LMFDB - Embedded newform 1412.1.r.a.1055.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1412,1,Mod(35,1412)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1412, base_ring=CyclotomicField(44))

chi = DirichletCharacter(H, H._module([22, 7]))

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

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

chi := DirichletCharacter("1412.35");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1412 = 2^{2} \cdot 353 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1412.r (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:	$0.704679797838$
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:	$\Z[a_1, \ldots, a_{5}]$
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			1055.1
Root			$-0.755750 + 0.654861i$ of defining polynomial
Character	$\chi$	$=$	1412.1055
Dual form			1412.1.r.a.1147.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.841254 + 0.540641i) q^{2} +(0.415415 - 0.909632i) q^{4} +(-0.203743 - 0.936593i) q^{5} +(0.142315 + 0.989821i) q^{8} +(-0.909632 + 0.415415i) q^{9} +O(q^{10})$	\(q+(-0.841254 + 0.540641i) q^{2} +(0.415415 - 0.909632i) q^{4} +(-0.203743 - 0.936593i) q^{5} +(0.142315 + 0.989821i) q^{8} +(-0.909632 + 0.415415i) q^{9} +(0.677760 + 0.677760i) q^{10} +(0.139418 - 1.94931i) q^{13} +(-0.654861 - 0.755750i) q^{16} +(0.118239 + 0.822373i) q^{17} +(0.540641 - 0.841254i) q^{18} +(-0.936593 - 0.203743i) q^{20} +(0.0739364 - 0.0337656i) q^{25} +(0.936593 + 1.71524i) q^{26} +(-0.627899 - 1.37491i) q^{29} +(0.959493 + 0.281733i) q^{32} +(-0.544078 - 0.627899i) q^{34} +1.00000i q^{36} +(-1.28173 - 0.959493i) q^{37} +(0.898064 - 0.334961i) q^{40} +(-0.0801894 - 0.273100i) q^{41} +(0.574406 + 0.767317i) q^{45} -1.00000i q^{49} +(-0.0439442 + 0.0683785i) q^{50} +(-1.71524 - 0.936593i) q^{52} +(-0.682956 + 0.148568i) q^{53} +(1.27155 + 0.817178i) q^{58} +(-0.989821 - 1.14231i) q^{61} +(-0.959493 + 0.281733i) q^{64} +(-1.85412 + 0.266582i) q^{65} +(0.797176 + 0.234072i) q^{68} +(-0.540641 - 0.841254i) q^{72} +(1.59700 + 0.114220i) q^{74} +(-0.574406 + 0.767317i) q^{80} +(0.654861 - 0.755750i) q^{81} +(0.215109 + 0.186393i) q^{82} +(0.746139 - 0.278295i) q^{85} +(0.767317 - 0.574406i) q^{89} +(-0.898064 - 0.334961i) q^{90} +(-0.239446 + 1.66538i) q^{97} +(0.540641 + 0.841254i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) $ 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8	$ 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8} + 2 q^{10} - 2 q^{13} - 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{26} + 2 q^{32} - 4 q^{34} - 20 q^{37} + 2 q^{40} + 2 q^{45} - 22 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{64} + 4 q^{68} - 2 q^{74} - 2 q^{80} + 2 q^{81} + 4 q^{85} - 2 q^{89} - 2 q^{90} - 4 q^{97}+O(q^{100}) $ 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8 + 2 * q^10 - 2 * q^13 - 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^26 + 2 * q^32 - 4 * q^34 - 20 * q^37 + 2 * q^40 + 2 * q^45 - 22 * q^50 - 2 * q^52 - 2 * q^53 - 2 * q^64 + 4 * q^68 - 2 * q^74 - 2 * q^80 + 2 * q^81 + 4 * q^85 - 2 * q^89 - 2 * q^90 - 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1412\mathbb{Z}\right)^\times$.

$n$	$707$	$709$
$\chi(n)$	$-1$	$e\left(\frac{19}{44}\right)$

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 $

$2$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$2$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$3$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$3$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$4$	0.415415	−	0.909632i	0.415415	−	0.909632i
$5$	−0.203743	−	0.936593i	−0.203743	−	0.936593i	−0.959493	−	0.281733i	$-0.909091\pi$
$5$	−0.203743	−	0.936593i	−0.203743	−	0.936593i	0.755750	−	0.654861i	$-0.227273\pi$
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	0.142315	+	0.989821i	0.142315	+	0.989821i
$9$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$10$	0.677760	+	0.677760i	0.677760	+	0.677760i
$11$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$11$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$12$		0			0
$13$	0.139418	−	1.94931i	0.139418	−	1.94931i	−0.142315	−	0.989821i	$-0.545455\pi$
$13$	0.139418	−	1.94931i	0.139418	−	1.94931i	0.281733	−	0.959493i	$-0.409091\pi$
$14$		0			0
$15$		0			0
$16$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$17$	0.118239	+	0.822373i	0.118239	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$17$	0.118239	+	0.822373i	0.118239	+	0.822373i	−0.841254	+	0.540641i	$0.818182\pi$
$18$	0.540641	−	0.841254i	0.540641	−	0.841254i
$19$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$19$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$20$	−0.936593	−	0.203743i	−0.936593	−	0.203743i
$21$		0			0
$22$		0			0
$23$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$23$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$24$		0			0
$25$	0.0739364	−	0.0337656i	0.0739364	−	0.0337656i
$26$	0.936593	+	1.71524i	0.936593	+	1.71524i
$27$		0			0
$28$		0			0
$29$	−0.627899	−	1.37491i	−0.627899	−	1.37491i	−0.909632	−	0.415415i	$-0.863636\pi$
$29$	−0.627899	−	1.37491i	−0.627899	−	1.37491i	0.281733	−	0.959493i	$-0.409091\pi$
$30$		0			0
$31$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$31$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$32$	0.959493	+	0.281733i	0.959493	+	0.281733i
$33$		0			0
$34$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$35$		0			0
$36$			1.00000i			1.00000i
$37$	−1.28173	−	0.959493i	−1.28173	−	0.959493i	−0.281733	−	0.959493i	$-0.590909\pi$
$37$	−1.28173	−	0.959493i	−1.28173	−	0.959493i	−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	0.898064	−	0.334961i	0.898064	−	0.334961i
