LMFDB - Embedded newform 1587.1.h.c.647.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1587,1,Mod(170,1587)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1587, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 10]))

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

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

chi := DirichletCharacter("1587.170");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1587 = 3 \cdot 23^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1587.h (of order $22$, degree $10$, not 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:	$0.792016175049$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	20.0.5969915757478328440239161344.6
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 2x^{18} + 4x^{16} + 8x^{14} + 16x^{12} + 32x^{10} + 64x^{8} + 128x^{6} + 256x^{4} + 512x^{2} + 1024 $ x^20 + 2x^18 + 4x^16 + 8x^14 + 16x^12 + 32x^10 + 64x^8 + 128x^6 + 256x^4 + 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{4}$
Projective field:	Galois closure of 4.0.36501.1
Artin image:	$C_{11}\times D_8$
Artin field:	Galois closure of $\mathbb{Q}[x]/(x^{88} - \cdots)$

Embedding invariants

Embedding label			647.1
Root			$-0.201264 - 1.39982i$ of defining polynomial
Character	$\chi$	$=$	1587.647
Dual form			1587.1.h.c.1457.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.415415 + 0.909632i) q^{3} +(-0.959493 + 0.281733i) q^{4} +(-1.18971 + 0.764582i) q^{7} +(-0.654861 - 0.755750i) q^{9} +(0.142315 - 0.989821i) q^{12} +(0.841254 - 0.540641i) q^{16} +(-1.35693 + 0.398430i) q^{19} +(-0.201264 - 1.39982i) q^{21} +(-0.142315 - 0.989821i) q^{25} +(0.959493 - 0.281733i) q^{27} +(0.926113 - 1.06879i) q^{28} +(0.841254 + 0.540641i) q^{36} +(-0.926113 - 1.06879i) q^{37} +(0.587486 - 1.28641i) q^{43} +(0.142315 + 0.989821i) q^{48} +(0.415415 - 0.909632i) q^{49} +(0.201264 - 1.39982i) q^{57} +(-0.587486 - 1.28641i) q^{61} +(1.35693 + 0.398430i) q^{63} +(-0.654861 + 0.755750i) q^{64} +(0.201264 + 1.39982i) q^{67} +(0.959493 + 0.281733i) q^{75} +(1.18971 - 0.764582i) q^{76} +(-1.18971 - 0.764582i) q^{79} +(-0.142315 + 0.989821i) q^{81} +(0.587486 + 1.28641i) q^{84} +(-0.926113 + 1.06879i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} - 2 q^{4} - 2 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{25} + 2 q^{27} - 2 q^{36} + 2 q^{48} - 2 q^{49} - 2 q^{64} + 2 q^{75} - 2 q^{81}+O(q^{100}) $ 20 * q + 2 * q^3 - 2 * q^4 - 2 * q^9 + 2 * q^12 - 2 * q^16 - 2 * q^25 + 2 * q^27 - 2 * q^36 + 2 * q^48 - 2 * q^49 - 2 * q^64 + 2 * q^75 - 2 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1587\mathbb{Z}\right)^\times$.

$n$	$530$	$1063$
$\chi(n)$	$-1$	$e\left(\frac{8}{11}\right)$

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		−0.142315	−	0.989821i	$-0.545455\pi$
$2$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$3$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$3$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$4$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$5$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$5$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$6$		0			0
$7$	−1.18971	+	0.764582i	−1.18971	+	0.764582i	−0.977147	−	0.212565i	$-0.931818\pi$
$7$	−1.18971	+	0.764582i	−1.18971	+	0.764582i	−0.212565	+	0.977147i	$0.568182\pi$
$8$		0			0
$9$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$10$		0			0
$11$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$11$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$12$	0.142315	−	0.989821i	0.142315	−	0.989821i
$13$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$13$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$14$		0			0
$15$		0			0
$16$	0.841254	−	0.540641i	0.841254	−	0.540641i
$17$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$17$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$18$		0			0
$19$	−1.35693	+	0.398430i	−1.35693	+	0.398430i	−0.877679	−	0.479249i	$-0.840909\pi$
$19$	−1.35693	+	0.398430i	−1.35693	+	0.398430i	−0.479249	+	0.877679i	$0.659091\pi$
$20$		0			0
$21$	−0.201264	−	1.39982i	−0.201264	−	1.39982i
$22$		0			0
$23$		0			0
$23$		0			0
$24$		0			0
$25$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$26$		0			0
$27$	0.959493	−	0.281733i	0.959493	−	0.281733i
$28$	0.926113	−	1.06879i	0.926113	−	1.06879i
$29$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$29$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$30$		0			0
$31$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$31$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.841254	+	0.540641i	0.841254	+	0.540641i
$37$	−0.926113	−	1.06879i	−0.926113	−	1.06879i	−0.997452	−	0.0713392i	$-0.977273\pi$
