LMFDB - Embedded newform 3204.1.cd.b.1223.1

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			1223.1
Root			$-0.212565 + 0.977147i$ of defining polynomial
Character	$\chi$	$=$	3204.1223
Dual form			3204.1.cd.b.503.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.909632 - 0.415415i) q^{2} +(0.654861 - 0.755750i) q^{4} +(-1.73749 + 0.948742i) q^{5} +(0.281733 - 0.959493i) q^{8} +O(q^{10})$	\(q+(0.909632 - 0.415415i) q^{2} +(0.654861 - 0.755750i) q^{4} +(-1.73749 + 0.948742i) q^{5} +(0.281733 - 0.959493i) q^{8} +(-1.18636 + 1.58479i) q^{10} +(0.156199 + 0.469302i) q^{13} +(-0.142315 - 0.989821i) q^{16} +(0.682956 + 1.83107i) q^{17} +(-0.420803 + 1.93440i) q^{20} +(1.57812 - 2.45561i) q^{25} +(0.337038 + 0.362005i) q^{26} +(1.53254 + 1.23500i) q^{29} +(-0.540641 - 0.841254i) q^{32} +(1.38189 + 1.38189i) q^{34} +(-0.521535 + 1.25910i) q^{37} +(0.420803 + 1.93440i) q^{40} +(-0.199459 + 0.599278i) q^{41} +(0.977147 - 0.212565i) q^{49} +(0.415415 - 2.88927i) q^{50} +(0.456963 + 0.189280i) q^{52} +(-0.697148 - 0.0498610i) q^{53} +(1.90708 + 0.486754i) q^{58} +(1.63343 + 1.13411i) q^{61} +(-0.841254 - 0.540641i) q^{64} +(-0.716640 - 0.667215i) q^{65} +(1.83107 + 0.682956i) q^{68} +(-1.29639 + 0.186393i) q^{73} +(0.0486428 + 1.36197i) q^{74} +(1.18636 + 1.58479i) q^{80} +(0.0675149 + 0.627980i) q^{82} +(-2.92385 - 2.53353i) q^{85} +(0.479249 - 0.877679i) q^{89} +(0.136899 + 0.0401971i) q^{97} +(0.800541 - 0.599278i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/3204\mathbb{Z}\right)^\times$.

$n$	$181$	$713$	$1603$
$\chi(n)$	$e\left(\frac{13}{88}\right)$	$-1$	$-1$

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
$5$	−1.73749	+	0.948742i	−1.73749	+	0.948742i	−0.800541	+	0.599278i	$0.795455\pi$
$5$	−1.73749	+	0.948742i	−1.73749	+	0.948742i	−0.936950	+	0.349464i	$0.886364\pi$
$6$		0			0
$7$		0			0		0.994270	−	0.106895i	$-0.0340909\pi$
$7$		0			0		−0.994270	+	0.106895i	$0.965909\pi$
$8$	0.281733	−	0.959493i	0.281733	−	0.959493i
$9$		0			0
$10$	−1.18636	+	1.58479i	−1.18636	+	1.58479i
$11$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$11$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$12$		0			0
$13$	0.156199	+	0.469302i	0.156199	+	0.469302i	0.997452	−	0.0713392i	$-0.0227273\pi$
$13$	0.156199	+	0.469302i	0.156199	+	0.469302i	−0.841254	+	0.540641i	$0.818182\pi$
$14$		0			0
$15$		0			0
$16$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$17$	0.682956	+	1.83107i	0.682956	+	1.83107i	0.540641	+	0.841254i	$0.318182\pi$
$17$	0.682956	+	1.83107i	0.682956	+	1.83107i	0.142315	+	0.989821i	$0.454545\pi$
$18$		0			0
$19$		0			0		−0.860014	−	0.510270i	$-0.829545\pi$
$19$		0			0		0.860014	+	0.510270i	$0.170455\pi$
$20$	−0.420803	+	1.93440i	−0.420803	+	1.93440i
$21$		0			0
$22$		0			0
$23$		0			0		0.968937	−	0.247307i	$-0.0795455\pi$
$23$		0			0		−0.968937	+	0.247307i	$0.920455\pi$
$24$		0			0
$25$	1.57812	−	2.45561i	1.57812	−	2.45561i
$26$	0.337038	+	0.362005i	0.337038	+	0.362005i
$27$		0			0
$28$		0			0
$29$	1.53254	+	1.23500i	1.53254	+	1.23500i	0.877679	+	0.479249i	$0.159091\pi$
$29$	1.53254	+	1.23500i	1.53254	+	1.23500i	0.654861	+	0.755750i	$0.272727\pi$
$30$		0			0
$31$		0			0		0.247307	−	0.968937i	$-0.420455\pi$
$31$		0			0		−0.247307	+	0.968937i	$0.579545\pi$
$32$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$33$		0			0
$34$	1.38189	+	1.38189i	1.38189	+	1.38189i
$35$		0			0
$36$		0			0
$37$	−0.521535	+	1.25910i	−0.521535	+	1.25910i	0.415415	+	0.909632i	$0.363636\pi$
$37$	−0.521535	+	1.25910i	−0.521535	+	1.25910i	−0.936950	+	0.349464i	$0.886364\pi$
$38$		0			0
$39$		0			0
$40$	0.420803	+	1.93440i	0.420803	+	1.93440i
$41$	−0.199459	+	0.599278i	−0.199459	+	0.599278i	0.800541	+	0.599278i	$0.204545\pi$
$41$	−0.199459	+	0.599278i	−0.199459	+	0.599278i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$43$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$47$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$48$		0			0
$49$	0.977147	−	0.212565i	0.977147	−	0.212565i
$50$	0.415415	−	2.88927i	0.415415	−	2.88927i
$51$		0			0
$52$	0.456963	+	0.189280i	0.456963	+	0.189280i
$53$	−0.697148	−	0.0498610i	−0.697148	−	0.0498610i	−0.281733	−	0.959493i	$-0.590909\pi$
$53$	−0.697148	−	0.0498610i	−0.697148	−	0.0498610i	−0.415415	+	0.909632i	$0.636364\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.90708	+	0.486754i	1.90708	+	0.486754i
$59$		0			0		0.894225	−	0.447617i	$-0.147727\pi$
$59$		0			0		−0.894225	+	0.447617i	$0.852273\pi$
$60$		0			0
$61$	1.63343	+	1.13411i	1.63343	+	1.13411i	0.877679	+	0.479249i	$0.159091\pi$
$61$	1.63343	+	1.13411i	1.63343	+	1.13411i	0.755750	+	0.654861i	$0.227273\pi$
$62$		0			0
$63$		0			0
$64$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$65$	−0.716640	−	0.667215i	−0.716640	−	0.667215i
$66$		0			0
$67$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$67$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$68$	1.83107	+	0.682956i	1.83107	+	0.682956i
$69$		0			0
$70$		0			0
$71$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$71$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$72$		0			0
