LMFDB - Embedded newform 759.1.r.a.329.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [759,1,Mod(65,759)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(759, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 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("759.65");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$759 = 3 \cdot 11 \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	759.r (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:	$0.378790344588$
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			329.1
Root			$0.654861 - 0.755750i$ of defining polynomial
Character	$\chi$	$=$	759.329
Dual form			759.1.r.a.263.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.959493 - 0.281733i) q^{3} +(0.142315 + 0.989821i) q^{4} +(-0.544078 - 1.19136i) q^{5} +(0.841254 - 0.540641i) q^{9} +(0.654861 + 0.755750i) q^{11} +(0.415415 + 0.909632i) q^{12} +(-0.857685 - 0.989821i) q^{15} +(-0.959493 + 0.281733i) q^{16} +(1.10181 - 0.708089i) q^{20} +(-0.654861 - 0.755750i) q^{23} +(-0.468468 + 0.540641i) q^{25} +(0.654861 - 0.755750i) q^{27} +(0.698939 + 0.449181i) q^{31} +(0.841254 + 0.540641i) q^{33} +(0.654861 + 0.755750i) q^{36} +(-1.80075 - 0.822373i) q^{37} +(-0.654861 + 0.755750i) q^{44} +(-1.10181 - 0.708089i) q^{45} +1.81926i q^{47} +(-0.841254 + 0.540641i) q^{48} +(-0.841254 + 0.540641i) q^{49} +(-0.273100 + 0.0801894i) q^{53} +(0.544078 - 1.19136i) q^{55} +(-0.557730 + 1.89945i) q^{59} +(0.857685 - 0.989821i) q^{60} +(-0.415415 - 0.909632i) q^{64} +(-0.425839 - 0.368991i) q^{67} +(-0.841254 - 0.540641i) q^{69} +(-1.14231 - 0.989821i) q^{71} +(-0.297176 + 0.650724i) q^{75} +(0.857685 + 0.989821i) q^{80} +(0.415415 - 0.909632i) q^{81} +(1.61435 - 1.03748i) q^{89} +(0.654861 - 0.755750i) q^{92} +(0.797176 + 0.234072i) q^{93} +(-1.65486 + 0.755750i) q^{97} +(0.959493 + 0.281733i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$10 q + q^{3} + q^{4} - 2 q^{5} - q^{9} + q^{11} - q^{12} - 9 q^{15} - q^{16} + 2 q^{20} - q^{23} - 3 q^{25} + q^{27} - 2 q^{31} - q^{33} + q^{36} - q^{44} - 2 q^{45} + q^{48} + q^{49} + 2 q^{53}+ \cdots + q^{99}+O(q^{100})$ 10 * q + q^3 + q^4 - 2 * q^5 - q^9 + q^11 - q^12 - 9 * q^15 - q^16 + 2 * q^20 - q^23 - 3 * q^25 + q^27 - 2 * q^31 - q^33 + q^36 - q^44 - 2 * q^45 + q^48 + q^49 + 2 * q^53 + 2 * q^55 + 9 * q^60 + q^64 + q^69 - 11 * q^71 + 3 * q^75 + 9 * q^80 - q^81 + 2 * q^89 + q^92 + 2 * q^93 - 11 * q^97 + q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/759\mathbb{Z}\right)^\times$ .

