LMFDB - Embedded newform 3963.1.x.a.1568.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3963,1,Mod(383,3963)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3963, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 15]))

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

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

chi := DirichletCharacter("3963.383");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3963 = 3 \cdot 1321$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3963.x (of order $22$ , degree $10$ , 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:	$1.97779464506$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} - \cdots)$

Embedding invariants

Embedding label			1568.1
Root			$0.142315 - 0.989821i$ of defining polynomial
Character	$\chi$	$=$	3963.1568
Dual form			3963.1.x.a.599.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.654861 - 0.755750i) q^{3} +1.00000 q^{4} +(-0.512546 + 1.74557i) q^{7} +(-0.142315 + 0.989821i) q^{9} +(-0.654861 - 0.755750i) q^{12} +(-1.37491 + 1.19136i) q^{13} +1.00000 q^{16} +(-0.425839 - 1.45027i) q^{19} +(1.65486 - 0.755750i) q^{21} +(-0.959493 - 0.281733i) q^{25} +(0.841254 - 0.540641i) q^{27} +(-0.512546 + 1.74557i) q^{28} +(-0.0405070 + 0.281733i) q^{31} +(-0.142315 + 0.989821i) q^{36} +(-0.345139 + 0.755750i) q^{37} +(1.80075 + 0.258908i) q^{39} +(-1.25667 + 1.45027i) q^{43} +(-0.654861 - 0.755750i) q^{48} +(-1.94306 - 1.24873i) q^{49} +(-1.37491 + 1.19136i) q^{52} +(-0.817178 + 1.27155i) q^{57} +(-0.304632 + 0.474017i) q^{61} +(-1.65486 - 0.755750i) q^{63} +1.00000 q^{64} +(-1.80075 + 0.822373i) q^{67} +(1.95949 + 0.281733i) q^{73} +(0.415415 + 0.909632i) q^{75} +(-0.425839 - 1.45027i) q^{76} +(-0.983568 + 1.53046i) q^{79} +(-0.959493 - 0.281733i) q^{81} +(1.65486 - 0.755750i) q^{84} +(-1.37491 - 3.01063i) q^{91} +(0.239446 - 0.153882i) q^{93} +(-0.557730 - 0.0801894i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q - q^{3} + 10 q^{4} - q^{9} - q^{12} + 10 q^{16} + 11 q^{21} - q^{25} - q^{27} - 9 q^{31} - q^{36} - 9 q^{37} + 2 q^{43} - q^{48} - q^{49} - 11 q^{63} + 10 q^{64} + 11 q^{73} - q^{75} - q^{81}+ \cdots + 2 q^{93}+O(q^{100})$ 10 * q - q^3 + 10 * q^4 - q^9 - q^12 + 10 * q^16 + 11 * q^21 - q^25 - q^27 - 9 * q^31 - q^36 - 9 * q^37 + 2 * q^43 - q^48 - q^49 - 11 * q^63 + 10 * q^64 + 11 * q^73 - q^75 - q^81 + 11 * q^84 + 2 * q^93

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3963\mathbb{Z}\right)^\times$ .

