LMFDB - Embedded newform 3204.1.cd.b.1439.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3204,1,Mod(35,3204)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3204, base_ring=CyclotomicField(88))

chi = DirichletCharacter(H, H._module([44, 44, 63]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3204.35");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3204 = 2^{2} \cdot 3^{2} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3204.cd (of order $88$, degree $40$, 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:	$1.59900430048$
Analytic rank:	$0$
Dimension:	$40$
Coefficient field:	$\Q(\zeta_{88})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{40} - x^{36} + x^{32} - x^{28} + x^{24} - x^{20} + x^{16} - x^{12} + x^{8} - x^{4} + 1 $ x^40 - x^36 + x^32 - x^28 + x^24 - x^20 + x^16 - x^12 + x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{88}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{88} - \cdots)$

Embedding invariants

Embedding label			1439.1
Root			$-0.800541 + 0.599278i$ of defining polynomial
Character	$\chi$	$=$	3204.1439
Dual form			3204.1.cd.b.2231.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.281733 - 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-1.81463 + 0.129785i) q^{5} +(0.755750 + 0.654861i) q^{8} +O(q^{10})$	\(q+(-0.281733 - 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-1.81463 + 0.129785i) q^{5} +(0.755750 + 0.654861i) q^{8} +(0.635768 + 1.70456i) q^{10} +(0.354880 - 1.96697i) q^{13} +(0.415415 - 0.909632i) q^{16} +(0.574406 + 1.05195i) q^{17} +(1.45640 - 1.09024i) q^{20} +(2.28621 - 0.328708i) q^{25} +(-1.98727 + 0.213654i) q^{26} +(-1.83871 - 0.611980i) q^{29} +(-0.989821 - 0.142315i) q^{32} +(0.847507 - 0.847507i) q^{34} +(-1.83717 + 0.760982i) q^{37} +(-1.45640 - 1.09024i) q^{40} +(-0.0630503 - 0.349464i) q^{41} +(-0.599278 + 0.800541i) q^{49} +(-0.959493 - 2.10100i) q^{50} +(0.764879 + 1.84658i) q^{52} +(0.203743 + 0.936593i) q^{53} +(-0.0691673 + 1.93664i) q^{58} +(-0.456811 + 0.769914i) q^{61} +(0.142315 + 0.989821i) q^{64} +(-0.388693 + 3.61538i) q^{65} +(-1.05195 - 0.574406i) q^{68} +(-1.53046 - 0.698939i) q^{73} +(1.24775 + 1.54836i) q^{74} +(-0.635768 + 1.70456i) q^{80} +(-0.317545 + 0.158952i) q^{82} +(-1.17886 - 1.83434i) q^{85} +(0.0713392 - 0.997452i) q^{89} +(-1.27979 + 1.47696i) q^{97} +(0.936950 + 0.349464i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 4 q^{4}+O(q^{10}) $ 40 * q + 4 * q^4	$ 40 q + 4 q^{4} + 4 q^{13} - 4 q^{16} + 4 q^{17} + 4 q^{29} - 4 q^{34} - 4 q^{37} - 40 q^{41} - 4 q^{50} - 4 q^{52} + 4 q^{53} + 4 q^{64} - 4 q^{68}+O(q^{100}) $ 40 * q + 4 * q^4 + 4 * q^13 - 4 * q^16 + 4 * q^17 + 4 * q^29 - 4 * q^34 - 4 * q^37 - 40 * q^41 - 4 * q^50 - 4 * q^52 + 4 * q^53 + 4 * q^64 - 4 * q^68

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times$.

$n$	$181$	$713$	$1603$
$\chi(n)$	$e\left(\frac{71}{88}\right)$	$-1$	$-1$

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 $

$2$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$2$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$3$		0			0
$3$		0			0
$4$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$5$	−1.81463	+	0.129785i	−1.81463	+	0.129785i	−0.936950	−	0.349464i	$-0.886364\pi$
$5$	−1.81463	+	0.129785i	−1.81463	+	0.129785i	−0.877679	+	0.479249i	$0.840909\pi$
$6$		0			0
$7$		0			0		−0.447617	−	0.894225i	$-0.647727\pi$
$7$		0			0		0.447617	+	0.894225i	$0.352273\pi$
$8$	0.755750	+	0.654861i	0.755750	+	0.654861i
$9$		0			0
$10$	0.635768	+	1.70456i	0.635768	+	1.70456i
$11$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$11$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$12$		0			0
$13$	0.354880	−	1.96697i	0.354880	−	1.96697i	0.142315	−	0.989821i	$-0.454545\pi$
$13$	0.354880	−	1.96697i	0.354880	−	1.96697i	0.212565	−	0.977147i	$-0.431818\pi$
$14$		0			0
$15$		0			0
$16$	0.415415	−	0.909632i	0.415415	−	0.909632i
$17$	0.574406	+	1.05195i	0.574406	+	1.05195i	0.989821	+	0.142315i	$0.0454545\pi$
$17$	0.574406	+	1.05195i	0.574406	+	1.05195i	−0.415415	+	0.909632i	$0.636364\pi$
$18$		0			0
$19$		0			0		−0.731895	−	0.681418i	$-0.761364\pi$
$19$		0			0		0.731895	+	0.681418i	$0.238636\pi$
$20$	1.45640	−	1.09024i	1.45640	−	1.09024i
$21$		0			0
$22$		0			0
$23$		0			0		−0.0356923	−	0.999363i	$-0.511364\pi$
$23$		0			0		0.0356923	+	0.999363i	$0.488636\pi$
$24$		0			0
$25$	2.28621	−	0.328708i	2.28621	−	0.328708i
$26$	−1.98727	+	0.213654i	−1.98727	+	0.213654i
$27$		0			0
$28$		0			0
$29$	−1.83871	−	0.611980i	−1.83871	−	0.611980i	−0.997452	−	0.0713392i	$-0.977273\pi$
$29$	−1.83871	−	0.611980i	−1.83871	−	0.611980i	−0.841254	−	0.540641i	$-0.818182\pi$
$30$		0			0
$31$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$31$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$32$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$33$		0			0
$34$	0.847507	−	0.847507i	0.847507	−	0.847507i
$35$		0			0
$36$		0			0
$37$	−1.83717	+	0.760982i	−1.83717	+	0.760982i	−0.877679	+	0.479249i	$0.840909\pi$
$37$	−1.83717	+	0.760982i	−1.83717	+	0.760982i	−0.959493	+	0.281733i	$0.909091\pi$
$38$		0			0
$39$		0			0
$40$	−1.45640	−	1.09024i	−1.45640	−	1.09024i
$41$	−0.0630503	−	0.349464i	−0.0630503	−	0.349464i	0.936950	−	0.349464i	$-0.113636\pi$
$41$	−0.0630503	−	0.349464i	−0.0630503	−	0.349464i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.315800	−	0.948826i	$-0.602273\pi$
