LMFDB - Embedded newform 3204.1.cd.b.431.1

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			431.1
Root			$-0.977147 + 0.212565i$ of defining polynomial
Character	$\chi$	$=$	3204.431
Dual form			3204.1.cd.b.1799.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.909632 - 0.415415i) q^{2} +(0.654861 + 0.755750i) q^{4} +(0.948742 - 1.73749i) q^{5} +(-0.281733 - 0.959493i) q^{8} +O(q^{10})$	\(q+(-0.909632 - 0.415415i) q^{2} +(0.654861 + 0.755750i) q^{4} +(0.948742 - 1.73749i) q^{5} +(-0.281733 - 0.959493i) q^{8} +(-1.58479 + 1.18636i) q^{10} +(-0.912593 + 0.456811i) q^{13} +(-0.142315 + 0.989821i) q^{16} +(-0.398326 - 0.148568i) q^{17} +(1.93440 - 0.420803i) q^{20} +(-1.57812 - 2.45561i) q^{25} +(1.01989 - 0.0364255i) q^{26} +(1.13411 + 0.121929i) q^{29} +(0.540641 - 0.841254i) q^{32} +(0.300613 + 0.300613i) q^{34} +(0.764879 - 1.84658i) q^{37} +(-1.93440 - 0.420803i) q^{40} +(-1.59928 - 0.800541i) q^{41} +(0.212565 - 0.977147i) q^{49} +(0.415415 + 2.88927i) q^{50} +(-0.942856 - 0.390544i) q^{52} +(-0.133682 - 1.86912i) q^{53} +(-0.980971 - 0.582037i) q^{58} +(-0.276501 + 1.53254i) q^{61} +(-0.841254 + 0.540641i) q^{64} +(-0.0721094 + 2.01902i) q^{65} +(-0.148568 - 0.398326i) q^{68} +(1.29639 + 0.186393i) q^{73} +(-1.46286 + 1.36197i) q^{74} +(1.58479 + 1.18636i) q^{80} +(1.12220 + 1.39256i) q^{82} +(-0.636044 + 0.551135i) q^{85} +(0.877679 - 0.479249i) q^{89} +(-1.91410 + 0.562029i) q^{97} +(-0.599278 + 0.800541i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times$.

$n$	$181$	$713$	$1603$
$\chi(n)$	$e\left(\frac{53}{88}\right)$	$-1$	$-1$

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 $

$2$	−0.909632	−	0.415415i	−0.909632	−	0.415415i
$2$	−0.909632	−	0.415415i	−0.909632	−	0.415415i
$3$		0			0
$3$		0			0
$4$	0.654861	+	0.755750i	0.654861	+	0.755750i
$5$	0.948742	−	1.73749i	0.948742	−	1.73749i	0.349464	−	0.936950i	$-0.386364\pi$
$5$	0.948742	−	1.73749i	0.948742	−	1.73749i	0.599278	−	0.800541i	$-0.295455\pi$
$6$		0			0
$7$		0			0		0.778642	−	0.627469i	$-0.215909\pi$
$7$		0			0		−0.778642	+	0.627469i	$0.784091\pi$
$8$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$9$		0			0
$10$	−1.58479	+	1.18636i	−1.58479	+	1.18636i
$11$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$11$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$12$		0			0
$13$	−0.912593	+	0.456811i	−0.912593	+	0.456811i	−0.841254	−	0.540641i	$-0.818182\pi$
$13$	−0.912593	+	0.456811i	−0.912593	+	0.456811i	−0.0713392	+	0.997452i	$0.522727\pi$
$14$		0			0
$15$		0			0
$16$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$17$	−0.398326	−	0.148568i	−0.398326	−	0.148568i	0.142315	−	0.989821i	$-0.454545\pi$
$17$	−0.398326	−	0.148568i	−0.398326	−	0.148568i	−0.540641	+	0.841254i	$0.681818\pi$
$18$		0			0
$19$		0			0		−0.968937	−	0.247307i	$-0.920455\pi$
$19$		0			0		0.968937	+	0.247307i	$0.0795455\pi$
$20$	1.93440	−	0.420803i	1.93440	−	0.420803i
$21$		0			0
$22$		0			0
$23$		0			0		0.860014	−	0.510270i	$-0.170455\pi$
$23$		0			0		−0.860014	+	0.510270i	$0.829545\pi$
$24$		0			0
$25$	−1.57812	−	2.45561i	−1.57812	−	2.45561i
$26$	1.01989	−	0.0364255i	1.01989	−	0.0364255i
$27$		0			0
$28$		0			0
$29$	1.13411	+	0.121929i	1.13411	+	0.121929i	0.654861	−	0.755750i	$-0.272727\pi$
$29$	1.13411	+	0.121929i	1.13411	+	0.121929i	0.479249	+	0.877679i	$0.340909\pi$
$30$		0			0
$31$		0			0		0.510270	−	0.860014i	$-0.329545\pi$
$31$		0			0		−0.510270	+	0.860014i	$0.670455\pi$
$32$	0.540641	−	0.841254i	0.540641	−	0.841254i
$33$		0			0
$34$	0.300613	+	0.300613i	0.300613	+	0.300613i
$35$		0			0
$36$		0			0
$37$	0.764879	−	1.84658i	0.764879	−	1.84658i	0.349464	−	0.936950i	$-0.386364\pi$
$37$	0.764879	−	1.84658i	0.764879	−	1.84658i	0.415415	−	0.909632i	$-0.363636\pi$
$38$		0			0
$39$		0			0
$40$	−1.93440	−	0.420803i	−1.93440	−	0.420803i
$41$	−1.59928	−	0.800541i	−1.59928	−	0.800541i	−0.599278	−	0.800541i	$-0.704545\pi$
$41$	−1.59928	−	0.800541i	−1.59928	−	0.800541i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$43$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$47$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$48$		0			0
$49$	0.212565	−	0.977147i	0.212565	−	0.977147i
$50$	0.415415	+	2.88927i	0.415415	+	2.88927i
$51$		0			0
$52$	−0.942856	−	0.390544i	−0.942856	−	0.390544i
$53$	−0.133682	−	1.86912i	−0.133682	−	1.86912i	−0.415415	−	0.909632i	$-0.636364\pi$
$53$	−0.133682	−	1.86912i	−0.133682	−	1.86912i	0.281733	−	0.959493i	$-0.409091\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.980971	−	0.582037i	−0.980971	−	0.582037i
$59$		0			0		−0.315800	−	0.948826i	$-0.602273\pi$
$59$		0			0		0.315800	+	0.948826i	$0.397727\pi$
$60$		0			0
$61$	−0.276501	+	1.53254i	−0.276501	+	1.53254i	0.479249	+	0.877679i	$0.340909\pi$
$61$	−0.276501	+	1.53254i	−0.276501	+	1.53254i	−0.755750	+	0.654861i	$0.772727\pi$
$62$		0			0
$63$		0			0
$64$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$65$	−0.0721094	+	2.01902i	−0.0721094	+	2.01902i
$66$		0			0
$67$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$67$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$68$	−0.148568	−	0.398326i	−0.148568	−	0.398326i
$69$		0			0
$70$		0			0
$71$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$71$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$72$		0			0