$41$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	0.909632	−	0.415415i	$-0.136364\pi$
$41$	−0.0801894	−	0.273100i	−0.0801894	−	0.273100i	−0.989821	+	0.142315i	$0.954545\pi$
$42$		0			0
$43$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$43$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$44$		0			0
$45$	0.574406	+	0.767317i	0.574406	+	0.767317i
$46$		0			0
$47$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$47$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$48$		0			0
$49$		−	1.00000i		−	1.00000i
$50$	−0.0439442	+	0.0683785i	−0.0439442	+	0.0683785i
$51$		0			0
$52$	−1.71524	−	0.936593i	−1.71524	−	0.936593i
$53$	−0.682956	+	0.148568i	−0.682956	+	0.148568i	−0.540641	−	0.841254i	$-0.681818\pi$
$53$	−0.682956	+	0.148568i	−0.682956	+	0.148568i	−0.142315	+	0.989821i	$0.545455\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.27155	+	0.817178i	1.27155	+	0.817178i
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	−0.989821	−	1.14231i	−0.989821	−	1.14231i	−0.989821	−	0.142315i	$-0.954545\pi$
$61$	−0.989821	−	1.14231i	−0.989821	−	1.14231i		−	1.00000i	$-0.5\pi$
$62$		0			0
$63$		0			0
$64$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$65$	−1.85412	+	0.266582i	−1.85412	+	0.266582i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	0.797176	+	0.234072i	0.797176	+	0.234072i
$69$		0			0
$70$		0			0
$71$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$71$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$72$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$73$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$73$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$74$	1.59700	+	0.114220i	1.59700	+	0.114220i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$79$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$80$	−0.574406	+	0.767317i	−0.574406	+	0.767317i
$81$	0.654861	−	0.755750i	0.654861	−	0.755750i
$82$	0.215109	+	0.186393i	0.215109	+	0.186393i
$83$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$83$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$84$		0			0
$85$	0.746139	−	0.278295i	0.746139	−	0.278295i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.767317	−	0.574406i	0.767317	−	0.574406i	−0.142315	−	0.989821i	$-0.545455\pi$
$89$	0.767317	−	0.574406i	0.767317	−	0.574406i	0.909632	+	0.415415i	$0.136364\pi$
$90$	−0.898064	−	0.334961i	−0.898064	−	0.334961i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	0.415415	+	0.909632i	$0.363636\pi$
$97$	−0.239446	+	1.66538i	−0.239446	+	1.66538i	−0.654861	+	0.755750i	$0.727273\pi$
$98$	0.540641	+	0.841254i	0.540641	+	0.841254i
$99$		0			0
$100$		−	0.0812816i		−	0.0812816i
$101$	1.38189	+	1.38189i	1.38189	+	1.38189i	0.841254	+	0.540641i	$0.181818\pi$
$101$	1.38189	+	1.38189i	1.38189	+	1.38189i	0.540641	+	0.841254i	$0.318182\pi$
$102$		0			0
$103$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$103$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$104$	1.94931	−	0.139418i	1.94931	−	0.139418i
$105$		0			0
$106$	0.494217	−	0.494217i	0.494217	−	0.494217i
$107$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$107$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$108$		0			0
$109$	1.84125	−	0.540641i	1.84125	−	0.540641i	0.841254	−	0.540641i	$-0.181818\pi$
$109$	1.84125	−	0.540641i	1.84125	−	0.540641i	1.00000			$0$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.755750	+	0.345139i	0.755750	+	0.345139i	0.755750	−	0.654861i	$-0.227273\pi$
$113$	0.755750	+	0.345139i	0.755750	+	0.345139i			1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	−1.51150			−1.51150
$117$	0.682956	+	1.83107i	0.682956	+	1.83107i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.841254	−	0.540641i	0.841254	−	0.540641i
$122$	1.45027	+	0.425839i	1.45027	+	0.425839i
$123$		0			0
$124$		0			0
$125$	−0.621095	−	0.829686i	−0.621095	−	0.829686i
$126$		0			0
$127$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$127$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$128$	0.654861	−	0.755750i	0.654861	−	0.755750i
$129$		0			0
$130$	1.41566	−	1.22668i	1.41566	−	1.22668i
$131$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$131$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$137$	−0.100889	+	0.100889i	−0.100889	+	0.100889i	−0.755750	−	0.654861i	$-0.772727\pi$
$137$	−0.100889	+	0.100889i	−0.100889	+	0.100889i	0.654861	+	0.755750i	$0.272727\pi$
$138$		0			0
$139$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$139$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.909632	+	0.415415i	0.909632	+	0.415415i
$145$	−1.15980	+	0.868215i	−1.15980	+	0.868215i
$146$		0			0
$147$		0			0
$148$	−1.40524	+	0.767317i	−1.40524	+	0.767317i
$149$	−0.0101786	+	0.142315i	−0.0101786	+	0.142315i	0.989821	+	0.142315i	$0.0454545\pi$
$149$	−0.0101786	+	0.142315i	−0.0101786	+	0.142315i	−1.00000			$\pi$
$150$		0			0
$151$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$151$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$152$		0			0
$153$	−0.449181	−	0.698939i	−0.449181	−	0.698939i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.989821	+	0.142315i	$0.0454545\pi$
$157$	1.74557	+	0.797176i	1.74557	+	0.797176i	0.755750	+	0.654861i	$0.227273\pi$
$158$		0			0
$159$		0			0
$160$	0.0683785	−	0.956056i	0.0683785	−	0.956056i
$161$		0			0
$162$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$163$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$163$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$164$	−0.281733	−	0.0405070i	−0.281733	−	0.0405070i
$165$		0			0
$166$		0			0
