LMFDB - Embedded newform 3963.1.bi.a.2666.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3963,1,Mod(71,3963)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3963, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 1]))

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

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

chi := DirichletCharacter("3963.71");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3963 = 3 \cdot 1321 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3963.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:	$1.97779464506$
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:	$\Z[a_1, \ldots, a_{7}]$
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			2666.1
Root			$0.989821 + 0.142315i$ of defining polynomial
Character	$\chi$	$=$	3963.2666
Dual form			3963.1.bi.a.3908.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.959493 - 0.281733i) q^{3} -1.00000 q^{4} +(-0.418852 - 1.12299i) q^{7} +(0.841254 - 0.540641i) q^{9} +O(q^{10})$	$q+(0.959493 - 0.281733i) q^{3} -1.00000 q^{4} +(-0.418852 - 1.12299i) q^{7} +(0.841254 - 0.540641i) q^{9} +(-0.959493 + 0.281733i) q^{12} +(-0.574406 + 1.05195i) q^{13} +1.00000 q^{16} +(0.613435 - 1.64468i) q^{19} +(-0.718267 - 0.959493i) q^{21} +(-0.415415 + 0.909632i) q^{25} +(0.654861 - 0.755750i) q^{27} +(0.418852 + 1.12299i) q^{28} +(0.909632 - 0.584585i) q^{31} +(-0.841254 + 0.540641i) q^{36} +(-0.281733 - 1.95949i) q^{37} +(-0.254771 + 1.17116i) q^{39} +(-1.74557 - 0.512546i) q^{43} +(0.959493 - 0.281733i) q^{48} +(-0.329911 + 0.285870i) q^{49} +(0.574406 - 1.05195i) q^{52} +(0.125226 - 1.75089i) q^{57} +(0.133682 - 1.86912i) q^{61} +(-0.959493 - 0.718267i) q^{63} -1.00000 q^{64} +(1.17116 + 1.56449i) q^{67} +(-1.90963 - 0.415415i) q^{73} +(-0.142315 + 0.989821i) q^{75} +(-0.613435 + 1.64468i) q^{76} +(0.0855040 - 1.19550i) q^{79} +(0.415415 - 0.909632i) q^{81} +(0.718267 + 0.959493i) q^{84} +(1.42191 + 0.204440i) q^{91} +(0.708089 - 0.817178i) q^{93} +(-0.148568 + 0.682956i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} - 20 q^{4} - 2 q^{7} - 2 q^{9}+O(q^{10}) $ 20 * q + 2 * q^3 - 20 * q^4 - 2 * q^7 - 2 * q^9	$ 20 q + 2 q^{3} - 20 q^{4} - 2 q^{7} - 2 q^{9} - 2 q^{12} - 2 q^{13} + 20 q^{16} - 2 q^{19} - 20 q^{21} + 2 q^{25} + 2 q^{27} + 2 q^{28} + 2 q^{36} + 2 q^{39} + 2 q^{48} + 2 q^{52} + 2 q^{57} - 2 q^{61} - 2 q^{63} - 20 q^{64} - 2 q^{67} - 20 q^{73} - 2 q^{75} + 2 q^{76} - 2 q^{79} - 2 q^{81} + 20 q^{84} - 2 q^{97}+O(q^{100}) $ 20 * q + 2 * q^3 - 20 * q^4 - 2 * q^7 - 2 * q^9 - 2 * q^12 - 2 * q^13 + 20 * q^16 - 2 * q^19 - 20 * q^21 + 2 * q^25 + 2 * q^27 + 2 * q^28 + 2 * q^36 + 2 * q^39 + 2 * q^48 + 2 * q^52 + 2 * q^57 - 2 * q^61 - 2 * q^63 - 20 * q^64 - 2 * q^67 - 20 * q^73 - 2 * q^75 + 2 * q^76 - 2 * q^79 - 2 * q^81 + 20 * q^84 - 2 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3963\mathbb{Z}\right)^\times$.

$n$	$13$	$1322$
$\chi(n)$	$e\left(\frac{29}{44}\right)$	$-1$

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$	0.959493	−	0.281733i	0.959493	−	0.281733i
$3$	0.959493	−	0.281733i	0.959493	−	0.281733i
$4$	−1.00000			−1.00000
$5$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$5$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$6$		0			0
$7$	−0.418852	−	1.12299i	−0.418852	−	1.12299i	−0.959493	−	0.281733i	$-0.909091\pi$
$7$	−0.418852	−	1.12299i	−0.418852	−	1.12299i	0.540641	−	0.841254i	$-0.318182\pi$
$8$		0			0
$9$	0.841254	−	0.540641i	0.841254	−	0.540641i
$10$		0			0
$11$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$11$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$12$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$13$	−0.574406	+	1.05195i	−0.574406	+	1.05195i	0.415415	+	0.909632i	$0.363636\pi$
$13$	−0.574406	+	1.05195i	−0.574406	+	1.05195i	−0.989821	+	0.142315i	$0.954545\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			1.00000
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$	0.613435	−	1.64468i	0.613435	−	1.64468i	−0.142315	−	0.989821i	$-0.545455\pi$
$19$	0.613435	−	1.64468i	0.613435	−	1.64468i	0.755750	−	0.654861i	$-0.227273\pi$
$20$		0			0
$21$	−0.718267	−	0.959493i	−0.718267	−	0.959493i
$22$		0			0
$23$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$23$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$24$		0			0
$25$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$26$		0			0
$27$	0.654861	−	0.755750i	0.654861	−	0.755750i
$28$	0.418852	+	1.12299i	0.418852	+	1.12299i
$29$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$29$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$30$		0			0
$31$	0.909632	−	0.584585i	0.909632	−	0.584585i		−	1.00000i	$-0.5\pi$
$31$	0.909632	−	0.584585i	0.909632	−	0.584585i	0.909632	+	0.415415i	$0.136364\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$37$	−0.281733	−	1.95949i	−0.281733	−	1.95949i	−0.281733	−	0.959493i	$-0.590909\pi$
$37$	−0.281733	−	1.95949i	−0.281733	−	1.95949i		−	1.00000i	$-0.5\pi$
$38$		0			0
$39$	−0.254771	+	1.17116i	−0.254771	+	1.17116i
$40$		0			0
$41$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$41$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$42$		0			0