$n$	$13$	$1322$
$\chi(n)$	$e\left(\frac{17}{22}\right)$	$-1$

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			0		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$3$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$4$	1.00000			1.00000
$5$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$5$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$6$		0			0
$7$	−0.512546	+	1.74557i	−0.512546	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$7$	−0.512546	+	1.74557i	−0.512546	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$8$		0			0
$9$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$10$		0			0
$11$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$11$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$12$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$13$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$13$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$19$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$20$		0			0
$21$	1.65486	−	0.755750i	1.65486	−	0.755750i
$22$		0			0
$23$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$23$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$24$		0			0
$25$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$26$		0			0
$27$	0.841254	−	0.540641i	0.841254	−	0.540641i
$28$	−0.512546	+	1.74557i	−0.512546	+	1.74557i
$29$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$29$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$30$		0			0
$31$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	0.959493	+	0.281733i	$0.0909091\pi$
$31$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	−1.00000			$1.00000\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$37$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	0.654861	+	0.755750i	$0.272727\pi$
$37$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	−1.00000			$\pi$
$38$		0			0
$39$	1.80075	+	0.258908i	1.80075	+	0.258908i
$40$		0			0
$41$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$41$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$42$		0			0
$43$	−1.25667	+	1.45027i	−1.25667	+	1.45027i	−0.415415	+	0.909632i	$0.636364\pi$
$43$	−1.25667	+	1.45027i	−1.25667	+	1.45027i	−0.841254	+	0.540641i	$0.818182\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$47$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$48$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$49$	−1.94306	−	1.24873i	−1.94306	−	1.24873i
$50$		0			0
$51$		0			0
$52$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$53$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$53$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$58$		0			0
$59$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$59$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$60$		0			0
$61$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$61$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$62$		0			0
$63$	−1.65486	−	0.755750i	−1.65486	−	0.755750i
$64$	1.00000			1.00000
$65$		0			0
$66$		0			0
$67$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$67$	−1.80075	+	0.822373i	−1.80075	+	0.822373i	−0.959493	+	0.281733i	$0.909091\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$71$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$72$		0			0
$73$	1.95949	+	0.281733i	1.95949	+	0.281733i	1.00000			$0$
$73$	1.95949	+	0.281733i	1.95949	+	0.281733i	0.959493	+	0.281733i	$0.0909091\pi$
$74$		0			0
$75$	0.415415	+	0.909632i	0.415415	+	0.909632i
$76$	−0.425839	−	1.45027i	−0.425839	−	1.45027i
$77$		0			0
$78$		0			0
$79$	−0.983568	+	1.53046i	−0.983568	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$79$	−0.983568	+	1.53046i	−0.983568	+	1.53046i	−0.841254	+	0.540641i	$0.818182\pi$
$80$		0			0
$81$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$82$		0			0
$83$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$83$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$84$	1.65486	−	0.755750i	1.65486	−	0.755750i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$89$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$90$		0			0
$91$	−1.37491	−	3.01063i	−1.37491	−	3.01063i
$92$		0			0
$93$	0.239446	−	0.153882i	0.239446	−	0.153882i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$97$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i	−0.415415	+	0.909632i	$0.636364\pi$
$98$		0			0
$99$		0			0
$100$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$101$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$101$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$102$		0			0
$103$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.654861	−	0.755750i	$-0.272727\pi$
$103$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$104$		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
$109$			1.81926i			1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$109$			1.81926i			1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$110$		0			0
$111$	0.797176	−	0.234072i	0.797176	−	0.234072i
$112$	−0.512546	+	1.74557i	−0.512546	+	1.74557i
$113$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$113$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.983568	−	1.53046i	−0.983568	−	1.53046i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.415415	+	0.909632i	0.415415	+	0.909632i
$122$		0			0
$123$		0			0
$124$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i
$125$		0			0
$126$		0			0
