LMFDB - Embedded newform 3204.1.cd.b.1907.1

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			1907.1
Root			$0.349464 - 0.936950i$ of defining polynomial
Character	$\chi$	$=$	3204.1907
Dual form			3204.1.cd.b.2663.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.755750 - 0.654861i) q^{2} +(0.142315 - 0.989821i) q^{4} +(-0.550588 - 0.119773i) q^{5} +(-0.540641 - 0.841254i) q^{8} +O(q^{10})$	\(q+(0.755750 - 0.654861i) q^{2} +(0.142315 - 0.989821i) q^{4} +(-0.550588 - 0.119773i) q^{5} +(-0.540641 - 0.841254i) q^{8} +(-0.494541 + 0.270040i) q^{10} +(-1.01469 - 1.71017i) q^{13} +(-0.959493 - 0.281733i) q^{16} +(1.86912 - 0.133682i) q^{17} +(-0.196911 + 0.527938i) q^{20} +(-0.620830 - 0.283524i) q^{25} +(-1.88678 - 0.627980i) q^{26} +(-0.834832 + 1.20239i) q^{29} +(-0.909632 + 0.415415i) q^{32} +(1.32505 - 1.32505i) q^{34} +(-0.726200 - 1.75320i) q^{37} +(0.196911 + 0.527938i) q^{40} +(-0.520751 + 0.877679i) q^{41} +(0.936950 - 0.349464i) q^{49} +(-0.654861 + 0.192284i) q^{50} +(-1.83717 + 0.760982i) q^{52} +(1.19550 - 1.59700i) q^{53} +(0.156472 + 1.45540i) q^{58} +(-1.96697 + 0.0702505i) q^{61} +(-0.415415 + 0.909632i) q^{64} +(0.353845 + 1.06313i) q^{65} +(0.133682 - 1.86912i) q^{68} +(0.0801894 - 0.273100i) q^{73} +(-1.69693 - 0.849422i) q^{74} +(0.494541 + 0.270040i) q^{80} +(0.181200 + 1.00432i) q^{82} +(-1.04513 - 0.150267i) q^{85} +(-0.212565 - 0.977147i) q^{89} +(1.34692 - 0.865611i) q^{97} +(0.479249 - 0.877679i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times$.

$n$	$181$	$713$	$1603$
$\chi(n)$	$e\left(\frac{51}{88}\right)$	$-1$	$-1$

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 $

$2$	0.755750	−	0.654861i	0.755750	−	0.654861i
$2$	0.755750	−	0.654861i	0.755750	−	0.654861i
$3$		0			0
$3$		0			0
$4$	0.142315	−	0.989821i	0.142315	−	0.989821i
$5$	−0.550588	−	0.119773i	−0.550588	−	0.119773i	−0.0713392	−	0.997452i	$-0.522727\pi$
$5$	−0.550588	−	0.119773i	−0.550588	−	0.119773i	−0.479249	+	0.877679i	$0.659091\pi$
$6$		0			0
$7$		0			0		0.984111	−	0.177553i	$-0.0568182\pi$
$7$		0			0		−0.984111	+	0.177553i	$0.943182\pi$
$8$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$9$		0			0
$10$	−0.494541	+	0.270040i	−0.494541	+	0.270040i
$11$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$11$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$12$		0			0
$13$	−1.01469	−	1.71017i	−1.01469	−	1.71017i	−0.599278	−	0.800541i	$-0.704545\pi$
$13$	−1.01469	−	1.71017i	−1.01469	−	1.71017i	−0.415415	−	0.909632i	$-0.636364\pi$
$14$		0			0
$15$		0			0
$16$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$17$	1.86912	−	0.133682i	1.86912	−	0.133682i	0.909632	−	0.415415i	$-0.136364\pi$
$17$	1.86912	−	0.133682i	1.86912	−	0.133682i	0.959493	+	0.281733i	$0.0909091\pi$
$18$		0			0
$19$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$19$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$20$	−0.196911	+	0.527938i	−0.196911	+	0.527938i
$21$		0			0
$22$		0			0
$23$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$23$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$24$		0			0
$25$	−0.620830	−	0.283524i	−0.620830	−	0.283524i
$26$	−1.88678	−	0.627980i	−1.88678	−	0.627980i
$27$		0			0
$28$		0			0
$29$	−0.834832	+	1.20239i	−0.834832	+	1.20239i	0.142315	+	0.989821i	$0.454545\pi$
$29$	−0.834832	+	1.20239i	−0.834832	+	1.20239i	−0.977147	+	0.212565i	$0.931818\pi$
$30$		0			0
$31$		0			0		0.994270	−	0.106895i	$-0.0340909\pi$
$31$		0			0		−0.994270	+	0.106895i	$0.965909\pi$
$32$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$33$		0			0
$34$	1.32505	−	1.32505i	1.32505	−	1.32505i
$35$		0			0
$36$		0			0
$37$	−0.726200	−	1.75320i	−0.726200	−	1.75320i	−0.654861	−	0.755750i	$-0.727273\pi$
$37$	−0.726200	−	1.75320i	−0.726200	−	1.75320i	−0.0713392	−	0.997452i	$-0.522727\pi$
$38$		0			0
$39$		0			0
$40$	0.196911	+	0.527938i	0.196911	+	0.527938i
$41$	−0.520751	+	0.877679i	−0.520751	+	0.877679i	0.479249	+	0.877679i	$0.340909\pi$
$41$	−0.520751	+	0.877679i	−0.520751	+	0.877679i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$43$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$47$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$48$		0			0
$49$	0.936950	−	0.349464i	0.936950	−	0.349464i
$50$	−0.654861	+	0.192284i	−0.654861	+	0.192284i
$51$		0			0
$52$	−1.83717	+	0.760982i	−1.83717	+	0.760982i
$53$	1.19550	−	1.59700i	1.19550	−	1.59700i	0.540641	−	0.841254i	$-0.318182\pi$
$53$	1.19550	−	1.59700i	1.19550	−	1.59700i	0.654861	−	0.755750i	$-0.272727\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.156472	+	1.45540i	0.156472	+	1.45540i
$59$		0			0		−0.247307	−	0.968937i	$-0.579545\pi$
$59$		0			0		0.247307	+	0.968937i	$0.420455\pi$
$60$		0			0
$61$	−1.96697	+	0.0702505i	−1.96697	+	0.0702505i	−0.989821	−	0.142315i	$-0.954545\pi$
$61$	−1.96697	+	0.0702505i	−1.96697	+	0.0702505i	−0.977147	+	0.212565i	$0.931818\pi$
$62$		0			0
$63$		0			0
$64$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$65$	0.353845	+	1.06313i	0.353845	+	1.06313i
$66$		0			0
$67$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$67$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$68$	0.133682	−	1.86912i	0.133682	−	1.86912i
$69$		0			0
$70$		0			0
