LMFDB - Embedded newform 4004.2.m.c.2157.22

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4004,2,Mod(2157,4004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4004, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4004.2157");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4004.m (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:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2157.22
Character	$\chi$	$=$	4004.2157
Dual form			4004.2.m.c.2157.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.334006 q^{3} +1.98967i q^{5} -1.00000i q^{7} -2.88844 q^{9} +O(q^{10})$	$q+0.334006 q^{3} +1.98967i q^{5} -1.00000i q^{7} -2.88844 q^{9} +1.00000i q^{11} +(-2.44303 - 2.65172i) q^{13} +0.664561i q^{15} -5.84704 q^{17} +2.86896i q^{19} -0.334006i q^{21} +0.564296 q^{23} +1.04122 q^{25} -1.96677 q^{27} +8.26319 q^{29} -1.77046i q^{31} +0.334006i q^{33} +1.98967 q^{35} +0.265522i q^{37} +(-0.815987 - 0.885689i) q^{39} -5.63133i q^{41} -0.932502 q^{43} -5.74704i q^{45} -10.4636i q^{47} -1.00000 q^{49} -1.95295 q^{51} +11.6043 q^{53} -1.98967 q^{55} +0.958248i q^{57} +8.63659i q^{59} +10.1344 q^{61} +2.88844i q^{63} +(5.27604 - 4.86082i) q^{65} +1.98939i q^{67} +0.188478 q^{69} -10.5033i q^{71} -3.74473i q^{73} +0.347772 q^{75} +1.00000 q^{77} +4.24966 q^{79} +8.00841 q^{81} -8.15399i q^{83} -11.6337i q^{85} +2.75995 q^{87} -18.0493i q^{89} +(-2.65172 + 2.44303i) q^{91} -0.591343i q^{93} -5.70827 q^{95} -12.1084i q^{97} -2.88844i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) $ 36 * q - 4 * q^3 + 40 * q^9	$ 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) $ 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times$.

$n$	$365$	$925$	$2003$	$3433$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.334006			0.192838			0.0964192	−	0.995341i	$-0.469261\pi$
$3$	0.334006			0.192838			0.0964192	+	0.995341i	$0.469261\pi$
$4$		0			0
$5$			1.98967i			0.889807i	0.895578	+	0.444904i	$0.146762\pi$
$5$			1.98967i			0.889807i	−0.895578	+	0.444904i	$0.853238\pi$
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	−2.88844			−0.962813
$10$		0			0
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		0			0
$13$	−2.44303	−	2.65172i	−0.677575	−	0.735454i
$13$	−2.44303	−	2.65172i	−0.677575	−	0.735454i
$14$		0			0
$15$			0.664561i			0.171589i
$16$		0			0
$17$	−5.84704			−1.41812			−0.709058	−	0.705150i	$-0.750880\pi$
$17$	−5.84704			−1.41812			−0.709058	+	0.705150i	$0.750880\pi$
$18$		0			0
$19$			2.86896i			0.658183i	0.944298	+	0.329092i	$0.106743\pi$
$19$			2.86896i			0.658183i	−0.944298	+	0.329092i	$0.893257\pi$
$20$		0			0
$21$		−	0.334006i		−	0.0728861i
$22$		0			0
$23$	0.564296			0.117664			0.0588319	−	0.998268i	$-0.481262\pi$
$23$	0.564296			0.117664			0.0588319	+	0.998268i	$0.481262\pi$
$24$		0			0
$25$	1.04122			0.208243
$26$		0			0
$27$	−1.96677			−0.378506
$28$		0			0
$29$	8.26319			1.53444			0.767218	−	0.641387i	$-0.221640\pi$
$29$	8.26319			1.53444			0.767218	+	0.641387i	$0.221640\pi$
$30$		0			0
$31$		−	1.77046i		−	0.317983i	−0.987280	−	0.158992i	$-0.949176\pi$
$31$		−	1.77046i		−	0.317983i	0.987280	−	0.158992i	$-0.0508243\pi$
$32$		0			0
$33$			0.334006i			0.0581430i
$34$		0			0
$35$	1.98967			0.336315
$36$		0			0
$37$			0.265522i			0.0436515i	0.999762	+	0.0218257i	$0.00694790\pi$
$37$			0.265522i			0.0436515i	−0.999762	+	0.0218257i	$0.993052\pi$
$38$		0			0
$39$	−0.815987	−	0.885689i	−0.130662	−	0.141824i
$40$		0			0
$41$		−	5.63133i		−	0.879465i	−0.898129	−	0.439733i	$-0.855073\pi$
$41$		−	5.63133i		−	0.879465i	0.898129	−	0.439733i	$-0.144927\pi$
$42$		0			0
$43$	−0.932502			−0.142205			−0.0711026	−	0.997469i	$-0.522652\pi$
$43$	−0.932502			−0.142205			−0.0711026	+	0.997469i	$0.522652\pi$
$44$		0			0
$45$		−	5.74704i		−	0.856718i
$46$		0			0
$47$		−	10.4636i		−	1.52628i	−0.646236	−	0.763138i	$-0.723658\pi$
$47$		−	10.4636i		−	1.52628i	0.646236	−	0.763138i	$-0.276342\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−1.95295			−0.273467
$52$		0			0
$53$	11.6043			1.59398			0.796989	−	0.603994i	$-0.206425\pi$
$53$	11.6043			1.59398			0.796989	+	0.603994i	$0.206425\pi$
$54$		0			0
$55$	−1.98967			−0.268287
$56$		0			0
$57$			0.958248i			0.126923i
$58$		0			0
$59$			8.63659i			1.12439i	0.827005	+	0.562194i	$0.190043\pi$
$59$			8.63659i			1.12439i	−0.827005	+	0.562194i	$0.809957\pi$
$60$		0			0
$61$	10.1344			1.29758			0.648789	−	0.760968i	$-0.275276\pi$
$61$	10.1344			1.29758			0.648789	+	0.760968i	$0.275276\pi$
$62$		0			0
$63$			2.88844i			0.363909i
$64$		0			0
$65$	5.27604	−	4.86082i	0.654412	−	0.602911i
$66$		0			0
$67$			1.98939i			0.243043i	0.992589	+	0.121521i	$0.0387773\pi$
$67$			1.98939i			0.243043i	−0.992589	+	0.121521i	$0.961223\pi$
$68$		0			0
$69$	0.188478			0.0226901
$70$		0			0
$71$		−	10.5033i		−	1.24651i	−0.782017	−	0.623257i	$-0.785809\pi$
$71$		−	10.5033i		−	1.24651i	0.782017	−	0.623257i	$-0.214191\pi$
$72$		0			0
$73$		−	3.74473i		−	0.438287i	−0.975693	−	0.219144i	$-0.929674\pi$
$73$		−	3.74473i		−	0.438287i	0.975693	−	0.219144i	$-0.0703263\pi$
$74$		0			0
$75$	0.347772			0.0401573
$76$		0			0
$77$	1.00000			0.113961
$78$		0			0
$79$	4.24966			0.478125			0.239062	−	0.971004i	$-0.423160\pi$
$79$	4.24966			0.478125			0.239062	+	0.971004i	$0.423160\pi$
$80$		0			0