$167$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$167$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$168$		0			0
$169$	−2.79057	−	0.401223i	−2.79057	−	0.401223i
$170$	−0.477234	+	0.637510i	−0.477234	+	0.637510i
$171$		0			0
$172$		0			0
$173$	−1.71524	+	0.373128i	−1.71524	+	0.373128i	−0.959493	−	0.281733i	$-0.909091\pi$
$173$	−1.71524	+	0.373128i	−1.71524	+	0.373128i	−0.755750	+	0.654861i	$0.772727\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−0.334961	+	0.898064i	−0.334961	+	0.898064i
$179$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$179$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$180$	0.936593	−	0.203743i	0.936593	−	0.203743i
$181$	1.07028	+	0.153882i	1.07028	+	0.153882i	0.654861	−	0.755750i	$-0.272727\pi$
$181$	1.07028	+	0.153882i	1.07028	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.637510	+	1.39595i	−0.637510	+	1.39595i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$191$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$192$		0			0
$193$	−0.100889	+	1.41061i	−0.100889	+	1.41061i	0.654861	+	0.755750i	$0.272727\pi$
$193$	−0.100889	+	1.41061i	−0.100889	+	1.41061i	−0.755750	+	0.654861i	$0.772727\pi$
$194$	−0.698939	−	1.53046i	−0.698939	−	1.53046i
$195$		0			0
$196$	−0.909632	−	0.415415i	−0.909632	−	0.415415i
$197$	−0.474017	+	1.61435i	−0.474017	+	1.61435i	0.281733	+	0.959493i	$0.409091\pi$
$197$	−0.474017	+	1.61435i	−0.474017	+	1.61435i	−0.755750	+	0.654861i	$0.772727\pi$
$198$		0			0
$199$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$199$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$200$	0.0439442	+	0.0683785i	0.0439442	+	0.0683785i
$201$		0			0
$202$	−1.90963	−	0.415415i	−1.90963	−	0.415415i
$203$		0			0
$204$		0			0
$205$	−0.239446	+	0.130747i	−0.239446	+	0.130747i
$206$		0			0
$207$		0			0
$208$	−1.56449	+	1.17116i	−1.56449	+	1.17116i
$209$		0			0
$210$		0			0
$211$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$211$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$212$	−0.148568	+	0.682956i	−0.148568	+	0.682956i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.25667	+	1.45027i	−1.25667	+	1.45027i
$219$		0			0
$220$		0			0
$221$	1.61955	−	0.115832i	1.61955	−	0.115832i
$222$		0			0
$223$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$223$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$224$		0			0
$225$	−0.0532282	+	0.0614286i	−0.0532282	+	0.0614286i
$226$	−0.822373	+	0.118239i	−0.822373	+	0.118239i
$227$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$227$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$228$		0			0
$229$	1.13214	−	0.847507i	1.13214	−	0.847507i	0.142315	−	0.989821i	$-0.454545\pi$
$229$	1.13214	−	0.847507i	1.13214	−	0.847507i	0.989821	+	0.142315i	$0.0454545\pi$
$230$		0			0
$231$		0			0
$232$	1.27155	−	0.817178i	1.27155	−	0.817178i
$233$	−1.07028	−	1.66538i	−1.07028	−	1.66538i	−0.654861	−	0.755750i	$-0.727273\pi$
$233$	−1.07028	−	1.66538i	−1.07028	−	1.66538i	−0.415415	−	0.909632i	$-0.636364\pi$
$234$	−1.56449	−	1.17116i	−1.56449	−	1.17116i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$239$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$240$		0			0
$241$	−0.697148	−	0.0498610i	−0.697148	−	0.0498610i	−0.281733	−	0.959493i	$-0.590909\pi$
$241$	−0.697148	−	0.0498610i	−0.697148	−	0.0498610i	−0.415415	+	0.909632i	$0.636364\pi$
$242$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$243$		0			0
$244$	−1.45027	+	0.425839i	−1.45027	+	0.425839i
$245$	−0.936593	+	0.203743i	−0.936593	+	0.203743i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.971061	+	0.362187i	0.971061	+	0.362187i
$251$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$251$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$257$	−0.125226	+	1.75089i	−0.125226	+	1.75089i	0.415415	+	0.909632i	$0.363636\pi$
$257$	−0.125226	+	1.75089i	−0.125226	+	1.75089i	−0.540641	+	0.841254i	$0.681818\pi$
$258$		0			0
$259$		0			0
$260$	−0.527738	+	1.79731i	−0.527738	+	1.79731i
$261$	1.14231	+	0.989821i	1.14231	+	0.989821i
$262$		0			0
$263$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$263$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$264$		0			0
$265$	0.278295	+	0.609382i	0.278295	+	0.609382i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.474017	−	0.304632i	0.474017	−	0.304632i	−0.281733	−	0.959493i	$-0.590909\pi$
$269$	0.474017	−	0.304632i	0.474017	−	0.304632i	0.755750	+	0.654861i	$0.227273\pi$
$270$		0			0
$271$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$271$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$272$	0.544078	−	0.627899i	0.544078	−	0.627899i
$273$		0			0
$274$	0.0303285	−	0.139418i	0.0303285	−	0.139418i
$275$		0			0
$276$		0			0
$277$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$277$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.304632	+	0.474017i	0.304632	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$281$	0.304632	+	0.474017i	0.304632	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$289$	0.297176	−	0.0872586i	0.297176	−	0.0872586i
$290$	0.506293	−	1.35742i	0.506293	−	1.35742i
$291$		0			0
$292$		0			0
$293$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$293$			2.00000i			2.00000i			1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$	0.767317	−	1.40524i	0.767317	−	1.40524i
$297$		0			0
$298$	−0.0683785	−	0.125226i	−0.0683785	−	0.125226i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.868215	+	1.15980i	−0.868215	+	1.15980i
$306$	0.755750	+	0.345139i	0.755750	+	0.345139i
$307$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$307$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	1.50013	−	0.559521i	1.50013	−	0.559521i	0.540641	−	0.841254i	$-0.318182\pi$