$127$	0.817178	+	0.708089i	0.817178	+	0.708089i	0.959493	−	0.281733i	$-0.0909091\pi$
$127$	0.817178	+	0.708089i	0.817178	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$128$		0			0
$129$	1.91899			1.91899
$130$		0			0
$131$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$131$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$132$		0			0
$133$	2.74982			2.74982
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$137$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$138$		0			0
$139$	0.512546	−	1.74557i	0.512546	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$139$	0.512546	−	1.74557i	0.512546	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$145$		0			0
$146$		0			0
$147$	0.328708	+	2.28621i	0.328708	+	2.28621i
$148$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.142315	−	0.989821i	$-0.454545\pi$
$151$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.80075	+	0.258908i	1.80075	+	0.258908i
$157$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$157$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$163$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$167$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$168$		0			0
$169$	0.328708	−	2.28621i	0.328708	−	2.28621i
$170$		0			0
$171$	1.49611	−	0.215109i	1.49611	−	0.215109i
$172$	−1.25667	+	1.45027i	−1.25667	+	1.45027i
$173$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$173$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$174$		0			0
$175$	0.983568	−	1.53046i	0.983568	−	1.53046i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$179$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$180$		0			0
$181$	−1.37491	−	1.19136i	−1.37491	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$181$	−1.37491	−	1.19136i	−1.37491	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$182$		0			0
$183$	0.557730	−	0.0801894i	0.557730	−	0.0801894i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	0.512546	+	1.74557i	0.512546	+	1.74557i
$190$		0			0
$191$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$191$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$192$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$193$	0.817178	+	0.708089i	0.817178	+	0.708089i	0.959493	−	0.281733i	$-0.0909091\pi$
$193$	0.817178	+	0.708089i	0.817178	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$194$		0			0
$195$		0			0
$196$	−1.94306	−	1.24873i	−1.94306	−	1.24873i
$197$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$197$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$198$		0			0
$199$	0.557730	−	0.0801894i	0.557730	−	0.0801894i	0.142315	−	0.989821i	$-0.454545\pi$
$199$	0.557730	−	0.0801894i	0.557730	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$200$		0			0
$201$	1.80075	+	0.822373i	1.80075	+	0.822373i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$209$		0			0
$210$		0			0
$211$	0.118239	+	0.822373i	0.118239	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$211$	0.118239	+	0.822373i	0.118239	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−0.471022	−	0.215109i	−0.471022	−	0.215109i
$218$		0			0
$219$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.841254	−	0.540641i	$-0.818182\pi$
$223$	−1.49611	−	1.29639i	−1.49611	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$224$		0			0
$225$	0.415415	−	0.909632i	0.415415	−	0.909632i
$226$		0			0
$227$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$227$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$228$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$229$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$229$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	−0.841254	+	0.540641i	$0.818182\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	1.80075	−	0.258908i	1.80075	−	0.258908i
$238$		0			0
$239$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$239$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$240$		0			0
$241$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.654861	+	0.755750i	$0.272727\pi$
$241$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$242$		0			0
$243$	0.415415	+	0.909632i	0.415415	+	0.909632i
$244$	−0.304632	+	0.474017i	−0.304632	+	0.474017i
$245$		0			0
$246$		0			0
$247$	2.31329	+	1.48666i	2.31329	+	1.48666i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$251$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$252$	−1.65486	−	0.755750i	−1.65486	−	0.755750i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$260$		0			0
$261$		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
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.80075	+	0.822373i	−1.80075	+	0.822373i
$269$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$269$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$270$		0			0
$271$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$271$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$272$		0			0
$273$	−1.37491	+	3.01063i	−1.37491	+	3.01063i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i	0.959493	−	0.281733i	$-0.0909091\pi$
$277$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i	−1.00000			$\pi$
$278$		0			0
$279$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$280$		0			0
$281$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$281$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$282$		0			0
