$363$		−	12.1033i		−	0.635260i
$364$		0			0
$365$			8.04529i			0.421110i
$366$		0			0
$367$	5.21796			0.272375			0.136188	−	0.990683i	$-0.456515\pi$
$367$	5.21796			0.272375			0.136188	+	0.990683i	$0.456515\pi$
$368$		0			0
$369$	5.17202			0.269245
$370$		0			0
$371$			1.90955i			0.0991391i
$372$		0			0
$373$		−	24.8539i		−	1.28689i	−0.765493	−	0.643444i	$-0.777505\pi$
$373$		−	24.8539i		−	1.28689i	0.765493	−	0.643444i	$-0.222495\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	40.1614			2.06842
$378$		0			0
$379$	14.2322			0.731058			0.365529	−	0.930800i	$-0.380888\pi$
$379$	14.2322			0.731058			0.365529	+	0.930800i	$0.380888\pi$
$380$		0			0
$381$	−3.74112			−0.191663
$382$		0			0
$383$	23.6762			1.20980			0.604898	−	0.796303i	$-0.293214\pi$
$383$	23.6762			1.20980			0.604898	+	0.796303i	$0.293214\pi$
$384$		0			0
$385$		−	0.687974i		−	0.0350624i
$386$		0			0
$387$	1.03505			0.0526144
$388$		0			0
$389$		−	6.73328i		−	0.341391i	−0.985324	−	0.170695i	$-0.945399\pi$
$389$		−	6.73328i		−	0.341391i	0.985324	−	0.170695i	$-0.0546014\pi$
$390$		0			0
$391$	22.6579	−	5.61530i	1.14586	−	0.283978i
$392$		0			0
$393$	11.9550			0.603048
$394$		0			0
$395$		−	1.53132i		−	0.0770489i
$396$		0			0
$397$	9.27378			0.465438			0.232719	−	0.972544i	$-0.425238\pi$
$397$	9.27378			0.465438			0.232719	+	0.972544i	$0.425238\pi$
$398$		0			0
$399$		−	0.0542955i		−	0.00271817i
$400$		0			0
$401$		−	38.6681i		−	1.93099i	−0.260421	−	0.965495i	$-0.583861\pi$
$401$		−	38.6681i		−	1.93099i	0.260421	−	0.965495i	$-0.416139\pi$
$402$		0			0
$403$			12.2398i			0.609709i
$404$		0			0
$405$		−	1.00000i		−	0.0496904i
$406$		0			0
$407$			26.7382i			1.32536i
$408$		0			0
$409$	−23.1274			−1.14358			−0.571789	−	0.820401i	$-0.693750\pi$
$409$	−23.1274			−1.14358			−0.571789	+	0.820401i	$0.693750\pi$
$410$		0			0
$411$	8.16863			0.402929
$412$		0			0
$413$			0.940859i			0.0462967i
$414$		0			0
$415$			14.3973i			0.706738i
$416$		0			0
$417$	0.870322			0.0426199
$418$		0			0
$419$	−2.95569			−0.144395			−0.0721974	−	0.997390i	$-0.523001\pi$
$419$	−2.95569			−0.144395			−0.0721974	+	0.997390i	$0.523001\pi$
$420$		0			0
$421$			4.13502i			0.201528i	0.994910	+	0.100764i	$0.0321288\pi$
$421$			4.13502i			0.201528i	−0.994910	+	0.100764i	$0.967871\pi$
$422$		0			0
$423$			8.93486i			0.434428i
$424$		0			0
$425$		−	4.86743i		−	0.236105i
$426$		0			0
$427$		−	1.07325i		−	0.0519383i
$428$		0			0
$429$			22.9827i			1.10962i
$430$		0			0
$431$	−8.46802			−0.407890			−0.203945	−	0.978982i	$-0.565376\pi$
$431$	−8.46802			−0.407890			−0.203945	+	0.978982i	$0.565376\pi$
$432$		0			0
$433$			17.3572i			0.834132i	0.908876	+	0.417066i	$0.136942\pi$
$433$			17.3572i			0.834132i	−0.908876	+	0.417066i	$0.863058\pi$
$434$		0			0
$435$	−8.39933			−0.402717
$436$		0			0
$437$	0.437625	+	1.76583i	0.0209344	+	0.0844712i
$438$		0			0
$439$		−	24.8336i		−	1.18524i	−0.805481	−	0.592622i	$-0.798093\pi$
$439$		−	24.8336i		−	1.18524i	0.805481	−	0.592622i	$-0.201907\pi$
$440$		0			0
$441$	6.97951			0.332358
$442$		0			0
$443$			26.8894i			1.27756i	0.769391	+	0.638778i	$0.220560\pi$