$43$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.755750	−	0.654861i	$-0.772727\pi$
$43$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.989821	+	0.142315i	$0.954545\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$47$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$48$	0.959493	−	0.281733i	0.959493	−	0.281733i
$49$	−0.329911	+	0.285870i	−0.329911	+	0.285870i
$50$		0			0
$51$		0			0
$52$	0.574406	−	1.05195i	0.574406	−	1.05195i
$53$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$53$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.125226	−	1.75089i	0.125226	−	1.75089i
$58$		0			0
$59$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$59$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$60$		0			0
$61$	0.133682	−	1.86912i	0.133682	−	1.86912i	−0.281733	−	0.959493i	$-0.590909\pi$
$61$	0.133682	−	1.86912i	0.133682	−	1.86912i	0.415415	−	0.909632i	$-0.363636\pi$
$62$		0			0
$63$	−0.959493	−	0.718267i	−0.959493	−	0.718267i
$64$	−1.00000			−1.00000
$65$		0			0
$66$		0			0
$67$	1.17116	+	1.56449i	1.17116	+	1.56449i	0.755750	+	0.654861i	$0.227273\pi$
$67$	1.17116	+	1.56449i	1.17116	+	1.56449i	0.415415	+	0.909632i	$0.363636\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$71$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$72$		0			0
$73$	−1.90963	−	0.415415i	−1.90963	−	0.415415i	−0.909632	−	0.415415i	$-0.863636\pi$
$73$	−1.90963	−	0.415415i	−1.90963	−	0.415415i	−1.00000			$\pi$
$74$		0			0
$75$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$76$	−0.613435	+	1.64468i	−0.613435	+	1.64468i
$77$		0			0
$78$		0			0
$79$	0.0855040	−	1.19550i	0.0855040	−	1.19550i	−0.755750	−	0.654861i	$-0.772727\pi$
$79$	0.0855040	−	1.19550i	0.0855040	−	1.19550i	0.841254	−	0.540641i	$-0.181818\pi$
$80$		0			0
$81$	0.415415	−	0.909632i	0.415415	−	0.909632i
$82$		0			0
$83$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$83$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$84$	0.718267	+	0.959493i	0.718267	+	0.959493i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$89$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$90$		0			0
$91$	1.42191	+	0.204440i	1.42191	+	0.204440i
$92$		0			0
$93$	0.708089	−	0.817178i	0.708089	−	0.817178i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.148568	+	0.682956i	−0.148568	+	0.682956i	0.841254	+	0.540641i	$0.181818\pi$
$97$	−0.148568	+	0.682956i	−0.148568	+	0.682956i	−0.989821	+	0.142315i	$0.954545\pi$
$98$		0			0
$99$		0			0
$100$	0.415415	−	0.909632i	0.415415	−	0.909632i
$101$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$101$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$102$		0			0
$103$	0.936593	−	1.71524i	0.936593	−	1.71524i	0.281733	−	0.959493i	$-0.409091\pi$
$103$	0.936593	−	1.71524i	0.936593	−	1.71524i	0.654861	−	0.755750i	$-0.272727\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$107$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$108$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$109$	−1.13214	+	1.13214i	−1.13214	+	1.13214i	−0.142315	+	0.989821i	$0.545455\pi$
$109$	−1.13214	+	1.13214i	−1.13214	+	1.13214i	−0.989821	+	0.142315i	$0.954545\pi$
$110$		0			0
$111$	−0.822373	−	1.80075i	−0.822373	−	1.80075i
$112$	−0.418852	−	1.12299i	−0.418852	−	1.12299i
$113$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$113$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.0855040	+	1.19550i	0.0855040	+	1.19550i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$122$		0			0
$123$		0			0
$124$	−0.909632	+	0.584585i	−0.909632	+	0.584585i
$125$		0			0
$126$		0			0
$127$	0.0683785	+	0.125226i	0.0683785	+	0.125226i	0.909632	−	0.415415i	$-0.136364\pi$
$127$	0.0683785	+	0.125226i	0.0683785	+	0.125226i	−0.841254	+	0.540641i	$0.818182\pi$
$128$		0			0
$129$	−1.81926			−1.81926
$130$		0			0
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$	−2.10389			−2.10389
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$137$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$138$		0			0
$139$	0.559521	+	1.50013i	0.559521	+	1.50013i	0.841254	+	0.540641i	$0.181818\pi$
$139$	0.559521	+	1.50013i	0.559521	+	1.50013i	−0.281733	+	0.959493i	$0.590909\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.841254	−	0.540641i	0.841254	−	0.540641i
$145$		0			0
$146$		0			0
$147$	−0.236009	+	0.367237i	−0.236009	+	0.367237i
$148$	0.281733	+	1.95949i	0.281733	+	1.95949i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	1.19550	+	1.59700i	1.19550	+	1.59700i	0.654861	+	0.755750i	$0.272727\pi$
$151$	1.19550	+	1.59700i	1.19550	+	1.59700i	0.540641	+	0.841254i	$0.318182\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	0.254771	−	1.17116i	0.254771	−	1.17116i
$157$	−0.114220	−	1.59700i	−0.114220	−	1.59700i	−0.654861	−	0.755750i	$-0.727273\pi$
$157$	−0.114220	−	1.59700i	−0.114220	−	1.59700i	0.540641	−	0.841254i	$-0.318182\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.300613	+	0.300613i	−0.300613	+	0.300613i	−0.841254	−	0.540641i	$-0.818182\pi$
$163$	−0.300613	+	0.300613i	−0.300613	+	0.300613i	0.540641	+	0.841254i	$0.318182\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$167$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$168$		0			0
$169$	−0.236009	−	0.367237i	−0.236009	−	0.367237i
$170$		0			0