$n$	$166$	$254$	$277$
$\chi(n)$	$e\left(\frac{19}{22}\right)$	$-1$	$-1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$2$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$3$	0.959493	−	0.281733i	0.959493	−	0.281733i
$3$	0.959493	−	0.281733i	0.959493	−	0.281733i
$4$	0.142315	+	0.989821i	0.142315	+	0.989821i
$5$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$5$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$6$		0			0
$7$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$7$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$8$		0			0
$9$	0.841254	−	0.540641i	0.841254	−	0.540641i
$10$		0			0
$11$	0.654861	+	0.755750i	0.654861	+	0.755750i
$11$	0.654861	+	0.755750i	0.654861	+	0.755750i
$12$	0.415415	+	0.909632i	0.415415	+	0.909632i
$13$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$13$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$14$		0			0
$15$	−0.857685	−	0.989821i	−0.857685	−	0.989821i
$16$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$17$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$17$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$18$		0			0
$19$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$19$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$20$	1.10181	−	0.708089i	1.10181	−	0.708089i
$21$		0			0
$22$		0			0
$23$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$23$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$24$		0			0
$25$	−0.468468	+	0.540641i	−0.468468	+	0.540641i
$26$		0			0
$27$	0.654861	−	0.755750i	0.654861	−	0.755750i
$28$		0			0
$29$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$29$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$30$		0			0
$31$	0.698939	+	0.449181i	0.698939	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$31$	0.698939	+	0.449181i	0.698939	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$32$		0			0
$33$	0.841254	+	0.540641i	0.841254	+	0.540641i
$34$		0			0
$35$		0			0
$36$	0.654861	+	0.755750i	0.654861	+	0.755750i
$37$	−1.80075	−	0.822373i	−1.80075	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$37$	−1.80075	−	0.822373i	−1.80075	−	0.822373i	−0.841254	−	0.540641i	$-0.818182\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$41$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$42$		0			0
$43$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$43$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$44$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$45$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$46$		0			0
$47$			1.81926i			1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$47$			1.81926i			1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$48$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$49$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$53$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$54$		0			0
$55$	0.544078	−	1.19136i	0.544078	−	1.19136i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$59$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$60$	0.857685	−	0.989821i	0.857685	−	0.989821i
$61$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$61$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$62$		0			0
$63$		0			0
$64$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$65$		0			0
$66$		0			0
$67$	−0.425839	−	0.368991i	−0.425839	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$67$	−0.425839	−	0.368991i	−0.425839	−	0.368991i	−0.841254	+	0.540641i	$0.818182\pi$
$68$		0			0
$69$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$70$		0			0
$71$	−1.14231	−	0.989821i	−1.14231	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$71$	−1.14231	−	0.989821i	−1.14231	−	0.989821i	−1.00000			$\pi$
$72$		0			0
$73$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$73$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$74$		0			0
$75$	−0.297176	+	0.650724i	−0.297176	+	0.650724i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$79$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$80$	0.857685	+	0.989821i	0.857685	+	0.989821i
$81$	0.415415	−	0.909632i	0.415415	−	0.909632i
$82$		0			0
$83$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$83$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.61435	−	1.03748i	1.61435	−	1.03748i	0.654861	−	0.755750i	$-0.272727\pi$
$89$	1.61435	−	1.03748i	1.61435	−	1.03748i	0.959493	−	0.281733i	$-0.0909091\pi$
$90$		0			0
$91$		0			0
$92$	0.654861	−	0.755750i	0.654861	−	0.755750i
$93$	0.797176	+	0.234072i	0.797176	+	0.234072i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$97$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−1.00000			$\pi$
$98$		0			0
$99$	0.959493	+	0.281733i	0.959493	+	0.281733i
$100$	−0.601808	−	0.386758i	−0.601808	−	0.386758i
$101$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$101$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$102$		0			0
$103$	0.817178	−	0.708089i	0.817178	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$103$	0.817178	−	0.708089i	0.817178	−	0.708089i	0.959493	+	0.281733i	$0.0909091\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$107$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$108$	0.841254	+	0.540641i	0.841254	+	0.540641i
$109$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$109$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$110$		0			0
$111$	−1.95949	−	0.281733i	−1.95949	−	0.281733i
$112$		0			0
$113$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$113$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$114$		0			0
$115$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$122$		0			0
$123$		0			0
$124$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$125$	−0.357685	−	0.105026i	−0.357685	−	0.105026i
$126$		0			0
