LMFDB - Embedded newform 979.1.v.a.780.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [979,1,Mod(10,979)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(979, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 43]))

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

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

chi := DirichletCharacter("979.10");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 979 = 11 \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	979.v (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:	$0.488584647368$
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			780.1
Root			$-0.540641 - 0.841254i$ of defining polynomial
Character	$\chi$	$=$	979.780
Dual form			979.1.v.a.428.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.415415 - 1.90963i) q^{3} +(0.841254 - 0.540641i) q^{4} +(1.27155 + 1.10181i) q^{5} +(-2.56449 + 1.17116i) q^{9} +O(q^{10})$	\(q+(-0.415415 - 1.90963i) q^{3} +(0.841254 - 0.540641i) q^{4} +(1.27155 + 1.10181i) q^{5} +(-2.56449 + 1.17116i) q^{9} +(-0.654861 - 0.755750i) q^{11} +(-1.38189 - 1.38189i) q^{12} +(1.57582 - 2.88591i) q^{15} +(0.415415 - 0.909632i) q^{16} +(1.66538 + 0.239446i) q^{20} +(-0.0498610 + 0.133682i) q^{23} +(0.260554 + 1.81219i) q^{25} +(2.13066 + 2.84623i) q^{27} +(-0.613435 - 1.64468i) q^{31} +(-1.17116 + 1.56449i) q^{33} +(-1.52421 + 2.37172i) q^{36} +(-0.300613 + 0.300613i) q^{37} +(-0.959493 - 0.281733i) q^{44} +(-4.55129 - 1.33638i) q^{45} +(1.07028 + 1.66538i) q^{47} +(-1.90963 - 0.415415i) q^{48} +(0.989821 - 0.142315i) q^{49} +(-0.708089 + 1.10181i) q^{53} -1.68251i q^{55} +(-0.148568 + 0.682956i) q^{59} +(-0.234571 - 3.27974i) q^{60} +(-0.142315 - 0.989821i) q^{64} +(0.474017 + 0.304632i) q^{67} +(0.275997 + 0.0396824i) q^{69} +(-1.45027 + 1.25667i) q^{71} +(3.35239 - 1.25038i) q^{75} +(1.53046 - 0.698939i) q^{80} +(2.70391 - 3.12048i) q^{81} +(0.654861 - 0.755750i) q^{89} +(0.0303285 + 0.139418i) q^{92} +(-2.88591 + 1.85466i) q^{93} +(-0.989821 + 1.14231i) q^{97} +(2.56449 + 1.17116i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{3} - 2 q^{4} - 22 q^{9}+O(q^{10}) $ 20 * q + 2 * q^3 - 2 * q^4 - 22 * q^9	$ 20 q + 2 q^{3} - 2 q^{4} - 22 q^{9} - 2 q^{11} + 2 q^{12} + 4 q^{15} - 2 q^{16} - 2 q^{23} + 6 q^{25} + 2 q^{31} + 2 q^{33} + 2 q^{37} - 2 q^{44} - 4 q^{45} - 20 q^{48} - 2 q^{59} + 4 q^{60} - 2 q^{64} - 6 q^{75} + 20 q^{81} + 2 q^{89} - 2 q^{92} - 4 q^{93} + 22 q^{99}+O(q^{100}) $ 20 * q + 2 * q^3 - 2 * q^4 - 22 * q^9 - 2 * q^11 + 2 * q^12 + 4 * q^15 - 2 * q^16 - 2 * q^23 + 6 * q^25 + 2 * q^31 + 2 * q^33 + 2 * q^37 - 2 * q^44 - 4 * q^45 - 20 * q^48 - 2 * q^59 + 4 * q^60 - 2 * q^64 - 6 * q^75 + 20 * q^81 + 2 * q^89 - 2 * q^92 - 4 * q^93 + 22 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/979\mathbb{Z}\right)^\times$.

$n$	$90$	$804$
$\chi(n)$	$-1$	$e\left(\frac{19}{44}\right)$

Coefficient data

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

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

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

$2$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$2$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$3$	−0.415415	−	1.90963i	−0.415415	−	1.90963i	−0.415415	−	0.909632i	$-0.636364\pi$
$3$	−0.415415	−	1.90963i	−0.415415	−	1.90963i		−	1.00000i	$-0.5\pi$
$4$	0.841254	−	0.540641i	0.841254	−	0.540641i
$5$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.989821	+	0.142315i	$0.0454545\pi$
$5$	1.27155	+	1.10181i	1.27155	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$6$		0			0
$7$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$7$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$8$		0			0
$9$	−2.56449	+	1.17116i	−2.56449	+	1.17116i
$10$		0			0
$11$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$11$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$12$	−1.38189	−	1.38189i	−1.38189	−	1.38189i
$13$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$13$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$14$		0			0
$15$	1.57582	−	2.88591i	1.57582	−	2.88591i
$16$	0.415415	−	0.909632i	0.415415	−	0.909632i
$17$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$17$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$18$		0			0
$19$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$19$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$20$	1.66538	+	0.239446i	1.66538	+	0.239446i
$21$		0			0
$22$		0			0
$23$	−0.0498610	+	0.133682i	−0.0498610	+	0.133682i	−0.959493	−	0.281733i	$-0.909091\pi$
$23$	−0.0498610	+	0.133682i	−0.0498610	+	0.133682i	0.909632	+	0.415415i	$0.136364\pi$
$24$		0			0
$25$	0.260554	+	1.81219i	0.260554	+	1.81219i
$26$		0			0
$27$	2.13066	+	2.84623i	2.13066	+	2.84623i
$28$		0			0
$29$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$29$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$30$		0			0
$31$	−0.613435	−	1.64468i	−0.613435	−	1.64468i	−0.755750	−	0.654861i	$-0.772727\pi$
$31$	−0.613435	−	1.64468i	−0.613435	−	1.64468i	0.142315	−	0.989821i	$-0.454545\pi$
$32$		0			0
$33$	−1.17116	+	1.56449i	−1.17116	+	1.56449i
$34$		0			0
$35$		0			0
$36$	−1.52421	+	2.37172i	−1.52421	+	2.37172i
$37$	−0.300613	+	0.300613i	−0.300613	+	0.300613i	−0.841254	−	0.540641i	$-0.818182\pi$
