LMFDB - Embedded newform 1265.1.bi.a.373.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1265,1,Mod(43,1265)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1265, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([33, 22, 10]))

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

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

chi := DirichletCharacter("1265.43");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1265 = 5 \cdot 11 \cdot 23 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1265.bi (of order $44$, degree $20$, 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:	$0.631317240981$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{44}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{44} - \cdots)$

Embedding invariants

Embedding label			373.1
Root			$0.989821 + 0.142315i$ of defining polynomial
Character	$\chi$	$=$	1265.373
Dual form			1265.1.bi.a.1187.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.86912 + 0.133682i) q^{3} +(0.540641 - 0.841254i) q^{4} +(-0.989821 + 0.142315i) q^{5} +(2.48594 - 0.357424i) q^{9} +O(q^{10})$	\(q+(-1.86912 + 0.133682i) q^{3} +(0.540641 - 0.841254i) q^{4} +(-0.989821 + 0.142315i) q^{5} +(2.48594 - 0.357424i) q^{9} +(0.281733 + 0.959493i) q^{11} +(-0.898064 + 1.64468i) q^{12} +(1.83107 - 0.398326i) q^{15} +(-0.415415 - 0.909632i) q^{16} +(-0.415415 + 0.909632i) q^{20} +(-0.281733 - 0.959493i) q^{23} +(0.959493 - 0.281733i) q^{25} +(-2.76767 + 0.602069i) q^{27} +(-1.29639 - 1.49611i) q^{31} +(-0.654861 - 1.75575i) q^{33} +(1.04331 - 2.28454i) q^{36} +(1.17116 - 1.56449i) q^{37} +(0.959493 + 0.281733i) q^{44} +(-2.40977 + 0.707571i) q^{45} +(0.847507 - 0.847507i) q^{47} +(0.898064 + 1.64468i) q^{48} +(-0.755750 - 0.654861i) q^{49} +(-0.398326 - 0.148568i) q^{53} +(-0.415415 - 0.909632i) q^{55} +(0.983568 + 0.449181i) q^{59} +(0.654861 - 1.75575i) q^{60} +(-0.989821 - 0.142315i) q^{64} +(0.334961 + 0.613435i) q^{67} +(0.654861 + 1.75575i) q^{69} +(-1.84125 - 0.540641i) q^{71} +(-1.75575 + 0.654861i) q^{75} +(0.540641 + 0.841254i) q^{80} +(2.68287 - 0.787761i) q^{81} +(1.19136 - 1.37491i) q^{89} +(-0.959493 - 0.281733i) q^{92} +(2.62312 + 2.62312i) q^{93} +(1.28173 - 0.959493i) q^{97} +(1.04331 + 2.28454i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{3}+O(q^{10}) $ 20 * q - 2 * q^3	$ 20 q - 2 q^{3} - 2 q^{12} - 2 q^{15} + 2 q^{16} + 2 q^{20} + 2 q^{25} - 2 q^{33} - 2 q^{36} - 2 q^{37} + 2 q^{44} - 2 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{53} + 2 q^{55} + 2 q^{60} - 2 q^{67} + 2 q^{69} - 18 q^{71} - 20 q^{75} - 2 q^{81} - 2 q^{92} + 20 q^{97} - 2 q^{99}+O(q^{100}) $ 20 * q - 2 * q^3 - 2 * q^12 - 2 * q^15 + 2 * q^16 + 2 * q^20 + 2 * q^25 - 2 * q^33 - 2 * q^36 - 2 * q^37 + 2 * q^44 - 2 * q^45 - 2 * q^47 + 2 * q^48 + 2 * q^53 + 2 * q^55 + 2 * q^60 - 2 * q^67 + 2 * q^69 - 18 * q^71 - 20 * q^75 - 2 * q^81 - 2 * q^92 + 20 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1265\mathbb{Z}\right)^\times$.

$n$	$166$	$507$	$1036$
$\chi(n)$	$e\left(\frac{1}{22}\right)$	$e\left(\frac{3}{4}\right)$	$-1$

Coefficient data

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

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

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

$2$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$2$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$3$	−1.86912	+	0.133682i	−1.86912	+	0.133682i	−0.959493	−	0.281733i	$-0.909091\pi$
$3$	−1.86912	+	0.133682i	−1.86912	+	0.133682i	−0.909632	+	0.415415i	$0.863636\pi$
$4$	0.540641	−	0.841254i	0.540641	−	0.841254i
$5$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$5$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$6$		0			0
$7$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$7$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$8$		0			0
$9$	2.48594	−	0.357424i	2.48594	−	0.357424i
$10$		0			0
$11$	0.281733	+	0.959493i	0.281733	+	0.959493i
$11$	0.281733	+	0.959493i	0.281733	+	0.959493i
$12$	−0.898064	+	1.64468i	−0.898064	+	1.64468i
$13$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$13$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$14$		0			0
$15$	1.83107	−	0.398326i	1.83107	−	0.398326i
$16$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$17$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$17$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$18$		0			0
$19$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$19$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$20$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$21$		0			0
$22$		0			0
$23$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$23$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$24$		0			0
$25$	0.959493	−	0.281733i	0.959493	−	0.281733i
$26$		0			0
$27$	−2.76767	+	0.602069i	−2.76767	+	0.602069i
$28$		0			0
$29$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$29$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$30$		0			0
$31$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.755750	−	0.654861i	$-0.772727\pi$
$31$	−1.29639	−	1.49611i	−1.29639	−	1.49611i	−0.540641	−	0.841254i	$-0.681818\pi$
$32$		0			0
$33$	−0.654861	−	1.75575i	−0.654861	−	1.75575i
$34$		0			0
$35$		0			0