$283$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$283$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000			−1.00000
$290$		0			0
$291$	0.304632	+	0.474017i	0.304632	+	0.474017i
$292$	1.95949	+	0.281733i	1.95949	+	0.281733i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.415415	+	0.909632i	0.415415	+	0.909632i
$301$	−1.88745	−	2.93694i	−1.88745	−	2.93694i
$302$		0			0
$303$		0			0
$304$	−0.425839	−	1.45027i	−0.425839	−	1.45027i
$305$		0			0
$306$		0			0
$307$	0.239446	−	0.153882i	0.239446	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$307$	0.239446	−	0.153882i	0.239446	−	0.153882i	0.654861	+	0.755750i	$0.272727\pi$
$308$		0			0
$309$	−1.95949	−	0.281733i	−1.95949	−	0.281733i
$310$		0			0
$311$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$311$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$312$		0			0
$313$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$313$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$314$		0			0
$315$		0			0
$316$	−0.983568	+	1.53046i	−0.983568	+	1.53046i
$317$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$317$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$325$	1.65486	−	0.755750i	1.65486	−	0.755750i
$326$		0			0
$327$	1.37491	−	1.19136i	1.37491	−	1.19136i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$331$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$332$		0			0
$333$	−0.698939	−	0.449181i	−0.698939	−	0.449181i
$334$		0			0
$335$		0			0
$336$	1.65486	−	0.755750i	1.65486	−	0.755750i
$337$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$337$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	1.80075	−	1.56036i	1.80075	−	1.56036i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$347$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$348$		0			0
$349$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$349$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$350$		0			0
$351$	−0.512546	+	1.74557i	−0.512546	+	1.74557i
$352$		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		−0.281733	−	0.959493i	$-0.590909\pi$
$359$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$360$		0			0
$361$	−1.08070	+	0.694523i	−1.08070	+	0.694523i
$362$		0			0
$363$	0.415415	−	0.909632i	0.415415	−	0.909632i
$364$	−1.37491	−	3.01063i	−1.37491	−	3.01063i
$365$		0			0
$366$		0			0
$367$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$367$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	0.239446	−	0.153882i	0.239446	−	0.153882i
$373$	−1.49611	+	0.215109i	−1.49611	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$373$	−1.49611	+	0.215109i	−1.49611	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.158746	−	0.540641i	0.158746	−	0.540641i	−0.841254	−	0.540641i	$-0.818182\pi$
$379$	0.158746	−	0.540641i	0.158746	−	0.540641i	1.00000			$0$
$380$		0			0
$381$		−	1.08128i		−	1.08128i
$382$		0			0
$383$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$383$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.25667	−	1.45027i	−1.25667	−	1.45027i
$388$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	1.80075	+	0.258908i	1.80075	+	0.258908i	0.959493	−	0.281733i	$-0.0909091\pi$
$397$	1.80075	+	0.258908i	1.80075	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$398$		0			0
$399$	−1.80075	−	2.07817i	−1.80075	−	2.07817i
$400$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$401$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$401$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$402$		0			0
$403$	−0.279953	−	0.435615i	−0.279953	−	0.435615i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.07028	+	1.66538i	−1.07028	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$409$	−1.07028	+	1.66538i	−1.07028	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$410$		0			0
$411$		0			0
$412$	1.49611	−	1.29639i	1.49611	−	1.29639i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.65486	+	0.755750i	−1.65486	+	0.755750i
$418$		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
$421$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$421$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−0.671292	−	0.774713i	−0.671292	−	0.774713i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$431$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$432$	0.841254	−	0.540641i	0.841254	−	0.540641i
$433$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$433$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.959493	+	0.281733i	$0.0909091\pi$
$434$		0			0
$435$		0			0
$436$			1.81926i			1.81926i
$437$		0			0
$438$		0			0
$439$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$439$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$440$		0			0
$441$	1.51255	−	1.74557i	1.51255	−	1.74557i
$442$		0			0
$443$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$443$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$444$	0.797176	−	0.234072i	0.797176	−	0.234072i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.512546	+	1.74557i	−0.512546	+	1.74557i
$449$		0			0			−	1.00000i	$-0.5\pi$
$449$		0			0				1.00000i	$0.5\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−0.983568	−	0.449181i	−0.983568	−	0.449181i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$457$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$461$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$462$		0			0
