LMFDB - Embedded newform 3204.1.cd.b.647.1

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			647.1
Root			$0.599278 + 0.800541i$ of defining polynomial
Character	$\chi$	$=$	3204.647
Dual form			3204.1.cd.b.827.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.281733 + 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-0.129785 - 1.81463i) q^{5} +(-0.755750 - 0.654861i) q^{8} +O(q^{10})$	\(q+(0.281733 + 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-0.129785 - 1.81463i) q^{5} +(-0.755750 - 0.654861i) q^{8} +(1.70456 - 0.635768i) q^{10} +(1.11946 - 0.777256i) q^{13} +(0.415415 - 0.909632i) q^{16} +(-1.40524 + 0.767317i) q^{17} +(1.09024 + 1.45640i) q^{20} +(-2.28621 + 0.328708i) q^{25} +(1.06116 + 0.855137i) q^{26} +(-0.769914 - 1.53809i) q^{29} +(0.989821 + 0.142315i) q^{32} +(-1.13214 - 1.13214i) q^{34} +(-1.43874 - 0.595946i) q^{37} +(-1.09024 + 1.45640i) q^{40} +(-1.34946 - 0.936950i) q^{41} +(0.800541 + 0.599278i) q^{49} +(-0.959493 - 2.10100i) q^{50} +(-0.521535 + 1.25910i) q^{52} +(1.71524 - 0.373128i) q^{53} +(1.25888 - 1.17206i) q^{58} +(-0.469302 - 1.83871i) q^{61} +(0.142315 + 0.989821i) q^{64} +(-1.55572 - 1.93053i) q^{65} +(0.767317 - 1.40524i) q^{68} +(1.53046 + 0.698939i) q^{73} +(0.166466 - 1.54836i) q^{74} +(-1.70456 - 0.635768i) q^{80} +(0.518809 - 1.55877i) q^{82} +(1.57477 + 2.45040i) q^{85} +(-0.997452 - 0.0713392i) q^{89} +(-0.278401 + 0.321292i) q^{97} +(-0.349464 + 0.936950i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times$.

$n$	$181$	$713$	$1603$
$\chi(n)$	$e\left(\frac{49}{88}\right)$	$-1$	$-1$

Coefficient data

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

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

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

$2$	0.281733	+	0.959493i	0.281733	+	0.959493i
$2$	0.281733	+	0.959493i	0.281733	+	0.959493i
$3$		0			0
$3$		0			0
$4$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$5$	−0.129785	−	1.81463i	−0.129785	−	1.81463i	−0.479249	−	0.877679i	$-0.659091\pi$
$5$	−0.129785	−	1.81463i	−0.129785	−	1.81463i	0.349464	−	0.936950i	$-0.386364\pi$
$6$		0			0
$7$		0			0		−0.948826	−	0.315800i	$-0.897727\pi$
$7$		0			0		0.948826	+	0.315800i	$0.102273\pi$
$8$	−0.755750	−	0.654861i	−0.755750	−	0.654861i
$9$		0			0
$10$	1.70456	−	0.635768i	1.70456	−	0.635768i
$11$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$11$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$12$		0			0
$13$	1.11946	−	0.777256i	1.11946	−	0.777256i	0.142315	−	0.989821i	$-0.454545\pi$
$13$	1.11946	−	0.777256i	1.11946	−	0.777256i	0.977147	+	0.212565i	$0.0681818\pi$
$14$		0			0
$15$		0			0
$16$	0.415415	−	0.909632i	0.415415	−	0.909632i
$17$	−1.40524	+	0.767317i	−1.40524	+	0.767317i	−0.989821	−	0.142315i	$-0.954545\pi$
$17$	−1.40524	+	0.767317i	−1.40524	+	0.767317i	−0.415415	+	0.909632i	$0.636364\pi$
$18$		0			0
$19$		0			0		−0.0356923	−	0.999363i	$-0.511364\pi$
$19$		0			0		0.0356923	+	0.999363i	$0.488636\pi$
$20$	1.09024	+	1.45640i	1.09024	+	1.45640i
$21$		0			0
$22$		0			0
$23$		0			0		−0.731895	−	0.681418i	$-0.761364\pi$
$23$		0			0		0.731895	+	0.681418i	$0.238636\pi$
$24$		0			0
$25$	−2.28621	+	0.328708i	−2.28621	+	0.328708i
$26$	1.06116	+	0.855137i	1.06116	+	0.855137i
$27$		0			0
$28$		0			0
$29$	−0.769914	−	1.53809i	−0.769914	−	1.53809i	−0.841254	−	0.540641i	$-0.818182\pi$
$29$	−0.769914	−	1.53809i	−0.769914	−	1.53809i	0.0713392	−	0.997452i	$-0.477273\pi$
$30$		0			0
$31$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$31$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$32$	0.989821	+	0.142315i	0.989821	+	0.142315i
$33$		0			0
$34$	−1.13214	−	1.13214i	−1.13214	−	1.13214i
$35$		0			0
$36$		0			0
$37$	−1.43874	−	0.595946i	−1.43874	−	0.595946i	−0.479249	−	0.877679i	$-0.659091\pi$
$37$	−1.43874	−	0.595946i	−1.43874	−	0.595946i	−0.959493	+	0.281733i	$0.909091\pi$
$38$		0			0
$39$		0			0
$40$	−1.09024	+	1.45640i	−1.09024	+	1.45640i
$41$	−1.34946	−	0.936950i	−1.34946	−	0.936950i	−0.349464	−	0.936950i	$-0.613636\pi$
$41$	−1.34946	−	0.936950i	−1.34946	−	0.936950i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.894225	−	0.447617i	$-0.852273\pi$
$43$		0			0		0.894225	+	0.447617i	$0.147727\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$47$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$48$		0			0
$49$	0.800541	+	0.599278i	0.800541	+	0.599278i
$50$	−0.959493	−	2.10100i	−0.959493	−	2.10100i
$51$		0			0
$52$	−0.521535	+	1.25910i	−0.521535	+	1.25910i
$53$	1.71524	−	0.373128i	1.71524	−	0.373128i	0.755750	−	0.654861i	$-0.227273\pi$
$53$	1.71524	−	0.373128i	1.71524	−	0.373128i	0.959493	+	0.281733i	$0.0909091\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.25888	−	1.17206i	1.25888	−	1.17206i
$59$		0			0		−0.177553	−	0.984111i	$-0.556818\pi$
$59$		0			0		0.177553	+	0.984111i	$0.443182\pi$
$60$		0			0
$61$	−0.469302	−	1.83871i	−0.469302	−	1.83871i	−0.540641	−	0.841254i	$-0.681818\pi$
$61$	−0.469302	−	1.83871i	−0.469302	−	1.83871i	0.0713392	−	0.997452i	$-0.477273\pi$
$62$		0			0
$63$		0			0
$64$	0.142315	+	0.989821i	0.142315	+	0.989821i
$65$	−1.55572	−	1.93053i	−1.55572	−	1.93053i
$66$		0			0
$67$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$67$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$68$	0.767317	−	1.40524i	0.767317	−	1.40524i