$313$	1.50013	−	0.559521i	1.50013	−	0.559521i	0.959493	+	0.281733i	$0.0909091\pi$
$314$	−1.89945	+	0.273100i	−1.89945	+	0.273100i
$315$		0			0
$316$		0			0
$317$		−	1.68251i		−	1.68251i	−0.540641	−	0.841254i	$-0.681818\pi$
$317$		−	1.68251i		−	1.68251i	0.540641	−	0.841254i	$-0.318182\pi$
$318$		0			0
$319$		0			0
$320$	0.459359	+	0.841254i	0.459359	+	0.841254i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$325$	−0.0555118	−	0.148833i	−0.0555118	−	0.148833i
$326$		0			0
$327$		0			0
$328$	0.258908	−	0.118239i	0.258908	−	0.118239i
$329$		0			0
$330$		0			0
$331$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$331$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$332$		0			0
$333$	1.56449	+	0.340335i	1.56449	+	0.340335i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$337$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$338$	2.56449	−	1.17116i	2.56449	−	1.17116i
$339$		0			0
$340$	0.0568109	−	0.794320i	0.0568109	−	0.794320i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.24123	−	1.24123i	1.24123	−	1.24123i
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	−0.822373	+	1.80075i	−0.822373	+	1.80075i	−0.281733	+	0.959493i	$0.590909\pi$
$349$	−0.822373	+	1.80075i	−0.822373	+	1.80075i	−0.540641	+	0.841254i	$0.681818\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$353$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$354$		0			0
$355$		0			0
$356$	−0.203743	−	0.936593i	−0.203743	−	0.936593i
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$	−0.677760	+	0.677760i	−0.677760	+	0.677760i
$361$	0.142315	+	0.989821i	0.142315	+	0.989821i
$362$	−0.983568	+	0.449181i	−0.983568	+	0.449181i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$367$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$368$		0			0
$369$	0.186393	+	0.215109i	0.186393	+	0.215109i
$370$	−0.218401	−	1.51901i	−0.218401	−	1.51901i
$371$		0			0
$372$		0			0
$373$	−1.38189	−	0.300613i	−1.38189	−	0.300613i	−0.540641	−	0.841254i	$-0.681818\pi$
$373$	−1.38189	−	0.300613i	−1.38189	−	0.300613i	−0.841254	+	0.540641i	$0.818182\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.76767	+	1.03229i	−2.76767	+	1.03229i
$378$		0			0
$379$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$379$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$383$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$384$		0			0
$385$		0			0
$386$	−0.677760	−	1.24123i	−0.677760	−	1.24123i
$387$		0			0
$388$	1.41542	+	0.909632i	1.41542	+	0.909632i
$389$		−	1.81926i		−	1.81926i	−0.415415	−	0.909632i	$-0.636364\pi$
$389$		−	1.81926i		−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$390$		0			0
$391$		0			0
$392$	0.989821	−	0.142315i	0.989821	−	0.142315i
$393$		0			0
$394$	−0.474017	−	1.61435i	−0.474017	−	1.61435i
$395$		0			0
$396$		0			0
$397$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$397$	−0.273100	+	1.89945i	−0.273100	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$398$		0			0
$399$		0			0
$400$	−0.0739364	−	0.0337656i	−0.0739364	−	0.0337656i
$401$	0.847507	−	1.13214i	0.847507	−	1.13214i	−0.142315	−	0.989821i	$-0.545455\pi$
$401$	0.847507	−	1.13214i	0.847507	−	1.13214i	0.989821	−	0.142315i	$-0.0454545\pi$
$402$		0			0
$403$		0			0
$404$	1.83107	−	0.682956i	1.83107	−	0.682956i
$405$	−0.841254	−	0.459359i	−0.841254	−	0.459359i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.959493	−	0.718267i	−0.959493	−	0.718267i		−	1.00000i	$-0.5\pi$
$409$	−0.959493	−	0.718267i	−0.959493	−	0.718267i	−0.959493	+	0.281733i	$0.909091\pi$
$410$	0.130747	−	0.239446i	0.130747	−	0.239446i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.682956	−	1.83107i	0.682956	−	1.83107i
$417$		0			0
$418$		0			0
$419$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$419$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$420$		0			0
$421$	1.74557	+	0.512546i	1.74557	+	0.512546i	0.989821	−	0.142315i	$-0.0454545\pi$
$421$	1.74557	+	0.512546i	1.74557	+	0.512546i	0.755750	+	0.654861i	$0.227273\pi$
$422$		0			0
$423$		0			0
$424$	−0.244250	−	0.654861i	−0.244250	−	0.654861i
$425$	0.0365101	+	0.0568109i	0.0365101	+	0.0568109i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$431$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$432$		0			0
$433$	0.254771	−	0.340335i	0.254771	−	0.340335i	−0.654861	−	0.755750i	$-0.727273\pi$
$433$	0.254771	−	0.340335i	0.254771	−	0.340335i	0.909632	+	0.415415i	$0.136364\pi$
$434$		0			0
$435$		0			0
$436$	0.273100	−	1.89945i	0.273100	−	1.89945i
$437$		0			0
$438$		0			0
$439$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$439$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$440$		0			0
$441$	0.415415	+	0.909632i	0.415415	+	0.909632i
$442$	−1.29983	+	0.973039i	−1.29983	+	0.973039i
$443$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$443$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$444$		0			0
$445$	−0.694321	−	0.601633i	−0.694321	−	0.601633i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	−0.909632	−	0.415415i	$-0.863636\pi$
$449$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	0.959493	−	0.281733i	$-0.0909091\pi$
$450$	0.0115676	−	0.0804543i	0.0115676	−	0.0804543i
$451$		0			0
$452$	0.627899	−	0.544078i	0.627899	−	0.544078i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.86912	−	0.133682i	1.86912	−	0.133682i	0.909632	−	0.415415i	$-0.136364\pi$
$457$	1.86912	−	0.133682i	1.86912	−	0.133682i	0.959493	+	0.281733i	$0.0909091\pi$
$458$	−0.494217	+	1.32505i	−0.494217	+	1.32505i
$459$		0			0
$460$		0			0
