LMFDB - Embedded newform 253.1.k.a.87.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [253,1,Mod(32,253)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(253, base_ring=CyclotomicField(22))

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

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

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

chi := DirichletCharacter("253.32");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 253 = 11 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	253.k (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:	$0.126263448196$
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:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{11}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{11} - \cdots)$

Embedding invariants

Embedding label			87.1
Root			$-0.415415 + 0.909632i$ of defining polynomial
Character	$\chi$	$=$	253.87
Dual form			253.1.k.a.32.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.544078 + 0.627899i) q^{3} +(0.841254 + 0.540641i) q^{4} +(0.273100 - 1.89945i) q^{5} +(0.0440780 + 0.306569i) q^{9} +(-0.959493 + 0.281733i) q^{11} +(-0.797176 + 0.234072i) q^{12} +(1.04408 + 1.20493i) q^{15} +(0.415415 + 0.909632i) q^{16} +(1.25667 - 1.45027i) q^{20} +(-0.959493 + 0.281733i) q^{23} +(-2.57385 - 0.755750i) q^{25} +(-0.915415 - 0.588302i) q^{27} +(0.186393 + 0.215109i) q^{31} +(0.345139 - 0.755750i) q^{33} +(-0.128663 + 0.281733i) q^{36} +(-0.239446 - 1.66538i) q^{37} +(-0.959493 - 0.281733i) q^{44} +0.594351 q^{45} -0.284630 q^{47} +(-0.797176 - 0.234072i) q^{48} +(-0.654861 + 0.755750i) q^{49} +(0.698939 + 1.53046i) q^{53} +(0.273100 + 1.89945i) q^{55} +(0.698939 - 1.53046i) q^{59} +(0.226900 + 1.57812i) q^{60} +(-0.142315 + 0.989821i) q^{64} +(-0.797176 - 0.234072i) q^{67} +(0.345139 - 0.755750i) q^{69} +(1.84125 + 0.540641i) q^{71} +(1.87491 - 1.20493i) q^{75} +(1.84125 - 0.540641i) q^{80} +(0.570276 - 0.167448i) q^{81} +(-0.544078 + 0.627899i) q^{89} +(-0.959493 - 0.281733i) q^{92} -0.236479 q^{93} +(0.0405070 - 0.281733i) q^{97} +(-0.128663 - 0.281733i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{9} - q^{11} - 2 q^{12} + 7 q^{15} - q^{16} - 2 q^{20} - q^{23} - 3 q^{25} - 4 q^{27} - 2 q^{31} + 9 q^{33} - 3 q^{36} - 2 q^{37} - q^{44} - 6 q^{45} - 2 q^{47}+ \cdots - 3 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - q^4 - 2 * q^5 - 3 * q^9 - q^11 - 2 * q^12 + 7 * q^15 - q^16 - 2 * q^20 - q^23 - 3 * q^25 - 4 * q^27 - 2 * q^31 + 9 * q^33 - 3 * q^36 - 2 * q^37 - q^44 - 6 * q^45 - 2 * q^47 - 2 * q^48 - q^49 - 2 * q^53 - 2 * q^55 - 2 * q^59 + 7 * q^60 - q^64 - 2 * q^67 + 9 * q^69 + 9 * q^71 + 5 * q^75 + 9 * q^80 - 5 * q^81 - 2 * q^89 - q^92 - 4 * q^93 + 9 * q^97 - 3 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/253\mathbb{Z}\right)^\times$.