$69$		0			0
$70$		0			0
$71$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$71$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$72$		0			0
$73$	1.53046	+	0.698939i	1.53046	+	0.698939i	0.989821	−	0.142315i	$-0.0454545\pi$
$73$	1.53046	+	0.698939i	1.53046	+	0.698939i	0.540641	+	0.841254i	$0.318182\pi$
$74$	0.166466	−	1.54836i	0.166466	−	1.54836i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$79$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$80$	−1.70456	−	0.635768i	−1.70456	−	0.635768i
$81$		0			0
$82$	0.518809	−	1.55877i	0.518809	−	1.55877i
$83$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$83$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$84$		0			0
$85$	1.57477	+	2.45040i	1.57477	+	2.45040i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.997452	−	0.0713392i	−0.997452	−	0.0713392i
$89$	−0.997452	−	0.0713392i	−0.997452	−	0.0713392i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.278401	+	0.321292i	−0.278401	+	0.321292i	−0.877679	−	0.479249i	$-0.840909\pi$
$97$	−0.278401	+	0.321292i	−0.278401	+	0.321292i	0.599278	+	0.800541i	$0.295455\pi$
$98$	−0.349464	+	0.936950i	−0.349464	+	0.936950i
$99$		0			0
$100$	1.74557	−	1.51255i	1.74557	−	1.51255i
$101$	0.328076	−	0.135893i	0.328076	−	0.135893i	−0.212565	−	0.977147i	$-0.568182\pi$
$101$	0.328076	−	0.135893i	0.328076	−	0.135893i	0.540641	+	0.841254i	$0.318182\pi$
$102$		0			0
$103$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$103$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$104$	−1.35503	−	0.145681i	−1.35503	−	0.145681i
$105$		0			0
$106$	0.841254	+	1.54064i	0.841254	+	1.54064i
$107$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$107$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$108$		0			0
$109$	0.865611	−	0.647988i	0.865611	−	0.647988i	−0.0713392	−	0.997452i	$-0.522727\pi$
$109$	0.865611	−	0.647988i	0.865611	−	0.647988i	0.936950	+	0.349464i	$0.113636\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.977147	−	0.787435i	0.977147	−	0.787435i		−	1.00000i	$-0.5\pi$
$113$	0.977147	−	0.787435i	0.977147	−	0.787435i	0.977147	+	0.212565i	$0.0681818\pi$
$114$		0			0
$115$		0			0
$116$	1.47925	+	0.877679i	1.47925	+	0.877679i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$122$	1.63201	−	0.968315i	1.63201	−	0.968315i
$123$		0			0
$124$		0			0
$125$	0.506486	+	2.32828i	0.506486	+	2.32828i
$126$		0			0
$127$		0			0		−0.315800	−	0.948826i	$-0.602273\pi$
$127$		0			0		0.315800	+	0.948826i	$0.397727\pi$
$128$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$129$		0			0
$130$	1.41403	−	2.03660i	1.41403	−	2.03660i
$131$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$131$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.56449	+	0.340335i	1.56449	+	0.340335i
$137$	0.282089	−	0.406285i	0.282089	−	0.406285i	−0.654861	−	0.755750i	$-0.727273\pi$
$137$	0.282089	−	0.406285i	0.282089	−	0.406285i	0.936950	+	0.349464i	$0.113636\pi$
$138$		0			0
$139$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$139$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−2.69114	+	1.59673i	−2.69114	+	1.59673i
$146$	−0.239446	+	1.66538i	−0.239446	+	1.66538i
$147$		0			0
$148$	1.53254	−	0.276501i	1.53254	−	0.276501i
$149$	0.684410	−	0.342591i	0.684410	−	0.342591i	−0.0713392	−	0.997452i	$-0.522727\pi$
$149$	0.684410	−	0.342591i	0.684410	−	0.342591i	0.755750	+	0.654861i	$0.227273\pi$
$150$		0			0
$151$		0			0		−0.860014	−	0.510270i	$-0.829545\pi$
$151$		0			0		0.860014	+	0.510270i	$0.170455\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.68425	−	0.919672i	−1.68425	−	0.919672i	−0.977147	−	0.212565i	$-0.931818\pi$
$157$	−1.68425	−	0.919672i	−1.68425	−	0.919672i	−0.707107	−	0.707107i	$-0.750000\pi$
$158$		0			0
$159$		0			0
$160$	0.129785	−	1.81463i	0.129785	−	1.81463i
$161$		0			0
$162$		0			0
$163$		0			0		−0.994270	−	0.106895i	$-0.965909\pi$
$163$		0			0		0.994270	+	0.106895i	$0.0340909\pi$
$164$	1.64179	+	0.0586368i	1.64179	+	0.0586368i
$165$		0			0
$166$		0			0
$167$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$167$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$168$		0			0
$169$	0.299603	−	0.803267i	0.299603	−	0.803267i
$170$	−1.90747	+	2.20134i	−1.90747	+	2.20134i
$171$		0			0
$172$		0			0
$173$	−1.50765	+	0.107829i	−1.50765	+	0.107829i	−0.800541	−	0.599278i	$-0.795455\pi$
$173$	−1.50765	+	0.107829i	−1.50765	+	0.107829i	−0.707107	+	0.707107i	$0.750000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.212565	−	0.977147i	−0.212565	−	0.977147i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	−0.317545	+	1.76003i	−0.317545	+	1.76003i	0.281733	+	0.959493i	$0.409091\pi$
$181$	−0.317545	+	1.76003i	−0.317545	+	1.76003i	−0.599278	+	0.800541i	$0.704545\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.894695	+	2.68813i	−0.894695	+	2.68813i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.570323	−	0.821421i	$-0.693182\pi$
$191$		0			0		0.570323	+	0.821421i	$0.306818\pi$
$192$		0			0
$193$	0.0956962	−	0.890105i	0.0956962	−	0.890105i	−0.841254	−	0.540641i	$-0.818182\pi$
$193$	0.0956962	−	0.890105i	0.0956962	−	0.890105i	0.936950	−	0.349464i	$-0.113636\pi$
$194$	−0.386712	−	0.176606i	−0.386712	−	0.176606i