$461$	−0.139418	+	0.0303285i	−0.139418	+	0.0303285i	−0.281733	−	0.959493i	$-0.590909\pi$
$461$	−0.139418	+	0.0303285i	−0.139418	+	0.0303285i	0.142315	+	0.989821i	$0.454545\pi$
$462$		0			0
$463$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$463$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$464$	−0.627899	+	1.37491i	−0.627899	+	1.37491i
$465$		0			0
$466$	1.80075	+	0.822373i	1.80075	+	0.822373i
$467$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$467$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$468$	1.94931	+	0.139418i	1.94931	+	0.139418i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.559521	−	0.418852i	0.559521	−	0.418852i
$478$		0			0
$479$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$479$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$480$		0			0
$481$	−2.04905	+	2.36473i	−2.04905	+	2.36473i
$482$	0.613435	−	0.334961i	0.613435	−	0.334961i
$483$		0			0
$484$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$485$	1.60857	−	0.115047i	1.60857	−	0.115047i
$486$		0			0
$487$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$487$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$488$	0.989821	−	1.14231i	0.989821	−	1.14231i
$489$		0			0
$490$	0.677760	−	0.677760i	0.677760	−	0.677760i
$491$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$491$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$492$		0			0
$493$	1.05645	−	0.678936i	1.05645	−	0.678936i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$499$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$500$	−1.01272	+	0.220304i	−1.01272	+	0.220304i
$501$		0			0
$502$		0			0
$503$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$503$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$504$		0			0
$505$	1.01272	−	1.57582i	1.01272	−	1.57582i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.142315	−	0.989821i	$-0.454545\pi$
$509$	0.557730	−	1.89945i	0.557730	−	1.89945i	0.415415	−	0.909632i	$-0.363636\pi$
$510$		0			0
$511$		0			0
$512$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$513$		0			0
$514$	−0.841254	−	1.54064i	−0.841254	−	1.54064i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−0.527738	−	1.79731i	−0.527738	−	1.79731i
$521$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.281733	−	0.959493i	$-0.409091\pi$
$521$	0.822373	−	1.80075i	0.822373	−	1.80075i	0.540641	−	0.841254i	$-0.318182\pi$
$522$	−1.49611	−	0.215109i	−1.49611	−	0.215109i
$523$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$523$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$530$	−0.563574	−	0.362187i	−0.563574	−	0.362187i
$531$		0			0
$532$		0			0
$533$	−0.543538	+	0.118239i	−0.543538	+	0.118239i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.234072	+	0.512546i	−0.234072	+	0.512546i
$539$		0			0
$540$		0			0
$541$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$541$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$542$		0			0
$543$		0			0
$544$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$545$	−0.881504	−	1.61435i	−0.881504	−	1.61435i
$546$		0			0
$547$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$547$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$548$	0.0498610	+	0.133682i	0.0498610	+	0.133682i
$549$	1.37491	+	0.627899i	1.37491	+	0.627899i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−1.07028	+	1.66538i	−1.07028	+	1.66538i
$555$		0			0
$556$		0			0
$557$	0.142315	−	1.98982i	0.142315	−	1.98982i		−	1.00000i	$-0.5\pi$
$557$	0.142315	−	1.98982i	0.142315	−	1.98982i	0.142315	−	0.989821i	$-0.454545\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$563$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$563$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$564$		0			0
$565$	0.169276	−	0.778150i	0.169276	−	0.778150i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	1.32505	−	1.32505i	1.32505	−	1.32505i	0.415415	−	0.909632i	$-0.363636\pi$
$569$	1.32505	−	1.32505i	1.32505	−	1.32505i	0.909632	−	0.415415i	$-0.136364\pi$
$570$		0			0
$571$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$571$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.755750	−	0.654861i	0.755750	−	0.654861i
$577$	0.125226	−	0.0683785i	0.125226	−	0.0683785i	−0.415415	−	0.909632i	$-0.636364\pi$
$577$	0.125226	−	0.0683785i	0.125226	−	0.0683785i	0.540641	+	0.841254i	$0.318182\pi$
$578$	−0.202824	+	0.234072i	−0.202824	+	0.234072i
$579$		0			0
$580$	0.307958	+	1.41566i	0.307958	+	1.41566i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	1.57582	−	1.01272i	1.57582	−	1.01272i
$586$	−1.08128	−	1.68251i	−1.08128	−	1.68251i
$587$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$587$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.114220	+	1.59700i	0.114220	+	1.59700i
$593$	0.755750	+	0.345139i	0.755750	+	0.345139i	0.755750	−	0.654861i	$-0.227273\pi$
$593$	0.755750	+	0.345139i	0.755750	+	0.345139i			1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	0.125226	+	0.0683785i	0.125226	+	0.0683785i
$597$		0			0
$598$		0			0
$599$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$599$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$600$		0			0
$601$	−0.559521	+	1.50013i	−0.559521	+	1.50013i	0.281733	+	0.959493i	$0.409091\pi$
$601$	−0.559521	+	1.50013i	−0.559521	+	1.50013i	−0.841254	+	0.540641i	$0.818182\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.677760	−	0.677760i	−0.677760	−	0.677760i
$606$		0			0
$607$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$607$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$608$		0			0
$609$		0			0
$610$	0.103354	−	1.44508i	0.103354	−	1.44508i
$611$		0			0