$171$	−0.373128	−	1.71524i	−0.373128	−	1.71524i
$172$	1.74557	+	0.512546i	1.74557	+	0.512546i
$173$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$173$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$174$		0			0
$175$	1.19550	+	0.0855040i	1.19550	+	0.0855040i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$179$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$180$		0			0
$181$	1.40524	−	0.767317i	1.40524	−	0.767317i	0.415415	−	0.909632i	$-0.363636\pi$
$181$	1.40524	−	0.767317i	1.40524	−	0.767317i	0.989821	+	0.142315i	$0.0454545\pi$
$182$		0			0
$183$	−0.398326	−	1.83107i	−0.398326	−	1.83107i
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.12299	−	0.418852i	−1.12299	−	0.418852i
$190$		0			0
$191$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$191$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$192$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$193$	0.0683785	+	0.125226i	0.0683785	+	0.125226i	0.909632	−	0.415415i	$-0.136364\pi$
$193$	0.0683785	+	0.125226i	0.0683785	+	0.125226i	−0.841254	+	0.540641i	$0.818182\pi$
$194$		0			0
$195$		0			0
$196$	0.329911	−	0.285870i	0.329911	−	0.285870i
$197$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$197$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$198$		0			0
$199$	0.682956	−	0.148568i	0.682956	−	0.148568i	0.142315	−	0.989821i	$-0.454545\pi$
$199$	0.682956	−	0.148568i	0.682956	−	0.148568i	0.540641	+	0.841254i	$0.318182\pi$
$200$		0			0
$201$	1.56449	+	1.17116i	1.56449	+	1.17116i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−0.574406	+	1.05195i	−0.574406	+	1.05195i
$209$		0			0
$210$		0			0
$211$	1.66538	+	1.07028i	1.66538	+	1.07028i	0.909632	+	0.415415i	$0.136364\pi$
$211$	1.66538	+	1.07028i	1.66538	+	1.07028i	0.755750	+	0.654861i	$0.227273\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.03748	−	0.776649i	−1.03748	−	0.776649i
$218$		0			0
$219$	−1.94931	+	0.139418i	−1.94931	+	0.139418i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.755750	−	0.654861i	$-0.227273\pi$
$223$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.959493	−	0.281733i	$-0.0909091\pi$
$224$		0			0
$225$	0.142315	+	0.989821i	0.142315	+	0.989821i
$226$		0			0
$227$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$227$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$228$	−0.125226	+	1.75089i	−0.125226	+	1.75089i
$229$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.540641	−	0.841254i	$-0.318182\pi$
$229$	1.29639	−	0.186393i	1.29639	−	0.186393i	0.755750	+	0.654861i	$0.227273\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$233$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−0.254771	−	1.17116i	−0.254771	−	1.17116i
$238$		0			0
$239$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$239$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$240$		0			0
$241$	−0.258908	+	1.80075i	−0.258908	+	1.80075i	0.281733	+	0.959493i	$0.409091\pi$
$241$	−0.258908	+	1.80075i	−0.258908	+	1.80075i	−0.540641	+	0.841254i	$0.681818\pi$
$242$		0			0
$243$	0.142315	−	0.989821i	0.142315	−	0.989821i
$244$	−0.133682	+	1.86912i	−0.133682	+	1.86912i
$245$		0			0
$246$		0			0
$247$	1.37776	+	1.59002i	1.37776	+	1.59002i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$251$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$252$	0.959493	+	0.718267i	0.959493	+	0.718267i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.00000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	−2.08248	+	1.13712i	−2.08248	+	1.13712i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$263$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.17116	−	1.56449i	−1.17116	−	1.56449i
$269$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$269$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$270$		0			0
$271$	0.373128	+	1.71524i	0.373128	+	1.71524i	0.654861	+	0.755750i	$0.272727\pi$
$271$	0.373128	+	1.71524i	0.373128	+	1.71524i	−0.281733	+	0.959493i	$0.590909\pi$
$272$		0			0
$273$	1.42191	−	0.204440i	1.42191	−	0.204440i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	0.909632	−	1.41542i	0.909632	−	1.41542i		−	1.00000i	$-0.5\pi$
$277$	0.909632	−	1.41542i	0.909632	−	1.41542i	0.909632	−	0.415415i	$-0.136364\pi$
$278$		0			0
$279$	0.449181	−	0.983568i	0.449181	−	0.983568i
$280$		0			0
$281$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$281$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$282$		0			0
$283$	−0.898064	−	0.334961i	−0.898064	−	0.334961i	−0.142315	−	0.989821i	$-0.545455\pi$
$283$	−0.898064	−	0.334961i	−0.898064	−	0.334961i	−0.755750	+	0.654861i	$0.772727\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$			1.00000i			1.00000i
$290$		0			0
$291$	0.0498610	+	0.697148i	0.0498610	+	0.697148i
$292$	1.90963	+	0.415415i	1.90963	+	0.415415i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	0.142315	−	0.989821i	0.142315	−	0.989821i
$301$	0.155554	+	2.17493i	0.155554	+	2.17493i
$302$		0			0
$303$		0			0
$304$	0.613435	−	1.64468i	0.613435	−	1.64468i
$305$		0			0
$306$		0			0
$307$	−0.708089	+	0.817178i	−0.708089	+	0.817178i	−0.989821	−	0.142315i	$-0.954545\pi$
$307$	−0.708089	+	0.817178i	−0.708089	+	0.817178i	0.281733	+	0.959493i	$0.409091\pi$
$308$		0			0