$195$		0			0
$196$	−0.997452	−	0.0713392i	−0.997452	−	0.0713392i
$197$	1.78731	−	0.0638340i	1.78731	−	0.0638340i	0.877679	−	0.479249i	$-0.159091\pi$
$197$	1.78731	−	0.0638340i	1.78731	−	0.0638340i	0.909632	+	0.415415i	$0.136364\pi$
$198$		0			0
$199$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$199$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$200$	1.94306	+	1.24873i	1.94306	+	1.24873i
$201$		0			0
$202$	0.222818	+	0.276501i	0.222818	+	0.276501i
$203$		0			0
$204$		0			0
$205$	−1.52508	+	2.57038i	−1.52508	+	2.57038i
$206$		0			0
$207$		0			0
$208$	−0.241976	−	1.34118i	−0.241976	−	1.34118i
$209$		0			0
$210$		0			0
$211$		0			0		−0.510270	−	0.860014i	$-0.670455\pi$
$211$		0			0		0.510270	+	0.860014i	$0.329545\pi$
$212$	−1.24123	+	1.24123i	−1.24123	+	1.24123i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.865611	+	0.647988i	0.865611	+	0.647988i
$219$		0			0
$220$		0			0
$221$	−0.976706	+	1.95121i	−0.976706	+	1.95121i
$222$		0			0
$223$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$223$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$224$		0			0
$225$		0			0
$226$	1.03083	+	0.715720i	1.03083	+	0.715720i
$227$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$227$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$228$		0			0
$229$	−0.202850	+	0.0675149i	−0.202850	+	0.0675149i	−0.415415	−	0.909632i	$-0.636364\pi$
$229$	−0.202850	+	0.0675149i	−0.202850	+	0.0675149i	0.212565	+	0.977147i	$0.431818\pi$
$230$		0			0
$231$		0			0
$232$	−0.425374	+	1.66660i	−0.425374	+	1.66660i
$233$	−0.201264	−	0.201264i	−0.201264	−	0.201264i	0.599278	−	0.800541i	$-0.295455\pi$
$233$	−0.201264	−	0.201264i	−0.201264	−	0.201264i	−0.800541	+	0.599278i	$0.795455\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.968937	−	0.247307i	$-0.0795455\pi$
$239$		0			0		−0.968937	+	0.247307i	$0.920455\pi$
$240$		0			0
$241$	0.595946	+	0.480244i	0.595946	+	0.480244i	0.877679	−	0.479249i	$-0.159091\pi$
$241$	0.595946	+	0.480244i	0.595946	+	0.480244i	−0.281733	+	0.959493i	$0.590909\pi$
$242$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$243$		0			0
$244$	1.38888	+	1.29309i	1.38888	+	1.29309i
$245$	0.983568	−	1.53046i	0.983568	−	1.53046i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−2.09127	+	1.14192i	−2.09127	+	1.14192i
$251$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$251$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$257$	1.79799	−	0.670617i	1.79799	−	0.670617i	0.800541	−	0.599278i	$-0.204545\pi$
$257$	1.79799	−	0.670617i	1.79799	−	0.670617i	0.997452	−	0.0713392i	$-0.0227273\pi$
$258$		0			0
$259$		0			0
$260$	2.35248	+	0.782980i	2.35248	+	0.782980i
$261$		0			0
$262$		0			0
$263$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$263$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$264$		0			0
$265$	−0.899702	−	3.06410i	−0.899702	−	3.06410i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$269$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$270$		0			0
$271$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$271$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$272$	0.114220	+	1.59700i	0.114220	+	1.59700i
$273$		0			0
$274$	0.469302	+	0.156199i	0.469302	+	0.156199i
$275$		0			0
$276$		0			0
$277$	1.86912	−	0.697148i	1.86912	−	0.697148i	0.909632	−	0.415415i	$-0.136364\pi$
$277$	1.86912	−	0.697148i	1.86912	−	0.697148i	0.959493	−	0.281733i	$-0.0909091\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.0407123	+	1.13992i	−0.0407123	+	1.13992i	0.800541	+	0.599278i	$0.204545\pi$
$281$	−0.0407123	+	1.13992i	−0.0407123	+	1.13992i	−0.841254	+	0.540641i	$0.818182\pi$
$282$		0			0
$283$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$283$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.845273	−	1.31527i	0.845273	−	1.31527i
$290$	−2.29023	−	2.13228i	−2.29023	−	2.13228i
$291$		0			0
$292$	−1.66538	+	0.239446i	−1.66538	+	0.239446i
$293$	0.794635	+	0.640357i	0.794635	+	0.640357i	0.936950	−	0.349464i	$-0.113636\pi$
$293$	0.794635	+	0.640357i	0.794635	+	0.640357i	−0.142315	+	0.989821i	$0.545455\pi$
$294$		0			0
$295$		0			0
$296$	0.697067	+	1.39256i	0.697067	+	1.39256i
$297$		0			0
$298$	0.521535	+	0.560168i	0.521535	+	0.560168i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.27566	+	1.09024i	−3.27566	+	1.09024i
$306$		0			0
$307$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$307$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$311$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$312$		0			0
$313$	0.867426	−	1.73290i	0.867426	−	1.73290i	0.212565	−	0.977147i	$-0.431818\pi$
$313$	0.867426	−	1.73290i	0.867426	−	1.73290i	0.654861	−	0.755750i	$-0.272727\pi$
$314$	0.407910	−	1.87513i	0.407910	−	1.87513i
$315$		0			0
$316$		0			0
$317$	0.828713	+	1.81463i	0.828713	+	1.81463i	0.479249	+	0.877679i	$0.340909\pi$
$317$	0.828713	+	1.81463i	0.828713	+	1.81463i	0.349464	+	0.936950i	$0.386364\pi$
$318$		0			0
$319$		0			0
$320$	1.77769	−	0.386712i	1.77769	−	0.386712i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−2.30384	+	2.14495i	−2.30384	+	2.14495i
$326$		0			0
$327$		0			0
$328$	0.406285	+	1.59181i	0.406285	+	1.59181i
$329$		0			0
$330$		0			0