$37$	−0.300613	+	0.300613i	−0.300613	+	0.300613i	0.540641	+	0.841254i	$0.318182\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$41$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$42$		0			0
$43$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$43$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$44$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$45$	−4.55129	−	1.33638i	−4.55129	−	1.33638i
$46$		0			0
$47$	1.07028	+	1.66538i	1.07028	+	1.66538i	0.654861	+	0.755750i	$0.272727\pi$
$47$	1.07028	+	1.66538i	1.07028	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$48$	−1.90963	−	0.415415i	−1.90963	−	0.415415i
$49$	0.989821	−	0.142315i	0.989821	−	0.142315i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	0.281733	+	0.959493i	$0.409091\pi$
$53$	−0.708089	+	1.10181i	−0.708089	+	1.10181i	−0.989821	+	0.142315i	$0.954545\pi$
$54$		0			0
$55$		−	1.68251i		−	1.68251i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−0.148568	+	0.682956i	−0.148568	+	0.682956i	0.841254	+	0.540641i	$0.181818\pi$
$59$	−0.148568	+	0.682956i	−0.148568	+	0.682956i	−0.989821	+	0.142315i	$0.954545\pi$
$60$	−0.234571	−	3.27974i	−0.234571	−	3.27974i
$61$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$61$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$62$		0			0
$63$		0			0
$64$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$65$		0			0
$66$		0			0
$67$	0.474017	+	0.304632i	0.474017	+	0.304632i	0.755750	−	0.654861i	$-0.227273\pi$
$67$	0.474017	+	0.304632i	0.474017	+	0.304632i	−0.281733	+	0.959493i	$0.590909\pi$
$68$		0			0
$69$	0.275997	+	0.0396824i	0.275997	+	0.0396824i
$70$		0			0
$71$	−1.45027	+	1.25667i	−1.45027	+	1.25667i	−0.540641	+	0.841254i	$0.681818\pi$
$71$	−1.45027	+	1.25667i	−1.45027	+	1.25667i	−0.909632	+	0.415415i	$0.863636\pi$
$72$		0			0
$73$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$73$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$74$		0			0
$75$	3.35239	−	1.25038i	3.35239	−	1.25038i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$79$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$80$	1.53046	−	0.698939i	1.53046	−	0.698939i
$81$	2.70391	−	3.12048i	2.70391	−	3.12048i
$82$		0			0
$83$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$83$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.654861	−	0.755750i	0.654861	−	0.755750i
$89$	0.654861	−	0.755750i	0.654861	−	0.755750i
$90$		0			0
$91$		0			0
$92$	0.0303285	+	0.139418i	0.0303285	+	0.139418i
$93$	−2.88591	+	1.85466i	−2.88591	+	1.85466i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.989821	+	1.14231i	−0.989821	+	1.14231i			1.00000i	$0.5\pi$
$97$	−0.989821	+	1.14231i	−0.989821	+	1.14231i	−0.989821	+	0.142315i	$0.954545\pi$
$98$		0			0
$99$	2.56449	+	1.17116i	2.56449	+	1.17116i
$100$	1.19894	+	1.38365i	1.19894	+	1.38365i
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$		0			0
$103$	−1.64468	+	0.613435i	−1.64468	+	0.613435i	−0.989821	−	0.142315i	$-0.954545\pi$
$103$	−1.64468	+	0.613435i	−1.64468	+	0.613435i	−0.654861	+	0.755750i	$0.727273\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$107$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$108$	3.33121	+	1.24248i	3.33121	+	1.24248i
$109$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$109$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$110$		0			0
$111$	0.698939	+	0.449181i	0.698939	+	0.449181i
$112$		0			0
$113$	0.125226	+	0.0683785i	0.125226	+	0.0683785i	0.540641	−	0.841254i	$-0.318182\pi$
$113$	0.125226	+	0.0683785i	0.125226	+	0.0683785i	−0.415415	+	0.909632i	$0.636364\pi$
$114$		0			0
$115$	−0.210693	+	0.115047i	−0.210693	+	0.115047i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$122$		0			0
$123$		0			0
$124$	−1.40524	−	1.05195i	−1.40524	−	1.05195i
$125$	−0.755750	+	1.17597i	−0.755750	+	1.17597i
$126$		0			0
$127$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$127$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$131$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$132$	−0.139418	+	1.94931i	−0.139418	+	1.94931i
$133$		0			0
$134$		0			0
$135$	−0.426747	+	5.96670i	−0.426747	+	5.96670i
$136$		0			0
$137$	1.56449	+	0.340335i	1.56449	+	0.340335i	0.909632	−	0.415415i	$-0.136364\pi$
$137$	1.56449	+	0.340335i	1.56449	+	0.340335i	0.654861	+	0.755750i	$0.272727\pi$
$138$		0			0
$139$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$139$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$140$		0			0
$141$	2.73566	−	2.73566i	2.73566	−	2.73566i
$142$		0			0
$143$		0			0
$144$			2.81926i			2.81926i
$145$		0			0
$146$		0			0
$147$	−0.682956	−	1.83107i	−0.682956	−	1.83107i
$148$	−0.0903680	+	0.415415i	−0.0903680	+	0.415415i
$149$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$149$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$150$		0			0
$151$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$151$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.03211	−	2.76719i	1.03211	−	2.76719i
$156$		0			0
$157$	−0.304632	−	1.03748i	−0.304632	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$157$	−0.304632	−	1.03748i	−0.304632	−	1.03748i	0.654861	−	0.755750i	$-0.272727\pi$
$158$		0			0
$159$	2.39820	+	0.894482i	2.39820	+	0.894482i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.677760	−	1.24123i	0.677760	−	1.24123i	−0.281733	−	0.959493i	$-0.590909\pi$
$163$	0.677760	−	1.24123i	0.677760	−	1.24123i	0.959493	−	0.281733i	$-0.0909091\pi$
$164$		0			0
