LMFDB - Embedded newform 1521.1.bm.a.694.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,1,Mod(73,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(52))

chi = DirichletCharacter(H, H._module([0, 17]))

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

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

chi := DirichletCharacter("1521.73");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1521.bm (of order $52$, degree $24$, 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:	$0.759077884215$
Analytic rank:	$0$
Dimension:	$24$
Coefficient field:	$\Q(\zeta_{52})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{52}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{52} - \cdots)$

Embedding invariants

Embedding label			694.1
Root			$-0.239316 - 0.970942i$ of defining polynomial
Character	$\chi$	$=$	1521.694
Dual form			1521.1.bm.a.892.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.822984 + 0.568065i) q^{4} +(0.814480 - 0.638104i) q^{7} +O(q^{10})$	$q+(0.822984 + 0.568065i) q^{4} +(0.814480 - 0.638104i) q^{7} -1.00000i q^{13} +(0.354605 + 0.935016i) q^{16} +(-0.420733 + 0.420733i) q^{19} +(0.663123 - 0.748511i) q^{25} +(1.03279 - 0.0624722i) q^{28} +(0.0217671 - 0.359852i) q^{31} +(-1.57149 - 0.0950579i) q^{37} +(-1.31658 + 1.48611i) q^{43} +(0.0168846 - 0.0685035i) q^{49} +(0.568065 - 0.822984i) q^{52} +(0.198399 + 1.63397i) q^{61} +(-0.239316 + 0.970942i) q^{64} +(1.96365 + 0.359852i) q^{67} +(-0.244198 - 0.783659i) q^{73} +(-0.585260 + 0.107253i) q^{76} +(0.271894 + 0.393906i) q^{79} +(-0.638104 - 0.814480i) q^{91} +(-1.28962 - 0.580411i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 2 q^{7}+O(q^{10}) $ 24 * q + 2 * q^7	$ 24 q + 2 q^{7} + 2 q^{16} + 2 q^{19} - 2 q^{28} - 2 q^{31} - 2 q^{37} - 2 q^{52} + 2 q^{67} + 2 q^{73} + 2 q^{76} + 2 q^{91} - 2 q^{97}+O(q^{100}) $ 24 * q + 2 * q^7 + 2 * q^16 + 2 * q^19 - 2 * q^28 - 2 * q^31 - 2 * q^37 - 2 * q^52 + 2 * q^67 + 2 * q^73 + 2 * q^76 + 2 * q^91 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$e\left(\frac{31}{52}\right)$

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			0		−0.954721	−	0.297503i	$-0.903846\pi$
$2$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$3$		0			0
$3$		0			0
$4$	0.822984	+	0.568065i	0.822984	+	0.568065i
$5$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$5$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$6$		0			0
$7$	0.814480	−	0.638104i	0.814480	−	0.638104i	−0.120537	−	0.992709i	$-0.538462\pi$
$7$	0.814480	−	0.638104i	0.814480	−	0.638104i	0.935016	+	0.354605i	$0.115385\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$11$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$12$		0			0
$13$		−	1.00000i		−	1.00000i
$13$		−	1.00000i		−	1.00000i
$14$		0			0
$15$		0			0
$16$	0.354605	+	0.935016i	0.354605	+	0.935016i
$17$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$17$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$18$		0			0
$19$	−0.420733	+	0.420733i	−0.420733	+	0.420733i	−0.885456	−	0.464723i	$-0.846154\pi$
$19$	−0.420733	+	0.420733i	−0.420733	+	0.420733i	0.464723	+	0.885456i	$0.346154\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	0.663123	−	0.748511i	0.663123	−	0.748511i
$26$		0			0
$27$		0			0
$28$	1.03279	−	0.0624722i	1.03279	−	0.0624722i
$29$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$29$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$30$		0			0
$31$	0.0217671	−	0.359852i	0.0217671	−	0.359852i	−0.970942	−	0.239316i	$-0.923077\pi$
$31$	0.0217671	−	0.359852i	0.0217671	−	0.359852i	0.992709	−	0.120537i	$-0.0384615\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−1.57149	−	0.0950579i	−1.57149	−	0.0950579i	−0.748511	−	0.663123i	$-0.769231\pi$
$37$	−1.57149	−	0.0950579i	−1.57149	−	0.0950579i	−0.822984	+	0.568065i	$0.807692\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$41$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$42$		0			0
$43$	−1.31658	+	1.48611i	−1.31658	+	1.48611i	−0.568065	+	0.822984i	$0.692308\pi$
$43$	−1.31658	+	1.48611i	−1.31658	+	1.48611i	−0.748511	+	0.663123i	$0.769231\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$47$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$48$		0			0
$49$	0.0168846	−	0.0685035i	0.0168846	−	0.0685035i
$50$		0			0
$51$		0			0
$52$	0.568065	−	0.822984i	0.568065	−	0.822984i