$71$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$71$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$72$		0			0
$73$	0.0801894	−	0.273100i	0.0801894	−	0.273100i	−0.909632	−	0.415415i	$-0.863636\pi$
$73$	0.0801894	−	0.273100i	0.0801894	−	0.273100i	0.989821	+	0.142315i	$0.0454545\pi$
$74$	−1.69693	−	0.849422i	−1.69693	−	0.849422i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$79$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$80$	0.494541	+	0.270040i	0.494541	+	0.270040i
$81$		0			0
$82$	0.181200	+	1.00432i	0.181200	+	1.00432i
$83$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$83$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$84$		0			0
$85$	−1.04513	−	0.150267i	−1.04513	−	0.150267i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.212565	−	0.977147i	−0.212565	−	0.977147i
$89$	−0.212565	−	0.977147i	−0.212565	−	0.977147i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.34692	−	0.865611i	1.34692	−	0.865611i	0.349464	−	0.936950i	$-0.386364\pi$
$97$	1.34692	−	0.865611i	1.34692	−	0.865611i	0.997452	+	0.0713392i	$0.0227273\pi$
$98$	0.479249	−	0.877679i	0.479249	−	0.877679i
$99$		0			0
$100$	−0.368991	+	0.574161i	−0.368991	+	0.574161i
$101$	0.189280	−	0.456963i	0.189280	−	0.456963i	−0.800541	−	0.599278i	$-0.795455\pi$
$101$	0.189280	−	0.456963i	0.189280	−	0.456963i	0.989821	+	0.142315i	$0.0454545\pi$
$102$		0			0
$103$		0			0		−0.778642	−	0.627469i	$-0.784091\pi$
$103$		0			0		0.778642	+	0.627469i	$0.215909\pi$
$104$	−0.890105	+	1.77820i	−0.890105	+	1.77820i
$105$		0			0
$106$	−0.142315	−	1.98982i	−0.142315	−	1.98982i
$107$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$107$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$108$		0			0
$109$	1.85483	+	0.691814i	1.85483	+	0.691814i	0.977147	+	0.212565i	$0.0681818\pi$
$109$	1.85483	+	0.691814i	1.85483	+	0.691814i	0.877679	+	0.479249i	$0.159091\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.599278	+	0.199459i	−0.599278	+	0.199459i	−0.599278	−	0.800541i	$-0.704545\pi$
$113$	−0.599278	+	0.199459i	−0.599278	+	0.199459i			1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	1.07134	+	0.997452i	1.07134	+	0.997452i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.415415	+	0.909632i	0.415415	+	0.909632i
$122$	−1.44053	+	1.34118i	−1.44053	+	1.34118i
$123$		0			0
$124$		0			0
$125$	0.758940	+	0.568136i	0.758940	+	0.568136i
$126$		0			0
$127$		0			0		0.177553	−	0.984111i	$-0.443182\pi$
$127$		0			0		−0.177553	+	0.984111i	$0.556818\pi$
$128$	0.281733	+	0.959493i	0.281733	+	0.959493i
$129$		0			0
$130$	0.963623	+	0.571744i	0.963623	+	0.571744i
$131$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$131$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−1.12299	−	1.50013i	−1.12299	−	1.50013i
$137$	1.71893	+	1.01989i	1.71893	+	1.01989i	0.877679	+	0.479249i	$0.159091\pi$
$137$	1.71893	+	1.01989i	1.71893	+	1.01989i	0.841254	+	0.540641i	$0.181818\pi$
$138$		0			0
$139$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$139$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	0.603662	−	0.562029i	0.603662	−	0.562029i
$146$	−0.118239	−	0.258908i	−0.118239	−	0.258908i
$147$		0			0
$148$	−1.83871	+	0.469302i	−1.83871	+	0.469302i
$149$	1.51779	+	1.05382i	1.51779	+	1.05382i	0.977147	+	0.212565i	$0.0681818\pi$
$149$	1.51779	+	1.05382i	1.51779	+	1.05382i	0.540641	+	0.841254i	$0.318182\pi$
$150$		0			0
$151$		0			0		−0.731895	−	0.681418i	$-0.761364\pi$
$151$		0			0		0.731895	+	0.681418i	$0.238636\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.30638	+	0.0934345i	1.30638	+	0.0934345i	0.707107	−	0.707107i	$-0.250000\pi$
$157$	1.30638	+	0.0934345i	1.30638	+	0.0934345i	0.599278	+	0.800541i	$0.295455\pi$
$158$		0			0
$159$		0			0
$160$	0.550588	−	0.119773i	0.550588	−	0.119773i
$161$		0			0
$162$		0			0
$163$		0			0		0.447617	−	0.894225i	$-0.352273\pi$
$163$		0			0		−0.447617	+	0.894225i	$0.647727\pi$
$164$	0.794635	+	0.640357i	0.794635	+	0.640357i
$165$		0			0
$166$		0			0
$167$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$167$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$168$		0			0
$169$	−1.41584	+	2.59292i	−1.41584	+	2.59292i
$170$	−0.888260	+	0.570850i	−0.888260	+	0.570850i
$171$		0			0
$172$		0			0
$173$	−0.229843	+	1.05657i	−0.229843	+	1.05657i	0.707107	+	0.707107i	$0.250000\pi$
$173$	−0.229843	+	1.05657i	−0.229843	+	1.05657i	−0.936950	+	0.349464i	$0.886364\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.800541	−	0.599278i	−0.800541	−	0.599278i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	0.406285	−	1.59181i	0.406285	−	1.59181i	−0.349464	−	0.936950i	$-0.613636\pi$
$181$	0.406285	−	1.59181i	0.406285	−	1.59181i	0.755750	−	0.654861i	$-0.227273\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.189851	+	1.05227i	0.189851	+	1.05227i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.860014	−	0.510270i	$-0.170455\pi$
$191$		0			0		−0.860014	+	0.510270i	$0.829545\pi$
$192$		0			0
$193$	1.01999	+	0.510572i	1.01999	+	0.510572i	0.877679	−	0.479249i	$-0.159091\pi$
$193$	1.01999	+	0.510572i	1.01999	+	0.510572i	0.142315	+	0.989821i	$0.454545\pi$
$194$	0.451077	−	1.53623i	0.451077	−	1.53623i
$195$		0			0
$196$	−0.212565	−	0.977147i	−0.212565	−	0.977147i