$331$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$331$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.0713392	−	0.00254789i	0.0713392	−	0.00254789i		−	1.00000i	$-0.5\pi$
$337$	0.0713392	−	0.00254789i	0.0713392	−	0.00254789i	0.0713392	+	0.997452i	$0.477273\pi$
$338$	0.855137	+	0.0611606i	0.855137	+	0.0611606i
$339$		0			0
$340$	−2.64957	−	1.21002i	−2.64957	−	1.21002i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.528215	−	1.41620i	−0.528215	−	1.41620i
$347$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$347$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$348$		0			0
$349$	−0.322286	+	0.968315i	−0.322286	+	0.968315i	0.654861	+	0.755750i	$0.272727\pi$
$349$	−0.322286	+	0.968315i	−0.322286	+	0.968315i	−0.977147	+	0.212565i	$0.931818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.349464	+	1.93695i	−0.349464	+	1.93695i			1.00000i	$0.5\pi$
$353$	−0.349464	+	1.93695i	−0.349464	+	1.93695i	−0.349464	+	0.936950i	$0.613636\pi$
$354$		0			0
$355$		0			0
$356$	0.877679	−	0.479249i	0.877679	−	0.479249i
$357$		0			0
$358$		0			0
$359$		0			0		−0.984111	−	0.177553i	$-0.943182\pi$
$359$		0			0		0.984111	+	0.177553i	$0.0568182\pi$
$360$		0			0
$361$	−0.997452	+	0.0713392i	−0.997452	+	0.0713392i
$362$	−1.77820	+	0.191177i	−1.77820	+	0.191177i
$363$		0			0
$364$		0			0
$365$	1.06968	−	2.86793i	1.06968	−	2.86793i
$366$		0			0
$367$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$367$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$368$		0			0
$369$		0			0
$370$	−2.83130	−	0.101120i	−2.83130	−	0.101120i
$371$		0			0
$372$		0			0
$373$	0.574406	+	1.05195i	0.574406	+	1.05195i	0.989821	+	0.142315i	$0.0454545\pi$
$373$	0.574406	+	1.05195i	0.574406	+	1.05195i	−0.415415	+	0.909632i	$0.636364\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.05738	−	1.12342i	−2.05738	−	1.12342i
$378$		0			0
$379$		0			0		0.681418	−	0.731895i	$-0.261364\pi$
$379$		0			0		−0.681418	+	0.731895i	$0.738636\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.860014	−	0.510270i	$-0.829545\pi$
$383$		0			0		0.860014	+	0.510270i	$0.170455\pi$
$384$		0			0
$385$		0			0
$386$	0.881010	−	0.158952i	0.881010	−	0.158952i
$387$		0			0
$388$	0.0605024	−	0.420803i	0.0605024	−	0.420803i
$389$	1.66660	−	0.988839i	1.66660	−	0.988839i	0.707107	−	0.707107i	$-0.250000\pi$
$389$	1.66660	−	0.988839i	1.66660	−	0.988839i	0.959493	−	0.281733i	$-0.0909091\pi$
$390$		0			0
$391$		0			0
$392$	−0.212565	−	0.977147i	−0.212565	−	0.977147i
$393$		0			0
$394$	0.564792	+	1.69693i	0.564792	+	1.69693i
$395$		0			0
$396$		0			0
$397$	0.0407123	−	0.0586368i	0.0407123	−	0.0586368i	−0.800541	−	0.599278i	$-0.795455\pi$
$397$	0.0407123	−	0.0586368i	0.0407123	−	0.0586368i	0.841254	+	0.540641i	$0.181818\pi$
$398$		0			0
$399$		0			0
$400$	−0.650724	+	2.21616i	−0.650724	+	2.21616i
$401$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$401$	−0.304632	+	1.03748i	−0.304632	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$402$		0			0
$403$		0			0
$404$	−0.202525	+	0.291692i	−0.202525	+	0.291692i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.321292	−	1.47696i	−0.321292	−	1.47696i	−0.800541	−	0.599278i	$-0.795455\pi$
$409$	−0.321292	−	1.47696i	−0.321292	−	1.47696i	0.479249	−	0.877679i	$-0.340909\pi$
$410$	−2.89592	−	0.739140i	−2.89592	−	0.739140i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	1.21868	−	0.610029i	1.21868	−	0.610029i
$417$		0			0
$418$		0			0
$419$		0			0		0.627469	−	0.778642i	$-0.284091\pi$
$419$		0			0		−0.627469	+	0.778642i	$0.715909\pi$
$420$		0			0
$421$	1.55629	−	1.25414i	1.55629	−	1.25414i	0.755750	−	0.654861i	$-0.227273\pi$
$421$	1.55629	−	1.25414i	1.55629	−	1.25414i	0.800541	−	0.599278i	$-0.204545\pi$
$422$		0			0
$423$		0			0
$424$	−1.54064	−	0.841254i	−1.54064	−	0.841254i
$425$	2.96045	−	2.21616i	2.96045	−	2.21616i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$431$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$432$		0			0
$433$	1.15941	−	0.480244i	1.15941	−	0.480244i	0.281733	−	0.959493i	$-0.409091\pi$
$433$	1.15941	−	0.480244i	1.15941	−	0.480244i	0.877679	+	0.479249i	$0.159091\pi$
$434$		0			0
$435$		0			0
$436$	−0.377869	+	1.01311i	−0.377869	+	1.01311i
$437$		0			0
$438$		0			0
$439$		0			0		0.994270	−	0.106895i	$-0.0340909\pi$
$439$		0			0		−0.994270	+	0.106895i	$0.965909\pi$
$440$		0			0
$441$		0			0
$442$	−2.14734	−	0.387423i	−2.14734	−	0.387423i
$443$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$443$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$444$		0			0
$445$			1.81926i			1.81926i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.518203	−	0.806340i	−0.518203	−	0.806340i	0.479249	−	0.877679i	$-0.340909\pi$
$449$	−0.518203	−	0.806340i	−0.518203	−	0.806340i	−0.997452	+	0.0713392i	$0.977273\pi$
$450$		0			0
$451$		0			0
$452$	−0.396309	+	1.19072i	−0.396309	+	1.19072i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.135893	−	0.328076i	−0.135893	−	0.328076i	0.841254	−	0.540641i	$-0.181818\pi$
$457$	−0.135893	−	0.328076i	−0.135893	−	0.328076i	−0.977147	+	0.212565i	$0.931818\pi$
$458$	−0.121929	−	0.175612i	−0.121929	−	0.175612i
$459$		0			0
$460$		0			0
$461$	−1.37491	−	0.627899i	−1.37491	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$461$	−1.37491	−	0.627899i	−1.37491	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$462$		0			0