$53$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$53$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.410413	−	0.911900i	$-0.634615\pi$
$59$		0			0		0.410413	+	0.911900i	$0.365385\pi$
$60$		0			0
$61$	0.198399	+	1.63397i	0.198399	+	1.63397i	0.663123	+	0.748511i	$0.269231\pi$
$61$	0.198399	+	1.63397i	0.198399	+	1.63397i	−0.464723	+	0.885456i	$0.653846\pi$
$62$		0			0
$63$		0			0
$64$	−0.239316	+	0.970942i	−0.239316	+	0.970942i
$65$		0			0
$66$		0			0
$67$	1.96365	+	0.359852i	1.96365	+	0.359852i	0.992709	+	0.120537i	$0.0384615\pi$
$67$	1.96365	+	0.359852i	1.96365	+	0.359852i	0.970942	+	0.239316i	$0.0769231\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.855781	−	0.517338i	$-0.826923\pi$
$71$		0			0		0.855781	+	0.517338i	$0.173077\pi$
$72$		0			0
$73$	−0.244198	−	0.783659i	−0.244198	−	0.783659i	−0.992709	−	0.120537i	$-0.961538\pi$
$73$	−0.244198	−	0.783659i	−0.244198	−	0.783659i	0.748511	−	0.663123i	$-0.230769\pi$
$74$		0			0
$75$		0			0
$76$	−0.585260	+	0.107253i	−0.585260	+	0.107253i
$77$		0			0
$78$		0			0
$79$	0.271894	+	0.393906i	0.271894	+	0.393906i	0.935016	−	0.354605i	$-0.115385\pi$
$79$	0.271894	+	0.393906i	0.271894	+	0.393906i	−0.663123	+	0.748511i	$0.730769\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		−0.517338	−	0.855781i	$-0.673077\pi$
$83$		0			0		0.517338	+	0.855781i	$0.326923\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$89$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$90$		0			0
$91$	−0.638104	−	0.814480i	−0.638104	−	0.814480i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.28962	−	0.580411i	−1.28962	−	0.580411i	−0.354605	−	0.935016i	$-0.615385\pi$
$97$	−1.28962	−	0.580411i	−1.28962	−	0.580411i	−0.935016	+	0.354605i	$0.884615\pi$
$98$		0			0
$99$		0			0
$100$	0.970942	−	0.239316i	0.970942	−	0.239316i
$101$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$101$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$102$		0			0
$103$	−0.447528	−	1.81569i	−0.447528	−	1.81569i	−0.568065	−	0.822984i	$-0.692308\pi$
$103$	−0.447528	−	1.81569i	−0.447528	−	1.81569i	0.120537	−	0.992709i	$-0.461538\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$107$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$108$		0			0
$109$	−1.82047	+	0.110118i	−1.82047	+	0.110118i	−0.935016	−	0.354605i	$-0.884615\pi$
$109$	−1.82047	+	0.110118i	−1.82047	+	0.110118i	−0.885456	+	0.464723i	$0.846154\pi$
$110$		0			0
$111$		0			0
$112$	0.885456	+	0.535277i	0.885456	+	0.535277i
$113$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$113$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.822984	−	0.568065i	0.822984	−	0.568065i
$122$		0			0
$123$		0			0
$124$	0.222333	−	0.283788i	0.222333	−	0.283788i
$125$		0			0
$126$		0			0
$127$	−0.583668	+	0.402877i	−0.583668	+	0.402877i	−0.822984	−	0.568065i	$-0.807692\pi$
$127$	−0.583668	+	0.402877i	−0.583668	+	0.402877i	0.239316	+	0.970942i	$0.423077\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$131$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$132$		0			0
$133$	−0.0742070	+	0.611150i	−0.0742070	+	0.611150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$137$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$138$		0			0
$139$	0.530851	+	1.39974i	0.530851	+	1.39974i	0.885456	+	0.464723i	$0.153846\pi$
$139$	0.530851	+	1.39974i	0.530851	+	1.39974i	−0.354605	+	0.935016i	$0.615385\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−1.23932	−	0.970942i	−1.23932	−	0.970942i
$149$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$149$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$150$		0			0
$151$	−0.118779	+	0.0217671i	−0.118779	+	0.0217671i	−0.239316	−	0.970942i	$-0.576923\pi$
$151$	−0.118779	+	0.0217671i	−0.118779	+	0.0217671i	0.120537	+	0.992709i	$0.461538\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.402877	−	1.06230i	0.402877	−	1.06230i	−0.568065	−	0.822984i	$-0.692308\pi$
$157$	0.402877	−	1.06230i	0.402877	−	1.06230i	0.970942	−	0.239316i	$-0.0769231\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.120537	+	0.00729113i	0.120537	+	0.00729113i	0.120537	−	0.992709i	$-0.461538\pi$
$163$	0.120537	+	0.00729113i	0.120537	+	0.00729113i			1.00000i	$0.5\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.297503	−	0.954721i	$-0.403846\pi$
$167$		0			0		−0.297503	+	0.954721i	$0.596154\pi$