$43$		0			0		0.315800	+	0.948826i	$0.397727\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$47$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$48$		0			0
$49$	−0.599278	+	0.800541i	−0.599278	+	0.800541i
$50$	−0.959493	−	2.10100i	−0.959493	−	2.10100i
$51$		0			0
$52$	0.764879	+	1.84658i	0.764879	+	1.84658i
$53$	0.203743	+	0.936593i	0.203743	+	0.936593i	0.959493	+	0.281733i	$0.0909091\pi$
$53$	0.203743	+	0.936593i	0.203743	+	0.936593i	−0.755750	+	0.654861i	$0.772727\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.0691673	+	1.93664i	−0.0691673	+	1.93664i
$59$		0			0		−0.821421	−	0.570323i	$-0.806818\pi$
$59$		0			0		0.821421	+	0.570323i	$0.193182\pi$
$60$		0			0
$61$	−0.456811	+	0.769914i	−0.456811	+	0.769914i	−0.997452	−	0.0713392i	$-0.977273\pi$
$61$	−0.456811	+	0.769914i	−0.456811	+	0.769914i	0.540641	+	0.841254i	$0.318182\pi$
$62$		0			0
$63$		0			0
$64$	0.142315	+	0.989821i	0.142315	+	0.989821i
$65$	−0.388693	+	3.61538i	−0.388693	+	3.61538i
$66$		0			0
$67$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$67$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$68$	−1.05195	−	0.574406i	−1.05195	−	0.574406i
$69$		0			0
$70$		0			0
$71$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$71$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$72$		0			0
$73$	−1.53046	−	0.698939i	−1.53046	−	0.698939i	−0.540641	−	0.841254i	$-0.681818\pi$
$73$	−1.53046	−	0.698939i	−1.53046	−	0.698939i	−0.989821	+	0.142315i	$0.954545\pi$
$74$	1.24775	+	1.54836i	1.24775	+	1.54836i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$79$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$80$	−0.635768	+	1.70456i	−0.635768	+	1.70456i
$81$		0			0
$82$	−0.317545	+	0.158952i	−0.317545	+	0.158952i
$83$		0			0		0.627469	−	0.778642i	$-0.284091\pi$
$83$		0			0		−0.627469	+	0.778642i	$0.715909\pi$
$84$		0			0
$85$	−1.17886	−	1.83434i	−1.17886	−	1.83434i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.0713392	−	0.997452i	0.0713392	−	0.997452i
$89$	0.0713392	−	0.997452i	0.0713392	−	0.997452i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.27979	+	1.47696i	−1.27979	+	1.47696i	−0.479249	+	0.877679i	$0.659091\pi$
$97$	−1.27979	+	1.47696i	−1.27979	+	1.47696i	−0.800541	+	0.599278i	$0.795455\pi$
$98$	0.936950	+	0.349464i	0.936950	+	0.349464i
$99$		0			0
$100$	−1.74557	+	1.51255i	−1.74557	+	1.51255i
$101$	−1.51779	−	0.628688i	−1.51779	−	0.628688i	−0.540641	−	0.841254i	$-0.681818\pi$
$101$	−1.51779	−	0.628688i	−1.51779	−	0.628688i	−0.977147	+	0.212565i	$0.931818\pi$
$102$		0			0
$103$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$103$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$104$	1.55629	−	1.25414i	1.55629	−	1.25414i
$105$		0			0
$106$	0.841254	−	0.459359i	0.841254	−	0.459359i
$107$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$107$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$108$		0			0
$109$	0.647988	+	0.865611i	0.647988	+	0.865611i	0.997452	−	0.0713392i	$-0.0227273\pi$
$109$	0.647988	+	0.865611i	0.647988	+	0.865611i	−0.349464	+	0.936950i	$0.613636\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.212565	+	0.0228531i	0.212565	+	0.0228531i	0.212565	−	0.977147i	$-0.431818\pi$
$113$	0.212565	+	0.0228531i	0.212565	+	0.0228531i			1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	1.87768	−	0.479249i	1.87768	−	0.479249i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$122$	0.867426	+	0.221397i	0.867426	+	0.221397i
$123$		0			0
$124$		0			0
$125$	−2.32828	+	0.506486i	−2.32828	+	0.506486i
$126$		0			0
$127$		0			0		−0.894225	−	0.447617i	$-0.852273\pi$
$127$		0			0		0.894225	+	0.447617i	$0.147727\pi$
$128$	0.909632	−	0.415415i	0.909632	−	0.415415i
$129$		0			0
$130$	3.57843	−	0.645621i	3.57843	−	0.645621i
$131$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$131$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.254771	+	1.17116i	−0.254771	+	1.17116i
$137$	−1.00432	+	0.181200i	−1.00432	+	0.181200i	−0.654861	−	0.755750i	$-0.727273\pi$
$137$	−1.00432	+	0.181200i	−1.00432	+	0.181200i	−0.349464	+	0.936950i	$0.613636\pi$
$138$		0			0
$139$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$139$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	3.41599	+	0.871880i	3.41599	+	0.871880i
$146$	−0.239446	+	1.66538i	−0.239446	+	1.66538i
$147$		0			0
$148$	1.13411	−	1.63343i	1.13411	−	1.63343i
$149$	0.241703	−	0.726200i	0.241703	−	0.726200i	−0.755750	−	0.654861i	$-0.772727\pi$
$149$	0.241703	−	0.726200i	0.241703	−	0.726200i	0.997452	−	0.0713392i	$-0.0227273\pi$
$150$		0			0
$151$		0			0		0.968937	−	0.247307i	$-0.0795455\pi$
$151$		0			0		−0.968937	+	0.247307i	$0.920455\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.919672	+	1.68425i	−0.919672	+	1.68425i	−0.212565	+	0.977147i	$0.568182\pi$
$157$	−0.919672	+	1.68425i	−0.919672	+	1.68425i	−0.707107	+	0.707107i	$0.750000\pi$
$158$		0			0
$159$		0			0
$160$	1.81463	+	0.129785i	1.81463	+	0.129785i
$161$		0			0
$162$		0			0
$163$		0			0		0.778642	−	0.627469i	$-0.215909\pi$
$163$		0			0		−0.778642	+	0.627469i	$0.784091\pi$
$164$	0.241976	+	0.259900i	0.241976	+	0.259900i
$165$		0			0
$166$		0			0
$167$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$167$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$168$		0			0
$169$	−2.80607	−	1.04661i	−2.80607	−	1.04661i
$170$	−1.42792	+	1.64790i	−1.42792	+	1.64790i
$171$		0			0
$172$		0			0
$173$	−0.107829	−	1.50765i	−0.107829	−	1.50765i	−0.707107	−	0.707107i	$-0.750000\pi$