$443$			26.8894i			1.27756i	−0.769391	+	0.638778i	$0.779440\pi$
$444$		0			0
$445$	−12.6001			−0.597302
$446$		0			0
$447$	−11.4050			−0.539440
$448$		0			0
$449$	−33.4193			−1.57716			−0.788578	−	0.614935i	$-0.789182\pi$
$449$	−33.4193			−1.57716			−0.788578	+	0.614935i	$0.789182\pi$
$450$		0			0
$451$	24.8598			1.17060
$452$		0			0
$453$	−5.66516			−0.266172
$454$		0			0
$455$			0.684383i			0.0320844i
$456$		0			0
$457$			7.03968i			0.329302i	0.986352	+	0.164651i	$0.0526498\pi$
$457$			7.03968i			0.329302i	−0.986352	+	0.164651i	$0.947350\pi$
$458$		0			0
$459$	−4.86743			−0.227192
$460$		0			0
$461$	−32.0115			−1.49092			−0.745462	−	0.666548i	$-0.767771\pi$
$461$	−32.0115			−1.49092			−0.745462	+	0.666548i	$0.767771\pi$
$462$		0			0
$463$		−	16.1154i		−	0.748945i	−0.927238	−	0.374472i	$-0.877824\pi$
$463$		−	16.1154i		−	0.748945i	0.927238	−	0.374472i	$-0.122176\pi$
$464$		0			0
$465$		−	2.55983i		−	0.118709i
$466$		0			0
$467$	−40.8400			−1.88985			−0.944926	−	0.327285i	$-0.893866\pi$
$467$	−40.8400			−1.88985			−0.944926	+	0.327285i	$0.893866\pi$
$468$		0			0
$469$	0.711203			0.0328403
$470$		0			0
$471$	5.94964			0.274145
$472$		0			0
$473$	4.97505			0.228753
$474$		0			0
$475$	0.379340			0.0174053
$476$		0			0
$477$			13.3413i			0.610855i
$478$		0			0
$479$	−12.5812			−0.574848			−0.287424	−	0.957803i	$-0.592799\pi$
$479$	−12.5812			−0.574848			−0.287424	+	0.957803i	$0.592799\pi$
$480$		0			0
$481$		−	26.5986i		−	1.21279i
$482$		0			0
$483$	0.666278	−	0.165123i	0.0303167	−	0.00751336i
$484$		0			0
$485$	5.67311			0.257603
$486$		0			0
$487$			31.0193i			1.40562i	0.711378	+	0.702810i	$0.248072\pi$
$487$			31.0193i			1.40562i	−0.711378	+	0.702810i	$0.751928\pi$
$488$		0			0
$489$	12.0905			0.546749
$490$		0			0
$491$		−	35.8601i		−	1.61835i	−0.587571	−	0.809173i	$-0.699916\pi$
$491$		−	35.8601i		−	1.61835i	0.587571	−	0.809173i	$-0.300084\pi$
$492$		0			0
$493$			40.8832i			1.84129i
$494$		0			0
$495$		−	4.80659i		−	0.216040i
$496$		0			0
$497$		−	0.969550i		−	0.0434903i
$498$		0			0
$499$			10.0149i			0.448330i	0.974551	+	0.224165i	$0.0719654\pi$
$499$			10.0149i			0.448330i	−0.974551	+	0.224165i	$0.928035\pi$
$500$		0			0
$501$	−14.1105			−0.630413
$502$		0			0
$503$	−38.5120			−1.71716			−0.858582	−	0.512676i	$-0.828654\pi$
$503$	−38.5120			−1.71716			−0.858582	+	0.512676i	$0.828654\pi$
$504$		0			0
$505$			17.3244i			0.770925i
$506$		0			0
$507$		−	9.86279i		−	0.438022i
$508$		0			0
$509$	31.4747			1.39509			0.697546	−	0.716540i	$-0.254275\pi$
$509$	31.4747			1.39509			0.697546	+	0.716540i	$0.254275\pi$
$510$		0			0
$511$	1.15153			0.0509409
$512$		0			0
$513$		−	0.379340i		−	0.0167483i
$514$		0			0
$515$			12.2942i			0.541746i
$516$		0			0
$517$			42.9462i			1.88877i
$518$		0			0
$519$		−	3.68070i		−	0.161565i
$520$		0			0
$521$		−	21.5620i		−	0.944647i	−0.881425	−	0.472324i	$-0.843415\pi$
$521$		−	21.5620i		−	0.944647i	0.881425	−	0.472324i	$-0.156585\pi$
$522$		0			0
$523$	40.7270			1.78087			0.890434	−	0.455112i	$-0.150401\pi$
$523$	40.7270			1.78087			0.890434	+	0.455112i	$0.150401\pi$
$524$		0			0
$525$		−	0.143131i		−	0.00624677i
$526$		0			0