$197$	−1.27918	+	1.03083i	−1.27918	+	1.03083i	−0.281733	+	0.959493i	$0.590909\pi$
$197$	−1.27918	+	1.03083i	−1.27918	+	1.03083i	−0.997452	+	0.0713392i	$0.977273\pi$
$198$		0			0
$199$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$199$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$200$	0.0971309	+	0.675560i	0.0971309	+	0.675560i
$201$		0			0
$202$	−0.156199	−	0.469302i	−0.156199	−	0.469302i
$203$		0			0
$204$		0			0
$205$	0.391842	−	0.420868i	0.391842	−	0.420868i
$206$		0			0
$207$		0			0
$208$	0.491779	+	1.92677i	0.491779	+	1.92677i
$209$		0			0
$210$		0			0
$211$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$211$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$212$	−1.41061	−	1.41061i	−1.41061	−	1.41061i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.85483	−	0.691814i	1.85483	−	0.691814i
$219$		0			0
$220$		0			0
$221$	−2.12521	−	3.06088i	−2.12521	−	3.06088i
$222$		0			0
$223$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$223$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$224$		0			0
$225$		0			0
$226$	−0.322286	+	0.543184i	−0.322286	+	0.543184i
$227$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$227$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$228$		0			0
$229$	1.76003	+	0.317545i	1.76003	+	0.317545i	0.959493	−	0.281733i	$-0.0909091\pi$
$229$	1.76003	+	0.317545i	1.76003	+	0.317545i	0.800541	+	0.599278i	$0.204545\pi$
$230$		0			0
$231$		0			0
$232$	1.46286	+	0.0522460i	1.46286	+	0.0522460i
$233$	−0.587486	+	0.587486i	−0.587486	+	0.587486i	−0.936950	−	0.349464i	$-0.886364\pi$
$233$	−0.587486	+	0.587486i	−0.587486	+	0.587486i	0.349464	+	0.936950i	$0.386364\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.0356923	−	0.999363i	$-0.511364\pi$
$239$		0			0		0.0356923	+	0.999363i	$0.488636\pi$
$240$		0			0
$241$	−1.75320	−	0.583522i	−1.75320	−	0.583522i	−0.755750	−	0.654861i	$-0.772727\pi$
$241$	−1.75320	−	0.583522i	−1.75320	−	0.583522i	−0.997452	+	0.0713392i	$0.977273\pi$
$242$	0.909632	+	0.415415i	0.909632	+	0.415415i
$243$		0			0
$244$	−0.210393	+	1.95695i	−0.210393	+	1.95695i
$245$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.945619	−	0.0676320i	0.945619	−	0.0676320i
$251$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$251$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.841254	+	0.540641i	0.841254	+	0.540641i
$257$	1.14952	−	0.627683i	1.14952	−	0.627683i	0.212565	−	0.977147i	$-0.431818\pi$
$257$	1.14952	−	0.627683i	1.14952	−	0.627683i	0.936950	+	0.349464i	$0.113636\pi$
$258$		0			0
$259$		0			0
$260$	1.10267	−	0.198944i	1.10267	−	0.198944i
$261$		0			0
$262$		0			0
$263$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$263$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$264$		0			0
$265$	−0.849507	+	0.736102i	−0.849507	+	0.736102i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$269$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$270$		0			0
$271$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$271$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$272$	−1.83107	−	0.398326i	−1.83107	−	0.398326i
$273$		0			0
$274$	1.96697	−	0.354880i	1.96697	−	0.354880i
$275$		0			0
$276$		0			0
$277$	0.373128	−	0.203743i	0.373128	−	0.203743i	−0.281733	−	0.959493i	$-0.590909\pi$
$277$	0.373128	−	0.203743i	0.373128	−	0.203743i	0.654861	+	0.755750i	$0.272727\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.07926	−	1.33929i	1.07926	−	1.33929i	0.142315	−	0.989821i	$-0.454545\pi$
$281$	1.07926	−	1.33929i	1.07926	−	1.33929i	0.936950	−	0.349464i	$-0.113636\pi$
$282$		0			0
$283$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$283$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.48594	−	0.357424i	2.48594	−	0.357424i
$290$	0.0881665	−	0.820068i	0.0881665	−	0.820068i
$291$		0			0
$292$	−0.258908	−	0.118239i	−0.258908	−	0.118239i
$293$	1.29309	+	0.430383i	1.29309	+	0.430383i	0.877679	−	0.479249i	$-0.159091\pi$
$293$	1.29309	+	0.430383i	1.29309	+	0.430383i	0.415415	+	0.909632i	$0.363636\pi$
$294$		0			0
$295$		0			0
$296$	−1.08227	+	1.55877i	−1.08227	+	1.55877i
$297$		0			0
$298$	1.83717	−	0.197516i	1.83717	−	0.197516i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1.09140	+	0.196911i	1.09140	+	0.196911i
$306$		0			0
$307$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$307$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$311$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$312$		0			0
$313$	−0.0407123	−	0.0586368i	−0.0407123	−	0.0586368i	0.800541	−	0.599278i	$-0.204545\pi$
$313$	−0.0407123	−	0.0586368i	−0.0407123	−	0.0586368i	−0.841254	+	0.540641i	$0.818182\pi$
$314$	1.04849	−	0.784887i	1.04849	−	0.784887i
$315$		0			0
$316$		0			0
$317$	−0.407910	+	0.119773i	−0.407910	+	0.119773i	−0.479249	−	0.877679i	$-0.659091\pi$
$317$	−0.407910	+	0.119773i	−0.407910	+	0.119773i	0.0713392	+	0.997452i	$0.477273\pi$
$318$		0			0
$319$		0			0
$320$	0.337672	−	0.451077i	0.337672	−	0.451077i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	0.145077	+	1.34942i	0.145077	+	1.34942i
$326$		0			0
$327$		0			0
$328$	1.01989	−	0.0364255i	1.01989	−	0.0364255i
$329$		0			0
$330$		0			0
$331$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$331$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.977147	+	0.787435i	−0.977147	+	0.787435i	−0.977147	−	0.212565i	$-0.931818\pi$