$37$	−0.926113	−	1.06879i	−0.926113	−	1.06879i	0.0713392	−	0.997452i	$-0.477273\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$41$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$42$		0			0
$43$	0.587486	−	1.28641i	0.587486	−	1.28641i	−0.349464	−	0.936950i	$-0.613636\pi$
$43$	0.587486	−	1.28641i	0.587486	−	1.28641i	0.936950	−	0.349464i	$-0.113636\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	0.142315	+	0.989821i	0.142315	+	0.989821i
$49$	0.415415	−	0.909632i	0.415415	−	0.909632i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$53$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.201264	−	1.39982i	0.201264	−	1.39982i
$58$		0			0
$59$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$59$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$60$		0			0
$61$	−0.587486	−	1.28641i	−0.587486	−	1.28641i	−0.936950	−	0.349464i	$-0.886364\pi$
$61$	−0.587486	−	1.28641i	−0.587486	−	1.28641i	0.349464	−	0.936950i	$-0.386364\pi$
$62$		0			0
$63$	1.35693	+	0.398430i	1.35693	+	0.398430i
$64$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$65$		0			0
$66$		0			0
$67$	0.201264	+	1.39982i	0.201264	+	1.39982i	0.800541	+	0.599278i	$0.204545\pi$
$67$	0.201264	+	1.39982i	0.201264	+	1.39982i	−0.599278	+	0.800541i	$0.704545\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$71$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$72$		0			0
$73$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$73$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$74$		0			0
$75$	0.959493	+	0.281733i	0.959493	+	0.281733i
$76$	1.18971	−	0.764582i	1.18971	−	0.764582i
$77$		0			0
$78$		0			0
$79$	−1.18971	−	0.764582i	−1.18971	−	0.764582i	−0.212565	−	0.977147i	$-0.568182\pi$
$79$	−1.18971	−	0.764582i	−1.18971	−	0.764582i	−0.977147	+	0.212565i	$0.931818\pi$
$80$		0			0
$81$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$82$		0			0
$83$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$83$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$84$	0.587486	+	1.28641i	0.587486	+	1.28641i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$89$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.926113	+	1.06879i	−0.926113	+	1.06879i	0.0713392	+	0.997452i	$0.477273\pi$
$97$	−0.926113	+	1.06879i	−0.926113	+	1.06879i	−0.997452	+	0.0713392i	$0.977273\pi$
$98$		0			0
$99$		0			0
$100$	0.415415	+	0.909632i	0.415415	+	0.909632i
$101$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$101$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$102$		0			0
$103$	−0.201264	+	1.39982i	−0.201264	+	1.39982i	0.599278	+	0.800541i	$0.295455\pi$
$103$	−0.201264	+	1.39982i	−0.201264	+	1.39982i	−0.800541	+	0.599278i	$0.795455\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$107$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$108$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$109$	−1.35693	−	0.398430i	−1.35693	−	0.398430i	−0.479249	−	0.877679i	$-0.659091\pi$
$109$	−1.35693	−	0.398430i	−1.35693	−	0.398430i	−0.877679	+	0.479249i	$0.840909\pi$
$110$		0			0
$111$	1.35693	−	0.398430i	1.35693	−	0.398430i
$112$	−0.587486	+	1.28641i	−0.587486	+	1.28641i
$113$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$113$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$127$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$128$		0			0
$129$	0.926113	+	1.06879i	0.926113	+	1.06879i
$130$		0			0
$131$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$131$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$132$		0			0
$133$	1.30972	−	1.51150i	1.30972	−	1.51150i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$	−2.00000			−2.00000			−1.00000			$\pi$
$139$	−2.00000			−2.00000			−1.00000			$\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$145$		0			0
$146$		0			0
$147$	0.654861	+	0.755750i	0.654861	+	0.755750i
$148$	1.18971	+	0.764582i	1.18971	+	0.764582i
$149$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$149$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$150$		0			0
$151$	−1.68251	−	1.08128i	−1.68251	−	1.08128i	−0.841254	−	0.540641i	$-0.818182\pi$
$151$	−1.68251	−	1.08128i	−1.68251	−	1.08128i	−0.841254	−	0.540641i	$-0.818182\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.35693	−	0.398430i	1.35693	−	0.398430i	0.479249	−	0.877679i	$-0.340909\pi$
$157$	1.35693	−	0.398430i	1.35693	−	0.398430i	0.877679	+	0.479249i	$0.159091\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$163$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$167$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$168$		0			0
$169$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$170$		0			0
$171$	1.18971	+	0.764582i	1.18971	+	0.764582i
$172$	−0.201264	+	1.39982i	−0.201264	+	1.39982i