$463$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$463$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$464$	−1.71893	+	0.0613918i	−1.71893	+	0.0613918i
$465$		0			0
$466$	0.136408	−	0.249813i	0.136408	−	0.249813i
$467$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$467$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$	−2.07382	+	0.451132i	−2.07382	+	0.451132i
$482$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$483$		0			0
$484$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$485$	0.619158	+	0.463496i	0.619158	+	0.463496i
$486$		0			0
$487$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$487$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$488$	−0.849422	+	1.69693i	−0.849422	+	1.69693i
$489$		0			0
$490$	1.74557	+	0.512546i	1.74557	+	0.512546i
$491$		0			0		−0.894225	−	0.447617i	$-0.852273\pi$
$491$		0			0		0.894225	+	0.447617i	$0.147727\pi$
$492$		0			0
$493$	2.26212	+	1.57062i	2.26212	+	1.57062i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.247307	−	0.968937i	$-0.420455\pi$
$499$		0			0		−0.247307	+	0.968937i	$0.579545\pi$
$500$	−1.68484	−	1.68484i	−1.68484	−	1.68484i
$501$		0			0
$502$		0			0
$503$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$503$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$504$		0			0
$505$	−0.289175	−	0.577698i	−0.289175	−	0.577698i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.66538	+	0.239446i	−1.66538	+	0.239446i	−0.909632	−	0.415415i	$-0.863636\pi$
$509$	−1.66538	+	0.239446i	−1.66538	+	0.239446i	−0.755750	+	0.654861i	$0.772727\pi$
$510$		0			0
$511$		0			0
$512$	0.540641	−	0.841254i	0.540641	−	0.841254i
$513$		0			0
$514$	1.15001	+	1.53623i	1.15001	+	1.53623i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.0884941	+	2.47778i	−0.0884941	+	2.47778i
$521$	1.20239	−	0.834832i	1.20239	−	0.834832i	0.212565	−	0.977147i	$-0.431818\pi$
$521$	1.20239	−	0.834832i	1.20239	−	0.834832i	0.989821	+	0.142315i	$0.0454545\pi$
$522$		0			0
$523$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$523$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.0713392	+	0.997452i	0.0713392	+	0.997452i
$530$	2.68651	−	1.72651i	2.68651	−	1.72651i
$531$		0			0
$532$		0			0
$533$	−2.23892			−2.23892
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.66538	+	1.07028i	−1.66538	+	1.07028i
$539$		0			0
$540$		0			0
$541$	−0.849422	−	0.282715i	−0.849422	−	0.282715i	−0.142315	−	0.989821i	$-0.545455\pi$
$541$	−0.849422	−	0.282715i	−0.849422	−	0.282715i	−0.707107	+	0.707107i	$0.750000\pi$
$542$		0			0
$543$		0			0
$544$	−1.50013	+	0.559521i	−1.50013	+	0.559521i
$545$	−1.28820	−	1.48666i	−1.28820	−	1.48666i
$546$		0			0
$547$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$547$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$548$	−0.0176539	+	0.494298i	−0.0176539	+	0.494298i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.19550	+	1.59700i	1.19550	+	1.59700i
$555$		0			0
$556$		0			0
$557$	1.07134	+	0.997452i	1.07134	+	0.997452i	1.00000			$0$
$557$	1.07134	+	0.997452i	1.07134	+	0.997452i	0.0713392	+	0.997452i	$0.477273\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.10521	+	0.282089i	−1.10521	+	0.282089i
$563$		0			0		−0.447617	−	0.894225i	$-0.647727\pi$
$563$		0			0		0.447617	+	0.894225i	$0.352273\pi$
$564$		0			0
$565$	−1.55572	−	1.67096i	−1.55572	−	1.67096i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−0.442295	+	1.73290i	−0.442295	+	1.73290i	0.212565	+	0.977147i	$0.431818\pi$
$569$	−0.442295	+	1.73290i	−0.442295	+	1.73290i	−0.654861	+	0.755750i	$0.727273\pi$
$570$		0			0
$571$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$571$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.65231	−	0.827089i	−1.65231	−	0.827089i	−0.997452	−	0.0713392i	$-0.977273\pi$
$577$	−1.65231	−	0.827089i	−1.65231	−	0.827089i	−0.654861	−	0.755750i	$-0.727273\pi$
$578$	1.50013	+	0.440479i	1.50013	+	0.440479i
$579$		0			0
$580$	1.40068	−	2.79820i	1.40068	−	2.79820i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.698939	−	1.53046i	−0.698939	−	1.53046i
$585$		0			0
$586$	−0.390544	+	0.942856i	−0.390544	+	0.942856i
$587$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$587$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.13977	+	1.06116i	−1.13977	+	1.06116i
$593$	−0.158952	−	0.881010i	−0.158952	−	0.881010i	−0.959493	−	0.281733i	$-0.909091\pi$
$593$	−0.158952	−	0.881010i	−0.158952	−	0.881010i	0.800541	−	0.599278i	$-0.204545\pi$
$594$		0			0
$595$		0			0
$596$	−0.390544	+	0.658226i	−0.390544	+	0.658226i
$597$		0			0
$598$		0			0
$599$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$599$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$600$		0			0
$601$	0.474017	+	0.304632i	0.474017	+	0.304632i	0.755750	−	0.654861i	$-0.227273\pi$
$601$	0.474017	+	0.304632i	0.474017	+	0.304632i	−0.281733	+	0.959493i	$0.590909\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.81463	+	0.129785i	1.81463	+	0.129785i
$606$		0			0
$607$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$607$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$608$		0			0
$609$		0			0
$610$	−1.96894	−	2.83582i	−1.96894	−	2.83582i
$611$		0			0
$612$		0			0
