LMFDB - Embedded newform 1725.1.bc.a.1301.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1725,1,Mod(26,1725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1725, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 0, 16]))

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

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

chi := DirichletCharacter("1725.26");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1725 = 3 \cdot 5^{2} \cdot 23$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1725.bc (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:	$0.860887146792$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1$ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 345)
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} - \cdots)$

Embedding invariants

Embedding label			1301.1
Root			$-0.540641 - 0.841254i$ of defining polynomial
Character	$\chi$	$=$	1725.1301
Dual form			1725.1.bc.a.476.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.27155 + 1.10181i) q^{2} +(-0.540641 + 0.841254i) q^{3} +(0.260554 - 1.81219i) q^{4} +(-0.239446 - 1.66538i) q^{6} +(0.755750 + 1.17597i) q^{8} +(-0.415415 - 0.909632i) q^{9} +(1.38365 + 1.19894i) q^{12} +(-0.500000 - 0.146813i) q^{16} +(0.281733 - 0.0405070i) q^{17} +(1.53046 + 0.698939i) q^{18} +(-0.273100 + 1.89945i) q^{19} +(-0.755750 - 0.654861i) q^{23} -1.39788 q^{24} +(0.989821 + 0.142315i) q^{27} +(-1.61435 + 1.03748i) q^{31} +(-0.474017 + 0.216476i) q^{32} +(-0.313607 + 0.361922i) q^{34} +(-1.75667 + 0.515804i) q^{36} +(-1.74557 - 2.71616i) q^{38} +1.68251 q^{46} +1.30972i q^{47} +(0.393828 - 0.341254i) q^{48} +(-0.841254 - 0.540641i) q^{49} +(-0.118239 + 0.258908i) q^{51} +(-0.368991 + 1.25667i) q^{53} +(-1.41542 + 0.909632i) q^{54} +(-1.45027 - 1.25667i) q^{57} +(-1.10181 + 0.708089i) q^{61} +(0.909632 - 3.09792i) q^{62} +(0.580699 - 1.27155i) q^{64} -0.521109i q^{68} +(0.959493 - 0.281733i) q^{69} +(0.755750 - 1.17597i) q^{72} +(3.37102 + 0.989821i) q^{76} +(0.797176 - 0.234072i) q^{79} +(-0.654861 + 0.755750i) q^{81} +(-1.53046 + 0.698939i) q^{83} +(-1.38365 + 1.19894i) q^{92} -1.91899i q^{93} +(-1.44306 - 1.66538i) q^{94} +(0.0741615 - 0.515804i) q^{96} +(1.66538 - 0.239446i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$20 q + 6 q^{4} - 4 q^{6} + 2 q^{9} - 10 q^{16} + 4 q^{19} + 8 q^{24} - 4 q^{31} - 14 q^{34} - 6 q^{36} - 4 q^{46} + 2 q^{49} - 4 q^{51} - 18 q^{54} - 4 q^{61} - 8 q^{64} + 2 q^{69} + 10 q^{76} + 4 q^{79}+ \cdots + 10 q^{96}+O(q^{100})$ 20 * q + 6 * q^4 - 4 * q^6 + 2 * q^9 - 10 * q^16 + 4 * q^19 + 8 * q^24 - 4 * q^31 - 14 * q^34 - 6 * q^36 - 4 * q^46 + 2 * q^49 - 4 * q^51 - 18 * q^54 - 4 * q^61 - 8 * q^64 + 2 * q^69 + 10 * q^76 + 4 * q^79 - 2 * q^81 + 8 * q^94 + 10 * q^96

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1725\mathbb{Z}\right)^\times$ .

$n$	$277$	$1151$	$1201$
$\chi(n)$	$1$	$-1$	$e\left(\frac{7}{11}\right)$

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

)