$165$	−3.21297	+	0.698939i	−3.21297	+	0.698939i
$166$		0			0
$167$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$167$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$168$		0			0
$169$	0.909632	−	0.415415i	0.909632	−	0.415415i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$173$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$174$		0			0
$175$		0			0
$176$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$177$	1.36591			1.36591
$178$		0			0
$179$	−1.08128			−1.08128			−0.540641	−	0.841254i	$-0.681818\pi$
$179$	−1.08128			−1.08128			−0.540641	+	0.841254i	$0.681818\pi$
$180$	−4.55129	+	1.33638i	−4.55129	+	1.33638i
$181$	−0.424047	−	1.94931i	−0.424047	−	1.94931i	−0.281733	−	0.959493i	$-0.590909\pi$
$181$	−0.424047	−	1.94931i	−0.424047	−	1.94931i	−0.142315	−	0.989821i	$-0.545455\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.713463	+	0.0510279i	−0.713463	+	0.0510279i
$186$		0			0
$187$		0			0
$188$	1.80075	+	0.822373i	1.80075	+	0.822373i
$189$		0			0
$190$		0			0
$191$	1.17116	−	0.254771i	1.17116	−	0.254771i	0.415415	−	0.909632i	$-0.363636\pi$
$191$	1.17116	−	0.254771i	1.17116	−	0.254771i	0.755750	+	0.654861i	$0.227273\pi$
$192$	−1.83107	+	0.682956i	−1.83107	+	0.682956i
$193$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$193$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$194$		0			0
$195$		0			0
$196$	0.755750	−	0.654861i	0.755750	−	0.654861i
$197$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$197$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$198$		0			0
$199$	0.234072	+	0.797176i	0.234072	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$199$	0.234072	+	0.797176i	0.234072	+	0.797176i	−0.755750	+	0.654861i	$0.772727\pi$
$200$		0			0
$201$	0.384822	−	1.03175i	0.384822	−	1.03175i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−0.0286960	−	0.401223i	−0.0286960	−	0.401223i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$211$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$212$			1.30972i			1.30972i
$213$	3.00224	+	2.24745i	3.00224	+	2.24745i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−0.909632	−	1.41542i	−0.909632	−	1.41542i
$221$		0			0
$222$		0			0
$223$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$223$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$224$		0			0
$225$	−2.79057	−	4.34221i	−2.79057	−	4.34221i
$226$		0			0
$227$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$227$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$228$		0			0
$229$	−1.86912	−	0.133682i	−1.86912	−	0.133682i	−0.909632	−	0.415415i	$-0.863636\pi$
$229$	−1.86912	−	0.133682i	−1.86912	−	0.133682i	−0.959493	+	0.281733i	$0.909091\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0			−	1.00000i	$-0.5\pi$
$233$		0			0				1.00000i	$0.5\pi$
$234$		0			0
$235$	−0.474017	+	3.29686i	−0.474017	+	3.29686i
$236$	0.244250	+	0.654861i	0.244250	+	0.654861i
$237$		0			0
$238$		0			0
$239$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$239$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$240$	−1.97049	−	2.63227i	−1.97049	−	2.63227i
$241$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$241$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$242$		0			0
$243$	−3.96173	−	2.16327i	−3.96173	−	2.16327i
$244$		0			0
$245$	1.41542	+	0.909632i	1.41542	+	0.909632i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.142315	−	0.989821i	$-0.454545\pi$
$251$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$252$		0			0
$253$	0.133682	−	0.0498610i	0.133682	−	0.0498610i
$254$		0			0
$255$		0			0
$256$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$257$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−0.654861	−	0.755750i	$-0.727273\pi$
$257$	−1.65486	−	0.755750i	−1.65486	−	0.755750i	−1.00000			$\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$263$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$264$		0			0
$265$	−2.11435	+	0.620830i	−2.11435	+	0.620830i
$266$		0			0
$267$	−1.71524	−	0.936593i	−1.71524	−	0.936593i
$268$	0.563465			0.563465
$269$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$269$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.415415	+	0.909632i	$0.636364\pi$
$270$		0			0
$271$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$271$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.19894	−	1.38365i	1.19894	−	1.38365i
$276$	0.253638	−	0.115832i	0.253638	−	0.115832i
$277$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$277$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$278$		0			0
$279$	3.49934	+	3.49934i	3.49934	+	3.49934i
$280$		0			0
$281$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$281$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$282$		0			0
$283$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$283$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$284$	−0.540641	+	1.84125i	−0.540641	+	1.84125i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$290$		0			0
$291$	2.59259	+	1.41566i	2.59259	+	1.41566i
$292$		0			0
$293$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$293$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$294$		0			0
$295$	−0.941398	+	0.704722i	−0.941398	+	0.704722i
$296$		0			0
$297$	0.755750	−	3.47412i	0.755750	−	3.47412i
$298$		0			0
$299$		0			0
$300$	2.14420	−	2.86432i	2.14420	−	2.86432i