$612$	−0.822373	+	0.118239i	−0.822373	+	0.118239i
$613$	−0.540641	+	1.84125i	−0.540641	+	1.84125i			1.00000i	$0.5\pi$
$613$	−0.540641	+	1.84125i	−0.540641	+	1.84125i	−0.540641	+	0.841254i	$0.681818\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.56449	−	1.17116i	1.56449	−	1.17116i	0.654861	−	0.755750i	$-0.272727\pi$
$617$	1.56449	−	1.17116i	1.56449	−	1.17116i	0.909632	−	0.415415i	$-0.136364\pi$
$618$		0			0
$619$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$619$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.597306	+	0.689328i	−0.597306	+	0.689328i
$626$	−0.959493	+	1.28173i	−0.959493	+	1.28173i
$627$		0			0
$628$	1.45027	−	1.25667i	1.45027	−	1.25667i
$629$	0.637510	−	1.16751i	0.637510	−	1.16751i
$630$		0			0
$631$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$631$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$632$		0			0
$633$		0			0
$634$	0.909632	+	1.41542i	0.909632	+	1.41542i
$635$		0			0
$636$		0			0
$637$	−1.94931	−	0.139418i	−1.94931	−	0.139418i
$638$		0			0
$639$		0			0
$640$	−0.841254	−	0.459359i	−0.841254	−	0.459359i
$641$	−1.89945	+	0.273100i	−1.89945	+	0.273100i	−0.989821	−	0.142315i	$-0.954545\pi$
$641$	−1.89945	+	0.273100i	−1.89945	+	0.273100i	−0.909632	+	0.415415i	$0.863636\pi$
$642$		0			0
$643$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$643$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$647$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$648$	0.841254	+	0.540641i	0.841254	+	0.540641i
$649$		0			0
$650$	0.127165	+	0.0951942i	0.127165	+	0.0951942i
$651$		0			0
$652$		0			0
$653$	1.38189	−	0.300613i	1.38189	−	0.300613i	0.540641	−	0.841254i	$-0.318182\pi$
$653$	1.38189	−	0.300613i	1.38189	−	0.300613i	0.841254	+	0.540641i	$0.181818\pi$
$654$		0			0
$655$		0			0
$656$	−0.153882	+	0.239446i	−0.153882	+	0.239446i
$657$		0			0
$658$		0			0
$659$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$659$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$660$		0			0
$661$	−0.254771	−	0.340335i	−0.254771	−	0.340335i	0.654861	−	0.755750i	$-0.272727\pi$
$661$	−0.254771	−	0.340335i	−0.254771	−	0.340335i	−0.909632	+	0.415415i	$0.863636\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.50013	+	0.559521i	−1.50013	+	0.559521i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.898064	−	1.64468i	−0.898064	−	1.64468i	−0.755750	−	0.654861i	$-0.772727\pi$
$673$	−0.898064	−	1.64468i	−0.898064	−	1.64468i	−0.142315	−	0.989821i	$-0.545455\pi$
$674$	0.273100	+	0.0801894i	0.273100	+	0.0801894i
$675$		0			0
$676$	−1.52421	+	2.37172i	−1.52421	+	2.37172i
$677$	0.698939	+	1.53046i	0.698939	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$677$	0.698939	+	1.53046i	0.698939	+	1.53046i	−0.142315	+	0.989821i	$0.545455\pi$
$678$		0			0
$679$		0			0
$680$	0.381649	+	0.698939i	0.381649	+	0.698939i
$681$		0			0
$682$		0			0
$683$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$683$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$684$		0			0
$685$	0.115047	+	0.0739364i	0.115047	+	0.0739364i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.194389	+	1.35201i	0.194389	+	1.35201i
$690$		0			0
$691$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$691$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$692$	−0.373128	+	1.71524i	−0.373128	+	1.71524i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.215109	−	0.0982369i	0.215109	−	0.0982369i
$698$	−0.281733	−	1.95949i	−0.281733	−	1.95949i
$699$		0			0
$700$		0			0
$701$	0.203743	+	0.936593i	0.203743	+	0.936593i	0.959493	+	0.281733i	$0.0909091\pi$
$701$	0.203743	+	0.936593i	0.203743	+	0.936593i	−0.755750	+	0.654861i	$0.772727\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$707$		0			0
$708$		0			0
$709$	0.340335	+	1.56449i	0.340335	+	1.56449i	0.755750	+	0.654861i	$0.227273\pi$
$709$	0.340335	+	1.56449i	0.340335	+	1.56449i	−0.415415	+	0.909632i	$0.636364\pi$
$710$		0			0
$711$		0			0
$712$	0.677760	+	0.677760i	0.677760	+	0.677760i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$719$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$720$	0.203743	−	0.936593i	0.203743	−	0.936593i
$721$		0			0
$722$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$723$		0			0
$724$	0.584585	−	0.909632i	0.584585	−	0.909632i
$725$	−0.0928492	−	0.0804543i	−0.0928492	−	0.0804543i
$726$		0			0
$727$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$727$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$728$		0			0
$729$	−0.281733	+	0.959493i	−0.281733	+	0.959493i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−1.19550	−	1.59700i	−1.19550	−	1.59700i	−0.654861	−	0.755750i	$-0.727273\pi$
$733$	−1.19550	−	1.59700i	−1.19550	−	1.59700i	−0.540641	−	0.841254i	$-0.681818\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$739$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$739$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$740$	1.00497	+	1.15980i	1.00497	+	1.15980i
$741$		0			0
$742$		0			0
$743$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$743$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$744$		0			0
$745$	0.135365	−	0.0194625i	0.135365	−	0.0194625i
$746$	1.32505	−	0.494217i	1.32505	−	0.494217i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$751$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$752$		0			0
$753$		0			0
$754$	1.77021	−	2.36473i	1.77021	−	2.36473i
$755$		0			0
$756$		0			0
$757$	1.75575	−	0.654861i	1.75575	−	0.654861i	0.755750	−	0.654861i	$-0.227273\pi$
$757$	1.75575	−	0.654861i	1.75575	−	0.654861i	1.00000			$0$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.574406	+	1.05195i	0.574406	+	1.05195i	0.989821	+	0.142315i	$0.0454545\pi$