$36$	1.04331	−	2.28454i	1.04331	−	2.28454i
$37$	1.17116	−	1.56449i	1.17116	−	1.56449i	0.415415	−	0.909632i	$-0.363636\pi$
$37$	1.17116	−	1.56449i	1.17116	−	1.56449i	0.755750	−	0.654861i	$-0.227273\pi$
$38$		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		−0.0713392	−	0.997452i	$-0.522727\pi$
$43$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$44$	0.959493	+	0.281733i	0.959493	+	0.281733i
$45$	−2.40977	+	0.707571i	−2.40977	+	0.707571i
$46$		0			0
$47$	0.847507	−	0.847507i	0.847507	−	0.847507i	−0.142315	−	0.989821i	$-0.545455\pi$
$47$	0.847507	−	0.847507i	0.847507	−	0.847507i	0.989821	+	0.142315i	$0.0454545\pi$
$48$	0.898064	+	1.64468i	0.898064	+	1.64468i
$49$	−0.755750	−	0.654861i	−0.755750	−	0.654861i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.398326	−	0.148568i	−0.398326	−	0.148568i	0.142315	−	0.989821i	$-0.454545\pi$
$53$	−0.398326	−	0.148568i	−0.398326	−	0.148568i	−0.540641	+	0.841254i	$0.681818\pi$
$54$		0			0
$55$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$59$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$60$	0.654861	−	1.75575i	0.654861	−	1.75575i
$61$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$61$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$62$		0			0
$63$		0			0
$64$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$65$		0			0
$66$		0			0
$67$	0.334961	+	0.613435i	0.334961	+	0.613435i	0.989821	−	0.142315i	$-0.0454545\pi$
$67$	0.334961	+	0.613435i	0.334961	+	0.613435i	−0.654861	+	0.755750i	$0.727273\pi$
$68$		0			0
$69$	0.654861	+	1.75575i	0.654861	+	1.75575i
$70$		0			0
$71$	−1.84125	−	0.540641i	−1.84125	−	0.540641i	−0.841254	−	0.540641i	$-0.818182\pi$
$71$	−1.84125	−	0.540641i	−1.84125	−	0.540641i	−1.00000			$\pi$
$72$		0			0
$73$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$73$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$74$		0			0
$75$	−1.75575	+	0.654861i	−1.75575	+	0.654861i
$76$		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	$0.636364\pi$
$80$	0.540641	+	0.841254i	0.540641	+	0.841254i
$81$	2.68287	−	0.787761i	2.68287	−	0.787761i
$82$		0			0
$83$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$83$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.19136	−	1.37491i	1.19136	−	1.37491i	0.281733	−	0.959493i	$-0.409091\pi$
$89$	1.19136	−	1.37491i	1.19136	−	1.37491i	0.909632	−	0.415415i	$-0.136364\pi$
$90$		0			0
$91$		0			0
$92$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$93$	2.62312	+	2.62312i	2.62312	+	2.62312i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.28173	−	0.959493i	1.28173	−	0.959493i	0.281733	−	0.959493i	$-0.409091\pi$
$97$	1.28173	−	0.959493i	1.28173	−	0.959493i	1.00000			$0$
$98$		0			0
$99$	1.04331	+	2.28454i	1.04331	+	2.28454i
$100$	0.281733	−	0.959493i	0.281733	−	0.959493i
$101$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$101$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$102$		0			0
$103$	0.0683785	−	0.125226i	0.0683785	−	0.125226i	−0.841254	−	0.540641i	$-0.818182\pi$
$103$	0.0683785	−	0.125226i	0.0683785	−	0.125226i	0.909632	+	0.415415i	$0.136364\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$107$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$108$	−0.989821	+	2.65381i	−0.989821	+	2.65381i
$109$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$109$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$110$		0			0
$111$	−1.97991	+	3.08080i	−1.97991	+	3.08080i
$112$		0			0
$113$	−1.05195	+	0.574406i	−1.05195	+	0.574406i	−0.909632	−	0.415415i	$-0.863636\pi$
$113$	−1.05195	+	0.574406i	−1.05195	+	0.574406i	−0.142315	+	0.989821i	$0.545455\pi$
$114$		0			0
$115$	0.415415	+	0.909632i	0.415415	+	0.909632i
$116$		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
$124$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$125$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$126$		0			0
$127$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$127$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$128$		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	$0.363636\pi$
$132$	−1.83107	−	0.398326i	−1.83107	−	0.398326i
$133$		0			0
$134$		0			0
$135$	2.65381	−	0.989821i	2.65381	−	0.989821i
$136$		0			0
$137$	1.41061	+	1.41061i	1.41061	+	1.41061i	0.755750	+	0.654861i	$0.227273\pi$
$137$	1.41061	+	1.41061i	1.41061	+	1.41061i	0.654861	+	0.755750i	$0.272727\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$	−1.47080	+	1.69739i	−1.47080	+	1.69739i
$142$		0			0
$143$		0			0
$144$	−1.35782	−	2.11281i	−1.35782	−	2.11281i
$145$		0			0
$146$		0			0
$147$	1.50013	+	1.12299i	1.50013	+	1.12299i
$148$	−0.682956	−	1.83107i	−0.682956	−	1.83107i
$149$		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
$155$	1.49611	+	1.29639i	1.49611	+	1.29639i
$156$		0			0
$157$	−0.300613	−	1.38189i	−0.300613	−	1.38189i	−0.841254	−	0.540641i	$-0.818182\pi$
$157$	−0.300613	−	1.38189i	−0.300613	−	1.38189i	0.540641	−	0.841254i	$-0.318182\pi$
$158$		0			0
$159$	0.764382	+	0.224443i	0.764382	+	0.224443i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.0683785	+	0.125226i	0.0683785	+	0.125226i	0.909632	−	0.415415i	$-0.136364\pi$
$163$	0.0683785	+	0.125226i	0.0683785	+	0.125226i	−0.841254	+	0.540641i	$0.818182\pi$
$164$		0			0
$165$	0.898064	+	1.64468i	0.898064	+	1.64468i