$73$	1.29639	+	0.186393i	1.29639	+	0.186393i	0.755750	−	0.654861i	$-0.227273\pi$
$73$	1.29639	+	0.186393i	1.29639	+	0.186393i	0.540641	+	0.841254i	$0.318182\pi$
$74$	−1.46286	+	1.36197i	−1.46286	+	1.36197i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$79$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$80$	1.58479	+	1.18636i	1.58479	+	1.18636i
$81$		0			0
$82$	1.12220	+	1.39256i	1.12220	+	1.39256i
$83$		0			0		−0.731895	−	0.681418i	$-0.761364\pi$
$83$		0			0		0.731895	+	0.681418i	$0.238636\pi$
$84$		0			0
$85$	−0.636044	+	0.551135i	−0.636044	+	0.551135i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.877679	−	0.479249i	0.877679	−	0.479249i
$89$	0.877679	−	0.479249i	0.877679	−	0.479249i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.91410	+	0.562029i	−1.91410	+	0.562029i	−0.936950	+	0.349464i	$0.886364\pi$
$97$	−1.91410	+	0.562029i	−1.91410	+	0.562029i	−0.977147	+	0.212565i	$0.931818\pi$
$98$	−0.599278	+	0.800541i	−0.599278	+	0.800541i
$99$		0			0
$100$	0.822373	−	2.80075i	0.822373	−	2.80075i
$101$	−0.241703	−	0.583522i	−0.241703	−	0.583522i	0.755750	−	0.654861i	$-0.227273\pi$
$101$	−0.241703	−	0.583522i	−0.241703	−	0.583522i	−0.997452	+	0.0713392i	$0.977273\pi$
$102$		0			0
$103$		0			0		−0.247307	−	0.968937i	$-0.579545\pi$
$103$		0			0		0.247307	+	0.968937i	$0.420455\pi$
$104$	0.695414	+	0.746928i	0.695414	+	0.746928i
$105$		0			0
$106$	−0.654861	+	1.75575i	−0.654861	+	1.75575i
$107$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$107$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$108$		0			0
$109$	0.321292	+	1.47696i	0.321292	+	1.47696i	0.800541	+	0.599278i	$0.204545\pi$
$109$	0.321292	+	1.47696i	0.321292	+	1.47696i	−0.479249	+	0.877679i	$0.659091\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.0713392	−	0.00254789i	−0.0713392	−	0.00254789i		−	1.00000i	$-0.5\pi$
$113$	−0.0713392	−	0.00254789i	−0.0713392	−	0.00254789i	−0.0713392	+	0.997452i	$0.522727\pi$
$114$		0			0
$115$		0			0
$116$	0.650536	+	0.936950i	0.650536	+	0.936950i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.841254	+	0.540641i	0.841254	+	0.540641i
$122$	0.888154	−	1.27918i	0.888154	−	1.27918i
$123$		0			0
$124$		0			0
$125$	−3.78923	+	0.271011i	−3.78923	+	0.271011i
$126$		0			0
$127$		0			0		0.627469	−	0.778642i	$-0.284091\pi$
$127$		0			0		−0.627469	+	0.778642i	$0.715909\pi$
$128$	0.989821	−	0.142315i	0.989821	−	0.142315i
$129$		0			0
$130$	0.904323	−	1.80661i	0.904323	−	1.80661i
$131$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$131$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.0303285	+	0.424047i	−0.0303285	+	0.424047i
$137$	−0.158952	+	0.317545i	−0.158952	+	0.317545i	−0.959493	−	0.281733i	$-0.909091\pi$
$137$	−0.158952	+	0.317545i	−0.158952	+	0.317545i	0.800541	+	0.599278i	$0.204545\pi$
$138$		0			0
$139$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$139$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	1.28783	−	1.85483i	1.28783	−	1.85483i
$146$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$147$		0			0
$148$	1.89644	−	0.631197i	1.89644	−	0.631197i
$149$	−0.197516	+	1.83717i	−0.197516	+	1.83717i	0.281733	+	0.959493i	$0.409091\pi$
$149$	−0.197516	+	1.83717i	−0.197516	+	1.83717i	−0.479249	+	0.877679i	$0.659091\pi$
$150$		0			0
$151$		0			0		−0.570323	−	0.821421i	$-0.693182\pi$
$151$		0			0		0.570323	+	0.821421i	$0.306818\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.778446	−	0.290345i	0.778446	−	0.290345i	0.0713392	−	0.997452i	$-0.477273\pi$
$157$	0.778446	−	0.290345i	0.778446	−	0.290345i	0.707107	+	0.707107i	$0.250000\pi$
$158$		0			0
$159$		0			0
$160$	−0.948742	−	1.73749i	−0.948742	−	1.73749i
$161$		0			0
$162$		0			0
$163$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$163$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$164$	−0.442295	−	1.73290i	−0.442295	−	1.73290i
$165$		0			0
$166$		0			0
$167$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$167$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$168$		0			0
$169$	0.0248712	−	0.0332241i	0.0248712	−	0.0332241i
$170$	0.807516	−	0.237108i	0.807516	−	0.237108i
$171$		0			0
$172$		0			0
$173$	0.494541	+	0.270040i	0.494541	+	0.270040i	0.707107	−	0.707107i	$-0.250000\pi$
$173$	0.494541	+	0.270040i	0.494541	+	0.270040i	−0.212565	+	0.977147i	$0.568182\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.997452	+	0.0713392i	−0.997452	+	0.0713392i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	0.0675149	−	0.202850i	0.0675149	−	0.202850i	−0.909632	−	0.415415i	$-0.863636\pi$
$181$	0.0675149	−	0.202850i	0.0675149	−	0.202850i	0.977147	+	0.212565i	$0.0681818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.48275	−	3.08090i	−2.48275	−	3.08090i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.447617	−	0.894225i	$-0.647727\pi$
$191$		0			0		0.447617	+	0.894225i	$0.352273\pi$
$192$		0			0
$193$	1.45540	−	1.35503i	1.45540	−	1.35503i	0.654861	−	0.755750i	$-0.272727\pi$
$193$	1.45540	−	1.35503i	1.45540	−	1.35503i	0.800541	−	0.599278i	$-0.204545\pi$
$194$	1.97460	+	0.283904i	1.97460	+	0.283904i
$195$		0			0
$196$	0.877679	−	0.479249i	0.877679	−	0.479249i
