LMFDB - Embedded newform 371.1.o.a.356.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [371,1,Mod(6,371)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(371, base_ring=CyclotomicField(26))

chi = DirichletCharacter(H, H._module([13, 9]))

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

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

chi := DirichletCharacter("371.6");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 371 = 7 \cdot 53 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	371.o (of order $26$, degree $12$, 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:	$0.185153119687$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\Q(\zeta_{26})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^12 - x^11 + 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:	$D_{26}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{26} - \cdots)$

Embedding invariants

Embedding label			356.1
Root			$0.354605 - 0.935016i$ of defining polynomial
Character	$\chi$	$=$	371.356
Dual form			371.1.o.a.272.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.09148 - 1.23202i) q^{2} +(-0.206022 - 1.69675i) q^{4} +(-0.748511 - 0.663123i) q^{7} +(-0.960699 - 0.663123i) q^{8} +(-0.885456 + 0.464723i) q^{9} +O(q^{10})$	\(q+(1.09148 - 1.23202i) q^{2} +(-0.206022 - 1.69675i) q^{4} +(-0.748511 - 0.663123i) q^{7} +(-0.960699 - 0.663123i) q^{8} +(-0.885456 + 0.464723i) q^{9} +(0.627974 + 1.65583i) q^{11} +(-1.63397 + 0.198399i) q^{14} +(-0.206022 + 0.0507800i) q^{16} +(-0.393906 + 1.59814i) q^{18} +(2.72545 + 1.03363i) q^{22} -1.87003i q^{23} +(-0.568065 + 0.822984i) q^{25} +(-0.970942 + 1.40665i) q^{28} +(-0.402877 + 1.06230i) q^{29} +(0.380181 - 0.724375i) q^{32} +(0.970942 + 1.40665i) q^{36} +(0.234068 - 0.0576926i) q^{37} +(-1.88546 - 0.464723i) q^{43} +(2.68015 - 1.40665i) q^{44} +(-2.30393 - 2.04110i) q^{46} +(0.120537 + 0.992709i) q^{49} +(0.393906 + 1.59814i) q^{50} +(0.885456 + 0.464723i) q^{53} +(0.279362 + 1.13342i) q^{56} +(0.869047 + 1.65583i) q^{58} +(0.970942 + 0.239316i) q^{63} +(-0.552731 - 1.45743i) q^{64} +(-0.922670 + 0.112032i) q^{67} +(0.317391 - 1.28771i) q^{71} +(1.15883 + 0.140707i) q^{72} +(0.184402 - 0.351348i) q^{74} +(0.627974 - 1.65583i) q^{77} +(-0.879463 - 0.992709i) q^{79} +(0.568065 - 0.822984i) q^{81} +(-2.63049 + 1.81569i) q^{86} +(0.494725 - 2.00718i) q^{88} +(-3.17297 + 0.385269i) q^{92} +(1.35460 + 0.935016i) q^{98} +(-1.32555 - 1.17433i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - q^{4} - q^{7} - 13 q^{8} + q^{9}+O(q^{10}) $ 12 * q - q^4 - q^7 - 13 * q^8 + q^9	$ 12 q - q^{4} - q^{7} - 13 q^{8} + q^{9} + 2 q^{11} - q^{16} + q^{25} - q^{28} - 2 q^{29} + q^{36} + 2 q^{37} - 11 q^{43} + 15 q^{44} - q^{49} - q^{53} - 13 q^{56} + q^{63} + 12 q^{64} - 13 q^{74} + 2 q^{77} - 13 q^{79} - q^{81} + 13 q^{88} + 13 q^{98} - 2 q^{99}+O(q^{100}) $ 12 * q - q^4 - q^7 - 13 * q^8 + q^9 + 2 * q^11 - q^16 + q^25 - q^28 - 2 * q^29 + q^36 + 2 * q^37 - 11 * q^43 + 15 * q^44 - q^49 - q^53 - 13 * q^56 + q^63 + 12 * q^64 - 13 * q^74 + 2 * q^77 - 13 * q^79 - q^81 + 13 * q^88 + 13 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/371\mathbb{Z}\right)^\times$.

$n$	$213$	$267$
$\chi(n)$	$-1$	$e\left(\frac{19}{26}\right)$

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 $

$2$	1.09148	−	1.23202i	1.09148	−	1.23202i	0.120537	−	0.992709i	$-0.461538\pi$
$2$	1.09148	−	1.23202i	1.09148	−	1.23202i	0.970942	−	0.239316i	$-0.0769231\pi$
$3$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$3$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$4$	−0.206022	−	1.69675i	−0.206022	−	1.69675i
$5$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$5$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$6$		0			0
$7$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$7$	−0.748511	−	0.663123i	−0.748511	−	0.663123i
$8$	−0.960699	−	0.663123i	−0.960699	−	0.663123i
$9$	−0.885456	+	0.464723i	−0.885456	+	0.464723i
$10$		0			0
$11$	0.627974	+	1.65583i	0.627974	+	1.65583i	0.748511	+	0.663123i	$0.230769\pi$
$11$	0.627974	+	1.65583i	0.627974	+	1.65583i	−0.120537	+	0.992709i	$0.538462\pi$
$12$		0			0
$13$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$13$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$14$	−1.63397	+	0.198399i	−1.63397	+	0.198399i
$15$		0			0
$16$	−0.206022	+	0.0507800i	−0.206022	+	0.0507800i
$17$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$17$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$18$	−0.393906	+	1.59814i	−0.393906	+	1.59814i
$19$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$19$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$20$		0			0
$21$		0			0
$22$	2.72545	+	1.03363i	2.72545	+	1.03363i
$23$		−	1.87003i		−	1.87003i	−0.354605	−	0.935016i	$-0.615385\pi$
$23$		−	1.87003i		−	1.87003i	0.354605	−	0.935016i	$-0.384615\pi$
$24$		0			0
$25$	−0.568065	+	0.822984i	−0.568065	+	0.822984i
$26$		0			0
$27$		0			0
$28$	−0.970942	+	1.40665i	−0.970942	+	1.40665i
$29$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	0.568065	+	0.822984i	$0.307692\pi$
$29$	−0.402877	+	1.06230i	−0.402877	+	1.06230i	−0.970942	+	0.239316i	$0.923077\pi$
$30$		0			0
$31$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$31$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$32$	0.380181	−	0.724375i	0.380181	−	0.724375i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	0.970942	+	1.40665i	0.970942	+	1.40665i
$37$	0.234068	−	0.0576926i	0.234068	−	0.0576926i	−0.120537	−	0.992709i	$-0.538462\pi$
$37$	0.234068	−	0.0576926i	0.234068	−	0.0576926i	0.354605	+	0.935016i	$0.384615\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$41$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$42$		0			0