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$307$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$308$		0			0
$309$	1.85466	+	2.88591i	1.85466	+	2.88591i
$310$		0			0
$311$	−1.45027	−	0.425839i	−1.45027	−	0.425839i	−0.540641	−	0.841254i	$-0.681818\pi$
$311$	−1.45027	−	0.425839i	−1.45027	−	0.425839i	−0.909632	+	0.415415i	$0.863636\pi$
$312$		0			0
$313$	0.142315	−	1.98982i	0.142315	−	1.98982i		−	1.00000i	$-0.5\pi$
$313$	0.142315	−	1.98982i	0.142315	−	1.98982i	0.142315	−	0.989821i	$-0.454545\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$317$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$318$		0			0
$319$		0			0
$320$	0.909632	−	1.41542i	0.909632	−	1.41542i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.587616	−	4.08696i	0.587616	−	4.08696i
$325$		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	$0.0454545\pi$
$331$	0.0801894	+	0.557730i	0.0801894	+	0.557730i	−0.909632	+	0.415415i	$0.863636\pi$
$332$		0			0
$333$	0.418852	−	1.12299i	0.418852	−	1.12299i
$334$		0			0
$335$	0.267092	+	0.909632i	0.267092	+	0.909632i
$336$		0			0
$337$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$337$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$338$		0			0
$339$	0.0785570	−	0.267541i	0.0785570	−	0.267541i
$340$		0			0
$341$	−0.841254	+	1.54064i	−0.841254	+	1.54064i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	0.307223	+	0.354554i	0.307223	+	0.354554i
$346$		0			0
$347$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$347$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$348$		0			0
$349$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$349$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.254771	+	1.17116i	0.254771	+	1.17116i	0.909632	+	0.415415i	$0.136364\pi$
$353$	0.254771	+	1.17116i	0.254771	+	1.17116i	−0.654861	+	0.755750i	$0.727273\pi$
$354$		0			0
$355$	−3.22871			−3.22871
$356$	0.142315	−	0.989821i	0.142315	−	0.989821i
$357$		0			0
$358$		0			0
$359$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$359$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$360$		0			0
$361$	0.755750	+	0.654861i	0.755750	+	0.654861i
$362$		0			0
$363$	1.94931	−	0.139418i	1.94931	−	0.139418i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	0.186393	+	0.215109i	0.186393	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$367$	0.186393	+	0.215109i	0.186393	+	0.215109i	−0.654861	+	0.755750i	$0.727273\pi$
$368$	0.100889	+	0.100889i	0.100889	+	0.100889i
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.42508	+	3.12048i	−1.42508	+	3.12048i
$373$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$373$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$374$		0			0
$375$	2.55962	+	0.954688i	2.55962	+	0.954688i
$376$		0			0
$377$		0			0
$378$		0			0
$379$	0.418852	−	1.12299i	0.418852	−	1.12299i	−0.540641	−	0.841254i	$-0.681818\pi$
$379$	0.418852	−	1.12299i	0.418852	−	1.12299i	0.959493	−	0.281733i	$-0.0909091\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.12299	−	1.50013i	−1.12299	−	1.50013i	−0.841254	−	0.540641i	$-0.818182\pi$
$383$	−1.12299	−	1.50013i	−1.12299	−	1.50013i	−0.281733	−	0.959493i	$-0.590909\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.215109	+	1.49611i	−0.215109	+	1.49611i
$389$	0.574406	−	0.767317i	0.574406	−	0.767317i	−0.415415	−	0.909632i	$-0.636364\pi$
$389$	0.574406	−	0.767317i	0.574406	−	0.767317i	0.989821	+	0.142315i	$0.0454545\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$	2.79057	−	0.401223i	2.79057	−	0.401223i
$397$	−0.682956	−	0.148568i	−0.682956	−	0.148568i	−0.142315	−	0.989821i	$-0.545455\pi$
$397$	−0.682956	−	0.148568i	−0.682956	−	0.148568i	−0.540641	+	0.841254i	$0.681818\pi$
$398$		0			0
$399$		0			0
$400$	1.75667	+	0.515804i	1.75667	+	0.515804i
$401$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$401$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	6.87634	−	0.988668i	6.87634	−	0.988668i
$406$		0			0
$407$	0.424047	+	0.0303285i	0.424047	+	0.0303285i
$408$		0			0
$409$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$409$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$410$		0			0
$411$		−	3.12899i		−	3.12899i
$412$	−1.05195	+	1.40524i	−1.05195	+	1.40524i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.841254	+	0.540641i	$0.818182\pi$
$419$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.909632	+	0.415415i	$0.863636\pi$
$420$		0			0
$421$	0.841254	+	0.459359i	0.841254	+	0.459359i	0.841254	−	0.540641i	$-0.181818\pi$
$421$	0.841254	+	0.459359i	0.841254	+	0.459359i			1.00000i	$0.5\pi$
$422$		0			0
$423$	−4.69515	−	3.01739i	−4.69515	−	3.01739i
$424$		0			0
$425$		0			0
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$431$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$432$	3.47412	−	0.755750i	3.47412	−	0.755750i
$433$	0.494217	+	0.494217i	0.494217	+	0.494217i	0.909632	−	0.415415i	$-0.136364\pi$
$433$	0.494217	+	0.494217i	0.494217	+	0.494217i	−0.415415	+	0.909632i	$0.636364\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$439$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$440$		0			0
$441$	−2.37172	+	1.52421i	−2.37172	+	1.52421i
$442$		0			0
$443$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$443$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$444$	0.830830			0.830830