$337$	−0.977147	+	0.787435i	−0.977147	+	0.787435i			1.00000i	$0.5\pi$
$338$	0.627980	+	2.88678i	0.627980	+	2.88678i
$339$		0			0
$340$	−0.297475	+	1.01311i	−0.297475	+	1.01311i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	0.518203	+	0.949018i	0.518203	+	0.949018i
$347$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$347$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$348$		0			0
$349$	−0.241976	−	1.34118i	−0.241976	−	1.34118i	−0.841254	−	0.540641i	$-0.818182\pi$
$349$	−0.241976	−	1.34118i	−0.241976	−	1.34118i	0.599278	−	0.800541i	$-0.295455\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.479249	−	1.87768i	0.479249	−	1.87768i		−	1.00000i	$-0.5\pi$
$353$	0.479249	−	1.87768i	0.479249	−	1.87768i	0.479249	−	0.877679i	$-0.340909\pi$
$354$		0			0
$355$		0			0
$356$	−0.997452	+	0.0713392i	−0.997452	+	0.0713392i
$357$		0			0
$358$		0			0
$359$		0			0		−0.968937	−	0.247307i	$-0.920455\pi$
$359$		0			0		0.968937	+	0.247307i	$0.0795455\pi$
$360$		0			0
$361$	−0.212565	+	0.977147i	−0.212565	+	0.977147i
$362$	−0.735364	−	1.46907i	−0.735364	−	1.46907i
$363$		0			0
$364$		0			0
$365$	−0.0768614	+	0.140761i	−0.0768614	+	0.140761i
$366$		0			0
$367$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$367$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$368$		0			0
$369$		0			0
$370$	0.832571	+	0.670928i	0.832571	+	0.670928i
$371$		0			0
$372$		0			0
$373$	0.0498610	+	0.697148i	0.0498610	+	0.697148i	0.959493	+	0.281733i	$0.0909091\pi$
$373$	0.0498610	+	0.697148i	0.0498610	+	0.697148i	−0.909632	+	0.415415i	$0.863636\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.90339	+	0.207654i	2.90339	+	0.207654i
$378$		0			0
$379$		0			0		−0.994270	−	0.106895i	$-0.965909\pi$
$379$		0			0		0.994270	+	0.106895i	$0.0340909\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.731895	−	0.681418i	$-0.761364\pi$
$383$		0			0		0.731895	+	0.681418i	$0.238636\pi$
$384$		0			0
$385$		0			0
$386$	1.10521	−	0.282089i	1.10521	−	0.282089i
$387$		0			0
$388$	−0.665114	−	1.45640i	−0.665114	−	1.45640i
$389$	−0.0522460	+	0.0486428i	−0.0522460	+	0.0486428i	−0.707107	−	0.707107i	$-0.750000\pi$
$389$	−0.0522460	+	0.0486428i	−0.0522460	+	0.0486428i	0.654861	+	0.755750i	$0.272727\pi$
$390$		0			0
$391$		0			0
$392$	−0.800541	−	0.599278i	−0.800541	−	0.599278i
$393$		0			0
$394$	−0.291692	+	1.61674i	−0.291692	+	1.61674i
$395$		0			0
$396$		0			0
$397$	−1.07926	−	0.640357i	−1.07926	−	0.640357i	−0.142315	−	0.989821i	$-0.545455\pi$
$397$	−1.07926	−	0.640357i	−1.07926	−	0.640357i	−0.936950	+	0.349464i	$0.886364\pi$
$398$		0			0
$399$		0			0
$400$	0.515804	+	0.446947i	0.515804	+	0.446947i
$401$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.841254	−	0.540641i	$-0.818182\pi$
$401$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$402$		0			0
$403$		0			0
$404$	−0.425374	−	0.252386i	−0.425374	−	0.252386i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.865611	−	0.647988i	−0.865611	−	0.647988i	0.0713392	−	0.997452i	$-0.477273\pi$
$409$	−0.865611	−	0.647988i	−0.865611	−	0.647988i	−0.936950	+	0.349464i	$0.886364\pi$
$410$	0.0205245	−	0.574672i	0.0205245	−	0.574672i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.63343	+	1.13411i	1.63343	+	1.13411i
$417$		0			0
$418$		0			0
$419$		0			0		0.315800	−	0.948826i	$-0.397727\pi$
$419$		0			0		−0.315800	+	0.948826i	$0.602273\pi$
$420$		0			0
$421$	1.47759	−	0.491789i	1.47759	−	0.491789i	0.540641	−	0.841254i	$-0.318182\pi$
$421$	1.47759	−	0.491789i	1.47759	−	0.491789i	0.936950	+	0.349464i	$0.113636\pi$
$422$		0			0
$423$		0			0
$424$	−1.98982	−	0.142315i	−1.98982	−	0.142315i
$425$	−1.19831	−	0.446947i	−1.19831	−	0.446947i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.778642	−	0.627469i	$-0.784091\pi$
$431$		0			0		0.778642	+	0.627469i	$0.215909\pi$
$432$		0			0
$433$	−0.241703	+	0.583522i	−0.241703	+	0.583522i	−0.997452	−	0.0713392i	$-0.977273\pi$
$433$	−0.241703	+	0.583522i	−0.241703	+	0.583522i	0.755750	+	0.654861i	$0.227273\pi$
$434$		0			0
$435$		0			0
$436$	0.948742	−	1.73749i	0.948742	−	1.73749i
$437$		0			0
$438$		0			0
$439$		0			0		−0.447617	−	0.894225i	$-0.647727\pi$
$439$		0			0		0.447617	+	0.894225i	$0.352273\pi$
$440$		0			0
$441$		0			0
$442$	−3.61058	−	0.921544i	−3.61058	−	0.921544i
$443$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$443$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$444$		0			0
$445$			0.563465i			0.563465i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.141226	−	0.0203052i	−0.141226	−	0.0203052i	0.0713392	−	0.997452i	$-0.477273\pi$
$449$	−0.141226	−	0.0203052i	−0.141226	−	0.0203052i	−0.212565	+	0.977147i	$0.568182\pi$
$450$		0			0
$451$		0			0
$452$	0.112142	+	0.621564i	0.112142	+	0.621564i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.456963	+	0.189280i	0.456963	+	0.189280i	0.599278	−	0.800541i	$-0.295455\pi$
$457$	0.456963	+	0.189280i	0.456963	+	0.189280i	−0.142315	+	0.989821i	$0.545455\pi$
$458$	1.53809	−	0.912593i	1.53809	−	0.912593i
$459$		0			0
$460$		0			0
$461$	0.304632	−	1.03748i	0.304632	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$461$	0.304632	−	1.03748i	0.304632	−	1.03748i	0.959493	−	0.281733i	$-0.0909091\pi$