$173$	−0.107829	−	1.50765i	−0.107829	−	1.50765i	0.599278	−	0.800541i	$-0.295455\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.977147	+	0.212565i	−0.977147	+	0.212565i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	0.518809	−	0.360215i	0.518809	−	0.360215i	−0.281733	−	0.959493i	$-0.590909\pi$
$181$	0.518809	−	0.360215i	0.518809	−	0.360215i	0.800541	+	0.599278i	$0.204545\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	3.23502	−	1.61934i	3.23502	−	1.61934i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.984111	−	0.177553i	$-0.943182\pi$
$191$		0			0		0.984111	+	0.177553i	$0.0568182\pi$
$192$		0			0
$193$	−1.19072	−	1.47759i	−1.19072	−	1.47759i	−0.841254	−	0.540641i	$-0.818182\pi$
$193$	−1.19072	−	1.47759i	−1.19072	−	1.47759i	−0.349464	−	0.936950i	$-0.613636\pi$
$194$	1.77769	+	0.811843i	1.77769	+	0.811843i
$195$		0			0
$196$	0.0713392	−	0.997452i	0.0713392	−	0.997452i
$197$	−0.430383	+	0.462264i	−0.430383	+	0.462264i	−0.909632	−	0.415415i	$-0.863636\pi$
$197$	−0.430383	+	0.462264i	−0.430383	+	0.462264i	0.479249	+	0.877679i	$0.340909\pi$
$198$		0			0
$199$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$199$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$200$	1.94306	+	1.24873i	1.94306	+	1.24873i
$201$		0			0
$202$	−0.175612	+	1.63343i	−0.175612	+	1.63343i
$203$		0			0
$204$		0			0
$205$	0.159768	+	0.625965i	0.159768	+	0.625965i
$206$		0			0
$207$		0			0
$208$	−1.64179	−	1.13992i	−1.64179	−	1.13992i
$209$		0			0
$210$		0			0
$211$		0			0		0.247307	−	0.968937i	$-0.420455\pi$
$211$		0			0		−0.247307	+	0.968937i	$0.579545\pi$
$212$	−0.677760	−	0.677760i	−0.677760	−	0.677760i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.647988	−	0.865611i	0.647988	−	0.865611i
$219$		0			0
$220$		0			0
$221$	2.27299	−	0.756524i	2.27299	−	0.756524i
$222$		0			0
$223$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$223$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$224$		0			0
$225$		0			0
$226$	−0.0379591	−	0.210393i	−0.0379591	−	0.210393i
$227$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$227$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$228$		0			0
$229$	0.561732	−	1.12220i	0.561732	−	1.12220i	−0.415415	−	0.909632i	$-0.636364\pi$
$229$	0.561732	−	1.12220i	0.561732	−	1.12220i	0.977147	−	0.212565i	$-0.0681818\pi$
$230$		0			0
$231$		0			0
$232$	−0.988839	−	1.66660i	−0.988839	−	1.66660i
$233$	−0.201264	+	0.201264i	−0.201264	+	0.201264i	−0.800541	−	0.599278i	$-0.795455\pi$
$233$	−0.201264	+	0.201264i	−0.201264	+	0.201264i	0.599278	+	0.800541i	$0.295455\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.860014	−	0.510270i	$-0.829545\pi$
$239$		0			0		0.860014	+	0.510270i	$0.170455\pi$
$240$		0			0
$241$	0.760982	−	0.0818140i	0.760982	−	0.0818140i	0.281733	−	0.959493i	$-0.409091\pi$
$241$	0.760982	−	0.0818140i	0.760982	−	0.0818140i	0.479249	+	0.877679i	$0.340909\pi$
$242$	0.989821	−	0.142315i	0.989821	−	0.142315i
$243$		0			0
$244$	−0.0319530	−	0.894664i	−0.0319530	−	0.894664i
$245$	0.983568	−	1.53046i	0.983568	−	1.53046i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	1.14192	+	2.09127i	1.14192	+	2.09127i
$251$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$251$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$257$	−0.670617	−	1.79799i	−0.670617	−	1.79799i	−0.599278	−	0.800541i	$-0.704545\pi$
$257$	−0.670617	−	1.79799i	−0.670617	−	1.79799i	−0.0713392	−	0.997452i	$-0.522727\pi$
$258$		0			0
$259$		0			0
$260$	−1.62763	−	3.25159i	−1.62763	−	3.25159i
$261$		0			0
$262$		0			0
$263$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$263$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$264$		0			0
$265$	−0.491274	−	1.67313i	−0.491274	−	1.67313i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$269$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$270$		0			0
$271$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$271$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$272$	1.19550	−	0.0855040i	1.19550	−	0.0855040i
$273$		0			0
$274$	0.456811	+	0.912593i	0.456811	+	0.912593i
$275$		0			0
$276$		0			0
$277$	0.0498610	+	0.133682i	0.0498610	+	0.133682i	0.959493	−	0.281733i	$-0.0909091\pi$
$277$	0.0498610	+	0.133682i	0.0498610	+	0.133682i	−0.909632	+	0.415415i	$0.863636\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.44053	+	1.34118i	−1.44053	+	1.34118i	−0.599278	+	0.800541i	$0.704545\pi$
$281$	−1.44053	+	1.34118i	−1.44053	+	1.34118i	−0.841254	+	0.540641i	$0.818182\pi$
$282$		0			0
$283$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$283$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.236009	+	0.367237i	−0.236009	+	0.367237i
$290$	−0.125834	−	3.52326i	−0.125834	−	3.52326i
$291$		0			0
$292$	1.66538	−	0.239446i	1.66538	−	0.239446i
$293$	−0.491779	+	0.0528717i	−0.491779	+	0.0528717i	−0.349464	−	0.936950i	$-0.613636\pi$
$293$	−0.491779	+	0.0528717i	−0.491779	+	0.0528717i	−0.142315	+	0.989821i	$0.545455\pi$
$294$		0			0
$295$		0			0
$296$	−1.88678	−	0.627980i	−1.88678	−	0.627980i
$297$		0			0
$298$	−0.764879	−	0.0273177i	−0.764879	−	0.0273177i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.729020	−	1.45640i	0.729020	−	1.45640i
$306$		0			0
$307$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$307$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$311$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$312$		0			0
$313$	1.63201	−	0.543184i	1.63201	−	0.543184i	0.654861	−	0.755750i	$-0.272727\pi$
$313$	1.63201	−	0.543184i	1.63201	−	0.543184i	0.977147	+	0.212565i	$0.0681818\pi$
