LMFDB - Embedded newform 712.1.y.a.403.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [712,1,Mod(99,712)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(712, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 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("712.99");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 712 = 2^{3} \cdot 89 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	712.y (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:	$0.355334288995$
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:	$\Z[a_1, a_2, a_3]$
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			403.1
Root			$0.281733 - 0.959493i$ of defining polynomial
Character	$\chi$	$=$	712.403
Dual form			712.1.y.a.659.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.841254 + 0.540641i) q^{2} +(0.613435 - 1.64468i) q^{3} +(0.415415 - 0.909632i) q^{4} +(0.373128 + 1.71524i) q^{6} +(0.142315 + 0.989821i) q^{8} +(-1.57293 - 1.36295i) q^{9} +O(q^{10})$	\(q+(-0.841254 + 0.540641i) q^{2} +(0.613435 - 1.64468i) q^{3} +(0.415415 - 0.909632i) q^{4} +(0.373128 + 1.71524i) q^{6} +(0.142315 + 0.989821i) q^{8} +(-1.57293 - 1.36295i) q^{9} +(0.281733 - 1.95949i) q^{11} +(-1.24123 - 1.24123i) q^{12} +(-0.654861 - 0.755750i) q^{16} +(-0.909632 + 1.41542i) q^{17} +(2.06010 + 0.296197i) q^{18} +(0.0855040 + 1.19550i) q^{19} +(0.822373 + 1.80075i) q^{22} +(1.71524 + 0.373128i) q^{24} +(0.959493 - 0.281733i) q^{25} +(-1.66587 + 0.909632i) q^{27} +(0.959493 + 0.281733i) q^{32} +(-3.04992 - 1.66538i) q^{33} -1.68251i q^{34} +(-1.89320 + 0.864596i) q^{36} +(-0.718267 - 0.959493i) q^{38} +(-1.32505 + 0.494217i) q^{41} +(1.56449 - 1.17116i) q^{43} +(-1.66538 - 1.07028i) q^{44} +(-1.64468 + 0.613435i) q^{48} +(-0.281733 - 0.959493i) q^{49} +(-0.654861 + 0.755750i) q^{50} +(1.76991 + 2.36432i) q^{51} +(0.909632 - 1.66587i) q^{54} +(2.01867 + 0.592735i) q^{57} +(-0.148568 - 0.398326i) q^{59} +(-0.959493 + 0.281733i) q^{64} +(3.46613 - 0.247902i) q^{66} +(0.627899 + 1.37491i) q^{67} +(0.909632 + 1.41542i) q^{68} +(1.12523 - 1.75089i) q^{72} +(0.368991 + 0.425839i) q^{73} +(0.125226 - 1.75089i) q^{75} +(1.12299 + 0.418852i) q^{76} +(0.177958 + 1.23772i) q^{81} +(0.847507 - 1.13214i) q^{82} +(0.148568 + 0.682956i) q^{83} +(-0.682956 + 1.83107i) q^{86} +1.97964 q^{88} +(0.959493 + 0.281733i) q^{89} +(1.05195 - 1.40524i) q^{96} +(0.186393 + 1.29639i) q^{97} +(0.755750 + 0.654861i) q^{98} +(-3.11384 + 2.69815i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8}+O(q^{10}) $ 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8	$ 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} - 2 q^{12} - 2 q^{16} - 2 q^{19} + 2 q^{24} + 2 q^{25} - 22 q^{27} + 2 q^{32} - 20 q^{38} + 2 q^{41} + 2 q^{43} - 2 q^{48} - 2 q^{50} + 4 q^{51} - 4 q^{57} - 2 q^{59} - 2 q^{64} + 22 q^{72} + 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{86} + 2 q^{89} + 2 q^{96} - 4 q^{97}+O(q^{100}) $ 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8 - 2 * q^12 - 2 * q^16 - 2 * q^19 + 2 * q^24 + 2 * q^25 - 22 * q^27 + 2 * q^32 - 20 * q^38 + 2 * q^41 + 2 * q^43 - 2 * q^48 - 2 * q^50 + 4 * q^51 - 4 * q^57 - 2 * q^59 - 2 * q^64 + 22 * q^72 + 2 * q^75 - 2 * q^76 - 2 * q^81 - 2 * q^82 + 2 * q^83 - 2 * q^86 + 2 * q^89 + 2 * q^96 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/712\mathbb{Z}\right)^\times$.

$n$	$357$	$535$	$537$
$\chi(n)$	$-1$	$-1$	$e\left(\frac{27}{44}\right)$

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 $

$2$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$2$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$3$	0.613435	−	1.64468i	0.613435	−	1.64468i	−0.142315	−	0.989821i	$-0.545455\pi$
$3$	0.613435	−	1.64468i	0.613435	−	1.64468i	0.755750	−	0.654861i	$-0.227273\pi$
$4$	0.415415	−	0.909632i	0.415415	−	0.909632i
$5$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$5$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$6$	0.373128	+	1.71524i	0.373128	+	1.71524i
$7$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$7$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$8$	0.142315	+	0.989821i	0.142315	+	0.989821i
$9$	−1.57293	−	1.36295i	−1.57293	−	1.36295i
$10$		0			0
$11$	0.281733	−	1.95949i	0.281733	−	1.95949i		−	1.00000i	$-0.5\pi$
$11$	0.281733	−	1.95949i	0.281733	−	1.95949i	0.281733	−	0.959493i	$-0.409091\pi$
$12$	−1.24123	−	1.24123i	−1.24123	−	1.24123i
$13$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$13$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$14$		0			0
$15$		0			0
$16$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$17$	−0.909632	+	1.41542i	−0.909632	+	1.41542i			1.00000i	$0.5\pi$
$17$	−0.909632	+	1.41542i	−0.909632	+	1.41542i	−0.909632	+	0.415415i	$0.863636\pi$
$18$	2.06010	+	0.296197i	2.06010	+	0.296197i
$19$	0.0855040	+	1.19550i	0.0855040	+	1.19550i	0.841254	+	0.540641i	$0.181818\pi$
$19$	0.0855040	+	1.19550i	0.0855040	+	1.19550i	−0.755750	+	0.654861i	$0.772727\pi$
$20$		0			0
$21$		0			0
$22$	0.822373	+	1.80075i	0.822373	+	1.80075i
$23$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$23$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$24$	1.71524	+	0.373128i	1.71524	+	0.373128i
$25$	0.959493	−	0.281733i	0.959493	−	0.281733i
$26$		0			0
$27$	−1.66587	+	0.909632i	−1.66587	+	0.909632i
$28$		0			0
$29$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$29$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$30$		0			0
$31$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$31$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$32$	0.959493	+	0.281733i	0.959493	+	0.281733i
$33$	−3.04992	−	1.66538i	−3.04992	−	1.66538i
$34$		−	1.68251i		−	1.68251i
$35$		0			0
$36$	−1.89320	+	0.864596i	−1.89320	+	0.864596i
$37$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$37$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$38$	−0.718267	−	0.959493i	−0.718267	−	0.959493i
$39$		0			0
$40$		0			0
$41$	−1.32505	+	0.494217i	−1.32505	+	0.494217i	−0.909632	−	0.415415i	$-0.863636\pi$