$73$	−1.29639	+	0.186393i	−1.29639	+	0.186393i	−0.755750	−	0.654861i	$-0.772727\pi$
$73$	−1.29639	+	0.186393i	−1.29639	+	0.186393i	−0.540641	+	0.841254i	$0.681818\pi$
$74$	0.0486428	+	1.36197i	0.0486428	+	1.36197i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$79$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$80$	1.18636	+	1.58479i	1.18636	+	1.58479i
$81$		0			0
$82$	0.0675149	+	0.627980i	0.0675149	+	0.627980i
$83$		0			0		0.0356923	−	0.999363i	$-0.488636\pi$
$83$		0			0		−0.0356923	+	0.999363i	$0.511364\pi$
$84$		0			0
$85$	−2.92385	−	2.53353i	−2.92385	−	2.53353i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.479249	−	0.877679i	0.479249	−	0.877679i
$89$	0.479249	−	0.877679i	0.479249	−	0.877679i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.136899	+	0.0401971i	0.136899	+	0.0401971i	0.349464	−	0.936950i	$-0.386364\pi$
$97$	0.136899	+	0.0401971i	0.136899	+	0.0401971i	−0.212565	+	0.977147i	$0.568182\pi$
$98$	0.800541	−	0.599278i	0.800541	−	0.599278i
$99$		0			0
$100$	−0.822373	−	2.80075i	−0.822373	−	2.80075i
$101$	−0.684410	−	1.65231i	−0.684410	−	1.65231i	−0.755750	−	0.654861i	$-0.772727\pi$
$101$	−0.684410	−	1.65231i	−0.684410	−	1.65231i	0.0713392	−	0.997452i	$-0.477273\pi$
$102$		0			0
$103$		0			0		−0.510270	−	0.860014i	$-0.670455\pi$
$103$		0			0		0.510270	+	0.860014i	$0.329545\pi$
$104$	0.494298	−	0.0176539i	0.494298	−	0.0176539i
$105$		0			0
$106$	−0.654861	+	0.244250i	−0.654861	+	0.244250i
$107$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$107$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$108$		0			0
$109$	−1.47696	−	0.321292i	−1.47696	−	0.321292i	−0.599278	−	0.800541i	$-0.704545\pi$
$109$	−1.47696	−	0.321292i	−1.47696	−	0.321292i	−0.877679	+	0.479249i	$0.840909\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.997452	−	1.07134i	0.997452	−	1.07134i		−	1.00000i	$-0.5\pi$
$113$	0.997452	−	1.07134i	0.997452	−	1.07134i	0.997452	−	0.0713392i	$-0.0227273\pi$
$114$		0			0
$115$		0			0
$116$	1.93695	−	0.349464i	1.93695	−	0.349464i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.841254	−	0.540641i	0.841254	−	0.540641i
$122$	1.95695	+	0.353072i	1.95695	+	0.353072i
$123$		0			0
$124$		0			0
$125$	−0.271011	+	3.78923i	−0.271011	+	3.78923i
$126$		0			0
$127$		0			0		0.106895	−	0.994270i	$-0.465909\pi$
$127$		0			0		−0.106895	+	0.994270i	$0.534091\pi$
$128$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$129$		0			0
$130$	−0.929050	−	0.309217i	−0.929050	−	0.309217i
$131$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$131$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.94931	−	0.139418i	1.94931	−	0.139418i
$137$	−1.55877	−	0.518809i	−1.55877	−	0.518809i	−0.599278	−	0.800541i	$-0.704545\pi$
$137$	−1.55877	−	0.518809i	−1.55877	−	0.518809i	−0.959493	+	0.281733i	$0.909091\pi$
$138$		0			0
$139$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$139$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−3.83447	−	0.691814i	−3.83447	−	0.691814i
$146$	−1.10181	+	0.708089i	−1.10181	+	0.708089i
$147$		0			0
$148$	0.610029	+	1.21868i	0.610029	+	1.21868i
$149$	−1.15941	+	1.43874i	−1.15941	+	1.43874i	−0.281733	+	0.959493i	$0.590909\pi$
$149$	−1.15941	+	1.43874i	−1.15941	+	1.43874i	−0.877679	+	0.479249i	$0.840909\pi$
$150$		0			0
$151$		0			0		0.984111	−	0.177553i	$-0.0568182\pi$
$151$		0			0		−0.984111	+	0.177553i	$0.943182\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.290345	+	0.778446i	−0.290345	+	0.778446i	0.707107	+	0.707107i	$0.250000\pi$
$157$	−0.290345	+	0.778446i	−0.290345	+	0.778446i	−0.997452	+	0.0713392i	$0.977273\pi$
$158$		0			0
$159$		0			0
$160$	1.73749	+	0.948742i	1.73749	+	0.948742i
$161$		0			0
$162$		0			0
$163$		0			0		0.999363	−	0.0356923i	$-0.0113636\pi$
$163$		0			0		−0.999363	+	0.0356923i	$0.988636\pi$
$164$	0.322286	+	0.543184i	0.322286	+	0.543184i
$165$		0			0
$166$		0			0
$167$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$167$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$168$		0			0
$169$	0.604695	−	0.452669i	0.604695	−	0.452669i
$170$	−3.71209	−	1.08997i	−3.71209	−	1.08997i
$171$		0			0
$172$		0			0
$173$	−0.270040	−	0.494541i	−0.270040	−	0.494541i	0.707107	−	0.707107i	$-0.250000\pi$
$173$	−0.270040	−	0.494541i	−0.270040	−	0.494541i	−0.977147	+	0.212565i	$0.931818\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.0713392	−	0.997452i	0.0713392	−	0.997452i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$		0			0
$181$	1.12220	+	0.561732i	1.12220	+	0.561732i	0.909632	−	0.415415i	$-0.136364\pi$
$181$	1.12220	+	0.561732i	1.12220	+	0.561732i	0.212565	+	0.977147i	$0.431818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.288395	−	2.68247i	−0.288395	−	2.68247i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.948826	−	0.315800i	$-0.102273\pi$
$191$		0			0		−0.948826	+	0.315800i	$0.897727\pi$
$192$		0			0
$193$	0.0555831	+	1.55629i	0.0555831	+	1.55629i	0.654861	+	0.755750i	$0.272727\pi$
$193$	0.0555831	+	1.55629i	0.0555831	+	1.55629i	−0.599278	+	0.800541i	$0.704545\pi$