$166$		0			0
$167$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$167$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$168$		0			0
$169$	0.755750	−	0.654861i	0.755750	−	0.654861i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$173$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$174$		0			0
$175$		0			0
$176$	0.755750	−	0.654861i	0.755750	−	0.654861i
$177$	−1.89846	−	0.708089i	−1.89846	−	0.708089i
$178$		0			0
$179$	−1.29639	−	0.186393i	−1.29639	−	0.186393i	−0.540641	−	0.841254i	$-0.681818\pi$
$179$	−1.29639	−	0.186393i	−1.29639	−	0.186393i	−0.755750	+	0.654861i	$0.772727\pi$
$180$	−0.707571	+	2.40977i	−0.707571	+	2.40977i
$181$	0.989821	+	0.857685i	0.989821	+	0.857685i	0.989821	−	0.142315i	$-0.0454545\pi$
$181$	0.989821	+	0.857685i	0.989821	+	0.857685i			1.00000i	$0.5\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.936593	+	1.71524i	−0.936593	+	1.71524i
$186$		0			0
$187$		0			0
$188$	−0.254771	−	1.17116i	−0.254771	−	1.17116i
$189$		0			0
$190$		0			0
$191$	0.755750	−	0.345139i	0.755750	−	0.345139i		−	1.00000i	$-0.5\pi$
$191$	0.755750	−	0.345139i	0.755750	−	0.345139i	0.755750	+	0.654861i	$0.227273\pi$
$192$	1.86912	+	0.133682i	1.86912	+	0.133682i
$193$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$193$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$194$		0			0
$195$		0			0
$196$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$197$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$197$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$198$		0			0
$199$	0.544078	+	0.627899i	0.544078	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$199$	0.544078	+	0.627899i	0.544078	+	0.627899i	−0.415415	+	0.909632i	$0.636364\pi$
$200$		0			0
$201$	−0.708089	−	1.10181i	−0.708089	−	1.10181i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.04331	−	2.28454i	−1.04331	−	2.28454i
$208$		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	$0.181818\pi$
$212$	−0.340335	+	0.254771i	−0.340335	+	0.254771i
$213$	3.51381	+	0.764382i	3.51381	+	0.764382i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$221$		0			0
$222$		0			0
$223$	−0.697148	+	1.86912i	−0.697148	+	1.86912i	−0.281733	+	0.959493i	$0.590909\pi$
$223$	−0.697148	+	1.86912i	−0.697148	+	1.86912i	−0.415415	+	0.909632i	$0.636364\pi$
$224$		0			0
$225$	2.28454	−	1.04331i	2.28454	−	1.04331i
$226$		0			0
$227$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$227$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$228$		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		0.997452	−	0.0713392i	$-0.0227273\pi$
$233$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$234$		0			0
$235$	−0.718267	+	0.959493i	−0.718267	+	0.959493i
$236$	0.909632	−	0.584585i	0.909632	−	0.584585i
$237$		0			0
$238$		0			0
$239$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$239$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$240$	−1.12299	−	1.50013i	−1.12299	−	1.50013i
$241$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$241$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$242$		0			0
$243$	−2.25549	+	0.841254i	−2.25549	+	0.841254i
$244$		0			0
$245$	0.841254	+	0.540641i	0.841254	+	0.540641i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.304632	−	1.03748i	0.304632	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$251$	0.304632	−	1.03748i	0.304632	−	1.03748i	0.959493	−	0.281733i	$-0.0909091\pi$
$252$		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
$257$	−0.415415	+	0.0903680i	−0.415415	+	0.0903680i	−0.415415	−	0.909632i	$-0.636364\pi$
$257$	−0.415415	+	0.0903680i	−0.415415	+	0.0903680i			1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$263$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$264$		0			0
$265$	0.415415	+	0.0903680i	0.415415	+	0.0903680i
$266$		0			0
$267$	−2.04301	+	2.72914i	−2.04301	+	2.72914i
$268$	0.697148	+	0.0498610i	0.697148	+	0.0498610i
$269$	−0.512546	+	0.234072i	−0.512546	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$269$	−0.512546	+	0.234072i	−0.512546	+	0.234072i	0.142315	+	0.989821i	$0.454545\pi$
$270$		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
$275$	0.540641	+	0.841254i	0.540641	+	0.841254i
$276$	1.83107	+	0.398326i	1.83107	+	0.398326i
$277$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$277$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$278$		0			0
$279$	−3.75749	−	3.25588i	−3.75749	−	3.25588i
$280$		0			0
$281$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$281$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$282$		0			0
$283$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$283$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$284$	−1.45027	+	1.25667i	−1.45027	+	1.25667i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.909632	+	0.415415i	0.909632	+	0.415415i
$290$		0			0
$291$	−2.26745	+	1.96476i	−2.26745	+	1.96476i
$292$		0			0
$293$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$293$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$294$		0			0
$295$	−1.03748	−	0.304632i	−1.03748	−	0.304632i
$296$		0			0
$297$	−1.35742	−	2.48594i	−1.35742	−	2.48594i
$298$		0			0
$299$		0			0
$300$	−0.398326	+	1.83107i	−0.398326	+	1.83107i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$307$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$308$		0			0