$41$	−1.32505	+	0.494217i	−1.32505	+	0.494217i	−0.415415	+	0.909632i	$0.636364\pi$
$42$		0			0
$43$	1.56449	−	1.17116i	1.56449	−	1.17116i	0.654861	−	0.755750i	$-0.272727\pi$
$43$	1.56449	−	1.17116i	1.56449	−	1.17116i	0.909632	−	0.415415i	$-0.136364\pi$
$44$	−1.66538	−	1.07028i	−1.66538	−	1.07028i
$45$		0			0
$46$		0			0
$47$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$47$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$48$	−1.64468	+	0.613435i	−1.64468	+	0.613435i
$49$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$50$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$51$	1.76991	+	2.36432i	1.76991	+	2.36432i
$52$		0			0
$53$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$53$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$54$	0.909632	−	1.66587i	0.909632	−	1.66587i
$55$		0			0
$56$		0			0
$57$	2.01867	+	0.592735i	2.01867	+	0.592735i
$58$		0			0
$59$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	0.841254	−	0.540641i	$-0.181818\pi$
$59$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	−0.989821	+	0.142315i	$0.954545\pi$
$60$		0			0
$61$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$61$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$62$		0			0
$63$		0			0
$64$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$65$		0			0
$66$	3.46613	−	0.247902i	3.46613	−	0.247902i
$67$	0.627899	+	1.37491i	0.627899	+	1.37491i	0.909632	+	0.415415i	$0.136364\pi$
$67$	0.627899	+	1.37491i	0.627899	+	1.37491i	−0.281733	+	0.959493i	$0.590909\pi$
$68$	0.909632	+	1.41542i	0.909632	+	1.41542i
$69$		0			0
$70$		0			0
$71$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$71$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$72$	1.12523	−	1.75089i	1.12523	−	1.75089i
$73$	0.368991	+	0.425839i	0.368991	+	0.425839i	0.909632	−	0.415415i	$-0.136364\pi$
$73$	0.368991	+	0.425839i	0.368991	+	0.425839i	−0.540641	+	0.841254i	$0.681818\pi$
$74$		0			0
$75$	0.125226	−	1.75089i	0.125226	−	1.75089i
$76$	1.12299	+	0.418852i	1.12299	+	0.418852i
$77$		0			0
$78$		0			0
$79$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$79$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$80$		0			0
$81$	0.177958	+	1.23772i	0.177958	+	1.23772i
$82$	0.847507	−	1.13214i	0.847507	−	1.13214i
$83$	0.148568	+	0.682956i	0.148568	+	0.682956i	0.989821	+	0.142315i	$0.0454545\pi$
$83$	0.148568	+	0.682956i	0.148568	+	0.682956i	−0.841254	+	0.540641i	$0.818182\pi$
$84$		0			0
$85$		0			0
$86$	−0.682956	+	1.83107i	−0.682956	+	1.83107i
$87$		0			0
$88$	1.97964			1.97964
$89$	0.959493	+	0.281733i	0.959493	+	0.281733i
$89$	0.959493	+	0.281733i	0.959493	+	0.281733i
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	1.05195	−	1.40524i	1.05195	−	1.40524i
$97$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$97$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$98$	0.755750	+	0.654861i	0.755750	+	0.654861i
$99$	−3.11384	+	2.69815i	−3.11384	+	2.69815i
$100$	0.142315	−	0.989821i	0.142315	−	0.989821i
$101$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$101$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$102$	−2.76719	−	1.03211i	−2.76719	−	1.03211i
$103$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$103$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$104$		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	$0.545455\pi$
$108$	0.135404	+	1.89320i	0.135404	+	1.89320i
$109$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$109$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.682956	+	0.148568i	0.682956	+	0.148568i	0.540641	−	0.841254i	$-0.318182\pi$
$113$	0.682956	+	0.148568i	0.682956	+	0.148568i	0.142315	+	0.989821i	$0.454545\pi$
$114$	−2.01867	+	0.592735i	−2.01867	+	0.592735i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	0.340335	+	0.254771i	0.340335	+	0.254771i
$119$		0			0
$120$		0			0
$121$	−2.80075	−	0.822373i	−2.80075	−	0.822373i
$122$		0			0
$123$			2.48245i			2.48245i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$127$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$128$	0.654861	−	0.755750i	0.654861	−	0.755750i
$129$	−0.966479	−	3.29153i	−0.966479	−	3.29153i
$130$		0			0
$131$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$131$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.755750	+	0.654861i	$0.227273\pi$
$132$	−2.78187	+	2.08248i	−2.78187	+	2.08248i
$133$		0			0
$134$	−1.27155	−	0.817178i	−1.27155	−	0.817178i
$135$		0			0
$136$	−1.53046	−	0.698939i	−1.53046	−	0.698939i
$137$	0.398326	−	0.148568i	0.398326	−	0.148568i	−0.142315	−	0.989821i	$-0.545455\pi$
$137$	0.398326	−	0.148568i	0.398326	−	0.148568i	0.540641	+	0.841254i	$0.318182\pi$
$138$		0			0
$139$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	−0.841254	−	0.540641i	$-0.818182\pi$
$139$	−0.186393	+	0.215109i	−0.186393	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$			2.08128i			2.08128i
$145$		0			0
$146$	−0.540641	−	0.158746i	−0.540641	−	0.158746i
$147$	−1.75089	−	0.125226i	−1.75089	−	0.125226i
$148$		0			0
$149$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$149$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$150$	0.841254	+	1.54064i	0.841254	+	1.54064i
$151$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$151$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$152$	−1.17116	+	0.254771i	−1.17116	+	0.254771i
$153$	3.35992	−	0.986563i	3.35992	−	0.986563i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$157$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	−0.818872	−	0.945028i	−0.818872	−	0.945028i
$163$	0.203743	−	0.936593i	0.203743	−	0.936593i	−0.755750	−	0.654861i	$-0.772727\pi$
$163$	0.203743	−	0.936593i	0.203743	−	0.936593i	0.959493	−	0.281733i	$-0.0909091\pi$
$164$	−0.100889	+	1.41061i	−0.100889	+	1.41061i
$165$		0			0
$166$	−0.494217	−	0.494217i	−0.494217	−	0.494217i
$167$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$167$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$168$		0			0
$169$	0.755750	+	0.654861i	0.755750	+	0.654861i