$194$	0.141226	−	0.0203052i	0.141226	−	0.0203052i
$195$		0			0
$196$	0.479249	−	0.877679i	0.479249	−	0.877679i
$197$	0.640357	−	1.07926i	0.640357	−	1.07926i	−0.349464	−	0.936950i	$-0.613636\pi$
$197$	0.640357	−	1.07926i	0.640357	−	1.07926i	0.989821	−	0.142315i	$-0.0454545\pi$
$198$		0			0
$199$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$199$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$200$	−1.91153	−	2.20602i	−1.91153	−	2.20602i
$201$		0			0
$202$	−1.30896	−	1.21868i	−1.30896	−	1.21868i
$203$		0			0
$204$		0			0
$205$	−0.222002	−	1.23047i	−0.222002	−	1.23047i
$206$		0			0
$207$		0			0
$208$	0.442295	−	0.221397i	0.442295	−	0.221397i
$209$		0			0
$210$		0			0
$211$		0			0		0.177553	−	0.984111i	$-0.443182\pi$
$211$		0			0		−0.177553	+	0.984111i	$0.556818\pi$
$212$	−0.494217	+	0.494217i	−0.494217	+	0.494217i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.47696	+	0.321292i	−1.47696	+	0.321292i
$219$		0			0
$220$		0			0
$221$	−0.752650	+	0.606524i	−0.752650	+	0.606524i
$222$		0			0
$223$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$223$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$224$		0			0
$225$		0			0
$226$	0.462264	−	1.38888i	0.462264	−	1.38888i
$227$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$227$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$228$		0			0
$229$	0.0709757	+	0.00763067i	0.0709757	+	0.00763067i	0.142315	−	0.989821i	$-0.454545\pi$
$229$	0.0709757	+	0.00763067i	0.0709757	+	0.00763067i	−0.0713392	+	0.997452i	$0.522727\pi$
$230$		0			0
$231$		0			0
$232$	1.61674	−	1.12252i	1.61674	−	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		0.570323	−	0.821421i	$-0.306818\pi$
$239$		0			0		−0.570323	+	0.821421i	$0.693182\pi$
$240$		0			0
$241$	−1.25910	−	1.35236i	−1.25910	−	1.35236i	−0.909632	−	0.415415i	$-0.863636\pi$
$241$	−1.25910	−	1.35236i	−1.25910	−	1.35236i	−0.349464	−	0.936950i	$-0.613636\pi$
$242$	0.540641	−	0.841254i	0.540641	−	0.841254i
$243$		0			0
$244$	1.92677	−	0.491779i	1.92677	−	0.491779i
$245$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	1.32758	+	3.55938i	1.32758	+	3.55938i
$251$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$251$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$257$	0.497898	−	0.665114i	0.497898	−	0.665114i	−0.479249	−	0.877679i	$-0.659091\pi$
$257$	0.497898	−	0.665114i	0.497898	−	0.665114i	0.977147	+	0.212565i	$0.0681818\pi$
$258$		0			0
$259$		0			0
$260$	−0.973547	+	0.104667i	−0.973547	+	0.104667i
$261$		0			0
$262$		0			0
$263$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$263$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$264$		0			0
$265$	1.25859	−	0.574780i	1.25859	−	0.574780i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$269$	−0.983568	+	0.449181i	−0.983568	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$270$		0			0
$271$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$271$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$272$	1.71524	−	0.936593i	1.71524	−	0.936593i
$273$		0			0
$274$	−1.63343	+	0.175612i	−1.63343	+	0.175612i
$275$		0			0
$276$		0			0
$277$	0.574406	−	0.767317i	0.574406	−	0.767317i	−0.415415	−	0.909632i	$-0.636364\pi$
$277$	0.574406	−	0.767317i	0.574406	−	0.767317i	0.989821	+	0.142315i	$0.0454545\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.63201	−	0.968315i	1.63201	−	0.968315i	0.654861	−	0.755750i	$-0.272727\pi$
$281$	1.63201	−	0.968315i	1.63201	−	0.968315i	0.977147	−	0.212565i	$-0.0681818\pi$
$282$		0			0
$283$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$283$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.13066	+	1.84623i	−2.13066	+	1.84623i
$290$	−3.77535	+	0.963599i	−3.77535	+	0.963599i
$291$		0			0
$292$	−0.708089	+	1.10181i	−0.708089	+	1.10181i
$293$	0.241976	+	0.259900i	0.241976	+	0.259900i	0.841254	−	0.540641i	$-0.181818\pi$
$293$	0.241976	+	0.259900i	0.241976	+	0.259900i	−0.599278	+	0.800541i	$0.704545\pi$
$294$		0			0
$295$		0			0
$296$	1.06116	+	0.855137i	1.06116	+	0.855137i
$297$		0			0
$298$	−0.456963	+	1.79036i	−0.456963	+	1.79036i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−3.91404	−	0.420803i	−3.91404	−	0.420803i
$306$		0			0
$307$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$307$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$311$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$312$		0			0
$313$	0.888154	−	0.715720i	0.888154	−	0.715720i	−0.0713392	−	0.997452i	$-0.522727\pi$
$313$	0.888154	−	0.715720i	0.888154	−	0.715720i	0.959493	+	0.281733i	$0.0909091\pi$
$314$	0.0592707	+	0.828713i	0.0592707	+	0.828713i
$315$		0			0
$316$		0			0
$317$	0.136408	−	0.948742i	0.136408	−	0.948742i	−0.800541	−	0.599278i	$-0.795455\pi$
$317$	0.136408	−	0.948742i	0.136408	−	0.948742i	0.936950	−	0.349464i	$-0.113636\pi$
$318$		0			0
$319$		0			0
$320$	1.97460	+	0.141226i	1.97460	+	0.141226i
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.39892	+	0.357053i	1.39892	+	0.357053i
$326$		0			0
$327$		0			0
$328$	0.518809	+	0.360215i	0.518809	+	0.360215i
$329$		0			0
$330$		0			0