$613$	0.0994679	+	0.266684i	0.0994679	+	0.266684i	0.977147	−	0.212565i	$-0.0681818\pi$
$613$	0.0994679	+	0.266684i	0.0994679	+	0.266684i	−0.877679	+	0.479249i	$0.840909\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.00763067	−	0.0709757i	−0.00763067	−	0.0709757i	0.989821	−	0.142315i	$-0.0454545\pi$
$617$	−0.00763067	−	0.0709757i	−0.00763067	−	0.0709757i	−0.997452	+	0.0713392i	$0.977273\pi$
$618$		0			0
$619$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$619$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1.94306	−	0.570534i	1.94306	−	0.570534i
$626$	1.90708	+	0.344076i	1.90708	+	0.344076i
$627$		0			0
$628$	1.91410	−	0.136899i	1.91410	−	0.136899i
$629$	2.47905	−	0.266526i	2.47905	−	0.266526i
$630$		0			0
$631$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$631$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$632$		0			0
$633$		0			0
$634$	−1.50765	+	1.30638i	−1.50765	+	1.30638i
$635$		0			0
$636$		0			0
$637$	1.36197	+	0.0486428i	1.36197	+	0.0486428i
$638$		0			0
$639$		0			0
$640$	0.871880	+	1.59673i	0.871880	+	1.59673i
$641$	0.0401971	−	0.562029i	0.0401971	−	0.562029i	−0.936950	−	0.349464i	$-0.886364\pi$
$641$	0.0401971	−	0.562029i	0.0401971	−	0.562029i	0.977147	−	0.212565i	$-0.0681818\pi$
$642$		0			0
$643$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$643$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.778642	−	0.627469i	$-0.215909\pi$
$647$		0			0		−0.778642	+	0.627469i	$0.784091\pi$
$648$		0			0
$649$		0			0
$650$	−2.70713	−	1.60621i	−2.70713	−	1.60621i
$651$		0			0
$652$		0			0
$653$	−1.23500	+	0.222818i	−1.23500	+	0.222818i	−0.755750	−	0.654861i	$-0.772727\pi$
$653$	−1.23500	+	0.222818i	−1.23500	+	0.222818i	−0.479249	+	0.877679i	$0.659091\pi$
$654$		0			0
$655$		0			0
$656$	−1.41287	+	0.838293i	−1.41287	+	0.838293i
$657$		0			0
$658$		0			0
$659$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$659$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$660$		0			0
$661$	−0.199459	−	0.599278i	−0.199459	−	0.599278i	0.800541	−	0.599278i	$-0.204545\pi$
$661$	−0.199459	−	0.599278i	−0.199459	−	0.599278i	−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
$673$	1.59673	−	0.729202i	1.59673	−	0.729202i	0.599278	−	0.800541i	$-0.295455\pi$
$673$	1.59673	−	0.729202i	1.59673	−	0.729202i	0.997452	+	0.0713392i	$0.0227273\pi$
$674$	0.0225432	+	0.0677316i	0.0225432	+	0.0677316i
$675$		0			0
$676$	0.182237	+	0.837729i	0.182237	+	0.837729i
$677$	0.741593	+	0.189280i	0.741593	+	0.189280i	0.599278	−	0.800541i	$-0.295455\pi$
$677$	0.741593	+	0.189280i	0.741593	+	0.189280i	0.142315	+	0.989821i	$0.454545\pi$
$678$		0			0
$679$		0			0
$680$	0.414533	−	2.88314i	0.414533	−	2.88314i
$681$		0			0
$682$		0			0
$683$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$683$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$684$		0			0
$685$	−0.773868	−	0.459157i	−0.773868	−	0.459157i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.63013	−	1.75089i	1.63013	−	1.75089i
$690$		0			0
$691$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$691$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$692$	1.21002	−	0.905808i	1.21002	−	0.905808i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	2.61525	+	0.281169i	2.61525	+	0.281169i
$698$	−1.01989	−	0.0364255i	−1.01989	−	0.0364255i
$699$		0			0
$700$		0			0
$701$	0.107829	−	0.0934345i	0.107829	−	0.0934345i	−0.599278	−	0.800541i	$-0.704545\pi$
$701$	0.107829	−	0.0934345i	0.107829	−	0.0934345i	0.707107	+	0.707107i	$0.250000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−1.95695	+	0.210393i	−1.95695	+	0.210393i
$707$		0			0
$708$		0			0
$709$	−1.44053	−	0.259900i	−1.44053	−	0.259900i	−0.599278	−	0.800541i	$-0.704545\pi$
$709$	−1.44053	−	0.259900i	−1.44053	−	0.259900i	−0.841254	+	0.540641i	$0.818182\pi$
$710$		0			0
$711$		0			0
$712$	0.707107	+	0.707107i	0.707107	+	0.707107i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.315800	−	0.948826i	$-0.397727\pi$
$719$		0			0		−0.315800	+	0.948826i	$0.602273\pi$
$720$		0			0
$721$		0			0
$722$	−0.349464	−	0.936950i	−0.349464	−	0.936950i
$723$		0			0
$724$	−0.684410	−	1.65231i	−0.684410	−	1.65231i
$725$	2.26577	+	3.26333i	2.26577	+	3.26333i
$726$		0			0
$727$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$727$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$728$		0			0
$729$		0			0
$730$	3.05313	+	0.218364i	3.05313	+	0.218364i
$731$		0			0
$732$		0			0
$733$	0.724384	−	1.32661i	0.724384	−	1.32661i	−0.212565	−	0.977147i	$-0.568182\pi$
$733$	0.724384	−	1.32661i	0.724384	−	1.32661i	0.936950	−	0.349464i	$-0.113636\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		0.510270	−	0.860014i	$-0.329545\pi$
$739$		0			0		−0.510270	+	0.860014i	$0.670455\pi$
$740$	−0.700646	−	2.74511i	−0.700646	−	2.74511i
$741$		0			0
$742$		0			0
$743$		0			0		0.731895	−	0.681418i	$-0.238636\pi$
$743$		0			0		−0.731895	+	0.681418i	$0.761364\pi$
$744$		0			0
$745$	−0.710502	−	1.19749i	−0.710502	−	1.19749i
$746$	−0.847507	+	0.847507i	−0.847507	+	0.847507i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$751$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$752$		0			0
$753$		0			0
$754$	0.498278	−	2.29055i	0.498278	−	2.29055i
$755$		0			0
$756$		0			0