$761$	0.574406	+	1.05195i	0.574406	+	1.05195i	−0.415415	+	0.909632i	$0.636364\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.563104	+	0.563104i	−0.563104	+	0.563104i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.494217	+	1.32505i	−0.494217	+	1.32505i	0.415415	+	0.909632i	$0.363636\pi$
$769$	−0.494217	+	1.32505i	−0.494217	+	1.32505i	−0.909632	+	0.415415i	$0.863636\pi$
$770$		0			0
$771$		0			0
$772$	1.24123	+	0.677760i	1.24123	+	0.677760i
$773$	−1.32505	−	1.32505i	−1.32505	−	1.32505i	−0.909632	−	0.415415i	$-0.863636\pi$
$773$	−1.32505	−	1.32505i	−1.32505	−	1.32505i	−0.415415	−	0.909632i	$-0.636364\pi$
$774$		0			0
$775$		0			0
$776$	−1.68251			−1.68251
$777$		0			0
$778$	0.983568	+	1.53046i	0.983568	+	1.53046i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$785$	0.390981	−	1.79731i	0.390981	−	1.79731i
$786$		0			0
$787$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$787$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$788$	1.27155	+	1.10181i	1.27155	+	1.10181i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−2.36473	+	1.77021i	−2.36473	+	1.77021i
$794$	−0.797176	−	1.74557i	−0.797176	−	1.74557i
$795$		0			0
$796$		0			0
$797$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$797$	−1.10181	−	1.27155i	−1.10181	−	1.27155i	−0.142315	−	0.989821i	$-0.545455\pi$
$798$		0			0
$799$		0			0
$800$	0.0804543	−	0.0115676i	0.0804543	−	0.0115676i
$801$	−0.459359	+	0.841254i	−0.459359	+	0.841254i
$802$	−0.100889	+	1.41061i	−0.100889	+	1.41061i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.17116	+	1.56449i	−1.17116	+	1.56449i
$809$	1.83107	+	0.682956i	1.83107	+	0.682956i	0.989821	+	0.142315i	$0.0454545\pi$
$809$	1.83107	+	0.682956i	1.83107	+	0.682956i	0.841254	+	0.540641i	$0.181818\pi$
$810$	0.956056	−	0.0683785i	0.956056	−	0.0683785i
$811$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$811$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.19550	+	0.0855040i	1.19550	+	0.0855040i
$819$		0			0
$820$	0.0194625	+	0.272122i	0.0194625	+	0.272122i
$821$	−1.75089	−	0.125226i	−1.75089	−	0.125226i	−0.841254	−	0.540641i	$-0.818182\pi$
$821$	−1.75089	−	0.125226i	−1.75089	−	0.125226i	−0.909632	+	0.415415i	$0.863636\pi$
$822$		0			0
$823$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$823$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$827$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$828$		0			0
$829$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.755750	−	0.654861i	$-0.227273\pi$
$829$	1.71524	−	0.936593i	1.71524	−	0.936593i	0.959493	−	0.281733i	$-0.0909091\pi$
$830$		0			0
$831$		0			0
$832$	0.415415	+	1.90963i	0.415415	+	1.90963i
$833$	0.822373	−	0.118239i	0.822373	−	0.118239i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$839$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$840$		0			0
$841$	−0.841254	+	0.970858i	−0.841254	+	0.970858i
$842$	−1.74557	+	0.512546i	−1.74557	+	0.512546i
$843$		0			0
$844$		0			0
$845$	0.192777	+	2.69537i	0.192777	+	2.69537i
$846$		0			0
$847$		0			0
$848$	0.559521	+	0.418852i	0.559521	+	0.418852i
$849$		0			0
$850$	−0.0614286	−	0.0280535i	−0.0614286	−	0.0280535i
$851$		0			0
$852$		0			0
$853$	−0.936593	+	0.203743i	−0.936593	+	0.203743i	−0.654861	−	0.755750i	$-0.727273\pi$
$853$	−0.936593	+	0.203743i	−0.936593	+	0.203743i	−0.281733	+	0.959493i	$0.590909\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−1.83107	−	0.398326i	−1.83107	−	0.398326i	−0.841254	−	0.540641i	$-0.818182\pi$
$857$	−1.83107	−	0.398326i	−1.83107	−	0.398326i	−0.989821	+	0.142315i	$0.954545\pi$
$858$		0			0
$859$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$859$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$863$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$864$		0			0
$865$	0.698939	+	1.53046i	0.698939	+	1.53046i
$866$	−0.0303285	+	0.424047i	−0.0303285	+	0.424047i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.797176	+	1.74557i	0.797176	+	1.74557i
$873$	−0.474017	−	1.61435i	−0.474017	−	1.61435i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	0.142315	−	0.989821i	$-0.454545\pi$
$877$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i	−0.415415	+	0.909632i	$0.636364\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	0.654861	−	1.75575i	0.654861	−	1.75575i		−	1.00000i	$-0.5\pi$
$881$	0.654861	−	1.75575i	0.654861	−	1.75575i	0.654861	−	0.755750i	$-0.272727\pi$
$882$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$883$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$883$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$884$	0.567420	−	1.52131i	0.567420	−	1.52131i
$885$		0			0
$886$		0			0
$887$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$887$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$888$		0			0
$889$		0			0
$890$	0.909367	+	0.130747i	0.909367	+	0.130747i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.334961	+	0.613435i	0.334961	+	0.613435i
$899$		0			0
$900$	0.0337656	+	0.0739364i	0.0337656	+	0.0739364i
$901$	−0.202931	−	0.544078i	−0.202931	−	0.544078i
$902$		0			0
$903$		0			0
$904$	−0.234072	+	0.797176i	−0.234072	+	0.797176i
$905$	−0.0739364	−	1.03377i	−0.0739364	−	1.03377i
$906$		0			0
$907$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$907$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$908$		0			0
$909$	−1.83107	−	0.682956i	−1.83107	−	0.682956i
$910$		0			0
$911$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$911$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$912$		0			0
$913$		0			0
$914$	−1.50013	+	1.12299i	−1.50013	+	1.12299i
$915$		0			0