$445$	1.66538	−	0.239446i	1.66538	−	0.239446i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$449$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$450$		0			0
$451$		0			0
$452$	0.142315	−	0.0101786i	0.142315	−	0.0101786i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$457$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$458$		0			0
$459$		0			0
$460$	−0.115047	+	0.210693i	−0.115047	+	0.210693i
$461$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$461$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$462$		0			0
$463$	−0.989821	+	0.857685i	−0.989821	+	0.857685i	−0.989821	−	0.142315i	$-0.954545\pi$
$463$	−0.989821	+	0.857685i	−0.989821	+	0.857685i			1.00000i	$0.5\pi$
$464$		0			0
$465$	−5.71307	−	0.821415i	−5.71307	−	0.821415i
$466$		0			0
$467$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	0.281733	−	0.959493i	$-0.409091\pi$
$467$	−0.474017	−	0.304632i	−0.474017	−	0.304632i	−0.755750	+	0.654861i	$0.772727\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1.85466	+	1.01272i	−1.85466	+	1.01272i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.525491	−	3.65487i	0.525491	−	3.65487i
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	0.415415	+	0.909632i	0.415415	+	0.909632i
$485$	−2.51722	+	0.361922i	−2.51722	+	0.361922i
$486$		0			0
$487$	0.584585	+	0.909632i	0.584585	+	0.909632i	1.00000			$0$
$487$	0.584585	+	0.909632i	0.584585	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$488$		0			0
$489$	−2.65184	−	0.778649i	−2.65184	−	0.778649i
$490$		0			0
$491$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$491$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$	1.97049	+	4.31478i	1.97049	+	4.31478i
$496$	−1.75089	−	0.125226i	−1.75089	−	0.125226i
$497$		0			0
$498$		0			0
$499$	−0.559521	−	0.418852i	−0.559521	−	0.418852i	0.281733	−	0.959493i	$-0.409091\pi$
$499$	−0.559521	−	0.418852i	−0.559521	−	0.418852i	−0.841254	+	0.540641i	$0.818182\pi$
$500$			1.39788i			1.39788i
$501$		0			0
$502$		0			0
$503$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$503$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.17116	−	1.56449i	−1.17116	−	1.56449i
$508$		0			0
$509$	−0.153882	−	1.07028i	−0.153882	−	1.07028i	−0.909632	−	0.415415i	$-0.863636\pi$
$509$	−0.153882	−	1.07028i	−0.153882	−	1.07028i	0.755750	−	0.654861i	$-0.227273\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.76719	−	1.03211i	−2.76719	−	1.03211i
$516$		0			0
$517$	0.557730	−	1.89945i	0.557730	−	1.89945i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.415415	−	0.0903680i	0.415415	−	0.0903680i		−	1.00000i	$-0.5\pi$
$521$	0.415415	−	0.0903680i	0.415415	−	0.0903680i	0.415415	+	0.909632i	$0.363636\pi$
$522$		0			0
$523$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$523$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.936593	+	1.71524i	0.936593	+	1.71524i
$529$	0.740365	+	0.641530i	0.740365	+	0.641530i
$530$		0			0
$531$	−0.418852	−	1.92543i	−0.418852	−	1.92543i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	0.449181	+	2.06485i	0.449181	+	2.06485i
$538$		0			0
$539$	−0.755750	−	0.654861i	−0.755750	−	0.654861i
$540$	2.86684	+	5.25023i	2.86684	+	5.25023i
$541$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$541$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$542$		0			0
$543$	−3.54632	+	1.61955i	−3.54632	+	1.61955i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$547$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$548$	1.50013	−	0.559521i	1.50013	−	0.559521i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$	0.393828	+	1.34125i	0.393828	+	1.34125i
$556$		0			0
$557$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$557$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$563$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$564$	0.822373	−	3.78039i	0.822373	−	3.78039i
$565$	0.0838914	+	0.224922i	0.0838914	+	0.224922i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$569$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$570$		0			0
$571$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$571$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$572$		0			0
$573$	−0.973039	−	2.13066i	−0.973039	−	2.13066i
$574$		0			0
$575$	−0.255250	−	0.0555263i	−0.255250	−	0.0555263i
$576$	1.52421	+	2.37172i	1.52421	+	2.37172i
$577$	−0.0683785	+	0.956056i	−0.0683785	+	0.956056i	0.841254	+	0.540641i	$0.181818\pi$
$577$	−0.0683785	+	0.956056i	−0.0683785	+	0.956056i	−0.909632	+	0.415415i	$0.863636\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	1.29639	−	0.186393i	1.29639	−	0.186393i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	0.817178	−	1.27155i	0.817178	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$587$	0.817178	−	1.27155i	0.817178	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$588$	−1.56449	−	1.17116i	−1.56449	−	1.17116i
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.148568	+	0.398326i	0.148568	+	0.398326i
$593$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$593$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.42508	−	0.778150i	1.42508	−	0.778150i
$598$		0			0
$599$	0.373128	+	0.203743i	0.373128	+	0.203743i	0.654861	−	0.755750i	$-0.272727\pi$
$599$	0.373128	+	0.203743i	0.373128	+	0.203743i	−0.281733	+	0.959493i	$0.590909\pi$
$600$		0			0
$601$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$601$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$602$		0			0