$127$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$127$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$131$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$132$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$133$		0			0
$134$		0			0
$135$	−1.25667	−	0.368991i	−1.25667	−	0.368991i
$136$		0			0
$137$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$137$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	0.512546	+	1.74557i	0.512546	+	1.74557i
$142$		0			0
$143$		0			0
$144$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$145$		0			0
$146$		0			0
$147$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$148$	0.557730	−	1.89945i	0.557730	−	1.89945i
$149$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$149$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$150$		0			0
$151$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$151$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.154861	−	1.07708i	0.154861	−	1.07708i
$156$		0			0
$157$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$157$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$158$		0			0
$159$	−0.239446	+	0.153882i	−0.239446	+	0.153882i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$163$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$164$		0			0
$165$	0.186393	−	1.29639i	0.186393	−	1.29639i
$166$		0			0
$167$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$167$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$168$		0			0
$169$	0.841254	+	0.540641i	0.841254	+	0.540641i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$173$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$174$		0			0
$175$		0			0
$176$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$177$			1.97964i			1.97964i
$178$		0			0
$179$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.142315	−	0.989821i	$-0.454545\pi$
$179$	0.983568	−	0.449181i	0.983568	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$180$	0.544078	−	1.19136i	0.544078	−	1.19136i
$181$	0.584585	+	0.909632i	0.584585	+	0.909632i	1.00000			$0$
$181$	0.584585	+	0.909632i	0.584585	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			2.59278i			2.59278i
$186$		0			0
$187$		0			0
$188$	−1.80075	+	0.258908i	−1.80075	+	0.258908i
$189$		0			0
$190$		0			0
$191$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$191$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$192$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$193$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$193$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$194$		0			0
$195$		0			0
$196$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$197$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$197$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$198$		0			0
$199$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$199$	−0.304632	+	0.474017i	−0.304632	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$200$		0			0
$201$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$211$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$212$	−0.118239	−	0.258908i	−0.118239	−	0.258908i
$213$	−1.37491	−	0.627899i	−1.37491	−	0.627899i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	1.25667	+	0.368991i	1.25667	+	0.368991i
$221$		0			0
$222$		0			0
$223$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$223$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.959493	+	0.281733i	$0.0909091\pi$
$224$		0			0
$225$	−0.101808	+	0.708089i	−0.101808	+	0.708089i
$226$		0			0
$227$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$227$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$228$		0			0
$229$			1.81926i			1.81926i	0.415415	+	0.909632i	$0.363636\pi$
$229$			1.81926i			1.81926i	−0.415415	+	0.909632i	$0.636364\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$233$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$234$		0			0
$235$	2.16741	−	0.989821i	2.16741	−	0.989821i
$236$	−1.95949	−	0.281733i	−1.95949	−	0.281733i
$237$		0			0
$238$		0			0
$239$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$239$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$240$	1.10181	+	0.708089i	1.10181	+	0.708089i
$241$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$241$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$242$		0			0
$243$	0.142315	−	0.989821i	0.142315	−	0.989821i
$244$		0			0
$245$	1.10181	+	0.708089i	1.10181	+	0.708089i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$251$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$252$		0			0
$253$	0.142315	−	0.989821i	0.142315	−	0.989821i
$254$		0			0
$255$		0			0
$256$	0.841254	−	0.540641i	0.841254	−	0.540641i
$257$	1.95949	−	0.281733i	1.95949	−	0.281733i	0.959493	−	0.281733i	$-0.0909091\pi$
$257$	1.95949	−	0.281733i	1.95949	−	0.281733i	1.00000			$0$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$263$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$264$		0			0
$265$	0.244123	+	0.281733i	0.244123	+	0.281733i
$266$		0			0
$267$	1.25667	−	1.45027i	1.25667	−	1.45027i
$268$	0.304632	−	0.474017i	0.304632	−	0.474017i
$269$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$269$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$270$		0			0
$271$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$271$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−0.715370			−0.715370
$276$	0.415415	−	0.909632i	0.415415	−	0.909632i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	0.830830			0.830830
$280$		0			0
$281$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$281$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$282$		0			0