$527$	−12.4598			−0.542757
$528$		0			0
$529$	−20.3382	+	10.7405i	−0.884269	+	0.466977i
$530$		0			0
$531$			6.57340i			0.285261i
$532$		0			0
$533$	−24.7300			−1.07118
$534$		0			0
$535$			1.02774i			0.0444332i
$536$		0			0
$537$	9.27393			0.400200
$538$		0			0
$539$	33.5477			1.44500
$540$		0			0
$541$	−38.8931			−1.67215			−0.836073	−	0.548618i	$-0.815154\pi$
$541$	−38.8931			−1.67215			−0.836073	+	0.548618i	$0.815154\pi$
$542$		0			0
$543$	5.10162			0.218931
$544$		0			0
$545$	16.9971			0.728074
$546$		0			0
$547$		−	0.658772i		−	0.0281671i	−0.999901	−	0.0140835i	$-0.995517\pi$
$547$		−	0.658772i		−	0.0281671i	0.999901	−	0.0140835i	$-0.00448308\pi$
$548$		0			0
$549$		−	7.49837i		−	0.320023i
$550$		0			0
$551$	−3.18620			−0.135737
$552$		0			0
$553$	−0.219179			−0.00932046
$554$		0			0
$555$			5.56282i			0.236128i
$556$		0			0
$557$			4.89839i			0.207551i	0.994601	+	0.103776i	$0.0330924\pi$
$557$			4.89839i			0.207551i	−0.994601	+	0.103776i	$0.966908\pi$
$558$		0			0
$559$	−4.94908			−0.209324
$560$		0			0
$561$	−23.3958			−0.987770
$562$		0			0
$563$	−8.22969			−0.346840			−0.173420	−	0.984848i	$-0.555482\pi$
$563$	−8.22969			−0.346840			−0.173420	+	0.984848i	$0.555482\pi$
$564$		0			0
$565$	0.637571			0.0268228
$566$		0			0
$567$	−0.143131			−0.00601095
$568$		0			0
$569$		−	4.84217i		−	0.202994i	−0.994836	−	0.101497i	$-0.967637\pi$
$569$		−	4.84217i		−	0.202994i	0.994836	−	0.101497i	$-0.0323632\pi$
$570$		0			0
$571$	−18.6944			−0.782335			−0.391167	−	0.920320i	$-0.627929\pi$
$571$	−18.6944			−0.782335			−0.391167	+	0.920320i	$0.627929\pi$
$572$		0			0
$573$			20.1633i			0.842336i
$574$		0			0
$575$	1.15365	+	4.65501i	0.0481104	+	0.194127i
$576$		0			0
$577$	37.9160			1.57847			0.789233	−	0.614094i	$-0.210479\pi$
$577$	37.9160			1.57847			0.789233	+	0.614094i	$0.210479\pi$
$578$		0			0
$579$		−	1.06748i		−	0.0443631i
$580$		0			0
$581$	2.06071			0.0854927
$582$		0			0
$583$			64.1260i			2.65583i
$584$		0			0
$585$			4.78151i			0.197691i
$586$		0			0
$587$		−	4.52386i		−	0.186720i	−0.995632	−	0.0933599i	$-0.970239\pi$
$587$		−	4.52386i		−	0.186720i	0.995632	−	0.0933599i	$-0.0297607\pi$
$588$		0			0
$589$		−	0.971046i		−	0.0400112i
$590$		0			0
$591$		−	4.31277i		−	0.177404i
$592$		0			0
$593$	21.4620			0.881338			0.440669	−	0.897670i	$-0.354741\pi$
$593$	21.4620			0.881338			0.440669	+	0.897670i	$0.354741\pi$
$594$		0			0
$595$	−0.696682			−0.0285612
$596$		0			0
$597$			9.35032i			0.382683i
$598$		0			0
$599$		−	43.6864i		−	1.78498i	−0.451069	−	0.892489i	$-0.648957\pi$
$599$		−	43.6864i		−	1.78498i	0.451069	−	0.892489i	$-0.351043\pi$
$600$		0			0
$601$	−43.6463			−1.78037			−0.890185	−	0.455599i	$-0.849425\pi$
$601$	−43.6463			−1.78037			−0.890185	+	0.455599i	$0.849425\pi$
$602$		0			0
$603$	4.96888			0.202349
$604$		0			0
$605$		−	12.1033i		−	0.492070i
$606$		0			0
$607$			13.8775i			0.563269i	0.959522	+	0.281635i	$0.0908766\pi$
$607$			13.8775i			0.563269i	−0.959522	+	0.281635i	$0.909123\pi$
$608$		0			0
$609$			1.20221i			0.0487159i
$610$		0			0
$611$		−	42.7221i		−	1.72835i
$612$		0			0
$613$			18.7574i			0.757606i	0.925477	+	0.378803i	$0.123664\pi$