$81$	8.00841			0.889823
$82$		0			0
$83$		−	8.15399i		−	0.895017i	−0.894280	−	0.447508i	$-0.852311\pi$
$83$		−	8.15399i		−	0.895017i	0.894280	−	0.447508i	$-0.147689\pi$
$84$		0			0
$85$		−	11.6337i		−	1.26185i
$86$		0			0
$87$	2.75995			0.295898
$88$		0			0
$89$		−	18.0493i		−	1.91323i	−0.291362	−	0.956613i	$-0.594108\pi$
$89$		−	18.0493i		−	1.91323i	0.291362	−	0.956613i	$-0.405892\pi$
$90$		0			0
$91$	−2.65172	+	2.44303i	−0.277975	+	0.256099i
$92$		0			0
$93$		−	0.591343i		−	0.0613194i
$94$		0			0
$95$	−5.70827			−0.585656
$96$		0			0
$97$		−	12.1084i		−	1.22942i	−0.788752	−	0.614711i	$-0.789273\pi$
$97$		−	12.1084i		−	1.22942i	0.788752	−	0.614711i	$-0.210727\pi$
$98$		0			0
$99$		−	2.88844i		−	0.290299i
$100$		0			0
$101$	−16.0095			−1.59300			−0.796501	−	0.604638i	$-0.793318\pi$
$101$	−16.0095			−1.59300			−0.796501	+	0.604638i	$0.793318\pi$
$102$		0			0
$103$	16.9288			1.66805			0.834023	−	0.551729i	$-0.186032\pi$
$103$	16.9288			1.66805			0.834023	+	0.551729i	$0.186032\pi$
$104$		0			0
$105$	0.664561			0.0648545
$106$		0			0
$107$	−13.9699			−1.35053			−0.675263	−	0.737577i	$-0.735970\pi$
$107$	−13.9699			−1.35053			−0.675263	+	0.737577i	$0.735970\pi$
$108$		0			0
$109$			8.44134i			0.808533i	0.914641	+	0.404267i	$0.132473\pi$
$109$			8.44134i			0.808533i	−0.914641	+	0.404267i	$0.867527\pi$
$110$		0			0
$111$			0.0886858i			0.00841768i
$112$		0			0
$113$	−2.06873			−0.194610			−0.0973051	−	0.995255i	$-0.531022\pi$
$113$	−2.06873			−0.194610			−0.0973051	+	0.995255i	$0.531022\pi$
$114$		0			0
$115$			1.12276i			0.104698i
$116$		0			0
$117$	7.05655	+	7.65933i	0.652378	+	0.708105i
$118$		0			0
$119$			5.84704i			0.535998i
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$		−	1.88090i		−	0.169595i
$124$		0			0
$125$			12.0200i			1.07510i
$126$		0			0
$127$	15.9509			1.41542			0.707708	−	0.706505i	$-0.249729\pi$
$127$	15.9509			1.41542			0.707708	+	0.706505i	$0.249729\pi$
$128$		0			0
$129$	−0.311461			−0.0274226
$130$		0			0
$131$	11.3928			0.995390			0.497695	−	0.867352i	$-0.334180\pi$
$131$	11.3928			0.995390			0.497695	+	0.867352i	$0.334180\pi$
$132$		0			0
$133$	2.86896			0.248770
$134$		0			0
$135$		−	3.91323i		−	0.336797i
$136$		0			0
$137$			2.06067i			0.176055i	0.996118	+	0.0880276i	$0.0280564\pi$
$137$			2.06067i			0.176055i	−0.996118	+	0.0880276i	$0.971944\pi$
$138$		0			0
$139$	−14.7361			−1.24990			−0.624950	−	0.780665i	$-0.714881\pi$
$139$	−14.7361			−1.24990			−0.624950	+	0.780665i	$0.714881\pi$
$140$		0			0
$141$		−	3.49491i		−	0.294325i
$142$		0			0
$143$	2.65172	−	2.44303i	0.221748	−	0.204296i
$144$		0			0
$145$			16.4410i			1.36535i
$146$		0			0
$147$	−0.334006			−0.0275483
$148$		0			0
$149$			18.9665i			1.55380i	0.629625	+	0.776899i	$0.283209\pi$
$149$			18.9665i			1.55380i	−0.629625	+	0.776899i	$0.716791\pi$
$150$		0			0
$151$		−	0.201196i		−	0.0163731i	−0.999966	−	0.00818653i	$-0.997394\pi$
$151$		−	0.201196i		−	0.0163731i	0.999966	−	0.00818653i	$-0.00260588\pi$
$152$		0			0
$153$	16.8888			1.36538
$154$		0			0
$155$	3.52262			0.282944
$156$		0			0
$157$	10.9308			0.872370			0.436185	−	0.899857i	$-0.356329\pi$
$157$	10.9308			0.872370			0.436185	+	0.899857i	$0.356329\pi$
$158$		0			0
$159$	3.87592			0.307380
$160$		0			0
$161$		−	0.564296i		−	0.0444728i
$162$		0			0
$163$		−	12.1721i		−	0.953392i	−0.879068	−	0.476696i	$-0.841834\pi$
$163$		−	12.1721i		−	0.953392i	0.879068	−	0.476696i	$-0.158166\pi$
$164$		0			0
$165$	−0.664561			−0.0517360
$166$		0			0
$167$		−	19.2181i		−	1.48714i	−0.668659	−	0.743569i	$-0.733131\pi$
$167$		−	19.2181i		−	1.48714i	0.668659	−	0.743569i	$-0.266869\pi$
$168$		0			0
$169$	−1.06321	+	12.9564i	−0.0817852	+	0.996650i
$170$		0			0
$171$		−	8.28681i		−	0.633708i
$172$		0			0
$173$	−20.6675			−1.57132			−0.785659	−	0.618660i	$-0.787676\pi$
$173$	−20.6675			−1.57132			−0.785659	+	0.618660i	$0.787676\pi$
$174$		0			0
$175$		−	1.04122i		−	0.0787085i
$176$		0			0
$177$			2.88467i			0.216825i
$178$		0			0
$179$	−4.89482			−0.365856			−0.182928	−	0.983126i	$-0.558558\pi$
$179$	−4.89482			−0.365856			−0.182928	+	0.983126i	$0.558558\pi$
$180$		0			0
$181$	17.9975			1.33775			0.668873	−	0.743377i	$-0.266777\pi$
$181$	17.9975			1.33775			0.668873	+	0.743377i	$0.266777\pi$
$182$		0			0
$183$	3.38495			0.250223
$184$		0			0
$185$	−0.528300			−0.0388414
$186$		0			0
$187$		−	5.84704i		−	0.427578i
$188$		0			0
$189$			1.96677i			0.143062i
$190$		0			0
$191$	−18.3828			−1.33013			−0.665066	−	0.746785i	$-0.731596\pi$
$191$	−18.3828			−1.33013			−0.665066	+	0.746785i	$0.731596\pi$
$192$		0			0
$193$		−	9.75303i		−	0.702038i	−0.936368	−	0.351019i	$-0.885835\pi$
$193$		−	9.75303i		−	0.702038i	0.936368	−	0.351019i	$-0.114165\pi$
$194$		0			0
$195$	1.76223	−	1.62354i	0.126196	−	0.116264i
$196$		0			0
$197$		−	11.6023i		−	0.826629i	−0.910588	−	0.413314i	$-0.864371\pi$
$197$		−	11.6023i		−	0.826629i	0.910588	−	0.413314i	$-0.135629\pi$
$198$		0			0
$199$	18.2180			1.29144			0.645721	−	0.763574i	$-0.276557\pi$
$199$	18.2180			1.29144			0.645721	+	0.763574i	$0.276557\pi$
$200$		0			0
$201$			0.664468i			0.0468680i
$202$		0			0
$203$		−	8.26319i		−	0.579962i
$204$		0			0
$205$	11.2045			0.782555
$206$		0			0
$207$	−1.62994			−0.113288
$208$		0			0
$209$	−2.86896			−0.198450
$210$		0			0