$331$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$331$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.877679	−	1.47925i	0.877679	−	1.47925i		−	1.00000i	$-0.5\pi$
$337$	0.877679	−	1.47925i	0.877679	−	1.47925i	0.877679	−	0.479249i	$-0.159091\pi$
$338$	0.362005	−	0.662962i	0.362005	−	0.662962i
$339$		0			0
$340$	−3.82942	+	0.550588i	−3.82942	+	0.550588i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.451077	−	0.337672i	−0.451077	−	0.337672i
$347$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$347$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$348$		0			0
$349$	−0.0379591	−	0.353072i	−0.0379591	−	0.353072i	−0.997452	−	0.0713392i	$-0.977273\pi$
$349$	−0.0379591	−	0.353072i	−0.0379591	−	0.353072i	0.959493	−	0.281733i	$-0.0909091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.800541	+	0.400722i	0.800541	+	0.400722i	0.800541	−	0.599278i	$-0.204545\pi$
$353$	0.800541	+	0.400722i	0.800541	+	0.400722i			1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$	−0.349464	−	0.936950i	−0.349464	−	0.936950i
$357$		0			0
$358$		0			0
$359$		0			0		0.447617	−	0.894225i	$-0.352273\pi$
$359$		0			0		−0.447617	+	0.894225i	$0.647727\pi$
$360$		0			0
$361$	0.479249	+	0.877679i	0.479249	+	0.877679i
$362$	1.25414	+	0.0447917i	1.25414	+	0.0447917i
$363$		0			0
$364$		0			0
$365$	2.07563	−	1.55380i	2.07563	−	1.55380i
$366$		0			0
$367$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$367$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$368$		0			0
$369$		0			0
$370$	−1.37667	−	2.32026i	−1.37667	−	2.32026i
$371$		0			0
$372$		0			0
$373$	−0.398326	+	0.148568i	−0.398326	+	0.148568i	−0.540641	−	0.841254i	$-0.681818\pi$
$373$	−0.398326	+	0.148568i	−0.398326	+	0.148568i	0.142315	+	0.989821i	$0.454545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.340206	+	0.912128i	−0.340206	+	0.912128i
$378$		0			0
$379$		0			0		−0.247307	−	0.968937i	$-0.579545\pi$
$379$		0			0		0.247307	+	0.968937i	$0.420455\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.984111	−	0.177553i	$-0.0568182\pi$
$383$		0			0		−0.984111	+	0.177553i	$0.943182\pi$
$384$		0			0
$385$		0			0
$386$	0.697067	+	1.39256i	0.697067	+	1.39256i
$387$		0			0
$388$	0.120029	−	0.0771377i	0.120029	−	0.0771377i
$389$	−1.12252	−	0.202525i	−1.12252	−	0.202525i	−0.415415	−	0.909632i	$-0.636364\pi$
$389$	−1.12252	−	0.202525i	−1.12252	−	0.202525i	−0.707107	+	0.707107i	$0.750000\pi$
$390$		0			0
$391$		0			0
$392$	0.0713392	−	0.997452i	0.0713392	−	0.997452i
$393$		0			0
$394$	0.134147	−	1.24775i	0.134147	−	1.24775i
$395$		0			0
$396$		0			0
$397$	−1.63201	−	0.543184i	−1.63201	−	0.543184i	−0.654861	−	0.755750i	$-0.727273\pi$
$397$	−1.63201	−	0.543184i	−1.63201	−	0.543184i	−0.977147	+	0.212565i	$0.931818\pi$
$398$		0			0
$399$		0			0
$400$	−2.65520	−	1.21259i	−2.65520	−	1.21259i
$401$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$401$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$402$		0			0
$403$		0			0
$404$	−1.69693	−	0.564792i	−1.69693	−	0.564792i
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.0401971	+	0.562029i	−0.0401971	+	0.562029i	0.936950	+	0.349464i	$0.113636\pi$
$409$	−0.0401971	+	0.562029i	−0.0401971	+	0.562029i	−0.977147	+	0.212565i	$0.931818\pi$
$410$	−0.713098	−	1.02706i	−0.713098	−	1.02706i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.310354	−	0.385126i	0.310354	−	0.385126i
$417$		0			0
$418$		0			0
$419$		0			0		0.731895	−	0.681418i	$-0.238636\pi$
$419$		0			0		−0.731895	+	0.681418i	$0.761364\pi$
$420$		0			0
$421$	0.695414	−	0.746928i	0.695414	−	0.746928i	−0.281733	−	0.959493i	$-0.590909\pi$
$421$	0.695414	−	0.746928i	0.695414	−	0.746928i	0.977147	+	0.212565i	$0.0681818\pi$
$422$		0			0
$423$		0			0
$424$	−0.244250	+	0.654861i	−0.244250	+	0.654861i
$425$	5.57419	+	1.21259i	5.57419	+	1.21259i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.510270	−	0.860014i	$-0.670455\pi$
$431$		0			0		0.510270	+	0.860014i	$0.329545\pi$
$432$		0			0
$433$	0.560168	+	1.35236i	0.560168	+	1.35236i	0.909632	+	0.415415i	$0.136364\pi$
$433$	0.560168	+	1.35236i	0.560168	+	1.35236i	−0.349464	+	0.936950i	$0.613636\pi$
$434$		0			0
$435$		0			0
$436$	−1.21002	+	0.905808i	−1.21002	+	0.905808i
$437$		0			0
$438$		0			0
$439$		0			0		−0.999363	−	0.0356923i	$-0.988636\pi$
$439$		0			0		0.999363	+	0.0356923i	$0.0113636\pi$
$440$		0			0
$441$		0			0
$442$	−0.432675	+	0.864375i	−0.432675	+	0.864375i
$443$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$443$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$444$		0			0
$445$			1.97964i			1.97964i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.41620	+	1.22714i	1.41620	+	1.22714i	0.936950	+	0.349464i	$0.113636\pi$
$449$	1.41620	+	1.22714i	1.41620	+	1.22714i	0.479249	+	0.877679i	$0.340909\pi$
$450$		0			0
$451$		0			0
$452$	−0.156472	−	1.45540i	−0.156472	−	1.45540i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.65231	+	0.684410i	−1.65231	+	0.684410i	−0.997452	−	0.0713392i	$-0.977273\pi$
$457$	−1.65231	+	0.684410i	−1.65231	+	0.684410i	−0.654861	+	0.755750i	$0.727273\pi$
$458$	0.0677316	−	0.0225432i	0.0677316	−	0.0225432i