$173$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$173$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$174$		0			0
$175$	0.926113	+	1.06879i	0.926113	+	1.06879i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$179$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$180$		0			0
$181$	−0.587486	+	1.28641i	−0.587486	+	1.28641i	0.349464	+	0.936950i	$0.386364\pi$
$181$	−0.587486	+	1.28641i	−0.587486	+	1.28641i	−0.936950	+	0.349464i	$0.886364\pi$
$182$		0			0
$183$	1.41421			1.41421
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−0.926113	+	1.06879i	−0.926113	+	1.06879i
$190$		0			0
$191$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$191$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$192$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$193$	1.30972	+	1.51150i	1.30972	+	1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$193$	1.30972	+	1.51150i	1.30972	+	1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$194$		0			0
$195$		0			0
$196$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$197$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$197$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$198$		0			0
$199$	0.587486	+	1.28641i	0.587486	+	1.28641i	0.936950	+	0.349464i	$0.113636\pi$
$199$	0.587486	+	1.28641i	0.587486	+	1.28641i	−0.349464	+	0.936950i	$0.613636\pi$
$200$		0			0
$201$	−1.35693	−	0.398430i	−1.35693	−	0.398430i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$211$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.68251	+	1.08128i	−1.68251	+	1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$223$	−1.68251	+	1.08128i	−1.68251	+	1.08128i	−0.841254	+	0.540641i	$0.818182\pi$
$224$		0			0
$225$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$226$		0			0
$227$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$227$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$228$	0.201264	+	1.39982i	0.201264	+	1.39982i
$229$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$229$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$233$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.18971	−	0.764582i	1.18971	−	0.764582i
$238$		0			0
$239$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$239$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$240$		0			0
$241$	0.201264	−	1.39982i	0.201264	−	1.39982i	−0.599278	−	0.800541i	$-0.704545\pi$
$241$	0.201264	−	1.39982i	0.201264	−	1.39982i	0.800541	−	0.599278i	$-0.204545\pi$
$242$		0			0
$243$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$244$	0.926113	+	1.06879i	0.926113	+	1.06879i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$251$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$252$	−1.41421			−1.41421
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.415415	−	0.909632i	0.415415	−	0.909632i
$257$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$257$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$258$		0			0
$259$	1.91899	+	0.563465i	1.91899	+	0.563465i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$263$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−0.587486	−	1.28641i	−0.587486	−	1.28641i
$269$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$269$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$270$		0			0
$271$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$271$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$281$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$282$		0			0
$283$	1.18971	−	0.764582i	1.18971	−	0.764582i	0.212565	−	0.977147i	$-0.431818\pi$
$283$	1.18971	−	0.764582i	1.18971	−	0.764582i	0.977147	+	0.212565i	$0.0681818\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.841254	+	0.540641i	0.841254	+	0.540641i
$290$		0			0
$291$	−0.587486	−	1.28641i	−0.587486	−	1.28641i
$292$		0			0
$293$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$293$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1.00000			−1.00000
$301$	0.284630	+	1.97964i	0.284630	+	1.97964i
$302$		0			0
$303$		0			0
$304$	−0.926113	+	1.06879i	−0.926113	+	1.06879i
$305$		0			0
$306$		0			0
$307$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$307$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$308$		0			0
$309$	−1.18971	−	0.764582i	−1.18971	−	0.764582i
$310$		0			0
$311$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$311$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$312$		0			0
$313$	−0.926113	−	1.06879i	−0.926113	−	1.06879i	−0.997452	−	0.0713392i	$-0.977273\pi$
$313$	−0.926113	−	1.06879i	−0.926113	−	1.06879i	0.0713392	−	0.997452i	$-0.477273\pi$
$314$		0			0
$315$		0			0
$316$	1.35693	+	0.398430i	1.35693	+	0.398430i
$317$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$317$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$325$		0			0
$326$		0			0