$462$		0			0
$463$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$463$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$464$	1.13977	−	0.918482i	1.13977	−	0.918482i
$465$		0			0
$466$	−0.0592707	+	0.828713i	−0.0592707	+	0.828713i
$467$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$467$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$468$		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
$481$	−2.26141	+	3.02089i	−2.26141	+	3.02089i
$482$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$483$		0			0
$484$	0.959493	−	0.281733i	0.959493	−	0.281733i
$485$	−0.845273	+	0.315271i	−0.845273	+	0.315271i
$486$		0			0
$487$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$487$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$488$	1.12252	+	1.61674i	1.12252	+	1.61674i
$489$		0			0
$490$	−0.368991	+	0.425839i	−0.368991	+	0.425839i
$491$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$491$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$492$		0			0
$493$	−1.39967	+	2.35901i	−1.39967	+	2.35901i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$499$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$500$	0.670361	−	0.670361i	0.670361	−	0.670361i
$501$		0			0
$502$		0			0
$503$		0			0		0.994270	−	0.106895i	$-0.0340909\pi$
$503$		0			0		−0.994270	+	0.106895i	$0.965909\pi$
$504$		0			0
$505$	−0.158947	+	0.228928i	−0.158947	+	0.228928i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.258908	−	0.118239i	−0.258908	−	0.118239i	0.281733	−	0.959493i	$-0.409091\pi$
$509$	−0.258908	−	0.118239i	−0.258908	−	0.118239i	−0.540641	+	0.841254i	$0.681818\pi$
$510$		0			0
$511$		0			0
$512$	0.989821	−	0.142315i	0.989821	−	0.142315i
$513$		0			0
$514$	0.457701	−	1.22714i	0.457701	−	1.22714i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.703062	−	0.872447i	0.703062	−	0.872447i
$521$	−0.109091	−	0.183863i	−0.109091	−	0.183863i	0.800541	−	0.599278i	$-0.204545\pi$
$521$	−0.109091	−	0.183863i	−0.109091	−	0.183863i	−0.909632	+	0.415415i	$0.863636\pi$
$522$		0			0
$523$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$523$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.977147	−	0.212565i	−0.977147	−	0.212565i
$530$	−0.159970	+	1.11262i	−0.159970	+	1.11262i
$531$		0			0
$532$		0			0
$533$	2.02939			2.02939
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.258908	+	1.80075i	−0.258908	+	1.80075i
$539$		0			0
$540$		0			0
$541$	1.12252	−	0.202525i	1.12252	−	0.202525i	0.415415	−	0.909632i	$-0.363636\pi$
$541$	1.12252	−	0.202525i	1.12252	−	0.202525i	0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$		0			0
$544$	−1.64468	+	0.898064i	−1.64468	+	0.898064i
$545$	−0.938384	−	0.603063i	−0.938384	−	0.603063i
$546$		0			0
$547$		0			0		−0.510270	−	0.860014i	$-0.670455\pi$
$547$		0			0		0.510270	+	0.860014i	$0.329545\pi$
$548$	1.25414	−	1.55629i	1.25414	−	1.55629i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.148568	−	0.398326i	0.148568	−	0.398326i
$555$		0			0
$556$		0			0
$557$	0.0228531	−	0.212565i	0.0228531	−	0.212565i	−0.977147	−	0.212565i	$-0.931818\pi$
$557$	0.0228531	−	0.212565i	0.0228531	−	0.212565i	1.00000			$0$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.0613918	−	1.71893i	−0.0613918	−	1.71893i
$563$		0			0		0.570323	−	0.821421i	$-0.306818\pi$
$563$		0			0		−0.570323	+	0.821421i	$0.693182\pi$
$564$		0			0
$565$	0.353845	−	0.0380423i	0.353845	−	0.0380423i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.64179	+	0.0586368i	1.64179	+	0.0586368i	0.841254	−	0.540641i	$-0.181818\pi$
$569$	1.64179	+	0.0586368i	1.64179	+	0.0586368i	0.800541	+	0.599278i	$0.204545\pi$
$570$		0			0
$571$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$571$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	0.628688	−	0.436506i	0.628688	−	0.436506i	−0.212565	−	0.977147i	$-0.568182\pi$
$577$	0.628688	−	0.436506i	0.628688	−	0.436506i	0.841254	+	0.540641i	$0.181818\pi$
$578$	1.64468	−	1.89806i	1.64468	−	1.89806i
$579$		0			0
$580$	−0.470399	−	0.677503i	−0.470399	−	0.677503i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i
$585$		0			0
$586$	1.25910	−	0.521535i	1.25910	−	0.521535i
$587$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$587$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.202850	+	1.88678i	0.202850	+	1.88678i
$593$	0.282089	+	1.10521i	0.282089	+	1.10521i	0.936950	+	0.349464i	$0.113636\pi$
$593$	0.282089	+	1.10521i	0.282089	+	1.10521i	−0.654861	+	0.755750i	$0.727273\pi$
$594$		0			0
$595$		0			0
$596$	1.25910	−	1.35236i	1.25910	−	1.35236i
$597$		0			0
$598$		0			0
$599$		0			0		−0.315800	−	0.948826i	$-0.602273\pi$
$599$		0			0		0.315800	+	0.948826i	$0.397727\pi$
$600$		0			0
$601$	−0.215109	−	1.49611i	−0.215109	−	1.49611i	−0.755750	−	0.654861i	$-0.772727\pi$
$601$	−0.215109	−	1.49611i	−0.215109	−	1.49611i	0.540641	−	0.841254i	$-0.318182\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.119773	−	0.550588i	−0.119773	−	0.550588i
$606$		0			0
$607$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$607$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$608$		0			0
$609$		0			0
$610$	0.953777	−	0.565902i	0.953777	−	0.565902i
$611$		0			0
$612$		0			0