$43$	−1.88546	−	0.464723i	−1.88546	−	0.464723i	−0.885456	−	0.464723i	$-0.846154\pi$
$43$	−1.88546	−	0.464723i	−1.88546	−	0.464723i	−1.00000			$\pi$
$44$	2.68015	−	1.40665i	2.68015	−	1.40665i
$45$		0			0
$46$	−2.30393	−	2.04110i	−2.30393	−	2.04110i
$47$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$47$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$48$		0			0
$49$	0.120537	+	0.992709i	0.120537	+	0.992709i
$50$	0.393906	+	1.59814i	0.393906	+	1.59814i
$51$		0			0
$52$		0			0
$53$	0.885456	+	0.464723i	0.885456	+	0.464723i
$53$	0.885456	+	0.464723i	0.885456	+	0.464723i
$54$		0			0
$55$		0			0
$56$	0.279362	+	1.13342i	0.279362	+	1.13342i
$57$		0			0
$58$	0.869047	+	1.65583i	0.869047	+	1.65583i
$59$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$59$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$60$		0			0
$61$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$61$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$62$		0			0
$63$	0.970942	+	0.239316i	0.970942	+	0.239316i
$64$	−0.552731	−	1.45743i	−0.552731	−	1.45743i
$65$		0			0
$66$		0			0
$67$	−0.922670	+	0.112032i	−0.922670	+	0.112032i	−0.568065	−	0.822984i	$-0.692308\pi$
$67$	−0.922670	+	0.112032i	−0.922670	+	0.112032i	−0.354605	+	0.935016i	$0.615385\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	0.317391	−	1.28771i	0.317391	−	1.28771i	−0.568065	−	0.822984i	$-0.692308\pi$
$71$	0.317391	−	1.28771i	0.317391	−	1.28771i	0.885456	−	0.464723i	$-0.153846\pi$
$72$	1.15883	+	0.140707i	1.15883	+	0.140707i
$73$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$73$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$74$	0.184402	−	0.351348i	0.184402	−	0.351348i
$75$		0			0
$76$		0			0
$77$	0.627974	−	1.65583i	0.627974	−	1.65583i
$78$		0			0
$79$	−0.879463	−	0.992709i	−0.879463	−	0.992709i	0.120537	−	0.992709i	$-0.461538\pi$
$79$	−0.879463	−	0.992709i	−0.879463	−	0.992709i	−1.00000			$\pi$
$80$		0			0
$81$	0.568065	−	0.822984i	0.568065	−	0.822984i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$		0			0
$86$	−2.63049	+	1.81569i	−2.63049	+	1.81569i
$87$		0			0
$88$	0.494725	−	2.00718i	0.494725	−	2.00718i
$89$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$89$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$90$		0			0
$91$		0			0
$92$	−3.17297	+	0.385269i	−3.17297	+	0.385269i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$97$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$98$	1.35460	+	0.935016i	1.35460	+	0.935016i
$99$	−1.32555	−	1.17433i	−1.32555	−	1.17433i
$100$	1.51343	+	0.794309i	1.51343	+	0.794309i
$101$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$101$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$102$		0			0
$103$		0			0		−0.239316	−	0.970942i	$-0.576923\pi$
$103$		0			0		0.239316	+	0.970942i	$0.423077\pi$
$104$		0			0
$105$		0			0
$106$	1.53901	−	0.583668i	1.53901	−	0.583668i
$107$	1.94188			1.94188			0.970942	−	0.239316i	$-0.0769231\pi$
$107$	1.94188			1.94188			0.970942	+	0.239316i	$0.0769231\pi$
$108$		0			0
$109$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$109$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$110$		0			0
$111$		0			0
$112$	0.187883	+	0.0986088i	0.187883	+	0.0986088i
$113$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$113$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$114$		0			0
$115$		0			0
$116$	1.88546	+	0.464723i	1.88546	+	0.464723i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.59892	+	1.41652i	−1.59892	+	1.41652i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$	1.35460	−	0.935016i	1.35460	−	0.935016i
$127$	−0.869047	+	1.65583i	−0.869047	+	1.65583i	−0.120537	+	0.992709i	$0.538462\pi$
$127$	−0.869047	+	1.65583i	−0.869047	+	1.65583i	−0.748511	+	0.663123i	$0.769231\pi$
$128$	−1.63397	−	0.619682i	−1.63397	−	0.619682i
$129$		0			0
$130$		0			0
$131$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$131$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$132$		0			0
$133$		0			0
$134$	−0.869047	+	1.25903i	−0.869047	+	1.25903i
$135$		0			0
$136$		0			0
$137$	−0.447528	−	0.169725i	−0.447528	−	0.169725i	0.120537	−	0.992709i	$-0.461538\pi$
$137$	−0.447528	−	0.169725i	−0.447528	−	0.169725i	−0.568065	+	0.822984i	$0.692308\pi$
$138$		0			0
$139$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$139$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$140$		0			0
$141$		0			0
$142$	−1.24006	−	1.79654i	−1.24006	−	1.79654i
$143$		0			0
$144$	0.158825	−	0.140707i	0.158825	−	0.140707i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.146113	−	0.385269i	−0.146113	−	0.385269i
$149$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.885456	−	0.464723i	$-0.153846\pi$
$149$	1.45352	+	0.358261i	1.45352	+	0.358261i	0.568065	+	0.822984i	$0.307692\pi$
$150$		0			0
$151$	0.393906	+	0.271894i	0.393906	+	0.271894i	0.748511	−	0.663123i	$-0.230769\pi$
$151$	0.393906	+	0.271894i	0.393906	+	0.271894i	−0.354605	+	0.935016i	$0.615385\pi$
$152$		0			0
$153$		0			0
$154$	−1.35460	−	2.58098i	−1.35460	−	2.58098i
$155$		0			0
$156$		0			0
$157$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$157$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$158$	−2.18296			−2.18296
$159$		0			0
$160$		0			0
$161$	−1.24006	+	1.39974i	−1.24006	+	1.39974i
$162$	−0.393906	−	1.59814i	−0.393906	−	1.59814i
$163$	−0.180446	−	1.48611i	−0.180446	−	1.48611i	−0.748511	−	0.663123i	$-0.769231\pi$
$163$	−0.180446	−	1.48611i	−0.180446	−	1.48611i	0.568065	−	0.822984i	$-0.307692\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$167$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$168$		0			0