$211$	3.19929			0.220248			0.110124	−	0.993918i	$-0.464875\pi$
$211$	3.19929			0.220248			0.110124	+	0.993918i	$0.464875\pi$
$212$		0			0
$213$		−	3.50817i		−	0.240376i
$214$		0			0
$215$		−	1.85537i		−	0.126535i
$216$		0			0
$217$	−1.77046			−0.120186
$218$		0			0
$219$		−	1.25076i		−	0.0845186i
$220$		0			0
$221$	14.2845	+	15.5047i	0.960880	+	1.04296i
$222$		0			0
$223$			0.869972i			0.0582576i	0.999576	+	0.0291288i	$0.00927330\pi$
$223$			0.869972i			0.0582576i	−0.999576	+	0.0291288i	$0.990727\pi$
$224$		0			0
$225$	−3.00749			−0.200499
$226$		0			0
$227$			18.6612i			1.23859i	0.785160	+	0.619293i	$0.212581\pi$
$227$			18.6612i			1.23859i	−0.785160	+	0.619293i	$0.787419\pi$
$228$		0			0
$229$		−	14.1185i		−	0.932979i	−0.884527	−	0.466489i	$-0.845519\pi$
$229$		−	14.1185i		−	0.932979i	0.884527	−	0.466489i	$-0.154481\pi$
$230$		0			0
$231$	0.334006			0.0219760
$232$		0			0
$233$	3.44163			0.225469			0.112734	−	0.993625i	$-0.464039\pi$
$233$	3.44163			0.225469			0.112734	+	0.993625i	$0.464039\pi$
$234$		0			0
$235$	20.8191			1.35809
$236$		0			0
$237$	1.41941			0.0922008
$238$		0			0
$239$		−	14.3694i		−	0.929482i	−0.885447	−	0.464741i	$-0.846147\pi$
$239$		−	14.3694i		−	0.929482i	0.885447	−	0.464741i	$-0.153853\pi$
$240$		0			0
$241$			9.29418i			0.598691i	0.954145	+	0.299345i	$0.0967683\pi$
$241$			9.29418i			0.598691i	−0.954145	+	0.299345i	$0.903232\pi$
$242$		0			0
$243$	8.57518			0.550098
$244$		0			0
$245$		−	1.98967i		−	0.127115i
$246$		0			0
$247$	7.60766	−	7.00894i	0.484064	−	0.445968i
$248$		0			0
$249$		−	2.72348i		−	0.172594i
$250$		0			0
$251$	8.28430			0.522900			0.261450	−	0.965217i	$-0.415799\pi$
$251$	8.28430			0.522900			0.261450	+	0.965217i	$0.415799\pi$
$252$		0			0
$253$			0.564296i			0.0354770i
$254$		0			0
$255$		−	3.88572i		−	0.243333i
$256$		0			0
$257$	3.31704			0.206911			0.103456	−	0.994634i	$-0.467010\pi$
$257$	3.31704			0.206911			0.103456	+	0.994634i	$0.467010\pi$
$258$		0			0
$259$	0.265522			0.0164987
$260$		0			0
$261$	−23.8677			−1.47738
$262$		0			0
$263$	0.951054			0.0586445			0.0293223	−	0.999570i	$-0.490665\pi$
$263$	0.951054			0.0586445			0.0293223	+	0.999570i	$0.490665\pi$
$264$		0			0
$265$			23.0888i			1.41833i
$266$		0			0
$267$		−	6.02858i		−	0.368943i
$268$		0			0
$269$	−1.14975			−0.0701013			−0.0350506	−	0.999386i	$-0.511159\pi$
$269$	−1.14975			−0.0701013			−0.0350506	+	0.999386i	$0.511159\pi$
$270$		0			0
$271$		−	10.7915i		−	0.655540i	−0.944758	−	0.327770i	$-0.893703\pi$
$271$		−	10.7915i		−	0.655540i	0.944758	−	0.327770i	$-0.106297\pi$
$272$		0			0
$273$	−0.885689	+	0.815987i	−0.0536044	+	0.0493858i
$274$		0			0
$275$			1.04122i			0.0627877i
$276$		0			0
$277$	−16.4190			−0.986524			−0.493262	−	0.869881i	$-0.664196\pi$
$277$	−16.4190			−0.986524			−0.493262	+	0.869881i	$0.664196\pi$
$278$		0			0
$279$			5.11386i			0.306159i
$280$		0			0
$281$			17.5696i			1.04811i	0.851683	+	0.524057i	$0.175582\pi$
$281$			17.5696i			1.04811i	−0.851683	+	0.524057i	$0.824418\pi$
$282$		0			0
$283$	8.42629			0.500891			0.250446	−	0.968131i	$-0.419423\pi$
$283$	8.42629			0.500891			0.250446	+	0.968131i	$0.419423\pi$
$284$		0			0
$285$	−1.90660			−0.112937
$286$		0			0
$287$	−5.63133			−0.332407
$288$		0			0
$289$	17.1879			1.01105
$290$		0			0
$291$		−	4.04428i		−	0.237080i
$292$		0			0
$293$			3.67682i			0.214802i	0.994216	+	0.107401i	$0.0342529\pi$
$293$			3.67682i			0.214802i	−0.994216	+	0.107401i	$0.965747\pi$
$294$		0			0
$295$	−17.1840			−1.00049
$296$		0			0
$297$		−	1.96677i		−	0.114124i
$298$		0			0
$299$	−1.37859	−	1.49635i	−0.0797260	−	0.0865363i
$300$		0			0
$301$			0.932502i			0.0537485i
$302$		0			0
$303$	−5.34726			−0.307192
$304$		0			0
$305$			20.1641i			1.15459i
$306$		0			0
$307$		−	19.8627i		−	1.13363i	−0.823846	−	0.566813i	$-0.808176\pi$
$307$		−	19.8627i		−	1.13363i	0.823846	−	0.566813i	$-0.191824\pi$
$308$		0			0
$309$	5.65433			0.321663
$310$		0			0
$311$	−6.65730			−0.377501			−0.188750	−	0.982025i	$-0.560444\pi$
$311$	−6.65730			−0.377501			−0.188750	+	0.982025i	$0.560444\pi$
$312$		0			0
$313$	6.96921			0.393923			0.196962	−	0.980411i	$-0.436893\pi$
$313$	6.96921			0.393923			0.196962	+	0.980411i	$0.436893\pi$
$314$		0			0
$315$	−5.74704			−0.323809
$316$		0			0
$317$		−	9.91644i		−	0.556963i	−0.960442	−	0.278481i	$-0.910169\pi$
$317$		−	9.91644i		−	0.556963i	0.960442	−	0.278481i	$-0.0898310\pi$
$318$		0			0
$319$			8.26319i			0.462650i
$320$		0			0
$321$	−4.66604			−0.260433
$322$		0			0
$323$		−	16.7749i		−	0.933381i
$324$		0			0
$325$	−2.54372	−	2.76101i	−0.141100	−	0.153153i
$326$		0			0
$327$			2.81946i			0.155916i
$328$		0			0
$329$	−10.4636			−0.576878
$330$		0			0
$331$		−	8.79505i		−	0.483420i	−0.970349	−	0.241710i	$-0.922292\pi$
$331$		−	8.79505i		−	0.483420i	0.970349	−	0.241710i	$-0.0777082\pi$
$332$		0			0
$333$		−	0.766943i		−	0.0420282i
$334$		0			0
$335$	−3.95823			−0.216261
$336$		0			0
$337$	−35.6009			−1.93930			−0.969652	−	0.244491i	$-0.921379\pi$
$337$	−35.6009			−1.93930			−0.969652	+	0.244491i	$0.921379\pi$
$338$		0			0
$339$	−0.690970			−0.0375283
$340$		0			0
$341$	1.77046			0.0958756
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$			0.375009i			0.0201898i
$346$		0			0
$347$	33.3591			1.79081			0.895405	−	0.445252i	$-0.146886\pi$