$168$		0			0
$169$	−1.00000			−1.00000
$170$		0			0
$171$		0			0
$172$	−1.92773	+	0.475142i	−1.92773	+	0.475142i
$173$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$173$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$174$		0			0
$175$	0.0624722	−	1.03279i	0.0624722	−	1.03279i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$179$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$180$		0			0
$181$	−1.24006	−	0.470293i	−1.24006	−	0.470293i	−0.354605	−	0.935016i	$-0.615385\pi$
$181$	−1.24006	−	0.470293i	−1.24006	−	0.470293i	−0.885456	+	0.464723i	$0.846154\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−0.760684	+	0.970942i	−0.760684	+	0.970942i	0.239316	+	0.970942i	$0.423077\pi$
$193$	−0.760684	+	0.970942i	−0.760684	+	0.970942i	−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$	0.0528102	−	0.0467857i	0.0528102	−	0.0467857i
$197$		0			0		−0.0603785	−	0.998176i	$-0.519231\pi$
$197$		0			0		0.0603785	+	0.998176i	$0.480769\pi$
$198$		0			0
$199$	−1.65583	+	0.627974i	−1.65583	+	0.627974i	−0.992709	−	0.120537i	$-0.961538\pi$
$199$	−1.65583	+	0.627974i	−1.65583	+	0.627974i	−0.663123	+	0.748511i	$0.730769\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.935016	−	0.354605i	0.935016	−	0.354605i
$209$		0			0
$210$		0			0
$211$	0.753393	+	1.09148i	0.753393	+	1.09148i	0.992709	+	0.120537i	$0.0384615\pi$
$211$	0.753393	+	1.09148i	0.753393	+	1.09148i	−0.239316	+	0.970942i	$0.576923\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.211894	−	0.306982i	−0.211894	−	0.306982i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.580411	−	1.28962i	−0.580411	−	1.28962i	−0.935016	−	0.354605i	$-0.884615\pi$
$223$	−0.580411	−	1.28962i	−0.580411	−	1.28962i	0.354605	−	0.935016i	$-0.384615\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$227$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$228$		0			0
$229$	−0.110118	−	1.82047i	−0.110118	−	1.82047i	−0.464723	−	0.885456i	$-0.653846\pi$
$229$	−0.110118	−	1.82047i	−0.110118	−	1.82047i	0.354605	−	0.935016i	$-0.384615\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$233$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$239$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$240$		0			0
$241$	0.807380	−	1.79393i	0.807380	−	1.79393i	0.239316	−	0.970942i	$-0.423077\pi$
$241$	0.807380	−	1.79393i	0.807380	−	1.79393i	0.568065	−	0.822984i	$-0.307692\pi$
$242$		0			0
$243$		0			0
$244$	−0.764919	+	1.45743i	−0.764919	+	1.45743i
$245$		0			0
$246$		0			0
$247$	0.420733	+	0.420733i	0.420733	+	0.420733i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$251$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.748511	+	0.663123i	−0.748511	+	0.663123i
$257$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$257$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$258$		0			0
$259$	−1.34061	+	0.925354i	−1.34061	+	0.925354i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$263$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.41163	+	1.41163i	1.41163	+	1.41163i
$269$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$269$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$270$		0			0
$271$	−0.354605	+	1.93502i	−0.354605	+	1.93502i			1.00000i	$0.5\pi$
$271$	−0.354605	+	1.93502i	−0.354605	+	1.93502i	−0.354605	+	0.935016i	$0.615385\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.475142	+	0.0576926i	−0.475142	+	0.0576926i	−0.354605	−	0.935016i	$-0.615385\pi$
$277$	−0.475142	+	0.0576926i	−0.475142	+	0.0576926i	−0.120537	+	0.992709i	$0.538462\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.517338	−	0.855781i	$-0.326923\pi$
$281$		0			0		−0.517338	+	0.855781i	$0.673077\pi$
$282$		0			0
$283$	0.470293	+	0.530851i	0.470293	+	0.530851i	0.935016	−	0.354605i	$-0.115385\pi$
$283$	0.470293	+	0.530851i	0.470293	+	0.530851i	−0.464723	+	0.885456i	$0.653846\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$290$		0			0
$291$		0			0
$292$	0.244198	−	0.783659i	0.244198	−	0.783659i
$293$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$293$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−0.124033	+	2.05052i	−0.124033	+	2.05052i
$302$		0			0
$303$		0			0
$304$	−0.542586	−	0.244198i	−0.542586	−	0.244198i
$305$		0			0
$306$		0			0
$307$	1.23119	+	0.0744731i	1.23119	+	0.0744731i	0.663123	−	0.748511i	$-0.269231\pi$