$170$		0			0
$171$	1.49492	−	1.99698i	1.49492	−	1.99698i
$172$	−0.415415	−	1.90963i	−0.415415	−	1.90963i
$173$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$173$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$174$		0			0
$175$		0			0
$176$	−1.66538	+	1.07028i	−1.66538	+	1.07028i
$177$	−0.746256			−0.746256
$178$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$179$	−1.30972			−1.30972			−0.654861	−	0.755750i	$-0.727273\pi$
$179$	−1.30972			−1.30972			−0.654861	+	0.755750i	$0.727273\pi$
$180$		0			0
$181$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$181$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.51722	+	2.18119i	2.51722	+	2.18119i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$191$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$192$	−0.125226	+	1.75089i	−0.125226	+	1.75089i
$193$	−0.424047	+	1.94931i	−0.424047	+	1.94931i	−0.142315	+	0.989821i	$0.545455\pi$
$193$	−0.424047	+	1.94931i	−0.424047	+	1.94931i	−0.281733	+	0.959493i	$0.590909\pi$
$194$	−0.857685	−	0.989821i	−0.857685	−	0.989821i
$195$		0			0
$196$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$197$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$197$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$198$	1.16079	−	3.95330i	1.16079	−	3.95330i
$199$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$199$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$200$	0.415415	+	0.909632i	0.415415	+	0.909632i
$201$	2.64646	−	0.189279i	2.64646	−	0.189279i
$202$		0			0
$203$		0			0
$204$	2.88591	−	0.627791i	2.88591	−	0.627791i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.36667	+	0.169267i	2.36667	+	0.169267i
$210$		0			0
$211$	−1.71524	−	0.936593i	−1.71524	−	0.936593i	−0.959493	−	0.281733i	$-0.909091\pi$
$211$	−1.71524	−	0.936593i	−1.71524	−	0.936593i	−0.755750	−	0.654861i	$-0.772727\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$	−1.13745	−	1.51946i	−1.13745	−	1.51946i
$217$		0			0
$218$		0			0
$219$	0.926721	−	0.345649i	0.926721	−	0.345649i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$223$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$224$		0			0
$225$	−1.89320	−	0.864596i	−1.89320	−	0.864596i
$226$	−0.654861	+	0.244250i	−0.654861	+	0.244250i
$227$	−0.158746	−	0.540641i	−0.158746	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$227$	−0.158746	−	0.540641i	−0.158746	−	0.540641i	−1.00000			$\pi$
$228$	1.37776	−	1.59002i	1.37776	−	1.59002i
$229$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$229$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			0.830830i			0.830830i	0.909632	+	0.415415i	$0.136364\pi$
$233$			0.830830i			0.830830i	−0.909632	+	0.415415i	$0.863636\pi$
$234$		0			0
$235$		0			0
$236$	−0.424047	−	0.0303285i	−0.424047	−	0.0303285i
$237$		0			0
$238$		0			0
$239$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$239$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$240$		0			0
$241$	−1.83107	+	0.398326i	−1.83107	+	0.398326i	−0.989821	−	0.142315i	$-0.954545\pi$
$241$	−1.83107	+	0.398326i	−1.83107	+	0.398326i	−0.841254	+	0.540641i	$0.818182\pi$
$242$	2.80075	−	0.822373i	2.80075	−	0.822373i
$243$	0.290168	+	0.0631221i	0.290168	+	0.0631221i
$244$		0			0
$245$		0			0
$246$	−1.34211	−	2.08837i	−1.34211	−	2.08837i
$247$		0			0
$248$		0			0
$249$	1.21438	+	0.174602i	1.21438	+	0.174602i
$250$		0			0
$251$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$251$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$257$	−0.215109	+	0.186393i	−0.215109	+	0.186393i	−0.755750	−	0.654861i	$-0.772727\pi$
$257$	−0.215109	+	0.186393i	−0.215109	+	0.186393i	0.540641	+	0.841254i	$0.318182\pi$
$258$	2.59259	+	2.24649i	2.59259	+	2.24649i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	−1.89945	+	0.273100i	−1.89945	+	0.273100i
$263$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$263$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$264$	1.21438	−	3.25588i	1.21438	−	3.25588i
$265$		0			0
$266$		0			0
$267$	1.05195	−	1.40524i	1.05195	−	1.40524i
$268$	1.51150			1.51150
$269$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$269$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$270$		0			0
$271$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$271$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$272$	1.66538	−	0.239446i	1.66538	−	0.239446i
$273$		0			0
$274$	−0.254771	+	0.340335i	−0.254771	+	0.340335i
$275$	−0.281733	−	1.95949i	−0.281733	−	1.95949i
$276$		0			0
$277$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$277$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$278$	0.0405070	−	0.281733i	0.0405070	−	0.281733i
$279$		0			0
$280$		0			0
$281$	0.114220	−	1.59700i	0.114220	−	1.59700i	−0.540641	−	0.841254i	$-0.681818\pi$
$281$	0.114220	−	1.59700i	0.114220	−	1.59700i	0.654861	−	0.755750i	$-0.272727\pi$
$282$		0			0
$283$	−1.19136	−	1.37491i	−1.19136	−	1.37491i	−0.909632	−	0.415415i	$-0.863636\pi$
$283$	−1.19136	−	1.37491i	−1.19136	−	1.37491i	−0.281733	−	0.959493i	$-0.590909\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.12523	−	1.75089i	−1.12523	−	1.75089i
$289$	−0.760554	−	1.66538i	−0.760554	−	1.66538i
$290$		0			0
$291$	2.24649	+	0.488694i	2.24649	+	0.488694i
$292$	0.540641	−	0.158746i	0.540641	−	0.158746i
$293$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$293$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$294$	1.54064	−	0.841254i	1.54064	−	0.841254i
$295$		0			0
$296$		0			0
$297$	1.31309	+	3.52053i	1.31309	+	3.52053i
$298$		0			0
$299$		0			0
$300$	−1.54064	−	0.841254i	−1.54064	−	0.841254i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	0.847507	−	0.847507i	0.847507	−	0.847507i
$305$		0			0
$306$	−2.29317	+	2.64646i	−2.29317	+	2.64646i
$307$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	−0.989821	−	0.142315i	$-0.954545\pi$
$307$	−0.234072	−	0.797176i	−0.234072	−	0.797176i	0.755750	−	0.654861i	$-0.227273\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$311$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$312$		0			0