$n$	$24$	$166$
$\chi(n)$	$-1$	$e\left(\frac{6}{11}\right)$

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$2$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$3$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	−0.959493	−	0.281733i	$-0.909091\pi$
$3$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$4$	0.841254	+	0.540641i	0.841254	+	0.540641i
$5$	0.273100	−	1.89945i	0.273100	−	1.89945i	−0.142315	−	0.989821i	$-0.545455\pi$
$5$	0.273100	−	1.89945i	0.273100	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$6$		0			0
$7$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$7$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$8$		0			0
$9$	0.0440780	+	0.306569i	0.0440780	+	0.306569i
$10$		0			0
$11$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$11$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$12$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$13$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$13$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$14$		0			0
$15$	1.04408	+	1.20493i	1.04408	+	1.20493i
$16$	0.415415	+	0.909632i	0.415415	+	0.909632i
$17$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$17$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$18$		0			0
$19$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$19$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$20$	1.25667	−	1.45027i	1.25667	−	1.45027i
$21$		0			0
$22$		0			0
$23$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$23$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$24$		0			0
$25$	−2.57385	−	0.755750i	−2.57385	−	0.755750i
$26$		0			0
$27$	−0.915415	−	0.588302i	−0.915415	−	0.588302i
$28$		0			0
$29$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$29$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$30$		0			0
$31$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$31$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$32$		0			0
$33$	0.345139	−	0.755750i	0.345139	−	0.755750i
$34$		0			0
$35$		0			0
$36$	−0.128663	+	0.281733i	−0.128663	+	0.281733i
$37$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	−0.654861	−	0.755750i	$-0.727273\pi$
$37$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	0.415415	−	0.909632i	$-0.363636\pi$
$38$		0			0
$39$		0			0
$40$		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
$43$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$43$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$44$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$45$	0.594351			0.594351
$46$		0			0
$47$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$47$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$48$	−0.797176	−	0.234072i	−0.797176	−	0.234072i
$49$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$53$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$54$		0			0
$55$	0.273100	+	1.89945i	0.273100	+	1.89945i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.698939	−	1.53046i	0.698939	−	1.53046i	−0.142315	−	0.989821i	$-0.545455\pi$
$59$	0.698939	−	1.53046i	0.698939	−	1.53046i	0.841254	−	0.540641i	$-0.181818\pi$
$60$	0.226900	+	1.57812i	0.226900	+	1.57812i
$61$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$61$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$62$		0			0
$63$		0			0
$64$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$65$		0			0
$66$		0			0
$67$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$67$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$68$		0			0
$69$	0.345139	−	0.755750i	0.345139	−	0.755750i
$70$		0			0
$71$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$71$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$72$		0			0
$73$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$73$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$74$		0			0
$75$	1.87491	−	1.20493i	1.87491	−	1.20493i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$79$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$80$	1.84125	−	0.540641i	1.84125	−	0.540641i
$81$	0.570276	−	0.167448i	0.570276	−	0.167448i
$82$		0			0
$83$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$83$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	−0.959493	−	0.281733i	$-0.909091\pi$
$89$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$90$		0			0
$91$		0			0
$92$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$93$	−0.236479			−0.236479
$94$		0			0
$95$		0			0
$96$		0			0
$97$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$97$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$98$		0			0
$99$	−0.128663	−	0.281733i	−0.128663	−	0.281733i
$100$	−1.75667	−	2.02730i	−1.75667	−	2.02730i
$101$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$101$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$102$		0			0
$103$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$103$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$107$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$108$	−0.452036	−	0.989821i	−0.452036	−	0.989821i
$109$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$109$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$110$		0			0
$111$	1.17597	+	0.755750i	1.17597	+	0.755750i
$112$		0			0