$197$	−0.0528717	+	0.207149i	−0.0528717	+	0.207149i	−0.989821	−	0.142315i	$-0.954545\pi$
$197$	−0.0528717	+	0.207149i	−0.0528717	+	0.207149i	0.936950	+	0.349464i	$0.113636\pi$
$198$		0			0
$199$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$199$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$200$	−1.91153	+	2.20602i	−1.91153	+	2.20602i
$201$		0			0
$202$	−0.0225432	+	0.631197i	−0.0225432	+	0.631197i
$203$		0			0
$204$		0			0
$205$	−2.90823	+	2.01922i	−2.90823	+	2.01922i
$206$		0			0
$207$		0			0
$208$	−0.322286	−	0.968315i	−0.322286	−	0.968315i
$209$		0			0
$210$		0			0
$211$		0			0		−0.821421	−	0.570323i	$-0.806818\pi$
$211$		0			0		0.821421	+	0.570323i	$0.193182\pi$
$212$	1.32505	−	1.32505i	1.32505	−	1.32505i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.321292	−	1.47696i	0.321292	−	1.47696i
$219$		0			0
$220$		0			0
$221$	0.431377	−	0.0463778i	0.431377	−	0.0463778i
$222$		0			0
$223$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$223$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$224$		0			0
$225$		0			0
$226$	0.0638340	+	0.0319530i	0.0638340	+	0.0319530i
$227$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$227$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$228$		0			0
$229$	1.13977	+	0.918482i	1.13977	+	0.918482i	0.997452	−	0.0713392i	$-0.0227273\pi$
$229$	1.13977	+	0.918482i	1.13977	+	0.918482i	0.142315	+	0.989821i	$0.454545\pi$
$230$		0			0
$231$		0			0
$232$	−0.202525	−	1.12252i	−0.202525	−	1.12252i
$233$	−1.18971	−	1.18971i	−1.18971	−	1.18971i	−0.977147	−	0.212565i	$-0.931818\pi$
$233$	−1.18971	−	1.18971i	−1.18971	−	1.18971i	−0.212565	−	0.977147i	$-0.568182\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.984111	−	0.177553i	$-0.943182\pi$
$239$		0			0		0.984111	+	0.177553i	$0.0568182\pi$
$240$		0			0
$241$	1.84658	−	0.0659508i	1.84658	−	0.0659508i	0.909632	−	0.415415i	$-0.136364\pi$
$241$	1.84658	−	0.0659508i	1.84658	−	0.0659508i	0.936950	+	0.349464i	$0.113636\pi$
$242$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$243$		0			0
$244$	−1.33929	+	0.794635i	−1.33929	+	0.794635i
$245$	−1.49611	−	1.29639i	−1.49611	−	1.29639i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	3.55938	+	1.32758i	3.55938	+	1.32758i
$251$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$251$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$257$	−0.665114	+	0.497898i	−0.665114	+	0.497898i	−0.877679	−	0.479249i	$-0.840909\pi$
$257$	−0.665114	+	0.497898i	−0.665114	+	0.497898i	0.212565	+	0.977147i	$0.431818\pi$
$258$		0			0
$259$		0			0
$260$	−1.57309	+	1.26768i	−1.57309	+	1.26768i
$261$		0			0
$262$		0			0
$263$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$263$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$264$		0			0
$265$	−3.37442	−	1.54105i	−3.37442	−	1.54105i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.983568	−	0.449181i	−0.983568	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$269$	−0.983568	−	0.449181i	−0.983568	−	0.449181i	−0.841254	+	0.540641i	$0.818182\pi$
$270$		0			0
$271$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$271$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$272$	0.203743	−	0.373128i	0.203743	−	0.373128i
$273$		0			0
$274$	0.276501	−	0.222818i	0.276501	−	0.222818i
$275$		0			0
$276$		0			0
$277$	−1.40524	+	1.05195i	−1.40524	+	1.05195i	−0.415415	+	0.909632i	$0.636364\pi$
$277$	−1.40524	+	1.05195i	−1.40524	+	1.05195i	−0.989821	+	0.142315i	$0.954545\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.867426	−	0.221397i	0.867426	−	0.221397i	0.212565	−	0.977147i	$-0.431818\pi$
$281$	0.867426	−	0.221397i	0.867426	−	0.221397i	0.654861	+	0.755750i	$0.272727\pi$
$282$		0			0
$283$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$283$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.619158	−	0.536504i	−0.619158	−	0.536504i
$290$	−1.94197	+	1.15223i	−1.94197	+	1.15223i
$291$		0			0
$292$	0.708089	+	1.10181i	0.708089	+	1.10181i
$293$	1.64179	−	0.0586368i	1.64179	−	0.0586368i	0.800541	−	0.599278i	$-0.204545\pi$
$293$	1.64179	−	0.0586368i	1.64179	−	0.0586368i	0.841254	+	0.540641i	$0.181818\pi$
$294$		0			0
$295$		0			0
$296$	−1.98727	−	0.213654i	−1.98727	−	0.213654i
$297$		0			0
$298$	0.942856	−	1.58910i	0.942856	−	1.58910i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.40045	+	1.93440i	2.40045	+	1.93440i
$306$		0			0
$307$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$307$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$311$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$312$		0			0
$313$	1.95695	−	0.210393i	1.95695	−	0.210393i	0.959493	−	0.281733i	$-0.0909091\pi$
$313$	1.95695	−	0.210393i	1.95695	−	0.210393i	0.997452	+	0.0713392i	$0.0227273\pi$
$314$	−0.828713	−	0.0592707i	−0.828713	−	0.0592707i
$315$		0			0
$316$		0			0
$317$	0.249813	+	1.73749i	0.249813	+	1.73749i	0.599278	+	0.800541i	$0.295455\pi$
$317$	0.249813	+	1.73749i	0.249813	+	1.73749i	−0.349464	+	0.936950i	$0.613636\pi$
$318$		0			0
$319$		0			0
$320$	0.141226	+	1.97460i	0.141226	+	1.97460i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	2.56193	+	1.52006i	2.56193	+	1.52006i
$326$		0			0
$327$		0			0
$328$	−0.317545	+	1.76003i	−0.317545	+	1.76003i
$329$		0			0
$330$		0			0