$307$	1.23119	+	0.0744731i	1.23119	+	0.0744731i	0.568065	+	0.822984i	$0.307692\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$311$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$312$		0			0
$313$	−0.358261	−	0.317391i	−0.358261	−	0.317391i	0.464723	−	0.885456i	$-0.346154\pi$
$313$	−0.358261	−	0.317391i	−0.358261	−	0.317391i	−0.822984	+	0.568065i	$0.807692\pi$
$314$		0			0
$315$		0			0
$316$			0.478631i			0.478631i
$317$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$317$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.23119	−	1.57149i	−1.23119	−	1.57149i	−0.663123	−	0.748511i	$-0.730769\pi$
$331$	−1.23119	−	1.57149i	−1.23119	−	1.57149i	−0.568065	−	0.822984i	$-0.692308\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$			0.241073i			0.241073i	0.992709	+	0.120537i	$0.0384615\pi$
$337$			0.241073i			0.241073i	−0.992709	+	0.120537i	$0.961538\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.394684	+	0.876952i	0.394684	+	0.876952i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$347$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$348$		0			0
$349$	−0.420733	−	1.35018i	−0.420733	−	1.35018i	−0.885456	−	0.464723i	$-0.846154\pi$
$349$	−0.420733	−	1.35018i	−0.420733	−	1.35018i	0.464723	−	0.885456i	$-0.346154\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.787183	−	0.616719i	$-0.211538\pi$
$353$		0			0		−0.787183	+	0.616719i	$0.788462\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.787183	−	0.616719i	$-0.788462\pi$
$359$		0			0		0.787183	+	0.616719i	$0.211538\pi$
$360$		0			0
$361$			0.645968i			0.645968i
$362$		0			0
$363$		0			0
$364$	−0.0624722	−	1.03279i	−0.0624722	−	1.03279i
$365$		0			0
$366$		0			0
$367$	1.71945	−	0.902438i	1.71945	−	0.902438i	0.748511	−	0.663123i	$-0.230769\pi$
$367$	1.71945	−	0.902438i	1.71945	−	0.902438i	0.970942	−	0.239316i	$-0.0769231\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.45743	+	0.764919i	1.45743	+	0.764919i	0.992709	−	0.120537i	$-0.0384615\pi$
$373$	1.45743	+	0.764919i	1.45743	+	0.764919i	0.464723	+	0.885456i	$0.346154\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.987826	+	1.63406i	−0.987826	+	1.63406i	−0.239316	+	0.970942i	$0.576923\pi$
$379$	−0.987826	+	1.63406i	−0.987826	+	1.63406i	−0.748511	+	0.663123i	$0.769231\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$383$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.731626	−	1.21026i	−0.731626	−	1.21026i
$389$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$389$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.79393	−	0.807380i	1.79393	−	0.807380i	0.822984	−	0.568065i	$-0.192308\pi$
$397$	1.79393	−	0.807380i	1.79393	−	0.807380i	0.970942	−	0.239316i	$-0.0769231\pi$
$398$		0			0
$399$		0			0
$400$	0.935016	+	0.354605i	0.935016	+	0.354605i
$401$		0			0		−0.911900	−	0.410413i	$-0.865385\pi$
$401$		0			0		0.911900	+	0.410413i	$0.134615\pi$
$402$		0			0
$403$	−0.359852	−	0.0217671i	−0.359852	−	0.0217671i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.987826	−	1.63406i	0.987826	−	1.63406i	0.239316	−	0.970942i	$-0.423077\pi$
$409$	0.987826	−	1.63406i	0.987826	−	1.63406i	0.748511	−	0.663123i	$-0.230769\pi$
$410$		0			0
$411$		0			0
$412$	0.663123	−	1.74851i	0.663123	−	1.74851i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$419$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$420$		0			0
$421$	0.702447	−	0.424644i	0.702447	−	0.424644i	−0.120537	−	0.992709i	$-0.538462\pi$
$421$	0.702447	−	0.424644i	0.702447	−	0.424644i	0.822984	+	0.568065i	$0.192308\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.20423	+	1.20423i	1.20423	+	1.20423i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$431$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$432$		0			0
$433$	−1.65583	−	0.627974i	−1.65583	−	0.627974i	−0.663123	−	0.748511i	$-0.730769\pi$
$433$	−1.65583	−	0.627974i	−1.65583	−	0.627974i	−0.992709	+	0.120537i	$0.961538\pi$
$434$		0			0
$435$		0			0
$436$	−1.56077	−	0.943521i	−1.56077	−	0.943521i
$437$		0			0
$438$		0			0
$439$	−0.616337	+	0.695701i	−0.616337	+	0.695701i	−0.970942	−	0.239316i	$-0.923077\pi$
$439$	−0.616337	+	0.695701i	−0.616337	+	0.695701i	0.354605	+	0.935016i	$0.384615\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$443$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.424644	+	0.943521i	0.424644	+	0.943521i