$327$	0.926113	−	1.06879i	0.926113	−	1.06879i
$328$		0			0
$329$		0			0
$330$		0			0
$331$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$331$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$332$		0			0
$333$	−0.201264	+	1.39982i	−0.201264	+	1.39982i
$334$		0			0
$335$		0			0
$336$	−0.926113	−	1.06879i	−0.926113	−	1.06879i
$337$	0.587486	+	1.28641i	0.587486	+	1.28641i	0.936950	+	0.349464i	$0.113636\pi$
$337$	0.587486	+	1.28641i	0.587486	+	1.28641i	−0.349464	+	0.936950i	$0.613636\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$347$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$348$		0			0
$349$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$349$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$353$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$359$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$360$		0			0
$361$	0.841254	−	0.540641i	0.841254	−	0.540641i
$362$		0			0
$363$	0.654861	−	0.755750i	0.654861	−	0.755750i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$367$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.926113	−	1.06879i	0.926113	−	1.06879i	−0.0713392	−	0.997452i	$-0.522727\pi$
$373$	0.926113	−	1.06879i	0.926113	−	1.06879i	0.997452	−	0.0713392i	$-0.0227273\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.201264	−	1.39982i	0.201264	−	1.39982i	−0.599278	−	0.800541i	$-0.704545\pi$
$379$	0.201264	−	1.39982i	0.201264	−	1.39982i	0.800541	−	0.599278i	$-0.204545\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$383$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.35693	+	0.398430i	−1.35693	+	0.398430i
$388$	0.587486	−	1.28641i	0.587486	−	1.28641i
$389$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$389$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−1.91899	−	0.563465i	−1.91899	−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$397$	−1.91899	−	0.563465i	−1.91899	−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$398$		0			0
$399$	0.830830	+	1.81926i	0.830830	+	1.81926i
$400$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$401$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$401$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.30972	+	1.51150i	−1.30972	+	1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$409$	−1.30972	+	1.51150i	−1.30972	+	1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$410$		0			0
$411$		0			0
$412$	−0.201264	−	1.39982i	−0.201264	−	1.39982i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.830830	−	1.81926i	0.830830	−	1.81926i
$418$		0			0
$419$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$419$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$420$		0			0
$421$	−1.18971	+	0.764582i	−1.18971	+	0.764582i	−0.977147	−	0.212565i	$-0.931818\pi$
$421$	−1.18971	+	0.764582i	−1.18971	+	0.764582i	−0.212565	+	0.977147i	$0.568182\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.68251	+	1.08128i	1.68251	+	1.08128i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$431$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$432$	0.654861	−	0.755750i	0.654861	−	0.755750i
$433$	1.35693	−	0.398430i	1.35693	−	0.398430i	0.479249	−	0.877679i	$-0.340909\pi$
$433$	1.35693	−	0.398430i	1.35693	−	0.398430i	0.877679	+	0.479249i	$0.159091\pi$
$434$		0			0
$435$		0			0
$436$	1.41421			1.41421
$437$		0			0
$438$		0			0
$439$	−0.284630	−	1.97964i	−0.284630	−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$439$	−0.284630	−	1.97964i	−0.284630	−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$440$		0			0
$441$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$442$		0			0
$443$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$443$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$444$	−1.18971	+	0.764582i	−1.18971	+	0.764582i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.201264	−	1.39982i	0.201264	−	1.39982i
$449$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$449$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.68251	−	1.08128i	1.68251	−	1.08128i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.587486	−	1.28641i	0.587486	−	1.28641i	−0.349464	−	0.936950i	$-0.613636\pi$
$457$	0.587486	−	1.28641i	0.587486	−	1.28641i	0.936950	−	0.349464i	$-0.113636\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−0.830830	+	1.81926i	−0.830830	+	1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$463$	−0.830830	+	1.81926i	−0.830830	+	1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$467$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$468$		0			0
$469$	−1.30972	−	1.51150i	−1.30972	−	1.51150i
$470$		0			0