$113$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$113$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$114$		0			0
$115$	0.273100	+	1.89945i	0.273100	+	1.89945i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.841254	−	0.540641i	0.841254	−	0.540641i
$122$		0			0
$123$		0			0
$124$	0.0405070	+	0.281733i	0.0405070	+	0.281733i
$125$	−1.34125	+	2.93694i	−1.34125	+	2.93694i
$126$		0			0
$127$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$127$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$131$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$132$	0.698939	−	0.449181i	0.698939	−	0.449181i
$133$		0			0
$134$		0			0
$135$	−1.36745	+	1.57812i	−1.36745	+	1.57812i
$136$		0			0
$137$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$137$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	0.154861	−	0.178719i	0.154861	−	0.178719i
$142$		0			0
$143$		0			0
$144$	−0.260554	+	0.167448i	−0.260554	+	0.167448i
$145$		0			0
$146$		0			0
$147$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$148$	0.698939	−	1.53046i	0.698939	−	1.53046i
$149$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$149$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$150$		0			0
$151$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$151$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.459493	−	0.295298i	0.459493	−	0.295298i
$156$		0			0
$157$	1.68251	+	1.08128i	1.68251	+	1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$157$	1.68251	+	1.08128i	1.68251	+	1.08128i	0.841254	+	0.540641i	$0.181818\pi$
$158$		0			0
$159$	−1.34125	−	0.393828i	−1.34125	−	0.393828i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$163$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$164$		0			0
$165$	−1.34125	−	0.861971i	−1.34125	−	0.861971i
$166$		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
$169$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$173$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$174$		0			0
$175$		0			0
$176$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$177$	0.580699	+	1.27155i	0.580699	+	1.27155i
$178$		0			0
$179$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$179$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$180$	0.500000	+	0.321330i	0.500000	+	0.321330i
$181$	0.857685	−	0.989821i	0.857685	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$181$	0.857685	−	0.989821i	0.857685	−	0.989821i	1.00000			$0$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.22871			−3.22871
$186$		0			0
$187$		0			0
$188$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$189$		0			0
$190$		0			0
$191$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$191$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$192$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$193$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$193$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$194$		0			0
$195$		0			0
$196$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$197$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$197$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$198$		0			0
$199$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$199$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$200$		0			0
$201$	0.580699	−	0.373193i	0.580699	−	0.373193i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.128663	−	0.281733i	−0.128663	−	0.281733i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$211$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$212$	−0.239446	+	1.66538i	−0.239446	+	1.66538i
$213$	−1.34125	+	0.861971i	−1.34125	+	0.861971i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$221$		0			0
$222$		0			0
$223$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$223$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$224$		0			0
$225$	0.118239	−	0.822373i	0.118239	−	0.822373i
$226$		0			0
$227$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$227$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$228$		0			0
$229$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$229$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$233$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$234$		0			0
$235$	−0.0777324	+	0.540641i	−0.0777324	+	0.540641i
$236$	1.41542	−	0.909632i	1.41542	−	0.909632i
$237$		0			0
$238$		0			0
$239$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$239$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$240$	−0.662317	+	1.45027i	−0.662317	+	1.45027i
$241$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$241$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$242$		0			0
$243$	0.246902	−	0.540641i	0.246902	−	0.540641i
$244$		0			0
$245$	1.25667	+	1.45027i	1.25667	+	1.45027i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$251$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$252$		0			0
$253$	0.841254	−	0.540641i	0.841254	−	0.540641i
$254$		0			0
$255$		0			0
$256$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$257$	1.41542	+	0.909632i	1.41542	+	0.909632i	1.00000			$0$
$257$	1.41542	+	0.909632i	1.41542	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$263$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$264$		0			0
$265$	3.09792	−	0.909632i	3.09792	−	0.909632i
$266$		0			0