$347$	33.3591			1.79081			0.895405	+	0.445252i	$0.146886\pi$
$348$		0			0
$349$			4.72523i			0.252936i	0.991971	+	0.126468i	$0.0403641\pi$
$349$			4.72523i			0.252936i	−0.991971	+	0.126468i	$0.959636\pi$
$350$		0			0
$351$	4.80489	+	5.21533i	0.256466	+	0.278374i
$352$		0			0
$353$		−	34.5527i		−	1.83906i	−0.393025	−	0.919528i	$-0.628571\pi$
$353$		−	34.5527i		−	1.83906i	0.393025	−	0.919528i	$-0.371429\pi$
$354$		0			0
$355$	20.8981			1.10916
$356$		0			0
$357$			1.95295i			0.103361i
$358$		0			0
$359$		−	29.7215i		−	1.56864i	−0.620357	−	0.784319i	$-0.713012\pi$
$359$		−	29.7215i		−	1.56864i	0.620357	−	0.784319i	$-0.286988\pi$
$360$		0			0
$361$	10.7691			0.566795
$362$		0			0
$363$	−0.334006			−0.0175308
$364$		0			0
$365$	7.45077			0.389991
$366$		0			0
$367$	−12.6132			−0.658406			−0.329203	−	0.944259i	$-0.606780\pi$
$367$	−12.6132			−0.658406			−0.329203	+	0.944259i	$0.606780\pi$
$368$		0			0
$369$			16.2657i			0.846761i
$370$		0			0
$371$		−	11.6043i		−	0.602467i
$372$		0			0
$373$	−25.4010			−1.31522			−0.657608	−	0.753360i	$-0.728432\pi$
$373$	−25.4010			−1.31522			−0.657608	+	0.753360i	$0.728432\pi$
$374$		0			0
$375$			4.01476i			0.207321i
$376$		0			0
$377$	−20.1872	−	21.9116i	−1.03969	−	1.12851i
$378$		0			0
$379$		−	6.66771i		−	0.342497i	−0.985228	−	0.171249i	$-0.945220\pi$
$379$		−	6.66771i		−	0.342497i	0.985228	−	0.171249i	$-0.0547801\pi$
$380$		0			0
$381$	5.32770			0.272947
$382$		0			0
$383$		−	19.8188i		−	1.01269i	−0.862330	−	0.506346i	$-0.830996\pi$
$383$		−	19.8188i		−	1.01269i	0.862330	−	0.506346i	$-0.169004\pi$
$384$		0			0
$385$			1.98967i			0.101403i
$386$		0			0
$387$	2.69347			0.136917
$388$		0			0
$389$	−19.2777			−0.977419			−0.488710	−	0.872446i	$-0.662532\pi$
$389$	−19.2777			−0.977419			−0.488710	+	0.872446i	$0.662532\pi$
$390$		0			0
$391$	−3.29946			−0.166861
$392$		0			0
$393$	3.80525			0.191949
$394$		0			0
$395$			8.45543i			0.425439i
$396$		0			0
$397$			23.9898i			1.20402i	0.798490	+	0.602008i	$0.205632\pi$
$397$			23.9898i			1.20402i	−0.798490	+	0.602008i	$0.794368\pi$
$398$		0			0
$399$	0.958248			0.0479724
$400$		0			0
$401$			24.5641i			1.22667i	0.789822	+	0.613336i	$0.210173\pi$
$401$			24.5641i			1.22667i	−0.789822	+	0.613336i	$0.789827\pi$
$402$		0			0
$403$	−4.69475	+	4.32528i	−0.233862	+	0.215457i
$404$		0			0
$405$			15.9341i			0.791771i
$406$		0			0
$407$	−0.265522			−0.0131614
$408$		0			0
$409$		−	8.74397i		−	0.432361i	−0.976353	−	0.216181i	$-0.930640\pi$
$409$		−	8.74397i		−	0.432361i	0.976353	−	0.216181i	$-0.0693600\pi$
$410$		0			0
$411$			0.688277i			0.0339502i
$412$		0			0
$413$	8.63659			0.424979
$414$		0			0
$415$	16.2237			0.796392
$416$		0			0
$417$	−4.92194			−0.241029
$418$		0			0
$419$	9.88400			0.482865			0.241432	−	0.970418i	$-0.422383\pi$
$419$	9.88400			0.482865			0.241432	+	0.970418i	$0.422383\pi$
$420$		0			0
$421$		−	10.3897i		−	0.506361i	−0.967419	−	0.253181i	$-0.918523\pi$
$421$		−	10.3897i		−	0.506361i	0.967419	−	0.253181i	$-0.0814767\pi$
$422$		0			0
$423$			30.2235i			1.46952i
$424$		0			0
$425$	−6.08804			−0.295313
$426$		0			0
$427$		−	10.1344i		−	0.490438i
$428$		0			0
$429$	0.885689	−	0.815987i	0.0427615	−	0.0393962i
$430$		0			0
$431$			19.5651i			0.942416i	0.882022	+	0.471208i	$0.156182\pi$
$431$			19.5651i			0.942416i	−0.882022	+	0.471208i	$0.843818\pi$
$432$		0			0
$433$	−1.31448			−0.0631699			−0.0315849	−	0.999501i	$-0.510055\pi$
$433$	−1.31448			−0.0631699			−0.0315849	+	0.999501i	$0.510055\pi$
$434$		0			0
$435$			5.49140i			0.263292i
$436$		0			0
$437$			1.61894i			0.0774444i
$438$		0			0
$439$	5.81692			0.277626			0.138813	−	0.990319i	$-0.455671\pi$
$439$	5.81692			0.277626			0.138813	+	0.990319i	$0.455671\pi$
$440$		0			0
$441$	2.88844			0.137545
$442$		0			0
$443$	25.1932			1.19696			0.598481	−	0.801137i	$-0.295771\pi$
$443$	25.1932			1.19696			0.598481	+	0.801137i	$0.295771\pi$
$444$		0			0
$445$	35.9122			1.70240
$446$		0			0
$447$			6.33493i			0.299632i
$448$		0			0
$449$		−	17.8833i		−	0.843964i	−0.906604	−	0.421982i	$-0.861335\pi$
$449$		−	17.8833i		−	0.843964i	0.906604	−	0.421982i	$-0.138665\pi$
$450$		0			0
$451$	5.63133			0.265169
$452$		0			0
$453$		−	0.0672005i		−	0.00315736i
$454$		0			0
$455$	−4.86082	−	5.27604i	−0.227879	−	0.247345i
$456$		0			0
$457$			8.52780i			0.398914i	0.979907	+	0.199457i	$0.0639178\pi$
$457$			8.52780i			0.398914i	−0.979907	+	0.199457i	$0.936082\pi$
$458$		0			0
$459$	11.4998			0.536765
$460$		0			0
$461$		−	6.58140i		−	0.306526i	−0.988185	−	0.153263i	$-0.951022\pi$
$461$		−	6.58140i		−	0.306526i	0.988185	−	0.153263i	$-0.0489782\pi$
$462$		0			0
$463$		−	23.8669i		−	1.10919i	−0.832121	−	0.554594i	$-0.812874\pi$
$463$		−	23.8669i		−	1.10919i	0.832121	−	0.554594i	$-0.187126\pi$
$464$		0			0
$465$	1.17658			0.0545625
$466$		0			0
$467$	−18.7891			−0.869454			−0.434727	−	0.900562i	$-0.643155\pi$
$467$	−18.7891			−0.869454			−0.434727	+	0.900562i	$0.643155\pi$
$468$		0			0
$469$	1.98939			0.0918615
$470$		0			0
$471$	3.65094			0.168226
$472$		0			0
$473$		−	0.932502i		−	0.0428765i
$474$		0			0
$475$			2.98720i			0.137062i
$476$		0			0
$477$	−33.5184			−1.53470
$478$		0			0
$479$		−	8.10105i		−	0.370146i	−0.982725	−	0.185073i	$-0.940748\pi$
$479$		−	8.10105i		−	0.370146i	0.982725	−	0.185073i	$-0.0592522\pi$