$309$	−0.111067	+	0.243204i	−0.111067	+	0.243204i
$310$		0			0
$311$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$311$	1.61435	−	0.474017i	1.61435	−	0.474017i	0.959493	+	0.281733i	$0.0909091\pi$
$312$		0			0
$313$	1.40524	+	1.05195i	1.40524	+	1.05195i	0.989821	+	0.142315i	$0.0454545\pi$
$313$	1.40524	+	1.05195i	1.40524	+	1.05195i	0.415415	+	0.909632i	$0.363636\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.959493	−	1.28173i	−0.959493	−	1.28173i	−0.959493	−	0.281733i	$-0.909091\pi$
$317$	−0.959493	−	1.28173i	−0.959493	−	1.28173i		−	1.00000i	$-0.5\pi$
$318$		0			0
$319$		0			0
$320$	1.00000			1.00000
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.787761	−	2.68287i	0.787761	−	2.68287i
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.0801894	−	0.557730i	−0.0801894	−	0.557730i	−0.989821	−	0.142315i	$-0.954545\pi$
$331$	−0.0801894	−	0.557730i	−0.0801894	−	0.557730i	0.909632	−	0.415415i	$-0.136364\pi$
$332$		0			0
$333$	2.35225	−	4.30783i	2.35225	−	4.30783i
$334$		0			0
$335$	−0.418852	−	0.559521i	−0.418852	−	0.559521i
$336$		0			0
$337$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$337$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$338$		0			0
$339$	1.88943	−	1.21426i	1.88943	−	1.21426i
$340$		0			0
$341$	1.07028	−	1.66538i	1.07028	−	1.66538i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−0.898064	−	1.64468i	−0.898064	−	1.64468i
$346$		0			0
$347$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$347$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$348$		0			0
$349$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$349$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.0101786	−	0.142315i	0.0101786	−	0.142315i	−0.989821	−	0.142315i	$-0.954545\pi$
$353$	0.0101786	−	0.142315i	0.0101786	−	0.142315i	1.00000			$0$
$354$		0			0
$355$	1.89945	+	0.273100i	1.89945	+	0.273100i
$356$	−0.512546	−	1.74557i	−0.512546	−	1.74557i
$357$		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
$363$	1.50013	−	1.12299i	1.50013	−	1.12299i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.677760	+	0.677760i	0.677760	+	0.677760i	0.959493	−	0.281733i	$-0.0909091\pi$
$367$	0.677760	+	0.677760i	0.677760	+	0.677760i	−0.281733	+	0.959493i	$0.590909\pi$
$368$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	3.62487	−	0.788543i	3.62487	−	0.788543i
$373$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$373$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$374$		0			0
$375$	1.64468	−	0.898064i	1.64468	−	0.898064i
$376$		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
$383$	0.142315	+	0.0101786i	0.142315	+	0.0101786i	0.142315	−	0.989821i	$-0.454545\pi$
$383$	0.142315	+	0.0101786i	0.142315	+	0.0101786i			1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.114220	−	1.59700i	−0.114220	−	1.59700i
$389$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.415415	−	0.909632i	$-0.636364\pi$
$389$	−1.25667	−	0.368991i	−1.25667	−	0.368991i	−0.841254	+	0.540641i	$0.818182\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	2.48594	+	0.357424i	2.48594	+	0.357424i
$397$	0.373128	−	1.71524i	0.373128	−	1.71524i	−0.281733	−	0.959493i	$-0.590909\pi$
$397$	0.373128	−	1.71524i	0.373128	−	1.71524i	0.654861	−	0.755750i	$-0.272727\pi$
$398$		0			0
$399$		0			0
$400$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$401$	0.258908	+	0.118239i	0.258908	+	0.118239i	0.540641	−	0.841254i	$-0.318182\pi$
$401$	0.258908	+	0.118239i	0.258908	+	0.118239i	−0.281733	+	0.959493i	$0.590909\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	−2.54345	+	1.16155i	−2.54345	+	1.16155i
$406$		0			0
$407$	1.83107	+	0.682956i	1.83107	+	0.682956i
$408$		0			0
$409$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$409$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$410$		0			0
$411$	−2.82518	−	2.44803i	−2.82518	−	2.44803i
$412$	−0.0683785	−	0.125226i	−0.0683785	−	0.125226i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−0.215109	+	1.49611i	−0.215109	+	1.49611i	0.540641	+	0.841254i	$0.318182\pi$
$419$	−0.215109	+	1.49611i	−0.215109	+	1.49611i	−0.755750	+	0.654861i	$0.772727\pi$
$420$		0			0
$421$	−1.74557	+	0.797176i	−1.74557	+	0.797176i	−0.755750	+	0.654861i	$0.772727\pi$
$421$	−1.74557	+	0.797176i	−1.74557	+	0.797176i	−0.989821	+	0.142315i	$0.954545\pi$
$422$		0			0
$423$	1.80393	−	2.40977i	1.80393	−	2.40977i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$431$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$432$	1.69739	+	2.26745i	1.69739	+	2.26745i
$433$	−1.94931	+	0.424047i	−1.94931	+	0.424047i	−0.959493	+	0.281733i	$0.909091\pi$
$433$	−1.94931	+	0.424047i	−1.94931	+	0.424047i	−0.989821	+	0.142315i	$0.954545\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$439$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$440$		0			0
$441$	−2.11281	−	1.35782i	−2.11281	−	1.35782i
$442$		0			0
$443$	−1.71524	−	0.373128i	−1.71524	−	0.373128i	−0.755750	−	0.654861i	$-0.772727\pi$
$443$	−1.71524	−	0.373128i	−1.71524	−	0.373128i	−0.959493	+	0.281733i	$0.909091\pi$
$444$	1.52131	+	3.33121i	1.52131	+	3.33121i
$445$	−0.983568	+	1.53046i	−0.983568	+	1.53046i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.0801894	+	0.273100i	0.0801894	+	0.273100i	0.989821	−	0.142315i	$-0.0454545\pi$