$267$	−0.0982369	−	0.683252i	−0.0982369	−	0.683252i
$268$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$269$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$269$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$270$		0			0
$271$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$271$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.68251			2.68251
$276$	0.698939	−	0.449181i	0.698939	−	0.449181i
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		0			0
$279$	−0.0577299	+	0.0666238i	−0.0577299	+	0.0666238i
$280$		0			0
$281$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$281$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$282$		0			0
$283$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$283$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$284$	1.25667	+	1.45027i	1.25667	+	1.45027i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.415415	−	0.909632i	0.415415	−	0.909632i
$290$		0			0
$291$	0.154861	+	0.178719i	0.154861	+	0.178719i
$292$		0			0
$293$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$293$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$294$		0			0
$295$	−2.71616	−	1.74557i	−2.71616	−	1.74557i
$296$		0			0
$297$	1.04408	+	0.306569i	1.04408	+	0.306569i
$298$		0			0
$299$		0			0
$300$	2.22871			2.22871
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$307$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$308$		0			0
$309$	−0.452036	+	0.989821i	−0.452036	+	0.989821i
$310$		0			0
$311$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$311$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$312$		0			0
$313$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$313$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$317$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$318$		0			0
$319$		0			0
$320$	1.84125	+	0.540641i	1.84125	+	0.540641i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.570276	+	0.167448i	0.570276	+	0.167448i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.273100	+	1.89945i	0.273100	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$331$	0.273100	+	1.89945i	0.273100	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$332$		0			0
$333$	0.500000	−	0.146813i	0.500000	−	0.146813i
$334$		0			0
$335$	−0.662317	+	1.45027i	−0.662317	+	1.45027i
$336$		0			0
$337$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$337$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$338$		0			0
$339$	−0.198939	+	0.127850i	−0.198939	+	0.127850i
$340$		0			0
$341$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.34125	−	0.861971i	−1.34125	−	0.861971i
$346$		0			0
$347$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$347$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$348$		0			0
$349$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$349$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$353$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$354$		0			0
$355$	1.52977	−	3.34973i	1.52977	−	3.34973i
$356$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$357$		0			0
$358$		0			0
$359$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$359$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$360$		0			0
$361$	0.415415	+	0.909632i	0.415415	+	0.909632i
$362$		0			0
$363$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$367$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$368$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.198939	−	0.127850i	−0.198939	−	0.127850i
$373$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$373$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$374$		0			0
$375$	−1.11435	−	2.44009i	−1.11435	−	2.44009i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$379$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$383$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.186393	−	0.215109i	0.186393	−	0.215109i
$389$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$389$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	0.0440780	−	0.306569i	0.0440780	−	0.306569i
$397$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$397$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$398$		0			0
$399$		0			0
$400$	−0.381761	−	2.65520i	−0.381761	−	2.65520i
$401$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$401$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−0.162317	−	1.12894i	−0.162317	−	1.12894i
$406$		0			0
$407$	0.698939	+	1.53046i	0.698939	+	1.53046i
$408$		0			0
$409$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$409$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$410$		0			0
$411$	0.712591	−	0.822373i	0.712591	−	0.822373i
$412$	1.25667	+	0.368991i	1.25667	+	0.368991i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$419$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$420$		0			0
$421$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$421$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$422$		0			0
$423$	−0.0125459	−	0.0872586i	−0.0125459	−	0.0872586i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$431$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$432$	0.154861	−	1.07708i	0.154861	−	1.07708i