$313$	−1.59700	+	1.19550i	−1.59700	+	1.19550i	−0.755750	+	0.654861i	$0.772727\pi$
$313$	−1.59700	+	1.19550i	−1.59700	+	1.19550i	−0.841254	+	0.540641i	$0.818182\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$317$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.76991	−	0.966443i	−1.76991	−	0.966443i
$324$	1.19980	+	0.352293i	1.19980	+	0.352293i
$325$		0			0
$326$	0.334961	+	0.898064i	0.334961	+	0.898064i
$327$		0			0
$328$	−0.677760	−	1.24123i	−0.677760	−	1.24123i
$329$		0			0
$330$		0			0
$331$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$331$	−0.273100	+	0.0801894i	−0.273100	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$332$	0.682956	+	0.148568i	0.682956	+	0.148568i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.114220	+	1.59700i	0.114220	+	1.59700i	0.654861	+	0.755750i	$0.272727\pi$
$337$	0.114220	+	1.59700i	0.114220	+	1.59700i	−0.540641	+	0.841254i	$0.681818\pi$
$338$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$339$	0.663296	−	1.03211i	0.663296	−	1.03211i
$340$		0			0
$341$		0			0
$342$	−0.177958	+	2.48818i	−0.177958	+	2.48818i
$343$		0			0
$344$	1.38189	+	1.38189i	1.38189	+	1.38189i
$345$		0			0
$346$		0			0
$347$	−1.37491	−	1.19136i	−1.37491	−	1.19136i	−0.959493	−	0.281733i	$-0.909091\pi$
$347$	−1.37491	−	1.19136i	−1.37491	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$348$		0			0
$349$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$349$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$350$		0			0
$351$		0			0
$352$	0.822373	−	1.80075i	0.822373	−	1.80075i
$353$	−0.697148	+	1.86912i	−0.697148	+	1.86912i	−0.281733	+	0.959493i	$0.590909\pi$
$353$	−0.697148	+	1.86912i	−0.697148	+	1.86912i	−0.415415	+	0.909632i	$0.636364\pi$
$354$	0.627791	−	0.403457i	0.627791	−	0.403457i
$355$		0			0
$356$	0.654861	−	0.755750i	0.654861	−	0.755750i
$357$		0			0
$358$	1.10181	−	0.708089i	1.10181	−	0.708089i
$359$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$359$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$360$		0			0
$361$	−0.432092	+	0.0621254i	−0.432092	+	0.0621254i
$362$		0			0
$363$	−3.07062	+	4.10187i	−3.07062	+	4.10187i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$367$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$368$		0			0
$369$	2.75780	+	1.02860i	2.75780	+	1.02860i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$373$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$374$	−3.29686	−	0.474017i	−3.29686	−	0.474017i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−0.142315	+	0.0101786i	−0.142315	+	0.0101786i	−0.142315	−	0.989821i	$-0.545455\pi$
$379$	−0.142315	+	0.0101786i	−0.142315	+	0.0101786i			1.00000i	$0.5\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$383$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$384$	−0.841254	−	1.54064i	−0.841254	−	1.54064i
$385$		0			0
$386$	−0.697148	−	1.86912i	−0.697148	−	1.86912i
$387$	−4.05707	−	0.290168i	−4.05707	−	0.290168i
$388$	1.25667	+	0.368991i	1.25667	+	0.368991i
$389$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$389$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$390$		0			0
$391$		0			0
$392$	0.909632	−	0.415415i	0.909632	−	0.415415i
$393$	2.38189	−	2.38189i	2.38189	−	2.38189i
$394$		0			0
$395$		0			0
$396$	1.16079	+	3.95330i	1.16079	+	3.95330i
$397$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$397$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$398$		0			0
$399$		0			0
$400$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$401$	1.27155	+	0.817178i	1.27155	+	0.817178i	0.989821	−	0.142315i	$-0.0454545\pi$
$401$	1.27155	+	0.817178i	1.27155	+	0.817178i	0.281733	+	0.959493i	$0.409091\pi$
$402$	−2.12401	+	1.59002i	−2.12401	+	1.59002i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	−2.08837	+	2.08837i	−2.08837	+	2.08837i
$409$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.281733	−	0.959493i	$-0.409091\pi$
$409$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.909632	+	0.415415i	$0.136364\pi$
$410$		0			0
$411$		−	0.746256i		−	0.746256i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	0.239446	+	0.438512i	0.239446	+	0.438512i
$418$	−2.08248	+	1.13712i	−2.08248	+	1.13712i
$419$	1.94931	−	0.424047i	1.94931	−	0.424047i	0.959493	−	0.281733i	$-0.0909091\pi$
$419$	1.94931	−	0.424047i	1.94931	−	0.424047i	0.989821	−	0.142315i	$-0.0454545\pi$
$420$		0			0
$421$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$421$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$422$	1.94931	−	0.139418i	1.94931	−	0.139418i
$423$		0			0
$424$		0			0
$425$	−0.474017	+	1.61435i	−0.474017	+	1.61435i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$431$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$432$	1.77836	+	0.663296i	1.77836	+	0.663296i
$433$	−0.677760	−	0.677760i	−0.677760	−	0.677760i	0.281733	−	0.959493i	$-0.409091\pi$
$433$	−0.677760	−	0.677760i	−0.677760	−	0.677760i	−0.959493	+	0.281733i	$0.909091\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	−0.592735	+	0.791802i	−0.592735	+	0.791802i
$439$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$439$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$440$		0			0
$441$	−0.864596	+	1.89320i	−0.864596	+	1.89320i
$442$		0			0
$443$	−1.53046	+	0.983568i	−1.53046	+	0.983568i	−0.540641	+	0.841254i	$0.681818\pi$
$443$	−1.53046	+	0.983568i	−1.53046	+	0.983568i	−0.989821	+	0.142315i	$0.954545\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.755750	+	1.65486i	−0.755750	+	1.65486i			1.00000i	$0.5\pi$
$449$	−0.755750	+	1.65486i	−0.755750	+	1.65486i	−0.755750	+	0.654861i	$0.772727\pi$
$450$	2.06010	−	0.296197i	2.06010	−	0.296197i
$451$	0.595106	+	2.73566i	0.595106	+	2.73566i
$452$	0.418852	−	0.559521i	0.418852	−	0.559521i
$453$		0			0
$454$	0.425839	+	0.368991i	0.425839	+	0.368991i
$455$		0			0
$456$	−0.299415	+	2.08248i	−0.299415	+	2.08248i
$457$	0.100889	+	0.100889i	0.100889	+	0.100889i	0.755750	−	0.654861i	$-0.227273\pi$