$449$	0.0801894	+	0.273100i	0.0801894	+	0.273100i	−0.909632	+	0.415415i	$0.863636\pi$
$450$		0			0
$451$		0			0
$452$	−0.0855040	+	1.19550i	−0.0855040	+	1.19550i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$457$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$458$		0			0
$459$		0			0
$460$	0.989821	+	0.142315i	0.989821	+	0.142315i
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	1.59700	−	0.114220i	1.59700	−	0.114220i	0.755750	−	0.654861i	$-0.227273\pi$
$463$	1.59700	−	0.114220i	1.59700	−	0.114220i	0.841254	+	0.540641i	$0.181818\pi$
$464$		0			0
$465$	−2.96973	−	2.22311i	−2.96973	−	2.22311i
$466$		0			0
$467$	0.559521	−	1.50013i	0.559521	−	1.50013i	−0.281733	−	0.959493i	$-0.590909\pi$
$467$	0.559521	−	1.50013i	0.559521	−	1.50013i	0.841254	−	0.540641i	$-0.181818\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	0.746618	+	2.54275i	0.746618	+	2.54275i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.04331	−	0.226959i	−1.04331	−	0.226959i
$478$		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
$484$			1.00000i			1.00000i
$485$	−1.13214	+	1.13214i	−1.13214	+	1.13214i
$486$		0			0
$487$	−0.682956	+	0.148568i	−0.682956	+	0.148568i	−0.540641	−	0.841254i	$-0.681818\pi$
$487$	−0.682956	+	0.148568i	−0.682956	+	0.148568i	−0.142315	+	0.989821i	$0.545455\pi$
$488$		0			0
$489$	−0.144548	−	0.224922i	−0.144548	−	0.224922i
$490$		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
$495$	−1.35782	−	2.11281i	−1.35782	−	2.11281i
$496$	−0.822373	+	1.80075i	−0.822373	+	1.80075i
$497$		0			0
$498$		0			0
$499$	1.65486	−	0.755750i	1.65486	−	0.755750i	0.654861	−	0.755750i	$-0.272727\pi$
$499$	1.65486	−	0.755750i	1.65486	−	0.755750i	1.00000			$0$
$500$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$501$		0			0
$502$		0			0
$503$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$503$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.32505	+	1.32505i	−1.32505	+	1.32505i
$508$		0			0
$509$	1.37491	+	1.19136i	1.37491	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$509$	1.37491	+	1.19136i	1.37491	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−0.0498610	+	0.133682i	−0.0498610	+	0.133682i
$516$		0			0
$517$	1.05195	+	0.574406i	1.05195	+	0.574406i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.27155	+	1.10181i	−1.27155	+	1.10181i	−0.281733	+	0.959493i	$0.590909\pi$
$521$	−1.27155	+	1.10181i	−1.27155	+	1.10181i	−0.989821	+	0.142315i	$0.954545\pi$
$522$		0			0
$523$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$523$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−1.32505	+	1.32505i	−1.32505	+	1.32505i
$529$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$530$		0			0
$531$	2.60564	+	0.765084i	2.60564	+	0.765084i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	2.44803	+	0.175087i	2.44803	+	0.175087i
$538$		0			0
$539$	0.415415	−	0.909632i	0.415415	−	0.909632i
$540$	0.602069	−	2.76767i	0.602069	−	2.76767i
$541$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$541$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$542$		0			0
$543$	−1.96476	−	1.47080i	−1.96476	−	1.47080i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$547$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$548$	1.94931	−	0.424047i	1.94931	−	0.424047i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	1.52131	−	3.33121i	1.52131	−	3.33121i
$556$		0			0
$557$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$557$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$563$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$564$	0.632763	+	2.15499i	0.632763	+	2.15499i
$565$	0.959493	−	0.718267i	0.959493	−	0.718267i
$566$		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		0.540641	−	0.841254i	$-0.318182\pi$
$571$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$572$		0			0
$573$	−1.36645	+	0.746139i	−1.36645	+	0.746139i
$574$		0			0
$575$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$576$	−2.51150			−2.51150
$577$	−1.71524	+	0.936593i	−1.71524	+	0.936593i	−0.755750	+	0.654861i	$0.772727\pi$
$577$	−1.71524	+	0.936593i	−1.71524	+	0.936593i	−0.959493	+	0.281733i	$0.909091\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	0.0303285	−	0.424047i	0.0303285	−	0.424047i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.334961	+	0.613435i	−0.334961	+	0.613435i	−0.989821	−	0.142315i	$-0.954545\pi$
$587$	−0.334961	+	0.613435i	−0.334961	+	0.613435i	0.654861	+	0.755750i	$0.272727\pi$
$588$	1.75575	−	0.654861i	1.75575	−	0.654861i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.90963	−	0.415415i	−1.90963	−	0.415415i
$593$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$593$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.10089	−	1.10089i	−1.10089	−	1.10089i
$598$		0			0
$599$		−	0.830830i		−	0.830830i	−0.909632	−	0.415415i	$-0.863636\pi$
$599$		−	0.830830i		−	0.830830i	0.909632	−	0.415415i	$-0.136364\pi$
$600$		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
$603$	1.05195	+	1.40524i	1.05195	+	1.40524i
$604$		0			0
$605$	0.755750	−	0.654861i	0.755750	−	0.654861i
$606$		0			0