$309$	0.415415	−	1.90963i	0.415415	−	1.90963i
$310$		0			0
$311$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$311$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$312$		0			0
$313$	0.114220	+	0.0855040i	0.114220	+	0.0855040i	0.654861	−	0.755750i	$-0.272727\pi$
$313$	0.114220	+	0.0855040i	0.114220	+	0.0855040i	−0.540641	+	0.841254i	$0.681818\pi$
$314$		0			0
$315$		0			0
$316$	−0.0855040	+	1.19550i	−0.0855040	+	1.19550i
$317$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$317$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$325$	−0.718267	−	0.959493i	−0.718267	−	0.959493i
$326$		0			0
$327$	−0.767317	+	1.40524i	−0.767317	+	1.40524i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.959493	−	0.281733i	$-0.909091\pi$
$331$	−1.61435	+	0.474017i	−1.61435	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$332$		0			0
$333$	−1.29639	−	1.49611i	−1.29639	−	1.49611i
$334$		0			0
$335$		0			0
$336$	−0.718267	−	0.959493i	−0.718267	−	0.959493i
$337$	0.186393	−	0.215109i	0.186393	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$337$	0.186393	−	0.215109i	0.186393	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.592735	−	0.323658i	−0.592735	−	0.323658i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$347$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$348$		0			0
$349$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	0.654861	+	0.755750i	$0.272727\pi$
$349$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$350$		0			0
$351$	0.418852	+	1.12299i	0.418852	+	1.12299i
$352$		0			0
$353$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$353$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$359$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$360$		0			0
$361$	−1.57293	−	1.36295i	−1.57293	−	1.36295i
$362$		0			0
$363$	0.142315	+	0.989821i	0.142315	+	0.989821i
$364$	−1.42191	−	0.204440i	−1.42191	−	0.204440i
$365$		0			0
$366$		0			0
$367$	−1.07028	+	1.66538i	−1.07028	+	1.66538i	−0.415415	+	0.909632i	$0.636364\pi$
$367$	−1.07028	+	1.66538i	−1.07028	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−0.708089	+	0.817178i	−0.708089	+	0.817178i
$373$	0.203743	+	0.936593i	0.203743	+	0.936593i	0.959493	+	0.281733i	$0.0909091\pi$
$373$	0.203743	+	0.936593i	0.203743	+	0.936593i	−0.755750	+	0.654861i	$0.772727\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.75575	−	0.654861i	1.75575	−	0.654861i	0.755750	−	0.654861i	$-0.227273\pi$
$379$	1.75575	−	0.654861i	1.75575	−	0.654861i	1.00000			$0$
$380$		0			0
$381$	0.100889	+	0.100889i	0.100889	+	0.100889i
$382$		0			0
$383$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$383$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.74557	+	0.512546i	−1.74557	+	0.512546i
$388$	0.148568	−	0.682956i	0.148568	−	0.682956i
$389$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$389$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−0.254771	+	1.17116i	−0.254771	+	1.17116i	0.654861	+	0.755750i	$0.272727\pi$
$397$	−0.254771	+	1.17116i	−0.254771	+	1.17116i	−0.909632	+	0.415415i	$0.863636\pi$
$398$		0			0
$399$	−2.01867	+	0.592735i	−2.01867	+	0.592735i
$400$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$401$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$401$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$402$		0			0
$403$	0.0924539	+	1.29267i	0.0924539	+	1.29267i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.0303285	−	0.424047i	0.0303285	−	0.424047i	−0.959493	−	0.281733i	$-0.909091\pi$
$409$	0.0303285	−	0.424047i	0.0303285	−	0.424047i	0.989821	−	0.142315i	$-0.0454545\pi$
$410$		0			0
$411$		0			0
$412$	−0.936593	+	1.71524i	−0.936593	+	1.71524i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.959493	+	1.28173i	0.959493	+	1.28173i
$418$		0			0
$419$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$419$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$420$		0			0
$421$	−0.273100	−	1.89945i	−0.273100	−	1.89945i	−0.415415	−	0.909632i	$-0.636364\pi$
$421$	−0.273100	−	1.89945i	−0.273100	−	1.89945i	0.142315	−	0.989821i	$-0.454545\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.15499	+	0.632763i	−2.15499	+	0.632763i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$431$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$432$	0.654861	−	0.755750i	0.654861	−	0.755750i
$433$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.281733	−	0.959493i	$-0.409091\pi$
$433$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.909632	+	0.415415i	$0.136364\pi$
$434$		0			0
$435$		0			0
$436$	1.13214	−	1.13214i	1.13214	−	1.13214i
$437$		0			0
$438$		0			0
$439$	−1.89945	−	0.557730i	−1.89945	−	0.557730i	−0.989821	−	0.142315i	$-0.954545\pi$
$439$	−1.89945	−	0.557730i	−1.89945	−	0.557730i	−0.909632	−	0.415415i	$-0.863636\pi$
$440$		0			0
$441$	−0.122986	+	0.418852i	−0.122986	+	0.418852i
$442$		0			0
$443$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$443$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$444$	0.822373	+	1.80075i	0.822373	+	1.80075i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.418852	+	1.12299i	0.418852	+	1.12299i