$433$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$433$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$439$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$440$		0			0
$441$	−0.260554	−	0.167448i	−0.260554	−	0.167448i
$442$		0			0
$443$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.654861	+	0.755750i	$0.727273\pi$
$443$	−1.61435	+	1.03748i	−1.61435	+	1.03748i	−0.959493	+	0.281733i	$0.909091\pi$
$444$	0.580699	+	1.27155i	0.580699	+	1.27155i
$445$	1.04408	+	1.20493i	1.04408	+	1.20493i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$449$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$450$		0			0
$451$		0			0
$452$	0.186393	+	0.215109i	0.186393	+	0.215109i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$457$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$458$		0			0
$459$		0			0
$460$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	0.186393	−	0.215109i	0.186393	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$463$	0.186393	−	0.215109i	0.186393	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$464$		0			0
$465$	−0.0645824	+	0.449181i	−0.0645824	+	0.449181i
$466$		0			0
$467$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$467$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.59435	+	0.468144i	−1.59435	+	0.468144i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.438384	+	0.281733i	−0.438384	+	0.281733i
$478$		0			0
$479$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$479$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	1.00000			1.00000
$485$	−0.524075	−	0.153882i	−0.524075	−	0.153882i
$486$		0			0
$487$	0.698939	+	0.449181i	0.698939	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$487$	0.698939	+	0.449181i	0.698939	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$488$		0			0
$489$	−0.915415	+	0.588302i	−0.915415	+	0.588302i
$490$		0			0
$491$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$491$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	−0.570276	+	0.167448i	−0.570276	+	0.167448i
$496$	−0.118239	+	0.258908i	−0.118239	+	0.258908i
$497$		0			0
$498$		0			0
$499$	0.345139	+	0.755750i	0.345139	+	0.755750i	1.00000			$0$
$499$	0.345139	+	0.755750i	0.345139	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$500$	−2.71616	+	1.74557i	−2.71616	+	1.74557i
$501$		0			0
$502$		0			0
$503$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$503$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	0.830830			0.830830
$508$		0			0
$509$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	−0.959493	−	0.281733i	$-0.909091\pi$
$509$	−0.544078	+	0.627899i	−0.544078	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.357685	−	2.48775i	−0.357685	−	2.48775i
$516$		0			0
$517$	0.273100	−	0.0801894i	0.273100	−	0.0801894i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$521$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$522$		0			0
$523$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$523$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.830830			0.830830
$529$	0.841254	−	0.540641i	0.841254	−	0.540641i
$530$		0			0
$531$	0.500000	+	0.146813i	0.500000	+	0.146813i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.712591	+	0.822373i	0.712591	+	0.822373i
$538$		0			0
$539$	0.415415	−	0.909632i	0.415415	−	0.909632i
$540$	−2.00357	+	0.588302i	−2.00357	+	0.588302i
$541$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$541$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$542$		0			0
$543$	0.154861	+	1.07708i	0.154861	+	1.07708i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$547$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$548$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.75667	−	2.02730i	1.75667	−	2.02730i
$556$		0			0
$557$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$557$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$563$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$564$	0.226900	−	0.0666238i	0.226900	−	0.0666238i
$565$	0.226900	−	0.496841i	0.226900	−	0.496841i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$569$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$570$		0			0
$571$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$571$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$572$		0			0
$573$	−0.662317	−	0.194474i	−0.662317	−	0.194474i
$574$		0			0
$575$	2.68251			2.68251
$576$	−0.309721			−0.309721
$577$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$577$	−1.61435	−	0.474017i	−1.61435	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$587$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$588$	0.345139	−	0.755750i	0.345139	−	0.755750i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.41542	−	0.909632i	1.41542	−	0.909632i
$593$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$593$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.690279			0.690279
$598$		0			0
$599$	0.830830			0.830830			0.415415	−	0.909632i	$-0.363636\pi$
$599$	0.830830			0.830830			0.415415	+	0.909632i	$0.363636\pi$
$600$		0			0
$601$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$601$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$602$		0			0
$603$	0.0366213	−	0.254707i	0.0366213	−	0.254707i
$604$		0			0
$605$	−0.797176	−	1.74557i	−0.797176	−	1.74557i
$606$		0			0