$480$		0			0
$481$	0.704088	−	0.648677i	0.0321036	−	0.0295771i
$482$		0			0
$483$		−	0.188478i		−	0.00857606i
$484$		0			0
$485$	24.0917			1.09395
$486$		0			0
$487$			3.77850i			0.171220i	0.996329	+	0.0856100i	$0.0272839\pi$
$487$			3.77850i			0.171220i	−0.996329	+	0.0856100i	$0.972716\pi$
$488$		0			0
$489$		−	4.06555i		−	0.183851i
$490$		0			0
$491$	−34.9183			−1.57584			−0.787920	−	0.615778i	$-0.788842\pi$
$491$	−34.9183			−1.57584			−0.787920	+	0.615778i	$0.788842\pi$
$492$		0			0
$493$	−48.3152			−2.17601
$494$		0			0
$495$	5.74704			0.258310
$496$		0			0
$497$	−10.5033			−0.471138
$498$		0			0
$499$			23.6873i			1.06039i	0.847876	+	0.530194i	$0.177881\pi$
$499$			23.6873i			1.06039i	−0.847876	+	0.530194i	$0.822119\pi$
$500$		0			0
$501$		−	6.41895i		−	0.286777i
$502$		0			0
$503$	−2.98432			−0.133064			−0.0665322	−	0.997784i	$-0.521194\pi$
$503$	−2.98432			−0.133064			−0.0665322	+	0.997784i	$0.521194\pi$
$504$		0			0
$505$		−	31.8535i		−	1.41746i
$506$		0			0
$507$	−0.355118	+	4.32753i	−0.0157713	+	0.192192i
$508$		0			0
$509$		−	26.7956i		−	1.18770i	−0.804577	−	0.593848i	$-0.797608\pi$
$509$		−	26.7956i		−	1.18770i	0.804577	−	0.593848i	$-0.202392\pi$
$510$		0			0
$511$	−3.74473			−0.165657
$512$		0			0
$513$		−	5.64259i		−	0.249126i
$514$		0			0
$515$			33.6828i			1.48424i
$516$		0			0
$517$	10.4636			0.460189
$518$		0			0
$519$	−6.90305			−0.303010
$520$		0			0
$521$	30.4565			1.33432			0.667162	−	0.744912i	$-0.267509\pi$
$521$	30.4565			1.33432			0.667162	+	0.744912i	$0.267509\pi$
$522$		0			0
$523$	11.0064			0.481275			0.240637	−	0.970615i	$-0.422644\pi$
$523$	11.0064			0.481275			0.240637	+	0.970615i	$0.422644\pi$
$524$		0			0
$525$		−	0.347772i		−	0.0151780i
$526$		0			0
$527$			10.3519i			0.450938i
$528$		0			0
$529$	−22.6816			−0.986155
$530$		0			0
$531$		−	24.9463i		−	1.08258i
$532$		0			0
$533$	−14.9327	+	13.7575i	−0.646806	+	0.595904i
$534$		0			0
$535$		−	27.7956i		−	1.20171i
$536$		0			0
$537$	−1.63490			−0.0705511
$538$		0			0
$539$		−	1.00000i		−	0.0430730i
$540$		0			0
$541$		−	1.49229i		−	0.0641584i	−0.999485	−	0.0320792i	$-0.989787\pi$
$541$		−	1.49229i		−	0.0641584i	0.999485	−	0.0320792i	$-0.0102129\pi$
$542$		0			0
$543$	6.01128			0.257969
$544$		0			0
$545$	−16.7955			−0.719439
$546$		0			0
$547$	30.8115			1.31740			0.658701	−	0.752405i	$-0.271106\pi$
$547$	30.8115			1.31740			0.658701	+	0.752405i	$0.271106\pi$
$548$		0			0
$549$	−29.2726			−1.24933
$550$		0			0
$551$			23.7067i			1.00994i
$552$		0			0
$553$		−	4.24966i		−	0.180714i
$554$		0			0
$555$	−0.176455			−0.00749011
$556$		0			0
$557$			36.6201i			1.55164i	0.630953	+	0.775821i	$0.282664\pi$
$557$			36.6201i			1.55164i	−0.630953	+	0.775821i	$0.717336\pi$
$558$		0			0
$559$	2.27813	+	2.47273i	0.0963546	+	0.104585i
$560$		0			0
$561$		−	1.95295i		−	0.0824535i
$562$		0			0
$563$	−13.7271			−0.578530			−0.289265	−	0.957249i	$-0.593411\pi$
$563$	−13.7271			−0.578530			−0.289265	+	0.957249i	$0.593411\pi$
$564$		0			0
$565$		−	4.11610i		−	0.173166i
$566$		0			0
$567$		−	8.00841i		−	0.336321i
$568$		0			0
$569$	10.4710			0.438967			0.219483	−	0.975616i	$-0.429563\pi$
$569$	10.4710			0.438967			0.219483	+	0.975616i	$0.429563\pi$
$570$		0			0
$571$	21.8541			0.914567			0.457284	−	0.889321i	$-0.348822\pi$
$571$	21.8541			0.914567			0.457284	+	0.889321i	$0.348822\pi$
$572$		0			0
$573$	−6.13996			−0.256500
$574$		0			0
$575$	0.587554			0.0245027
$576$		0			0
$577$		−	24.0434i		−	1.00094i	−0.865754	−	0.500470i	$-0.833161\pi$
$577$		−	24.0434i		−	1.00094i	0.865754	−	0.500470i	$-0.166839\pi$
$578$		0			0
$579$		−	3.25757i		−	0.135380i
$580$		0			0
$581$	−8.15399			−0.338284
$582$		0			0
$583$			11.6043i			0.480603i
$584$		0			0
$585$	−15.2395	+	14.0402i	−0.630077	+	0.580491i
$586$		0			0
$587$			11.7847i			0.486407i	0.969975	+	0.243204i	$0.0781983\pi$
$587$			11.7847i			0.486407i	−0.969975	+	0.243204i	$0.921802\pi$
$588$		0			0
$589$	5.07936			0.209291
$590$		0			0
$591$		−	3.87523i		−	0.159406i
$592$		0			0
$593$		−	45.1460i		−	1.85392i	−0.375157	−	0.926961i	$-0.622411\pi$
$593$		−	45.1460i		−	1.85392i	0.375157	−	0.926961i	$-0.377589\pi$
$594$		0			0
$595$	−11.6337			−0.476935
$596$		0			0
$597$	6.08493			0.249039
$598$		0			0
$599$	−22.1887			−0.906607			−0.453303	−	0.891356i	$-0.649755\pi$
$599$	−22.1887			−0.906607			−0.453303	+	0.891356i	$0.649755\pi$
$600$		0			0
$601$	−1.53498			−0.0626130			−0.0313065	−	0.999510i	$-0.509967\pi$
$601$	−1.53498			−0.0626130			−0.0313065	+	0.999510i	$0.509967\pi$
$602$		0			0
$603$		−	5.74623i		−	0.234005i
$604$		0			0
$605$		−	1.98967i		−	0.0808916i
$606$		0			0
$607$	4.51246			0.183155			0.0915776	−	0.995798i	$-0.470809\pi$
$607$	4.51246			0.183155			0.0915776	+	0.995798i	$0.470809\pi$
$608$		0			0
$609$		−	2.75995i		−	0.111839i
$610$		0			0
$611$	−27.7466	+	25.5629i	−1.12251	+	1.03417i
$612$		0			0
$613$			13.8182i			0.558113i	0.960275	+	0.279057i	$0.0900217\pi$
$613$			13.8182i			0.558113i	−0.960275	+	0.279057i	$0.909978\pi$
$614$		0			0
$615$	3.74236			0.150907
$616$		0			0
$617$			43.1793i			1.73833i	0.494519	+	0.869167i	$0.335344\pi$
$617$			43.1793i			1.73833i	−0.494519	+	0.869167i	$0.664656\pi$
$618$		0			0
$619$		−	14.5834i		−	0.586157i	−0.956088	−	0.293078i	$-0.905320\pi$