$449$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$449$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.59700	+	1.19550i	1.59700	+	1.19550i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.37491	+	1.19136i	1.37491	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$457$	1.37491	+	1.19136i	1.37491	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$461$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$462$		0			0
$463$	0.936593	+	1.71524i	0.936593	+	1.71524i	0.654861	+	0.755750i	$0.272727\pi$
$463$	0.936593	+	1.71524i	0.936593	+	1.71524i	0.281733	+	0.959493i	$0.409091\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$467$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$468$	−0.0855040	−	1.19550i	−0.0855040	−	1.19550i
$469$	1.26636	−	1.97049i	1.26636	−	1.97049i
$470$		0			0
$471$	−0.559521	−	1.50013i	−0.559521	−	1.50013i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.24123	+	1.24123i	1.24123	+	1.24123i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$479$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$480$		0			0
$481$	2.22311	+	0.829178i	2.22311	+	0.829178i
$482$		0			0
$483$		0			0
$484$	0.142315	−	0.989821i	0.142315	−	0.989821i
$485$		0			0
$486$		0			0
$487$	−1.13214	+	0.847507i	−1.13214	+	0.847507i	−0.989821	−	0.142315i	$-0.954545\pi$
$487$	−1.13214	+	0.847507i	−1.13214	+	0.847507i	−0.142315	+	0.989821i	$0.545455\pi$
$488$		0			0
$489$	−0.203743	+	0.373128i	−0.203743	+	0.373128i
$490$		0			0
$491$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$491$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.909632	−	0.584585i	0.909632	−	0.584585i
$497$		0			0
$498$		0			0
$499$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	−0.841254	−	0.540641i	$-0.818182\pi$
$499$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$503$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−0.329911	−	0.285870i	−0.329911	−	0.285870i
$508$	−0.0683785	−	0.125226i	−0.0683785	−	0.125226i
$509$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$509$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$510$		0			0
$511$	0.333348	+	2.31849i	0.333348	+	2.31849i
$512$		0			0
$513$	−0.841254	−	1.54064i	−0.841254	−	1.54064i
$514$		0			0
$515$		0			0
$516$	1.81926			1.81926
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$521$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$522$		0			0
$523$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	−0.989821	−	0.142315i	$-0.954545\pi$
$523$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	0.755750	−	0.654861i	$-0.227273\pi$
$524$		0			0
$525$	1.17116	−	0.254771i	1.17116	−	0.254771i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.281733	+	0.959493i	−0.281733	+	0.959493i
$530$		0			0
$531$		0			0
$532$	2.10389			2.10389
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.398326	−	0.148568i	0.398326	−	0.148568i	−0.142315	−	0.989821i	$-0.545455\pi$
$541$	0.398326	−	0.148568i	0.398326	−	0.148568i	0.540641	+	0.841254i	$0.318182\pi$
$542$		0			0
$543$	1.13214	−	1.13214i	1.13214	−	1.13214i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.304632	−	1.03748i	−0.304632	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$547$	−0.304632	−	1.03748i	−0.304632	−	1.03748i	0.654861	−	0.755750i	$-0.272727\pi$
$548$		0			0
$549$	−0.898064	−	1.64468i	−0.898064	−	1.64468i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.37835	+	0.404719i	−1.37835	+	0.404719i
$554$		0			0
$555$		0			0
$556$	−0.559521	−	1.50013i	−0.559521	−	1.50013i
$557$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$557$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$558$		0			0
$559$	1.54184	−	1.54184i	1.54184	−	1.54184i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$563$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.19550	−	0.0855040i	−1.19550	−	0.0855040i
$568$		0			0
$569$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$569$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$570$		0			0
$571$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$571$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$577$	−0.215109	−	0.186393i	−0.215109	−	0.186393i	0.540641	−	0.841254i	$-0.318182\pi$
$577$	−0.215109	−	0.186393i	−0.215109	−	0.186393i	−0.755750	+	0.654861i	$0.772727\pi$
$578$		0			0
$579$	0.100889	+	0.100889i	0.100889	+	0.100889i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$587$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$588$	0.236009	−	0.367237i	0.236009	−	0.367237i
$589$	−0.403457	−	1.85466i	−0.403457	−	1.85466i
$590$		0			0
$591$		0			0
$592$	−0.281733	−	1.95949i	−0.281733	−	1.95949i
$593$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$593$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	0.613435	−	0.334961i	0.613435	−	0.334961i
$598$		0			0
$599$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$599$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$600$		0			0
$601$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$601$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$602$		0			0
$603$	1.83107	+	0.682956i	1.83107	+	0.682956i
$604$	−1.19550	−	1.59700i	−1.19550	−	1.59700i
$605$		0			0
$606$		0			0
$607$	0.822373	+	0.118239i	0.822373	+	0.118239i	0.540641	−	0.841254i	$-0.318182\pi$