$613$			18.7574i			0.757606i	−0.925477	+	0.378803i	$0.876336\pi$
$614$		0			0
$615$	5.17202			0.208556
$616$		0			0
$617$		−	28.7105i		−	1.15584i	−0.816092	−	0.577922i	$-0.803864\pi$
$617$		−	28.7105i		−	1.15584i	0.816092	−	0.577922i	$-0.196136\pi$
$618$		0			0
$619$	−0.458137			−0.0184141			−0.00920703	−	0.999958i	$-0.502931\pi$
$619$	−0.458137			−0.0184141			−0.00920703	+	0.999958i	$0.502931\pi$
$620$		0			0
$621$	4.65501	−	1.15365i	0.186799	−	0.0462943i
$622$		0			0
$623$			1.80347i			0.0722545i
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$		−	1.82333i		−	0.0728169i
$628$		0			0
$629$	27.0766			1.07962
$630$		0			0
$631$	−37.7755			−1.50382			−0.751910	−	0.659265i	$-0.770867\pi$
$631$	−37.7755			−1.50382			−0.751910	+	0.659265i	$0.770867\pi$
$632$		0			0
$633$	−19.8715			−0.789820
$634$		0			0
$635$	−3.74112			−0.148462
$636$		0			0
$637$	−33.3726			−1.32227
$638$		0			0
$639$		−	6.77385i		−	0.267969i
$640$		0			0
$641$		−	29.5885i		−	1.16868i	−0.811510	−	0.584338i	$-0.801354\pi$
$641$		−	29.5885i		−	1.16868i	0.811510	−	0.584338i	$-0.198646\pi$
$642$		0			0
$643$	−47.0949			−1.85724			−0.928620	−	0.371032i	$-0.879004\pi$
$643$	−47.0949			−1.85724			−0.928620	+	0.371032i	$0.879004\pi$
$644$		0			0
$645$	1.03505			0.0407549
$646$		0			0
$647$		−	26.6440i		−	1.04748i	−0.851877	−	0.523741i	$-0.824536\pi$
$647$		−	26.6440i		−	1.04748i	0.851877	−	0.523741i	$-0.175464\pi$
$648$		0			0
$649$			31.5956i			1.24024i
$650$		0			0
$651$	−0.366392			−0.0143600
$652$		0			0
$653$	32.1606			1.25854			0.629272	−	0.777185i	$-0.283353\pi$
$653$	32.1606			1.25854			0.629272	+	0.777185i	$0.283353\pi$
$654$		0			0
$655$	11.9550			0.467119
$656$		0			0
$657$	8.04529			0.313877
$658$		0			0
$659$	1.56508			0.0609670			0.0304835	−	0.999535i	$-0.490295\pi$
$659$	1.56508			0.0609670			0.0304835	+	0.999535i	$0.490295\pi$
$660$		0			0
$661$			11.9297i			0.464010i	0.972715	+	0.232005i	$0.0745285\pi$
$661$			11.9297i			0.464010i	−0.972715	+	0.232005i	$0.925471\pi$
$662$		0			0
$663$	23.2737			0.903874
$664$		0			0
$665$		−	0.0542955i		−	0.00210549i
$666$		0			0
$667$	−9.68987	−	39.0989i	−0.375193	−	1.51392i
$668$		0			0
$669$	13.4715			0.520837
$670$		0			0
$671$		−	36.0416i		−	1.39137i
$672$		0			0
$673$	27.9834			1.07868			0.539340	−	0.842088i	$-0.318674\pi$
$673$	27.9834			1.07868			0.539340	+	0.842088i	$0.318674\pi$
$674$		0			0
$675$		−	1.00000i		−	0.0384900i
$676$		0			0
$677$		−	49.5307i		−	1.90362i	−0.306691	−	0.951809i	$-0.599222\pi$
$677$		−	49.5307i		−	1.90362i	0.306691	−	0.951809i	$-0.400778\pi$
$678$		0			0
$679$		−	0.812001i		−	0.0311617i
$680$		0			0
$681$			3.56731i			0.136700i
$682$		0			0
$683$			18.2648i			0.698884i	0.936958	+	0.349442i	$0.113629\pi$
$683$			18.2648i			0.698884i	−0.936958	+	0.349442i	$0.886371\pi$
$684$		0			0
$685$	8.16863			0.312107
$686$		0			0
$687$	9.73343			0.371354
$688$		0			0
$689$		−	63.7913i		−	2.43026i
$690$		0			0
$691$		−	24.6191i		−	0.936554i	−0.883582	−	0.468277i	$-0.844875\pi$
$691$		−	24.6191i		−	0.936554i	0.883582	−	0.468277i	$-0.155125\pi$
$692$		0			0
$693$	−0.687974			−0.0261340
$694$		0			0