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

$2$	−1.27155	+	1.10181i	−1.27155	+	1.10181i	−0.281733	+	0.959493i	$0.590909\pi$
$2$	−1.27155	+	1.10181i	−1.27155	+	1.10181i	−0.989821	+	0.142315i	$0.954545\pi$
$3$	−0.540641	+	0.841254i	−0.540641	+	0.841254i
$3$	−0.540641	+	0.841254i	−0.540641	+	0.841254i
$4$	0.260554	−	1.81219i	0.260554	−	1.81219i
$5$		0			0
$5$		0			0
$6$	−0.239446	−	1.66538i	−0.239446	−	1.66538i
$7$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$7$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$8$	0.755750	+	1.17597i	0.755750	+	1.17597i
$9$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$10$		0			0
$11$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$11$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$12$	1.38365	+	1.19894i	1.38365	+	1.19894i
$13$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$13$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$14$		0			0
$15$		0			0
$16$	−0.500000	−	0.146813i	−0.500000	−	0.146813i
$17$	0.281733	−	0.0405070i	0.281733	−	0.0405070i		−	1.00000i	$-0.5\pi$
$17$	0.281733	−	0.0405070i	0.281733	−	0.0405070i	0.281733	+	0.959493i	$0.409091\pi$
$18$	1.53046	+	0.698939i	1.53046	+	0.698939i
$19$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$19$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.755750	−	0.654861i	−0.755750	−	0.654861i
$23$	−0.755750	−	0.654861i	−0.755750	−	0.654861i
$24$	−1.39788			−1.39788
$25$		0			0
$26$		0			0
$27$	0.989821	+	0.142315i	0.989821	+	0.142315i
$28$		0			0
$29$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$29$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$30$		0			0
$31$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$31$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$32$	−0.474017	+	0.216476i	−0.474017	+	0.216476i
$33$		0			0
$34$	−0.313607	+	0.361922i	−0.313607	+	0.361922i
$35$		0			0
$36$	−1.75667	+	0.515804i	−1.75667	+	0.515804i
$37$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$37$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$38$	−1.74557	−	2.71616i	−1.74557	−	2.71616i
$39$		0			0
$40$		0			0
$41$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$41$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$42$		0			0
$43$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$43$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$44$		0			0
$45$		0			0
$46$	1.68251			1.68251
$47$			1.30972i			1.30972i	0.755750	+	0.654861i	$0.227273\pi$
$47$			1.30972i			1.30972i	−0.755750	+	0.654861i	$0.772727\pi$
$48$	0.393828	−	0.341254i	0.393828	−	0.341254i
$49$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$50$		0			0
$51$	−0.118239	+	0.258908i	−0.118239	+	0.258908i
$52$		0			0
$53$	−0.368991	+	1.25667i	−0.368991	+	1.25667i	0.540641	+	0.841254i	$0.318182\pi$
$53$	−0.368991	+	1.25667i	−0.368991	+	1.25667i	−0.909632	+	0.415415i	$0.863636\pi$
$54$	−1.41542	+	0.909632i	−1.41542	+	0.909632i
$55$		0			0
$56$		0			0
$57$	−1.45027	−	1.25667i	−1.45027	−	1.25667i
$58$		0			0
$59$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$59$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$60$		0			0
$61$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$61$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$62$	0.909632	−	3.09792i	0.909632	−	3.09792i
$63$		0			0
$64$	0.580699	−	1.27155i	0.580699	−	1.27155i
$65$		0			0
$66$		0			0
$67$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$67$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$68$		−	0.521109i		−	0.521109i
$69$	0.959493	−	0.281733i	0.959493	−	0.281733i
$70$		0			0
$71$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$71$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$72$	0.755750	−	1.17597i	0.755750	−	1.17597i
$73$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$73$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$74$		0			0
$75$		0			0
$76$	3.37102	+	0.989821i	3.37102	+	0.989821i
$77$		0			0
$78$		0			0
$79$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$79$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$80$		0			0
$81$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$82$		0			0
$83$	−1.53046	+	0.698939i	−1.53046	+	0.698939i	−0.989821	−	0.142315i	$-0.954545\pi$
$83$	−1.53046	+	0.698939i	−1.53046	+	0.698939i	−0.540641	+	0.841254i	$0.681818\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$89$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$90$		0			0
$91$		0			0
$92$	−1.38365	+	1.19894i	−1.38365	+	1.19894i
$93$		−	1.91899i		−	1.91899i
$94$	−1.44306	−	1.66538i	−1.44306	−	1.66538i
$95$		0			0
$96$	0.0741615	−	0.515804i	0.0741615	−	0.515804i
$97$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$97$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$98$	1.66538	−	0.239446i	1.66538	−	0.239446i
$99$		0			0
$100$		0			0
$101$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$101$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$102$	−0.134919	−	0.459493i	−0.134919	−	0.459493i
$103$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$103$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$104$		0			0
$105$		0			0
$106$	−0.915415	−	2.00448i	−0.915415	−	2.00448i
$107$	−0.449181	−	0.698939i	−0.449181	−	0.698939i	0.540641	−	0.841254i	$-0.318182\pi$
$107$	−0.449181	−	0.698939i	−0.449181	−	0.698939i	−0.989821	+	0.142315i	$0.954545\pi$
$108$	0.515804	−	1.75667i	0.515804	−	1.75667i
$109$	0.118239	+	0.822373i	0.118239	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$109$	0.118239	+	0.822373i	0.118239	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.27155	−	1.10181i	1.27155	−	1.10181i	0.281733	−	0.959493i	$-0.409091\pi$
$113$	1.27155	−	1.10181i	1.27155	−	1.10181i	0.989821	−	0.142315i	$-0.0454545\pi$
$114$	3.22871			3.22871
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$122$	0.620830	−	2.11435i	0.620830	−	2.11435i
$123$		0			0
$124$	1.45949	+	3.19584i	1.45949	+	3.19584i
$125$		0			0
$126$		0			0
$127$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$127$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$128$	0.515804	+	1.75667i	0.515804	+	1.75667i
$129$		0			0
$130$		0			0
$131$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$131$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.260554	+	0.300696i	0.260554	+	0.300696i
$137$		−	0.830830i		−	0.830830i	−0.909632	−	0.415415i	$-0.863636\pi$