$757$	0.136899	+	0.0401971i	0.136899	+	0.0401971i	0.349464	−	0.936950i	$-0.386364\pi$
$757$	0.136899	+	0.0401971i	0.136899	+	0.0401971i	−0.212565	+	0.977147i	$0.568182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.574406	−	0.767317i	0.574406	−	0.767317i	−0.415415	−	0.909632i	$-0.636364\pi$
$761$	0.574406	−	0.767317i	0.574406	−	0.767317i	0.989821	+	0.142315i	$0.0454545\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.18636	+	0.170572i	1.18636	+	0.170572i	0.707107	−	0.707107i	$-0.250000\pi$
$769$	1.18636	+	0.170572i	1.18636	+	0.170572i	0.479249	+	0.877679i	$0.340909\pi$
$770$		0			0
$771$		0			0
$772$	0.400722	+	0.800541i	0.400722	+	0.800541i
$773$	−1.10521	+	0.282089i	−1.10521	+	0.282089i	−0.755750	−	0.654861i	$-0.772727\pi$
$773$	−1.10521	+	0.282089i	−1.10521	+	0.282089i	−0.349464	+	0.936950i	$0.613636\pi$
$774$		0			0
$775$		0			0
$776$	0.420803	−	0.0605024i	0.420803	−	0.0605024i
$777$		0			0
$778$	1.41832	+	1.32050i	1.41832	+	1.32050i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.877679	−	0.479249i	0.877679	−	0.479249i
$785$	−1.45027	+	3.17565i	−1.45027	+	3.17565i
$786$		0			0
$787$		0			0		0.0356923	−	0.999363i	$-0.488636\pi$
$787$		0			0		−0.0356923	+	0.999363i	$0.511364\pi$
$788$	−1.46907	+	1.01999i	−1.46907	+	1.01999i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.95451	−	1.69359i	−1.95451	−	1.69359i
$794$	0.0677316	+	0.0225432i	0.0677316	+	0.0225432i
$795$		0			0
$796$		0			0
$797$	−0.587976	+	0.377869i	−0.587976	+	0.377869i	−0.800541	−	0.599278i	$-0.795455\pi$
$797$	−0.587976	+	0.377869i	−0.587976	+	0.377869i	0.212565	+	0.977147i	$0.431818\pi$
$798$		0			0
$799$		0			0
$800$	−2.30972			−2.30972
$801$		0			0
$802$	−1.08128			−1.08128
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−0.336934	−	0.112142i	−0.336934	−	0.112142i
$809$	1.41620	+	1.22714i	1.41620	+	1.22714i	0.936950	+	0.349464i	$0.113636\pi$
$809$	1.41620	+	1.22714i	1.41620	+	1.22714i	0.479249	+	0.877679i	$0.340909\pi$
$810$		0			0
$811$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$811$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.32661	−	0.724384i	1.32661	−	0.724384i
$819$		0			0
$820$	−0.106676	−	2.98686i	−0.106676	−	2.98686i
$821$	1.12299	+	1.50013i	1.12299	+	1.50013i	0.841254	+	0.540641i	$0.181818\pi$
$821$	1.12299	+	1.50013i	1.12299	+	1.50013i	0.281733	+	0.959493i	$0.409091\pi$
$822$		0			0
$823$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$823$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.778642	−	0.627469i	$-0.784091\pi$
$827$		0			0		0.778642	+	0.627469i	$0.215909\pi$
$828$		0			0
$829$	−1.79036	+	0.456963i	−1.79036	+	0.456963i	−0.989821	−	0.142315i	$-0.954545\pi$
$829$	−1.79036	+	0.456963i	−1.79036	+	0.456963i	−0.800541	+	0.599278i	$0.795455\pi$
$830$		0			0
$831$		0			0
$832$	0.928661	+	0.997452i	0.928661	+	0.997452i
$833$	−1.58479	−	0.227858i	−1.58479	−	0.227858i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.948826	−	0.315800i	$-0.102273\pi$
$839$		0			0		−0.948826	+	0.315800i	$0.897727\pi$
$840$		0			0
$841$	−1.17368	+	1.56786i	−1.17368	+	1.56786i
$842$	1.64179	+	1.13992i	1.64179	+	1.13992i
$843$		0			0
$844$		0			0
$845$	−1.49652	−	0.439417i	−1.49652	−	0.439417i
$846$		0			0
$847$		0			0
$848$	0.373128	−	1.71524i	0.373128	−	1.71524i
$849$		0			0
$850$	2.96045	+	2.21616i	2.96045	+	2.21616i
$851$		0			0
$852$		0			0
$853$	0.707107	−	1.70711i	0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$853$	0.707107	−	1.70711i	0.707107	−	1.70711i	0.707107	−	0.707107i	$-0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.01469	+	1.71017i	1.01469	+	1.71017i	0.599278	+	0.800541i	$0.295455\pi$
$857$	1.01469	+	1.71017i	1.01469	+	1.71017i	0.415415	+	0.909632i	$0.363636\pi$
$858$		0			0
$859$		0			0		0.731895	−	0.681418i	$-0.238636\pi$
$859$		0			0		−0.731895	+	0.681418i	$0.761364\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.510270	−	0.860014i	$-0.329545\pi$
$863$		0			0		−0.510270	+	0.860014i	$0.670455\pi$
$864$		0			0
$865$	0.391340	+	2.72183i	0.391340	+	2.72183i
$866$	0.787435	+	0.977147i	0.787435	+	0.977147i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.07853	−	0.0771377i	−1.07853	−	0.0771377i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.936950	+	1.34946i	0.936950	+	1.34946i	0.936950	+	0.349464i	$0.113636\pi$
$877$	0.936950	+	1.34946i	0.936950	+	1.34946i			1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.12299	+	0.418852i	1.12299	+	0.418852i	0.841254	−	0.540641i	$-0.181818\pi$
$881$	1.12299	+	0.418852i	1.12299	+	0.418852i	0.281733	+	0.959493i	$0.409091\pi$
$882$		0			0
$883$		0			0		0.315800	−	0.948826i	$-0.397727\pi$
$883$		0			0		−0.315800	+	0.948826i	$0.602273\pi$
$884$	−0.233247	−	2.16951i	−0.233247	−	2.16951i
$885$		0			0
$886$		0			0
$887$		0			0		0.177553	−	0.984111i	$-0.443182\pi$
$887$		0			0		−0.177553	+	0.984111i	$0.556818\pi$
$888$		0			0
$889$		0			0
$890$	−1.74557	+	0.512546i	−1.74557	+	0.512546i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.627683	−	0.724384i	0.627683	−	0.724384i
$899$		0			0
$900$		0			0
$901$	−2.12401	+	1.84047i	−2.12401	+	1.84047i
$902$		0			0
$903$		0			0
$904$	−1.25414	−	0.0447917i	−1.25414	−	0.0447917i
$905$	3.23502	+	0.347801i	3.23502	+	0.347801i
$906$		0			0