$449$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$449$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.56077	−	0.943521i	1.56077	−	0.943521i	0.568065	−	0.822984i	$-0.307692\pi$
$457$	1.56077	−	0.943521i	1.56077	−	0.943521i	0.992709	−	0.120537i	$-0.0384615\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$461$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$462$		0			0
$463$	1.82298	+	0.568065i	1.82298	+	0.568065i	1.00000			$0$
$463$	1.82298	+	0.568065i	1.82298	+	0.568065i	0.822984	+	0.568065i	$0.192308\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$467$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$468$		0			0
$469$	1.82898	−	0.959921i	1.82898	−	0.959921i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.0359256	+	0.593921i	0.0359256	+	0.593921i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.998176	−	0.0603785i	$-0.0192308\pi$
$479$		0			0		−0.998176	+	0.0603785i	$0.980769\pi$
$480$		0			0
$481$	−0.0950579	+	1.57149i	−0.0950579	+	1.57149i
$482$		0			0
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$		0			0
$487$	1.93502	+	0.354605i	1.93502	+	0.354605i	1.00000			$0$
$487$	1.93502	+	0.354605i	1.93502	+	0.354605i	0.935016	+	0.354605i	$0.115385\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$491$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.344186	−	0.107253i	0.344186	−	0.107253i
$497$		0			0
$498$		0			0
$499$	−0.506219	−	0.646140i	−0.506219	−	0.646140i	0.464723	−	0.885456i	$-0.346154\pi$
$499$	−0.506219	−	0.646140i	−0.506219	−	0.646140i	−0.970942	+	0.239316i	$0.923077\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$503$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−0.709210			−0.709210
$509$		0			0		−0.954721	−	0.297503i	$-0.903846\pi$
$509$		0			0		0.954721	+	0.297503i	$0.0961538\pi$
$510$		0			0
$511$	−0.698951	−	0.482451i	−0.698951	−	0.482451i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$521$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$522$		0			0
$523$	0.627974	+	1.65583i	0.627974	+	1.65583i	0.748511	+	0.663123i	$0.230769\pi$
$523$	0.627974	+	1.65583i	0.627974	+	1.65583i	−0.120537	+	0.992709i	$0.538462\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.00000			1.00000
$530$		0			0
$531$		0			0
$532$	−0.408244	+	0.460812i	−0.408244	+	0.460812i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.0495602	−	0.110118i	0.0495602	−	0.110118i	−0.885456	−	0.464723i	$-0.846154\pi$
$541$	0.0495602	−	0.110118i	0.0495602	−	0.110118i	0.935016	+	0.354605i	$0.115385\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.885456	−	0.464723i	$-0.153846\pi$
$547$	1.00599	+	0.527986i	1.00599	+	0.527986i	0.120537	+	0.992709i	$0.461538\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.472805	+	0.147332i	0.472805	+	0.147332i
$554$		0			0
$555$		0			0
$556$	−0.358261	+	1.45352i	−0.358261	+	1.45352i
$557$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$557$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$558$		0			0
$559$	1.48611	+	1.31658i	1.48611	+	1.31658i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$563$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$569$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$570$		0			0
$571$	0.464723	−	1.88546i	0.464723	−	1.88546i		−	1.00000i	$-0.5\pi$
$571$	0.464723	−	1.88546i	0.464723	−	1.88546i	0.464723	−	0.885456i	$-0.346154\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−0.0853881	+	0.0853881i	−0.0853881	+	0.0853881i	−0.748511	−	0.663123i	$-0.769231\pi$
$577$	−0.0853881	+	0.0853881i	−0.0853881	+	0.0853881i	0.663123	+	0.748511i	$0.269231\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$587$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$588$		0			0
$589$	0.142244	+	0.160560i	0.142244	+	0.160560i
$590$		0			0
$591$		0			0
$592$	−0.468379	−	1.50308i	−0.468379	−	1.50308i
$593$		0			0		0.855781	−	0.517338i	$-0.173077\pi$
$593$		0			0		−0.855781	+	0.517338i	$0.826923\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$599$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$600$		0			0
$601$	1.75800	+	0.922670i	1.75800	+	0.922670i	0.935016	+	0.354605i	$0.115385\pi$
$601$	1.75800	+	0.922670i	1.75800	+	0.922670i	0.822984	+	0.568065i	$0.192308\pi$
$602$		0			0
$603$		0			0
$604$	−0.110118	−	0.0495602i	−0.110118	−	0.0495602i
$605$		0			0
$606$		0			0
$607$	0.234068	−	0.0576926i	0.234068	−	0.0576926i	−0.120537	−	0.992709i	$-0.538462\pi$