$916$	−0.300613	−	1.38189i	−0.300613	−	1.38189i
$917$		0			0
$918$		0			0
$919$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$919$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$920$		0			0
$921$		0			0
$922$	0.100889	−	0.100889i	0.100889	−	0.100889i
$923$		0			0
$924$		0			0
$925$	−0.127165	−	0.0276629i	−0.127165	−	0.0276629i
$926$		0			0
$927$		0			0
$928$	−0.215109	−	1.49611i	−0.215109	−	1.49611i
$929$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$929$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$930$		0			0
$931$		0			0
$932$	−1.95949	+	0.281733i	−1.95949	+	0.281733i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.71524	+	0.936593i	−1.71524	+	0.936593i
$937$	1.84125	+	0.540641i	1.84125	+	0.540641i	1.00000			$0$
$937$	1.84125	+	0.540641i	1.84125	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.0303285	+	0.139418i	−0.0303285	+	0.139418i	−0.989821	−	0.142315i	$-0.954545\pi$
$941$	−0.0303285	+	0.139418i	−0.0303285	+	0.139418i	0.959493	+	0.281733i	$0.0909091\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$947$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.13214	−	1.13214i	1.13214	−	1.13214i	0.142315	−	0.989821i	$-0.454545\pi$
$953$	1.13214	−	1.13214i	1.13214	−	1.13214i	0.989821	−	0.142315i	$-0.0454545\pi$
$954$	−0.244250	+	0.654861i	−0.244250	+	0.654861i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.540641	+	0.841254i	0.540641	+	0.841254i
$962$	0.445301	−	3.09714i	0.445301	−	3.09714i
$963$		0			0
$964$	−0.334961	+	0.613435i	−0.334961	+	0.613435i
$965$	1.34172	−	0.192911i	1.34172	−	0.192911i
$966$		0			0
$967$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$967$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$968$	0.654861	+	0.755750i	0.654861	+	0.755750i
$969$		0			0
$970$	−1.29102	+	0.966443i	−1.29102	+	0.966443i
$971$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$971$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.215109	+	1.49611i	−0.215109	+	1.49611i
$977$	−0.989821	−	0.857685i	−0.989821	−	0.857685i		−	1.00000i	$-0.5\pi$
$977$	−0.989821	−	0.857685i	−0.989821	−	0.857685i	−0.989821	+	0.142315i	$0.954545\pi$
$978$		0			0
$979$		0			0
$980$	−0.203743	+	0.936593i	−0.203743	+	0.936593i
$981$	−1.45027	+	1.25667i	−1.45027	+	1.25667i
$982$		0			0
$983$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$983$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$984$		0			0
$985$	1.60857	+	0.115047i	1.60857	+	0.115047i
$986$	−0.521678	+	1.14231i	−0.521678	+	1.14231i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$991$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.98982	−	0.142315i	1.98982	−	0.142315i	0.989821	−	0.142315i	$-0.0454545\pi$
$997$	1.98982	−	0.142315i	1.98982	−	0.142315i	1.00000			$0$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1412.1.r.a.1055.1	✓	20
4.3	odd	2	CM	1412.1.r.a.1055.1	✓	20
353.88	even	44	inner	1412.1.r.a.1147.1	yes	20
1412.1147	odd	44	inner	1412.1.r.a.1147.1	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1412.1.r.a.1055.1	✓	20	1.1	even	1	trivial
1412.1.r.a.1055.1	✓	20	4.3	odd	2	CM
1412.1.r.a.1147.1	yes	20	353.88	even	44	inner
1412.1.r.a.1147.1	yes	20	1412.1147	odd	44	inner

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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:	\( N \)	\(=\)	\( 1412 = 2^{2} \cdot 353 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1412.r (of order \(44\), degree \(20\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.841254 + 0.540641i) q^{2} +(0.415415 - 0.909632i) q^{4} +(-0.203743 - 0.936593i) q^{5} +(0.142315 + 0.989821i) q^{8} +(-0.909632 + 0.415415i) q^{9} +O(q^{10})\)	\(q+(-0.841254 + 0.540641i) q^{2} +(0.415415 - 0.909632i) q^{4} +(-0.203743 - 0.936593i) q^{5} +(0.142315 + 0.989821i) q^{8} +(-0.909632 + 0.415415i) q^{9} +(0.677760 + 0.677760i) q^{10} +(0.139418 - 1.94931i) q^{13} +(-0.654861 - 0.755750i) q^{16} +(0.118239 + 0.822373i) q^{17} +(0.540641 - 0.841254i) q^{18} +(-0.936593 - 0.203743i) q^{20} +(0.0739364 - 0.0337656i) q^{25} +(0.936593 + 1.71524i) q^{26} +(-0.627899 - 1.37491i) q^{29} +(0.959493 + 0.281733i) q^{32} +(-0.544078 - 0.627899i) q^{34} +1.00000i q^{36} +(-1.28173 - 0.959493i) q^{37} +(0.898064 - 0.334961i) q^{40} +(-0.0801894 - 0.273100i) q^{41} +(0.574406 + 0.767317i) q^{45} -1.00000i q^{49} +(-0.0439442 + 0.0683785i) q^{50} +(-1.71524 - 0.936593i) q^{52} +(-0.682956 + 0.148568i) q^{53} +(1.27155 + 0.817178i) q^{58} +(-0.989821 - 1.14231i) q^{61} +(-0.959493 + 0.281733i) q^{64} +(-1.85412 + 0.266582i) q^{65} +(0.797176 + 0.234072i) q^{68} +(-0.540641 - 0.841254i) q^{72} +(1.59700 + 0.114220i) q^{74} +(-0.574406 + 0.767317i) q^{80} +(0.654861 - 0.755750i) q^{81} +(0.215109 + 0.186393i) q^{82} +(0.746139 - 0.278295i) q^{85} +(0.767317 - 0.574406i) q^{89} +(-0.898064 - 0.334961i) q^{90} +(-0.239446 + 1.66538i) q^{97} +(0.540641 + 0.841254i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8	\( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8} + 2 q^{10} - 2 q^{13} - 2 q^{16} + 4 q^{17} - 2 q^{20} + 2 q^{26} + 2 q^{32} - 4 q^{34} - 20 q^{37} + 2 q^{40} + 2 q^{45} - 22 q^{50} - 2 q^{52} - 2 q^{53} - 2 q^{64} + 4 q^{68} - 2 q^{74} - 2 q^{80} + 2 q^{81} + 4 q^{85} - 2 q^{89} - 2 q^{90} - 4 q^{97}+O(q^{100}) \) 20 * q + 2 * q^2 - 2 * q^4 - 2 * q^5 + 2 * q^8 + 2 * q^10 - 2 * q^13 - 2 * q^16 + 4 * q^17 - 2 * q^20 + 2 * q^26 + 2 * q^32 - 4 * q^34 - 20 * q^37 + 2 * q^40 + 2 * q^45 - 22 * q^50 - 2 * q^52 - 2 * q^53 - 2 * q^64 + 4 * q^68 - 2 * q^74 - 2 * q^80 + 2 * q^81 + 4 * q^85 - 2 * q^89 - 2 * q^90 - 4 * q^97

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