$619$		−	14.5834i		−	0.586157i	0.956088	−	0.293078i	$-0.0946797\pi$
$620$		0			0
$621$	−1.10984			−0.0445364
$622$		0			0
$623$	−18.0493			−0.723131
$624$		0			0
$625$	−18.7098			−0.748392
$626$		0			0
$627$	−0.958248			−0.0382687
$628$		0			0
$629$		−	1.55252i		−	0.0619029i
$630$		0			0
$631$			44.7887i			1.78301i	0.453010	+	0.891505i	$0.350350\pi$
$631$			44.7887i			1.78301i	−0.453010	+	0.891505i	$0.649650\pi$
$632$		0			0
$633$	1.06858			0.0424724
$634$		0			0
$635$			31.7371i			1.25945i
$636$		0			0
$637$	2.44303	+	2.65172i	0.0967964	+	0.105065i
$638$		0			0
$639$			30.3382i			1.20016i
$640$		0			0
$641$	12.8449			0.507342			0.253671	−	0.967291i	$-0.418362\pi$
$641$	12.8449			0.507342			0.253671	+	0.967291i	$0.418362\pi$
$642$		0			0
$643$		−	33.7362i		−	1.33042i	−0.746654	−	0.665212i	$-0.768341\pi$
$643$		−	33.7362i		−	1.33042i	0.746654	−	0.665212i	$-0.231659\pi$
$644$		0			0
$645$		−	0.619705i		−	0.0244008i
$646$		0			0
$647$	10.9760			0.431512			0.215756	−	0.976447i	$-0.430778\pi$
$647$	10.9760			0.431512			0.215756	+	0.976447i	$0.430778\pi$
$648$		0			0
$649$	−8.63659			−0.339016
$650$		0			0
$651$	−0.591343			−0.0231766
$652$		0			0
$653$	23.7991			0.931332			0.465666	−	0.884961i	$-0.345815\pi$
$653$	23.7991			0.931332			0.465666	+	0.884961i	$0.345815\pi$
$654$		0			0
$655$			22.6678i			0.885705i
$656$		0			0
$657$			10.8164i			0.421989i
$658$		0			0
$659$	10.9135			0.425131			0.212566	−	0.977147i	$-0.431818\pi$
$659$	10.9135			0.425131			0.212566	+	0.977147i	$0.431818\pi$
$660$		0			0
$661$		−	15.0091i		−	0.583787i	−0.956451	−	0.291893i	$-0.905715\pi$
$661$		−	15.0091i		−	0.583787i	0.956451	−	0.291893i	$-0.0942852\pi$
$662$		0			0
$663$	4.77111	+	5.17866i	0.185295	+	0.201123i
$664$		0			0
$665$			5.70827i			0.221357i
$666$		0			0
$667$	4.66288			0.180548
$668$		0			0
$669$			0.290576i			0.0112343i
$670$		0			0
$671$			10.1344i			0.391234i
$672$		0			0
$673$	21.6701			0.835321			0.417660	−	0.908603i	$-0.362850\pi$
$673$	21.6701			0.835321			0.417660	+	0.908603i	$0.362850\pi$
$674$		0			0
$675$	−2.04784			−0.0788213
$676$		0			0
$677$	8.17539			0.314206			0.157103	−	0.987582i	$-0.449785\pi$
$677$	8.17539			0.314206			0.157103	+	0.987582i	$0.449785\pi$
$678$		0			0
$679$	−12.1084			−0.464678
$680$		0			0
$681$			6.23294i			0.238847i
$682$		0			0
$683$		−	0.675788i		−	0.0258583i	−0.999916	−	0.0129291i	$-0.995884\pi$
$683$		−	0.675788i		−	0.0258583i	0.999916	−	0.0129291i	$-0.00411559\pi$
$684$		0			0
$685$	−4.10006			−0.156655
$686$		0			0
$687$		−	4.71567i		−	0.179914i
$688$		0			0
$689$	−28.3497	−	30.7714i	−1.08004	−	1.17230i
$690$		0			0
$691$		−	19.0334i		−	0.724065i	−0.932165	−	0.362033i	$-0.882083\pi$
$691$		−	19.0334i		−	0.724065i	0.932165	−	0.362033i	$-0.117917\pi$
$692$		0			0
$693$	−2.88844			−0.109723
$694$		0			0
$695$		−	29.3200i		−	1.11217i
$696$		0			0
$697$			32.9266i			1.24718i
$698$		0			0
$699$	1.14953			0.0434791
$700$		0			0
$701$	−36.1332			−1.36473			−0.682365	−	0.731011i	$-0.739049\pi$
$701$	−36.1332			−1.36473			−0.682365	+	0.731011i	$0.739049\pi$
$702$		0			0
$703$	−0.761769			−0.0287307
$704$		0			0
$705$	6.95372			0.261892
$706$		0			0
$707$			16.0095i			0.602098i
$708$		0			0
$709$			29.0477i			1.09091i	0.838140	+	0.545455i	$0.183643\pi$
$709$			29.0477i			1.09091i	−0.838140	+	0.545455i	$0.816357\pi$
$710$		0			0
$711$	−12.2749			−0.460345
$712$		0			0
$713$		−	0.999062i		−	0.0374151i
$714$		0			0
$715$	4.86082	+	5.27604i	0.181784	+	0.197313i
$716$		0			0
$717$		−	4.79948i		−	0.179240i
$718$		0			0
$719$	35.3277			1.31750			0.658751	−	0.752361i	$-0.271085\pi$
$719$	35.3277			1.31750			0.658751	+	0.752361i	$0.271085\pi$
$720$		0			0
$721$		−	16.9288i		−	0.630462i
$722$		0			0
$723$			3.10431i			0.115451i
$724$		0			0
$725$	8.60377			0.319536
$726$		0			0
$727$	−45.5855			−1.69067			−0.845337	−	0.534234i	$-0.820600\pi$
$727$	−45.5855			−1.69067			−0.845337	+	0.534234i	$0.820600\pi$
$728$		0			0
$729$	−21.1611			−0.783743
$730$		0			0
$731$	5.45238			0.201664
$732$		0			0
$733$			23.1027i			0.853318i	0.904412	+	0.426659i	$0.140310\pi$
$733$			23.1027i			0.853318i	−0.904412	+	0.426659i	$0.859690\pi$
$734$		0			0
$735$		−	0.664561i		−	0.0245127i
$736$		0			0
$737$	−1.98939			−0.0732801
$738$		0			0
$739$		−	37.2903i		−	1.37175i	−0.727722	−	0.685873i	$-0.759421\pi$
$739$		−	37.2903i		−	1.37175i	0.727722	−	0.685873i	$-0.240579\pi$
$740$		0			0
$741$	2.54100	−	2.34103i	0.0933461	−	0.0859998i
$742$		0			0
$743$		−	13.0429i		−	0.478498i	−0.970958	−	0.239249i	$-0.923099\pi$
$743$		−	13.0429i		−	0.478498i	0.970958	−	0.239249i	$-0.0769012\pi$
$744$		0			0
$745$	−37.7371			−1.38258
$746$		0			0
$747$			23.5523i			0.861734i
$748$		0			0
$749$			13.9699i			0.510451i
$750$		0			0
$751$	1.33950			0.0488789			0.0244394	−	0.999701i	$-0.492220\pi$
$751$	1.33950			0.0488789			0.0244394	+	0.999701i	$0.492220\pi$
$752$		0			0
$753$	2.76700			0.100835
$754$		0			0
$755$	0.400313			0.0145689
$756$		0			0
$757$	−29.4875			−1.07174			−0.535870	−	0.844300i	$-0.680016\pi$
$757$	−29.4875			−1.07174			−0.535870	+	0.844300i	$0.680016\pi$
$758$		0			0
$759$			0.188478i			0.00684133i
$760$		0			0
$761$			44.3323i			1.60704i	0.595275	+	0.803522i	$0.297043\pi$
$761$			44.3323i			1.60704i	−0.595275	+	0.803522i	$0.702957\pi$
$762$		0			0