$169$	−0.970942	−	0.239316i	−0.970942	−	0.239316i
$170$		0			0
$171$		0			0
$172$	−0.400072	+	3.29489i	−0.400072	+	3.29489i
$173$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$173$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$174$		0			0
$175$	0.970942	−	0.239316i	0.970942	−	0.239316i
$176$	−0.213460	−	0.309250i	−0.213460	−	0.309250i
$177$		0			0
$178$		0			0
$179$	1.63397	−	1.12785i	1.63397	−	1.12785i	0.748511	−	0.663123i	$-0.230769\pi$
$179$	1.63397	−	1.12785i	1.63397	−	1.12785i	0.885456	−	0.464723i	$-0.153846\pi$
$180$		0			0
$181$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$181$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$182$		0			0
$183$		0			0
$184$	−1.24006	+	1.79654i	−1.24006	+	1.79654i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$191$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$192$		0			0
$193$	1.97094	+	0.239316i	1.97094	+	0.239316i	1.00000			$0$
$193$	1.97094	+	0.239316i	1.97094	+	0.239316i	0.970942	+	0.239316i	$0.0769231\pi$
$194$		0			0
$195$		0			0
$196$	1.65954	−	0.409041i	1.65954	−	0.409041i
$197$	−0.530851	+	0.470293i	−0.530851	+	0.470293i	−0.885456	−	0.464723i	$-0.846154\pi$
$197$	−0.530851	+	0.470293i	−0.530851	+	0.470293i	0.354605	+	0.935016i	$0.384615\pi$
$198$	−2.89361	+	0.351348i	−2.89361	+	0.351348i
$199$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$199$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$200$	1.09148	−	0.413943i	1.09148	−	0.413943i
$201$		0			0
$202$		0			0
$203$	1.00599	−	0.527986i	1.00599	−	0.527986i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.869047	+	1.65583i	0.869047	+	1.65583i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.709210			0.709210			0.354605	−	0.935016i	$-0.384615\pi$
$211$	0.709210			0.709210			0.354605	+	0.935016i	$0.384615\pi$
$212$	0.606094	−	1.59814i	0.606094	−	1.59814i
$213$		0			0
$214$	2.11952	−	2.39245i	2.11952	−	2.39245i
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$223$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$224$	−0.764919	+	0.290096i	−0.764919	+	0.290096i
$225$	0.120537	−	0.992709i	0.120537	−	0.992709i
$226$	−0.393906	+	0.0478288i	−0.393906	+	0.0478288i
$227$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$227$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$228$		0			0
$229$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$229$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$230$		0			0
$231$		0			0
$232$	1.09148	−	0.753393i	1.09148	−	0.753393i
$233$	0.431935	−	0.822984i	0.431935	−	0.822984i	−0.568065	−	0.822984i	$-0.692308\pi$
$233$	0.431935	−	0.822984i	0.431935	−	0.822984i	1.00000			$0$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.31658	+	1.48611i	1.31658	+	1.48611i	0.748511	+	0.663123i	$0.230769\pi$
$239$	1.31658	+	1.48611i	1.31658	+	1.48611i	0.568065	+	0.822984i	$0.307692\pi$
$240$		0			0
$241$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$241$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$242$			3.51600i			3.51600i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$251$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$252$	0.206022	−	1.69675i	0.206022	−	1.69675i
$253$	3.09646	−	1.17433i	3.09646	−	1.17433i
$254$	1.09148	+	2.87799i	1.09148	+	2.87799i
$255$		0			0
$256$	−1.16672	+	0.612343i	−1.16672	+	0.612343i
$257$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$257$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$258$		0			0
$259$	−0.213460	−	0.112032i	−0.213460	−	0.112032i
$260$		0			0
$261$	−0.136945	−	1.12785i	−0.136945	−	1.12785i
$262$		0			0
$263$	−0.616337	+	0.695701i	−0.616337	+	0.695701i	−0.970942	−	0.239316i	$-0.923077\pi$
$263$	−0.616337	+	0.695701i	−0.616337	+	0.695701i	0.354605	+	0.935016i	$0.384615\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	0.380181	+	1.54246i	0.380181	+	1.54246i
$269$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$269$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$270$		0			0
$271$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$271$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$272$		0			0
$273$		0			0
$274$	−0.697573	+	0.366114i	−0.697573	+	0.366114i
$275$	−1.71945	−	0.423807i	−1.71945	−	0.423807i
$276$		0			0
$277$	1.24006	−	0.470293i	1.24006	−	0.470293i	0.354605	−	0.935016i	$-0.384615\pi$
$277$	1.24006	−	0.470293i	1.24006	−	0.470293i	0.885456	+	0.464723i	$0.153846\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.88546	+	0.464723i	−1.88546	+	0.464723i	−0.885456	+	0.464723i	$0.846154\pi$
$281$	−1.88546	+	0.464723i	−1.88546	+	0.464723i	−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$283$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$284$	−2.25030	−	0.273236i	−2.25030	−	0.273236i
$285$		0			0
$286$		0			0
$287$		0			0
$288$			0.818081i			0.818081i
$289$	−0.354605	+	0.935016i	−0.354605	+	0.935016i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$293$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$294$		0			0
$295$		0			0
$296$	−0.263126	−	0.0997907i	−0.263126	−	0.0997907i
$297$		0			0
$298$	2.02787	−	1.39974i	2.02787	−	1.39974i
$299$		0			0
$300$		0			0
$301$	1.10312	+	1.59814i	1.10312	+	1.59814i
$302$	0.764919	−	0.188536i	0.764919	−	0.188536i
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$307$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$308$	−2.93891	−	0.724375i	−2.93891	−	0.724375i
$309$		0			0
$310$		0			0
$311$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$311$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$312$		0			0