$457$	0.100889	+	0.100889i	0.100889	+	0.100889i	−0.654861	+	0.755750i	$0.727273\pi$
$458$		0			0
$459$	0.227819	−	3.18532i	0.227819	−	3.18532i
$460$		0			0
$461$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$461$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$462$		0			0
$463$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$463$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$464$		0			0
$465$		0			0
$466$	−0.449181	−	0.698939i	−0.449181	−	0.698939i
$467$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	−0.841254	−	0.540641i	$-0.818182\pi$
$467$	−0.698939	−	1.53046i	−0.698939	−	1.53046i	0.142315	−	0.989821i	$-0.454545\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	0.373128	−	0.203743i	0.373128	−	0.203743i
$473$	−1.85412	−	3.39557i	−1.85412	−	3.39557i
$474$		0			0
$475$	0.418852	+	1.12299i	0.418852	+	1.12299i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$	1.32505	−	1.32505i	1.32505	−	1.32505i
$483$		0			0
$484$	−1.91153	+	2.20602i	−1.91153	+	2.20602i
$485$		0			0
$486$	−0.278231	+	0.103775i	−0.278231	+	0.103775i
$487$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$487$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$488$		0			0
$489$	−1.41542	−	0.909632i	−1.41542	−	0.909632i
$490$		0			0
$491$	1.28173	−	0.959493i	1.28173	−	0.959493i	0.281733	−	0.959493i	$-0.409091\pi$
$491$	1.28173	−	0.959493i	1.28173	−	0.959493i	1.00000			$0$
$492$	2.25812	+	1.03125i	2.25812	+	1.03125i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−1.11600	+	0.509660i	−1.11600	+	0.509660i
$499$	−0.898064	+	1.64468i	−0.898064	+	1.64468i	−0.142315	+	0.989821i	$0.545455\pi$
$499$	−0.898064	+	1.64468i	−0.898064	+	1.64468i	−0.755750	+	0.654861i	$0.772727\pi$
$500$		0			0
$501$		0			0
$502$	0.797176	+	0.234072i	0.797176	+	0.234072i
$503$		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
$507$	1.54064	−	0.841254i	1.54064	−	0.841254i
$508$		0			0
$509$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$509$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$510$		0			0
$511$		0			0
$512$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$513$	−1.22990	−	1.91377i	−1.22990	−	1.91377i
$514$	0.0801894	−	0.273100i	0.0801894	−	0.273100i
$515$		0			0
$516$	−3.39557	−	0.488209i	−3.39557	−	0.488209i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.83107	−	0.682956i	−1.83107	−	0.682956i	−0.989821	−	0.142315i	$-0.954545\pi$
$521$	−1.83107	−	0.682956i	−1.83107	−	0.682956i	−0.841254	−	0.540641i	$-0.818182\pi$
$522$		0			0
$523$	0.0801894	−	0.557730i	0.0801894	−	0.557730i	−0.909632	−	0.415415i	$-0.863636\pi$
$523$	0.0801894	−	0.557730i	0.0801894	−	0.557730i	0.989821	−	0.142315i	$-0.0454545\pi$
$524$	1.45027	−	1.25667i	1.45027	−	1.25667i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	0.738661	+	3.39557i	0.738661	+	3.39557i
$529$	0.989821	−	0.142315i	0.989821	−	0.142315i
$530$		0			0
$531$	−0.309212	+	0.829029i	−0.309212	+	0.829029i
$532$		0			0
$533$		0			0
$534$	−0.125226	+	1.75089i	−0.125226	+	1.75089i
$535$		0			0
$536$	−1.27155	+	0.817178i	−1.27155	+	0.817178i
$537$	−0.803429	+	2.15408i	−0.803429	+	2.15408i
$538$		0			0
$539$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$540$		0			0
$541$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$541$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$542$		0			0
$543$		0			0
$544$	−1.27155	+	1.10181i	−1.27155	+	1.10181i
$545$		0			0
$546$		0			0
$547$	−1.50013	−	0.559521i	−1.50013	−	0.559521i	−0.540641	−	0.841254i	$-0.681818\pi$
$547$	−1.50013	−	0.559521i	−1.50013	−	0.559521i	−0.959493	+	0.281733i	$0.909091\pi$
$548$	0.0303285	−	0.424047i	0.0303285	−	0.424047i
$549$		0			0
$550$	1.29639	+	1.49611i	1.29639	+	1.49611i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	0.118239	+	0.258908i	0.118239	+	0.258908i
$557$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$557$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	5.13151	−	2.80202i	5.13151	−	2.80202i
$562$	0.767317	+	1.40524i	0.767317	+	1.40524i
$563$	1.28173	+	0.959493i	1.28173	+	0.959493i	1.00000			$0$
$563$	1.28173	+	0.959493i	1.28173	+	0.959493i	0.281733	+	0.959493i	$0.409091\pi$
$564$		0			0
$565$		0			0
$566$	1.74557	+	0.512546i	1.74557	+	0.512546i
$567$		0			0
$568$		0			0
$569$	−0.0683785	+	0.125226i	−0.0683785	+	0.125226i	−0.909632	−	0.415415i	$-0.863636\pi$
$569$	−0.0683785	+	0.125226i	−0.0683785	+	0.125226i	0.841254	+	0.540641i	$0.181818\pi$
$570$		0			0
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i			1.00000i	$0.5\pi$
$571$	−1.00000	+	1.00000i	−1.00000	+	1.00000i	−1.00000			$\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.89320	+	0.864596i	1.89320	+	0.864596i
$577$	0.959493	−	0.718267i	0.959493	−	0.718267i		−	1.00000i	$-0.5\pi$
$577$	0.959493	−	0.718267i	0.959493	−	0.718267i	0.959493	+	0.281733i	$0.0909091\pi$
$578$	1.54019	+	0.989821i	1.54019	+	0.989821i
$579$	2.94588	+	1.89320i	2.94588	+	1.89320i
$580$		0			0
$581$		0			0
$582$	−2.15408	+	0.803429i	−2.15408	+	0.803429i
$583$		0			0
$584$	−0.368991	+	0.425839i	−0.368991	+	0.425839i
$585$		0			0
$586$		0			0
$587$	−0.258908	+	0.118239i	−0.258908	+	0.118239i	−0.540641	−	0.841254i	$-0.681818\pi$
$587$	−0.258908	+	0.118239i	−0.258908	+	0.118239i	0.281733	+	0.959493i	$0.409091\pi$
$588$	−0.841254	+	1.54064i	−0.841254	+	1.54064i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−0.0498610	−	0.133682i	−0.0498610	−	0.133682i	0.909632	−	0.415415i	$-0.136364\pi$
$593$	−0.0498610	−	0.133682i	−0.0498610	−	0.133682i	−0.959493	+	0.281733i	$0.909091\pi$
$594$	−3.00798	−	2.25175i	−3.00798	−	2.25175i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$599$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$600$	1.75089	−	0.125226i	1.75089	−	0.125226i
$601$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$601$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$602$		0			0