$607$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$607$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$613$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.148568	−	0.682956i	−0.148568	−	0.682956i	−0.989821	−	0.142315i	$-0.954545\pi$
$617$	−0.148568	−	0.682956i	−0.148568	−	0.682956i	0.841254	−	0.540641i	$-0.181818\pi$
$618$		0			0
$619$	−0.540641	−	0.158746i	−0.540641	−	0.158746i		−	1.00000i	$-0.5\pi$
$619$	−0.540641	−	0.158746i	−0.540641	−	0.158746i	−0.540641	+	0.841254i	$0.681818\pi$
$620$	1.89945	−	0.557730i	1.89945	−	0.557730i
$621$	1.35742	+	2.48594i	1.35742	+	2.48594i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.841254	−	0.540641i	0.841254	−	0.540641i
$626$		0			0
$627$		0			0
$628$	−1.32505	−	0.494217i	−1.32505	−	0.494217i
$629$		0			0
$630$		0			0
$631$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$631$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	0.602069	−	0.521696i	0.602069	−	0.521696i
$637$		0			0
$638$		0			0
$639$	−4.77048	−	0.685891i	−4.77048	−	0.685891i
$640$		0			0
$641$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.989821	+	0.142315i	$0.0454545\pi$
$641$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$642$		0			0
$643$	−0.847507	+	0.847507i	−0.847507	+	0.847507i	−0.989821	−	0.142315i	$-0.954545\pi$
$643$	−0.847507	+	0.847507i	−0.847507	+	0.847507i	0.142315	+	0.989821i	$0.454545\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	0.142315	+	1.98982i	0.142315	+	1.98982i	0.142315	+	0.989821i	$0.454545\pi$
$647$	0.142315	+	1.98982i	0.142315	+	1.98982i			1.00000i	$0.5\pi$
$648$		0			0
$649$	−0.153882	+	1.07028i	−0.153882	+	1.07028i
$650$		0			0
$651$		0			0
$652$	0.142315	+	0.0101786i	0.142315	+	0.0101786i
$653$	−1.19550	+	1.59700i	−1.19550	+	1.59700i	−0.540641	+	0.841254i	$0.681818\pi$
$653$	−1.19550	+	1.59700i	−1.19550	+	1.59700i	−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	$0.272727\pi$
$660$	1.86912	+	0.133682i	1.86912	+	0.133682i
$661$	−0.817178	−	1.27155i	−0.817178	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$661$	−0.817178	−	1.27155i	−0.817178	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.05319	−	3.58682i	1.05319	−	3.58682i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$673$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$674$		0			0
$675$	−2.48594	+	1.35742i	−2.48594	+	1.35742i
$676$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$677$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$677$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.697148	−	1.86912i	0.697148	−	1.86912i	0.281733	−	0.959493i	$-0.409091\pi$
$683$	0.697148	−	1.86912i	0.697148	−	1.86912i	0.415415	−	0.909632i	$-0.363636\pi$
$684$		0			0
$685$	−1.59700	−	1.19550i	−1.59700	−	1.19550i
$686$		0			0
$687$	0.532008	−	0.0380500i	0.532008	−	0.0380500i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	1.30972			1.30972			0.654861	−	0.755750i	$-0.272727\pi$
$691$	1.30972			1.30972			0.654861	+	0.755750i	$0.272727\pi$
$692$		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		−0.281733	−	0.959493i	$-0.590909\pi$
$701$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$702$		0			0
$703$		0			0
$704$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$705$	1.21426	−	1.88943i	1.21426	−	1.88943i
$706$		0			0
$707$		0			0
$708$	−1.62207	+	1.21426i	−1.62207	+	1.21426i
$709$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.989821	+	0.142315i	$0.0454545\pi$
$709$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.540641	+	0.841254i	$0.318182\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.07028	+	1.66538i	−1.07028	+	1.66538i
$714$		0			0
$715$		0			0
$716$	−0.857685	+	0.989821i	−0.857685	+	0.989821i
$717$		0			0
$718$		0			0
$719$	0.449181	+	0.698939i	0.449181	+	0.698939i	0.989821	−	0.142315i	$-0.0454545\pi$
$719$	0.449181	+	0.698939i	0.449181	+	0.698939i	−0.540641	+	0.841254i	$0.681818\pi$
$720$	1.64468	+	1.89806i	1.64468	+	1.89806i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	1.25667	−	0.368991i	1.25667	−	0.368991i
$725$		0			0
$726$		0			0
$727$	0.254771	−	0.340335i	0.254771	−	0.340335i	−0.654861	−	0.755750i	$-0.727273\pi$
$727$	0.254771	−	0.340335i	0.254771	−	0.340335i	0.909632	+	0.415415i	$0.136364\pi$
$728$		0			0
$729$	1.55988	−	0.712373i	1.55988	−	0.712373i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$733$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$734$		0			0
$735$	−1.64468	−	0.898064i	−1.64468	−	0.898064i
$736$		0			0
$737$	−0.494217	+	0.494217i	−0.494217	+	0.494217i
$738$		0			0
$739$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$739$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$740$	0.936593	+	1.71524i	0.936593	+	1.71524i
$741$		0			0
$742$		0			0
$743$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$743$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$751$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$752$	−1.12299	−	0.418852i	−1.12299	−	0.418852i
$753$	−0.430703	+	1.97991i	−0.430703	+	1.97991i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.956056	+	1.75089i	0.956056	+	1.75089i	0.540641	+	0.841254i	$0.318182\pi$
$757$	0.956056	+	1.75089i	0.956056	+	1.75089i	0.415415	+	0.909632i	$0.363636\pi$
$758$		0			0
$759$	−1.50013	+	1.12299i	−1.50013	+	1.12299i
$760$		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