$313$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$313$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$314$		0			0
$315$		0			0
$316$	−1.50319	+	1.69675i	−1.50319	+	1.69675i
$317$	1.49702			1.49702			0.748511	−	0.663123i	$-0.230769\pi$
$317$	1.49702			1.49702			0.748511	+	0.663123i	$0.230769\pi$
$318$		0			0
$319$	−2.01199			−2.01199
$320$		0			0
$321$		0			0
$322$	0.371013	+	3.05557i	0.371013	+	3.05557i
$323$		0			0
$324$	−1.51343	−	0.794309i	−1.51343	−	0.794309i
$325$		0			0
$326$	−2.02787	−	1.39974i	−2.02787	−	1.39974i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.180446	+	1.48611i	−0.180446	+	1.48611i	0.568065	+	0.822984i	$0.307692\pi$
$331$	−0.180446	+	1.48611i	−0.180446	+	1.48611i	−0.748511	+	0.663123i	$0.769231\pi$
$332$		0			0
$333$	−0.180446	+	0.159861i	−0.180446	+	0.159861i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.922670	−	0.112032i	−0.922670	−	0.112032i	−0.354605	−	0.935016i	$-0.615385\pi$
$337$	−0.922670	−	0.112032i	−0.922670	−	0.112032i	−0.568065	+	0.822984i	$0.692308\pi$
$338$	−1.35460	+	0.935016i	−1.35460	+	0.935016i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.568065	−	0.822984i	0.568065	−	0.822984i
$344$	1.50319	+	1.69675i	1.50319	+	1.69675i
$345$		0			0
$346$		0			0
$347$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	0.120537	+	0.992709i	$0.461538\pi$
$347$	−0.627974	+	1.65583i	−0.627974	+	1.65583i	−0.748511	+	0.663123i	$0.769231\pi$
$348$		0			0
$349$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$349$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$350$	0.764919	−	1.45743i	0.764919	−	1.45743i
$351$		0			0
$352$	1.43819	+	0.174628i	1.43819	+	0.174628i
$353$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$353$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	0.393906	−	3.24411i	0.393906	−	3.24411i
$359$	−1.53901	+	0.583668i	−1.53901	+	0.583668i	−0.970942	−	0.239316i	$-0.923077\pi$
$359$	−1.53901	+	0.583668i	−1.53901	+	0.583668i	−0.568065	+	0.822984i	$0.692308\pi$
$360$		0			0
$361$	0.970942	+	0.239316i	0.970942	+	0.239316i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$367$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$368$	0.0949602	+	0.385269i	0.0949602	+	0.385269i
$369$		0			0
$370$		0			0
$371$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$372$		0			0
$373$	−1.09148	+	1.23202i	−1.09148	+	1.23202i	−0.120537	+	0.992709i	$0.538462\pi$
$373$	−1.09148	+	1.23202i	−1.09148	+	1.23202i	−0.970942	+	0.239316i	$0.923077\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.764919	−	0.527986i	−0.764919	−	0.527986i	0.120537	−	0.992709i	$-0.461538\pi$
$379$	−0.764919	−	0.527986i	−0.764919	−	0.527986i	−0.885456	+	0.464723i	$0.846154\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$383$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$384$		0			0
$385$		0			0
$386$	2.44608	−	2.16704i	2.44608	−	2.16704i
$387$	1.88546	−	0.464723i	1.88546	−	0.464723i
$388$		0			0
$389$	0.222431	−	0.902438i	0.222431	−	0.902438i	−0.748511	−	0.663123i	$-0.769231\pi$
$389$	0.222431	−	0.902438i	0.222431	−	0.902438i	0.970942	−	0.239316i	$-0.0769231\pi$
$390$		0			0
$391$		0			0
$392$	0.542488	−	1.03363i	0.542488	−	1.03363i
$393$		0			0
$394$			1.16734i			1.16734i
$395$		0			0
$396$	−1.71945	+	2.49106i	−1.71945	+	2.49106i
$397$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$397$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$398$		0			0
$399$		0			0
$400$	0.0752430	−	0.198399i	0.0752430	−	0.198399i
$401$			1.98542i			1.98542i	0.120537	+	0.992709i	$0.461538\pi$
$401$			1.98542i			1.98542i	−0.120537	+	0.992709i	$0.538462\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$	0.447528	−	1.81569i	0.447528	−	1.81569i
$407$	0.242518	+	0.351348i	0.242518	+	0.351348i
$408$		0			0
$409$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$409$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	2.98857	+	0.736617i	2.98857	+	0.736617i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$419$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$420$		0			0
$421$	−0.475142	−	1.92773i	−0.475142	−	1.92773i	−0.354605	−	0.935016i	$-0.615385\pi$
$421$	−0.475142	−	1.92773i	−0.475142	−	1.92773i	−0.120537	−	0.992709i	$-0.538462\pi$
$422$	0.774087	−	0.873764i	0.774087	−	0.873764i
$423$		0			0
$424$	−0.542488	−	1.03363i	−0.542488	−	1.03363i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	−0.400072	−	3.29489i	−0.400072	−	3.29489i
$429$		0			0
$430$		0			0
$431$	−1.12054	−	0.992709i	−1.12054	−	0.992709i	−0.120537	−	0.992709i	$-0.538462\pi$
$431$	−1.12054	−	0.992709i	−1.12054	−	0.992709i	−1.00000			$\pi$
$432$		0			0
$433$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$433$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$439$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$440$		0			0
$441$	−0.568065	−	0.822984i	−0.568065	−	0.822984i
$442$		0			0
$443$	1.97094	+	0.239316i	1.97094	+	0.239316i	1.00000			$0$
$443$	1.97094	+	0.239316i	1.97094	+	0.239316i	0.970942	+	0.239316i	$0.0769231\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−0.552731	+	1.45743i	−0.552731	+	1.45743i
$449$	0.402877	−	0.583668i	0.402877	−	0.583668i	−0.568065	−	0.822984i	$-0.692308\pi$
$449$	0.402877	−	0.583668i	0.402877	−	0.583668i	0.970942	+	0.239316i	$0.0769231\pi$
$450$	−1.09148	−	1.23202i	−1.09148	−	1.23202i
$451$		0			0
$452$	−0.234068	+	0.339106i	−0.234068	+	0.339106i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.35460	+	0.935016i	−1.35460	+	0.935016i	−0.354605	+	0.935016i	$0.615385\pi$
$457$	−1.35460	+	0.935016i	−1.35460	+	0.935016i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$461$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$462$		0			0