$331$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$331$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.479249	−	1.87768i	0.479249	−	1.87768i		−	1.00000i	$-0.5\pi$
$337$	0.479249	−	1.87768i	0.479249	−	1.87768i	0.479249	−	0.877679i	$-0.340909\pi$
$338$	−0.0364255	+	0.0198898i	−0.0364255	+	0.0198898i
$339$		0			0
$340$	−0.833040	−	0.119773i	−0.833040	−	0.119773i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.337672	−	0.451077i	−0.337672	−	0.451077i
$347$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$347$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$348$		0			0
$349$	1.03083	+	1.27918i	1.03083	+	1.27918i	0.959493	+	0.281733i	$0.0909091\pi$
$349$	1.03083	+	1.27918i	1.03083	+	1.27918i	0.0713392	+	0.997452i	$0.477273\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.599278	+	1.80054i	−0.599278	+	1.80054i			1.00000i	$0.5\pi$
$353$	−0.599278	+	1.80054i	−0.599278	+	1.80054i	−0.599278	+	0.800541i	$0.704545\pi$
$354$		0			0
$355$		0			0
$356$	0.936950	+	0.349464i	0.936950	+	0.349464i
$357$		0			0
$358$		0			0
$359$		0			0		−0.948826	−	0.315800i	$-0.897727\pi$
$359$		0			0		0.948826	+	0.315800i	$0.102273\pi$
$360$		0			0
$361$	0.877679	+	0.479249i	0.877679	+	0.479249i
$362$	−0.145681	+	0.156472i	−0.145681	+	0.156472i
$363$		0			0
$364$		0			0
$365$	1.55380	−	2.07563i	1.55380	−	2.07563i
$366$		0			0
$367$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$367$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$368$		0			0
$369$		0			0
$370$	0.978533	+	3.83386i	0.978533	+	3.83386i
$371$		0			0
$372$		0			0
$373$	0.682956	−	1.83107i	0.682956	−	1.83107i	0.142315	−	0.989821i	$-0.454545\pi$
$373$	0.682956	−	1.83107i	0.682956	−	1.83107i	0.540641	−	0.841254i	$-0.318182\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.09068	+	0.406802i	−1.09068	+	0.406802i
$378$		0			0
$379$		0			0		−0.510270	−	0.860014i	$-0.670455\pi$
$379$		0			0		0.510270	+	0.860014i	$0.329545\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.570323	−	0.821421i	$-0.693182\pi$
$383$		0			0		0.570323	+	0.821421i	$0.306818\pi$
$384$		0			0
$385$		0			0
$386$	−1.88678	+	0.627980i	−1.88678	+	0.627980i
$387$		0			0
$388$	−1.67822	−	1.07853i	−1.67822	−	1.07853i
$389$	−1.12252	+	1.61674i	−1.12252	+	1.61674i	−0.415415	+	0.909632i	$0.636364\pi$
$389$	−1.12252	+	1.61674i	−1.12252	+	1.61674i	−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	−0.997452	+	0.0713392i	−0.997452	+	0.0713392i
$393$		0			0
$394$	0.134147	−	0.166466i	0.134147	−	0.166466i
$395$		0			0
$396$		0			0
$397$	−0.867426	+	1.73290i	−0.867426	+	1.73290i	−0.212565	+	0.977147i	$0.568182\pi$
$397$	−0.867426	+	1.73290i	−0.867426	+	1.73290i	−0.654861	+	0.755750i	$0.727273\pi$
$398$		0			0
$399$		0			0
$400$	2.65520	−	1.21259i	2.65520	−	1.21259i
$401$	1.37491	−	0.627899i	1.37491	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$401$	1.37491	−	0.627899i	1.37491	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$402$		0			0
$403$		0			0
$404$	0.282715	−	0.564792i	0.282715	−	0.564792i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.562029	+	0.0401971i	−0.562029	+	0.0401971i	−0.349464	−	0.936950i	$-0.613636\pi$
$409$	−0.562029	+	0.0401971i	−0.562029	+	0.0401971i	−0.212565	+	0.977147i	$0.568182\pi$
$410$	3.48424	−	0.628626i	3.48424	−	0.628626i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.109091	+	1.01469i	−0.109091	+	1.01469i
$417$		0			0
$418$		0			0
$419$		0			0		−0.0356923	−	0.999363i	$-0.511364\pi$
$419$		0			0		0.0356923	+	0.999363i	$0.488636\pi$
$420$		0			0
$421$	0.494298	+	0.0176539i	0.494298	+	0.0176539i	0.281733	−	0.959493i	$-0.409091\pi$
$421$	0.494298	+	0.0176539i	0.494298	+	0.0176539i	0.212565	+	0.977147i	$0.431818\pi$
$422$		0			0
$423$		0			0
$424$	−1.75575	+	0.654861i	−1.75575	+	0.654861i
$425$	0.263783	+	1.21259i	0.263783	+	1.21259i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.247307	−	0.968937i	$-0.579545\pi$
$431$		0			0		0.247307	+	0.968937i	$0.420455\pi$
$432$		0			0
$433$	0.0273177	+	0.0659508i	0.0273177	+	0.0659508i	0.936950	−	0.349464i	$-0.113636\pi$
$433$	0.0273177	+	0.0659508i	0.0273177	+	0.0659508i	−0.909632	+	0.415415i	$0.863636\pi$
$434$		0			0
$435$		0			0
$436$	−0.905808	+	1.21002i	−0.905808	+	1.21002i
$437$		0			0
$438$		0			0
$439$		0			0		0.681418	−	0.731895i	$-0.261364\pi$
$439$		0			0		−0.681418	+	0.731895i	$0.738636\pi$
$440$		0			0
$441$		0			0
$442$	−0.411660	−	0.137014i	−0.411660	−	0.137014i
$443$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$443$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$444$		0			0
$445$		−	1.97964i		−	1.97964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.528215	−	0.457701i	0.528215	−	0.457701i	−0.349464	−	0.936950i	$-0.613636\pi$
$449$	0.528215	−	0.457701i	0.528215	−	0.457701i	0.877679	+	0.479249i	$0.159091\pi$
$450$		0			0
$451$		0			0
$452$	−0.0447917	−	0.0555831i	−0.0447917	−	0.0555831i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.583522	+	0.241703i	−0.583522	+	0.241703i	−0.654861	−	0.755750i	$-0.727273\pi$
$457$	−0.583522	+	0.241703i	−0.583522	+	0.241703i	0.0713392	+	0.997452i	$0.477273\pi$
$458$	−0.655217	−	1.30896i	−0.655217	−	1.30896i
$459$		0			0
$460$		0			0