$613$	0.398174	+	0.729202i	0.398174	+	0.729202i	0.997452	−	0.0713392i	$-0.0227273\pi$
$613$	0.398174	+	0.729202i	0.398174	+	0.729202i	−0.599278	+	0.800541i	$0.704545\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.12220	+	0.561732i	−1.12220	+	0.561732i	−0.909632	−	0.415415i	$-0.863636\pi$
$617$	−1.12220	+	0.561732i	−1.12220	+	0.561732i	−0.212565	+	0.977147i	$0.568182\pi$
$618$		0			0
$619$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$619$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.0971309	+	0.112095i	0.0971309	+	0.112095i
$626$	−0.0691673	−	0.0176539i	−0.0691673	−	0.0176539i
$627$		0			0
$628$	0.278401	−	1.27979i	0.278401	−	1.27979i
$629$	−1.59173	−	3.17987i	−1.59173	−	3.17987i
$630$		0			0
$631$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$631$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$632$		0			0
$633$		0			0
$634$	−0.229843	+	0.357643i	−0.229843	+	0.357643i
$635$		0			0
$636$		0			0
$637$	−1.54836	−	1.24775i	−1.54836	−	1.24775i
$638$		0			0
$639$		0			0
$640$	−0.0401971	−	0.562029i	−0.0401971	−	0.562029i
$641$	−1.47696	+	0.321292i	−1.47696	+	0.321292i	−0.877679	−	0.479249i	$-0.840909\pi$
$641$	−1.47696	+	0.321292i	−1.47696	+	0.321292i	−0.599278	+	0.800541i	$0.704545\pi$
$642$		0			0
$643$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$643$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.948826	−	0.315800i	$-0.102273\pi$
$647$		0			0		−0.948826	+	0.315800i	$0.897727\pi$
$648$		0			0
$649$		0			0
$650$	0.993322	+	0.924816i	0.993322	+	0.924816i
$651$		0			0
$652$		0			0
$653$	−0.611980	+	0.156199i	−0.611980	+	0.156199i	−0.540641	−	0.841254i	$-0.681818\pi$
$653$	−0.611980	+	0.156199i	−0.611980	+	0.156199i	−0.0713392	+	0.997452i	$0.522727\pi$
$654$		0			0
$655$		0			0
$656$	0.746928	−	0.695414i	0.746928	−	0.695414i
$657$		0			0
$658$		0			0
$659$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$659$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$660$		0			0
$661$	−0.0630503	+	0.349464i	−0.0630503	+	0.349464i	0.936950	+	0.349464i	$0.113636\pi$
$661$	−0.0630503	+	0.349464i	−0.0630503	+	0.349464i	−1.00000			$1.00000\pi$
$662$		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
$673$	0.562029	+	1.91410i	0.562029	+	1.91410i	0.349464	+	0.936950i	$0.386364\pi$
$673$	0.562029	+	1.91410i	0.562029	+	1.91410i	0.212565	+	0.977147i	$0.431818\pi$
$674$	−0.222818	+	1.23500i	−0.222818	+	1.23500i
$675$		0			0
$676$	2.36503	+	1.77044i	2.36503	+	1.77044i
$677$	−0.0659508	+	1.84658i	−0.0659508	+	1.84658i	0.349464	+	0.936950i	$0.386364\pi$
$677$	−0.0659508	+	1.84658i	−0.0659508	+	1.84658i	−0.415415	+	0.909632i	$0.636364\pi$
$678$		0			0
$679$		0			0
$680$	0.438627	+	0.960459i	0.438627	+	0.960459i
$681$		0			0
$682$		0			0
$683$		0			0		−0.821421	−	0.570323i	$-0.806818\pi$
$683$		0			0		0.821421	+	0.570323i	$0.193182\pi$
$684$		0			0
$685$	−0.824269	−	0.767421i	−0.824269	−	0.767421i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−3.94422	−	0.424047i	−3.94422	−	0.424047i
$690$		0			0
$691$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$691$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$692$	1.01311	+	0.377869i	1.01311	+	0.377869i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.856018	+	1.71011i	−0.856018	+	1.71011i
$698$	−1.06116	−	0.855137i	−1.06116	−	0.855137i
$699$		0			0
$700$		0			0
$701$	−1.05657	+	1.64406i	−1.05657	+	1.64406i	−0.349464	+	0.936950i	$0.613636\pi$
$701$	−1.05657	+	1.64406i	−1.05657	+	1.64406i	−0.707107	+	0.707107i	$0.750000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.867426	−	1.73290i	−0.867426	−	1.73290i
$707$		0			0
$708$		0			0
$709$	−0.207149	−	0.0528717i	−0.207149	−	0.0528717i	0.142315	−	0.989821i	$-0.454545\pi$
$709$	−0.207149	−	0.0528717i	−0.207149	−	0.0528717i	−0.349464	+	0.936950i	$0.613636\pi$
$710$		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		−0.177553	−	0.984111i	$-0.556818\pi$
$719$		0			0		0.177553	+	0.984111i	$0.443182\pi$
$720$		0			0
$721$		0			0
$722$	0.479249	+	0.877679i	0.479249	+	0.877679i
$723$		0			0
$724$	−1.51779	−	0.628688i	−1.51779	−	0.628688i
$725$	0.859194	−	0.509783i	0.859194	−	0.509783i
$726$		0			0
$727$		0			0		−0.894225	−	0.447617i	$-0.852273\pi$
$727$		0			0		0.894225	+	0.447617i	$0.147727\pi$
$728$		0			0
$729$		0			0
$730$	0.0340910	+	0.156714i	0.0340910	+	0.156714i
$731$		0			0
$732$		0			0
$733$	0.0771377	−	1.07853i	0.0771377	−	1.07853i	−0.800541	−	0.599278i	$-0.795455\pi$
$733$	0.0771377	−	1.07853i	0.0771377	−	1.07853i	0.877679	−	0.479249i	$-0.159091\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.681418	−	0.731895i	$-0.261364\pi$
$739$		0			0		−0.681418	+	0.731895i	$0.738636\pi$
$740$	1.06858	−	0.0381644i	1.06858	−	0.0381644i
$741$		0			0
$742$		0			0
$743$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$743$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$744$		0			0
$745$	−0.709457	−	0.762010i	−0.709457	−	0.762010i
$746$	0.494217	+	0.494217i	0.494217	+	0.494217i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$751$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$752$		0			0
$753$		0			0
$754$	2.33022	−	1.74438i	2.33022	−	1.74438i
$755$		0			0
$756$		0			0
$757$	−1.27979	+	1.47696i	−1.27979	+	1.47696i	−0.479249	+	0.877679i	$0.659091\pi$