$471$	−0.201264	+	1.39982i	−0.201264	+	1.39982i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	0.587486	+	1.28641i	0.587486	+	1.28641i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$479$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.00000			1.00000
$485$		0			0
$486$		0			0
$487$	−1.91899	+	0.563465i	−1.91899	+	0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$487$	−1.91899	+	0.563465i	−1.91899	+	0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$491$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$499$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$503$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.00000			1.00000
$508$		0			0
$509$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$509$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−1.18971	+	0.764582i	−1.18971	+	0.764582i
$514$		0			0
$515$		0			0
$516$	−1.18971	−	0.764582i	−1.18971	−	0.764582i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$521$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$522$		0			0
$523$	1.35693	+	0.398430i	1.35693	+	0.398430i	0.877679	−	0.479249i	$-0.159091\pi$
$523$	1.35693	+	0.398430i	1.35693	+	0.398430i	0.479249	+	0.877679i	$0.340909\pi$
$524$		0			0
$525$	−1.35693	+	0.398430i	−1.35693	+	0.398430i
$526$		0			0
$527$		0			0
$528$		0			0
$529$		0			0
$530$		0			0
$531$		0			0
$532$	−0.830830	+	1.81926i	−0.830830	+	1.81926i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.284630	−	1.97964i	0.284630	−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$541$	0.284630	−	1.97964i	0.284630	−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$542$		0			0
$543$	−0.926113	−	1.06879i	−0.926113	−	1.06879i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$547$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$548$		0			0
$549$	−0.587486	+	1.28641i	−0.587486	+	1.28641i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.00000			2.00000
$554$		0			0
$555$		0			0
$556$	1.91899	−	0.563465i	1.91899	−	0.563465i
$557$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$557$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$563$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.587486	−	1.28641i	−0.587486	−	1.28641i
$568$		0			0
$569$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$569$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$570$		0			0
$571$	−1.35693	+	0.398430i	−1.35693	+	0.398430i	−0.877679	−	0.479249i	$-0.840909\pi$
$571$	−1.35693	+	0.398430i	−1.35693	+	0.398430i	−0.479249	+	0.877679i	$0.659091\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.00000			1.00000
$577$	0.284630	+	1.97964i	0.284630	+	1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$577$	0.284630	+	1.97964i	0.284630	+	1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$578$		0			0
$579$	−1.91899	+	0.563465i	−1.91899	+	0.563465i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$587$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$588$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.35693	−	0.398430i	−1.35693	−	0.398430i
$593$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$593$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.41421			−1.41421
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	−0.830830	+	1.81926i	−0.830830	+	1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$601$	−0.830830	+	1.81926i	−0.830830	+	1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$602$		0			0
$603$	0.926113	−	1.06879i	0.926113	−	1.06879i
$604$	1.91899	+	0.563465i	1.91899	+	0.563465i
$605$		0			0
$606$		0			0
$607$	−1.30972	−	1.51150i	−1.30972	−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$607$	−1.30972	−	1.51150i	−1.30972	−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.587486	−	1.28641i	−0.587486	−	1.28641i	−0.936950	−	0.349464i	$-0.886364\pi$
$613$	−0.587486	−	1.28641i	−0.587486	−	1.28641i	0.349464	−	0.936950i	$-0.386364\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$617$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$618$		0			0
$619$	−0.201264	−	1.39982i	−0.201264	−	1.39982i	−0.800541	−	0.599278i	$-0.795455\pi$
$619$	−0.201264	−	1.39982i	−0.201264	−	1.39982i	0.599278	−	0.800541i	$-0.295455\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$626$		0			0
$627$		0			0
$628$	−1.18971	+	0.764582i	−1.18971	+	0.764582i
$629$		0			0
$630$		0			0
$631$	1.18971	+	0.764582i	1.18971	+	0.764582i	0.977147	−	0.212565i	$-0.0681818\pi$
$631$	1.18971	+	0.764582i	1.18971	+	0.764582i	0.212565	+	0.977147i	$0.431818\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$641$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$642$		0			0
$643$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$643$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$647$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$653$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$659$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$660$		0			0