$463$	1.49611	+	1.29639i	1.49611	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$463$	1.49611	+	1.29639i	1.49611	+	1.29639i	0.654861	+	0.755750i	$0.272727\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$467$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$468$	−0.983568	−	1.53046i	−0.983568	−	1.53046i
$469$	−0.512546	−	3.56484i	−0.512546	−	3.56484i
$470$		0			0
$471$	0.512546	−	1.74557i	0.512546	−	1.74557i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			1.51150i			1.51150i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$479$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$480$		0			0
$481$	−0.425839	−	1.45027i	−0.425839	−	1.45027i
$482$		0			0
$483$		0			0
$484$	0.415415	+	0.909632i	0.415415	+	0.909632i
$485$		0			0
$486$		0			0
$487$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$487$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$488$		0			0
$489$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$490$		0			0
$491$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$491$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i
$497$		0			0
$498$		0			0
$499$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$499$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$503$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.94306	+	1.24873i	−1.94306	+	1.24873i
$508$	0.817178	+	0.708089i	0.817178	+	0.708089i
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−1.49611	+	3.27603i	−1.49611	+	3.27603i
$512$		0			0
$513$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$514$		0			0
$515$		0			0
$516$	1.91899			1.91899
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$521$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$522$		0			0
$523$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$523$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.415415	−	0.909632i	$-0.636364\pi$
$524$		0			0
$525$	−1.80075	+	0.258908i	−1.80075	+	0.258908i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.654861	−	0.755750i	0.654861	−	0.755750i
$530$		0			0
$531$		0			0
$532$	2.74982			2.74982
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.142315	−	0.989821i	$-0.454545\pi$
$541$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$542$		0			0
$543$			1.81926i			1.81926i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$547$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$548$		0			0
$549$	−0.425839	−	0.368991i	−0.425839	−	0.368991i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.16741	−	2.50132i	−2.16741	−	2.50132i
$554$		0			0
$555$		0			0
$556$	0.512546	−	1.74557i	0.512546	−	1.74557i
$557$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$557$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$558$		0			0
$559$		−	3.49114i		−	3.49114i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$563$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.983568	−	1.53046i	0.983568	−	1.53046i
$568$		0			0
$569$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$569$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$570$		0			0
$571$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$571$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$577$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$577$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$578$		0			0
$579$		−	1.08128i		−	1.08128i
$580$		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	$-0.545455\pi$
$587$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$588$	0.328708	+	2.28621i	0.328708	+	2.28621i
$589$	0.425839	−	0.0612263i	0.425839	−	0.0612263i
$590$		0			0
$591$		0			0
$592$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$593$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$593$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.425839	−	0.368991i	−0.425839	−	0.368991i
$598$		0			0
$599$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$599$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$600$		0			0
$601$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$601$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$602$		0			0
$603$	−0.557730	−	1.89945i	−0.557730	−	1.89945i
$604$	0.983568	−	0.449181i	0.983568	−	0.449181i
$605$		0			0
$606$		0			0
$607$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.654861	+	0.755750i	$0.272727\pi$
$607$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$613$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$617$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$618$		0			0
$619$	−0.425839	+	1.45027i	−0.425839	+	1.45027i	0.415415	+	0.909632i	$0.363636\pi$
$619$	−0.425839	+	1.45027i	−0.425839	+	1.45027i	−0.841254	+	0.540641i	$0.818182\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.80075	+	0.258908i	1.80075	+	0.258908i
$625$	0.841254	+	0.540641i	0.841254	+	0.540641i
$626$		0			0
$627$		0			0
$628$	0.983568	+	1.53046i	0.983568	+	1.53046i
$629$		0			0
$630$		0			0
$631$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$631$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$632$		0			0
$633$	0.544078	−	0.627899i	0.544078	−	0.627899i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	4.15922	−	0.598006i	4.15922	−	0.598006i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$641$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$642$		0			0
$643$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$643$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$647$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	0.145886	+	0.496841i	0.145886	+	0.496841i