$463$	1.97094	−	0.239316i	1.97094	−	0.239316i	0.970942	−	0.239316i	$-0.0769231\pi$
$463$	1.97094	−	0.239316i	1.97094	−	0.239316i	1.00000			$0$
$464$	0.0290582	−	0.239316i	0.0290582	−	0.239316i
$465$		0			0
$466$	−0.542488	−	1.43042i	−0.542488	−	1.43042i
$467$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$467$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$468$		0			0
$469$	0.764919	+	0.527986i	0.764919	+	0.527986i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.414514	−	3.41383i	−0.414514	−	3.41383i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.00000			−1.00000
$478$	3.26793			3.26793
$479$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$479$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	2.73288	+	2.42112i	2.73288	+	2.42112i
$485$		0			0
$486$		0			0
$487$	−1.45352	−	0.358261i	−1.45352	−	0.358261i	−0.568065	−	0.822984i	$-0.692308\pi$
$487$	−1.45352	−	0.358261i	−1.45352	−	0.358261i	−0.885456	+	0.464723i	$0.846154\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$491$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.09148	+	0.753393i	−1.09148	+	0.753393i
$498$		0			0
$499$	−1.53901	−	0.583668i	−1.53901	−	0.583668i	−0.568065	−	0.822984i	$-0.692308\pi$
$499$	−1.53901	−	0.583668i	−1.53901	−	0.583668i	−0.970942	+	0.239316i	$0.923077\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$503$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$504$	−0.774087	−	0.873764i	−0.774087	−	0.873764i
$505$		0			0
$506$	1.93291	−	5.09667i	1.93291	−	5.09667i
$507$		0			0
$508$	2.98857	+	1.13342i	2.98857	+	1.13342i
$509$		0			0		0.464723	−	0.885456i	$-0.346154\pi$
$509$		0			0		−0.464723	+	0.885456i	$0.653846\pi$
$510$		0			0
$511$		0			0
$512$	−0.100819	+	0.409041i	−0.100819	+	0.409041i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$	−0.371013	+	0.140707i	−0.371013	+	0.140707i
$519$		0			0
$520$		0			0
$521$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$521$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$522$	−1.53901	−	1.06230i	−1.53901	−	1.06230i
$523$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$523$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$524$		0			0
$525$		0			0
$526$	0.184402	+	1.51868i	0.184402	+	1.51868i
$527$		0			0
$528$		0			0
$529$	−2.49702			−2.49702
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	0.960699	+	0.504214i	0.960699	+	0.504214i
$537$		0			0
$538$		0			0
$539$	−1.56806	+	0.822984i	−1.56806	+	0.822984i
$540$		0			0
$541$	−0.688601	−	1.81569i	−0.688601	−	1.81569i	−0.568065	−	0.822984i	$-0.692308\pi$
$541$	−0.688601	−	1.81569i	−0.688601	−	1.81569i	−0.120537	−	0.992709i	$-0.538462\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.136945	−	0.198399i	−0.136945	−	0.198399i	0.748511	−	0.663123i	$-0.230769\pi$
$547$	−0.136945	−	0.198399i	−0.136945	−	0.198399i	−0.885456	+	0.464723i	$0.846154\pi$
$548$	−0.195780	+	0.794309i	−0.195780	+	0.794309i
$549$		0			0
$550$	−2.39889	+	1.65583i	−2.39889	+	1.65583i
$551$		0			0
$552$		0			0
$553$			1.32625i			1.32625i
$554$	0.774087	−	2.04110i	0.774087	−	2.04110i
$555$		0			0
$556$		0			0
$557$	−1.24006	−	1.39974i	−1.24006	−	1.39974i	−0.885456	−	0.464723i	$-0.846154\pi$
$557$	−1.24006	−	1.39974i	−1.24006	−	1.39974i	−0.354605	−	0.935016i	$-0.615385\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−1.48538	+	2.83016i	−1.48538	+	2.83016i
$563$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$563$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−0.970942	+	0.239316i	−0.970942	+	0.239316i
$568$	−1.15883	+	1.02663i	−1.15883	+	1.02663i
$569$	1.31658	−	0.159861i	1.31658	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$569$	1.31658	−	0.159861i	1.31658	−	0.159861i	0.748511	+	0.663123i	$0.230769\pi$
$570$		0			0
$571$	0.447528	−	0.169725i	0.447528	−	0.169725i	−0.120537	−	0.992709i	$-0.538462\pi$
$571$	0.447528	−	0.169725i	0.447528	−	0.169725i	0.568065	+	0.822984i	$0.307692\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.53901	+	1.06230i	1.53901	+	1.06230i
$576$	1.16672	+	1.03363i	1.16672	+	1.03363i
$577$		0			0		−0.885456	−	0.464723i	$-0.846154\pi$
$577$		0			0		0.885456	+	0.464723i	$0.153846\pi$
$578$	0.764919	+	1.45743i	0.764919	+	1.45743i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.213460	+	1.75800i	−0.213460	+	1.75800i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.120537	−	0.992709i	$-0.538462\pi$
$587$		0			0		0.120537	+	0.992709i	$0.461538\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.0452937	+	0.0237720i	−0.0452937	+	0.0237720i
$593$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$593$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$594$		0			0
$595$		0			0
$596$	0.308420	−	2.54007i	0.308420	−	2.54007i
$597$		0			0
$598$		0			0
$599$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i	−0.354605	−	0.935016i	$-0.615385\pi$
$599$	−0.234068	+	0.0576926i	−0.234068	+	0.0576926i	0.120537	+	0.992709i	$0.461538\pi$
$600$		0			0
$601$		0			0		0.239316	−	0.970942i	$-0.423077\pi$
$601$		0			0		−0.239316	+	0.970942i	$0.576923\pi$
$602$	3.17297	+	0.385269i	3.17297	+	0.385269i
$603$	0.764919	−	0.527986i	0.764919	−	0.527986i
$604$	0.380181	−	0.724375i	0.380181	−	0.724375i
$605$		0			0
$606$		0			0
$607$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$607$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$			0.478631i			0.478631i	0.970942	+	0.239316i	$0.0769231\pi$
$613$			0.478631i			0.478631i	−0.970942	+	0.239316i	$0.923077\pi$
$614$		0			0
$615$		0			0
$616$	−1.70131	+	1.17433i	−1.70131	+	1.17433i
$617$	0.475142	+	0.0576926i	0.475142	+	0.0576926i	0.354605	−	0.935016i	$-0.384615\pi$