$661$	1.35693	+	0.398430i	1.35693	+	0.398430i	0.877679	−	0.479249i	$-0.159091\pi$
$661$	1.35693	+	0.398430i	1.35693	+	0.398430i	0.479249	+	0.877679i	$0.340909\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−0.284630	−	1.97964i	−0.284630	−	1.97964i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$673$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$674$		0			0
$675$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$676$	0.654861	+	0.755750i	0.654861	+	0.755750i
$677$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$677$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$678$		0			0
$679$	0.284630	−	1.97964i	0.284630	−	1.97964i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$683$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$684$	−1.35693	−	0.398430i	−1.35693	−	0.398430i
$685$		0			0
$686$		0			0
$687$	0.587486	−	1.28641i	0.587486	−	1.28641i
$688$	−0.201264	−	1.39982i	−0.201264	−	1.39982i
$689$		0			0
$690$		0			0
$691$		0			0			−	1.00000i	$-0.5\pi$
$691$		0			0				1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.18971	−	0.764582i	−1.18971	−	0.764582i
$701$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$701$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$702$		0			0
$703$	1.68251	+	1.08128i	1.68251	+	1.08128i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.35693	+	0.398430i	−1.35693	+	0.398430i	−0.877679	−	0.479249i	$-0.840909\pi$
$709$	−1.35693	+	0.398430i	−1.35693	+	0.398430i	−0.479249	+	0.877679i	$0.659091\pi$
$710$		0			0
$711$	0.201264	+	1.39982i	0.201264	+	1.39982i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$719$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$720$		0			0
$721$	−0.830830	−	1.81926i	−0.830830	−	1.81926i
$722$		0			0
$723$	1.18971	+	0.764582i	1.18971	+	0.764582i
$724$	0.201264	−	1.39982i	0.201264	−	1.39982i
$725$		0			0
$726$		0			0
$727$	0.926113	+	1.06879i	0.926113	+	1.06879i	0.997452	+	0.0713392i	$0.0227273\pi$
$727$	0.926113	+	1.06879i	0.926113	+	1.06879i	−0.0713392	+	0.997452i	$0.522727\pi$
$728$		0			0
$729$	0.841254	−	0.540641i	0.841254	−	0.540641i
$730$		0			0
$731$		0			0
$732$	−1.35693	+	0.398430i	−1.35693	+	0.398430i
$733$	−0.587486	+	1.28641i	−0.587486	+	1.28641i	0.349464	+	0.936950i	$0.386364\pi$
$733$	−0.587486	+	1.28641i	−0.587486	+	1.28641i	−0.936950	+	0.349464i	$0.886364\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.830830	−	1.81926i	0.830830	−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$739$	0.830830	−	1.81926i	0.830830	−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$743$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.587486	+	1.28641i	0.587486	+	1.28641i	0.936950	+	0.349464i	$0.113636\pi$
$751$	0.587486	+	1.28641i	0.587486	+	1.28641i	−0.349464	+	0.936950i	$0.613636\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.587486	−	1.28641i	0.587486	−	1.28641i
$757$	−0.201264	−	1.39982i	−0.201264	−	1.39982i	−0.800541	−	0.599278i	$-0.795455\pi$
$757$	−0.201264	−	1.39982i	−0.201264	−	1.39982i	0.599278	−	0.800541i	$-0.295455\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$761$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$762$		0			0
$763$	1.91899	−	0.563465i	1.91899	−	0.563465i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.654861	+	0.755750i	0.654861	+	0.755750i
$769$	1.18971	+	0.764582i	1.18971	+	0.764582i	0.977147	−	0.212565i	$-0.0681818\pi$
$769$	1.18971	+	0.764582i	1.18971	+	0.764582i	0.212565	+	0.977147i	$0.431818\pi$
$770$		0			0
$771$		0			0
$772$	−1.68251	−	1.08128i	−1.68251	−	1.08128i
$773$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$773$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	−1.30972	+	1.51150i	−1.30972	+	1.51150i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$785$		0			0
$786$		0			0
$787$	0.926113	−	1.06879i	0.926113	−	1.06879i	−0.0713392	−	0.997452i	$-0.522727\pi$
$787$	0.926113	−	1.06879i	0.926113	−	1.06879i	0.997452	−	0.0713392i	$-0.0227273\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−0.926113	−	1.06879i	−0.926113	−	1.06879i
$797$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$797$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.41421			1.41421
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$809$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$810$		0			0
$811$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$811$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.284630	+	1.97964i	−0.284630	+	1.97964i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$821$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$822$		0			0