$137$		−	0.830830i		−	0.830830i	0.909632	−	0.415415i	$-0.136364\pi$
$138$	−0.909632	+	1.41542i	−0.909632	+	1.41542i
$139$	1.30972			1.30972			0.654861	−	0.755750i	$-0.272727\pi$
$139$	1.30972			1.30972			0.654861	+	0.755750i	$0.272727\pi$
$140$		0			0
$141$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$142$		0			0
$143$		0			0
$144$	0.0741615	+	0.515804i	0.0741615	+	0.515804i
$145$		0			0
$146$		0			0
$147$	0.909632	−	0.415415i	0.909632	−	0.415415i
$148$		0			0
$149$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$149$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$150$		0			0
$151$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$151$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$152$	−2.44009	+	1.11435i	−2.44009	+	1.11435i
$153$	−0.153882	−	0.239446i	−0.153882	−	0.239446i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$157$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$158$	−0.755750	+	1.17597i	−0.755750	+	1.17597i
$159$	−0.857685	−	0.989821i	−0.857685	−	0.989821i
$160$		0			0
$161$		0			0
$162$		−	1.68251i		−	1.68251i
$163$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$163$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$164$		0			0
$165$		0			0
$166$	1.17597	−	2.57501i	1.17597	−	2.57501i
$167$	−0.822373	+	0.118239i	−0.822373	+	0.118239i	−0.540641	−	0.841254i	$-0.681818\pi$
$167$	−0.822373	+	0.118239i	−0.822373	+	0.118239i	−0.281733	+	0.959493i	$0.590909\pi$
$168$		0			0
$169$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$170$		0			0
$171$	1.84125	−	0.540641i	1.84125	−	0.540641i
$172$		0			0
$173$	0.989821	+	0.857685i	0.989821	+	0.857685i	0.989821	−	0.142315i	$-0.0454545\pi$
$173$	0.989821	+	0.857685i	0.989821	+	0.857685i			1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$179$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$180$		0			0
$181$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	0.415415	−	0.909632i	$-0.363636\pi$
$181$	−0.239446	−	0.153882i	−0.239446	−	0.153882i	−0.654861	+	0.755750i	$0.727273\pi$
$182$		0			0
$183$		−	1.30972i		−	1.30972i
$184$	0.198939	−	1.38365i	0.198939	−	1.38365i
$185$		0			0
$186$	2.11435	+	2.44009i	2.11435	+	2.44009i
$187$		0			0
$188$	2.37347	+	0.341254i	2.37347	+	0.341254i
$189$		0			0
$190$		0			0
$191$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$191$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$192$	0.755750	+	1.17597i	0.755750	+	1.17597i
$193$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$193$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$194$		0			0
$195$		0			0
$196$	−1.19894	+	1.38365i	−1.19894	+	1.38365i
$197$	0.234072	+	0.797176i	0.234072	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$197$	0.234072	+	0.797176i	0.234072	+	0.797176i	−0.755750	+	0.654861i	$0.772727\pi$
$198$		0			0
$199$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$199$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−1.00000			$\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	0.438384	+	0.281733i	0.438384	+	0.281733i
$205$		0			0
$206$		0			0
$207$	−0.281733	+	0.959493i	−0.281733	+	0.959493i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$211$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$212$	2.18119	+	0.996114i	2.18119	+	0.996114i
$213$		0			0
$214$	1.34125	+	0.393828i	1.34125	+	0.393828i
$215$		0			0
$216$	0.580699	+	1.27155i	0.580699	+	1.27155i
$217$		0			0
$218$	−1.05645	−	0.915415i	−1.05645	−	0.915415i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$223$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$224$		0			0
$225$		0			0
$226$	−0.402869	+	2.80202i	−0.402869	+	2.80202i
$227$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$227$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	−0.989821	+	0.142315i	$0.954545\pi$
$228$	−2.65520	+	2.30075i	−2.65520	+	2.30075i
$229$	−1.68251			−1.68251			−0.841254	−	0.540641i	$-0.818182\pi$
$229$	−1.68251			−1.68251			−0.841254	+	0.540641i	$0.818182\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.449181	+	0.698939i	−0.449181	+	0.698939i	−0.989821	−	0.142315i	$-0.954545\pi$
$233$	−0.449181	+	0.698939i	−0.449181	+	0.698939i	0.540641	+	0.841254i	$0.318182\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.234072	+	0.797176i	−0.234072	+	0.797176i
$238$		0			0
$239$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$239$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$240$		0			0
$241$	0.186393	−	0.215109i	0.186393	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$241$	0.186393	−	0.215109i	0.186393	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$242$	1.27155	+	1.10181i	1.27155	+	1.10181i
$243$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$244$	0.996114	+	2.18119i	0.996114	+	2.18119i
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−2.44009	−	1.11435i	−2.44009	−	1.11435i
$249$	0.239446	−	1.66538i	0.239446	−	1.66538i
$250$		0			0
$251$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$251$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.41542	−	0.909632i	−1.41542	−	0.909632i
$257$	−0.281733	−	0.0405070i	−0.281733	−	0.0405070i		−	1.00000i	$-0.5\pi$
$257$	−0.281733	−	0.0405070i	−0.281733	−	0.0405070i	−0.281733	+	0.959493i	$0.590909\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	0.540641	+	1.84125i	0.540641	+	1.84125i	0.540641	+	0.841254i	$0.318182\pi$
$263$	0.540641	+	1.84125i	0.540641	+	1.84125i			1.00000i	$0.5\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$269$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$270$		0			0
$271$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$271$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$272$	−0.146813	−	0.0211086i	−0.146813	−	0.0211086i
$273$		0			0
$274$	0.915415	+	1.05645i	0.915415	+	1.05645i
$275$		0			0
$276$	−0.260554	−	1.81219i	−0.260554	−	1.81219i
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−1.66538	+	1.44306i	−1.66538	+	1.44306i
$279$	1.61435	+	1.03748i	1.61435	+	1.03748i
$280$		0			0
$281$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$281$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$282$	2.18119	−	0.313607i	2.18119	−	0.313607i
$283$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$283$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	0.393828	+	0.341254i	0.393828	+	0.341254i