$757$	−1.27979	+	1.47696i	−1.27979	+	1.47696i	−0.800541	+	0.599278i	$0.795455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.0498610	+	0.133682i	0.0498610	+	0.133682i	0.959493	−	0.281733i	$-0.0909091\pi$
$761$	0.0498610	+	0.133682i	0.0498610	+	0.133682i	−0.909632	+	0.415415i	$0.863636\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.635768	+	0.290345i	−0.635768	+	0.290345i	−0.707107	−	0.707107i	$-0.750000\pi$
$769$	−0.635768	+	0.290345i	−0.635768	+	0.290345i	0.0713392	+	0.997452i	$0.477273\pi$
$770$		0			0
$771$		0			0
$772$	0.650536	−	0.936950i	0.650536	−	0.936950i
$773$	−0.0613918	−	1.71893i	−0.0613918	−	1.71893i	−0.540641	−	0.841254i	$-0.681818\pi$
$773$	−0.0613918	−	1.71893i	−0.0613918	−	1.71893i	0.479249	−	0.877679i	$-0.340909\pi$
$774$		0			0
$775$		0			0
$776$	−1.45640	−	0.665114i	−1.45640	−	0.665114i
$777$		0			0
$778$	−0.00763067	+	0.0709757i	−0.00763067	+	0.0709757i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.997452	+	0.0713392i	−0.997452	+	0.0713392i
$785$	−0.708089	−	0.207914i	−0.708089	−	0.207914i
$786$		0			0
$787$		0			0		0.627469	−	0.778642i	$-0.284091\pi$
$787$		0			0		−0.627469	+	0.778642i	$0.715909\pi$
$788$	0.838293	+	1.41287i	0.838293	+	1.41287i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.11601	+	3.29257i	2.11601	+	3.29257i
$794$	−1.23500	+	0.222818i	−1.23500	+	0.222818i
$795$		0			0
$796$		0			0
$797$	−0.136408	+	0.948742i	−0.136408	+	0.948742i	0.800541	+	0.599278i	$0.204545\pi$
$797$	−0.136408	+	0.948742i	−0.136408	+	0.948742i	−0.936950	+	0.349464i	$0.886364\pi$
$798$		0			0
$799$		0			0
$800$	0.682507			0.682507
$801$		0			0
$802$	−1.97964			−1.97964
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.486754	+	0.0878201i	−0.486754	+	0.0878201i
$809$	0.949018	+	1.47670i	0.949018	+	1.47670i	0.877679	+	0.479249i	$0.159091\pi$
$809$	0.949018	+	1.47670i	0.949018	+	1.47670i	0.0713392	+	0.997452i	$0.477273\pi$
$810$		0			0
$811$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$811$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−1.07853	+	0.0771377i	−1.07853	+	0.0771377i
$819$		0			0
$820$	−0.360819	−	0.447749i	−0.360819	−	0.447749i
$821$	0.613435	−	1.64468i	0.613435	−	1.64468i	−0.142315	−	0.989821i	$-0.545455\pi$
$821$	0.613435	−	1.64468i	0.613435	−	1.64468i	0.755750	−	0.654861i	$-0.227273\pi$
$822$		0			0
$823$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$823$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.948826	−	0.315800i	$-0.897727\pi$
$827$		0			0		0.948826	+	0.315800i	$0.102273\pi$
$828$		0			0
$829$	−0.0273177	−	0.764879i	−0.0273177	−	0.764879i	−0.936950	−	0.349464i	$-0.886364\pi$
$829$	−0.0273177	−	0.764879i	−0.0273177	−	0.764879i	0.909632	−	0.415415i	$-0.136364\pi$
$830$		0			0
$831$		0			0
$832$	1.97715	−	0.212565i	1.97715	−	0.212565i
$833$	1.70456	−	0.778446i	1.70456	−	0.778446i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.984111	−	0.177553i	$-0.943182\pi$
$839$		0			0		0.984111	+	0.177553i	$0.0568182\pi$
$840$		0			0
$841$	−0.399325	−	1.07063i	−0.399325	−	1.07063i
$842$	0.794635	−	1.33929i	0.794635	−	1.33929i
$843$		0			0
$844$		0			0
$845$	1.09011	−	1.25805i	1.09011	−	1.25805i
$846$		0			0
$847$		0			0
$848$	−1.59700	+	1.19550i	−1.59700	+	1.19550i
$849$		0			0
$850$	−1.19831	+	0.446947i	−1.19831	+	0.446947i
$851$		0			0
$852$		0			0
$853$	−0.707107	+	0.292893i	−0.707107	+	0.292893i	−0.707107	−	0.707107i	$-0.750000\pi$
$853$	−0.707107	+	0.292893i	−0.707107	+	0.292893i			1.00000i	$0.5\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.610029	−	0.655217i	−0.610029	−	0.655217i	0.349464	−	0.936950i	$-0.386364\pi$
$857$	−0.610029	−	0.655217i	−0.610029	−	0.655217i	−0.959493	+	0.281733i	$0.909091\pi$
$858$		0			0
$859$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$859$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.681418	−	0.731895i	$-0.261364\pi$
$863$		0			0		−0.681418	+	0.731895i	$0.738636\pi$
$864$		0			0
$865$	0.253098	−	0.554206i	0.253098	−	0.554206i
$866$	0.199459	+	0.599278i	0.199459	+	0.599278i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.420803	−	1.93440i	−0.420803	−	1.93440i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.877679	−	0.520751i	0.877679	−	0.520751i		−	1.00000i	$-0.5\pi$
$877$	0.877679	−	0.520751i	0.877679	−	0.520751i	0.877679	+	0.479249i	$0.159091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.613435	+	0.334961i	0.613435	+	0.334961i	0.755750	−	0.654861i	$-0.227273\pi$
$881$	0.613435	+	0.334961i	0.613435	+	0.334961i	−0.142315	+	0.989821i	$0.545455\pi$
$882$		0			0
$883$		0			0		−0.177553	−	0.984111i	$-0.556818\pi$
$883$		0			0		0.177553	+	0.984111i	$0.443182\pi$
$884$	−3.33217	+	1.66797i	−3.33217	+	1.66797i
$885$		0			0
$886$		0			0
$887$		0			0		0.247307	−	0.968937i	$-0.420455\pi$
$887$		0			0		−0.247307	+	0.968937i	$0.579545\pi$
$888$		0			0
$889$		0			0
$890$	0.368991	+	0.425839i	0.368991	+	0.425839i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.120029	+	0.0771377i	−0.120029	+	0.0771377i
$899$		0			0
$900$		0			0
$901$	2.02105	−	3.14482i	2.02105	−	3.14482i
$902$		0			0
$903$		0			0
$904$	0.491789	+	0.396309i	0.491789	+	0.396309i
$905$	−0.414352	+	0.827770i	−0.414352	+	0.827770i
$906$		0			0