$823$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$823$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−2.00000			−2.00000			−1.00000			$\pi$
$829$	−2.00000			−2.00000			−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$839$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$840$		0			0
$841$	0.841254	+	0.540641i	0.841254	+	0.540641i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$		0			0
$847$	1.35693	−	0.398430i	1.35693	−	0.398430i
$848$		0			0
$849$	0.201264	+	1.39982i	0.201264	+	1.39982i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.284630	+	1.97964i	0.284630	+	1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$853$	0.284630	+	1.97964i	0.284630	+	1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$857$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$858$		0			0
$859$	0.830830	+	1.81926i	0.830830	+	1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$859$	0.830830	+	1.81926i	0.830830	+	1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$863$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.41421			1.41421
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$877$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$881$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$882$		0			0
$883$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$883$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$887$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$	1.30972	−	1.51150i	1.30972	−	1.51150i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.415415	−	0.909632i	0.415415	−	0.909632i
$901$		0			0
$902$		0			0
$903$	−1.91899	−	0.563465i	−1.91899	−	0.563465i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.18971	−	0.764582i	−1.18971	−	0.764582i	−0.212565	−	0.977147i	$-0.568182\pi$
$907$	−1.18971	−	0.764582i	−1.18971	−	0.764582i	−0.977147	+	0.212565i	$0.931818\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$911$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$912$	−0.587486	−	1.28641i	−0.587486	−	1.28641i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.35693	−	0.398430i	1.35693	−	0.398430i
$917$		0			0
$918$		0			0
$919$	−1.41421			−1.41421			−0.707107	−	0.707107i	$-0.750000\pi$
$919$	−1.41421			−1.41421			−0.707107	+	0.707107i	$0.750000\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.926113	+	1.06879i	−0.926113	+	1.06879i
$926$		0			0
$927$	1.18971	−	0.764582i	1.18971	−	0.764582i
$928$		0			0
$929$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$929$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$930$		0			0
$931$	−0.201264	+	1.39982i	−0.201264	+	1.39982i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.35693	+	0.398430i	1.35693	+	0.398430i	0.877679	−	0.479249i	$-0.159091\pi$
$937$	1.35693	+	0.398430i	1.35693	+	0.398430i	0.479249	+	0.877679i	$0.340909\pi$
$938$		0			0
$939$	1.35693	−	0.398430i	1.35693	−	0.398430i
$940$		0			0
$941$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$941$		0			0		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		0.959493	−	0.281733i	$-0.0909091\pi$
$947$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$948$	−0.926113	+	1.06879i	−0.926113	+	1.06879i
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$953$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.654861	−	0.755750i	0.654861	−	0.755750i
$962$		0			0
$963$		0			0
$964$	0.201264	+	1.39982i	0.201264	+	1.39982i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$971$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$972$	0.959493	+	0.281733i	0.959493	+	0.281733i
$973$	2.37942	−	1.52916i	2.37942	−	1.52916i
$974$		0			0
$975$		0			0
$976$	−1.18971	−	0.764582i	−1.18971	−	0.764582i
$977$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$977$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.587486	+	1.28641i	0.587486	+	1.28641i
$982$		0			0
$983$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$983$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$991$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.830830	+	1.81926i	0.830830	+	1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$997$	0.830830	+	1.81926i	0.830830	+	1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$998$		0			0
$999$	−1.18971	−	0.764582i	−1.18971	−	0.764582i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1587.1.h.c.647.1		20
3.2	odd	2	CM	1587.1.h.c.647.1		20
23.2	even	11	inner	1587.1.h.c.863.2		20
23.3	even	11	inner	1587.1.h.c.1016.2		20
23.4	even	11	inner	1587.1.h.c.1313.2		20
23.5	odd	22	inner	1587.1.h.c.995.1		20
23.6	even	11	inner	1587.1.h.c.1235.2		20
23.7	odd	22	inner	1587.1.h.c.170.2		20
23.8	even	11	inner	1587.1.h.c.1457.1		20
23.9	even	11	inner	1587.1.h.c.266.2		20
23.10	odd	22		1587.1.b.b.530.2	yes	2
23.11	odd	22	inner	1587.1.h.c.1559.2		20