$289$	−0.881761	+	0.258908i	−0.881761	+	0.258908i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.66538	−	0.239446i	1.66538	−	0.239446i	0.755750	−	0.654861i	$-0.227273\pi$
$293$	1.66538	−	0.239446i	1.66538	−	0.239446i	0.909632	+	0.415415i	$0.136364\pi$
$294$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−0.258908	+	0.402869i	−0.258908	+	0.402869i
$303$		0			0
$304$	0.415415	−	0.909632i	0.415415	−	0.909632i
$305$		0			0
$306$	0.459493	+	0.134919i	0.459493	+	0.134919i
$307$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$307$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$311$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$312$		0			0
$313$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$313$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$314$		0			0
$315$		0			0
$316$	−0.216476	−	1.50563i	−0.216476	−	1.50563i
$317$	0.258908	+	0.118239i	0.258908	+	0.118239i	0.540641	−	0.841254i	$-0.318182\pi$
$317$	0.258908	+	0.118239i	0.258908	+	0.118239i	−0.281733	+	0.959493i	$0.590909\pi$
$318$	2.18119	+	0.313607i	2.18119	+	0.313607i
$319$		0			0
$320$		0			0
$321$	0.830830			0.830830
$322$		0			0
$323$			0.546200i			0.546200i
$324$	1.19894	+	1.38365i	1.19894	+	1.38365i
$325$		0			0
$326$		0			0
$327$	−0.755750	−	0.345139i	−0.755750	−	0.345139i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$331$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$332$	0.867845	+	2.95561i	0.867845	+	2.95561i
$333$		0			0
$334$	0.915415	−	1.05645i	0.915415	−	1.05645i
$335$		0			0
$336$		0			0
$337$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$337$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$338$	0.474017	−	1.61435i	0.474017	−	1.61435i
$339$	0.239446	+	1.66538i	0.239446	+	1.66538i
$340$		0			0
$341$		0			0
$342$	−1.74557	+	2.71616i	−1.74557	+	2.71616i
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−2.20362			−2.20362
$347$	1.45027	−	1.25667i	1.45027	−	1.25667i	0.540641	−	0.841254i	$-0.318182\pi$
$347$	1.45027	−	1.25667i	1.45027	−	1.25667i	0.909632	−	0.415415i	$-0.136364\pi$
$348$		0			0
$349$	0.239446	−	1.66538i	0.239446	−	1.66538i	−0.415415	−	0.909632i	$-0.636364\pi$
$349$	0.239446	−	1.66538i	0.239446	−	1.66538i	0.654861	−	0.755750i	$-0.272727\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.03748	−	1.61435i	−1.03748	−	1.61435i	−0.755750	−	0.654861i	$-0.772727\pi$
$353$	−1.03748	−	1.61435i	−1.03748	−	1.61435i	−0.281733	−	0.959493i	$-0.590909\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$359$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$360$		0			0
$361$	−2.57385	−	0.755750i	−2.57385	−	0.755750i
$362$	0.474017	−	0.0681534i	0.474017	−	0.0681534i
$363$	0.909632	+	0.415415i	0.909632	+	0.415415i
$364$		0			0
$365$		0			0
$366$	1.44306	+	1.66538i	1.44306	+	1.66538i
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$	0.281733	+	0.438384i	0.281733	+	0.438384i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−3.47758	−	0.500000i	−3.47758	−	0.500000i
$373$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$373$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$374$		0			0
$375$		0			0
$376$	−1.54019	+	0.989821i	−1.54019	+	0.989821i
$377$		0			0
$378$		0			0
$379$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$379$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.909632	+	1.41542i	0.909632	+	1.41542i	0.909632	+	0.415415i	$0.136364\pi$
$383$	0.909632	+	1.41542i	0.909632	+	1.41542i			1.00000i	$0.5\pi$
$384$	−1.75667	−	0.515804i	−1.75667	−	0.515804i
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$389$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$390$		0			0
$391$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$392$		−	1.39788i		−	1.39788i
$393$		0			0
$394$	−1.17597	−	0.755750i	−1.17597	−	0.755750i
$395$		0			0
$396$		0			0
$397$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$397$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$398$	0.797537	−	2.71616i	0.797537	−	2.71616i
$399$		0			0
$400$		0			0
$401$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$401$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−0.393828	+	0.0566239i	−0.393828	+	0.0566239i
$409$	0.118239	−	0.258908i	0.118239	−	0.258908i	−0.841254	−	0.540641i	$-0.818182\pi$
$409$	0.118239	−	0.258908i	0.118239	−	0.258908i	0.959493	+	0.281733i	$0.0909091\pi$
$410$		0			0
$411$	0.698939	+	0.449181i	0.698939	+	0.449181i
$412$		0			0
$413$		0			0
$414$	−0.698939	−	1.53046i	−0.698939	−	1.53046i
$415$		0			0
$416$		0			0
$417$	−0.708089	+	1.10181i	−0.708089	+	1.10181i
$418$		0			0
$419$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$419$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$420$		0			0
$421$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$421$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$422$	0.258908	+	0.402869i	0.258908	+	0.402869i
$423$	1.19136	−	0.544078i	1.19136	−	0.544078i
$424$	−1.75667	+	0.515804i	−1.75667	+	0.515804i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−1.38365	+	0.631891i	−1.38365	+	0.631891i
$429$		0			0
$430$		0			0
$431$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$431$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$432$	−0.474017	−	0.216476i	−0.474017	−	0.216476i
$433$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$433$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$434$		0			0
$435$		0			0
$436$	1.52111			1.52111
$437$	1.45027	−	1.25667i	1.45027	−	1.25667i
$438$		0			0
$439$	1.10181	+	1.27155i	1.10181	+	1.27155i	0.959493	+	0.281733i	$0.0909091\pi$
$439$	1.10181	+	1.27155i	1.10181	+	1.27155i	0.142315	+	0.989821i	$0.454545\pi$
$440$		0			0
$441$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$442$		0			0
$443$	−0.281733	+	0.0405070i	−0.281733	+	0.0405070i	−0.281733	−	0.959493i	$-0.590909\pi$
$443$	−0.281733	+	0.0405070i	−0.281733	+	0.0405070i			1.00000i	$0.5\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$449$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$450$		0			0
$451$		0			0
$452$	−1.66538	−	2.59138i	−1.66538	−	2.59138i
$453$	−0.0801894	+	0.273100i	−0.0801894	+	0.273100i
$454$	−0.313607	−	2.18119i	−0.313607	−	2.18119i
$455$		0			0
$456$	0.381761	−	2.65520i	0.381761	−	2.65520i
$457$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$457$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$458$	2.13940	−	1.85380i	2.13940	−	1.85380i
$459$	0.284630			0.284630