$603$	−1.57239	−	0.226075i	−1.57239	−	0.226075i
$604$		0			0
$605$	−1.27155	+	1.10181i	−1.27155	+	1.10181i
$606$		0			0
$607$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$607$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$613$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.574406	−	1.05195i	−0.574406	−	1.05195i	−0.989821	−	0.142315i	$-0.954545\pi$
$617$	−0.574406	−	1.05195i	−0.574406	−	1.05195i	0.415415	−	0.909632i	$-0.363636\pi$
$618$		0			0
$619$	1.66538	−	1.07028i	1.66538	−	1.07028i	0.755750	−	0.654861i	$-0.227273\pi$
$619$	1.66538	−	1.07028i	1.66538	−	1.07028i	0.909632	−	0.415415i	$-0.136364\pi$
$620$	−0.627791	−	2.88591i	−0.627791	−	2.88591i
$621$	−0.486727	+	0.142916i	−0.486727	+	0.142916i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	+	0.146813i	−0.500000	+	0.146813i
$626$		0			0
$627$		0			0
$628$	−0.817178	−	0.708089i	−0.817178	−	0.708089i
$629$		0			0
$630$		0			0
$631$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$631$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$	2.50109	−	0.544078i	2.50109	−	0.544078i
$637$		0			0
$638$		0			0
$639$	2.24745	−	4.92123i	2.24745	−	4.92123i
$640$		0			0
$641$	−1.14231	+	0.989821i	−1.14231	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$641$	−1.14231	+	0.989821i	−1.14231	+	0.989821i	−1.00000			$\pi$
$642$		0			0
$643$	1.89945	+	0.273100i	1.89945	+	0.273100i	0.989821	−	0.142315i	$-0.0454545\pi$
$643$	1.89945	+	0.273100i	1.89945	+	0.273100i	0.909632	+	0.415415i	$0.136364\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−1.64468	−	0.898064i	−1.64468	−	0.898064i	−0.989821	−	0.142315i	$-0.954545\pi$
$647$	−1.64468	−	0.898064i	−1.64468	−	0.898064i	−0.654861	−	0.755750i	$-0.727273\pi$
$648$		0			0
$649$	0.613435	−	0.334961i	0.613435	−	0.334961i
$650$		0			0
$651$		0			0
$652$	−0.100889	−	1.41061i	−0.100889	−	1.41061i
$653$	−0.424047	+	1.94931i	−0.424047	+	1.94931i	−0.142315	+	0.989821i	$0.545455\pi$
$653$	−0.424047	+	1.94931i	−0.424047	+	1.94931i	−0.281733	+	0.959493i	$0.590909\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$659$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$660$	−2.32505	+	2.32505i	−2.32505	+	2.32505i
$661$	1.94931	+	0.139418i	1.94931	+	0.139418i	0.989821	−	0.142315i	$-0.0454545\pi$
$661$	1.94931	+	0.139418i	1.94931	+	0.139418i	0.959493	+	0.281733i	$0.0909091\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$673$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$674$		0			0
$675$	−4.60276	+	4.60276i	−4.60276	+	4.60276i
$676$	0.540641	−	0.841254i	0.540641	−	0.841254i
$677$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$677$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	0.139418	+	1.94931i	0.139418	+	1.94931i	0.281733	+	0.959493i	$0.409091\pi$
$683$	0.139418	+	1.94931i	0.139418	+	1.94931i	−0.142315	+	0.989821i	$0.545455\pi$
$684$		0			0
$685$	1.61435	+	2.15652i	1.61435	+	2.15652i
$686$		0			0
$687$	0.521178	+	3.62487i	0.521178	+	3.62487i
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$691$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$692$		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		−0.654861	−	0.755750i	$-0.727273\pi$
$701$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$702$		0			0
$703$		0			0
$704$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$705$	6.49270	−	0.464367i	6.49270	−	0.464367i
$706$		0			0
$707$		0			0
$708$	1.14908	−	0.738467i	1.14908	−	0.738467i
$709$	−0.340335	−	1.56449i	−0.340335	−	1.56449i	−0.755750	−	0.654861i	$-0.772727\pi$
$709$	−0.340335	−	1.56449i	−0.340335	−	1.56449i	0.415415	−	0.909632i	$-0.363636\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	0.250452			0.250452
$714$		0			0
$715$		0			0
$716$	−0.909632	+	0.584585i	−0.909632	+	0.584585i
$717$		0			0
$718$		0			0
$719$	−1.75089	+	0.125226i	−1.75089	+	0.125226i	−0.909632	−	0.415415i	$-0.863636\pi$
$719$	−1.75089	+	0.125226i	−1.75089	+	0.125226i	−0.841254	+	0.540641i	$0.818182\pi$
$720$	−3.10629	+	3.58485i	−3.10629	+	3.58485i
$721$		0			0
$722$		0			0
$723$		0			0
$724$	−1.41061	−	1.41061i	−1.41061	−	1.41061i
$725$		0			0
$726$		0			0
$727$	−0.956056	+	1.75089i	−0.956056	+	1.75089i	−0.415415	+	0.909632i	$0.636364\pi$
$727$	−0.956056	+	1.75089i	−0.956056	+	1.75089i	−0.540641	+	0.841254i	$0.681818\pi$
$728$		0			0
$729$	−1.32202	+	4.50237i	−1.32202	+	4.50237i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$733$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$734$		0			0
$735$	1.14908	−	3.08080i	1.14908	−	3.08080i
$736$		0			0
$737$	−0.0801894	−	0.557730i	−0.0801894	−	0.557730i
$738$		0			0
$739$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$739$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$740$	−0.572615	+	0.428654i	−0.572615	+	0.428654i
$741$		0			0
$742$		0			0
$743$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$743$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$751$	−0.118239	−	0.258908i	−0.118239	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$752$	1.95949	−	0.281733i	1.95949	−	0.281733i
$753$	−3.66456	−	0.797176i	−3.66456	−	0.797176i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.540641	+	0.158746i	0.540641	+	0.158746i	0.540641	−	0.841254i	$-0.318182\pi$
$757$	0.540641	+	0.158746i	0.540641	+	0.158746i			1.00000i	$0.5\pi$
$758$		0			0