$461$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$461$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$462$		0			0
$463$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$463$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$464$	−0.282089	+	1.10521i	−0.282089	+	1.10521i
$465$		0			0
$466$	0.587976	+	1.57642i	0.587976	+	1.57642i
$467$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$467$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$468$		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	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$	0.145516	+	2.03458i	0.145516	+	2.03458i
$482$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$483$		0			0
$484$	0.142315	+	0.989821i	0.142315	+	0.989821i
$485$	−0.839462	+	3.85895i	−0.839462	+	3.85895i
$486$		0			0
$487$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$487$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$488$	1.54836	−	0.166466i	1.54836	−	0.166466i
$489$		0			0
$490$	0.822373	+	1.80075i	0.822373	+	1.80075i
$491$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$491$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$492$		0			0
$493$	−0.433631	−	0.217060i	−0.433631	−	0.217060i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.177553	−	0.984111i	$-0.556818\pi$
$499$		0			0		0.177553	+	0.984111i	$0.443182\pi$
$500$	−2.68623	−	2.68623i	−2.68623	−	2.68623i
$501$		0			0
$502$		0			0
$503$		0			0		0.510270	−	0.860014i	$-0.329545\pi$
$503$		0			0		−0.510270	+	0.860014i	$0.670455\pi$
$504$		0			0
$505$	−1.24318	−	0.133655i	−1.24318	−	0.133655i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.708089	+	1.10181i	0.708089	+	1.10181i	0.989821	+	0.142315i	$0.0454545\pi$
$509$	0.708089	+	1.10181i	0.708089	+	1.10181i	−0.281733	+	0.959493i	$0.590909\pi$
$510$		0			0
$511$		0			0
$512$	0.755750	+	0.654861i	0.755750	+	0.654861i
$513$		0			0
$514$	0.811843	−	0.176606i	0.811843	−	0.176606i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.95755	−	0.499635i	1.95755	−	0.499635i
$521$	1.53809	−	0.769914i	1.53809	−	0.769914i	0.540641	−	0.841254i	$-0.318182\pi$
$521$	1.53809	−	0.769914i	1.53809	−	0.769914i	0.997452	+	0.0713392i	$0.0227273\pi$
$522$		0			0
$523$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$523$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.479249	−	0.877679i	0.479249	−	0.877679i
$530$	2.42931	+	2.80357i	2.42931	+	2.80357i
$531$		0			0
$532$		0			0
$533$	1.82519			1.82519
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.708089	+	0.817178i	0.708089	+	0.817178i
$539$		0			0
$540$		0			0
$541$	1.54836	−	1.24775i	1.54836	−	1.24775i	0.707107	−	0.707107i	$-0.250000\pi$
$541$	1.54836	−	1.24775i	1.54836	−	1.24775i	0.841254	−	0.540641i	$-0.181818\pi$
$542$		0			0
$543$		0			0
$544$	−0.340335	+	0.254771i	−0.340335	+	0.254771i
$545$	2.87102	+	0.843008i	2.87102	+	0.843008i
$546$		0			0
$547$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$547$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$548$	−0.344076	+	0.0878201i	−0.344076	+	0.0878201i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.71524	−	0.373128i	1.71524	−	0.373128i
$555$		0			0
$556$		0			0
$557$	1.47925	−	0.877679i	1.47925	−	0.877679i	0.479249	−	0.877679i	$-0.340909\pi$
$557$	1.47925	−	0.877679i	1.47925	−	0.877679i	1.00000			$0$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.881010	−	0.158952i	−0.881010	−	0.158952i
$563$		0			0		−0.994270	−	0.106895i	$-0.965909\pi$
$563$		0			0		0.994270	+	0.106895i	$0.0340909\pi$
$564$		0			0
$565$	−0.0721094	+	0.121534i	−0.0721094	+	0.121534i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.0379591	+	0.210393i	0.0379591	+	0.210393i	0.997452	−	0.0713392i	$-0.0227273\pi$
$569$	0.0379591	+	0.210393i	0.0379591	+	0.210393i	−0.959493	+	0.281733i	$0.909091\pi$
$570$		0			0
$571$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$571$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.0818140	−	0.760982i	−0.0818140	−	0.760982i	−0.959493	−	0.281733i	$-0.909091\pi$
$577$	−0.0818140	−	0.760982i	−0.0818140	−	0.760982i	0.877679	−	0.479249i	$-0.159091\pi$
$578$	0.340335	+	0.745229i	0.340335	+	0.745229i
$579$		0			0
$580$	2.24513	−	0.241377i	2.24513	−	0.241377i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.186393	−	1.29639i	−0.186393	−	1.29639i
$585$		0			0
$586$	−1.51779	−	0.628688i	−1.51779	−	0.628688i
$587$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$587$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.71893	+	1.01989i	1.71893	+	1.01989i
$593$	0.627980	+	1.88678i	0.627980	+	1.88678i	0.415415	+	0.909632i	$0.363636\pi$
$593$	0.627980	+	1.88678i	0.627980	+	1.88678i	0.212565	+	0.977147i	$0.431818\pi$
$594$		0			0
$595$		0			0
$596$	−1.51779	+	1.05382i	−1.51779	+	1.05382i
$597$		0			0
$598$		0			0
$599$		0			0		0.0356923	−	0.999363i	$-0.488636\pi$
$599$		0			0		−0.0356923	+	0.999363i	$0.511364\pi$
$600$		0			0
$601$	1.19136	−	1.37491i	1.19136	−	1.37491i	0.281733	−	0.959493i	$-0.409091\pi$
$601$	1.19136	−	1.37491i	1.19136	−	1.37491i	0.909632	−	0.415415i	$-0.136364\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.73749	−	0.948742i	1.73749	−	0.948742i
$606$		0			0
$607$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$607$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$608$		0			0