$607$	0.234068	−	0.0576926i	0.234068	−	0.0576926i	0.354605	+	0.935016i	$0.384615\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.43566	−	0.646140i	−1.43566	−	0.646140i	−0.464723	−	0.885456i	$-0.653846\pi$
$613$	−1.43566	−	0.646140i	−1.43566	−	0.646140i	−0.970942	+	0.239316i	$0.923077\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.911900	−	0.410413i	$-0.134615\pi$
$617$		0			0		−0.911900	+	0.410413i	$0.865385\pi$
$618$		0			0
$619$	1.34731	+	0.814480i	1.34731	+	0.814480i	0.992709	−	0.120537i	$-0.0384615\pi$
$619$	1.34731	+	0.814480i	1.34731	+	0.814480i	0.354605	+	0.935016i	$0.384615\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.120537	−	0.992709i	−0.120537	−	0.992709i
$626$		0			0
$627$		0			0
$628$	0.935016	−	0.645395i	0.935016	−	0.645395i
$629$		0			0
$630$		0			0
$631$	−0.760684	+	0.970942i	−0.760684	+	0.970942i	0.239316	+	0.970942i	$0.423077\pi$
$631$	−0.760684	+	0.970942i	−0.760684	+	0.970942i	−1.00000			$\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.0685035	−	0.0168846i	−0.0685035	−	0.0168846i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$641$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$642$		0			0
$643$	−1.05555	+	0.638104i	−1.05555	+	0.638104i	−0.935016	−	0.354605i	$-0.884615\pi$
$643$	−1.05555	+	0.638104i	−1.05555	+	0.638104i	−0.120537	+	0.992709i	$0.538462\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$647$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.0950579	+	0.0744731i	0.0950579	+	0.0744731i
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$659$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$660$		0			0
$661$	1.50308	−	1.17759i	1.50308	−	1.17759i	0.568065	−	0.822984i	$-0.307692\pi$
$661$	1.50308	−	1.17759i	1.50308	−	1.17759i	0.935016	−	0.354605i	$-0.115385\pi$
$662$		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
$673$	0.222431	+	0.902438i	0.222431	+	0.902438i	0.970942	+	0.239316i	$0.0769231\pi$
$673$	0.222431	+	0.902438i	0.222431	+	0.902438i	−0.748511	+	0.663123i	$0.769231\pi$
$674$		0			0
$675$		0			0
$676$	−0.822984	−	0.568065i	−0.822984	−	0.568065i
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−1.42073	+	0.350179i	−1.42073	+	0.350179i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.616719	−	0.787183i	$-0.711538\pi$
$683$		0			0		0.616719	+	0.787183i	$0.288462\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.85640	−	0.704039i	−1.85640	−	0.704039i
$689$		0			0
$690$		0			0
$691$	0.222333	+	0.283788i	0.222333	+	0.283788i	0.885456	−	0.464723i	$-0.153846\pi$
$691$	0.222333	+	0.283788i	0.222333	+	0.283788i	−0.663123	+	0.748511i	$0.730769\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.638104	−	0.814480i	0.638104	−	0.814480i
$701$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$701$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$702$		0			0
$703$	0.701173	−	0.621185i	0.701173	−	0.621185i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.03279	+	1.70844i	1.03279	+	1.70844i	0.568065	+	0.822984i	$0.307692\pi$
$709$	1.03279	+	1.70844i	1.03279	+	1.70844i	0.464723	+	0.885456i	$0.346154\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$719$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$720$		0			0
$721$	−1.52310	−	1.19328i	−1.52310	−	1.19328i
$722$		0			0
$723$		0			0
$724$	−0.753393	−	1.09148i	−0.753393	−	1.09148i
$725$		0			0
$726$		0			0
$727$	−0.239316	+	0.0290582i	−0.239316	+	0.0290582i	−0.239316	−	0.970942i	$-0.576923\pi$
$727$	−0.239316	+	0.0290582i	−0.239316	+	0.0290582i			1.00000i	$0.5\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.814480	+	1.34731i	−0.814480	+	1.34731i	0.120537	+	0.992709i	$0.461538\pi$
$733$	−0.814480	+	1.34731i	−0.814480	+	1.34731i	−0.935016	+	0.354605i	$0.884615\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−0.807380	+	1.79393i	−0.807380	+	1.79393i	−0.239316	+	0.970942i	$0.576923\pi$
$739$	−0.807380	+	1.79393i	−0.807380	+	1.79393i	−0.568065	+	0.822984i	$0.692308\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.180255	−	0.983620i	$-0.442308\pi$
$743$		0			0		−0.180255	+	0.983620i	$0.557692\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.869047	−	1.65583i	0.869047	−	1.65583i	0.120537	−	0.992709i	$-0.461538\pi$