$607$	0.822373	+	0.118239i	0.822373	+	0.118239i	0.281733	+	0.959493i	$0.409091\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.148568	−	0.682956i	−0.148568	−	0.682956i	−0.989821	−	0.142315i	$-0.954545\pi$
$613$	−0.148568	−	0.682956i	−0.148568	−	0.682956i	0.841254	−	0.540641i	$-0.181818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$617$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$618$		0			0
$619$	−0.898064	+	0.334961i	−0.898064	+	0.334961i	−0.755750	−	0.654861i	$-0.772727\pi$
$619$	−0.898064	+	0.334961i	−0.898064	+	0.334961i	−0.142315	+	0.989821i	$0.545455\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−0.254771	+	1.17116i	−0.254771	+	1.17116i
$625$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$626$		0			0
$627$		0			0
$628$	0.114220	+	1.59700i	0.114220	+	1.59700i
$629$		0			0
$630$		0			0
$631$	1.41061	+	0.100889i	1.41061	+	0.100889i	0.755750	−	0.654861i	$-0.227273\pi$
$631$	1.41061	+	0.100889i	1.41061	+	0.100889i	0.654861	+	0.755750i	$0.272727\pi$
$632$		0			0
$633$	1.89945	+	0.557730i	1.89945	+	0.557730i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−0.111217	−	0.511254i	−0.111217	−	0.511254i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$641$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$642$		0			0
$643$	1.50013	+	1.12299i	1.50013	+	1.12299i	0.959493	+	0.281733i	$0.0909091\pi$
$643$	1.50013	+	1.12299i	1.50013	+	1.12299i	0.540641	+	0.841254i	$0.318182\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$647$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.21426	−	0.452897i	−1.21426	−	0.452897i
$652$	0.300613	−	0.300613i	0.300613	−	0.300613i
$653$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$653$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.83107	+	0.682956i	−1.83107	+	0.682956i
$658$		0			0
$659$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$659$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$660$		0			0
$661$			1.97964i			1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$661$			1.97964i			1.97964i	−0.142315	+	0.989821i	$0.545455\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.38189	−	1.38189i	1.38189	−	1.38189i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.19136	−	0.544078i	−1.19136	−	0.544078i	−0.281733	−	0.959493i	$-0.590909\pi$
$673$	−1.19136	−	0.544078i	−1.19136	−	0.544078i	−0.909632	+	0.415415i	$0.863636\pi$
$674$		0			0
$675$	0.415415	+	0.909632i	0.415415	+	0.909632i
$676$	0.236009	+	0.367237i	0.236009	+	0.367237i
$677$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$677$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$678$		0			0
$679$	0.829178	−	0.119218i	0.829178	−	0.119218i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$683$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$684$	0.373128	+	1.71524i	0.373128	+	1.71524i
$685$		0			0
$686$		0			0
$687$	1.19136	−	0.544078i	1.19136	−	0.544078i
$688$	−1.74557	−	0.512546i	−1.74557	−	0.512546i
$689$		0			0
$690$		0			0
$691$	0.142315	+	0.0101786i	0.142315	+	0.0101786i	0.142315	−	0.989821i	$-0.454545\pi$
$691$	0.142315	+	0.0101786i	0.142315	+	0.0101786i			1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.19550	−	0.0855040i	−1.19550	−	0.0855040i
$701$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$701$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$702$		0			0
$703$	−3.39557	−	0.738661i	−3.39557	−	0.738661i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.936593	−	0.203743i	−0.936593	−	0.203743i	−0.281733	−	0.959493i	$-0.590909\pi$
$709$	−0.936593	−	0.203743i	−0.936593	−	0.203743i	−0.654861	+	0.755750i	$0.727273\pi$
$710$		0			0
$711$	−0.574406	−	1.05195i	−0.574406	−	1.05195i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$719$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$720$		0			0
$721$	−2.31849	−	0.333348i	−2.31849	−	0.333348i
$722$		0			0
$723$	0.258908	+	1.80075i	0.258908	+	1.80075i
$724$	−1.40524	+	0.767317i	−1.40524	+	0.767317i
$725$		0			0
$726$		0			0
$727$		−	1.97964i		−	1.97964i	−0.142315	−	0.989821i	$-0.545455\pi$
$727$		−	1.97964i		−	1.97964i	0.142315	−	0.989821i	$-0.454545\pi$
$728$		0			0
$729$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$730$		0			0
$731$		0			0
$732$	0.398326	+	1.83107i	0.398326	+	1.83107i
$733$	−1.03748	+	1.61435i	−1.03748	+	1.61435i	−0.281733	+	0.959493i	$0.590909\pi$
$733$	−1.03748	+	1.61435i	−1.03748	+	1.61435i	−0.755750	+	0.654861i	$0.772727\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.98982	−	0.142315i	1.98982	−	0.142315i	0.989821	−	0.142315i	$-0.0454545\pi$
$739$	1.98982	−	0.142315i	1.98982	−	0.142315i	1.00000			$0$
$740$		0			0
$741$	1.76991	+	1.13745i	1.76991	+	1.13745i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.898064	−	1.64468i	−0.898064	−	1.64468i	−0.755750	−	0.654861i	$-0.772727\pi$
$751$	−0.898064	−	1.64468i	−0.898064	−	1.64468i	−0.142315	−	0.989821i	$-0.545455\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	1.12299	+	0.418852i	1.12299	+	0.418852i
$757$	−1.24123	+	0.677760i	−1.24123	+	0.677760i	−0.959493	−	0.281733i	$-0.909091\pi$
$757$	−1.24123	+	0.677760i	−1.24123	+	0.677760i	−0.281733	+	0.959493i	$0.590909\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$761$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$762$		0			0