$314$	1.87513	+	0.407910i	1.87513	+	0.407910i
$315$		0			0
$316$		0			0
$317$	−0.0592707	−	0.129785i	−0.0592707	−	0.129785i	0.877679	−	0.479249i	$-0.159091\pi$
$317$	−0.0592707	−	0.129785i	−0.0592707	−	0.129785i	−0.936950	+	0.349464i	$0.886364\pi$
$318$		0			0
$319$		0			0
$320$	−0.386712	−	1.77769i	−0.386712	−	1.77769i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.164774	−	4.61356i	0.164774	−	4.61356i
$326$		0			0
$327$		0			0
$328$	0.181200	−	0.305397i	0.181200	−	0.305397i
$329$		0			0
$330$		0			0
$331$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$331$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.997452	+	1.07134i	−0.997452	+	1.07134i			1.00000i	$0.5\pi$
$337$	−0.997452	+	1.07134i	−0.997452	+	1.07134i	−0.997452	+	0.0713392i	$0.977273\pi$
$338$	−0.213654	+	2.98727i	−0.213654	+	2.98727i
$339$		0			0
$340$	1.98344	+	0.905808i	1.98344	+	0.905808i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−1.41620	+	0.528215i	−1.41620	+	0.528215i
$347$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$347$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$348$		0			0
$349$	0.442295	−	0.221397i	0.442295	−	0.221397i	−0.212565	−	0.977147i	$-0.568182\pi$
$349$	0.442295	−	0.221397i	0.442295	−	0.221397i	0.654861	+	0.755750i	$0.272727\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.936950	−	0.650536i	0.936950	−	0.650536i		−	1.00000i	$-0.5\pi$
$353$	0.936950	−	0.650536i	0.936950	−	0.650536i	0.936950	+	0.349464i	$0.113636\pi$
$354$		0			0
$355$		0			0
$356$	0.479249	+	0.877679i	0.479249	+	0.877679i
$357$		0			0
$358$		0			0
$359$		0			0		−0.570323	−	0.821421i	$-0.693182\pi$
$359$		0			0		0.570323	+	0.821421i	$0.306818\pi$
$360$		0			0
$361$	0.0713392	+	0.997452i	0.0713392	+	0.997452i
$362$	−0.491789	−	0.396309i	−0.491789	−	0.396309i
$363$		0			0
$364$		0			0
$365$	2.86793	+	1.06968i	2.86793	+	1.06968i
$366$		0			0
$367$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$367$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$368$		0			0
$369$		0			0
$370$	−2.46515	−	2.64776i	−2.46515	−	2.64776i
$371$		0			0
$372$		0			0
$373$	−1.40524	+	0.767317i	−1.40524	+	0.767317i	−0.989821	−	0.142315i	$-0.954545\pi$
$373$	−1.40524	+	0.767317i	−1.40524	+	0.767317i	−0.415415	+	0.909632i	$0.636364\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.85627	+	3.39950i	−1.85627	+	3.39950i
$378$		0			0
$379$		0			0		0.999363	−	0.0356923i	$-0.0113636\pi$
$379$		0			0		−0.999363	+	0.0356923i	$0.988636\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.968937	−	0.247307i	$-0.0795455\pi$
$383$		0			0		−0.968937	+	0.247307i	$0.920455\pi$
$384$		0			0
$385$		0			0
$386$	−1.08227	+	1.55877i	−1.08227	+	1.55877i
$387$		0			0
$388$	0.278125	−	1.93440i	0.278125	−	1.93440i
$389$	1.66660	+	0.425374i	1.66660	+	0.425374i	0.959493	−	0.281733i	$-0.0909091\pi$
$389$	1.66660	+	0.425374i	1.66660	+	0.425374i	0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$	−0.977147	+	0.212565i	−0.977147	+	0.212565i
$393$		0			0
$394$	0.564792	+	0.282715i	0.564792	+	0.282715i
$395$		0			0
$396$		0			0
$397$	1.44053	−	0.259900i	1.44053	−	0.259900i	0.599278	−	0.800541i	$-0.295455\pi$
$397$	1.44053	−	0.259900i	1.44053	−	0.259900i	0.841254	+	0.540641i	$0.181818\pi$
$398$		0			0
$399$		0			0
$400$	0.650724	−	2.21616i	0.650724	−	2.21616i
$401$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$401$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$402$		0			0
$403$		0			0
$404$	1.61674	−	0.291692i	1.61674	−	0.291692i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.47696	−	0.321292i	1.47696	−	0.321292i	0.599278	−	0.800541i	$-0.295455\pi$
$409$	1.47696	−	0.321292i	1.47696	−	0.321292i	0.877679	+	0.479249i	$0.159091\pi$
$410$	0.555597	−	0.329651i	0.555597	−	0.329651i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.631197	+	1.89644i	−0.631197	+	1.89644i
$417$		0			0
$418$		0			0
$419$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$419$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$420$		0			0
$421$	−1.35503	−	0.145681i	−1.35503	−	0.145681i	−0.599278	−	0.800541i	$-0.704545\pi$
$421$	−1.35503	−	0.145681i	−1.35503	−	0.145681i	−0.755750	+	0.654861i	$0.772727\pi$
$422$		0			0
$423$		0			0
$424$	−0.459359	+	0.841254i	−0.459359	+	0.841254i
$425$	1.65900	+	2.21616i	1.65900	+	2.21616i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$431$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$432$		0			0
$433$	0.197516	+	0.0818140i	0.197516	+	0.0818140i	0.479249	−	0.877679i	$-0.340909\pi$
$433$	0.197516	+	0.0818140i	0.197516	+	0.0818140i	−0.281733	+	0.959493i	$0.590909\pi$
$434$		0			0
$435$		0			0
$436$	−1.01311	−	0.377869i	−1.01311	−	0.377869i
$437$		0			0
$438$		0			0
$439$		0			0		−0.778642	−	0.627469i	$-0.784091\pi$
$439$		0			0		0.778642	+	0.627469i	$0.215909\pi$
$440$		0			0
$441$		0			0
$442$	−1.36626	−	1.96778i	−1.36626	−	1.96778i
$443$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$443$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$444$		0			0
$445$			1.81926i			1.81926i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.949018	+	1.47670i	0.949018	+	1.47670i	0.877679	+	0.479249i	$0.159091\pi$
$449$	0.949018	+	1.47670i	0.949018	+	1.47670i	0.0713392	+	0.997452i	$0.477273\pi$
$450$		0			0
$451$		0			0
$452$	−0.191177	+	0.0956962i	−0.191177	+	0.0956962i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.628688	−	1.51779i	0.628688	−	1.51779i	−0.212565	−	0.977147i	$-0.568182\pi$