$763$	8.44134			0.305597
$764$		0			0
$765$			33.6032i			1.21493i
$766$		0			0
$767$	22.9018	−	21.0994i	0.826936	−	0.761857i
$768$		0			0
$769$		−	0.481611i		−	0.0173673i	−0.999962	−	0.00868366i	$-0.997236\pi$
$769$		−	0.481611i		−	0.0173673i	0.999962	−	0.00868366i	$-0.00276413\pi$
$770$		0			0
$771$	1.10791			0.0399004
$772$		0			0
$773$			21.7007i			0.780519i	0.920705	+	0.390259i	$0.127615\pi$
$773$			21.7007i			0.780519i	−0.920705	+	0.390259i	$0.872385\pi$
$774$		0			0
$775$		−	1.84343i		−	0.0662179i
$776$		0			0
$777$	0.0886858			0.00318158
$778$		0			0
$779$	16.1560			0.578850
$780$		0			0
$781$	10.5033			0.375838
$782$		0			0
$783$	−16.2518			−0.580793
$784$		0			0
$785$			21.7486i			0.776241i
$786$		0			0
$787$			19.0929i			0.680590i	0.940319	+	0.340295i	$0.110527\pi$
$787$			19.0929i			0.680590i	−0.940319	+	0.340295i	$0.889473\pi$
$788$		0			0
$789$	0.317658			0.0113089
$790$		0			0
$791$			2.06873i			0.0735557i
$792$		0			0
$793$	−24.7587	−	26.8736i	−0.879206	−	0.954309i
$794$		0			0
$795$			7.71179i			0.273509i
$796$		0			0
$797$	−31.2799			−1.10799			−0.553995	−	0.832520i	$-0.686897\pi$
$797$	−31.2799			−1.10799			−0.553995	+	0.832520i	$0.686897\pi$
$798$		0			0
$799$			61.1812i			2.16444i
$800$		0			0
$801$			52.1344i			1.84208i
$802$		0			0
$803$	3.74473			0.132149
$804$		0			0
$805$	1.12276			0.0395722
$806$		0			0
$807$	−0.384022			−0.0135182
$808$		0			0
$809$	6.72171			0.236323			0.118161	−	0.992994i	$-0.462300\pi$
$809$	6.72171			0.236323			0.118161	+	0.992994i	$0.462300\pi$
$810$		0			0
$811$			31.3768i			1.10179i	0.834575	+	0.550894i	$0.185713\pi$
$811$			31.3768i			1.10179i	−0.834575	+	0.550894i	$0.814287\pi$
$812$		0			0
$813$		−	3.60444i		−	0.126413i
$814$		0			0
$815$	24.2184			0.848335
$816$		0			0
$817$		−	2.67531i		−	0.0935971i
$818$		0			0
$819$	7.65933	−	7.05655i	0.267639	−	0.246576i
$820$		0			0
$821$			4.16952i			0.145517i	0.997350	+	0.0727586i	$0.0231803\pi$
$821$			4.16952i			0.145517i	−0.997350	+	0.0727586i	$0.976820\pi$
$822$		0			0
$823$	−4.83339			−0.168481			−0.0842406	−	0.996445i	$-0.526846\pi$
$823$	−4.83339			−0.168481			−0.0842406	+	0.996445i	$0.526846\pi$
$824$		0			0
$825$			0.347772i			0.0121079i
$826$		0			0
$827$			1.46190i			0.0508353i	0.999677	+	0.0254177i	$0.00809157\pi$
$827$			1.46190i			0.0508353i	−0.999677	+	0.0254177i	$0.991908\pi$
$828$		0			0
$829$	23.3244			0.810092			0.405046	−	0.914296i	$-0.367256\pi$
$829$	23.3244			0.810092			0.405046	+	0.914296i	$0.367256\pi$
$830$		0			0
$831$	−5.48405			−0.190240
$832$		0			0
$833$	5.84704			0.202588
$834$		0			0
$835$	38.2376			1.32327
$836$		0			0
$837$			3.48209i			0.120359i
$838$		0			0
$839$		−	27.0800i		−	0.934906i	−0.884018	−	0.467453i	$-0.845172\pi$
$839$		−	27.0800i		−	0.934906i	0.884018	−	0.467453i	$-0.154828\pi$
$840$		0			0
$841$	39.2803			1.35449
$842$		0			0
$843$			5.86835i			0.202117i
$844$		0			0
$845$	−25.7791	−	2.11543i	−0.886826	−	0.0727730i
$846$		0			0
$847$			1.00000i			0.0343604i
$848$		0			0
$849$	2.81443			0.0965911
$850$		0			0
$851$			0.149833i			0.00513620i
$852$		0			0
$853$		−	30.6971i		−	1.05105i	−0.850778	−	0.525525i	$-0.823869\pi$
$853$		−	30.6971i		−	1.05105i	0.850778	−	0.525525i	$-0.176131\pi$
$854$		0			0
$855$	16.4880			0.563878
$856$		0			0
$857$	−33.5183			−1.14496			−0.572481	−	0.819918i	$-0.694019\pi$
$857$	−33.5183			−1.14496			−0.572481	+	0.819918i	$0.694019\pi$
$858$		0			0
$859$	42.1450			1.43797			0.718985	−	0.695026i	$-0.244607\pi$
$859$	42.1450			1.43797			0.718985	+	0.695026i	$0.244607\pi$
$860$		0			0
$861$	−1.88090			−0.0641008
$862$		0			0
$863$		−	23.8328i		−	0.811279i	−0.914033	−	0.405640i	$-0.867049\pi$
$863$		−	23.8328i		−	0.811279i	0.914033	−	0.405640i	$-0.132951\pi$
$864$		0			0
$865$		−	41.1214i		−	1.39817i
$866$		0			0
$867$	5.74087			0.194970
$868$		0			0
$869$			4.24966i			0.144160i
$870$		0			0
$871$	5.27530	−	4.86014i	0.178747	−	0.164680i
$872$		0			0
$873$			34.9744i			1.18370i
$874$		0			0
$875$	12.0200			0.406351
$876$		0			0
$877$			29.2567i			0.987929i	0.869482	+	0.493964i	$0.164453\pi$
$877$			29.2567i			0.987929i	−0.869482	+	0.493964i	$0.835547\pi$
$878$		0			0
$879$			1.22808i			0.0414221i
$880$		0			0
$881$	−42.2118			−1.42215			−0.711075	−	0.703116i	$-0.751791\pi$
$881$	−42.2118			−1.42215			−0.711075	+	0.703116i	$0.751791\pi$
$882$		0			0
$883$	48.2936			1.62521			0.812605	−	0.582815i	$-0.198049\pi$
$883$	48.2936			1.62521			0.812605	+	0.582815i	$0.198049\pi$
$884$		0			0
$885$	−5.73954			−0.192933
$886$		0			0
$887$	−34.1928			−1.14808			−0.574041	−	0.818827i	$-0.694625\pi$
$887$	−34.1928			−1.14808			−0.574041	+	0.818827i	$0.694625\pi$
$888$		0			0
$889$		−	15.9509i		−	0.534977i
$890$		0			0
$891$			8.00841i			0.268292i
$892$		0			0
$893$	30.0197			1.00457
$894$		0			0
$895$		−	9.73907i		−	0.325541i
$896$		0			0
$897$	−0.460458	−	0.499791i	−0.0153742	−	0.0166875i
$898$		0			0
$899$		−	14.6296i		−	0.487925i
$900$		0			0
$901$	−67.8511			−2.26045
$902$		0			0
$903$			0.311461i			0.0103648i
$904$		0			0
$905$			35.8091i			1.19034i
$906$		0			0
$907$	12.3773			0.410983			0.205491	−	0.978659i	$-0.434121\pi$
$907$	12.3773			0.410983			0.205491	+	0.978659i	$0.434121\pi$
$908$		0			0
$909$	46.2424			1.53376
$910$		0			0
$911$	−27.0070			−0.894782			−0.447391	−	0.894338i	$-0.647647\pi$