$617$	0.475142	+	0.0576926i	0.475142	+	0.0576926i	0.120537	+	0.992709i	$0.461538\pi$
$618$		0			0
$619$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$619$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.354605	−	0.935016i	−0.354605	−	0.935016i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.222431	−	0.423807i	−0.222431	−	0.423807i	0.748511	−	0.663123i	$-0.230769\pi$
$631$	−0.222431	−	0.423807i	−0.222431	−	0.423807i	−0.970942	+	0.239316i	$0.923077\pi$
$632$	0.186612	+	1.53689i	0.186612	+	1.53689i
$633$		0			0
$634$	1.63397	−	1.84437i	1.63397	−	1.84437i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−2.19604	+	2.47882i	−2.19604	+	2.47882i
$639$	0.317391	+	1.28771i	0.317391	+	1.28771i
$640$		0			0
$641$	−0.616337	−	1.17433i	−0.616337	−	1.17433i	−0.970942	−	0.239316i	$-0.923077\pi$
$641$	−0.616337	−	1.17433i	−0.616337	−	1.17433i	0.354605	−	0.935016i	$-0.384615\pi$
$642$		0			0
$643$		0			0		−0.748511	−	0.663123i	$-0.769231\pi$
$643$		0			0		0.748511	+	0.663123i	$0.230769\pi$
$644$	2.63049	+	1.81569i	2.63049	+	1.81569i
$645$		0			0
$646$		0			0
$647$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$647$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$648$	−1.09148	+	0.413943i	−1.09148	+	0.413943i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	−2.48437	+	0.612343i	−2.48437	+	0.612343i
$653$	−0.645395	−	0.935016i	−0.645395	−	0.935016i	0.354605	−	0.935016i	$-0.384615\pi$
$653$	−0.645395	−	0.935016i	−0.645395	−	0.935016i	−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$			1.87003i			1.87003i	0.354605	+	0.935016i	$0.384615\pi$
$659$			1.87003i			1.87003i	−0.354605	+	0.935016i	$0.615385\pi$
$660$		0			0
$661$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$661$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$662$	1.63397	+	1.84437i	1.63397	+	1.84437i
$663$		0			0
$664$		0			0
$665$		0			0
$666$			0.396799i			0.396799i
$667$	1.98653	+	0.753393i	1.98653	+	0.753393i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.45352	+	0.358261i	−1.45352	+	0.358261i	−0.885456	−	0.464723i	$-0.846154\pi$
$673$	−1.45352	+	0.358261i	−1.45352	+	0.358261i	−0.568065	+	0.822984i	$0.692308\pi$
$674$	−1.14510	+	1.01447i	−1.14510	+	1.01447i
$675$		0			0
$676$	−0.206022	+	1.69675i	−0.206022	+	1.69675i
$677$		0			0		0.935016	−	0.354605i	$-0.115385\pi$
$677$		0			0		−0.935016	+	0.354605i	$0.884615\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.00599	−	0.527986i	−1.00599	−	0.527986i	−0.120537	−	0.992709i	$-0.538462\pi$
$683$	−1.00599	−	0.527986i	−1.00599	−	0.527986i	−0.885456	+	0.464723i	$0.846154\pi$
$684$		0			0
$685$		0			0
$686$	−0.393906	−	1.59814i	−0.393906	−	1.59814i
$687$		0			0
$688$	0.412045			0.412045
$689$		0			0
$690$		0			0
$691$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$691$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$692$		0			0
$693$	0.213460	+	1.75800i	0.213460	+	1.75800i
$694$	1.35460	+	2.58098i	1.35460	+	2.58098i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−0.606094	−	1.59814i	−0.606094	−	1.59814i
$701$	−0.869047	+	0.329586i	−0.869047	+	0.329586i	−0.748511	−	0.663123i	$-0.769231\pi$
$701$	−0.869047	+	0.329586i	−0.869047	+	0.329586i	−0.120537	+	0.992709i	$0.538462\pi$
$702$		0			0
$703$		0			0
$704$	2.06616	−	1.83046i	2.06616	−	1.83046i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	0.393906	−	0.271894i	0.393906	−	0.271894i	−0.354605	−	0.935016i	$-0.615385\pi$
$709$	0.393906	−	0.271894i	0.393906	−	0.271894i	0.748511	+	0.663123i	$0.230769\pi$
$710$		0			0
$711$	1.24006	+	0.470293i	1.24006	+	0.470293i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−2.25030	−	2.54007i	−2.25030	−	2.54007i
$717$		0			0
$718$	−0.960699	+	2.53316i	−0.960699	+	2.53316i
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$	1.35460	−	0.935016i	1.35460	−	0.935016i
$723$		0			0
$724$		0			0
$725$	−0.645395	−	0.935016i	−0.645395	−	0.935016i
$726$		0			0
$727$		0			0		0.748511	−	0.663123i	$-0.230769\pi$
$727$		0			0		−0.748511	+	0.663123i	$0.769231\pi$
$728$		0			0
$729$	−0.120537	+	0.992709i	−0.120537	+	0.992709i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$733$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$734$		0			0
$735$		0			0
$736$	−1.35460	−	0.710951i	−1.35460	−	0.710951i
$737$	−0.764919	−	1.45743i	−0.764919	−	1.45743i
$738$		0			0
$739$	0.475142	+	1.92773i	0.475142	+	1.92773i	0.354605	+	0.935016i	$0.384615\pi$
$739$	0.475142	+	1.92773i	0.475142	+	1.92773i	0.120537	+	0.992709i	$0.461538\pi$
$740$		0			0
$741$		0			0
$742$	−1.53901	−	0.583668i	−1.53901	−	0.583668i
$743$	−1.49702			−1.49702			−0.748511	−	0.663123i	$-0.769231\pi$
$743$	−1.49702			−1.49702			−0.748511	+	0.663123i	$0.769231\pi$
$744$		0			0
$745$		0			0
$746$	0.326559	+	2.68946i	0.326559	+	2.68946i
$747$		0			0
$748$		0			0
$749$	−1.45352	−	1.28771i	−1.45352	−	1.28771i
$750$		0			0
$751$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.120537	−	0.992709i	$-0.461538\pi$
$751$	1.00599	−	0.527986i	1.00599	−	0.527986i	0.885456	+	0.464723i	$0.153846\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.180446	−	0.159861i	0.180446	−	0.159861i	−0.568065	−	0.822984i	$-0.692308\pi$
$757$	0.180446	−	0.159861i	0.180446	−	0.159861i	0.748511	+	0.663123i	$0.230769\pi$
$758$	−1.48538	+	0.366114i	−1.48538	+	0.366114i
$759$		0			0
$760$		0			0
$761$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$761$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.663123	−	0.748511i	$-0.730769\pi$
$769$		0			0		0.663123	+	0.748511i	$0.269231\pi$
$770$		0			0
$771$		0			0
$772$		−	3.39350i		−	3.39350i
$773$		0			0		−0.935016	−	0.354605i	$-0.884615\pi$