$603$	0.886290	−	3.01843i	0.886290	−	3.01843i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$607$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$608$	−0.254771	+	1.17116i	−0.254771	+	1.17116i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	0.498354	−	3.46613i	0.498354	−	3.46613i
$613$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$613$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$614$	0.627899	+	0.544078i	0.627899	+	0.544078i
$615$		0			0
$616$		0			0
$617$	−0.203743	−	0.936593i	−0.203743	−	0.936593i	−0.959493	−	0.281733i	$-0.909091\pi$
$617$	−0.203743	−	0.936593i	−0.203743	−	0.936593i	0.755750	−	0.654861i	$-0.227273\pi$
$618$		0			0
$619$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.142315	−	0.989821i	$-0.454545\pi$
$619$	0.797176	−	1.74557i	0.797176	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.841254	−	0.540641i	0.841254	−	0.540641i
$626$	0.697148	−	1.86912i	0.697148	−	1.86912i
$627$	1.73019	−	3.78858i	1.73019	−	3.78858i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$631$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$632$		0			0
$633$	−2.59259	+	2.24649i	−2.59259	+	2.24649i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.89945	+	0.273100i	1.89945	+	0.273100i	0.989821	−	0.142315i	$-0.0454545\pi$
$641$	1.89945	+	0.273100i	1.89945	+	0.273100i	0.909632	+	0.415415i	$0.136364\pi$
$642$		0			0
$643$	−0.368991	+	1.25667i	−0.368991	+	1.25667i	0.540641	+	0.841254i	$0.318182\pi$
$643$	−0.368991	+	1.25667i	−0.368991	+	1.25667i	−0.909632	+	0.415415i	$0.863636\pi$
$644$		0			0
$645$		0			0
$646$	2.01144	−	0.143861i	2.01144	−	0.143861i
$647$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$647$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$648$	−1.19980	+	0.352293i	−1.19980	+	0.352293i
$649$	−0.822373	+	0.178896i	−0.822373	+	0.178896i
$650$		0			0
$651$		0			0
$652$	−0.767317	−	0.574406i	−0.767317	−	0.574406i
$653$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$653$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$654$		0			0
$655$		0			0
$656$	1.24123	+	0.677760i	1.24123	+	0.677760i
$657$		−	1.17273i		−	1.17273i
$658$		0			0
$659$	1.74557	−	0.797176i	1.74557	−	0.797176i	0.755750	−	0.654861i	$-0.227273\pi$
$659$	1.74557	−	0.797176i	1.74557	−	0.797176i	0.989821	−	0.142315i	$-0.0454545\pi$
$660$		0			0
$661$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$661$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$662$	0.186393	−	0.215109i	0.186393	−	0.215109i
$663$		0			0
$664$	−0.654861	+	0.244250i	−0.654861	+	0.244250i
$665$		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		−0.755750	−	0.654861i	$-0.772727\pi$
$673$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$674$	−0.959493	−	1.28173i	−0.959493	−	1.28173i
$675$	−1.34211	+	1.34211i	−1.34211	+	1.34211i
$676$	0.909632	−	0.415415i	0.909632	−	0.415415i
$677$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$677$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$678$			1.22687i			1.22687i
$679$		0			0
$680$		0			0
$681$	−0.986563	−	0.0705604i	−0.986563	−	0.0705604i
$682$		0			0
$683$	0.767317	+	0.574406i	0.767317	+	0.574406i	0.909632	−	0.415415i	$-0.136364\pi$
$683$	0.767317	+	0.574406i	0.767317	+	0.574406i	−0.142315	+	0.989821i	$0.545455\pi$
$684$	−1.19550	−	2.18940i	−1.19550	−	2.18940i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.90963	−	0.415415i	−1.90963	−	0.415415i
$689$		0			0
$690$		0			0
$691$	−0.449181	−	0.698939i	−0.449181	−	0.698939i	0.540641	−	0.841254i	$-0.318182\pi$
$691$	−0.449181	−	0.698939i	−0.449181	−	0.698939i	−0.989821	+	0.142315i	$0.954545\pi$
$692$		0			0
$693$		0			0
$694$	1.80075	+	0.258908i	1.80075	+	0.258908i
$695$		0			0
$696$		0			0
$697$	0.505783	−	2.32505i	0.505783	−	2.32505i
$698$		0			0
$699$	1.36645	+	0.509660i	1.36645	+	0.509660i
$700$		0			0
$701$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$701$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$702$		0			0
$703$		0			0
$704$	0.281733	+	1.95949i	0.281733	+	1.95949i
$705$		0			0
$706$	−0.424047	−	1.94931i	−0.424047	−	1.94931i
$707$		0			0
$708$	−0.310006	+	0.678819i	−0.310006	+	0.678819i
$709$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$709$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$710$		0			0
$711$		0			0
$712$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$717$		0			0
$718$		0			0
$719$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$719$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$720$		0			0
$721$		0			0
$722$	0.329911	−	0.285870i	0.329911	−	0.285870i
$723$	−0.468125	+	3.25588i	−0.468125	+	3.25588i
$724$		0			0
$725$		0			0
$726$	0.365532	−	5.11081i	0.365532	−	5.11081i
$727$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$727$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$728$		0			0
$729$	−0.394231	+	0.613435i	−0.394231	+	0.613435i
$730$		0			0
$731$	0.234571	+	3.27974i	0.234571	+	3.27974i
$732$		0			0
$733$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$733$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2.87102	−	0.843008i	2.87102	−	0.843008i
$738$	−2.87611	+	0.625660i	−2.87611	+	0.625660i
$739$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.841254	+	0.540641i	$0.818182\pi$
$739$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.909632	+	0.415415i	$0.863636\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$743$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	0.697148	−	1.27673i	0.697148	−	1.27673i
$748$	3.02977	−	1.38365i	3.02977	−	1.38365i
$749$		0			0
$750$		0			0
$751$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$751$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$752$		0			0
$753$	−1.36645	+	0.509660i	−1.36645	+	0.509660i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$757$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$758$	0.114220	−	0.0855040i	0.114220	−	0.0855040i
$759$		0			0
$760$		0			0
$761$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$761$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.54064	+	0.841254i	1.54064	+	0.841254i