$283$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$283$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$284$	0.817178	−	1.27155i	0.817178	−	1.27155i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$290$		0			0
$291$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$292$		0			0
$293$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$293$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$294$		0			0
$295$	2.56639	−	0.368991i	2.56639	−	0.368991i
$296$		0			0
$297$	1.00000			1.00000
$298$		0			0
$299$		0			0
$300$	−0.686393	−	0.201543i	−0.686393	−	0.201543i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$307$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$308$		0			0
$309$	0.584585	−	0.909632i	0.584585	−	0.909632i
$310$		0			0
$311$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.654861	−	0.755750i	$-0.272727\pi$
$311$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$312$		0			0
$313$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$313$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$317$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$	−0.857685	+	0.989821i	−0.857685	+	0.989821i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.959493	+	0.281733i	0.959493	+	0.281733i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$331$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$332$		0			0
$333$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$334$		0			0
$335$	−0.207914	+	0.708089i	−0.207914	+	0.708089i
$336$		0			0
$337$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$337$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$338$		0			0
$339$	0.345139	−	0.755750i	0.345139	−	0.755750i
$340$		0			0
$341$	0.118239	+	0.822373i	0.118239	+	0.822373i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$346$		0			0
$347$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$347$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$348$		0			0
$349$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$349$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.584585	−	0.909632i	0.584585	−	0.909632i	−0.415415	−	0.909632i	$-0.636364\pi$
$353$	0.584585	−	0.909632i	0.584585	−	0.909632i	1.00000			$0$
$354$		0			0
$355$	−0.557730	+	1.89945i	−0.557730	+	1.89945i
$356$	1.25667	+	1.45027i	1.25667	+	1.45027i
$357$		0			0
$358$		0			0
$359$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$359$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$360$		0			0
$361$	0.959493	−	0.281733i	0.959493	−	0.281733i
$362$		0			0
$363$	0.142315	+	0.989821i	0.142315	+	0.989821i
$364$		0			0
$365$		0			0
$366$		0			0
$367$			1.51150i			1.51150i	0.654861	+	0.755750i	$0.272727\pi$
$367$			1.51150i			1.51150i	−0.654861	+	0.755750i	$0.727273\pi$
$368$	0.841254	+	0.540641i	0.841254	+	0.540641i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$373$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$373$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$374$		0			0
$375$	−0.372786			−0.372786
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.654861	−	0.755750i	$-0.272727\pi$
$379$	1.49611	−	1.29639i	1.49611	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	1.41542	+	0.909632i	1.41542	+	0.909632i	1.00000			$0$
$383$	1.41542	+	0.909632i	1.41542	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.983568	−	1.53046i	−0.983568	−	1.53046i
$389$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$389$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.959493	+	0.281733i	$0.909091\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$397$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$397$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$398$		0			0
$399$		0			0
$400$	0.297176	−	0.650724i	0.297176	−	0.650724i
$401$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$401$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−1.30972			−1.30972
$406$		0			0
$407$	−0.557730	−	1.89945i	−0.557730	−	1.89945i
$408$		0			0
$409$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$409$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$410$		0			0
$411$	−1.61435	+	0.474017i	−1.61435	+	0.474017i
$412$	0.817178	+	0.708089i	0.817178	+	0.708089i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	−0.841254	−	0.540641i	$-0.818182\pi$
$419$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$420$		0			0
$421$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$421$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$422$		0			0
$423$	0.983568	+	1.53046i	0.983568	+	1.53046i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$431$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$432$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$433$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$433$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$439$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$440$		0			0
$441$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$442$		0			0
$443$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$443$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$444$		−	1.97964i		−	1.97964i
$445$	−2.11435	−	1.35881i	−2.11435	−	1.35881i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$449$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$450$		0			0
$451$		0			0
$452$	0.698939	+	0.449181i	0.698939	+	0.449181i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$457$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$458$		0			0
$459$		0			0
$460$	−1.25667	−	0.368991i	−1.25667	−	0.368991i
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$463$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$464$		0			0
$465$	−0.154861	−	1.07708i	−0.154861	−	1.07708i
$466$		0			0