$609$		0			0
$610$	−1.37994	−	2.75678i	−1.37994	−	2.75678i
$611$		0			0
$612$		0			0
$613$	−1.00829	−	1.34692i	−1.00829	−	1.34692i	−0.936950	−	0.349464i	$-0.886364\pi$
$613$	−1.00829	−	1.34692i	−1.00829	−	1.34692i	−0.0713392	−	0.997452i	$-0.522727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.41832	+	1.32050i	1.41832	+	1.32050i	0.877679	+	0.479249i	$0.159091\pi$
$617$	1.41832	+	1.32050i	1.41832	+	1.32050i	0.540641	+	0.841254i	$0.318182\pi$
$618$		0			0
$619$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$619$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.91153	+	4.18567i	−1.91153	+	4.18567i
$626$	−1.86750	−	0.621564i	−1.86750	−	0.621564i
$627$		0			0
$628$	0.729202	+	0.398174i	0.729202	+	0.398174i
$629$	−0.579014	+	0.621905i	−0.579014	+	0.621905i
$630$		0			0
$631$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$631$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$632$		0			0
$633$		0			0
$634$	0.494541	−	1.68425i	0.494541	−	1.68425i
$635$		0			0
$636$		0			0
$637$	0.252386	+	0.988839i	0.252386	+	0.988839i
$638$		0			0
$639$		0			0
$640$	0.691814	−	1.85483i	0.691814	−	1.85483i
$641$	−0.871880	−	1.59673i	−0.871880	−	1.59673i	−0.800541	−	0.599278i	$-0.795455\pi$
$641$	−0.871880	−	1.59673i	−0.871880	−	1.59673i	−0.0713392	−	0.997452i	$-0.522727\pi$
$642$		0			0
$643$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$643$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$647$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$648$		0			0
$649$		0			0
$650$	−1.69896	−	2.44696i	−1.69896	−	2.44696i
$651$		0			0
$652$		0			0
$653$	0.0677316	−	0.0225432i	0.0677316	−	0.0225432i	−0.281733	−	0.959493i	$-0.590909\pi$
$653$	0.0677316	−	0.0225432i	0.0677316	−	0.0225432i	0.349464	+	0.936950i	$0.386364\pi$
$654$		0			0
$655$		0			0
$656$	1.01999	−	1.46907i	1.01999	−	1.46907i
$657$		0			0
$658$		0			0
$659$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$659$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$660$		0			0
$661$	−0.787435	+	0.977147i	−0.787435	+	0.977147i	0.212565	+	0.977147i	$0.431818\pi$
$661$	−0.787435	+	0.977147i	−0.787435	+	0.977147i	−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
$673$	−1.85483	+	0.266684i	−1.85483	+	0.266684i	−0.977147	−	0.212565i	$-0.931818\pi$
$673$	−1.85483	+	0.266684i	−1.85483	+	0.266684i	−0.877679	+	0.479249i	$0.840909\pi$
$674$	−1.21596	+	1.50891i	−1.21596	+	1.50891i
$675$		0			0
$676$	0.0413963	−	0.00296072i	0.0413963	−	0.00296072i
$677$	−1.81840	+	0.328076i	−1.81840	+	0.328076i	−0.977147	−	0.212565i	$-0.931818\pi$
$677$	−1.81840	+	0.328076i	−1.81840	+	0.328076i	−0.841254	+	0.540641i	$0.818182\pi$
$678$		0			0
$679$		0			0
$680$	0.708005	+	0.455007i	0.708005	+	0.455007i
$681$		0			0
$682$		0			0
$683$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$683$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$684$		0			0
$685$	0.400928	+	0.577446i	0.400928	+	0.577446i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.975835	+	1.64468i	0.975835	+	1.64468i
$690$		0			0
$691$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$691$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$692$	0.119773	+	0.550588i	0.119773	+	0.550588i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.518099	+	0.556478i	0.518099	+	0.556478i
$698$	−0.406285	−	1.59181i	−0.406285	−	1.59181i
$699$		0			0
$700$		0			0
$701$	0.270040	−	0.919672i	0.270040	−	0.919672i	−0.707107	−	0.707107i	$-0.750000\pi$
$701$	0.270040	−	0.919672i	0.270040	−	0.919672i	0.977147	−	0.212565i	$-0.0681818\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	1.29309	−	1.38888i	1.29309	−	1.38888i
$707$		0			0
$708$		0			0
$709$	1.63201	+	0.543184i	1.63201	+	0.543184i	0.977147	−	0.212565i	$-0.0681818\pi$
$709$	1.63201	+	0.543184i	1.63201	+	0.543184i	0.654861	+	0.755750i	$0.272727\pi$
$710$		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		−0.627469	−	0.778642i	$-0.715909\pi$
$719$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$720$		0			0
$721$		0			0
$722$	−0.599278	−	0.800541i	−0.599278	−	0.800541i
$723$		0			0
$724$	0.197516	−	0.0818140i	0.197516	−	0.0818140i
$725$	−1.49035	−	2.97735i	−1.49035	−	2.97735i
$726$		0			0
$727$		0			0		0.731895	−	0.681418i	$-0.238636\pi$
$727$		0			0		−0.731895	+	0.681418i	$0.761364\pi$
$728$		0			0
$729$		0			0
$730$	−2.27563	+	1.24259i	−2.27563	+	1.24259i
$731$		0			0
$732$		0			0
$733$	−0.196911	−	0.527938i	−0.196911	−	0.527938i	0.800541	−	0.599278i	$-0.204545\pi$
$733$	−0.196911	−	0.527938i	−0.196911	−	0.527938i	−0.997452	+	0.0713392i	$0.977273\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$739$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$740$	0.702536	−	3.89389i	0.702536	−	3.89389i
$741$		0			0
$742$		0			0
$743$		0			0		−0.860014	−	0.510270i	$-0.829545\pi$
$743$		0			0		0.860014	+	0.510270i	$0.170455\pi$
$744$		0			0
$745$	3.00468	+	2.08618i	3.00468	+	2.08618i
$746$	−1.38189	+	1.38189i	−1.38189	+	1.38189i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$751$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$752$		0			0
$753$		0			0
$754$	1.16111	+	0.0830441i	1.16111	+	0.0830441i
$755$		0			0
$756$		0			0
$757$	−0.398174	−	0.871880i	−0.398174	−	0.871880i	−0.997452	−	0.0713392i	$-0.977273\pi$