$907$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$907$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$911$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$912$		0			0
$913$		0			0
$914$	0.276501	−	0.222818i	0.276501	−	0.222818i
$915$		0			0
$916$	0.134147	−	0.166466i	0.134147	−	0.166466i
$917$		0			0
$918$		0			0
$919$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$919$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$920$		0			0
$921$		0			0
$922$	0.215109	−	1.49611i	0.215109	−	1.49611i
$923$		0			0
$924$		0			0
$925$	3.48516	+	0.889534i	3.48516	+	0.889534i
$926$		0			0
$927$		0			0
$928$	−0.543184	−	1.63201i	−0.543184	−	1.63201i
$929$	−0.635768	+	0.290345i	−0.635768	+	0.290345i	−0.707107	−	0.707107i	$-0.750000\pi$
$929$	−0.635768	+	0.290345i	−0.635768	+	0.290345i	0.0713392	+	0.997452i	$0.477273\pi$
$930$		0			0
$931$		0			0
$932$	0.278125	+	0.0605024i	0.278125	+	0.0605024i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.415415	+	0.0903680i	0.415415	+	0.0903680i	0.415415	−	0.909632i	$-0.363636\pi$
$937$	0.415415	+	0.0903680i	0.415415	+	0.0903680i			1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.627980	−	1.88678i	−0.627980	−	1.88678i	−0.415415	−	0.909632i	$-0.636364\pi$
$941$	−0.627980	−	1.88678i	−0.627980	−	1.88678i	−0.212565	−	0.977147i	$-0.568182\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$947$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$948$		0			0
$949$	2.25655	−	0.407126i	2.25655	−	0.407126i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.222818	−	0.276501i	0.222818	−	0.276501i	−0.654861	−	0.755750i	$-0.727273\pi$
$953$	0.222818	−	0.276501i	0.222818	−	0.276501i	0.877679	+	0.479249i	$0.159091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.0713392	+	0.997452i	−0.0713392	+	0.997452i
$962$	−1.01712	−	1.86272i	−1.01712	−	1.86272i
$963$		0			0
$964$	−0.760982	−	0.0818140i	−0.760982	−	0.0818140i
$965$	−1.62763	−	0.0581310i	−1.62763	−	0.0581310i
$966$		0			0
$967$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$967$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$968$	0.755750	−	0.654861i	0.755750	−	0.654861i
$969$		0			0
$970$	−0.270284	+	0.724660i	−0.270284	+	0.724660i
$971$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$971$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−1.86750	−	0.336934i	−1.86750	−	0.336934i
$977$	−1.79799	+	0.527938i	−1.79799	+	0.527938i	−0.997452	−	0.0713392i	$-0.977273\pi$
$977$	−1.79799	+	0.527938i	−1.79799	+	0.527938i	−0.800541	+	0.599278i	$0.795455\pi$
$978$		0			0
$979$		0			0
$980$			1.81926i			1.81926i
$981$		0			0
$982$		0			0
$983$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$983$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$984$		0			0
$985$	−0.347801	−	3.23502i	−0.347801	−	3.23502i
$986$	−0.869683	+	2.61298i	−0.869683	+	2.61298i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$991$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.94931	+	0.139418i	1.94931	+	0.139418i	0.989821	−	0.142315i	$-0.0454545\pi$
$997$	1.94931	+	0.139418i	1.94931	+	0.139418i	0.959493	+	0.281733i	$0.0909091\pi$
$998$		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.647.1	yes	40
3.2	odd	2		3204.1.cd.a.647.1	✓	40
4.3	odd	2	CM	3204.1.cd.b.647.1	yes	40
12.11	even	2		3204.1.cd.a.647.1	✓	40
89.26	odd	88		3204.1.cd.a.827.1	yes	40
267.26	even	88	inner	3204.1.cd.b.827.1	yes	40
356.115	even	88		3204.1.cd.a.827.1	yes	40
1068.827	odd	88	inner	3204.1.cd.b.827.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3204.1.cd.a.647.1	✓	40	3.2	odd	2
3204.1.cd.a.647.1	✓	40	12.11	even	2
3204.1.cd.a.827.1	yes	40	89.26	odd	88
3204.1.cd.a.827.1	yes	40	356.115	even	88
3204.1.cd.b.647.1	yes	40	1.1	even	1	trivial
3204.1.cd.b.647.1	yes	40	4.3	odd	2	CM
3204.1.cd.b.827.1	yes	40	267.26	even	88	inner
3204.1.cd.b.827.1	yes	40	1068.827	odd	88	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.281733 + 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-0.129785 - 1.81463i) q^{5} +(-0.755750 - 0.654861i) q^{8} +O(q^{10})\)	\(q+(0.281733 + 0.959493i) q^{2} +(-0.841254 + 0.540641i) q^{4} +(-0.129785 - 1.81463i) q^{5} +(-0.755750 - 0.654861i) q^{8} +(1.70456 - 0.635768i) q^{10} +(1.11946 - 0.777256i) q^{13} +(0.415415 - 0.909632i) q^{16} +(-1.40524 + 0.767317i) q^{17} +(1.09024 + 1.45640i) q^{20} +(-2.28621 + 0.328708i) q^{25} +(1.06116 + 0.855137i) q^{26} +(-0.769914 - 1.53809i) q^{29} +(0.989821 + 0.142315i) q^{32} +(-1.13214 - 1.13214i) q^{34} +(-1.43874 - 0.595946i) q^{37} +(-1.09024 + 1.45640i) q^{40} +(-1.34946 - 0.936950i) q^{41} +(0.800541 + 0.599278i) q^{49} +(-0.959493 - 2.10100i) q^{50} +(-0.521535 + 1.25910i) q^{52} +(1.71524 - 0.373128i) q^{53} +(1.25888 - 1.17206i) q^{58} +(-0.469302 - 1.83871i) q^{61} +(0.142315 + 0.989821i) q^{64} +(-1.55572 - 1.93053i) q^{65} +(0.767317 - 1.40524i) q^{68} +(1.53046 + 0.698939i) q^{73} +(0.166466 - 1.54836i) q^{74} +(-1.70456 - 0.635768i) q^{80} +(0.518809 - 1.55877i) q^{82} +(1.57477 + 2.45040i) q^{85} +(-0.997452 - 0.0713392i) q^{89} +(-0.278401 + 0.321292i) q^{97} +(-0.349464 + 0.936950i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{4}+O(q^{10}) \) 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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