$467$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$467$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.186393	+	0.215109i	−0.186393	+	0.215109i
$478$		0			0
$479$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$479$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	−1.00000			−1.00000
$485$	1.80075	+	1.56036i	1.80075	+	1.56036i
$486$		0			0
$487$	−0.273100	−	1.89945i	−0.273100	−	1.89945i	−0.415415	−	0.909632i	$-0.636364\pi$
$487$	−0.273100	−	1.89945i	−0.273100	−	1.89945i	0.142315	−	0.989821i	$-0.454545\pi$
$488$		0			0
$489$	0.698939	−	1.53046i	0.698939	−	1.53046i
$490$		0			0
$491$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$491$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.186393	−	1.29639i	−0.186393	−	1.29639i
$496$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$497$		0			0
$498$		0			0
$499$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−0.841254	+	0.540641i	$0.818182\pi$
$499$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−1.00000			$1.00000\pi$
$500$	0.0530529	−	0.368991i	0.0530529	−	0.368991i
$501$		0			0
$502$		0			0
$503$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$503$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.959493	+	0.281733i	0.959493	+	0.281733i
$508$		0			0
$509$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$509$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.28820	−	0.588302i	−1.28820	−	0.588302i
$516$		0			0
$517$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.239446	+	0.153882i	0.239446	+	0.153882i	0.654861	−	0.755750i	$-0.272727\pi$
$521$	0.239446	+	0.153882i	0.239446	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$522$		0			0
$523$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$523$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$529$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$530$		0			0
$531$	0.557730	+	1.89945i	0.557730	+	1.89945i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.817178	−	0.708089i	0.817178	−	0.708089i
$538$		0			0
$539$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$540$	0.186393	−	1.29639i	0.186393	−	1.29639i
$541$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$541$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$542$		0			0
$543$	0.817178	+	0.708089i	0.817178	+	0.708089i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$547$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$548$	−0.239446	−	1.66538i	−0.239446	−	1.66538i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.730471	+	2.48775i	0.730471	+	2.48775i
$556$		0			0
$557$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$557$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$563$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$564$	−1.65486	+	0.755750i	−1.65486	+	0.755750i
$565$	−1.04408	−	0.306569i	−1.04408	−	0.306569i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$569$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$570$		0			0
$571$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$571$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$572$		0			0
$573$	1.61435	−	1.03748i	1.61435	−	1.03748i
$574$		0			0
$575$	0.715370			0.715370
$576$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$577$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$577$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.425839	−	0.368991i	0.425839	−	0.368991i	−0.415415	−	0.909632i	$-0.636364\pi$
$587$	0.425839	−	0.368991i	0.425839	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$588$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.95949	+	0.281733i	1.95949	+	0.281733i
$593$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$593$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.158746	+	0.540641i	−0.158746	+	0.540641i
$598$		0			0
$599$			0.563465i			0.563465i	0.959493	+	0.281733i	$0.0909091\pi$
$599$			0.563465i			0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$600$		0			0
$601$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$601$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$602$		0			0
$603$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$604$		0			0
$605$	1.25667	−	0.368991i	1.25667	−	0.368991i
$606$		0			0
$607$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$607$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$613$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$617$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$618$		0			0
$619$	1.14231	+	0.989821i	1.14231	+	0.989821i	1.00000			$0$
$619$	1.14231	+	0.989821i	1.14231	+	0.989821i	0.142315	+	0.989821i	$0.454545\pi$
$620$	1.08816			1.08816
$621$	−1.00000			−1.00000
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.171292	+	1.19136i	0.171292	+	1.19136i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.512546	+	1.74557i	−0.512546	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$631$	−0.512546	+	1.74557i	−0.512546	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−0.186393	−	0.215109i	−0.186393	−	0.215109i
$637$		0			0
$638$		0			0
$639$	−1.49611	−	0.215109i	−1.49611	−	0.215109i
$640$		0			0
$641$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$641$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$642$		0			0
$643$		−	1.81926i		−	1.81926i	−0.415415	−	0.909632i	$-0.636364\pi$
$643$		−	1.81926i		−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$647$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	−1.00000			$\pi$
$648$		0			0
$649$	−1.80075	+	0.822373i	−1.80075	+	0.822373i
$650$		0			0
$651$		0			0
$652$	1.41542	+	0.909632i	1.41542	+	0.909632i
$653$	−0.983568	−	0.449181i	−0.983568	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$653$	−0.983568	−	0.449181i	−0.983568	−	0.449181i	−0.841254	+	0.540641i	$0.818182\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$659$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$660$	1.30972			1.30972