$757$	−0.398174	−	0.871880i	−0.398174	−	0.871880i	0.599278	−	0.800541i	$-0.295455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.682956	+	0.148568i	0.682956	+	0.148568i	0.540641	−	0.841254i	$-0.318182\pi$
$761$	0.682956	+	0.148568i	0.682956	+	0.148568i	0.142315	+	0.989821i	$0.454545\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.05657	+	1.64406i	−1.05657	+	1.64406i	−0.349464	+	0.936950i	$0.613636\pi$
$769$	−1.05657	+	1.64406i	−1.05657	+	1.64406i	−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$		0			0
$772$	1.97715	+	0.212565i	1.97715	+	0.212565i
$773$	−0.881010	−	0.158952i	−0.881010	−	0.158952i	−0.281733	−	0.959493i	$-0.590909\pi$
$773$	−0.881010	−	0.158952i	−0.881010	−	0.158952i	−0.599278	+	0.800541i	$0.704545\pi$
$774$		0			0
$775$		0			0
$776$	1.07853	+	1.67822i	1.07853	+	1.67822i
$777$		0			0
$778$	1.69270	−	1.00432i	1.69270	−	1.00432i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.936950	+	0.349464i	0.936950	+	0.349464i
$785$	0.234072	−	1.62801i	0.234072	−	1.62801i
$786$		0			0
$787$		0			0		0.968937	−	0.247307i	$-0.0795455\pi$
$787$		0			0		−0.968937	+	0.247307i	$0.920455\pi$
$788$	−0.191177	+	0.0956962i	−0.191177	+	0.0956962i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.447749	−	1.52489i	−0.447749	−	1.52489i
$794$	1.50891	−	1.21596i	1.50891	−	1.21596i
$795$		0			0
$796$		0			0
$797$	0.784887	+	0.905808i	0.784887	+	0.905808i	0.997452	−	0.0713392i	$-0.0227273\pi$
$797$	0.784887	+	0.905808i	0.784887	+	0.905808i	−0.212565	+	0.977147i	$0.568182\pi$
$798$		0			0
$799$		0			0
$800$	−2.91899			−2.91899
$801$		0			0
$802$	−1.51150			−1.51150
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.491789	+	0.396309i	−0.491789	+	0.396309i
$809$	0.451077	+	1.53623i	0.451077	+	1.53623i	0.800541	+	0.599278i	$0.204545\pi$
$809$	0.451077	+	1.53623i	0.451077	+	1.53623i	−0.349464	+	0.936950i	$0.613636\pi$
$810$		0			0
$811$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$811$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.527938	+	0.196911i	0.527938	+	0.196911i
$819$		0			0
$820$	−3.43052	−	0.875587i	−3.43052	−	0.875587i
$821$	−1.56449	+	0.340335i	−1.56449	+	0.340335i	−0.909632	−	0.415415i	$-0.863636\pi$
$821$	−1.56449	+	0.340335i	−1.56449	+	0.340335i	−0.654861	+	0.755750i	$0.727273\pi$
$822$		0			0
$823$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$823$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.999363	−	0.0356923i	$-0.0113636\pi$
$827$		0			0		−0.999363	+	0.0356923i	$0.988636\pi$
$828$		0			0
$829$	−0.753206	−	0.135893i	−0.753206	−	0.135893i	−0.212565	−	0.977147i	$-0.568182\pi$
$829$	−0.753206	−	0.135893i	−0.753206	−	0.135893i	−0.540641	+	0.841254i	$0.681818\pi$
$830$		0			0
$831$		0			0
$832$	0.520751	−	0.877679i	0.520751	−	0.877679i
$833$	−0.229843	+	0.357643i	−0.229843	+	0.357643i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.778642	−	0.627469i	$-0.784091\pi$
$839$		0			0		0.778642	+	0.627469i	$0.215909\pi$
$840$		0			0
$841$	0.294191	+	0.0639974i	0.294191	+	0.0639974i
$842$	−0.442295	−	0.221397i	−0.442295	−	0.221397i
$843$		0			0
$844$		0			0
$845$	−0.0341302	−	0.0747346i	−0.0341302	−	0.0747346i
$846$		0			0
$847$		0			0
$848$	1.86912	+	0.133682i	1.86912	+	0.133682i
$849$		0			0
$850$	0.263783	−	1.21259i	0.263783	−	1.21259i
$851$		0			0
$852$		0			0
$853$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$853$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.11946	−	0.777256i	−1.11946	−	0.777256i	−0.142315	−	0.989821i	$-0.545455\pi$
$857$	−1.11946	−	0.777256i	−1.11946	−	0.777256i	−0.977147	+	0.212565i	$0.931818\pi$
$858$		0			0
$859$		0			0		−0.860014	−	0.510270i	$-0.829545\pi$
$859$		0			0		0.860014	+	0.510270i	$0.170455\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$863$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$864$		0			0
$865$	0.938384	−	0.603063i	0.938384	−	0.603063i
$866$	0.00254789	−	0.0713392i	0.00254789	−	0.0713392i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.32661	−	0.724384i	1.32661	−	0.724384i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.800541	+	1.59928i	0.800541	+	1.59928i	0.800541	+	0.599278i	$0.204545\pi$
$877$	0.800541	+	1.59928i	0.800541	+	1.59928i			1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−1.56449	−	1.17116i	−1.56449	−	1.17116i	−0.909632	−	0.415415i	$-0.863636\pi$
$881$	−1.56449	−	1.17116i	−1.56449	−	1.17116i	−0.654861	−	0.755750i	$-0.727273\pi$
$882$		0			0
$883$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$883$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$884$	0.317542	+	0.295642i	0.317542	+	0.295642i
$885$		0			0
$886$		0			0
$887$		0			0		0.315800	−	0.948826i	$-0.397727\pi$
$887$		0			0		−0.315800	+	0.948826i	$0.602273\pi$
$888$		0			0
$889$		0			0
$890$	−0.822373	+	1.80075i	−0.822373	+	1.80075i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.670617	+	0.196911i	−0.670617	+	0.196911i
$899$		0			0
$900$		0			0
$901$	−0.224443	+	0.764382i	−0.224443	+	0.764382i
$902$		0			0
$903$		0			0
$904$	0.0176539	+	0.0691673i	0.0176539	+	0.0691673i
$905$	−0.288395	−	0.309758i	−0.288395	−	0.309758i
$906$		0			0