$751$	0.869047	−	1.65583i	0.869047	−	1.65583i	0.748511	−	0.663123i	$-0.230769\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.12785	−	1.63397i	1.12785	−	1.63397i	0.464723	−	0.885456i	$-0.346154\pi$
$757$	1.12785	−	1.63397i	1.12785	−	1.63397i	0.663123	−	0.748511i	$-0.269231\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.954721	−	0.297503i	$-0.0961538\pi$
$761$		0			0		−0.954721	+	0.297503i	$0.903846\pi$
$762$		0			0
$763$	−1.41247	+	1.25134i	−1.41247	+	1.25134i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	0.359852	+	1.96365i	0.359852	+	1.96365i	0.239316	+	0.970942i	$0.423077\pi$
$769$	0.359852	+	1.96365i	0.359852	+	1.96365i	0.120537	+	0.992709i	$0.461538\pi$
$770$		0			0
$771$		0			0
$772$	−1.17759	+	0.366951i	−1.17759	+	0.366951i
$773$		0			0		0.410413	−	0.911900i	$-0.365385\pi$
$773$		0			0		−0.410413	+	0.911900i	$0.634615\pi$
$774$		0			0
$775$	−0.254919	−	0.254919i	−0.254919	−	0.254919i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.0700392	−	0.00850430i	0.0700392	−	0.00850430i
$785$		0			0
$786$		0			0
$787$	−1.79393	+	0.328749i	−1.79393	+	0.328749i	−0.970942	−	0.239316i	$-0.923077\pi$
$787$	−1.79393	+	0.328749i	−1.79393	+	0.328749i	−0.822984	+	0.568065i	$0.807692\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.63397	−	0.198399i	1.63397	−	0.198399i
$794$		0			0
$795$		0			0
$796$	−1.71945	−	0.423807i	−1.71945	−	0.423807i
$797$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$797$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$809$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$810$		0			0
$811$	1.74122	+	0.783659i	1.74122	+	0.783659i	0.992709	+	0.120537i	$0.0384615\pi$
$811$	1.74122	+	0.783659i	1.74122	+	0.783659i	0.748511	+	0.663123i	$0.230769\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.0713273	−	1.17918i	−0.0713273	−	1.17918i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.616719	−	0.787183i	$-0.288462\pi$
$821$		0			0		−0.616719	+	0.787183i	$0.711538\pi$
$822$		0			0
$823$			0.929446i			0.929446i	0.885456	+	0.464723i	$0.153846\pi$
$823$			0.929446i			0.929446i	−0.885456	+	0.464723i	$0.846154\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.180255	−	0.983620i	$-0.557692\pi$
$827$		0			0		0.180255	+	0.983620i	$0.442308\pi$
$828$		0			0
$829$	−0.663123	+	0.251489i	−0.663123	+	0.251489i	−0.663123	−	0.748511i	$-0.730769\pi$
$829$	−0.663123	+	0.251489i	−0.663123	+	0.251489i			1.00000i	$0.5\pi$
$830$		0			0
$831$		0			0
$832$	0.970942	+	0.239316i	0.970942	+	0.239316i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$839$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$840$		0			0
$841$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$842$		0			0
$843$		0			0
$844$			1.32625i			1.32625i
$845$		0			0
$846$		0			0
$847$	0.307819	−	0.987826i	0.307819	−	0.987826i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−1.43566	+	1.12477i	−1.43566	+	1.12477i	−0.464723	+	0.885456i	$0.653846\pi$
$853$	−1.43566	+	1.12477i	−1.43566	+	1.12477i	−0.970942	+	0.239316i	$0.923077\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$857$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$858$		0			0
$859$	−0.169725	−	0.447528i	−0.169725	−	0.447528i	0.822984	−	0.568065i	$-0.192308\pi$
$859$	−0.169725	−	0.447528i	−0.169725	−	0.447528i	−0.992709	+	0.120537i	$0.961538\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.983620	−	0.180255i	$-0.0576923\pi$
$863$		0			0		−0.983620	+	0.180255i	$0.942308\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		−	0.373011i		−	0.373011i
$869$		0			0
$870$		0			0
$871$	0.359852	−	1.96365i	0.359852	−	1.96365i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.23119	−	0.0744731i	1.23119	−	0.0744731i	0.568065	−	0.822984i	$-0.307692\pi$
$877$	1.23119	−	0.0744731i	1.23119	−	0.0744731i	0.663123	+	0.748511i	$0.269231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$881$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$882$		0			0
$883$	−0.112032	−	0.213460i	−0.112032	−	0.213460i	0.822984	−	0.568065i	$-0.192308\pi$
$883$	−0.112032	−	0.213460i	−0.112032	−	0.213460i	−0.935016	+	0.354605i	$0.884615\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$887$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$888$		0			0
$889$	−0.218308	+	0.700576i	−0.218308	+	0.700576i
$890$		0			0
$891$		0			0
$892$	0.254919	−	1.39105i	0.254919	−	1.39105i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.39974	−	0.530851i	−1.39974	−	0.530851i	−0.464723	−	0.885456i	$-0.653846\pi$