$695$	0.870322			0.0330132
$696$		0			0
$697$		−	25.1745i		−	0.953550i
$698$		0			0
$699$		−	19.7008i		−	0.745154i
$700$		0			0
$701$			35.5895i			1.34420i	0.740461	+	0.672099i	$0.234607\pi$
$701$			35.5895i			1.34420i	−0.740461	+	0.672099i	$0.765393\pi$
$702$		0			0
$703$			2.11020i			0.0795877i
$704$		0			0
$705$			8.93486i			0.336506i
$706$		0			0
$707$	2.47966			0.0932574
$708$		0			0
$709$			8.00880i			0.300777i	0.988627	+	0.150388i	$0.0480524\pi$
$709$			8.00880i			0.300777i	−0.988627	+	0.150388i	$0.951948\pi$
$710$		0			0
$711$	−1.53132			−0.0574289
$712$		0			0
$713$	11.9160	−	2.95314i	0.446258	−	0.110596i
$714$		0			0
$715$			22.9827i			0.859506i
$716$		0			0
$717$	9.71151			0.362683
$718$		0			0
$719$			40.1847i			1.49864i	0.662210	+	0.749318i	$0.269619\pi$
$719$			40.1847i			1.49864i	−0.662210	+	0.749318i	$0.730381\pi$
$720$		0			0
$721$	1.75968			0.0655340
$722$		0			0
$723$	11.3349			0.421549
$724$		0			0
$725$	−8.39933			−0.311943
$726$		0			0
$727$	46.1258			1.71071			0.855355	−	0.518042i	$-0.173339\pi$
$727$	46.1258			1.71071			0.855355	+	0.518042i	$0.173339\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	5.03802i		−	0.186338i
$732$		0			0
$733$		−	11.4344i		−	0.422340i	−0.977449	−	0.211170i	$-0.932273\pi$
$733$		−	11.4344i		−	0.422340i	0.977449	−	0.211170i	$-0.0677274\pi$
$734$		0			0
$735$	6.97951			0.257443
$736$		0			0
$737$	23.8834			0.879756
$738$		0			0
$739$		−	10.7005i		−	0.393625i	−0.980441	−	0.196813i	$-0.936941\pi$
$739$		−	10.7005i		−	0.393625i	0.980441	−	0.196813i	$-0.0630591\pi$
$740$		0			0
$741$			1.81382i			0.0666322i
$742$		0			0
$743$	−44.7313			−1.64103			−0.820515	−	0.571624i	$-0.806313\pi$
$743$	−44.7313			−1.64103			−0.820515	+	0.571624i	$0.806313\pi$
$744$		0			0
$745$	−11.4050			−0.417848
$746$		0			0
$747$	14.3973			0.526771
$748$		0			0
$749$	0.147102			0.00537500
$750$		0			0
$751$	15.5462			0.567289			0.283644	−	0.958930i	$-0.408456\pi$
$751$	15.5462			0.567289			0.283644	+	0.958930i	$0.408456\pi$
$752$		0			0
$753$			12.4895i			0.455142i
$754$		0			0
$755$	−5.66516			−0.206176
$756$		0			0
$757$			15.3689i			0.558594i	0.960205	+	0.279297i	$0.0901014\pi$
$757$			15.3689i			0.558594i	−0.960205	+	0.279297i	$0.909899\pi$
$758$		0			0
$759$	22.3747	−	5.54511i	0.812151	−	0.201275i
$760$		0			0
$761$	−26.4309			−0.958121			−0.479060	−	0.877782i	$-0.659023\pi$
$761$	−26.4309			−0.958121			−0.479060	+	0.877782i	$0.659023\pi$
$762$		0			0
$763$		−	2.43281i		−	0.0880737i
$764$		0			0
$765$	−4.86743			−0.175982
$766$		0			0
$767$		−	31.4307i		−	1.13490i
$768$		0			0
$769$		−	51.7442i		−	1.86594i	−0.359948	−	0.932972i	$-0.617206\pi$
$769$		−	51.7442i		−	1.86594i	0.359948	−	0.932972i	$-0.382794\pi$
$770$		0			0
$771$			4.82467i			0.173756i
$772$		0			0
$773$		−	11.3165i		−	0.407026i	−0.979072	−	0.203513i	$-0.934764\pi$
$773$		−	11.3165i		−	0.407026i	0.979072	−	0.203513i	$-0.0652360\pi$
$774$		0			0
$775$		−	2.55983i		−	0.0919517i
$776$		0			0
$777$	0.796214			0.0285640
$778$		0			0
$779$	1.96195			0.0702943
$780$		0			0
$781$		−	32.5591i		−	1.16506i
$782$		0			0
$783$			8.39933i			0.300168i
$784$		0			0