$459$		0			0
$460$		0			0
$461$	0.557730	−	0.0801894i	0.557730	−	0.0801894i	0.142315	−	0.989821i	$-0.454545\pi$
$461$	0.557730	−	0.0801894i	0.557730	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$462$		0			0
$463$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$463$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$464$	1.00432	−	1.69270i	1.00432	−	1.69270i
$465$		0			0
$466$	−1.57642	−	0.587976i	−1.57642	−	0.587976i
$467$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$467$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$468$		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
$481$	−0.672359	−	0.0480881i	−0.672359	−	0.0480881i
$482$	−1.70711	−	0.707107i	−1.70711	−	0.707107i
$483$		0			0
$484$	0.142315	−	0.989821i	0.142315	−	0.989821i
$485$	−0.275997	+	0.0600395i	−0.275997	+	0.0600395i
$486$		0			0
$487$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$487$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$488$	1.54836	−	1.24775i	1.54836	−	1.24775i
$489$		0			0
$490$	−0.822373	+	1.80075i	−0.822373	+	1.80075i
$491$		0			0		−0.627469	−	0.778642i	$-0.715909\pi$
$491$		0			0		0.627469	+	0.778642i	$0.284091\pi$
$492$		0			0
$493$	−1.21472	+	3.64964i	−1.21472	+	3.64964i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.821421	−	0.570323i	$-0.193182\pi$
$499$		0			0		−0.821421	+	0.570323i	$0.806818\pi$
$500$	2.68623	+	2.68623i	2.68623	+	2.68623i
$501$		0			0
$502$		0			0
$503$		0			0		0.247307	−	0.968937i	$-0.420455\pi$
$503$		0			0		−0.247307	+	0.968937i	$0.579545\pi$
$504$		0			0
$505$	2.75678	+	2.22155i	2.75678	+	2.22155i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$509$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	−0.989821	+	0.142315i	$0.954545\pi$
$510$		0			0
$511$		0			0
$512$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$513$		0			0
$514$	0.176606	−	0.811843i	0.176606	−	0.811843i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.842089	+	0.499635i	−0.842089	+	0.499635i
$521$	−0.611980	−	1.83871i	−0.611980	−	1.83871i	−0.540641	−	0.841254i	$-0.681818\pi$
$521$	−0.611980	−	1.83871i	−0.611980	−	1.83871i	−0.0713392	−	0.997452i	$-0.522727\pi$
$522$		0			0
$523$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$523$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.877679	−	0.479249i	0.877679	−	0.479249i
$530$	0.906084	−	1.04568i	0.906084	−	1.04568i
$531$		0			0
$532$		0			0
$533$	−0.312397			−0.312397
$534$		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	−	0.166466i	1.54836	−	0.166466i	0.707107	−	0.707107i	$-0.250000\pi$
$541$	1.54836	−	0.166466i	1.54836	−	0.166466i	0.841254	+	0.540641i	$0.181818\pi$
$542$		0			0
$543$		0			0
$544$	1.17116	−	1.56449i	1.17116	−	1.56449i
$545$	2.87102	−	0.843008i	2.87102	−	0.843008i
$546$		0			0
$547$		0			0		−0.315800	−	0.948826i	$-0.602273\pi$
$547$		0			0		0.315800	+	0.948826i	$0.397727\pi$
$548$	−1.41287	+	0.838293i	−1.41287	+	0.838293i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.203743	−	0.936593i	0.203743	−	0.936593i
$555$		0			0
$556$		0			0
$557$	1.87768	−	0.479249i	1.87768	−	0.479249i	0.877679	−	0.479249i	$-0.159091\pi$
$557$	1.87768	−	0.479249i	1.87768	−	0.479249i	1.00000			$0$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.08227	−	1.55877i	1.08227	−	1.55877i
$563$		0			0		−0.778642	−	0.627469i	$-0.784091\pi$
$563$		0			0		0.778642	+	0.627469i	$0.215909\pi$
$564$		0			0
$565$	−0.716640	+	2.80777i	−0.716640	+	2.80777i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.03083	+	0.715720i	−1.03083	+	0.715720i	−0.959493	−	0.281733i	$-0.909091\pi$
$569$	−1.03083	+	0.715720i	−1.03083	+	0.715720i	−0.0713392	+	0.997452i	$0.522727\pi$
$570$		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
$577$	−0.480244	−	0.595946i	−0.480244	−	0.595946i	0.479249	−	0.877679i	$-0.340909\pi$
$577$	−0.480244	−	0.595946i	−0.480244	−	0.595946i	−0.959493	+	0.281733i	$0.909091\pi$
$578$	−1.17116	+	2.56449i	−1.17116	+	2.56449i
$579$		0			0
$580$	−3.03388	+	2.44486i	−3.03388	+	2.44486i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$585$		0			0
$586$	0.328076	+	0.135893i	0.328076	+	0.135893i
$587$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$587$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.32050	+	0.337038i	1.32050	+	0.337038i
$593$	1.39256	−	0.697067i	1.39256	−	0.697067i	0.415415	−	0.909632i	$-0.363636\pi$
$593$	1.39256	−	0.697067i	1.39256	−	0.697067i	0.977147	+	0.212565i	$0.0681818\pi$
$594$		0			0
$595$		0			0
$596$	0.328076	+	1.81840i	0.328076	+	1.81840i
$597$		0			0
$598$		0			0
$599$		0			0		−0.731895	−	0.681418i	$-0.761364\pi$
$599$		0			0		0.731895	+	0.681418i	$0.238636\pi$
$600$		0			0
$601$	−1.19136	−	1.37491i	−1.19136	−	1.37491i	−0.909632	−	0.415415i	$-0.863636\pi$
$601$	−1.19136	−	1.37491i	−1.19136	−	1.37491i	−0.281733	−	0.959493i	$-0.590909\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.948742	+	1.73749i	−0.948742	+	1.73749i
$606$		0			0
$607$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$607$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$608$		0			0