$768$	1.12299	−	1.50013i	1.12299	−	1.50013i
$769$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$769$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$770$		0			0
$771$	0.764382	−	0.224443i	0.764382	−	0.224443i
$772$		0			0
$773$	−1.28173	−	0.959493i	−1.28173	−	0.959493i	−0.281733	−	0.959493i	$-0.590909\pi$
$773$	−1.28173	−	0.959493i	−1.28173	−	0.959493i	−1.00000			$\pi$
$774$		0			0
$775$	−1.66538	−	1.07028i	−1.66538	−	1.07028i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		−	1.91899i		−	1.91899i
$782$		0			0
$783$		0			0
$784$	−0.281733	+	0.959493i	−0.281733	+	0.959493i
$785$	0.494217	+	1.32505i	0.494217	+	1.32505i
$786$		0			0
$787$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$787$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	−0.788543	−	0.113375i	−0.788543	−	0.113375i
$796$	0.822373	−	0.118239i	0.822373	−	0.118239i
$797$	−0.0498610	+	0.697148i	−0.0498610	+	0.697148i	0.909632	+	0.415415i	$0.136364\pi$
$797$	−0.0498610	+	0.697148i	−0.0498610	+	0.697148i	−0.959493	+	0.281733i	$0.909091\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	2.47023	−	3.84375i	2.47023	−	3.84375i
$802$		0			0
$803$		0			0
$804$	−1.30972			−1.30972
$805$		0			0
$806$		0			0
$807$	0.926721	−	0.506028i	0.926721	−	0.506028i
$808$		0			0
$809$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$809$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$810$		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
$815$	−0.0855040	−	0.114220i	−0.0855040	−	0.114220i
$816$		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
$823$	0.340335	−	0.254771i	0.340335	−	0.254771i	−0.415415	−	0.909632i	$-0.636364\pi$
$823$	0.340335	−	0.254771i	0.340335	−	0.254771i	0.755750	+	0.654861i	$0.227273\pi$
$824$		0			0
$825$	−1.12299	−	1.50013i	−1.12299	−	1.50013i
$826$		0			0
$827$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$827$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$828$	−2.48594	−	0.357424i	−2.48594	−	0.357424i
$829$			1.30972i			1.30972i	0.755750	+	0.654861i	$0.227273\pi$
$829$			1.30972i			1.30972i	−0.755750	+	0.654861i	$0.772727\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.48874	+	3.36023i	4.48874	+	3.36023i
$838$		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
$845$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$846$		0			0
$847$		0			0
$848$	0.0303285	+	0.424047i	0.0303285	+	0.424047i
$849$		0			0
$850$		0			0
$851$	−1.83107	−	0.682956i	−1.83107	−	0.682956i
$852$	2.54275	−	2.54275i	2.54275	−	2.54275i
$853$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$853$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$857$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$858$		0			0
$859$	−0.627899	+	0.544078i	−0.627899	+	0.544078i	−0.909632	−	0.415415i	$-0.863636\pi$
$859$	−0.627899	+	0.544078i	−0.627899	+	0.544078i	0.281733	+	0.959493i	$0.409091\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.54064	−	0.841254i	−1.54064	−	0.841254i	−0.540641	−	0.841254i	$-0.681818\pi$
$863$	−1.54064	−	0.841254i	−1.54064	−	0.841254i	−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−1.75575	−	0.654861i	−1.75575	−	0.654861i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	2.84336	−	2.84336i	2.84336	−	2.84336i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$877$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$878$		0			0
$879$		0			0
$880$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$881$	1.37491	−	0.627899i	1.37491	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$881$	1.37491	−	0.627899i	1.37491	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$882$		0			0
$883$	0.959493	−	1.28173i	0.959493	−	1.28173i		−	1.00000i	$-0.5\pi$
$883$	0.959493	−	1.28173i	0.959493	−	1.28173i	0.959493	−	0.281733i	$-0.0909091\pi$
$884$		0			0
$885$	1.97991	+	0.430703i	1.97991	+	0.430703i
$886$		0			0
$887$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$887$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	1.51170	+	2.35225i	1.51170	+	2.35225i
$892$	1.19550	+	1.59700i	1.19550	+	1.59700i
$893$		0			0
$894$		0			0
$895$	1.30972			1.30972
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	0.357424	−	2.48594i	0.357424	−	2.48594i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−1.10181	−	0.708089i	−1.10181	−	0.708089i
$906$		0			0
$907$	0.898064	−	0.334961i	0.898064	−	0.334961i	0.142315	−	0.989821i	$-0.454545\pi$
$907$	0.898064	−	0.334961i	0.898064	−	0.334961i	0.755750	+	0.654861i	$0.227273\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.540641	−	0.841254i	$-0.318182\pi$
$911$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.755750	+	0.654861i	$0.227273\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−0.153882	+	0.239446i	−0.153882	+	0.239446i
$917$		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
$925$	0.682956	−	1.83107i	0.682956	−	1.83107i
$926$		0			0
$927$	0.125226	−	0.335743i	0.125226	−	0.335743i
$928$		0			0
$929$	−1.80075	+	0.258908i	−1.80075	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$929$	−1.80075	+	0.258908i	−1.80075	+	0.258908i	−0.841254	+	0.540641i	$0.818182\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−2.95406	+	1.10181i	−2.95406	+	1.10181i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$937$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$938$		0			0