$457$	0.628688	−	1.51779i	0.628688	−	1.51779i	0.841254	−	0.540641i	$-0.181818\pi$
$458$	−1.23500	−	0.222818i	−1.23500	−	0.222818i
$459$		0			0
$460$		0			0
$461$	−1.37491	−	0.627899i	−1.37491	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$461$	−1.37491	−	0.627899i	−1.37491	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$462$		0			0
$463$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$463$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$464$	−1.32050	+	1.41832i	−1.32050	+	1.41832i
$465$		0			0
$466$	0.249813	+	0.136408i	0.249813	+	0.136408i
$467$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$467$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$479$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$480$		0			0
$481$	0.844851	+	3.88372i	0.844851	+	3.88372i
$482$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$483$		0			0
$484$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$485$	2.13066	−	2.84623i	2.13066	−	2.84623i
$486$		0			0
$487$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$487$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$488$	−0.849422	+	0.282715i	−0.849422	+	0.282715i
$489$		0			0
$490$	−1.74557	−	0.512546i	−1.74557	−	0.512546i
$491$		0			0		−0.315800	−	0.948826i	$-0.602273\pi$
$491$		0			0		0.315800	+	0.948826i	$0.397727\pi$
$492$		0			0
$493$	−0.412394	−	2.28575i	−0.412394	−	2.28575i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.510270	−	0.860014i	$-0.670455\pi$
$499$		0			0		0.510270	+	0.860014i	$0.329545\pi$
$500$	1.68484	−	1.68484i	1.68484	−	1.68484i
$501$		0			0
$502$		0			0
$503$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$503$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$504$		0			0
$505$	2.83582	+	0.943850i	2.83582	+	0.943850i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.66538	−	0.239446i	1.66538	−	0.239446i	0.755750	−	0.654861i	$-0.227273\pi$
$509$	1.66538	−	0.239446i	1.66538	−	0.239446i	0.909632	+	0.415415i	$0.136364\pi$
$510$		0			0
$511$		0			0
$512$	−0.540641	+	0.841254i	−0.540641	+	0.841254i
$513$		0			0
$514$	−1.53623	+	1.15001i	−1.53623	+	1.15001i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−2.66132	+	2.47778i	−2.66132	+	2.47778i
$521$	−0.0126746	+	0.0702505i	−0.0126746	+	0.0702505i	−0.989821	−	0.142315i	$-0.954545\pi$
$521$	−0.0126746	+	0.0702505i	−0.0126746	+	0.0702505i	0.977147	+	0.212565i	$0.0681818\pi$
$522$		0			0
$523$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$523$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.997452	+	0.0713392i	−0.997452	+	0.0713392i
$530$	−1.46694	+	0.942748i	−1.46694	+	0.942748i
$531$		0			0
$532$		0			0
$533$	−0.709760			−0.709760
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.66538	−	1.07028i	1.66538	−	1.07028i
$539$		0			0
$540$		0			0
$541$	−0.849422	−	1.69693i	−0.849422	−	1.69693i	−0.707107	−	0.707107i	$-0.750000\pi$
$541$	−0.849422	−	1.69693i	−0.849422	−	1.69693i	−0.142315	−	0.989821i	$-0.545455\pi$
$542$		0			0
$543$		0			0
$544$	−0.418852	−	1.12299i	−0.418852	−	1.12299i
$545$	−1.28820	−	1.48666i	−1.28820	−	1.48666i
$546$		0			0
$547$		0			0		0.177553	−	0.984111i	$-0.443182\pi$
$547$		0			0		−0.177553	+	0.984111i	$0.556818\pi$
$548$	0.746928	−	0.695414i	0.746928	−	0.695414i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.114220	−	0.0855040i	0.114220	−	0.0855040i
$555$		0			0
$556$		0			0
$557$	0.00254789	+	0.0713392i	0.00254789	+	0.0713392i	1.00000			$0$
$557$	0.00254789	+	0.0713392i	0.00254789	+	0.0713392i	−0.997452	+	0.0713392i	$0.977273\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.69270	+	1.00432i	1.69270	+	1.00432i
$563$		0			0		−0.948826	−	0.315800i	$-0.897727\pi$
$563$		0			0		0.948826	+	0.315800i	$0.102273\pi$
$564$		0			0
$565$	−0.388693	−	0.0138822i	−0.388693	−	0.0138822i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.322286	+	0.543184i	0.322286	+	0.543184i	0.977147	−	0.212565i	$-0.0681818\pi$
$569$	0.322286	+	0.543184i	0.322286	+	0.543184i	−0.654861	+	0.755750i	$0.727273\pi$
$570$		0			0
$571$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$571$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.583522	−	1.75320i	−0.583522	−	1.75320i	−0.654861	−	0.755750i	$-0.727273\pi$
$577$	−0.583522	−	1.75320i	−0.583522	−	1.75320i	0.0713392	−	0.997452i	$-0.477273\pi$
$578$	0.418852	+	0.122986i	0.418852	+	0.122986i
$579$		0			0
$580$	−3.34509	+	1.11335i	−3.34509	+	1.11335i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.698939	−	1.53046i	−0.698939	−	1.53046i
$585$		0			0
$586$	0.189280	+	0.456963i	0.189280	+	0.456963i
$587$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$587$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.0709757	+	1.98727i	−0.0709757	+	1.98727i
$593$	−1.55877	−	1.08227i	−1.55877	−	1.08227i	−0.959493	−	0.281733i	$-0.909091\pi$
$593$	−1.55877	−	1.08227i	−1.55877	−	1.08227i	−0.599278	−	0.800541i	$-0.704545\pi$
$594$		0			0
$595$		0			0
$596$	0.189280	+	0.741593i	0.189280	+	0.741593i
$597$		0			0
$598$		0			0
$599$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$599$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$600$		0			0
$601$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	0.281733	−	0.959493i	$-0.409091\pi$
$601$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	−0.755750	+	0.654861i	$0.772727\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.129785	−	1.81463i	0.129785	−	1.81463i
$606$		0			0