$609$		0			0
$610$	−3.73515	+	1.24318i	−3.73515	+	1.24318i
$611$		0			0
$612$		0			0
$613$	1.34692	+	1.00829i	1.34692	+	1.00829i	0.997452	+	0.0713392i	$0.0227273\pi$
$613$	1.34692	+	1.00829i	1.34692	+	1.00829i	0.349464	+	0.936950i	$0.386364\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.0613918	+	1.71893i	−0.0613918	+	1.71893i	0.479249	+	0.877679i	$0.340909\pi$
$617$	−0.0613918	+	1.71893i	−0.0613918	+	1.71893i	−0.540641	+	0.841254i	$0.681818\pi$
$618$		0			0
$619$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$619$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.91153	−	4.18567i	−1.91153	−	4.18567i
$626$	0.510572	−	1.01999i	0.510572	−	1.01999i
$627$		0			0
$628$	0.398174	+	0.729202i	0.398174	+	0.729202i
$629$	−2.66168	−	0.0950623i	−2.66168	−	0.0950623i
$630$		0			0
$631$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$631$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$632$		0			0
$633$		0			0
$634$	−0.270040	−	0.919672i	−0.270040	−	0.919672i
$635$		0			0
$636$		0			0
$637$	0.252386	+	0.425374i	0.252386	+	0.425374i
$638$		0			0
$639$		0			0
$640$	1.85483	−	0.691814i	1.85483	−	0.691814i
$641$	1.59673	+	0.871880i	1.59673	+	0.871880i	0.997452	+	0.0713392i	$0.0227273\pi$
$641$	1.59673	+	0.871880i	1.59673	+	0.871880i	0.599278	+	0.800541i	$0.295455\pi$
$642$		0			0
$643$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$643$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.681418	−	0.731895i	$-0.261364\pi$
$647$		0			0		−0.681418	+	0.731895i	$0.738636\pi$
$648$		0			0
$649$		0			0
$650$	1.42083	−	0.256346i	1.42083	−	0.256346i
$651$		0			0
$652$		0			0
$653$	−0.655217	−	1.30896i	−0.655217	−	1.30896i	−0.936950	−	0.349464i	$-0.886364\pi$
$653$	−0.655217	−	1.30896i	−0.655217	−	1.30896i	0.281733	−	0.959493i	$-0.409091\pi$
$654$		0			0
$655$		0			0
$656$	0.621564	+	0.112142i	0.621564	+	0.112142i
$657$		0			0
$658$		0			0
$659$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$659$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$660$		0			0
$661$	−0.0228531	+	0.212565i	−0.0228531	+	0.212565i	0.977147	+	0.212565i	$0.0681818\pi$
$661$	−0.0228531	+	0.212565i	−0.0228531	+	0.212565i	−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.691814	−	0.0994679i	−0.691814	−	0.0994679i	−0.212565	−	0.977147i	$-0.568182\pi$
$673$	−0.691814	−	0.0994679i	−0.691814	−	0.0994679i	−0.479249	+	0.877679i	$0.659091\pi$
$674$	0.183863	−	1.71017i	0.183863	−	1.71017i
$675$		0			0
$676$	0.0538866	−	0.753433i	0.0538866	−	0.753433i
$677$	−1.05382	−	1.51779i	−1.05382	−	1.51779i	−0.841254	−	0.540641i	$-0.818182\pi$
$677$	−1.05382	−	1.51779i	−1.05382	−	1.51779i	−0.212565	−	0.977147i	$-0.568182\pi$
$678$		0			0
$679$		0			0
$680$	−3.25464	+	2.09163i	−3.25464	+	2.09163i
$681$		0			0
$682$		0			0
$683$		0			0		0.627469	−	0.778642i	$-0.284091\pi$
$683$		0			0		−0.627469	+	0.778642i	$0.715909\pi$
$684$		0			0
$685$	3.20057	−	0.577446i	3.20057	−	0.577446i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.0854936	−	0.334961i	−0.0854936	−	0.334961i
$690$		0			0
$691$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$691$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$692$	−0.550588	−	0.119773i	−0.550588	−	0.119773i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.23354	+	0.0440561i	−1.23354	+	0.0440561i
$698$	−0.181200	−	0.305397i	−0.181200	−	0.305397i
$699$		0			0
$700$		0			0
$701$	−0.494541	−	1.68425i	−0.494541	−	1.68425i	−0.707107	−	0.707107i	$-0.750000\pi$
$701$	−0.494541	−	1.68425i	−0.494541	−	1.68425i	0.212565	−	0.977147i	$-0.431818\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.894664	+	0.0319530i	0.894664	+	0.0319530i
$707$		0			0
$708$		0			0
$709$	0.867426	−	1.73290i	0.867426	−	1.73290i	0.212565	−	0.977147i	$-0.431818\pi$
$709$	0.867426	−	1.73290i	0.867426	−	1.73290i	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		−0.106895	−	0.994270i	$-0.534091\pi$
$719$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$720$		0			0
$721$		0			0
$722$	0.800541	+	0.599278i	0.800541	+	0.599278i
$723$		0			0
$724$	1.15941	−	0.480244i	1.15941	−	0.480244i
$725$	5.45121	−	1.81434i	5.45121	−	1.81434i
$726$		0			0
$727$		0			0		−0.0356923	−	0.999363i	$-0.511364\pi$
$727$		0			0		0.0356923	+	0.999363i	$0.488636\pi$
$728$		0			0
$729$		0			0
$730$	1.24259	−	2.27563i	1.24259	−	2.27563i
$731$		0			0
$732$		0			0
$733$	−0.527938	−	0.196911i	−0.527938	−	0.196911i	0.0713392	−	0.997452i	$-0.477273\pi$
$733$	−0.527938	−	0.196911i	−0.527938	−	0.196911i	−0.599278	+	0.800541i	$0.704545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$		0			0		−0.177553	−	0.984111i	$-0.556818\pi$
$739$		0			0		0.177553	+	0.984111i	$0.443182\pi$
$740$	−2.21613	−	1.53869i	−2.21613	−	1.53869i
$741$		0			0
$742$		0			0
$743$		0			0		−0.968937	−	0.247307i	$-0.920455\pi$
$743$		0			0		0.968937	+	0.247307i	$0.0795455\pi$
$744$		0			0
$745$	0.649472	−	3.59978i	0.649472	−	3.59978i
$746$	−0.300613	+	0.300613i	−0.300613	+	0.300613i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$751$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$752$		0			0