$759$	−0.150750	−	0.234571i	−0.150750	−	0.234571i
$760$		0			0
$761$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$761$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$762$		0			0
$763$		0			0
$764$	0.847507	−	0.847507i	0.847507	−	0.847507i
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−1.17116	+	1.56449i	−1.17116	+	1.56449i
$769$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$769$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$770$		0			0
$771$	−0.755750	+	3.47412i	−0.755750	+	3.47412i
$772$		0			0
$773$	−0.767317	+	0.574406i	−0.767317	+	0.574406i	−0.909632	−	0.415415i	$-0.863636\pi$
$773$	−0.767317	+	0.574406i	−0.767317	+	0.574406i	0.142315	+	0.989821i	$0.454545\pi$
$774$		0			0
$775$	2.82065	−	1.54019i	2.82065	−	1.54019i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	1.89945	+	0.273100i	1.89945	+	0.273100i
$782$		0			0
$783$		0			0
$784$	0.281733	−	0.959493i	0.281733	−	0.959493i
$785$	0.755750	−	1.65486i	0.755750	−	1.65486i
$786$		0			0
$787$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$787$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$	2.06389	+	3.77974i	2.06389	+	3.77974i
$796$	0.627899	+	0.544078i	0.627899	+	0.544078i
$797$	1.41542	−	0.909632i	1.41542	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$797$	1.41542	−	0.909632i	1.41542	−	0.909632i	1.00000			$0$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−0.794278	+	2.70506i	−0.794278	+	2.70506i
$802$		0			0
$803$		0			0
$804$	−0.234072	−	1.07601i	−0.234072	−	1.07601i
$805$		0			0
$806$		0			0
$807$	1.22668	+	2.24649i	1.22668	+	2.24649i
$808$		0			0
$809$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$809$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$810$		0			0
$811$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$811$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.22940	−	0.831524i	2.22940	−	0.831524i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$821$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$822$		0			0
$823$	−0.909632	−	0.584585i	−0.909632	−	0.584585i		−	1.00000i	$-0.5\pi$
$823$	−0.909632	−	0.584585i	−0.909632	−	0.584585i	−0.909632	+	0.415415i	$0.863636\pi$
$824$		0			0
$825$	−3.14032	−	1.71474i	−3.14032	−	1.71474i
$826$		0			0
$827$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$827$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$828$	−0.241058	−	0.322016i	−0.241058	−	0.322016i
$829$	1.28173	−	0.959493i	1.28173	−	0.959493i	0.281733	−	0.959493i	$-0.409091\pi$
$829$	1.28173	−	0.959493i	1.28173	−	0.959493i	1.00000			$0$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	3.37412	−	5.25023i	3.37412	−	5.25023i
$838$		0			0
$839$	−1.59700	−	0.114220i	−1.59700	−	0.114220i	−0.755750	−	0.654861i	$-0.772727\pi$
$839$	−1.59700	−	0.114220i	−1.59700	−	0.114220i	−0.841254	+	0.540641i	$0.818182\pi$
$840$		0			0
$841$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.61435	+	0.474017i	1.61435	+	0.474017i
$846$		0			0
$847$		0			0
$848$	0.708089	+	1.10181i	0.708089	+	1.10181i
$849$		0			0
$850$		0			0
$851$	−0.0251978	−	0.0551755i	−0.0251978	−	0.0551755i
$852$	3.74071	+	0.267541i	3.74071	+	0.267541i
$853$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$853$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$857$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$858$		0			0
$859$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	0.841254	−	0.540641i	$-0.181818\pi$
$859$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	−0.989821	+	0.142315i	$0.954545\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.05195	+	1.40524i	1.05195	+	1.40524i	0.909632	+	0.415415i	$0.136364\pi$
$863$	1.05195	+	1.40524i	1.05195	+	1.40524i	0.142315	+	0.989821i	$0.454545\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−0.682956	+	1.83107i	−0.682956	+	1.83107i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.20055	−	4.08870i	1.20055	−	4.08870i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$877$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$878$		0			0
$879$		0			0
$880$	−1.53046	−	0.698939i	−1.53046	−	0.698939i
$881$	0.258908	−	0.118239i	0.258908	−	0.118239i	−0.281733	−	0.959493i	$-0.590909\pi$
$881$	0.258908	−	0.118239i	0.258908	−	0.118239i	0.540641	+	0.841254i	$0.318182\pi$
$882$		0			0
$883$	1.86912	−	0.133682i	1.86912	−	0.133682i	0.909632	−	0.415415i	$-0.136364\pi$
$883$	1.86912	−	0.133682i	1.86912	−	0.133682i	0.959493	+	0.281733i	$0.0909091\pi$
$884$		0			0
$885$	1.73683	+	1.50497i	1.73683	+	1.50497i
$886$		0			0
$887$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$887$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−4.12899			−4.12899
$892$		0			0
$893$		0			0
$894$		0			0
$895$	−1.37491	−	1.19136i	−1.37491	−	1.19136i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−4.69515	−	2.14420i	−4.69515	−	2.14420i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.60857	−	2.94588i	1.60857	−	2.94588i
$906$		0			0
$907$	0.0801894	−	0.273100i	0.0801894	−	0.273100i	−0.909632	−	0.415415i	$-0.863636\pi$
$907$	0.0801894	−	0.273100i	0.0801894	−	0.273100i	0.989821	+	0.142315i	$0.0454545\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	−0.989821	−	0.142315i	$-0.954545\pi$
$911$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	0.755750	−	0.654861i	$-0.227273\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.64468	+	0.898064i	−1.64468	+	0.898064i