$607$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$607$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$613$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.698939	+	0.449181i	0.698939	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$617$	0.698939	+	0.449181i	0.698939	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$618$		0			0
$619$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$619$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$620$	0.546200			0.546200
$621$	1.04408	+	0.306569i	1.04408	+	0.306569i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	2.95561	+	1.89945i	2.95561	+	1.89945i
$626$		0			0
$627$		0			0
$628$	0.830830	+	1.81926i	0.830830	+	1.81926i
$629$		0			0
$630$		0			0
$631$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$631$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	−0.915415	−	1.05645i	−0.915415	−	1.05645i
$637$		0			0
$638$		0			0
$639$	−0.0845850	+	0.588302i	−0.0845850	+	0.588302i
$640$		0			0
$641$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$641$	−1.10181	+	1.27155i	−1.10181	+	1.27155i	−0.959493	+	0.281733i	$0.909091\pi$
$642$		0			0
$643$	−0.284630			−0.284630			−0.142315	−	0.989821i	$-0.545455\pi$
$643$	−0.284630			−0.284630			−0.142315	+	0.989821i	$0.545455\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.857685	−	0.989821i	0.857685	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$647$	0.857685	−	0.989821i	0.857685	−	0.989821i	1.00000			$0$
$648$		0			0
$649$	−0.239446	+	1.66538i	−0.239446	+	1.66538i
$650$		0			0
$651$		0			0
$652$	0.857685	+	0.989821i	0.857685	+	0.989821i
$653$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$653$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$659$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$660$	−0.662317	−	1.45027i	−0.662317	−	1.45027i
$661$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$661$	−1.10181	+	0.708089i	−1.10181	+	0.708089i	−0.142315	+	0.989821i	$0.545455\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.04408	+	0.306569i	1.04408	+	0.306569i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$673$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$674$		0			0
$675$	1.91153	+	2.20602i	1.91153	+	2.20602i
$676$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$677$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$677$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$683$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$684$		0			0
$685$	−0.357685	+	2.48775i	−0.357685	+	2.48775i
$686$		0			0
$687$	0.154861	−	0.178719i	0.154861	−	0.178719i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$691$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$692$		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		0.959493	−	0.281733i	$-0.0909091\pi$
$701$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$702$		0			0
$703$		0			0
$704$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$705$	−0.297176	−	0.342959i	−0.297176	−	0.342959i
$706$		0			0
$707$		0			0
$708$	−0.198939	+	1.38365i	−0.198939	+	1.38365i
$709$	0.698939	+	0.449181i	0.698939	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$709$	0.698939	+	0.449181i	0.698939	+	0.449181i	−0.142315	+	0.989821i	$0.545455\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$714$		0			0
$715$		0			0
$716$	0.857685	−	0.989821i	0.857685	−	0.989821i
$717$		0			0
$718$		0			0
$719$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$719$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$720$	0.246902	+	0.540641i	0.246902	+	0.540641i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	1.25667	−	0.368991i	1.25667	−	0.368991i
$725$		0			0
$726$		0			0
$727$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	−0.654861	−	0.755750i	$-0.727273\pi$
$727$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	0.415415	−	0.909632i	$-0.363636\pi$
$728$		0			0
$729$	0.452036	+	0.989821i	0.452036	+	0.989821i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$733$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$734$		0			0
$735$	−1.59435			−1.59435
$736$		0			0
$737$	0.830830			0.830830
$738$		0			0
$739$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$739$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$740$	−2.71616	−	1.74557i	−2.71616	−	1.74557i
$741$		0			0
$742$		0			0
$743$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$743$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.30972	−	1.51150i	−1.30972	−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$751$	−1.30972	−	1.51150i	−1.30972	−	1.51150i	−0.654861	−	0.755750i	$-0.727273\pi$
$752$	−0.118239	−	0.258908i	−0.118239	−	0.258908i
$753$	1.17597	−	0.755750i	1.17597	−	0.755750i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.841254	−	0.540641i	$-0.181818\pi$
$757$	1.25667	+	0.368991i	1.25667	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$758$		0			0
$759$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$760$		0			0
$761$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$761$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$762$		0			0
$763$		0			0
$764$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$769$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$769$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$770$		0			0