$763$	1.74557	+	0.797176i	1.74557	+	0.797176i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	0.959493	−	0.281733i	0.959493	−	0.281733i
$769$	−0.959493	+	1.28173i	−0.959493	+	1.28173i			1.00000i	$0.5\pi$
$769$	−0.959493	+	1.28173i	−0.959493	+	1.28173i	−0.959493	+	0.281733i	$0.909091\pi$
$770$		0			0
$771$		0			0
$772$	−0.0683785	−	0.125226i	−0.0683785	−	0.125226i
$773$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$773$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$774$		0			0
$775$	0.153882	+	1.07028i	0.153882	+	1.07028i
$776$		0			0
$777$	−1.67776	+	1.67776i	−1.67776	+	1.67776i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.329911	+	0.285870i	−0.329911	+	0.285870i
$785$		0			0
$786$		0			0
$787$	1.94931	+	0.424047i	1.94931	+	0.424047i	0.989821	+	0.142315i	$0.0454545\pi$
$787$	1.94931	+	0.424047i	1.94931	+	0.424047i	0.959493	+	0.281733i	$0.0909091\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.88943	+	1.21426i	1.88943	+	1.21426i
$794$		0			0
$795$		0			0
$796$	−0.682956	+	0.148568i	−0.682956	+	0.148568i
$797$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$797$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.56449	−	1.17116i	−1.56449	−	1.17116i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0			−	1.00000i	$-0.5\pi$
$809$		0			0				1.00000i	$0.5\pi$
$810$		0			0
$811$	0.142315	−	0.0101786i	0.142315	−	0.0101786i		−	1.00000i	$-0.5\pi$
$811$	0.142315	−	0.0101786i	0.142315	−	0.0101786i	0.142315	+	0.989821i	$0.454545\pi$
$812$		0			0
$813$	0.841254	+	1.54064i	0.841254	+	1.54064i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.91377	+	2.55650i	−1.91377	+	2.55650i
$818$		0			0
$819$	1.30672	−	0.596758i	1.30672	−	0.596758i
$820$		0			0
$821$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$821$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$822$		0			0
$823$	−1.13214	+	0.847507i	−1.13214	+	0.847507i	−0.989821	−	0.142315i	$-0.954545\pi$
$823$	−1.13214	+	0.847507i	−1.13214	+	0.847507i	−0.142315	+	0.989821i	$0.545455\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$827$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$828$		0			0
$829$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.989821	+	0.142315i	$0.0454545\pi$
$829$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$830$		0			0
$831$	0.474017	−	1.61435i	0.474017	−	1.61435i
$832$	0.574406	−	1.05195i	0.574406	−	1.05195i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	0.153882	−	1.07028i	0.153882	−	1.07028i
$838$		0			0
$839$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$839$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$840$		0			0
$841$	0.654861	−	0.755750i	0.654861	−	0.755750i
$842$		0			0
$843$		0			0
$844$	−1.66538	−	1.07028i	−1.66538	−	1.07028i
$845$		0			0
$846$		0			0
$847$	1.17116	−	0.254771i	1.17116	−	0.254771i
$848$		0			0
$849$	−0.956056	−	0.0683785i	−0.956056	−	0.0683785i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−0.125226	−	1.75089i	−0.125226	−	1.75089i	−0.540641	−	0.841254i	$-0.681818\pi$
$853$	−0.125226	−	1.75089i	−0.125226	−	1.75089i	0.415415	−	0.909632i	$-0.363636\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$857$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$858$		0			0
$859$	0.682956	+	0.148568i	0.682956	+	0.148568i	0.540641	−	0.841254i	$-0.318182\pi$
$859$	0.682956	+	0.148568i	0.682956	+	0.148568i	0.142315	+	0.989821i	$0.454545\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$863$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	0.281733	+	0.959493i	0.281733	+	0.959493i
$868$	1.03748	+	0.776649i	1.03748	+	0.776649i
$869$		0			0
$870$		0			0
$871$	−2.31849	+	0.333348i	−2.31849	+	0.333348i
$872$		0			0
$873$	0.244250	+	0.654861i	0.244250	+	0.654861i
$874$		0			0
$875$		0			0
$876$	1.94931	−	0.139418i	1.94931	−	0.139418i
$877$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$877$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$881$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$882$		0			0
$883$	0.142315	+	1.98982i	0.142315	+	1.98982i	0.142315	+	0.989821i	$0.454545\pi$
$883$	0.142315	+	1.98982i	0.142315	+	1.98982i			1.00000i	$0.5\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$887$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$888$		0			0
$889$	0.111986	−	0.129239i	0.111986	−	0.129239i
$890$		0			0
$891$		0			0
$892$	−1.71524	+	0.936593i	−1.71524	+	0.936593i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$901$		0			0
$902$		0			0
$903$	0.762003	+	2.04301i	0.762003	+	2.04301i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−0.494217	+	1.32505i	−0.494217	+	1.32505i	0.415415	+	0.909632i	$0.363636\pi$
$907$	−0.494217	+	1.32505i	−0.494217	+	1.32505i	−0.909632	+	0.415415i	$0.863636\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$911$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$912$	0.125226	−	1.75089i	0.125226	−	1.75089i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.29639	+	0.186393i	−1.29639	+	0.186393i
$917$		0			0
$918$		0			0
$919$	−0.254771	+	0.340335i	−0.254771	+	0.340335i	−0.909632	−	0.415415i	$-0.863636\pi$
$919$	−0.254771	+	0.340335i	−0.254771	+	0.340335i	0.654861	+	0.755750i	$0.272727\pi$