$661$	−1.07028	−	0.153882i	−1.07028	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$661$	−1.07028	−	0.153882i	−1.07028	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.41542	−	0.909632i	1.41542	−	0.909632i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$673$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$674$		0			0
$675$	0.101808	+	0.708089i	0.101808	+	0.708089i
$676$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$677$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$677$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.304632	+	1.03748i	0.304632	+	1.03748i	0.959493	+	0.281733i	$0.0909091\pi$
$683$	0.304632	+	1.03748i	0.304632	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$684$		0			0
$685$	0.915415	+	2.00448i	0.915415	+	2.00448i
$686$		0			0
$687$	0.512546	+	1.74557i	0.512546	+	1.74557i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$691$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$692$		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			0
$701$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$701$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$702$		0			0
$703$		0			0
$704$	0.415415	−	0.909632i	0.415415	−	0.909632i
$705$	1.80075	−	1.56036i	1.80075	−	1.56036i
$706$		0			0
$707$		0			0
$708$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$709$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$709$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$714$		0			0
$715$		0			0
$716$	0.584585	+	0.909632i	0.584585	+	0.909632i
$717$		0			0
$718$		0			0
$719$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$719$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$720$	1.25667	+	0.368991i	1.25667	+	0.368991i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−0.817178	+	0.708089i	−0.817178	+	0.708089i
$725$		0			0
$726$		0			0
$727$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$727$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$728$		0			0
$729$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$733$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$734$		0			0
$735$	1.25667	+	0.368991i	1.25667	+	0.368991i
$736$		0			0
$737$		−	0.563465i		−	0.563465i
$738$		0			0
$739$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$739$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$740$	−2.56639	+	0.368991i	−2.56639	+	0.368991i
$741$		0			0
$742$		0			0
$743$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$743$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$751$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$752$	−0.512546	−	1.74557i	−0.512546	−	1.74557i
$753$	−0.118239	+	0.258908i	−0.118239	+	0.258908i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.817178	−	0.708089i	−0.817178	−	0.708089i	0.142315	−	0.989821i	$-0.454545\pi$
$757$	−0.817178	−	0.708089i	−0.817178	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$758$		0			0
$759$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$760$		0			0
$761$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$761$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$762$		0			0
$763$		0			0
$764$	0.797176	+	1.74557i	0.797176	+	1.74557i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.654861	−	0.755750i	0.654861	−	0.755750i
$769$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$769$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$770$		0			0
$771$	1.80075	−	0.822373i	1.80075	−	0.822373i
$772$		0			0
$773$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	0.654861	+	0.755750i	$0.272727\pi$
$773$	−0.345139	+	0.755750i	−0.345139	+	0.755750i	−1.00000			$\pi$
$774$		0			0
$775$	−0.570276	+	0.167448i	−0.570276	+	0.167448i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	1.51150i		−	1.51150i
$782$		0			0
$783$		0			0
$784$	0.654861	−	0.755750i	0.654861	−	0.755750i
$785$		0			0
$786$		0			0
$787$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$787$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	0.313607	+	0.201543i	0.313607	+	0.201543i
$796$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$797$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.959493	+	0.281733i	$0.0909091\pi$
$797$	1.61435	+	1.03748i	1.61435	+	1.03748i	0.654861	+	0.755750i	$0.272727\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	0.797176	−	1.74557i	0.797176	−	1.74557i
$802$		0			0
$803$		0			0
$804$	0.158746	−	0.540641i	0.158746	−	0.540641i
$805$		0			0
$806$		0			0
$807$	−0.817178	−	1.27155i	−0.817178	−	1.27155i
$808$		0			0
$809$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$809$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$810$		0			0
$811$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$811$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−2.11435	−	0.620830i	−2.11435	−	0.620830i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$821$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$822$		0			0
$823$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$823$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$824$		0			0
$825$	−0.686393	+	0.201543i	−0.686393	+	0.201543i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	0.142315	−	0.989821i	0.142315	−	0.989821i
$829$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$829$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.797176	−	0.234072i	0.797176	−	0.234072i
$838$		0			0
$839$	0.544078	+	0.627899i	0.544078	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$839$	0.544078	+	0.627899i	0.544078	+	0.627899i	−0.415415	+	0.909632i	$0.636364\pi$
$840$		0			0