$753$		0			0
$754$	0.0694493	+	0.971028i	0.0694493	+	0.971028i
$755$		0			0
$756$		0			0
$757$	−0.729202	+	1.59673i	−0.729202	+	1.59673i	0.0713392	+	0.997452i	$0.477273\pi$
$757$	−0.729202	+	1.59673i	−0.729202	+	1.59673i	−0.800541	+	0.599278i	$0.795455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.398326	−	1.83107i	−0.398326	−	1.83107i	−0.540641	−	0.841254i	$-0.681818\pi$
$761$	−0.398326	−	1.83107i	−0.398326	−	1.83107i	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
$769$	0.229843	+	0.357643i	0.229843	+	0.357643i	0.936950	−	0.349464i	$-0.113636\pi$
$769$	0.229843	+	0.357643i	0.229843	+	0.357643i	−0.707107	+	0.707107i	$0.750000\pi$
$770$		0			0
$771$		0			0
$772$	1.21257	+	0.977147i	1.21257	+	0.977147i
$773$	1.08227	−	1.55877i	1.08227	−	1.55877i	0.281733	−	0.959493i	$-0.409091\pi$
$773$	1.08227	−	1.55877i	1.08227	−	1.55877i	0.800541	−	0.599278i	$-0.204545\pi$
$774$		0			0
$775$		0			0
$776$	0.0771377	−	0.120029i	0.0771377	−	0.120029i
$777$		0			0
$778$	−1.10521	+	0.282089i	−1.10521	+	0.282089i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.349464	−	0.936950i	−0.349464	−	0.936950i
$785$	−0.234072	−	1.62801i	−0.234072	−	1.62801i
$786$		0			0
$787$		0			0		0.860014	−	0.510270i	$-0.170455\pi$
$787$		0			0		−0.860014	+	0.510270i	$0.829545\pi$
$788$	−0.396309	−	1.19072i	−0.396309	−	1.19072i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.277100	+	0.943717i	−0.277100	+	0.943717i
$794$	−1.71017	+	0.183863i	−1.71017	+	0.183863i
$795$		0			0
$796$		0			0
$797$	−1.04849	+	1.21002i	−1.04849	+	1.21002i	−0.0713392	+	0.997452i	$0.522727\pi$
$797$	−1.04849	+	1.21002i	−1.04849	+	1.21002i	−0.977147	+	0.212565i	$0.931818\pi$
$798$		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
$808$	−1.77820	+	0.191177i	−1.77820	+	0.191177i
$809$	0.337672	−	1.15001i	0.337672	−	1.15001i	−0.599278	−	0.800541i	$-0.704545\pi$
$809$	0.337672	−	1.15001i	0.337672	−	1.15001i	0.936950	−	0.349464i	$-0.113636\pi$
$810$		0			0
$811$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$811$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.196911	+	0.527938i	0.196911	+	0.527938i
$819$		0			0
$820$	−1.07531	−	0.638011i	−1.07531	−	0.638011i
$821$	0.254771	−	1.17116i	0.254771	−	1.17116i	−0.654861	−	0.755750i	$-0.727273\pi$
$821$	0.254771	−	1.17116i	0.254771	−	1.17116i	0.909632	−	0.415415i	$-0.136364\pi$
$822$		0			0
$823$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$823$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.681418	−	0.731895i	$-0.738636\pi$
$827$		0			0		0.681418	+	0.731895i	$0.261364\pi$
$828$		0			0
$829$	−0.436506	+	0.628688i	−0.436506	+	0.628688i	−0.977147	−	0.212565i	$-0.931818\pi$
$829$	−0.436506	+	0.628688i	−0.436506	+	0.628688i	0.540641	+	0.841254i	$0.318182\pi$
$830$		0			0
$831$		0			0
$832$	0.122321	−	0.479249i	0.122321	−	0.479249i
$833$	1.05657	+	1.64406i	1.05657	+	1.64406i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.994270	−	0.106895i	$-0.965909\pi$
$839$		0			0		0.994270	+	0.106895i	$0.0340909\pi$
$840$		0			0
$841$	0.610891	+	2.80822i	0.610891	+	2.80822i
$842$	0.322286	−	0.968315i	0.322286	−	0.968315i
$843$		0			0
$844$		0			0
$845$	−0.621186	+	1.36021i	−0.621186	+	1.36021i
$846$		0			0
$847$		0			0
$848$	0.0498610	+	0.697148i	0.0498610	+	0.697148i
$849$		0			0
$850$	5.57419	−	1.21259i	5.57419	−	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
$857$	−0.354880	+	1.96697i	−0.354880	+	1.96697i	−0.142315	+	0.989821i	$0.545455\pi$
$857$	−0.354880	+	1.96697i	−0.354880	+	1.96697i	−0.212565	+	0.977147i	$0.568182\pi$
$858$		0			0
$859$		0			0		−0.968937	−	0.247307i	$-0.920455\pi$
$859$		0			0		0.968937	+	0.247307i	$0.0795455\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.177553	−	0.984111i	$-0.556818\pi$
$863$		0			0		0.177553	+	0.984111i	$0.443182\pi$
$864$		0			0
$865$	0.938384	+	0.603063i	0.938384	+	0.603063i
$866$	1.07134	+	0.997452i	1.07134	+	0.997452i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.724384	+	1.32661i	−0.724384	+	1.32661i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.599278	+	0.199459i	−0.599278	+	0.199459i	−0.599278	−	0.800541i	$-0.704545\pi$
$877$	−0.599278	+	0.199459i	−0.599278	+	0.199459i			1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.254771	+	0.340335i	0.254771	+	0.340335i	0.909632	−	0.415415i	$-0.136364\pi$
$881$	0.254771	+	0.340335i	0.254771	+	0.340335i	−0.654861	+	0.755750i	$0.727273\pi$
$882$		0			0
$883$		0			0		−0.106895	−	0.994270i	$-0.534091\pi$
$883$		0			0		0.106895	+	0.994270i	$0.465909\pi$
$884$	−0.0345009	+	0.966003i	−0.0345009	+	0.966003i
$885$		0			0
$886$		0			0
$887$		0			0		−0.894225	−	0.447617i	$-0.852273\pi$
$887$		0			0		0.894225	+	0.447617i	$0.147727\pi$
$888$		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
$898$	1.79799	+	0.527938i	1.79799	+	0.527938i
$899$		0			0
$900$		0			0
$901$	−0.384822	−	1.31058i	−0.384822	−	1.31058i
$902$		0			0
$903$		0			0
$904$	−0.746928	−	1.25888i	−0.746928	−	1.25888i
$905$	−2.48275	+	0.0886715i	−2.48275	+	0.0886715i
$906$		0			0