$939$	−2.76719	−	1.77836i	−2.76719	−	1.77836i
$940$	0.418852	+	1.12299i	0.418852	+	1.12299i
$941$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$941$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$942$		0			0
$943$		0			0
$944$		−	1.08128i		−	1.08128i
$945$		0			0
$946$		0			0
$947$	0.139418	−	0.0303285i	0.139418	−	0.0303285i	−0.142315	−	0.989821i	$-0.545455\pi$
$947$	0.139418	−	0.0303285i	0.139418	−	0.0303285i	0.281733	+	0.959493i	$0.409091\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	1.96476	+	2.26745i	1.96476	+	2.26745i
$952$		0			0
$953$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$953$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$954$		0			0
$955$	−0.698939	+	0.449181i	−0.698939	+	0.449181i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−1.86912	+	0.133682i	−1.86912	+	0.133682i
$961$	−0.415415	+	2.88927i	−0.415415	+	2.88927i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$967$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.29639	+	0.186393i	1.29639	+	0.186393i	0.755750	−	0.654861i	$-0.227273\pi$
$971$	1.29639	+	0.186393i	1.29639	+	0.186393i	0.540641	+	0.841254i	$0.318182\pi$
$972$	−0.511701	+	2.35225i	−0.511701	+	2.35225i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.125226	+	0.0683785i	0.125226	+	0.0683785i	0.540641	−	0.841254i	$-0.318182\pi$
$977$	0.125226	+	0.0683785i	0.125226	+	0.0683785i	−0.415415	+	0.909632i	$0.636364\pi$
$978$		0			0
$979$	1.65486	+	0.755750i	1.65486	+	0.755750i
$980$	0.909632	−	0.415415i	0.909632	−	0.415415i
$981$		0			0
$982$		0			0
$983$	−0.415415	+	1.90963i	−0.415415	+	1.90963i			1.00000i	$0.5\pi$
$983$	−0.415415	+	1.90963i	−0.415415	+	1.90963i	−0.415415	+	0.909632i	$0.636364\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.540641	−	0.158746i	−0.540641	−	0.158746i		−	1.00000i	$-0.5\pi$
$991$	−0.540641	−	0.158746i	−0.540641	−	0.158746i	−0.540641	+	0.841254i	$0.681818\pi$
$992$		0			0
$993$	0.224443	+	1.03175i	0.224443	+	1.03175i
$994$		0			0
$995$	−0.627899	−	0.544078i	−0.627899	−	0.544078i
$996$		0			0
$997$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$997$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$998$		0			0
$999$	−2.29946	+	5.03512i	−2.29946	+	5.03512i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1265.1.bi.a.373.1	✓	20
5.2	odd	4		1265.1.bi.b.1132.1	yes	20
11.10	odd	2	CM	1265.1.bi.a.373.1	✓	20
23.14	odd	22		1265.1.bi.b.428.1	yes	20
55.32	even	4		1265.1.bi.b.1132.1	yes	20
115.37	even	44	inner	1265.1.bi.a.1187.1	yes	20
253.175	even	22		1265.1.bi.b.428.1	yes	20
1265.1187	odd	44	inner	1265.1.bi.a.1187.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1265.1.bi.a.373.1	✓	20	1.1	even	1	trivial
1265.1.bi.a.373.1	✓	20	11.10	odd	2	CM
1265.1.bi.a.1187.1	yes	20	115.37	even	44	inner
1265.1.bi.a.1187.1	yes	20	1265.1187	odd	44	inner
1265.1.bi.b.428.1	yes	20	23.14	odd	22
1265.1.bi.b.428.1	yes	20	253.175	even	22
1265.1.bi.b.1132.1	yes	20	5.2	odd	4
1265.1.bi.b.1132.1	yes	20	55.32	even	4

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:	\( N \)	\(=\)	\( 1265 = 5 \cdot 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1265.bi (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.86912 + 0.133682i) q^{3} +(0.540641 - 0.841254i) q^{4} +(-0.989821 + 0.142315i) q^{5} +(2.48594 - 0.357424i) q^{9} +O(q^{10})\)	\(q+(-1.86912 + 0.133682i) q^{3} +(0.540641 - 0.841254i) q^{4} +(-0.989821 + 0.142315i) q^{5} +(2.48594 - 0.357424i) q^{9} +(0.281733 + 0.959493i) q^{11} +(-0.898064 + 1.64468i) q^{12} +(1.83107 - 0.398326i) q^{15} +(-0.415415 - 0.909632i) q^{16} +(-0.415415 + 0.909632i) q^{20} +(-0.281733 - 0.959493i) q^{23} +(0.959493 - 0.281733i) q^{25} +(-2.76767 + 0.602069i) q^{27} +(-1.29639 - 1.49611i) q^{31} +(-0.654861 - 1.75575i) q^{33} +(1.04331 - 2.28454i) q^{36} +(1.17116 - 1.56449i) q^{37} +(0.959493 + 0.281733i) q^{44} +(-2.40977 + 0.707571i) q^{45} +(0.847507 - 0.847507i) q^{47} +(0.898064 + 1.64468i) q^{48} +(-0.755750 - 0.654861i) q^{49} +(-0.398326 - 0.148568i) q^{53} +(-0.415415 - 0.909632i) q^{55} +(0.983568 + 0.449181i) q^{59} +(0.654861 - 1.75575i) q^{60} +(-0.989821 - 0.142315i) q^{64} +(0.334961 + 0.613435i) q^{67} +(0.654861 + 1.75575i) q^{69} +(-1.84125 - 0.540641i) q^{71} +(-1.75575 + 0.654861i) q^{75} +(0.540641 + 0.841254i) q^{80} +(2.68287 - 0.787761i) q^{81} +(1.19136 - 1.37491i) q^{89} +(-0.959493 - 0.281733i) q^{92} +(2.62312 + 2.62312i) q^{93} +(1.28173 - 0.959493i) q^{97} +(1.04331 + 2.28454i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{3}+O(q^{10}) \) 20 * q - 2 * q^3	\( 20 q - 2 q^{3} - 2 q^{12} - 2 q^{15} + 2 q^{16} + 2 q^{20} + 2 q^{25} - 2 q^{33} - 2 q^{36} - 2 q^{37} + 2 q^{44} - 2 q^{45} - 2 q^{47} + 2 q^{48} + 2 q^{53} + 2 q^{55} + 2 q^{60} - 2 q^{67} + 2 q^{69} - 18 q^{71} - 20 q^{75} - 2 q^{81} - 2 q^{92} + 20 q^{97} - 2 q^{99}+O(q^{100}) \) 20 * q - 2 * q^3 - 2 * q^12 - 2 * q^15 + 2 * q^16 + 2 * q^20 + 2 * q^25 - 2 * q^33 - 2 * q^36 - 2 * q^37 + 2 * q^44 - 2 * q^45 - 2 * q^47 + 2 * q^48 + 2 * q^53 + 2 * q^55 + 2 * q^60 - 2 * q^67 + 2 * q^69 - 18 * q^71 - 20 * q^75 - 2 * q^81 - 2 * q^92 + 20 * q^97 - 2 * q^99

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