$911$	−27.0070			−0.894782			−0.447391	+	0.894338i	$0.647647\pi$
$912$		0			0
$913$	8.15399			0.269858
$914$		0			0
$915$			6.73493i			0.222650i
$916$		0			0
$917$		−	11.3928i		−	0.376222i
$918$		0			0
$919$	17.5042			0.577410			0.288705	−	0.957418i	$-0.406775\pi$
$919$	17.5042			0.577410			0.288705	+	0.957418i	$0.406775\pi$
$920$		0			0
$921$		−	6.63427i		−	0.218607i
$922$		0			0
$923$	−27.8518	+	25.6599i	−0.916754	+	0.844606i
$924$		0			0
$925$			0.276465i			0.00909012i
$926$		0			0
$927$	−48.8979			−1.60602
$928$		0			0
$929$			44.7525i			1.46828i	0.678997	+	0.734141i	$0.262415\pi$
$929$			44.7525i			1.46828i	−0.678997	+	0.734141i	$0.737585\pi$
$930$		0			0
$931$		−	2.86896i		−	0.0940262i
$932$		0			0
$933$	−2.22358			−0.0727966
$934$		0			0
$935$	11.6337			0.380462
$936$		0			0
$937$	−9.56561			−0.312495			−0.156247	−	0.987718i	$-0.549940\pi$
$937$	−9.56561			−0.312495			−0.156247	+	0.987718i	$0.549940\pi$
$938$		0			0
$939$	2.32776			0.0759635
$940$		0			0
$941$			44.2046i			1.44103i	0.693441	+	0.720514i	$0.256094\pi$
$941$			44.2046i			1.44103i	−0.693441	+	0.720514i	$0.743906\pi$
$942$		0			0
$943$		−	3.17774i		−	0.103481i
$944$		0			0
$945$	−3.91323			−0.127297
$946$		0			0
$947$			3.24462i			0.105436i	0.998609	+	0.0527180i	$0.0167884\pi$
$947$			3.24462i			0.105436i	−0.998609	+	0.0527180i	$0.983212\pi$
$948$		0			0
$949$	−9.92996	+	9.14848i	−0.322340	+	0.296972i
$950$		0			0
$951$		−	3.31215i		−	0.107404i
$952$		0			0
$953$	1.38700			0.0449292			0.0224646	−	0.999748i	$-0.492849\pi$
$953$	1.38700			0.0449292			0.0224646	+	0.999748i	$0.492849\pi$
$954$		0			0
$955$		−	36.5756i		−	1.18356i
$956$		0			0
$957$			2.75995i			0.0892167i
$958$		0			0
$959$	2.06067			0.0665426
$960$		0			0
$961$	27.8655			0.898887
$962$		0			0
$963$	40.3513			1.30030
$964$		0			0
$965$	19.4053			0.624679
$966$		0			0
$967$		−	55.3462i		−	1.77981i	−0.456144	−	0.889906i	$-0.650770\pi$
$967$		−	55.3462i		−	1.77981i	0.456144	−	0.889906i	$-0.349230\pi$
$968$		0			0
$969$		−	5.60292i		−	0.179992i
$970$		0			0
$971$	−44.5410			−1.42939			−0.714695	−	0.699436i	$-0.753434\pi$
$971$	−44.5410			−1.42939			−0.714695	+	0.699436i	$0.753434\pi$
$972$		0			0
$973$			14.7361i			0.472418i
$974$		0			0
$975$	−0.849618	−	0.922194i	−0.0272096	−	0.0295338i
$976$		0			0
$977$			16.3694i			0.523702i	0.965108	+	0.261851i	$0.0843329\pi$
$977$			16.3694i			0.523702i	−0.965108	+	0.261851i	$0.915667\pi$
$978$		0			0
$979$	18.0493			0.576859
$980$		0			0
$981$		−	24.3823i		−	0.778467i
$982$		0			0
$983$			8.18472i			0.261052i	0.991445	+	0.130526i	$0.0416666\pi$
$983$			8.18472i			0.261052i	−0.991445	+	0.130526i	$0.958333\pi$
$984$		0			0
$985$	23.0847			0.735540
$986$		0			0
$987$	−3.49491			−0.111244
$988$		0			0
$989$	−0.526207			−0.0167324
$990$		0			0
$991$	−6.91577			−0.219687			−0.109843	−	0.993949i	$-0.535035\pi$
$991$	−6.91577			−0.219687			−0.109843	+	0.993949i	$0.535035\pi$
$992$		0			0
$993$		−	2.93760i		−	0.0932219i
$994$		0			0
$995$			36.2478i			1.14913i
$996$		0			0
$997$	−17.0610			−0.540326			−0.270163	−	0.962815i	$-0.587078\pi$
$997$	−17.0610			−0.540326			−0.270163	+	0.962815i	$0.587078\pi$
$998$		0			0
$999$		−	0.522221i		−	0.0165223i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4004.2.m.c.2157.22	yes	36
13.12	even	2	inner	4004.2.m.c.2157.21	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4004.2.m.c.2157.21	✓	36	13.12	even	2	inner
4004.2.m.c.2157.22	yes	36	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4004.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.334006 q^{3} +1.98967i q^{5} -1.00000i q^{7} -2.88844 q^{9} +O(q^{10})\)	\(q+0.334006 q^{3} +1.98967i q^{5} -1.00000i q^{7} -2.88844 q^{9} +1.00000i q^{11} +(-2.44303 - 2.65172i) q^{13} +0.664561i q^{15} -5.84704 q^{17} +2.86896i q^{19} -0.334006i q^{21} +0.564296 q^{23} +1.04122 q^{25} -1.96677 q^{27} +8.26319 q^{29} -1.77046i q^{31} +0.334006i q^{33} +1.98967 q^{35} +0.265522i q^{37} +(-0.815987 - 0.885689i) q^{39} -5.63133i q^{41} -0.932502 q^{43} -5.74704i q^{45} -10.4636i q^{47} -1.00000 q^{49} -1.95295 q^{51} +11.6043 q^{53} -1.98967 q^{55} +0.958248i q^{57} +8.63659i q^{59} +10.1344 q^{61} +2.88844i q^{63} +(5.27604 - 4.86082i) q^{65} +1.98939i q^{67} +0.188478 q^{69} -10.5033i q^{71} -3.74473i q^{73} +0.347772 q^{75} +1.00000 q^{77} +4.24966 q^{79} +8.00841 q^{81} -8.15399i q^{83} -11.6337i q^{85} +2.75995 q^{87} -18.0493i q^{89} +(-2.65172 + 2.44303i) q^{91} -0.591343i q^{93} -5.70827 q^{95} -12.1084i q^{97} -2.88844i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 4 q^{3} + 40 q^{9}+O(q^{10}) \) 36 * q - 4 * q^3 + 40 * q^9	\( 36 q - 4 q^{3} + 40 q^{9} - 4 q^{17} + 8 q^{23} - 80 q^{25} + 8 q^{27} + 8 q^{29} - 24 q^{39} + 32 q^{43} - 36 q^{49} - 20 q^{51} + 12 q^{53} + 32 q^{61} - 24 q^{65} + 80 q^{69} - 36 q^{75} + 36 q^{77} + 16 q^{79} + 132 q^{81} + 8 q^{91} + 56 q^{95}+O(q^{100}) \) 36 * q - 4 * q^3 + 40 * q^9 - 4 * q^17 + 8 * q^23 - 80 * q^25 + 8 * q^27 + 8 * q^29 - 24 * q^39 + 32 * q^43 - 36 * q^49 - 20 * q^51 + 12 * q^53 + 32 * q^61 - 24 * q^65 + 80 * q^69 - 36 * q^75 + 36 * q^77 + 16 * q^79 + 132 * q^81 + 8 * q^91 + 56 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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