$785$	5.94964			0.212352
$786$		0			0
$787$	20.8240			0.742296			0.371148	−	0.928574i	$-0.378964\pi$
$787$	20.8240			0.742296			0.371148	+	0.928574i	$0.378964\pi$
$788$		0			0
$789$			6.75865i			0.240614i
$790$		0			0
$791$		−	0.0912564i		−	0.00324470i
$792$		0			0
$793$			35.8535i			1.27319i
$794$		0			0
$795$			13.3413i			0.473166i
$796$		0			0
$797$		−	34.5644i		−	1.22433i	−0.790729	−	0.612166i	$-0.790298\pi$
$797$		−	34.5644i		−	1.22433i	0.790729	−	0.612166i	$-0.209702\pi$
$798$		0			0
$799$	43.4898			1.53856
$800$		0			0
$801$			12.6001i			0.445203i
$802$		0			0
$803$	38.6704			1.36465
$804$		0			0
$805$	0.666278	−	0.165123i	0.0234832	−	0.00581983i
$806$		0			0
$807$		−	9.49855i		−	0.334365i
$808$		0			0
$809$	−14.4517			−0.508095			−0.254047	−	0.967192i	$-0.581762\pi$
$809$	−14.4517			−0.508095			−0.254047	+	0.967192i	$0.581762\pi$
$810$		0			0
$811$			24.2239i			0.850616i	0.905049	+	0.425308i	$0.139834\pi$
$811$			24.2239i			0.850616i	−0.905049	+	0.425308i	$0.860166\pi$
$812$		0			0
$813$	−17.6288			−0.618268
$814$		0			0
$815$	12.0905			0.423510
$816$		0			0
$817$	0.392635			0.0137365
$818$		0			0
$819$	0.684383			0.0239143
$820$		0			0
$821$	19.2778			0.672799			0.336400	−	0.941719i	$-0.390791\pi$
$821$	19.2778			0.672799			0.336400	+	0.941719i	$0.390791\pi$
$822$		0			0
$823$		−	32.9410i		−	1.14825i	−0.818767	−	0.574126i	$-0.805342\pi$
$823$		−	32.9410i		−	1.14825i	0.818767	−	0.574126i	$-0.194658\pi$
$824$		0			0
$825$		−	4.80659i		−	0.167344i
$826$		0			0
$827$	26.6343			0.926164			0.463082	−	0.886316i	$-0.346744\pi$
$827$	26.6343			0.926164			0.463082	+	0.886316i	$0.346744\pi$
$828$		0			0
$829$	−5.05373			−0.175523			−0.0877616	−	0.996142i	$-0.527971\pi$
$829$	−5.05373			−0.175523			−0.0877616	+	0.996142i	$0.527971\pi$
$830$		0			0
$831$		−	8.47253i		−	0.293909i
$832$		0			0
$833$		−	33.9723i		−	1.17707i
$834$		0			0
$835$	−14.1105			−0.488316
$836$		0			0
$837$	−2.55983			−0.0884806
$838$		0			0
$839$	11.1301			0.384254			0.192127	−	0.981370i	$-0.438461\pi$
$839$	11.1301			0.384254			0.192127	+	0.981370i	$0.438461\pi$
$840$		0			0
$841$	41.5487			1.43272
$842$		0			0
$843$	−10.6546			−0.366965
$844$		0			0
$845$		−	9.86279i		−	0.339290i
$846$		0			0
$847$	−1.73237			−0.0595248
$848$		0			0
$849$		−	9.84325i		−	0.337819i
$850$		0			0
$851$	−25.8950	+	6.41753i	−0.887668	+	0.219990i
$852$		0			0
$853$	17.8102			0.609809			0.304904	−	0.952383i	$-0.401375\pi$
$853$	17.8102			0.609809			0.304904	+	0.952383i	$0.401375\pi$
$854$		0			0
$855$		−	0.379340i		−	0.0129732i
$856$		0			0
$857$	−42.1930			−1.44129			−0.720643	−	0.693307i	$-0.756153\pi$
$857$	−42.1930			−1.44129			−0.720643	+	0.693307i	$0.756153\pi$
$858$		0			0
$859$			48.2853i			1.64747i	0.566972	+	0.823737i	$0.308115\pi$
$859$			48.2853i			1.64747i	−0.566972	+	0.823737i	$0.691885\pi$
$860$		0			0
$861$		−	0.740278i		−	0.0252286i
$862$		0			0
$863$			56.5320i			1.92437i	0.272392	+	0.962186i	$0.412185\pi$
$863$			56.5320i			1.92437i	−0.272392	+	0.962186i	$0.587815\pi$
$864$		0			0
$865$		−	3.68070i		−	0.125148i
$866$		0			0
$867$			6.69189i			0.227269i
$868$		0			0