$907$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$907$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$911$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$912$		0			0
$913$		0			0
$914$	0.631197	+	0.0225432i	0.631197	+	0.0225432i
$915$		0			0
$916$	0.0522460	+	1.46286i	0.0522460	+	1.46286i
$917$		0			0
$918$		0			0
$919$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$919$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$920$		0			0
$921$		0			0
$922$	−0.474017	−	0.304632i	−0.474017	−	0.304632i
$923$		0			0
$924$		0			0
$925$	−5.74155	+	1.03589i	−5.74155	+	1.03589i
$926$		0			0
$927$		0			0
$928$	0.715720	−	0.888154i	0.715720	−	0.888154i
$929$	1.18636	−	0.170572i	1.18636	−	0.170572i	0.479249	−	0.877679i	$-0.340909\pi$
$929$	1.18636	−	0.170572i	1.18636	−	0.170572i	0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$		0			0
$932$	0.120029	−	1.67822i	0.120029	−	1.67822i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.142315	+	1.98982i	−0.142315	+	1.98982i			1.00000i	$0.5\pi$
$937$	−0.142315	+	1.98982i	−0.142315	+	1.98982i	−0.142315	+	0.989821i	$0.545455\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.855137	+	1.06116i	−0.855137	+	1.06116i	0.142315	+	0.989821i	$0.454545\pi$
$941$	−0.855137	+	1.06116i	−0.855137	+	1.06116i	−0.997452	+	0.0713392i	$0.977273\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$947$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$948$		0			0
$949$	−1.26822	+	0.422105i	−1.26822	+	0.422105i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.0225432	−	0.631197i	−0.0225432	−	0.631197i	−0.959493	−	0.281733i	$-0.909091\pi$
$953$	−0.0225432	−	0.631197i	−0.0225432	−	0.631197i	0.936950	−	0.349464i	$-0.113636\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.479249	−	0.877679i	−0.479249	−	0.877679i
$962$	0.712830	−	1.91117i	0.712830	−	1.91117i
$963$		0			0
$964$	1.25910	+	1.35236i	1.25910	+	1.35236i
$965$	−0.973547	−	3.81432i	−0.973547	−	3.81432i
$966$		0			0
$967$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$967$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$968$	0.281733	−	0.959493i	0.281733	−	0.959493i
$969$		0			0
$970$	2.36667	−	3.16150i	2.36667	−	3.16150i
$971$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$971$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.47759	−	0.491789i	−1.47759	−	0.491789i
$977$	0.665114	−	1.45640i	0.665114	−	1.45640i	−0.212565	−	0.977147i	$-0.568182\pi$
$977$	0.665114	−	1.45640i	0.665114	−	1.45640i	0.877679	−	0.479249i	$-0.159091\pi$
$978$		0			0
$979$		0			0
$980$		−	1.97964i		−	1.97964i
$981$		0			0
$982$		0			0
$983$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$983$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$984$		0			0
$985$	0.309758	+	0.288395i	0.309758	+	0.288395i
$986$	0.304274	+	0.377581i	0.304274	+	0.377581i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$991$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.125226	−	0.0683785i	0.125226	−	0.0683785i	−0.415415	−	0.909632i	$-0.636364\pi$
$997$	0.125226	−	0.0683785i	0.125226	−	0.0683785i	0.540641	+	0.841254i	$0.318182\pi$
$998$		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.431.1	yes	40
3.2	odd	2		3204.1.cd.a.431.1	✓	40
4.3	odd	2	CM	3204.1.cd.b.431.1	yes	40
12.11	even	2		3204.1.cd.a.431.1	✓	40
89.19	odd	88		3204.1.cd.a.1799.1	yes	40
267.197	even	88	inner	3204.1.cd.b.1799.1	yes	40
356.19	even	88		3204.1.cd.a.1799.1	yes	40
1068.731	odd	88	inner	3204.1.cd.b.1799.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3204.1.cd.a.431.1	✓	40	3.2	odd	2
3204.1.cd.a.431.1	✓	40	12.11	even	2
3204.1.cd.a.1799.1	yes	40	89.19	odd	88
3204.1.cd.a.1799.1	yes	40	356.19	even	88
3204.1.cd.b.431.1	yes	40	1.1	even	1	trivial
3204.1.cd.b.431.1	yes	40	4.3	odd	2	CM
3204.1.cd.b.1799.1	yes	40	267.197	even	88	inner
3204.1.cd.b.1799.1	yes	40	1068.731	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)

\(f(q)\)	\(=\)	\(q+(-0.909632 - 0.415415i) q^{2} +(0.654861 + 0.755750i) q^{4} +(0.948742 - 1.73749i) q^{5} +(-0.281733 - 0.959493i) q^{8} +O(q^{10})\)	\(q+(-0.909632 - 0.415415i) q^{2} +(0.654861 + 0.755750i) q^{4} +(0.948742 - 1.73749i) q^{5} +(-0.281733 - 0.959493i) q^{8} +(-1.58479 + 1.18636i) q^{10} +(-0.912593 + 0.456811i) q^{13} +(-0.142315 + 0.989821i) q^{16} +(-0.398326 - 0.148568i) q^{17} +(1.93440 - 0.420803i) q^{20} +(-1.57812 - 2.45561i) q^{25} +(1.01989 - 0.0364255i) q^{26} +(1.13411 + 0.121929i) q^{29} +(0.540641 - 0.841254i) q^{32} +(0.300613 + 0.300613i) q^{34} +(0.764879 - 1.84658i) q^{37} +(-1.93440 - 0.420803i) q^{40} +(-1.59928 - 0.800541i) q^{41} +(0.212565 - 0.977147i) q^{49} +(0.415415 + 2.88927i) q^{50} +(-0.942856 - 0.390544i) q^{52} +(-0.133682 - 1.86912i) q^{53} +(-0.980971 - 0.582037i) q^{58} +(-0.276501 + 1.53254i) q^{61} +(-0.841254 + 0.540641i) q^{64} +(-0.0721094 + 2.01902i) q^{65} +(-0.148568 - 0.398326i) q^{68} +(1.29639 + 0.186393i) q^{73} +(-1.46286 + 1.36197i) q^{74} +(1.58479 + 1.18636i) q^{80} +(1.12220 + 1.39256i) q^{82} +(-0.636044 + 0.551135i) q^{85} +(0.877679 - 0.479249i) q^{89} +(-1.91410 + 0.562029i) q^{97} +(-0.599278 + 0.800541i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{4}+O(q^{10}) \) 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

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