$652$		−	1.97964i		−	1.97964i
$653$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$653$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−0.557730	+	1.89945i	−0.557730	+	1.89945i
$658$		0			0
$659$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$659$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$660$		0			0
$661$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$661$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$			1.97964i			1.97964i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$673$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$674$		0			0
$675$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$676$	0.328708	−	2.28621i	0.328708	−	2.28621i
$677$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$677$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$678$		0			0
$679$	0.425839	−	0.932456i	0.425839	−	0.932456i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$683$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$684$	1.49611	−	0.215109i	1.49611	−	0.215109i
$685$		0			0
$686$		0			0
$687$	1.61435	−	0.474017i	1.61435	−	0.474017i
$688$	−1.25667	+	1.45027i	−1.25667	+	1.45027i
$689$		0			0
$690$		0			0
$691$	−0.584585	+	0.909632i	−0.584585	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$691$	−0.584585	+	0.909632i	−0.584585	+	0.909632i	−1.00000			$1.00000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	0.983568	−	1.53046i	0.983568	−	1.53046i
$701$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$701$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$702$		0			0
$703$	1.24302	+	0.178719i	1.24302	+	0.178719i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.49611	+	0.215109i	1.49611	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$709$	1.49611	+	0.215109i	1.49611	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$710$		0			0
$711$	−1.37491	−	1.19136i	−1.37491	−	1.19136i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$719$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$720$		0			0
$721$	1.49611	+	3.27603i	1.49611	+	3.27603i
$722$		0			0
$723$	0.797176	−	1.74557i	0.797176	−	1.74557i
$724$	−1.37491	−	1.19136i	−1.37491	−	1.19136i
$725$		0			0
$726$		0			0
$727$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$727$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$728$		0			0
$729$	0.415415	−	0.909632i	0.415415	−	0.909632i
$730$		0			0
$731$		0			0
$732$	0.557730	−	0.0801894i	0.557730	−	0.0801894i
$733$	−0.186393	−	1.29639i	−0.186393	−	1.29639i	−0.841254	−	0.540641i	$-0.818182\pi$
$733$	−0.186393	−	1.29639i	−0.186393	−	1.29639i	0.654861	−	0.755750i	$-0.272727\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	0.584585	+	0.909632i	0.584585	+	0.909632i	1.00000			$0$
$739$	0.584585	+	0.909632i	0.584585	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$740$		0			0
$741$	−0.391340	−	2.72183i	−0.391340	−	2.72183i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.425839	−	0.368991i	−0.425839	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$751$	−0.425839	−	0.368991i	−0.425839	−	0.368991i	−0.841254	+	0.540641i	$0.818182\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	0.512546	+	1.74557i	0.512546	+	1.74557i
$757$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$757$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$761$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$762$		0			0
$763$	−3.17565	−	0.932456i	−3.17565	−	0.932456i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$769$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$769$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	0.817178	+	0.708089i	0.817178	+	0.708089i
$773$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$773$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$774$		0			0
$775$	0.118239	−	0.258908i	0.118239	−	0.258908i
$776$		0			0
$777$			1.51150i			1.51150i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.94306	−	1.24873i	−1.94306	−	1.24873i
$785$		0			0
$786$		0			0
$787$	−1.07028	−	0.153882i	−1.07028	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$787$	−1.07028	−	0.153882i	−1.07028	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.145886	−	1.01466i	−0.145886	−	1.01466i
$794$		0			0
$795$		0			0
$796$	0.557730	−	0.0801894i	0.557730	−	0.0801894i
$797$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$797$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.80075	+	0.822373i	1.80075	+	0.822373i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$811$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	−1.00000			$\pi$
$812$		0			0
$813$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	2.63843	+	1.20493i	2.63843	+	1.20493i
$818$		0			0
$819$	3.17565	−	0.932456i	3.17565	−	0.932456i
$820$		0			0
$821$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$821$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$822$		0			0
$823$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$823$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$827$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$828$		0			0
$829$	0.239446	−	0.153882i	0.239446	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$829$	0.239446	−	0.153882i	0.239446	−	0.153882i	0.654861	+	0.755750i	$0.272727\pi$
$830$		0			0
$831$	−0.186393	+	0.215109i	−0.186393	+	0.215109i