$869$	−7.36041			−0.249685
$870$		0			0
$871$	−23.7587			−0.805034
$872$		0			0
$873$		−	5.67311i		−	0.192006i
$874$		0			0
$875$		−	0.143131i		−	0.00483872i
$876$		0			0
$877$	29.6117			0.999915			0.499958	−	0.866050i	$-0.333349\pi$
$877$	29.6117			0.999915			0.499958	+	0.866050i	$0.333349\pi$
$878$		0			0
$879$	29.3775			0.990879
$880$		0			0
$881$			10.6754i			0.359662i	0.983698	+	0.179831i	$0.0575551\pi$
$881$			10.6754i			0.359662i	−0.983698	+	0.179831i	$0.942445\pi$
$882$		0			0
$883$			13.9342i			0.468922i	0.972126	+	0.234461i	$0.0753325\pi$
$883$			13.9342i			0.468922i	−0.972126	+	0.234461i	$0.924667\pi$
$884$		0			0
$885$			6.57340i			0.220962i
$886$		0			0
$887$		−	47.6768i		−	1.60083i	−0.599446	−	0.800415i	$-0.704613\pi$
$887$		−	47.6768i		−	1.60083i	0.599446	−	0.800415i	$-0.295387\pi$
$888$		0			0
$889$			0.535471i			0.0179591i
$890$		0			0
$891$	−4.80659			−0.161027
$892$		0			0
$893$			3.38935i			0.113420i
$894$		0			0
$895$	9.27393			0.309993
$896$		0			0
$897$	−22.2579	+	5.51617i	−0.743171	+	0.184180i
$898$		0			0
$899$			21.5008i			0.717093i
$900$		0			0
$901$	64.9377			2.16339
$902$		0			0
$903$		−	0.148148i		−	0.00493004i
$904$		0			0
$905$	5.10162			0.169583
$906$		0			0
$907$	−12.0207			−0.399141			−0.199571	−	0.979883i	$-0.563955\pi$
$907$	−12.0207			−0.399141			−0.199571	+	0.979883i	$0.563955\pi$
$908$		0			0
$909$	17.3244			0.574614
$910$		0			0
$911$	−9.73661			−0.322588			−0.161294	−	0.986906i	$-0.551567\pi$
$911$	−9.73661			−0.322588			−0.161294	+	0.986906i	$0.551567\pi$
$912$		0			0
$913$	69.2021			2.29026
$914$		0			0
$915$		−	7.49837i		−	0.247888i
$916$		0			0
$917$		−	1.71113i		−	0.0565065i
$918$		0			0
$919$	38.9231			1.28396			0.641978	−	0.766723i	$-0.278114\pi$
$919$	38.9231			1.28396			0.641978	+	0.766723i	$0.278114\pi$
$920$		0			0
$921$	−15.6395			−0.515338
$922$		0			0
$923$			32.3892i			1.06610i
$924$		0			0
$925$			5.56282i			0.182904i
$926$		0			0
$927$	12.2942			0.403794
$928$		0			0
$929$	43.8793			1.43963			0.719817	−	0.694164i	$-0.244226\pi$
$929$	43.8793			1.43963			0.719817	+	0.694164i	$0.244226\pi$
$930$		0			0
$931$	2.64761			0.0867719
$932$		0			0
$933$	−16.7393			−0.548021
$934$		0			0
$935$	−23.3958			−0.765123
$936$		0			0
$937$		−	39.3915i		−	1.28686i	−0.765504	−	0.643432i	$-0.777510\pi$
$937$		−	39.3915i		−	1.28686i	0.765504	−	0.643432i	$-0.222490\pi$
$938$		0			0
$939$	−17.0663			−0.556939
$940$		0			0
$941$		−	45.6529i		−	1.48824i	−0.668045	−	0.744120i	$-0.732869\pi$
$941$		−	45.6529i		−	1.48824i	0.668045	−	0.744120i	$-0.267131\pi$
$942$		0			0
$943$	5.96669	+	24.0758i	0.194302	+	0.784016i
$944$		0			0
$945$	−0.143131			−0.00465606
$946$		0			0
$947$			0.780609i			0.0253664i	0.999920	+	0.0126832i	$0.00403729\pi$
$947$			0.780609i			0.0253664i	−0.999920	+	0.0126832i	$0.995963\pi$
$948$		0			0
$949$	−38.4686			−1.24874
$950$		0			0
$951$			29.6407i			0.961166i
$952$		0			0
$953$			0.373697i			0.0121052i	0.999982	+	0.00605261i	$0.00192662\pi$
$953$			0.373697i			0.0121052i	−0.999982	+	0.00605261i	$0.998073\pi$
$954$		0			0
$955$			20.1633i			0.652471i
$956$		0			0
$957$			40.3721i			1.30505i