$920$		0			0
$921$	−0.449181	+	0.983568i	−0.449181	+	0.983568i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.89945	+	0.557730i	1.89945	+	0.557730i
$926$		0			0
$927$	−0.139418	−	1.94931i	−0.139418	−	1.94931i
$928$		0			0
$929$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$929$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$930$		0			0
$931$	0.267786	+	0.717961i	0.267786	+	0.717961i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	0.0801894	+	0.273100i	0.0801894	+	0.273100i	0.989821	−	0.142315i	$-0.0454545\pi$
$937$	0.0801894	+	0.273100i	0.0801894	+	0.273100i	−0.909632	+	0.415415i	$0.863636\pi$
$938$		0			0
$939$	0.133682	+	0.0498610i	0.133682	+	0.0498610i
$940$		0			0
$941$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$941$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$947$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$948$	0.254771	+	1.17116i	0.254771	+	1.17116i
$949$	1.53390	−	1.77021i	1.53390	−	1.77021i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$953$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.0702757	−	0.153882i	0.0702757	−	0.153882i
$962$		0			0
$963$		0			0
$964$	0.258908	−	1.80075i	0.258908	−	1.80075i
$965$		0			0
$966$		0			0
$967$	−1.38189	+	0.300613i	−1.38189	+	0.300613i	−0.841254	−	0.540641i	$-0.818182\pi$
$967$	−1.38189	+	0.300613i	−1.38189	+	0.300613i	−0.540641	+	0.841254i	$0.681818\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$971$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$972$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$973$	1.45027	−	1.25667i	1.45027	−	1.25667i
$974$		0			0
$975$	−0.959493	−	0.718267i	−0.959493	−	0.718267i
$976$	0.133682	−	1.86912i	0.133682	−	1.86912i
$977$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$977$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−0.340335	+	1.56449i	−0.340335	+	1.56449i
$982$		0			0
$983$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$983$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.37776	−	1.59002i	−1.37776	−	1.59002i
$989$		0			0
$990$		0			0
$991$	−0.830830			−0.830830			−0.415415	−	0.909632i	$-0.636364\pi$
$991$	−0.830830			−0.830830			−0.415415	+	0.909632i	$0.636364\pi$
$992$		0			0
$993$	−1.41542	+	0.909632i	−1.41542	+	0.909632i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.544078	−	1.19136i	0.544078	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$997$	0.544078	−	1.19136i	0.544078	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$998$		0			0
$999$	−1.66538	−	1.07028i	−1.66538	−	1.07028i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3963.1.bi.a.2666.1	✓	20
3.2	odd	2	CM	3963.1.bi.a.2666.1	✓	20
1321.1266	even	44	inner	3963.1.bi.a.3908.1	yes	20
3963.3908	odd	44	inner	3963.1.bi.a.3908.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3963.1.bi.a.2666.1	✓	20	1.1	even	1	trivial
3963.1.bi.a.2666.1	✓	20	3.2	odd	2	CM
3963.1.bi.a.3908.1	yes	20	1321.1266	even	44	inner
3963.1.bi.a.3908.1	yes	20	3963.3908	odd	44	inner

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:	\( N \)	\(=\)	\( 3963 = 3 \cdot 1321 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3963.bi (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(0.959493 - 0.281733i) q^{3} -1.00000 q^{4} +(-0.418852 - 1.12299i) q^{7} +(0.841254 - 0.540641i) q^{9} +O(q^{10})\)	\(q+(0.959493 - 0.281733i) q^{3} -1.00000 q^{4} +(-0.418852 - 1.12299i) q^{7} +(0.841254 - 0.540641i) q^{9} +(-0.959493 + 0.281733i) q^{12} +(-0.574406 + 1.05195i) q^{13} +1.00000 q^{16} +(0.613435 - 1.64468i) q^{19} +(-0.718267 - 0.959493i) q^{21} +(-0.415415 + 0.909632i) q^{25} +(0.654861 - 0.755750i) q^{27} +(0.418852 + 1.12299i) q^{28} +(0.909632 - 0.584585i) q^{31} +(-0.841254 + 0.540641i) q^{36} +(-0.281733 - 1.95949i) q^{37} +(-0.254771 + 1.17116i) q^{39} +(-1.74557 - 0.512546i) q^{43} +(0.959493 - 0.281733i) q^{48} +(-0.329911 + 0.285870i) q^{49} +(0.574406 - 1.05195i) q^{52} +(0.125226 - 1.75089i) q^{57} +(0.133682 - 1.86912i) q^{61} +(-0.959493 - 0.718267i) q^{63} -1.00000 q^{64} +(1.17116 + 1.56449i) q^{67} +(-1.90963 - 0.415415i) q^{73} +(-0.142315 + 0.989821i) q^{75} +(-0.613435 + 1.64468i) q^{76} +(0.0855040 - 1.19550i) q^{79} +(0.415415 - 0.909632i) q^{81} +(0.718267 + 0.959493i) q^{84} +(1.42191 + 0.204440i) q^{91} +(0.708089 - 0.817178i) q^{93} +(-0.148568 + 0.682956i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} - 20 q^{4} - 2 q^{7} - 2 q^{9}+O(q^{10}) \) 20 * q + 2 * q^3 - 20 * q^4 - 2 * q^7 - 2 * q^9	\( 20 q + 2 q^{3} - 20 q^{4} - 2 q^{7} - 2 q^{9} - 2 q^{12} - 2 q^{13} + 20 q^{16} - 2 q^{19} - 20 q^{21} + 2 q^{25} + 2 q^{27} + 2 q^{28} + 2 q^{36} + 2 q^{39} + 2 q^{48} + 2 q^{52} + 2 q^{57} - 2 q^{61} - 2 q^{63} - 20 q^{64} - 2 q^{67} - 20 q^{73} - 2 q^{75} + 2 q^{76} - 2 q^{79} - 2 q^{81} + 20 q^{84} - 2 q^{97}+O(q^{100}) \) 20 * q + 2 * q^3 - 20 * q^4 - 2 * q^7 - 2 * q^9 - 2 * q^12 - 2 * q^13 + 20 * q^16 - 2 * q^19 - 20 * q^21 + 2 * q^25 + 2 * q^27 + 2 * q^28 + 2 * q^36 + 2 * q^39 + 2 * q^48 + 2 * q^52 + 2 * q^57 - 2 * q^61 - 2 * q^63 - 20 * q^64 - 2 * q^67 - 20 * q^73 - 2 * q^75 + 2 * q^76 - 2 * q^79 - 2 * q^81 + 20 * q^84 - 2 * q^97

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