$907$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$907$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$911$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$912$		0			0
$913$		0			0
$914$	−1.21868	+	1.30896i	−1.21868	+	1.30896i
$915$		0			0
$916$	0.0522460	−	0.0486428i	0.0522460	−	0.0486428i
$917$		0			0
$918$		0			0
$919$		0			0		0.627469	−	0.778642i	$-0.284091\pi$
$919$		0			0		−0.627469	+	0.778642i	$0.715909\pi$
$920$		0			0
$921$		0			0
$922$	0.474017	−	0.304632i	0.474017	−	0.304632i
$923$		0			0
$924$		0			0
$925$	2.26880	+	3.26769i	2.26880	+	3.26769i
$926$		0			0
$927$		0			0
$928$	0.210393	−	1.95695i	0.210393	−	1.95695i
$929$	1.58479	+	0.227858i	1.58479	+	0.227858i	0.877679	−	0.479249i	$-0.159091\pi$
$929$	1.58479	+	0.227858i	1.58479	+	0.227858i	0.707107	+	0.707107i	$0.250000\pi$
$930$		0			0
$931$		0			0
$932$	−1.67822	+	0.120029i	−1.67822	+	0.120029i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.142315	+	0.0101786i	−0.142315	+	0.0101786i	−0.142315	−	0.989821i	$-0.545455\pi$
$937$	−0.142315	+	0.0101786i	−0.142315	+	0.0101786i			1.00000i	$0.5\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	0.213654	−	1.98727i	0.213654	−	1.98727i	0.0713392	−	0.997452i	$-0.477273\pi$
$941$	0.213654	−	1.98727i	0.213654	−	1.98727i	0.142315	−	0.989821i	$-0.454545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$947$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$948$		0			0
$949$	−0.289969	−	0.579284i	−0.289969	−	0.579284i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.30896	+	1.21868i	−1.30896	+	1.21868i	−0.349464	+	0.936950i	$0.613636\pi$
$953$	−1.30896	+	1.21868i	−1.30896	+	1.21868i	−0.959493	+	0.281733i	$0.909091\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.877679	−	0.479249i	−0.877679	−	0.479249i
$962$	−0.631576	+	0.235566i	−0.631576	+	0.235566i
$963$		0			0
$964$	−1.84658	+	0.0659508i	−1.84658	+	0.0659508i
$965$	−1.57309	−	2.65131i	−1.57309	−	2.65131i
$966$		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
$970$	−0.226115	+	0.169267i	−0.226115	+	0.169267i
$971$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$971$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.890105	−	1.77820i	0.890105	−	1.77820i
$977$	−0.497898	−	1.09024i	−0.497898	−	1.09024i	−0.977147	−	0.212565i	$-0.931818\pi$
$977$	−0.497898	−	1.09024i	−0.497898	−	1.09024i	0.479249	−	0.877679i	$-0.340909\pi$
$978$		0			0
$979$		0			0
$980$			1.97964i			1.97964i
$981$		0			0
$982$		0			0
$983$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$983$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$984$		0			0
$985$	−0.0886715	+	2.48275i	−0.0886715	+	2.48275i
$986$	0.411170	+	3.82445i	0.411170	+	3.82445i
$987$		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
$997$	−0.956056	+	1.75089i	−0.956056	+	1.75089i	−0.415415	+	0.909632i	$0.636364\pi$
$997$	−0.956056	+	1.75089i	−0.956056	+	1.75089i	−0.540641	+	0.841254i	$0.681818\pi$
$998$		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.1223.1	yes	40
3.2	odd	2		3204.1.cd.a.1223.1	yes	40
4.3	odd	2	CM	3204.1.cd.b.1223.1	yes	40
12.11	even	2		3204.1.cd.a.1223.1	yes	40
89.58	odd	88		3204.1.cd.a.503.1	✓	40
267.236	even	88	inner	3204.1.cd.b.503.1	yes	40
356.147	even	88		3204.1.cd.a.503.1	✓	40
1068.503	odd	88	inner	3204.1.cd.b.503.1	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3204.1.cd.a.503.1	✓	40	89.58	odd	88
3204.1.cd.a.503.1	✓	40	356.147	even	88
3204.1.cd.a.1223.1	yes	40	3.2	odd	2
3204.1.cd.a.1223.1	yes	40	12.11	even	2
3204.1.cd.b.503.1	yes	40	267.236	even	88	inner
3204.1.cd.b.503.1	yes	40	1068.503	odd	88	inner
3204.1.cd.b.1223.1	yes	40	1.1	even	1	trivial
3204.1.cd.b.1223.1	yes	40	4.3	odd	2	CM

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)

\(f(q)\)	\(=\)	\(q+(0.909632 - 0.415415i) q^{2} +(0.654861 - 0.755750i) q^{4} +(-1.73749 + 0.948742i) q^{5} +(0.281733 - 0.959493i) q^{8} +O(q^{10})\)	\(q+(0.909632 - 0.415415i) q^{2} +(0.654861 - 0.755750i) q^{4} +(-1.73749 + 0.948742i) q^{5} +(0.281733 - 0.959493i) q^{8} +(-1.18636 + 1.58479i) q^{10} +(0.156199 + 0.469302i) q^{13} +(-0.142315 - 0.989821i) q^{16} +(0.682956 + 1.83107i) q^{17} +(-0.420803 + 1.93440i) q^{20} +(1.57812 - 2.45561i) q^{25} +(0.337038 + 0.362005i) q^{26} +(1.53254 + 1.23500i) q^{29} +(-0.540641 - 0.841254i) q^{32} +(1.38189 + 1.38189i) q^{34} +(-0.521535 + 1.25910i) q^{37} +(0.420803 + 1.93440i) q^{40} +(-0.199459 + 0.599278i) q^{41} +(0.977147 - 0.212565i) q^{49} +(0.415415 - 2.88927i) q^{50} +(0.456963 + 0.189280i) q^{52} +(-0.697148 - 0.0498610i) q^{53} +(1.90708 + 0.486754i) q^{58} +(1.63343 + 1.13411i) q^{61} +(-0.841254 - 0.540641i) q^{64} +(-0.716640 - 0.667215i) q^{65} +(1.83107 + 0.682956i) q^{68} +(-1.29639 + 0.186393i) q^{73} +(0.0486428 + 1.36197i) q^{74} +(1.18636 + 1.58479i) q^{80} +(0.0675149 + 0.627980i) q^{82} +(-2.92385 - 2.53353i) q^{85} +(0.479249 - 0.877679i) q^{89} +(0.136899 + 0.0401971i) q^{97} +(0.800541 - 0.599278i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 4 q^{4}+O(q^{10}) \) 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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