$958$		0			0
$959$		−	1.16919i		−	0.0377550i
$960$		0			0
$961$	24.4473			0.788622
$962$		0			0
$963$	1.02774			0.0331186
$964$		0			0
$965$		−	1.06748i		−	0.0343635i
$966$		0			0
$967$			11.2527i			0.361862i	0.983496	+	0.180931i	$0.0579110\pi$
$967$			11.2527i			0.361862i	−0.983496	+	0.180931i	$0.942089\pi$
$968$		0			0
$969$	−1.84641			−0.0593153
$970$		0			0
$971$	61.0936			1.96059			0.980294	−	0.197544i	$-0.0632966\pi$
$971$	61.0936			1.96059			0.980294	+	0.197544i	$0.0632966\pi$
$972$		0			0
$973$		−	0.124570i		−	0.00399354i
$974$		0			0
$975$			4.78151i			0.153131i
$976$		0			0
$977$			10.4612i			0.334685i	0.985899	+	0.167342i	$0.0535185\pi$
$977$			10.4612i			0.334685i	−0.985899	+	0.167342i	$0.946481\pi$
$978$		0			0
$979$			60.5636i			1.93562i
$980$		0			0
$981$		−	16.9971i		−	0.542674i
$982$		0			0
$983$	−57.0974			−1.82112			−0.910562	−	0.413373i	$-0.864351\pi$
$983$	−57.0974			−1.82112			−0.910562	+	0.413373i	$0.864351\pi$
$984$		0			0
$985$		−	4.31277i		−	0.137416i
$986$		0			0
$987$	1.27886			0.0407065
$988$		0			0
$989$	1.19408	+	4.81815i	0.0379695	+	0.153208i
$990$		0			0
$991$		−	8.99297i		−	0.285671i	−0.989746	−	0.142836i	$-0.954378\pi$
$991$		−	8.99297i		−	0.285671i	0.989746	−	0.142836i	$-0.0456220\pi$
$992$		0			0
$993$	4.03408			0.128018
$994$		0			0
$995$			9.35032i			0.296425i
$996$		0			0
$997$	−39.7082			−1.25757			−0.628786	−	0.777578i	$-0.716448\pi$
$997$	−39.7082			−1.25757			−0.628786	+	0.777578i	$0.716448\pi$
$998$		0			0
$999$	5.56282			0.176000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.be.c.1471.17	✓	32
4.3	odd	2		5520.2.be.d.1471.18	yes	32
23.22	odd	2		5520.2.be.d.1471.17	yes	32
92.91	even	2	inner	5520.2.be.c.1471.18	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5520.2.be.c.1471.17	✓	32	1.1	even	1	trivial
5520.2.be.c.1471.18	yes	32	92.91	even	2	inner
5520.2.be.d.1471.17	yes	32	23.22	odd	2
5520.2.be.d.1471.18	yes	32	4.3	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)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} -1.00000i q^{5} -0.143131 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} -1.00000i q^{5} -0.143131 q^{7} -1.00000 q^{9} -4.80659 q^{11} +4.78151 q^{13} -1.00000 q^{15} +4.86743i q^{17} -0.379340 q^{19} +0.143131i q^{21} +(-1.15365 - 4.65501i) q^{23} -1.00000 q^{25} +1.00000i q^{27} +8.39933 q^{29} +2.55983i q^{31} +4.80659i q^{33} +0.143131i q^{35} -5.56282i q^{37} -4.78151i q^{39} -5.17202 q^{41} -1.03505 q^{43} +1.00000i q^{45} -8.93486i q^{47} -6.97951 q^{49} +4.86743 q^{51} -13.3413i q^{53} +4.80659i q^{55} +0.379340i q^{57} -6.57340i q^{59} +7.49837i q^{61} +0.143131 q^{63} -4.78151i q^{65} -4.96888 q^{67} +(-4.65501 + 1.15365i) q^{69} +6.77385i q^{71} -8.04529 q^{73} +1.00000i q^{75} +0.687974 q^{77} +1.53132 q^{79} +1.00000 q^{81} -14.3973 q^{83} +4.86743 q^{85} -8.39933i q^{87} -12.6001i q^{89} -0.684383 q^{91} +2.55983 q^{93} +0.379340i q^{95} +5.67311i q^{97} +4.80659 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 8 q^{7} - 32 q^{9}+O(q^{10}) \) 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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