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$463$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$464$		0			0
$465$		0			0
$466$	−0.198939	−	1.38365i	−0.198939	−	1.38365i
$467$	0.0801894	−	0.273100i	0.0801894	−	0.273100i	−0.909632	−	0.415415i	$-0.863636\pi$
$467$	0.0801894	−	0.273100i	0.0801894	−	0.273100i	0.989821	+	0.142315i	$0.0454545\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$	−0.580699	−	1.27155i	−0.580699	−	1.27155i
$475$		0			0
$476$		0			0
$477$	1.29639	−	0.186393i	1.29639	−	0.186393i
$478$		0			0
$479$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$479$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$480$		0			0
$481$		0			0
$482$			0.478891i			0.478891i
$483$		0			0
$484$	−1.83083			−1.83083
$485$		0			0
$486$	1.41542	+	0.909632i	1.41542	+	0.909632i
$487$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$487$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$488$	−1.66538	−	0.760554i	−1.66538	−	0.760554i
$489$		0			0
$490$		0			0
$491$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$491$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.959493	−	0.281733i	0.959493	−	0.281733i
$497$		0			0
$498$	1.53046	+	2.38145i	1.53046	+	2.38145i
$499$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.654861	−	0.755750i	$-0.272727\pi$
$499$	0.797176	+	0.234072i	0.797176	+	0.234072i	0.142315	+	0.989821i	$0.454545\pi$
$500$		0			0
$501$	0.345139	−	0.755750i	0.345139	−	0.755750i
$502$		0			0
$503$	1.03748	−	1.61435i	1.03748	−	1.61435i	0.281733	−	0.959493i	$-0.409091\pi$
$503$	1.03748	−	1.61435i	1.03748	−	1.61435i	0.755750	−	0.654861i	$-0.227273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		−	1.00000i		−	1.00000i
$508$		0			0
$509$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$509$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$510$		0			0
$511$		0			0
$512$	0.989821	−	0.142315i	0.989821	−	0.142315i
$513$	−0.540641	+	1.84125i	−0.540641	+	1.84125i
$514$	0.402869	−	0.258908i	0.402869	−	0.258908i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−1.25667	+	0.368991i	−1.25667	+	0.368991i
$520$		0			0
$521$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$521$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$522$		0			0
$523$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$523$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$524$		0			0
$525$		0			0
$526$	−2.71616	−	1.74557i	−2.71616	−	1.74557i
$527$	−0.412791	+	0.357685i	−0.412791	+	0.357685i
$528$		0			0
$529$	0.142315	+	0.989821i	0.142315	+	0.989821i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.25667	−	1.45027i	1.25667	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$541$	1.25667	−	1.45027i	1.25667	−	1.45027i	0.841254	−	0.540641i	$-0.181818\pi$
$542$	−0.620830	−	2.11435i	−0.620830	−	2.11435i
$543$	0.258908	−	0.118239i	0.258908	−	0.118239i
$544$	−0.124777	+	0.0801894i	−0.124777	+	0.0801894i
$545$		0			0
$546$		0			0
$547$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$547$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$548$	−1.50563	−	0.216476i	−1.50563	−	0.216476i
$549$	1.10181	+	0.708089i	1.10181	+	0.708089i
$550$		0			0
$551$		0			0
$552$	1.05645	+	0.915415i	1.05645	+	0.915415i
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.341254	−	2.37347i	0.341254	−	2.37347i
$557$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$557$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.755750	+	0.654861i	$0.227273\pi$
$558$	−3.19584	+	0.459493i	−3.19584	+	0.459493i
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	0.989821	+	0.857685i	0.989821	+	0.857685i	0.989821	−	0.142315i	$-0.0454545\pi$
$563$	0.989821	+	0.857685i	0.989821	+	0.857685i			1.00000i	$0.5\pi$
$564$	−1.57028	+	1.81219i	−1.57028	+	1.81219i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$569$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$570$		0			0
$571$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$571$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.39788			−1.39788
$577$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$577$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$578$	0.835939	−	1.30075i	0.835939	−	1.30075i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$	−1.85380	+	2.13940i	−1.85380	+	2.13940i
$587$	−1.45027	−	1.25667i	−1.45027	−	1.25667i	−0.909632	−	0.415415i	$-0.863636\pi$
$587$	−1.45027	−	1.25667i	−1.45027	−	1.25667i	−0.540641	−	0.841254i	$-0.681818\pi$
$588$	−0.515804	−	1.75667i	−0.515804	−	1.75667i
$589$	−1.52977	−	3.34973i	−1.52977	−	3.34973i
$590$		0			0
$591$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$592$		0			0
$593$	−1.74557	−	0.797176i	−1.74557	−	0.797176i	−0.989821	−	0.142315i	$-0.954545\pi$
$593$	−1.74557	−	0.797176i	−1.74557	−	0.797176i	−0.755750	−	0.654861i	$-0.772727\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	1.68251i		−	1.68251i
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	0.698939	+	0.449181i	0.698939	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$601$	0.698939	+	0.449181i	0.698939	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$602$		0			0
$603$		0			0
$604$	−0.0741615	−	0.515804i	−0.0741615	−	0.515804i
$605$		0			0
$606$		0			0
$607$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$607$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$608$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.474017	+	0.216476i	−0.474017	+	0.216476i
$613$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$613$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.29639	−	0.186393i	−1.29639	−	0.186393i	−0.540641	−	0.841254i	$-0.681818\pi$
$617$	−1.29639	−	0.186393i	−1.29639	−	0.186393i	−0.755750	+	0.654861i	$0.772727\pi$
$618$		0			0
$619$	0.544078	+	0.627899i	0.544078	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$619$	0.544078	+	0.627899i	0.544078	+	0.627899i	−0.415415	+	0.909632i	$0.636364\pi$
$620$		0			0
$621$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$622$		0			0
$623$		0			0
$624$		0			0
$625$		0			0
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$631$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$632$	0.877726	+	0.760554i	0.877726	+	0.760554i
$633$	0.215109	+	0.186393i	0.215109	+	0.186393i
$634$	−0.459493	+	0.134919i	−0.459493	+	0.134919i
$635$		0			0
$636$	−2.01722	+	1.29639i	−2.01722	+	1.29639i