$773$		0			0		0.935016	+	0.354605i	$0.115385\pi$
$774$	1.48538	−	2.83016i	1.48538	−	2.83016i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.869047	−	1.25903i	−0.869047	−	1.25903i
$779$		0			0
$780$		0			0
$781$	2.33154	−	0.283100i	2.33154	−	0.283100i
$782$		0			0
$783$		0			0
$784$	−0.0752430	−	0.198399i	−0.0752430	−	0.198399i
$785$		0			0
$786$		0			0
$787$		0			0		−0.822984	−	0.568065i	$-0.807692\pi$
$787$		0			0		0.822984	+	0.568065i	$0.192308\pi$
$788$	0.907336	+	0.803830i	0.907336	+	0.803830i
$789$		0			0
$790$		0			0
$791$	0.0290582	+	0.239316i	0.0290582	+	0.239316i
$792$	0.494725	+	2.00718i	0.494725	+	2.00718i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.663123	−	0.748511i	$-0.269231\pi$
$797$		0			0		−0.663123	+	0.748511i	$0.730769\pi$
$798$		0			0
$799$		0			0
$800$	0.380181	+	0.724375i	0.380181	+	0.724375i
$801$		0			0
$802$	2.44608	+	2.16704i	2.44608	+	2.16704i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1.97094	−	0.239316i	1.97094	−	0.239316i	0.970942	−	0.239316i	$-0.0769231\pi$
$809$	1.97094	−	0.239316i	1.97094	−	0.239316i	1.00000			$0$
$810$		0			0
$811$		0			0		0.970942	−	0.239316i	$-0.0769231\pi$
$811$		0			0		−0.970942	+	0.239316i	$0.923077\pi$
$812$	−1.10312	−	1.59814i	−1.10312	−	1.59814i
$813$		0			0
$814$	0.697573	+	0.0847007i	0.697573	+	0.0847007i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	0.616337	+	0.695701i	0.616337	+	0.695701i	0.970942	−	0.239316i	$-0.0769231\pi$
$821$	0.616337	+	0.695701i	0.616337	+	0.695701i	−0.354605	+	0.935016i	$0.615385\pi$
$822$		0			0
$823$	1.10312	−	1.59814i	1.10312	−	1.59814i	0.354605	−	0.935016i	$-0.384615\pi$
$823$	1.10312	−	1.59814i	1.10312	−	1.59814i	0.748511	−	0.663123i	$-0.230769\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−0.222431	+	0.423807i	−0.222431	+	0.423807i	−0.970942	−	0.239316i	$-0.923077\pi$
$827$	−0.222431	+	0.423807i	−0.222431	+	0.423807i	0.748511	+	0.663123i	$0.230769\pi$
$828$	2.63049	−	1.81569i	2.63049	−	1.81569i
$829$		0			0		−0.992709	−	0.120537i	$-0.961538\pi$
$829$		0			0		0.992709	+	0.120537i	$0.0384615\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$839$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$840$		0			0
$841$	−0.217660	−	0.192830i	−0.217660	−	0.192830i
$842$	−2.89361	−	1.51868i	−2.89361	−	1.51868i
$843$		0			0
$844$	−0.146113	−	1.20335i	−0.146113	−	1.20335i
$845$		0			0
$846$		0			0
$847$	2.13613			2.13613
$848$	−0.206022	−	0.0507800i	−0.206022	−	0.0507800i
$849$		0			0
$850$		0			0
$851$	−0.107887	−	0.437715i	−0.107887	−	0.437715i
$852$		0			0
$853$		0			0		−0.464723	−	0.885456i	$-0.653846\pi$
$853$		0			0		0.464723	+	0.885456i	$0.346154\pi$
$854$		0			0
$855$		0			0
$856$	−1.86557	−	1.28771i	−1.86557	−	1.28771i
$857$		0			0		0.885456	−	0.464723i	$-0.153846\pi$
$857$		0			0		−0.885456	+	0.464723i	$0.846154\pi$
$858$		0			0
$859$		0			0		−0.354605	−	0.935016i	$-0.615385\pi$
$859$		0			0		0.354605	+	0.935016i	$0.384615\pi$
$860$		0			0
$861$		0			0
$862$	−2.44608	+	0.297008i	−2.44608	+	0.297008i
$863$	−1.45352	+	1.28771i	−1.45352	+	1.28771i	−0.568065	+	0.822984i	$0.692308\pi$
$863$	−1.45352	+	1.28771i	−1.45352	+	1.28771i	−0.885456	+	0.464723i	$0.846154\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.09148	−	2.07964i	1.09148	−	2.07964i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i	−0.970942	−	0.239316i	$-0.923077\pi$
$877$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i	0.885456	+	0.464723i	$0.153846\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.822984	−	0.568065i	$-0.192308\pi$
$881$		0			0		−0.822984	+	0.568065i	$0.807692\pi$
$882$	−1.63397	−	0.198399i	−1.63397	−	0.198399i
$883$	−0.447528	+	1.81569i	−0.447528	+	1.81569i	0.120537	+	0.992709i	$0.461538\pi$
$883$	−0.447528	+	1.81569i	−0.447528	+	1.81569i	−0.568065	+	0.822984i	$0.692308\pi$
$884$		0			0
$885$		0			0
$886$	2.44608	−	2.16704i	2.44608	−	2.16704i
$887$		0			0		0.992709	−	0.120537i	$-0.0384615\pi$
$887$		0			0		−0.992709	+	0.120537i	$0.961538\pi$
$888$		0			0
$889$	1.74851	−	0.663123i	1.74851	−	0.663123i
$890$		0			0
$891$	1.71945	+	0.423807i	1.71945	+	0.423807i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$	0.812117	+	1.54736i	0.812117	+	1.54736i
$897$		0			0
$898$	−0.279362	−	1.13342i	−0.279362	−	1.13342i
$899$		0			0
$900$	−1.70921			−1.70921
$901$		0			0
$902$		0			0
$903$		0			0
$904$	0.0673467	+	0.273236i	0.0673467	+	0.273236i
$905$		0			0
$906$		0			0
$907$	1.56806	+	0.822984i	1.56806	+	0.822984i	1.00000			$0$
$907$	1.56806	+	0.822984i	1.56806	+	0.822984i	0.568065	+	0.822984i	$0.307692\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.688601	+	0.169725i	0.688601	+	0.169725i	0.568065	−	0.822984i	$-0.307692\pi$
$911$	0.688601	+	0.169725i	0.688601	+	0.169725i	0.120537	+	0.992709i	$0.461538\pi$
$912$		0			0
$913$		0			0
$914$	−0.326559	+	2.68946i	−0.326559	+	2.68946i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	0.475142	−	1.92773i	0.475142	−	1.92773i	0.120537	−	0.992709i	$-0.461538\pi$
$919$	0.475142	−	1.92773i	0.475142	−	1.92773i	0.354605	−	0.935016i	$-0.384615\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.0854858	+	0.225408i	−0.0854858	+	0.225408i
$926$	1.85640	−	2.68946i	1.85640	−	2.68946i
$927$		0			0
$928$	0.616337	+	0.695701i	0.616337	+	0.695701i
$929$		0			0		0.568065	−	0.822984i	$-0.307692\pi$
$929$		0			0		−0.568065	+	0.822984i	$0.692308\pi$
$930$		0			0
$931$		0			0
$932$	−1.48538	−	0.563332i	−1.48538	−	0.563332i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$937$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$938$	1.48538	−	0.366114i	1.48538	−	0.366114i
$939$		0			0
$940$		0			0
$941$		0			0		0.120537	−	0.992709i	$-0.461538\pi$