$917$		0			0
$918$		0			0
$919$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$919$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.623095	−	0.466443i	−0.623095	−	0.466443i
$926$		0			0
$927$	3.49934	−	3.49934i	3.49934	−	3.49934i
$928$		0			0
$929$	0.822373	+	1.80075i	0.822373	+	1.80075i	0.540641	+	0.841254i	$0.318182\pi$
$929$	0.822373	+	1.80075i	0.822373	+	1.80075i	0.281733	+	0.959493i	$0.409091\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−0.210730	+	2.94639i	−0.210730	+	2.94639i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$937$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$938$		0			0
$939$	−3.85895	+	0.554833i	−3.85895	+	0.554833i
$940$	1.38365	+	3.02977i	1.38365	+	3.02977i
$941$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$941$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$942$		0			0
$943$		0			0
$944$	0.559521	+	0.418852i	0.559521	+	0.418852i
$945$		0			0
$946$		0			0
$947$	0.118239	−	0.822373i	0.118239	−	0.822373i	−0.841254	−	0.540641i	$-0.818182\pi$
$947$	0.118239	−	0.822373i	0.118239	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$	−0.445301	+	0.333348i	−0.445301	+	0.333348i
$952$		0			0
$953$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$953$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$954$		0			0
$955$	1.76991	+	0.966443i	1.76991	+	0.966443i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$	−3.08080	−	1.14908i	−3.08080	−	1.14908i
$961$	−1.57293	+	1.36295i	−1.57293	+	1.36295i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	0.989821	−	1.14231i	0.989821	−	1.14231i		−	1.00000i	$-0.5\pi$
$971$	0.989821	−	1.14231i	0.989821	−	1.14231i	0.989821	−	0.142315i	$-0.0454545\pi$
$972$	−4.50237	+	0.322016i	−4.50237	+	0.322016i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.654861	−	0.755750i	$-0.727273\pi$
$977$	−0.797176	+	0.234072i	−0.797176	+	0.234072i	−0.142315	+	0.989821i	$0.545455\pi$
$978$		0			0
$979$	−1.00000			−1.00000
$980$	1.68251			1.68251
$981$		0			0
$982$		0			0
$983$	1.10181	−	0.708089i	1.10181	−	0.708089i	0.142315	−	0.989821i	$-0.454545\pi$
$983$	1.10181	−	0.708089i	1.10181	−	0.708089i	0.959493	+	0.281733i	$0.0909091\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	1.38189	+	1.38189i	1.38189	+	1.38189i	0.841254	+	0.540641i	$0.181818\pi$
$991$	1.38189	+	1.38189i	1.38189	+	1.38189i	0.540641	+	0.841254i	$0.318182\pi$
$992$		0			0
$993$	1.03175	−	0.384822i	1.03175	−	0.384822i
$994$		0			0
$995$	−0.580699	+	1.27155i	−0.580699	+	1.27155i
$996$		0			0
$997$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$997$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$998$		0			0
$999$	−1.49611	−	0.215109i	−1.49611	−	0.215109i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	979.1.v.a.780.1	yes	20
11.10	odd	2	CM	979.1.v.a.780.1	yes	20
89.72	even	44	inner	979.1.v.a.428.1	✓	20
979.428	odd	44	inner	979.1.v.a.428.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
979.1.v.a.428.1	✓	20	89.72	even	44	inner
979.1.v.a.428.1	✓	20	979.428	odd	44	inner
979.1.v.a.780.1	yes	20	1.1	even	1	trivial
979.1.v.a.780.1	yes	20	11.10	odd	2	CM

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:	\( N \)	\(=\)	\( 979 = 11 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	979.v (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.415415 - 1.90963i) q^{3} +(0.841254 - 0.540641i) q^{4} +(1.27155 + 1.10181i) q^{5} +(-2.56449 + 1.17116i) q^{9} +O(q^{10})\)	\(q+(-0.415415 - 1.90963i) q^{3} +(0.841254 - 0.540641i) q^{4} +(1.27155 + 1.10181i) q^{5} +(-2.56449 + 1.17116i) q^{9} +(-0.654861 - 0.755750i) q^{11} +(-1.38189 - 1.38189i) q^{12} +(1.57582 - 2.88591i) q^{15} +(0.415415 - 0.909632i) q^{16} +(1.66538 + 0.239446i) q^{20} +(-0.0498610 + 0.133682i) q^{23} +(0.260554 + 1.81219i) q^{25} +(2.13066 + 2.84623i) q^{27} +(-0.613435 - 1.64468i) q^{31} +(-1.17116 + 1.56449i) q^{33} +(-1.52421 + 2.37172i) q^{36} +(-0.300613 + 0.300613i) q^{37} +(-0.959493 - 0.281733i) q^{44} +(-4.55129 - 1.33638i) q^{45} +(1.07028 + 1.66538i) q^{47} +(-1.90963 - 0.415415i) q^{48} +(0.989821 - 0.142315i) q^{49} +(-0.708089 + 1.10181i) q^{53} -1.68251i q^{55} +(-0.148568 + 0.682956i) q^{59} +(-0.234571 - 3.27974i) q^{60} +(-0.142315 - 0.989821i) q^{64} +(0.474017 + 0.304632i) q^{67} +(0.275997 + 0.0396824i) q^{69} +(-1.45027 + 1.25667i) q^{71} +(3.35239 - 1.25038i) q^{75} +(1.53046 - 0.698939i) q^{80} +(2.70391 - 3.12048i) q^{81} +(0.654861 - 0.755750i) q^{89} +(0.0303285 + 0.139418i) q^{92} +(-2.88591 + 1.85466i) q^{93} +(-0.989821 + 1.14231i) q^{97} +(2.56449 + 1.17116i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{3} - 2 q^{4} - 22 q^{9}+O(q^{10}) \) 20 * q + 2 * q^3 - 2 * q^4 - 22 * q^9	\( 20 q + 2 q^{3} - 2 q^{4} - 22 q^{9} - 2 q^{11} + 2 q^{12} + 4 q^{15} - 2 q^{16} - 2 q^{23} + 6 q^{25} + 2 q^{31} + 2 q^{33} + 2 q^{37} - 2 q^{44} - 4 q^{45} - 20 q^{48} - 2 q^{59} + 4 q^{60} - 2 q^{64} - 6 q^{75} + 20 q^{81} + 2 q^{89} - 2 q^{92} - 4 q^{93} + 22 q^{99}+O(q^{100}) \) 20 * q + 2 * q^3 - 2 * q^4 - 22 * q^9 - 2 * q^11 + 2 * q^12 + 4 * q^15 - 2 * q^16 - 2 * q^23 + 6 * q^25 + 2 * q^31 + 2 * q^33 + 2 * q^37 - 2 * q^44 - 4 * q^45 - 20 * q^48 - 2 * q^59 + 4 * q^60 - 2 * q^64 - 6 * q^75 + 20 * q^81 + 2 * q^89 - 2 * q^92 - 4 * q^93 + 22 * q^99

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