$907$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$907$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$911$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$912$		0			0
$913$		0			0
$914$	0.469302	−	0.156199i	0.469302	−	0.156199i
$915$		0			0
$916$	0.564792	−	1.69693i	0.564792	−	1.69693i
$917$		0			0
$918$		0			0
$919$		0			0		−0.821421	−	0.570323i	$-0.806818\pi$
$919$		0			0		0.821421	+	0.570323i	$0.193182\pi$
$920$		0			0
$921$		0			0
$922$	−0.449181	−	0.983568i	−0.449181	−	0.983568i
$923$		0			0
$924$		0			0
$925$	−0.0462273	+	1.29434i	−0.0462273	+	1.29434i
$926$		0			0
$927$		0			0
$928$	0.259900	−	1.44053i	0.259900	−	1.44053i
$929$	−0.270040	−	0.919672i	−0.270040	−	0.919672i	−0.977147	−	0.212565i	$-0.931818\pi$
$929$	−0.270040	−	0.919672i	−0.270040	−	0.919672i	0.707107	−	0.707107i	$-0.250000\pi$
$930$		0			0
$931$		0			0
$932$	0.497898	+	0.665114i	0.497898	+	0.665114i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.959493	−	1.28173i	−0.959493	−	1.28173i	−0.959493	−	0.281733i	$-0.909091\pi$
$937$	−0.959493	−	1.28173i	−0.959493	−	1.28173i		−	1.00000i	$-0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.158952	−	0.881010i	0.158952	−	0.881010i	−0.800541	−	0.599278i	$-0.795455\pi$
$941$	0.158952	−	0.881010i	0.158952	−	0.881010i	0.959493	−	0.281733i	$-0.0909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$947$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$948$		0			0
$949$	−0.548416	+	0.139975i	−0.548416	+	0.139975i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.156199	+	0.469302i	−0.156199	+	0.469302i	−0.997452	−	0.0713392i	$-0.977273\pi$
$953$	−0.156199	+	0.469302i	−0.156199	+	0.469302i	0.841254	+	0.540641i	$0.181818\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.977147	−	0.212565i	0.977147	−	0.212565i
$962$	0.269203	+	3.76394i	0.269203	+	3.76394i
$963$		0			0
$964$	−0.827089	+	1.65231i	−0.827089	+	1.65231i
$965$	−0.500444	−	0.403283i	−0.500444	−	0.403283i
$966$		0			0
$967$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$967$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$968$	0.540641	−	0.841254i	0.540641	−	0.841254i
$969$		0			0
$970$	−0.432356	+	0.791802i	−0.432356	+	0.791802i
$971$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$971$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.90708	+	0.486754i	1.90708	+	0.486754i
$977$	−1.14952	−	1.32661i	−1.14952	−	1.32661i	−0.936950	−	0.349464i	$-0.886364\pi$
$977$	−1.14952	−	1.32661i	−1.14952	−	1.32661i	−0.212565	−	0.977147i	$-0.568182\pi$
$978$		0			0
$979$		0			0
$980$			0.563465i			0.563465i
$981$		0			0
$982$		0			0
$983$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$983$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$984$		0			0
$985$	0.827770	−	0.414352i	0.827770	−	0.414352i
$986$	0.487027	+	2.69941i	0.487027	+	2.69941i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$991$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−0.254771	−	1.17116i	−0.254771	−	1.17116i	−0.909632	−	0.415415i	$-0.863636\pi$
$997$	−0.254771	−	1.17116i	−0.254771	−	1.17116i	0.654861	−	0.755750i	$-0.272727\pi$
$998$		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.1907.1	yes	40
3.2	odd	2		3204.1.cd.a.1907.1	✓	40
4.3	odd	2	CM	3204.1.cd.b.1907.1	yes	40
12.11	even	2		3204.1.cd.a.1907.1	✓	40
89.82	odd	88		3204.1.cd.a.2663.1	yes	40
267.260	even	88	inner	3204.1.cd.b.2663.1	yes	40
356.171	even	88		3204.1.cd.a.2663.1	yes	40
1068.527	odd	88	inner	3204.1.cd.b.2663.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3204.1.cd.a.1907.1	✓	40	3.2	odd	2
3204.1.cd.a.1907.1	✓	40	12.11	even	2
3204.1.cd.a.2663.1	yes	40	89.82	odd	88
3204.1.cd.a.2663.1	yes	40	356.171	even	88
3204.1.cd.b.1907.1	yes	40	1.1	even	1	trivial
3204.1.cd.b.1907.1	yes	40	4.3	odd	2	CM
3204.1.cd.b.2663.1	yes	40	267.260	even	88	inner
3204.1.cd.b.2663.1	yes	40	1068.527	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)

\(f(q)\)	\(=\)	\(q+(0.755750 - 0.654861i) q^{2} +(0.142315 - 0.989821i) q^{4} +(-0.550588 - 0.119773i) q^{5} +(-0.540641 - 0.841254i) q^{8} +O(q^{10})\)	\(q+(0.755750 - 0.654861i) q^{2} +(0.142315 - 0.989821i) q^{4} +(-0.550588 - 0.119773i) q^{5} +(-0.540641 - 0.841254i) q^{8} +(-0.494541 + 0.270040i) q^{10} +(-1.01469 - 1.71017i) q^{13} +(-0.959493 - 0.281733i) q^{16} +(1.86912 - 0.133682i) q^{17} +(-0.196911 + 0.527938i) q^{20} +(-0.620830 - 0.283524i) q^{25} +(-1.88678 - 0.627980i) q^{26} +(-0.834832 + 1.20239i) q^{29} +(-0.909632 + 0.415415i) q^{32} +(1.32505 - 1.32505i) q^{34} +(-0.726200 - 1.75320i) q^{37} +(0.196911 + 0.527938i) q^{40} +(-0.520751 + 0.877679i) q^{41} +(0.936950 - 0.349464i) q^{49} +(-0.654861 + 0.192284i) q^{50} +(-1.83717 + 0.760982i) q^{52} +(1.19550 - 1.59700i) q^{53} +(0.156472 + 1.45540i) q^{58} +(-1.96697 + 0.0702505i) q^{61} +(-0.415415 + 0.909632i) q^{64} +(0.353845 + 1.06313i) q^{65} +(0.133682 - 1.86912i) q^{68} +(0.0801894 - 0.273100i) q^{73} +(-1.69693 - 0.849422i) q^{74} +(0.494541 + 0.270040i) q^{80} +(0.181200 + 1.00432i) q^{82} +(-1.04513 - 0.150267i) q^{85} +(-0.212565 - 0.977147i) q^{89} +(1.34692 - 0.865611i) q^{97} +(0.479249 - 0.877679i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{4}+O(q^{10}) \) 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