$769$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.755750	−	0.654861i	$-0.772727\pi$
$769$	−1.74557	−	0.512546i	−1.74557	−	0.512546i	−0.989821	+	0.142315i	$0.954545\pi$
$770$		0			0
$771$	0.174602	+	0.468125i	0.174602	+	0.468125i
$772$	1.59700	+	1.19550i	1.59700	+	1.19550i
$773$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$773$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$774$	3.56990	−	1.94931i	3.56990	−	1.94931i
$775$		0			0
$776$	−1.25667	+	0.368991i	−1.25667	+	0.368991i
$777$		0			0
$778$		0			0
$779$	−0.704134	−	1.54184i	−0.704134	−	1.54184i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.540641	+	0.841254i	−0.540641	+	0.841254i
$785$		0			0
$786$	−0.716028	+	3.29153i	−0.716028	+	3.29153i
$787$	−0.133682	+	1.86912i	−0.133682	+	1.86912i	0.281733	+	0.959493i	$0.409091\pi$
$787$	−0.133682	+	1.86912i	−0.133682	+	1.86912i	−0.415415	+	0.909632i	$0.636364\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−3.11384	−	2.69815i	−3.11384	−	2.69815i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$797$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$798$		0			0
$799$		0			0
$800$	1.00000			1.00000
$801$	−1.12523	−	1.75089i	−1.12523	−	1.75089i
$802$	−1.51150			−1.51150
$803$	0.938384	−	0.603063i	0.938384	−	0.603063i
$804$	0.927206	−	2.48594i	0.927206	−	2.48594i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	−0.654861	−	0.755750i	$-0.727273\pi$
$809$	−0.239446	−	1.66538i	−0.239446	−	1.66538i	0.415415	−	0.909632i	$-0.363636\pi$
$810$		0			0
$811$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$811$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	0.627791	−	2.88591i	0.627791	−	2.88591i
$817$	1.53390	+	1.77021i	1.53390	+	1.77021i
$818$	−0.708089	+	1.10181i	−0.708089	+	1.10181i
$819$		0			0
$820$		0			0
$821$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$821$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$822$	0.403457	+	0.627791i	0.403457	+	0.627791i
$823$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$823$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$824$		0			0
$825$	−3.39557	−	0.738661i	−3.39557	−	0.738661i
$826$		0			0
$827$	−1.83107	+	0.398326i	−1.83107	+	0.398326i	−0.989821	−	0.142315i	$-0.954545\pi$
$827$	−1.83107	+	0.398326i	−1.83107	+	0.398326i	−0.841254	+	0.540641i	$0.818182\pi$
$828$		0			0
$829$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$829$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.61435	+	0.474017i	1.61435	+	0.474017i
$834$	−0.438512	−	0.239446i	−0.438512	−	0.239446i
$835$		0			0
$836$	1.13712	−	2.08248i	1.13712	−	2.08248i
$837$		0			0
$838$	−1.41061	+	1.41061i	−1.41061	+	1.41061i
$839$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$839$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$840$		0			0
$841$	0.281733	+	0.959493i	0.281733	+	0.959493i
$842$		0			0
$843$	−2.55650	−	1.16751i	−2.55650	−	1.16751i
$844$	−1.56449	+	1.17116i	−1.56449	+	1.17116i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−2.99211	+	1.11600i	−2.99211	+	1.11600i
$850$	−0.474017	−	1.61435i	−0.474017	−	1.61435i
$851$		0			0
$852$		0			0
$853$		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
$857$	1.05195	+	0.574406i	1.05195	+	0.574406i	0.909632	−	0.415415i	$-0.136364\pi$
$857$	1.05195	+	0.574406i	1.05195	+	0.574406i	0.142315	+	0.989821i	$0.454545\pi$
$858$		0			0
$859$	0.697148	+	0.0498610i	0.697148	+	0.0498610i	0.415415	−	0.909632i	$-0.363636\pi$
$859$	0.697148	+	0.0498610i	0.697148	+	0.0498610i	0.281733	+	0.959493i	$0.409091\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$863$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$864$	−1.85466	+	0.403457i	−1.85466	+	0.403457i
$865$		0			0
$866$	0.936593	+	0.203743i	0.936593	+	0.203743i
$867$	−3.20557	+	0.229267i	−3.20557	+	0.229267i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	1.47373	−	2.29317i	1.47373	−	2.29317i
$874$		0			0
$875$		0			0
$876$	0.0705604	−	0.986563i	0.0705604	−	0.986563i
$877$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$877$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.817178	−	0.708089i	−0.817178	−	0.708089i	0.142315	−	0.989821i	$-0.454545\pi$
$881$	−0.817178	−	0.708089i	−0.817178	−	0.708089i	−0.959493	+	0.281733i	$0.909091\pi$
$882$	−0.296197	−	2.06010i	−0.296197	−	2.06010i
$883$	−1.12299	+	1.50013i	−1.12299	+	1.50013i	−0.281733	+	0.959493i	$0.590909\pi$
$883$	−1.12299	+	1.50013i	−1.12299	+	1.50013i	−0.841254	+	0.540641i	$0.818182\pi$
$884$		0			0
$885$		0			0
$886$	0.755750	−	1.65486i	0.755750	−	1.65486i
$887$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$887$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	2.47545			2.47545
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.258908	−	1.80075i	−0.258908	−	1.80075i
$899$		0			0
$900$	−1.57293	+	1.36295i	−1.57293	+	1.36295i
$901$		0			0
$902$	−1.97964	−	1.97964i	−1.97964	−	1.97964i
$903$		0			0
$904$	−0.0498610	+	0.697148i	−0.0498610	+	0.697148i
$905$		0			0
$906$		0			0
$907$	−0.584585	+	0.909632i	−0.584585	+	0.909632i	0.415415	+	0.909632i	$0.363636\pi$
$907$	−0.584585	+	0.909632i	−0.584585	+	0.909632i	−1.00000			$1.00000\pi$
$908$	−0.557730	−	0.0801894i	−0.557730	−	0.0801894i
$909$		0			0
$910$		0			0
$911$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$911$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$912$	−0.873989	−	1.91377i	−0.873989	−	1.91377i
$913$	1.38010	−	0.0987069i	1.38010	−	0.0987069i
$914$	−0.139418	−	0.0303285i	−0.139418	−	0.0303285i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	1.53046	+	2.80283i	1.53046	+	2.80283i
$919$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$919$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$920$		0			0
$921$	−1.45469	−	0.104041i	−1.45469	−	0.104041i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$929$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$930$		0			0
$931$	1.12299	−	0.418852i	1.12299	−	0.418852i
$932$	0.755750	+	0.345139i	0.755750	+	0.345139i