$832$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.118239	+	0.258908i	0.118239	+	0.258908i
$838$		0			0
$839$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$839$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$840$		0			0
$841$	0.841254	−	0.540641i	0.841254	−	0.540641i
$842$		0			0
$843$		0			0
$844$	0.118239	+	0.822373i	0.118239	+	0.822373i
$845$		0			0
$846$		0			0
$847$	−1.80075	+	0.258908i	−1.80075	+	0.258908i
$848$		0			0
$849$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.817178	−	1.27155i	−0.817178	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$853$	−0.817178	−	1.27155i	−0.817178	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$857$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$858$		0			0
$859$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$859$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$863$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.654861	+	0.755750i	0.654861	+	0.755750i
$868$	−0.471022	−	0.215109i	−0.471022	−	0.215109i
$869$		0			0
$870$		0			0
$871$	1.49611	−	3.27603i	1.49611	−	3.27603i
$872$		0			0
$873$	0.158746	−	0.540641i	0.158746	−	0.540641i
$874$		0			0
$875$		0			0
$876$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$877$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$877$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$881$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$882$		0			0
$883$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$883$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	−1.00000			$\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$887$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$888$		0			0
$889$	−1.65486	+	1.06351i	−1.65486	+	1.06351i
$890$		0			0
$891$		0			0
$892$	−1.49611	−	1.29639i	−1.49611	−	1.29639i
$893$		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
$903$	−0.983568	+	3.34973i	−0.983568	+	3.34973i
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$907$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$911$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$912$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$917$		0			0
$918$		0			0
$919$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$919$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$920$		0			0
$921$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	0.544078	−	0.627899i	0.544078	−	0.627899i
$926$		0			0
$927$	1.07028	+	1.66538i	1.07028	+	1.66538i
$928$		0			0
$929$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$929$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$930$		0			0
$931$	−0.983568	+	3.34973i	−0.983568	+	3.34973i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.544078	+	0.627899i	0.544078	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$937$	0.544078	+	0.627899i	0.544078	+	0.627899i	−0.415415	+	0.909632i	$0.636364\pi$
$938$		0			0
$939$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$940$		0			0
$941$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$941$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$947$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$948$	1.80075	−	0.258908i	1.80075	−	0.258908i
$949$	−3.02977	+	1.94711i	−3.02977	+	1.94711i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$953$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.881761	+	0.258908i	0.881761	+	0.258908i
$962$		0			0
$963$		0			0
$964$	0.797176	+	1.74557i	0.797176	+	1.74557i
$965$		0			0
$966$		0			0
$967$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$967$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$971$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$972$	0.415415	+	0.909632i	0.415415	+	0.909632i
$973$	2.78431	+	1.78937i	2.78431	+	1.78937i
$974$		0			0
$975$	−1.65486	−	0.755750i	−1.65486	−	0.755750i
$976$	−0.304632	+	0.474017i	−0.304632	+	0.474017i
$977$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$977$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−1.80075	−	0.258908i	−1.80075	−	0.258908i
$982$		0			0
$983$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$983$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	2.31329	+	1.48666i	2.31329	+	1.48666i
$989$		0			0
$990$		0			0
$991$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$991$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$992$		0			0
$993$	0.0405070	−	0.281733i	0.0405070	−	0.281733i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$997$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$998$		0			0
$999$	0.118239	+	0.822373i	0.118239	+	0.822373i

Display

a_p

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p

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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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3963.1.x.a.1568.1	yes	10
3.2	odd	2	CM	3963.1.x.a.1568.1	yes	10
1321.599	even	22	inner	3963.1.x.a.599.1	✓	10
3963.599	odd	22	inner	3963.1.x.a.599.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3963.1.x.a.599.1	✓	10	1321.599	even	22	inner
3963.1.x.a.599.1	✓	10	3963.599	odd	22	inner
3963.1.x.a.1568.1	yes	10	1.1	even	1	trivial
3963.1.x.a.1568.1	yes	10	3.2	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3963.1.x.a.1568.1

Level	$3963$
Weight	$1$
Character	3963.1568
Analytic conductor	$1.978$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{22}$
CM discriminant	-3
Inner twists	$4$