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$641$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$642$	−1.05645	+	0.915415i	−1.05645	+	0.915415i
$643$		0			0			−	1.00000i	$-0.5\pi$
$643$		0			0				1.00000i	$0.5\pi$
$644$		0			0
$645$		0			0
$646$	−0.601808	−	0.694523i	−0.601808	−	0.694523i
$647$	−0.153882	+	0.239446i	−0.153882	+	0.239446i	−0.909632	−	0.415415i	$-0.863636\pi$
$647$	−0.153882	+	0.239446i	−0.153882	+	0.239446i	0.755750	+	0.654861i	$0.227273\pi$
$648$	−1.38365	−	0.198939i	−1.38365	−	0.198939i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−0.755750	+	0.345139i	−0.755750	+	0.345139i	−0.755750	−	0.654861i	$-0.772727\pi$
$653$	−0.755750	+	0.345139i	−0.755750	+	0.345139i			1.00000i	$0.5\pi$
$654$	1.34125	−	0.393828i	1.34125	−	0.393828i
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$659$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$660$		0			0
$661$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$661$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$662$	−2.57501	−	1.17597i	−2.57501	−	1.17597i
$663$		0			0
$664$	−1.97858	−	1.27155i	−1.97858	−	1.27155i
$665$		0			0
$666$		0			0
$667$		0			0
$668$			1.52111i			1.52111i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$673$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$674$		0			0
$675$		0			0
$676$	0.760554	+	1.66538i	0.760554	+	1.66538i
$677$	0.234072	+	0.797176i	0.234072	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$677$	0.234072	+	0.797176i	0.234072	+	0.797176i	−0.755750	+	0.654861i	$0.772727\pi$
$678$	−2.13940	−	1.85380i	−2.13940	−	1.85380i
$679$		0			0
$680$		0			0
$681$	−0.544078	−	1.19136i	−0.544078	−	1.19136i
$682$		0			0
$683$	−0.368991	+	1.25667i	−0.368991	+	1.25667i	0.540641	+	0.841254i	$0.318182\pi$
$683$	−0.368991	+	1.25667i	−0.368991	+	1.25667i	−0.909632	+	0.415415i	$0.863636\pi$
$684$	−0.500000	−	3.47758i	−0.500000	−	3.47758i
$685$		0			0
$686$		0			0
$687$	0.909632	−	1.41542i	0.909632	−	1.41542i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.68251			1.68251			0.841254	−	0.540641i	$-0.181818\pi$
$691$	1.68251			1.68251			0.841254	+	0.540641i	$0.181818\pi$
$692$	1.81219	−	1.57028i	1.81219	−	1.57028i
$693$		0			0
$694$	−0.459493	+	3.19584i	−0.459493	+	3.19584i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	1.53046	+	2.38145i	1.53046	+	2.38145i
$699$	−0.345139	−	0.755750i	−0.345139	−	0.755750i
$700$		0			0
$701$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$701$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	3.09792	+	0.909632i	3.09792	+	0.909632i
$707$		0			0
$708$		0			0
$709$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	0.959493	+	0.281733i	$0.0909091\pi$
$709$	−0.0405070	+	0.281733i	−0.0405070	+	0.281733i	−1.00000			$1.00000\pi$
$710$		0			0
$711$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$712$		0			0
$713$	1.89945	+	0.273100i	1.89945	+	0.273100i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$719$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$720$		0			0
$721$		0			0
$722$	4.10548	−	1.87491i	4.10548	−	1.87491i
$723$	0.0801894	+	0.273100i	0.0801894	+	0.273100i
$724$	−0.341254	+	0.393828i	−0.341254	+	0.393828i
$725$		0			0
$726$	−1.61435	+	0.474017i	−1.61435	+	0.474017i
$727$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$727$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$728$		0			0
$729$	0.959493	+	0.281733i	0.959493	+	0.281733i
$730$		0			0
$731$		0			0
$732$	−2.37347	−	0.341254i	−2.37347	−	0.341254i
$733$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$733$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$734$		0			0
$735$		0			0
$736$	0.500000	+	0.146813i	0.500000	+	0.146813i
$737$		0			0
$738$		0			0
$739$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−0.415415	−	0.909632i	$-0.636364\pi$
$739$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	0.474017	−	1.61435i	0.474017	−	1.61435i	−0.281733	−	0.959493i	$-0.590909\pi$
$743$	0.474017	−	1.61435i	0.474017	−	1.61435i	0.755750	−	0.654861i	$-0.227273\pi$
$744$	2.25667	−	1.45027i	2.25667	−	1.45027i
$745$		0			0
$746$		0			0
$747$	1.27155	+	1.10181i	1.27155	+	1.10181i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$751$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$752$	0.192284	−	0.654861i	0.192284	−	0.654861i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$757$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$758$			2.83083i			2.83083i
$759$		0			0
$760$		0			0
$761$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$761$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$	−2.71616	−	0.797537i	−2.71616	−	0.797537i
$767$		0			0
$768$	1.53046	−	0.698939i	1.53046	−	0.698939i
$769$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$769$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$770$		0			0
$771$	0.186393	−	0.215109i	0.186393	−	0.215109i
$772$		0			0
$773$	1.74557	−	0.797176i	1.74557	−	0.797176i	0.755750	−	0.654861i	$-0.227273\pi$
$773$	1.74557	−	0.797176i	1.74557	−	0.797176i	0.989821	−	0.142315i	$-0.0454545\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	0.474017	−	0.0681534i	0.474017	−	0.0681534i
$783$		0			0
$784$	0.341254	+	0.393828i	0.341254	+	0.393828i
$785$		0			0
$786$		0			0
$787$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$787$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$788$	1.50563	−	0.216476i	1.50563	−	0.216476i
$789$	−1.84125	−	0.540641i	−1.84125	−	0.540641i
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	1.27964	+	2.80202i	1.27964	+	2.80202i
$797$	1.03748	+	1.61435i	1.03748	+	1.61435i	0.755750	+	0.654861i	$0.227273\pi$
$797$	1.03748	+	1.61435i	1.03748	+	1.61435i	0.281733	+	0.959493i	$0.409091\pi$
$798$		0			0
$799$	0.0530529	+	0.368991i	0.0530529	+	0.368991i
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$809$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$810$		0			0
$811$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$811$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$812$		0			0
$813$	−0.708089	−	1.10181i	−0.708089	−	1.10181i
$814$		0			0
$815$		0			0
$816$	0.0971309	−	0.112095i	0.0971309	−	0.112095i
$817$		0			0
$818$	0.134919	+	0.459493i	0.134919	+	0.459493i
$819$		0			0
$820$		0			0
$821$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$821$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$822$	−1.38365	+	0.198939i	−1.38365	+	0.198939i