$933$		0			0
$934$	1.41542	+	0.909632i	1.41542	+	0.909632i
$935$		0			0
$936$		0			0
$937$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$937$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$938$		0			0
$939$	0.986563	+	3.35992i	0.986563	+	3.35992i
$940$		0			0
$941$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$941$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$942$		0			0
$943$		0			0
$944$	−0.203743	+	0.373128i	−0.203743	+	0.373128i
$945$		0			0
$946$	3.39557	+	1.85412i	3.39557	+	1.85412i
$947$	1.45027	+	0.425839i	1.45027	+	0.425839i	0.909632	−	0.415415i	$-0.136364\pi$
$947$	1.45027	+	0.425839i	1.45027	+	0.425839i	0.540641	+	0.841254i	$0.318182\pi$
$948$		0			0
$949$		0			0
$950$	−0.959493	−	0.718267i	−0.959493	−	0.718267i
$951$		0			0
$952$		0			0
$953$	−0.936593	+	0.203743i	−0.936593	+	0.203743i	−0.654861	−	0.755750i	$-0.727273\pi$
$953$	−0.936593	+	0.203743i	−0.936593	+	0.203743i	−0.281733	+	0.959493i	$0.590909\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.989821	+	0.142315i	0.989821	+	0.142315i
$962$		0			0
$963$		0			0
$964$	−0.398326	+	1.83107i	−0.398326	+	1.83107i
$965$		0			0
$966$		0			0
$967$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$967$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$968$	0.415415	−	2.88927i	0.415415	−	2.88927i
$969$	−2.67521	+	2.31809i	−2.67521	+	2.31809i
$970$		0			0
$971$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$971$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$972$	0.177958	−	0.237724i	0.177958	−	0.237724i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.474017	−	0.304632i	0.474017	−	0.304632i	−0.281733	−	0.959493i	$-0.590909\pi$
$977$	0.474017	−	0.304632i	0.474017	−	0.304632i	0.755750	+	0.654861i	$0.227273\pi$
$978$	1.68251			1.68251
$979$	0.822373	−	1.80075i	0.822373	−	1.80075i
$980$		0			0
$981$		0			0
$982$	−0.559521	+	1.50013i	−0.559521	+	1.50013i
$983$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$983$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$984$	−2.45718	+	0.353290i	−2.45718	+	0.353290i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$991$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$992$		0			0
$993$	−0.0356430	+	0.498354i	−0.0356430	+	0.498354i
$994$		0			0
$995$		0			0
$996$	0.663296	−	1.03211i	0.663296	−	1.03211i
$997$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$997$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$998$	−0.133682	−	1.86912i	−0.133682	−	1.86912i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	712.1.y.a.403.1	✓	20
4.3	odd	2		2848.1.cc.a.47.1		20
8.3	odd	2	CM	712.1.y.a.403.1	✓	20
8.5	even	2		2848.1.cc.a.47.1		20
89.36	even	44	inner	712.1.y.a.659.1	yes	20
356.303	odd	44		2848.1.cc.a.303.1		20
712.125	even	44		2848.1.cc.a.303.1		20
712.659	odd	44	inner	712.1.y.a.659.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
712.1.y.a.403.1	✓	20	1.1	even	1	trivial
712.1.y.a.403.1	✓	20	8.3	odd	2	CM
712.1.y.a.659.1	yes	20	89.36	even	44	inner
712.1.y.a.659.1	yes	20	712.659	odd	44	inner
2848.1.cc.a.47.1		20	4.3	odd	2
2848.1.cc.a.47.1		20	8.5	even	2
2848.1.cc.a.303.1		20	356.303	odd	44
2848.1.cc.a.303.1		20	712.125	even	44

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:	\( N \)	\(=\)	\( 712 = 2^{3} \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	712.y (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.841254 + 0.540641i) q^{2} +(0.613435 - 1.64468i) q^{3} +(0.415415 - 0.909632i) q^{4} +(0.373128 + 1.71524i) q^{6} +(0.142315 + 0.989821i) q^{8} +(-1.57293 - 1.36295i) q^{9} +O(q^{10})\)	\(q+(-0.841254 + 0.540641i) q^{2} +(0.613435 - 1.64468i) q^{3} +(0.415415 - 0.909632i) q^{4} +(0.373128 + 1.71524i) q^{6} +(0.142315 + 0.989821i) q^{8} +(-1.57293 - 1.36295i) q^{9} +(0.281733 - 1.95949i) q^{11} +(-1.24123 - 1.24123i) q^{12} +(-0.654861 - 0.755750i) q^{16} +(-0.909632 + 1.41542i) q^{17} +(2.06010 + 0.296197i) q^{18} +(0.0855040 + 1.19550i) q^{19} +(0.822373 + 1.80075i) q^{22} +(1.71524 + 0.373128i) q^{24} +(0.959493 - 0.281733i) q^{25} +(-1.66587 + 0.909632i) q^{27} +(0.959493 + 0.281733i) q^{32} +(-3.04992 - 1.66538i) q^{33} -1.68251i q^{34} +(-1.89320 + 0.864596i) q^{36} +(-0.718267 - 0.959493i) q^{38} +(-1.32505 + 0.494217i) q^{41} +(1.56449 - 1.17116i) q^{43} +(-1.66538 - 1.07028i) q^{44} +(-1.64468 + 0.613435i) q^{48} +(-0.281733 - 0.959493i) q^{49} +(-0.654861 + 0.755750i) q^{50} +(1.76991 + 2.36432i) q^{51} +(0.909632 - 1.66587i) q^{54} +(2.01867 + 0.592735i) q^{57} +(-0.148568 - 0.398326i) q^{59} +(-0.959493 + 0.281733i) q^{64} +(3.46613 - 0.247902i) q^{66} +(0.627899 + 1.37491i) q^{67} +(0.909632 + 1.41542i) q^{68} +(1.12523 - 1.75089i) q^{72} +(0.368991 + 0.425839i) q^{73} +(0.125226 - 1.75089i) q^{75} +(1.12299 + 0.418852i) q^{76} +(0.177958 + 1.23772i) q^{81} +(0.847507 - 1.13214i) q^{82} +(0.148568 + 0.682956i) q^{83} +(-0.682956 + 1.83107i) q^{86} +1.97964 q^{88} +(0.959493 + 0.281733i) q^{89} +(1.05195 - 1.40524i) q^{96} +(0.186393 + 1.29639i) q^{97} +(0.755750 + 0.654861i) q^{98} +(-3.11384 + 2.69815i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8}+O(q^{10}) \) 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8	\( 20 q + 2 q^{2} - 2 q^{3} - 2 q^{4} + 2 q^{6} + 2 q^{8} - 2 q^{12} - 2 q^{16} - 2 q^{19} + 2 q^{24} + 2 q^{25} - 22 q^{27} + 2 q^{32} - 20 q^{38} + 2 q^{41} + 2 q^{43} - 2 q^{48} - 2 q^{50} + 4 q^{51} - 4 q^{57} - 2 q^{59} - 2 q^{64} + 22 q^{72} + 2 q^{75} - 2 q^{76} - 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{86} + 2 q^{89} + 2 q^{96} - 4 q^{97}+O(q^{100}) \) 20 * q + 2 * q^2 - 2 * q^3 - 2 * q^4 + 2 * q^6 + 2 * q^8 - 2 * q^12 - 2 * q^16 - 2 * q^19 + 2 * q^24 + 2 * q^25 - 22 * q^27 + 2 * q^32 - 20 * q^38 + 2 * q^41 + 2 * q^43 - 2 * q^48 - 2 * q^50 + 4 * q^51 - 4 * q^57 - 2 * q^59 - 2 * q^64 + 22 * q^72 + 2 * q^75 - 2 * q^76 - 2 * q^81 - 2 * q^82 + 2 * q^83 - 2 * q^86 + 2 * q^89 + 2 * q^96 - 4 * q^97

Introduction

L-functions

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