$941$		0			0		−0.120537	+	0.992709i	$0.538462\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−4.65836	−	3.21543i	−4.65836	−	3.21543i
$947$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	0.568065	−	0.822984i	$-0.307692\pi$
$947$	−0.180446	−	0.159861i	−0.180446	−	0.159861i	−0.748511	+	0.663123i	$0.769231\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.94188			1.94188			0.970942	−	0.239316i	$-0.0769231\pi$
$953$	1.94188			1.94188			0.970942	+	0.239316i	$0.0769231\pi$
$954$	−1.09148	+	1.23202i	−1.09148	+	1.23202i
$955$		0			0
$956$	2.25030	−	2.54007i	2.25030	−	2.54007i
$957$		0			0
$958$		0			0
$959$	0.222431	+	0.423807i	0.222431	+	0.423807i
$960$		0			0
$961$	0.748511	+	0.663123i	0.748511	+	0.663123i
$962$		0			0
$963$	−1.71945	+	0.902438i	−1.71945	+	0.902438i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	0.136945	−	1.12785i	0.136945	−	1.12785i	−0.748511	−	0.663123i	$-0.769231\pi$
$967$	0.136945	−	1.12785i	0.136945	−	1.12785i	0.885456	−	0.464723i	$-0.153846\pi$
$968$	2.47540	−	0.300568i	2.47540	−	0.300568i
$969$		0			0
$970$		0			0
$971$		0			0		−0.568065	−	0.822984i	$-0.692308\pi$
$971$		0			0		0.568065	+	0.822984i	$0.307692\pi$
$972$		0			0
$973$		0			0
$974$	−2.02787	+	1.39974i	−2.02787	+	1.39974i
$975$		0			0
$976$		0			0
$977$			0.929446i			0.929446i	0.885456	+	0.464723i	$0.153846\pi$
$977$			0.929446i			0.929446i	−0.885456	+	0.464723i	$0.846154\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.354605	−	0.935016i	$-0.384615\pi$
$983$		0			0		−0.354605	+	0.935016i	$0.615385\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−0.869047	+	3.52586i	−0.869047	+	3.52586i
$990$		0			0
$991$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.970942	−	0.239316i	$-0.923077\pi$
$991$	−1.71945	+	0.423807i	−1.71945	+	0.423807i	−0.748511	+	0.663123i	$0.769231\pi$
$992$		0			0
$993$		0			0
$994$	−0.263126	+	2.16704i	−0.263126	+	2.16704i
$995$		0			0
$996$		0			0
$997$		0			0		−0.970942	−	0.239316i	$-0.923077\pi$
$997$		0			0		0.970942	+	0.239316i	$0.0769231\pi$
$998$	−2.39889	+	1.25903i	−2.39889	+	1.25903i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	371.1.o.a.356.1	yes	12
3.2	odd	2		3339.1.cm.a.2953.1		12
7.2	even	3		2597.1.be.a.1893.1		24
7.3	odd	6		2597.1.be.a.1734.1		24
7.4	even	3		2597.1.be.a.1734.1		24
7.5	odd	6		2597.1.be.a.1893.1		24
7.6	odd	2	CM	371.1.o.a.356.1	yes	12
21.20	even	2		3339.1.cm.a.2953.1		12
53.7	even	26	inner	371.1.o.a.272.1	✓	12
159.113	odd	26		3339.1.cm.a.1756.1		12
371.60	even	78		2597.1.be.a.166.1		24
371.166	odd	78		2597.1.be.a.325.1		24
371.219	even	78		2597.1.be.a.325.1		24
371.272	odd	26	inner	371.1.o.a.272.1	✓	12
371.325	odd	78		2597.1.be.a.166.1		24
1113.272	even	26		3339.1.cm.a.1756.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
371.1.o.a.272.1	✓	12	53.7	even	26	inner
371.1.o.a.272.1	✓	12	371.272	odd	26	inner
371.1.o.a.356.1	yes	12	1.1	even	1	trivial
371.1.o.a.356.1	yes	12	7.6	odd	2	CM
2597.1.be.a.166.1		24	371.60	even	78
2597.1.be.a.166.1		24	371.325	odd	78
2597.1.be.a.325.1		24	371.166	odd	78
2597.1.be.a.325.1		24	371.219	even	78
2597.1.be.a.1734.1		24	7.3	odd	6
2597.1.be.a.1734.1		24	7.4	even	3
2597.1.be.a.1893.1		24	7.2	even	3
2597.1.be.a.1893.1		24	7.5	odd	6
3339.1.cm.a.1756.1		12	159.113	odd	26
3339.1.cm.a.1756.1		12	1113.272	even	26
3339.1.cm.a.2953.1		12	3.2	odd	2
3339.1.cm.a.2953.1		12	21.20	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 371 = 7 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	371.o (of order \(26\), degree \(12\), minimal)

\(f(q)\)	\(=\)	\(q+(1.09148 - 1.23202i) q^{2} +(-0.206022 - 1.69675i) q^{4} +(-0.748511 - 0.663123i) q^{7} +(-0.960699 - 0.663123i) q^{8} +(-0.885456 + 0.464723i) q^{9} +O(q^{10})\)	\(q+(1.09148 - 1.23202i) q^{2} +(-0.206022 - 1.69675i) q^{4} +(-0.748511 - 0.663123i) q^{7} +(-0.960699 - 0.663123i) q^{8} +(-0.885456 + 0.464723i) q^{9} +(0.627974 + 1.65583i) q^{11} +(-1.63397 + 0.198399i) q^{14} +(-0.206022 + 0.0507800i) q^{16} +(-0.393906 + 1.59814i) q^{18} +(2.72545 + 1.03363i) q^{22} -1.87003i q^{23} +(-0.568065 + 0.822984i) q^{25} +(-0.970942 + 1.40665i) q^{28} +(-0.402877 + 1.06230i) q^{29} +(0.380181 - 0.724375i) q^{32} +(0.970942 + 1.40665i) q^{36} +(0.234068 - 0.0576926i) q^{37} +(-1.88546 - 0.464723i) q^{43} +(2.68015 - 1.40665i) q^{44} +(-2.30393 - 2.04110i) q^{46} +(0.120537 + 0.992709i) q^{49} +(0.393906 + 1.59814i) q^{50} +(0.885456 + 0.464723i) q^{53} +(0.279362 + 1.13342i) q^{56} +(0.869047 + 1.65583i) q^{58} +(0.970942 + 0.239316i) q^{63} +(-0.552731 - 1.45743i) q^{64} +(-0.922670 + 0.112032i) q^{67} +(0.317391 - 1.28771i) q^{71} +(1.15883 + 0.140707i) q^{72} +(0.184402 - 0.351348i) q^{74} +(0.627974 - 1.65583i) q^{77} +(-0.879463 - 0.992709i) q^{79} +(0.568065 - 0.822984i) q^{81} +(-2.63049 + 1.81569i) q^{86} +(0.494725 - 2.00718i) q^{88} +(-3.17297 + 0.385269i) q^{92} +(1.35460 + 0.935016i) q^{98} +(-1.32555 - 1.17433i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - q^{4} - q^{7} - 13 q^{8} + q^{9}+O(q^{10}) \) 12 * q - q^4 - q^7 - 13 * q^8 + q^9	\( 12 q - q^{4} - q^{7} - 13 q^{8} + q^{9} + 2 q^{11} - q^{16} + q^{25} - q^{28} - 2 q^{29} + q^{36} + 2 q^{37} - 11 q^{43} + 15 q^{44} - q^{49} - q^{53} - 13 q^{56} + q^{63} + 12 q^{64} - 13 q^{74} + 2 q^{77} - 13 q^{79} - q^{81} + 13 q^{88} + 13 q^{98} - 2 q^{99}+O(q^{100}) \) 12 * q - q^4 - q^7 - 13 * q^8 + q^9 + 2 * q^11 - q^16 + q^25 - q^28 - 2 * q^29 + q^36 + 2 * q^37 - 11 * q^43 + 15 * q^44 - q^49 - q^53 - 13 * q^56 + q^63 + 12 * q^64 - 13 * q^74 + 2 * q^77 - 13 * q^79 - q^81 + 13 * q^88 + 13 * q^98 - 2 * q^99

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