$823$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$823$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1.91899i			1.91899i	0.281733	+	0.959493i	$0.409091\pi$
$827$			1.91899i			1.91899i	−0.281733	+	0.959493i	$0.590909\pi$
$828$	1.66538	+	0.760554i	1.66538	+	0.760554i
$829$	−0.830830			−0.830830			−0.415415	−	0.909632i	$-0.636364\pi$
$829$	−0.830830			−0.830830			−0.415415	+	0.909632i	$0.636364\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−0.258908	−	0.118239i	−0.258908	−	0.118239i
$834$	−0.313607	−	2.18119i	−0.313607	−	2.18119i
$835$		0			0
$836$		0			0
$837$	−1.74557	+	0.797176i	−1.74557	+	0.797176i
$838$		0			0
$839$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$839$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$840$		0			0
$841$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$842$	−2.93694	+	1.34125i	−2.93694	+	1.34125i
$843$		0			0
$844$	−0.500000	−	0.146813i	−0.500000	−	0.146813i
$845$		0			0
$846$	−0.915415	+	2.00448i	−0.915415	+	2.00448i
$847$		0			0
$848$	0.368991	−	0.574161i	0.368991	−	0.574161i
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$853$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$854$		0			0
$855$		0			0
$856$	0.482462	−	1.05645i	0.482462	−	1.05645i
$857$	1.89945	−	0.273100i	1.89945	−	0.273100i	0.909632	−	0.415415i	$-0.136364\pi$
$857$	1.89945	−	0.273100i	1.89945	−	0.273100i	0.989821	+	0.142315i	$0.0454545\pi$
$858$		0			0
$859$	0.239446	−	0.153882i	0.239446	−	0.153882i	−0.415415	−	0.909632i	$-0.636364\pi$
$859$	0.239446	−	0.153882i	0.239446	−	0.153882i	0.654861	+	0.755750i	$0.272727\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.627899	−	0.544078i	−0.627899	−	0.544078i	0.281733	−	0.959493i	$-0.409091\pi$
$863$	−0.627899	−	0.544078i	−0.627899	−	0.544078i	−0.909632	+	0.415415i	$0.863636\pi$
$864$	−0.500000	+	0.146813i	−0.500000	+	0.146813i
$865$		0			0
$866$		0			0
$867$	0.258908	−	0.881761i	0.258908	−	0.881761i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−0.877726	+	0.760554i	−0.877726	+	0.760554i
$873$		0			0
$874$	−0.459493	+	3.19584i	−0.459493	+	3.19584i
$875$		0			0
$876$		0			0
$877$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$877$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$878$	−2.80202	−	0.402869i	−2.80202	−	0.402869i
$879$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$880$		0			0
$881$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$881$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$882$	−0.909632	−	1.41542i	−0.909632	−	1.41542i
$883$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$883$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$884$		0			0
$885$		0			0
$886$	0.313607	−	0.361922i	0.313607	−	0.361922i
$887$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	0.909632	−	0.415415i	$-0.136364\pi$
$887$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	−0.989821	+	0.142315i	$0.954545\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.48775	−	0.357685i	−2.48775	−	0.357685i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	−0.0530529	+	0.368991i	−0.0530529	+	0.368991i
$902$		0			0
$903$		0			0
$904$	2.25667	+	0.662618i	2.25667	+	0.662618i
$905$		0			0
$906$	−0.198939	−	0.435615i	−0.198939	−	0.435615i
$907$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$907$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$908$	1.81219	+	1.57028i	1.81219	+	1.57028i
$909$		0			0
$910$		0			0
$911$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$911$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$912$	0.540641	+	0.841254i	0.540641	+	0.841254i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.438384	+	3.04903i	−0.438384	+	3.04903i
$917$		0			0
$918$	−0.361922	+	0.313607i	−0.361922	+	0.313607i
$919$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$919$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$929$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$930$		0			0
$931$	1.25667	−	1.45027i	1.25667	−	1.45027i
$932$	1.14958	+	0.996114i	1.14958	+	0.996114i
$933$		0			0
$934$	0.198939	+	0.435615i	0.198939	+	0.435615i
$935$		0			0
$936$		0			0
$937$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$937$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$941$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−1.29639	−	0.186393i	−1.29639	−	0.186393i	−0.540641	−	0.841254i	$-0.681818\pi$
$947$	−1.29639	−	0.186393i	−1.29639	−	0.186393i	−0.755750	+	0.654861i	$0.772727\pi$
$948$	1.38365	+	0.631891i	1.38365	+	0.631891i
$949$		0			0
$950$		0			0
$951$	−0.239446	+	0.153882i	−0.239446	+	0.153882i
$952$		0			0
$953$	0.368991	+	1.25667i	0.368991	+	1.25667i	0.909632	+	0.415415i	$0.136364\pi$
$953$	0.368991	+	1.25667i	0.368991	+	1.25667i	−0.540641	+	0.841254i	$0.681818\pi$
$954$	−1.44306	+	1.66538i	−1.44306	+	1.66538i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.11435	−	2.44009i	1.11435	−	2.44009i
$962$		0			0
$963$	−0.449181	+	0.698939i	−0.449181	+	0.698939i
$964$	−0.341254	−	0.393828i	−0.341254	−	0.393828i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	1.05645	−	0.915415i	1.05645	−	0.915415i
$969$	−0.459493	−	0.295298i	−0.459493	−	0.295298i
$970$		0			0
$971$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$971$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$972$	−1.81219	+	0.260554i	−1.81219	+	0.260554i
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.654861	−	0.192284i	0.654861	−	0.192284i
$977$	0.627899	+	0.544078i	0.627899	+	0.544078i	0.909632	−	0.415415i	$-0.136364\pi$
$977$	0.627899	+	0.544078i	0.627899	+	0.544078i	−0.281733	+	0.959493i	$0.590909\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.698939	−	0.449181i	0.698939	−	0.449181i
$982$		0			0
$983$	0.822373	−	0.118239i	0.822373	−	0.118239i	0.281733	−	0.959493i	$-0.409091\pi$
$983$	0.822373	−	0.118239i	0.822373	−	0.118239i	0.540641	+	0.841254i	$0.318182\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$991$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$992$	0.540641	−	0.841254i	0.540641	−	0.841254i
$993$	−1.66538	−	0.239446i	−1.66538	−	0.239446i
$994$		0			0
$995$		0			0
$996$	−2.95561	−	0.867845i	−2.95561	−	0.867845i
$997$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$997$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$998$	−1.27155	+	0.580699i	−1.27155	+	0.580699i
$999$		0			0