$771$	−1.34125	+	0.393828i	−1.34125	+	0.393828i
$772$		0			0
$773$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$773$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$774$		0			0
$775$	−0.317178	−	0.694523i	−0.317178	−	0.694523i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−1.91899			−1.91899
$782$		0			0
$783$		0			0
$784$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$785$	2.51334	−	2.90055i	2.51334	−	2.90055i
$786$		0			0
$787$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$787$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−1.11435	+	2.44009i	−1.11435	+	2.44009i
$796$	−0.118239	−	0.822373i	−0.118239	−	0.822373i
$797$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$797$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.216476	−	0.139121i	−0.216476	−	0.139121i
$802$		0			0
$803$		0			0
$804$	0.690279			0.690279
$805$		0			0
$806$		0			0
$807$	1.52977	+	0.449181i	1.52977	+	0.449181i
$808$		0			0
$809$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$809$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$810$		0			0
$811$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$811$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.04408	−	2.28621i	1.04408	−	2.28621i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$821$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$822$		0			0
$823$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$823$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$824$		0			0
$825$	−1.45949	+	1.68434i	−1.45949	+	1.68434i
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$	0.0440780	−	0.306569i	0.0440780	−	0.306569i
$829$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$829$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−0.0440780	−	0.306569i	−0.0440780	−	0.306569i
$838$		0			0
$839$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	−0.142315	−	0.989821i	$-0.545455\pi$
$839$	0.273100	−	0.0801894i	0.273100	−	0.0801894i	0.415415	+	0.909632i	$0.363636\pi$
$840$		0			0
$841$	0.415415	−	0.909632i	0.415415	−	0.909632i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1.61435	+	1.03748i	−1.61435	+	1.03748i
$846$		0			0
$847$		0			0
$848$	−1.10181	+	1.27155i	−1.10181	+	1.27155i
$849$		0			0
$850$		0			0
$851$	0.698939	+	1.53046i	0.698939	+	1.53046i
$852$	−1.59435			−1.59435
$853$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$853$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$857$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$858$		0			0
$859$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$859$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$863$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.345139	+	0.755750i	0.345139	+	0.755750i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.0881559			0.0881559
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$877$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$878$		0			0
$879$		0			0
$880$	−1.61435	+	1.03748i	−1.61435	+	1.03748i
$881$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$881$	−0.544078	−	1.19136i	−0.544078	−	1.19136i	0.415415	−	0.909632i	$-0.363636\pi$
$882$		0			0
$883$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$883$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$884$		0			0
$885$	2.57385	−	0.755750i	2.57385	−	0.755750i
$886$		0			0
$887$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$887$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.500000	+	0.321330i	−0.500000	+	0.321330i
$892$	0.186393	−	1.29639i	0.186393	−	1.29639i
$893$		0			0
$894$		0			0
$895$	−2.41153	−	0.708089i	−2.41153	−	0.708089i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.544078	−	0.627899i	0.544078	−	0.627899i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.64589	−	1.89945i	−1.64589	−	1.89945i
$906$		0			0
$907$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$907$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$911$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$917$		0			0
$918$		0			0
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.642315	+	4.46740i	−0.642315	+	4.46740i
$926$		0			0
$927$	0.168513	+	0.368991i	0.168513	+	0.368991i
$928$		0			0
$929$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$929$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	0.580699	−	1.27155i	0.580699	−	1.27155i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$937$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$938$		0			0
$939$	−1.34125	−	0.861971i	−1.34125	−	0.861971i
$940$	−0.357685	+	0.412791i	−0.357685	+	0.412791i
$941$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$941$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$942$		0			0
$943$		0			0
$944$	1.68251			1.68251
$945$		0			0
$946$		0			0
$947$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.142315	−	0.989821i	$-0.545455\pi$
$947$	−1.10181	−	0.708089i	−1.10181	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	0.154861	+	0.178719i	0.154861	+	0.178719i
$952$		0			0
$953$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$953$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$954$		0			0