$607$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$607$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$608$		0			0
$609$		0			0
$610$	−1.60279	−	0.289175i	−1.60279	−	0.289175i
$611$		0			0
$612$		0			0
$613$	−0.266684	+	0.0994679i	−0.266684	+	0.0994679i	−0.479249	−	0.877679i	$-0.659091\pi$
$613$	−0.266684	+	0.0994679i	−0.266684	+	0.0994679i	0.212565	+	0.977147i	$0.431818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.918482	+	1.13977i	−0.918482	+	1.13977i	0.0713392	+	0.997452i	$0.477273\pi$
$617$	−0.918482	+	1.13977i	−0.918482	+	1.13977i	−0.989821	+	0.142315i	$0.954545\pi$
$618$		0			0
$619$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$619$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.94306	−	0.570534i	1.94306	−	0.570534i
$626$	−0.980971	−	1.41287i	−0.980971	−	1.41287i
$627$		0			0
$628$	−0.136899	−	1.91410i	−0.136899	−	1.91410i
$629$	−1.85580	−	1.49549i	−1.85580	−	1.49549i
$630$		0			0
$631$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$631$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$632$		0			0
$633$		0			0
$634$	−0.107829	+	0.0934345i	−0.107829	+	0.0934345i
$635$		0			0
$636$		0			0
$637$	1.36197	+	1.46286i	1.36197	+	1.46286i
$638$		0			0
$639$		0			0
$640$	−1.59673	+	0.871880i	−1.59673	+	0.871880i
$641$	0.562029	+	0.0401971i	0.562029	+	0.0401971i	0.349464	−	0.936950i	$-0.386364\pi$
$641$	0.562029	+	0.0401971i	0.562029	+	0.0401971i	0.212565	+	0.977147i	$0.431818\pi$
$642$		0			0
$643$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$643$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.994270	−	0.106895i	$-0.965909\pi$
$647$		0			0		0.994270	+	0.106895i	$0.0340909\pi$
$648$		0			0
$649$		0			0
$650$	−4.47310	+	1.14169i	−4.47310	+	1.14169i
$651$		0			0
$652$		0			0
$653$	−0.121929	+	0.175612i	−0.121929	+	0.175612i	−0.877679	−	0.479249i	$-0.840909\pi$
$653$	−0.121929	+	0.175612i	−0.121929	+	0.175612i	0.755750	+	0.654861i	$0.227273\pi$
$654$		0			0
$655$		0			0
$656$	−0.344076	−	0.0878201i	−0.344076	−	0.0878201i
$657$		0			0
$658$		0			0
$659$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$659$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$660$		0			0
$661$	−1.59928	−	0.800541i	−1.59928	−	0.800541i	−0.599278	−	0.800541i	$-0.704545\pi$
$661$	−1.59928	−	0.800541i	−1.59928	−	0.800541i	−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.871880	+	0.398174i	−0.871880	+	0.398174i	−0.800541	−	0.599278i	$-0.795455\pi$
$673$	−0.871880	+	0.398174i	−0.871880	+	0.398174i	−0.0713392	+	0.997452i	$0.522727\pi$
$674$	1.30896	+	0.655217i	1.30896	+	0.655217i
$675$		0			0
$676$	2.92646	−	0.636613i	2.92646	−	0.636613i
$677$	−0.658226	+	0.390544i	−0.658226	+	0.390544i	−0.800541	−	0.599278i	$-0.795455\pi$
$677$	−0.658226	+	0.390544i	−0.658226	+	0.390544i	0.142315	+	0.989821i	$0.454545\pi$
$678$		0			0
$679$		0			0
$680$	0.310316	−	2.15829i	0.310316	−	2.15829i
$681$		0			0
$682$		0			0
$683$		0			0		0.315800	−	0.948826i	$-0.397727\pi$
$683$		0			0		−0.315800	+	0.948826i	$0.602273\pi$
$684$		0			0
$685$	1.79896	−	0.459157i	1.79896	−	0.459157i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.91455	−	0.0683785i	1.91455	−	0.0683785i
$690$		0			0
$691$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$691$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$692$	0.905808	+	1.21002i	0.905808	+	1.21002i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.331401	−	0.267060i	0.331401	−	0.267060i
$698$	−0.337038	−	0.362005i	−0.337038	−	0.362005i
$699$		0			0
$700$		0			0
$701$	1.50765	−	1.30638i	1.50765	−	1.30638i	0.707107	−	0.707107i	$-0.250000\pi$
$701$	1.50765	−	1.30638i	1.50765	−	1.30638i	0.800541	−	0.599278i	$-0.204545\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.888154	−	0.715720i	−0.888154	−	0.715720i
$707$		0			0
$708$		0			0
$709$	−0.0407123	−	0.0586368i	−0.0407123	−	0.0586368i	0.800541	−	0.599278i	$-0.204545\pi$
$709$	−0.0407123	−	0.0586368i	−0.0407123	−	0.0586368i	−0.841254	+	0.540641i	$0.818182\pi$
$710$		0			0
$711$		0			0
$712$	0.707107	−	0.707107i	0.707107	−	0.707107i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$719$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$720$		0			0
$721$		0			0
$722$	0.936950	−	0.349464i	0.936950	−	0.349464i
$723$		0			0
$724$	−0.241703	+	0.583522i	−0.241703	+	0.583522i
$725$	−4.40483	−	0.794719i	−4.40483	−	0.794719i
$726$		0			0
$727$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$727$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$728$		0			0
$729$		0			0
$730$	0.218364	−	3.05313i	0.218364	−	3.05313i
$731$		0			0
$732$		0			0
$733$	−1.32661	−	0.724384i	−1.32661	−	0.724384i	−0.349464	−	0.936950i	$-0.613636\pi$
$733$	−1.32661	−	0.724384i	−1.32661	−	0.724384i	−0.977147	+	0.212565i	$0.931818\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.247307	−	0.968937i	$-0.579545\pi$
$739$		0			0		0.247307	+	0.968937i	$0.420455\pi$
$740$	−1.84599	+	3.11126i	−1.84599	+	3.11126i
$741$		0			0
$742$		0			0
$743$		0			0		0.0356923	−	0.999363i	$-0.488636\pi$
$743$		0			0		−0.0356923	+	0.999363i	$0.511364\pi$
$744$		0			0
$745$	−0.344351	+	1.34915i	−0.344351	+	1.34915i
$746$	1.13214	+	1.13214i	1.13214	+	1.13214i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$751$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$752$		0			0
$753$		0			0
$754$	3.78476	+	0.823325i	3.78476	+	0.823325i
$755$		0			0
$756$		0			0
$757$	−1.91410	−	0.562029i	−1.91410	−	0.562029i	−0.977147	−	0.212565i	$-0.931818\pi$