$841$	0.959493	+	0.281733i	0.959493	+	0.281733i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	0.186393	−	1.29639i	0.186393	−	1.29639i
$846$		0			0
$847$		0			0
$848$	0.239446	−	0.153882i	0.239446	−	0.153882i
$849$		0			0
$850$		0			0
$851$	0.557730	+	1.89945i	0.557730	+	1.89945i
$852$	0.425839	−	1.45027i	0.425839	−	1.45027i
$853$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$853$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$857$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$858$		0			0
$859$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$859$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.14231	−	0.989821i	1.14231	−	0.989821i	0.142315	−	0.989821i	$-0.454545\pi$
$863$	1.14231	−	0.989821i	1.14231	−	0.989821i	1.00000			$0$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.00000			−1.00000
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−0.983568	+	1.53046i	−0.983568	+	1.53046i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$877$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$878$		0			0
$879$		0			0
$880$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$881$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$881$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$882$		0			0
$883$	0.345139	−	0.755750i	0.345139	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$883$	0.345139	−	0.755750i	0.345139	−	0.755750i	1.00000			$0$
$884$		0			0
$885$	2.35848	−	1.07708i	2.35848	−	1.07708i
$886$		0			0
$887$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$887$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.959493	−	0.281733i	0.959493	−	0.281733i
$892$	0.698939	+	1.53046i	0.698939	+	1.53046i
$893$		0			0
$894$		0			0
$895$	−1.07028	−	0.927399i	−1.07028	−	0.927399i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.715370			−0.715370
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	0.765644	−	1.19136i	0.765644	−	1.19136i
$906$		0			0
$907$	0.425839	−	1.45027i	0.425839	−	1.45027i	−0.415415	−	0.909632i	$-0.636364\pi$
$907$	0.425839	−	1.45027i	0.425839	−	1.45027i	0.841254	−	0.540641i	$-0.181818\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$911$	−0.698939	+	1.53046i	−0.698939	+	1.53046i	−0.841254	+	0.540641i	$0.818182\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.80075	+	0.258908i	−1.80075	+	0.258908i
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.28820	−	0.588302i	1.28820	−	0.588302i
$926$		0			0
$927$	0.304632	−	1.03748i	0.304632	−	1.03748i
$928$		0			0
$929$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$929$	−0.512546	−	0.234072i	−0.512546	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	1.07028	−	1.66538i	1.07028	−	1.66538i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$937$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$938$		0			0
$939$	1.49611	+	0.215109i	1.49611	+	0.215109i
$940$	1.28820	+	2.00448i	1.28820	+	2.00448i
$941$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$941$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$942$		0			0
$943$		0			0
$944$		−	1.97964i		−	1.97964i
$945$		0			0
$946$		0			0
$947$	1.07028	−	0.153882i	1.07028	−	0.153882i	0.415415	−	0.909632i	$-0.363636\pi$
$947$	1.07028	−	0.153882i	1.07028	−	0.153882i	0.654861	+	0.755750i	$0.272727\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$952$		0			0
$953$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$953$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$954$		0			0
$955$	−1.64589	−	1.89945i	−1.64589	−	1.89945i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$961$	−0.128663	−	0.281733i	−0.128663	−	0.281733i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$971$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$972$	1.00000			1.00000
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$977$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$978$		0			0
$979$	1.84125	+	0.540641i	1.84125	+	0.540641i
$980$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$981$		0			0
$982$		0			0
$983$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	0.959493	+	0.281733i	$0.0909091\pi$
$983$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	−1.00000			$1.00000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.857685	+	0.989821i	−0.857685	+	0.989821i	0.142315	+	0.989821i	$0.454545\pi$
$991$	−0.857685	+	0.989821i	−0.857685	+	0.989821i	−1.00000			$\pi$
$992$		0			0
$993$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$994$		0			0
$995$	0.730471	+	0.105026i	0.730471	+	0.105026i
$996$		0			0
$997$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$997$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$998$		0			0
$999$	−1.80075	+	0.822373i	−1.80075	+	0.822373i

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	759.1.r.a.329.1	yes	10
3.2	odd	2		759.1.r.b.329.1	yes	10
11.10	odd	2	CM	759.1.r.a.329.1	yes	10
23.10	odd	22		759.1.r.b.263.1	yes	10
33.32	even	2		759.1.r.b.329.1	yes	10
69.56	even	22	inner	759.1.r.a.263.1	✓	10
253.10	even	22		759.1.r.b.263.1	yes	10
759.263	odd	22	inner	759.1.r.a.263.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
759.1.r.a.263.1	✓	10	69.56	even	22	inner
759.1.r.a.263.1	✓	10	759.263	odd	22	inner
759.1.r.a.329.1	yes	10	1.1	even	1	trivial
759.1.r.a.329.1	yes	10	11.10	odd	2	CM
759.1.r.b.263.1	yes	10	23.10	odd	22
759.1.r.b.263.1	yes	10	253.10	even	22
759.1.r.b.329.1	yes	10	3.2	odd	2
759.1.r.b.329.1	yes	10	33.32	even	2

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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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	759.1.r.a.329.1

Level	$759$
Weight	$1$
Character	759.329
Analytic conductor	$0.379$
Analytic rank	$0$
Dimension	$10$
Projective image	$D_{22}$
CM discriminant	-11
Inner twists	$4$