$955$	1.52977	−	0.449181i	1.52977	−	0.449181i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−1.34125	+	0.861971i	−1.34125	+	0.861971i
$961$	0.130785	−	0.909632i	0.130785	−	0.909632i
$962$		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
$971$	0.186393	−	1.29639i	0.186393	−	1.29639i	−0.654861	−	0.755750i	$-0.727273\pi$
$971$	0.186393	−	1.29639i	0.186393	−	1.29639i	0.841254	−	0.540641i	$-0.181818\pi$
$972$	0.500000	−	0.321330i	0.500000	−	0.321330i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.415415	−	0.909632i	$-0.363636\pi$
$977$	1.25667	−	0.368991i	1.25667	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$978$		0			0
$979$	0.345139	−	0.755750i	0.345139	−	0.755750i
$980$	0.273100	+	1.89945i	0.273100	+	1.89945i
$981$		0			0
$982$		0			0
$983$	1.41542	−	0.909632i	1.41542	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$983$	1.41542	−	0.909632i	1.41542	−	0.909632i	1.00000			$0$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$991$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$992$		0			0
$993$	−1.34125	−	0.861971i	−1.34125	−	0.861971i
$994$		0			0
$995$	−1.34125	+	0.861971i	−1.34125	+	0.861971i
$996$		0			0
$997$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$997$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$998$		0			0
$999$	−0.760554	+	1.66538i	−0.760554	+	1.66538i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	253.1.k.a.87.1	yes	10
3.2	odd	2		2277.1.be.a.1099.1		10
11.2	odd	10		2783.1.dh.a.524.1		40
11.3	even	5		2783.1.dh.a.1927.1		40
11.4	even	5		2783.1.dh.a.2272.1		40
11.5	even	5		2783.1.dh.a.1812.1		40
11.6	odd	10		2783.1.dh.a.1812.1		40
11.7	odd	10		2783.1.dh.a.2272.1		40
11.8	odd	10		2783.1.dh.a.1927.1		40
11.9	even	5		2783.1.dh.a.524.1		40
11.10	odd	2	CM	253.1.k.a.87.1	yes	10
23.9	even	11	inner	253.1.k.a.32.1	✓	10
33.32	even	2		2277.1.be.a.1099.1		10
69.32	odd	22		2277.1.be.a.1297.1		10
253.9	even	55		2783.1.dh.a.1734.1		40
253.32	odd	22	inner	253.1.k.a.32.1	✓	10
253.101	odd	110		2783.1.dh.a.1734.1		40
253.124	even	55		2783.1.dh.a.354.1		40
253.147	even	55		2783.1.dh.a.699.1		40
253.170	even	55		2783.1.dh.a.239.1		40
253.193	odd	110		2783.1.dh.a.239.1		40
253.216	odd	110		2783.1.dh.a.699.1		40
253.239	odd	110		2783.1.dh.a.354.1		40
759.32	even	22		2277.1.be.a.1297.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
253.1.k.a.32.1	✓	10	23.9	even	11	inner
253.1.k.a.32.1	✓	10	253.32	odd	22	inner
253.1.k.a.87.1	yes	10	1.1	even	1	trivial
253.1.k.a.87.1	yes	10	11.10	odd	2	CM
2277.1.be.a.1099.1		10	3.2	odd	2
2277.1.be.a.1099.1		10	33.32	even	2
2277.1.be.a.1297.1		10	69.32	odd	22
2277.1.be.a.1297.1		10	759.32	even	22
2783.1.dh.a.239.1		40	253.170	even	55
2783.1.dh.a.239.1		40	253.193	odd	110
2783.1.dh.a.354.1		40	253.124	even	55
2783.1.dh.a.354.1		40	253.239	odd	110
2783.1.dh.a.524.1		40	11.2	odd	10
2783.1.dh.a.524.1		40	11.9	even	5
2783.1.dh.a.699.1		40	253.147	even	55
2783.1.dh.a.699.1		40	253.216	odd	110
2783.1.dh.a.1734.1		40	253.9	even	55
2783.1.dh.a.1734.1		40	253.101	odd	110
2783.1.dh.a.1812.1		40	11.5	even	5
2783.1.dh.a.1812.1		40	11.6	odd	10
2783.1.dh.a.1927.1		40	11.3	even	5
2783.1.dh.a.1927.1		40	11.8	odd	10
2783.1.dh.a.2272.1		40	11.4	even	5
2783.1.dh.a.2272.1		40	11.7	odd	10

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 253 = 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	253.k (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.544078 + 0.627899i) q^{3} +(0.841254 + 0.540641i) q^{4} +(0.273100 - 1.89945i) q^{5} +(0.0440780 + 0.306569i) q^{9} +(-0.959493 + 0.281733i) q^{11} +(-0.797176 + 0.234072i) q^{12} +(1.04408 + 1.20493i) q^{15} +(0.415415 + 0.909632i) q^{16} +(1.25667 - 1.45027i) q^{20} +(-0.959493 + 0.281733i) q^{23} +(-2.57385 - 0.755750i) q^{25} +(-0.915415 - 0.588302i) q^{27} +(0.186393 + 0.215109i) q^{31} +(0.345139 - 0.755750i) q^{33} +(-0.128663 + 0.281733i) q^{36} +(-0.239446 - 1.66538i) q^{37} +(-0.959493 - 0.281733i) q^{44} +0.594351 q^{45} -0.284630 q^{47} +(-0.797176 - 0.234072i) q^{48} +(-0.654861 + 0.755750i) q^{49} +(0.698939 + 1.53046i) q^{53} +(0.273100 + 1.89945i) q^{55} +(0.698939 - 1.53046i) q^{59} +(0.226900 + 1.57812i) q^{60} +(-0.142315 + 0.989821i) q^{64} +(-0.797176 - 0.234072i) q^{67} +(0.345139 - 0.755750i) q^{69} +(1.84125 + 0.540641i) q^{71} +(1.87491 - 1.20493i) q^{75} +(1.84125 - 0.540641i) q^{80} +(0.570276 - 0.167448i) q^{81} +(-0.544078 + 0.627899i) q^{89} +(-0.959493 - 0.281733i) q^{92} -0.236479 q^{93} +(0.0405070 - 0.281733i) q^{97} +(-0.128663 - 0.281733i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - q^{4} - 2 q^{5} - 3 q^{9} - q^{11} - 2 q^{12} + 7 q^{15} - q^{16} - 2 q^{20} - q^{23} - 3 q^{25} - 4 q^{27} - 2 q^{31} + 9 q^{33} - 3 q^{36} - 2 q^{37} - q^{44} - 6 q^{45} - 2 q^{47}+ \cdots - 3 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - q^4 - 2 * q^5 - 3 * q^9 - q^11 - 2 * q^12 + 7 * q^15 - q^16 - 2 * q^20 - q^23 - 3 * q^25 - 4 * q^27 - 2 * q^31 + 9 * q^33 - 3 * q^36 - 2 * q^37 - q^44 - 6 * q^45 - 2 * q^47 - 2 * q^48 - q^49 - 2 * q^53 - 2 * q^55 - 2 * q^59 + 7 * q^60 - q^64 - 2 * q^67 + 9 * q^69 + 9 * q^71 + 5 * q^75 + 9 * q^80 - 5 * q^81 - 2 * q^89 - q^92 - 4 * q^93 + 9 * q^97 - 3 * q^99

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