$757$	−1.91410	−	0.562029i	−1.91410	−	0.562029i	−0.936950	−	0.349464i	$-0.886364\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.40524	−	1.05195i	−1.40524	−	1.05195i	−0.989821	−	0.142315i	$-0.954545\pi$
$761$	−1.40524	−	1.05195i	−1.40524	−	1.05195i	−0.415415	−	0.909632i	$-0.636364\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.58479	+	0.227858i	1.58479	+	0.227858i	0.877679	−	0.479249i	$-0.159091\pi$
$769$	1.58479	+	0.227858i	1.58479	+	0.227858i	0.707107	+	0.707107i	$0.250000\pi$
$770$		0			0
$771$		0			0
$772$	1.80054	+	0.599278i	1.80054	+	0.599278i
$773$	1.69270	+	1.00432i	1.69270	+	1.00432i	0.936950	+	0.349464i	$0.113636\pi$
$773$	1.69270	+	1.00432i	1.69270	+	1.00432i	0.755750	+	0.654861i	$0.227273\pi$
$774$		0			0
$775$		0			0
$776$	−1.93440	+	0.278125i	−1.93440	+	0.278125i
$777$		0			0
$778$	−0.0613918	−	1.71893i	−0.0613918	−	1.71893i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.479249	+	0.877679i	0.479249	+	0.877679i
$785$	1.45027	−	3.17565i	1.45027	−	3.17565i
$786$		0			0
$787$		0			0		0.731895	−	0.681418i	$-0.238636\pi$
$787$		0			0		−0.731895	+	0.681418i	$0.761364\pi$
$788$	0.112142	−	0.621564i	0.112142	−	0.621564i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.35228	+	1.17176i	1.35228	+	1.17176i
$794$	−0.655217	−	1.30896i	−0.655217	−	1.30896i
$795$		0			0
$796$		0			0
$797$	1.57642	−	1.01311i	1.57642	−	1.01311i	0.599278	−	0.800541i	$-0.295455\pi$
$797$	1.57642	−	1.01311i	1.57642	−	1.01311i	0.977147	−	0.212565i	$-0.0681818\pi$
$798$		0			0
$799$		0			0
$800$	−2.30972			−2.30972
$801$		0			0
$802$	1.08128			1.08128
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.735364	−	1.46907i	−0.735364	−	1.46907i
$809$	0.528215	+	0.457701i	0.528215	+	0.457701i	0.877679	−	0.479249i	$-0.159091\pi$
$809$	0.528215	+	0.457701i	0.528215	+	0.457701i	−0.349464	+	0.936950i	$0.613636\pi$
$810$		0			0
$811$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$811$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.724384	−	1.32661i	−0.724384	−	1.32661i
$819$		0			0
$820$	−0.472827	−	0.440218i	−0.472827	−	0.440218i
$821$	0.559521	−	0.418852i	0.559521	−	0.418852i	−0.281733	−	0.959493i	$-0.590909\pi$
$821$	0.559521	−	0.418852i	0.559521	−	0.418852i	0.841254	+	0.540641i	$0.181818\pi$
$822$		0			0
$823$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$823$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.994270	−	0.106895i	$-0.0340909\pi$
$827$		0			0		−0.994270	+	0.106895i	$0.965909\pi$
$828$		0			0
$829$	1.58910	+	0.942856i	1.58910	+	0.942856i	0.989821	+	0.142315i	$0.0454545\pi$
$829$	1.58910	+	0.942856i	1.58910	+	0.942856i	0.599278	+	0.800541i	$0.295455\pi$
$830$		0			0
$831$		0			0
$832$	1.99745	+	0.0713392i	1.99745	+	0.0713392i
$833$	−1.18636	−	0.170572i	−1.18636	−	0.170572i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.447617	−	0.894225i	$-0.352273\pi$
$839$		0			0		−0.447617	+	0.894225i	$0.647727\pi$
$840$		0			0
$841$	2.20578	+	1.65122i	2.20578	+	1.65122i
$842$	0.241976	+	1.34118i	0.241976	+	1.34118i
$843$		0			0
$844$		0			0
$845$	5.22782	+	1.53503i	5.22782	+	1.53503i
$846$		0			0
$847$		0			0
$848$	0.936593	+	0.203743i	0.936593	+	0.203743i
$849$		0			0
$850$	1.65900	−	2.21616i	1.65900	−	2.21616i
$851$		0			0
$852$		0			0
$853$	0.707107	+	1.70711i	0.707107	+	1.70711i	0.707107	+	0.707107i	$0.250000\pi$
$853$	0.707107	+	1.70711i	0.707107	+	1.70711i			1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.385126	+	1.50891i	−0.385126	+	1.50891i	0.415415	+	0.909632i	$0.363636\pi$
$857$	−0.385126	+	1.50891i	−0.385126	+	1.50891i	−0.800541	+	0.599278i	$0.795455\pi$
$858$		0			0
$859$		0			0		0.0356923	−	0.999363i	$-0.488636\pi$
$859$		0			0		−0.0356923	+	0.999363i	$0.511364\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.247307	−	0.968937i	$-0.579545\pi$
$863$		0			0		0.247307	+	0.968937i	$0.420455\pi$
$864$		0			0
$865$	0.391340	+	2.72183i	0.391340	+	2.72183i
$866$	0.0228531	−	0.212565i	0.0228531	−	0.212565i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.0771377	+	1.07853i	−0.0771377	+	1.07853i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.349464	−	0.0630503i	−0.349464	−	0.0630503i		−	1.00000i	$-0.5\pi$
$877$	−0.349464	−	0.0630503i	−0.349464	−	0.0630503i	−0.349464	+	0.936950i	$0.613636\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.559521	−	1.50013i	0.559521	−	1.50013i	−0.281733	−	0.959493i	$-0.590909\pi$
$881$	0.559521	−	1.50013i	0.559521	−	1.50013i	0.841254	−	0.540641i	$-0.181818\pi$
$882$		0			0
$883$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$883$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$884$	−1.50315	+	1.86530i	−1.50315	+	1.86530i
$885$		0			0
$886$		0			0
$887$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$887$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$888$		0			0
$889$		0			0
$890$	1.74557	−	0.512546i	1.74557	−	0.512546i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.14952	−	1.32661i	1.14952	−	1.32661i
$899$		0			0
$900$		0			0
$901$	−0.868215	+	0.752312i	−0.868215	+	0.752312i
$902$		0			0
$903$		0			0
$904$	0.145681	+	0.156472i	0.145681	+	0.156472i
$905$	−0.894695	+	0.720991i	−0.894695	+	0.720991i
$906$		0			0
$907$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$907$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$911$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$912$		0			0
$913$		0			0
$914$	−1.63343	−	0.175612i	−1.63343	−	0.175612i
$915$		0			0