$907$	−1.39974	−	0.530851i	−1.39974	−	0.530851i	−0.935016	+	0.354605i	$0.884615\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$911$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.943521	−	1.56077i	0.943521	−	1.56077i
$917$		0			0
$918$		0			0
$919$	0.470293	−	1.24006i	0.470293	−	1.24006i	−0.464723	−	0.885456i	$-0.653846\pi$
$919$	0.470293	−	1.24006i	0.470293	−	1.24006i	0.935016	−	0.354605i	$-0.115385\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.11325	+	1.11325i	−1.11325	+	1.11325i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$929$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$930$		0			0
$931$	0.0217178	+	0.0359256i	0.0217178	+	0.0359256i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.527986	−	0.764919i	0.527986	−	0.764919i	−0.464723	−	0.885456i	$-0.653846\pi$
$937$	0.527986	−	0.764919i	0.527986	−	0.764919i	0.992709	+	0.120537i	$0.0384615\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.297503	−	0.954721i	$-0.596154\pi$
$941$		0			0		0.297503	+	0.954721i	$0.403846\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.983620	−	0.180255i	$-0.942308\pi$
$947$		0			0		0.983620	+	0.180255i	$0.0576923\pi$
$948$		0			0
$949$	−0.783659	+	0.244198i	−0.783659	+	0.244198i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$953$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.863689	+	0.104871i	0.863689	+	0.104871i
$962$		0			0
$963$		0			0
$964$	1.68353	−	1.01773i	1.68353	−	1.01773i
$965$		0			0
$966$		0			0
$967$	0.354605	+	0.0649838i	0.354605	+	0.0649838i	0.354605	−	0.935016i	$-0.384615\pi$
$967$	0.354605	+	0.0649838i	0.354605	+	0.0649838i			1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$971$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$972$		0			0
$973$	1.32555	+	0.801321i	1.32555	+	0.801321i
$974$		0			0
$975$		0			0
$976$	−1.45743	+	0.764919i	−1.45743	+	0.764919i
$977$		0			0		−0.998176	−	0.0603785i	$-0.980769\pi$
$977$		0			0		0.998176	+	0.0603785i	$0.0192308\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.0603785	−	0.998176i	$-0.480769\pi$
$983$		0			0		−0.0603785	+	0.998176i	$0.519231\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	0.107253	+	0.585260i	0.107253	+	0.585260i
$989$		0			0
$990$		0			0
$991$	1.77091			1.77091			0.885456	−	0.464723i	$-0.153846\pi$
$991$	1.77091			1.77091			0.885456	+	0.464723i	$0.153846\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.180446	−	1.48611i	0.180446	−	1.48611i	−0.568065	−	0.822984i	$-0.692308\pi$
$997$	0.180446	−	1.48611i	0.180446	−	1.48611i	0.748511	−	0.663123i	$-0.230769\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.1.bm.a.694.1	✓	24
3.2	odd	2	CM	1521.1.bm.a.694.1	✓	24
169.47	odd	52	inner	1521.1.bm.a.892.1	yes	24
507.47	even	52	inner	1521.1.bm.a.892.1	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.1.bm.a.694.1	✓	24	1.1	even	1	trivial
1521.1.bm.a.694.1	✓	24	3.2	odd	2	CM
1521.1.bm.a.892.1	yes	24	169.47	odd	52	inner
1521.1.bm.a.892.1	yes	24	507.47	even	52	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1521.bm (of order \(52\), degree \(24\), minimal)

\(f(q)\)	\(=\)	\(q+(0.822984 + 0.568065i) q^{4} +(0.814480 - 0.638104i) q^{7} +O(q^{10})\)	\(q+(0.822984 + 0.568065i) q^{4} +(0.814480 - 0.638104i) q^{7} -1.00000i q^{13} +(0.354605 + 0.935016i) q^{16} +(-0.420733 + 0.420733i) q^{19} +(0.663123 - 0.748511i) q^{25} +(1.03279 - 0.0624722i) q^{28} +(0.0217671 - 0.359852i) q^{31} +(-1.57149 - 0.0950579i) q^{37} +(-1.31658 + 1.48611i) q^{43} +(0.0168846 - 0.0685035i) q^{49} +(0.568065 - 0.822984i) q^{52} +(0.198399 + 1.63397i) q^{61} +(-0.239316 + 0.970942i) q^{64} +(1.96365 + 0.359852i) q^{67} +(-0.244198 - 0.783659i) q^{73} +(-0.585260 + 0.107253i) q^{76} +(0.271894 + 0.393906i) q^{79} +(-0.638104 - 0.814480i) q^{91} +(-1.28962 - 0.580411i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 2 q^{7}+O(q^{10}) \) 24 * q + 2 * q^7	\( 24 q + 2 q^{7} + 2 q^{16} + 2 q^{19} - 2 q^{28} - 2 q^{31} - 2 q^{37} - 2 q^{52} + 2 q^{67} + 2 q^{73} + 2 q^{76} + 2 q^{91} - 2 q^{97}+O(q^{100}) \) 24 * q + 2 * q^7 + 2 * q^16 + 2 * q^19 - 2 * q^28 - 2 * q^31 - 2 * q^37 - 2 * q^52 + 2 * q^67 + 2 * q^73 + 2 * q^76 + 2 * q^91 - 2 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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