23.12	even	11	inner	1587.1.h.c.1559.1		20
23.13	even	11		1587.1.b.b.530.1	✓	2
23.14	odd	22	inner	1587.1.h.c.266.1		20
23.15	odd	22	inner	1587.1.h.c.1457.2		20
23.16	even	11	inner	1587.1.h.c.170.1		20
23.17	odd	22	inner	1587.1.h.c.1235.1		20
23.18	even	11	inner	1587.1.h.c.995.2		20
23.19	odd	22	inner	1587.1.h.c.1313.1		20
23.20	odd	22	inner	1587.1.h.c.1016.1		20
23.21	odd	22	inner	1587.1.h.c.863.1		20
23.22	odd	2	inner	1587.1.h.c.647.2		20
69.2	odd	22	inner	1587.1.h.c.863.2		20
69.5	even	22	inner	1587.1.h.c.995.1		20
69.8	odd	22	inner	1587.1.h.c.1457.1		20
69.11	even	22	inner	1587.1.h.c.1559.2		20
69.14	even	22	inner	1587.1.h.c.266.1		20
69.17	even	22	inner	1587.1.h.c.1235.1		20
69.20	even	22	inner	1587.1.h.c.1016.1		20
69.26	odd	22	inner	1587.1.h.c.1016.2		20
69.29	odd	22	inner	1587.1.h.c.1235.2		20
69.32	odd	22	inner	1587.1.h.c.266.2		20
69.35	odd	22	inner	1587.1.h.c.1559.1		20
69.38	even	22	inner	1587.1.h.c.1457.2		20
69.41	odd	22	inner	1587.1.h.c.995.2		20
69.44	even	22	inner	1587.1.h.c.863.1		20
69.50	odd	22	inner	1587.1.h.c.1313.2		20
69.53	even	22	inner	1587.1.h.c.170.2		20
69.56	even	22		1587.1.b.b.530.2	yes	2
69.59	odd	22		1587.1.b.b.530.1	✓	2
69.62	odd	22	inner	1587.1.h.c.170.1		20
69.65	even	22	inner	1587.1.h.c.1313.1		20
69.68	even	2	inner	1587.1.h.c.647.2		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1587.1.b.b.530.1	✓	2	23.13	even	11
1587.1.b.b.530.1	✓	2	69.59	odd	22
1587.1.b.b.530.2	yes	2	23.10	odd	22
1587.1.b.b.530.2	yes	2	69.56	even	22
1587.1.h.c.170.1		20	23.16	even	11	inner
1587.1.h.c.170.1		20	69.62	odd	22	inner
1587.1.h.c.170.2		20	23.7	odd	22	inner
1587.1.h.c.170.2		20	69.53	even	22	inner
1587.1.h.c.266.1		20	23.14	odd	22	inner
1587.1.h.c.266.1		20	69.14	even	22	inner
1587.1.h.c.266.2		20	23.9	even	11	inner
1587.1.h.c.266.2		20	69.32	odd	22	inner
1587.1.h.c.647.1		20	1.1	even	1	trivial
1587.1.h.c.647.1		20	3.2	odd	2	CM
1587.1.h.c.647.2		20	23.22	odd	2	inner
1587.1.h.c.647.2		20	69.68	even	2	inner
1587.1.h.c.863.1		20	23.21	odd	22	inner
1587.1.h.c.863.1		20	69.44	even	22	inner
1587.1.h.c.863.2		20	23.2	even	11	inner
1587.1.h.c.863.2		20	69.2	odd	22	inner
1587.1.h.c.995.1		20	23.5	odd	22	inner
1587.1.h.c.995.1		20	69.5	even	22	inner
1587.1.h.c.995.2		20	23.18	even	11	inner
1587.1.h.c.995.2		20	69.41	odd	22	inner
1587.1.h.c.1016.1		20	23.20	odd	22	inner
1587.1.h.c.1016.1		20	69.20	even	22	inner
1587.1.h.c.1016.2		20	23.3	even	11	inner
1587.1.h.c.1016.2		20	69.26	odd	22	inner
1587.1.h.c.1235.1		20	23.17	odd	22	inner
1587.1.h.c.1235.1		20	69.17	even	22	inner
1587.1.h.c.1235.2		20	23.6	even	11	inner
1587.1.h.c.1235.2		20	69.29	odd	22	inner
1587.1.h.c.1313.1		20	23.19	odd	22	inner
1587.1.h.c.1313.1		20	69.65	even	22	inner
1587.1.h.c.1313.2		20	23.4	even	11	inner
1587.1.h.c.1313.2		20	69.50	odd	22	inner
1587.1.h.c.1457.1		20	23.8	even	11	inner
1587.1.h.c.1457.1		20	69.8	odd	22	inner
1587.1.h.c.1457.2		20	23.15	odd	22	inner
1587.1.h.c.1457.2		20	69.38	even	22	inner
1587.1.h.c.1559.1		20	23.12	even	11	inner
1587.1.h.c.1559.1		20	69.35	odd	22	inner
1587.1.h.c.1559.2		20	23.11	odd	22	inner
1587.1.h.c.1559.2		20	69.11	even	22	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:	\( N \)	\(=\)	\( 1587 = 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1587.h (of order \(22\), degree \(10\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.415415 + 0.909632i) q^{3} +(-0.959493 + 0.281733i) q^{4} +(-1.18971 + 0.764582i) q^{7} +(-0.654861 - 0.755750i) q^{9} +(0.142315 - 0.989821i) q^{12} +(0.841254 - 0.540641i) q^{16} +(-1.35693 + 0.398430i) q^{19} +(-0.201264 - 1.39982i) q^{21} +(-0.142315 - 0.989821i) q^{25} +(0.959493 - 0.281733i) q^{27} +(0.926113 - 1.06879i) q^{28} +(0.841254 + 0.540641i) q^{36} +(-0.926113 - 1.06879i) q^{37} +(0.587486 - 1.28641i) q^{43} +(0.142315 + 0.989821i) q^{48} +(0.415415 - 0.909632i) q^{49} +(0.201264 - 1.39982i) q^{57} +(-0.587486 - 1.28641i) q^{61} +(1.35693 + 0.398430i) q^{63} +(-0.654861 + 0.755750i) q^{64} +(0.201264 + 1.39982i) q^{67} +(0.959493 + 0.281733i) q^{75} +(1.18971 - 0.764582i) q^{76} +(-1.18971 - 0.764582i) q^{79} +(-0.142315 + 0.989821i) q^{81} +(0.587486 + 1.28641i) q^{84} +(-0.926113 + 1.06879i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} - 2 q^{4} - 2 q^{9} + 2 q^{12} - 2 q^{16} - 2 q^{25} + 2 q^{27} - 2 q^{36} + 2 q^{48} - 2 q^{49} - 2 q^{64} + 2 q^{75} - 2 q^{81}+O(q^{100}) \) 20 * q + 2 * q^3 - 2 * q^4 - 2 * q^9 + 2 * q^12 - 2 * q^16 - 2 * q^25 + 2 * q^27 - 2 * q^36 + 2 * q^48 - 2 * q^49 - 2 * q^64 + 2 * q^75 - 2 * q^81

Introduction

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Modular forms

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Representations

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