$916$	0.134147	+	1.24775i	0.134147	+	1.24775i
$917$		0			0
$918$		0			0
$919$		0			0		0.315800	−	0.948826i	$-0.397727\pi$
$919$		0			0		−0.315800	+	0.948826i	$0.602273\pi$
$920$		0			0
$921$		0			0
$922$	−0.215109	+	1.49611i	−0.215109	+	1.49611i
$923$		0			0
$924$		0			0
$925$	−3.95002	+	2.34366i	−3.95002	+	2.34366i
$926$		0			0
$927$		0			0
$928$	1.73290	+	0.867426i	1.73290	+	0.867426i
$929$	−1.70456	+	0.778446i	−1.70456	+	0.778446i	−0.707107	+	0.707107i	$0.750000\pi$
$929$	−1.70456	+	0.778446i	−1.70456	+	0.778446i	−0.997452	+	0.0713392i	$0.977273\pi$
$930$		0			0
$931$		0			0
$932$	0.0605024	−	0.278125i	0.0605024	−	0.278125i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.415415	−	1.90963i	0.415415	−	1.90963i		−	1.00000i	$-0.5\pi$
$937$	0.415415	−	1.90963i	0.415415	−	1.90963i	0.415415	−	0.909632i	$-0.363636\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1.39256	−	0.697067i	−1.39256	−	0.697067i	−0.415415	−	0.909632i	$-0.636364\pi$
$941$	−1.39256	−	0.697067i	−1.39256	−	0.697067i	−0.977147	+	0.212565i	$0.931818\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$947$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$	−1.91792	+	2.76233i	−1.91792	+	2.76233i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.175612	−	1.63343i	−0.175612	−	1.63343i	−0.654861	−	0.755750i	$-0.727273\pi$
$953$	−0.175612	−	1.63343i	−0.175612	−	1.63343i	0.479249	−	0.877679i	$-0.340909\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.997452	+	0.0713392i	0.997452	+	0.0713392i
$962$	3.48838	−	1.90480i	3.48838	−	1.90480i
$963$		0			0
$964$	−0.595946	+	0.480244i	−0.595946	+	0.480244i
$965$	2.35248	+	2.52674i	2.35248	+	2.52674i
$966$		0			0
$967$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$967$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$968$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$969$		0			0
$970$	−3.33121	−	1.24248i	−3.33121	−	1.24248i
$971$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$971$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.510572	+	0.735364i	0.510572	+	0.735364i
$977$	0.670617	−	0.196911i	0.670617	−	0.196911i	0.0713392	−	0.997452i	$-0.477273\pi$
$977$	0.670617	−	0.196911i	0.670617	−	0.196911i	0.599278	+	0.800541i	$0.295455\pi$
$978$		0			0
$979$		0			0
$980$			1.81926i			1.81926i
$981$		0			0
$982$		0			0
$983$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$983$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$984$		0			0
$985$	0.720991	−	0.894695i	0.720991	−	0.894695i
$986$	−2.07697	+	1.03966i	−2.07697	+	1.03966i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$991$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.0303285	+	0.424047i	−0.0303285	+	0.424047i	0.959493	+	0.281733i	$0.0909091\pi$
$997$	−0.0303285	+	0.424047i	−0.0303285	+	0.424047i	−0.989821	+	0.142315i	$0.954545\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3204.1.cd.b.1439.1	yes	40
3.2	odd	2		3204.1.cd.a.1439.1	✓	40
4.3	odd	2	CM	3204.1.cd.b.1439.1	yes	40
12.11	even	2		3204.1.cd.a.1439.1	✓	40
89.6	odd	88		3204.1.cd.a.2231.1	yes	40
267.95	even	88	inner	3204.1.cd.b.2231.1	yes	40
356.95	even	88		3204.1.cd.a.2231.1	yes	40
1068.95	odd	88	inner	3204.1.cd.b.2231.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3204.1.cd.a.1439.1	✓	40	3.2	odd	2
3204.1.cd.a.1439.1	✓	40	12.11	even	2
3204.1.cd.a.2231.1	yes	40	89.6	odd	88
3204.1.cd.a.2231.1	yes	40	356.95	even	88
3204.1.cd.b.1439.1	yes	40	1.1	even	1	trivial
3204.1.cd.b.1439.1	yes	40	4.3	odd	2	CM
3204.1.cd.b.2231.1	yes	40	267.95	even	88	inner
3204.1.cd.b.2231.1	yes	40	1068.95	odd	88	inner

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:	\( N \)	\(=\)	\( 3204 = 2^{2} \cdot 3^{2} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3204.cd (of order \(88\), degree \(40\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.281733 - 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-1.81463 + 0.129785i) q^{5} +(0.755750 + 0.654861i) q^{8} +O(q^{10})\)	\(q+(-0.281733 - 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-1.81463 + 0.129785i) q^{5} +(0.755750 + 0.654861i) q^{8} +(0.635768 + 1.70456i) q^{10} +(0.354880 - 1.96697i) q^{13} +(0.415415 - 0.909632i) q^{16} +(0.574406 + 1.05195i) q^{17} +(1.45640 - 1.09024i) q^{20} +(2.28621 - 0.328708i) q^{25} +(-1.98727 + 0.213654i) q^{26} +(-1.83871 - 0.611980i) q^{29} +(-0.989821 - 0.142315i) q^{32} +(0.847507 - 0.847507i) q^{34} +(-1.83717 + 0.760982i) q^{37} +(-1.45640 - 1.09024i) q^{40} +(-0.0630503 - 0.349464i) q^{41} +(-0.599278 + 0.800541i) q^{49} +(-0.959493 - 2.10100i) q^{50} +(0.764879 + 1.84658i) q^{52} +(0.203743 + 0.936593i) q^{53} +(-0.0691673 + 1.93664i) q^{58} +(-0.456811 + 0.769914i) q^{61} +(0.142315 + 0.989821i) q^{64} +(-0.388693 + 3.61538i) q^{65} +(-1.05195 - 0.574406i) q^{68} +(-1.53046 - 0.698939i) q^{73} +(1.24775 + 1.54836i) q^{74} +(-0.635768 + 1.70456i) q^{80} +(-0.317545 + 0.158952i) q^{82} +(-1.17886 - 1.83434i) q^{85} +(0.0713392 - 0.997452i) q^{89} +(-1.27979 + 1.47696i) q^{97} +(0.936950 + 0.349464i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{4}+O(q^{10}) \) 40 * q + 4 * q^4	\( 40 q + 4 q^{4} + 4 q^{13} - 4 q^{16} + 4 q^{17} + 4 q^{29} - 4 q^{34} - 4 q^{37} - 40 q^{41} - 4 q^{50} - 4 q^{52} + 4 q^{53} + 4 q^{64} - 4 q^{68}+O(q^{100}) \) 40 * q + 4 * q^4 + 4 * q^13 - 4 * q^16 + 4 * q^17 + 4 * q^29 - 4 * q^34 - 4 * q^37 - 40 * q^41 - 4 * q^50 - 4 * q^52 + 4 * q^53 + 4 * q^64 - 4 * q^68

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more