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	1725.1.bc.a.1301.1		20
3.2	odd	2	inner	1725.1.bc.a.1301.2		20
5.2	odd	4		345.1.p.a.59.1	✓	10
5.3	odd	4		345.1.p.b.59.1	yes	10
5.4	even	2	inner	1725.1.bc.a.1301.2		20
15.2	even	4		345.1.p.b.59.1	yes	10
15.8	even	4		345.1.p.a.59.1	✓	10
15.14	odd	2	CM	1725.1.bc.a.1301.1		20
23.16	even	11	inner	1725.1.bc.a.476.2		20
69.62	odd	22	inner	1725.1.bc.a.476.1		20
115.39	even	22	inner	1725.1.bc.a.476.1		20
115.62	odd	44		345.1.p.a.269.1	yes	10
115.108	odd	44		345.1.p.b.269.1	yes	10
345.62	even	44		345.1.p.b.269.1	yes	10
345.269	odd	22	inner	1725.1.bc.a.476.2		20
345.338	even	44		345.1.p.a.269.1	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.1.p.a.59.1	✓	10	5.2	odd	4
345.1.p.a.59.1	✓	10	15.8	even	4
345.1.p.a.269.1	yes	10	115.62	odd	44
345.1.p.a.269.1	yes	10	345.338	even	44
345.1.p.b.59.1	yes	10	5.3	odd	4
345.1.p.b.59.1	yes	10	15.2	even	4
345.1.p.b.269.1	yes	10	115.108	odd	44
345.1.p.b.269.1	yes	10	345.62	even	44
1725.1.bc.a.476.1		20	69.62	odd	22	inner
1725.1.bc.a.476.1		20	115.39	even	22	inner
1725.1.bc.a.476.2		20	23.16	even	11	inner
1725.1.bc.a.476.2		20	345.269	odd	22	inner
1725.1.bc.a.1301.1		20	1.1	even	1	trivial
1725.1.bc.a.1301.1		20	15.14	odd	2	CM
1725.1.bc.a.1301.2		20	3.2	odd	2	inner
1725.1.bc.a.1301.2		20	5.4	even	2	inner

Overview	Random
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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Newspace parameters

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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	1725.1.bc.a.1301.1

Level	$1725$
Weight	$1$
Character	1725.1301
Analytic conductor	$0.861$
Analytic rank	$0$
Dimension	$20$
Projective image	$D_{11}$
CM discriminant	-15
Inner twists	$8$