LMFDB - Embedded newform 1412.1.r.a.1291.1

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			1291.1
Root			$0.909632 + 0.415415i$ of defining polynomial
Character	$\chi$	$=$	1412.1291
Dual form			1412.1.r.a.35.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.959493 - 0.281733i) q^{2} +(0.841254 - 0.540641i) q^{4} +(-1.05195 + 0.574406i) q^{5} +(0.654861 - 0.755750i) q^{8} +(0.540641 - 0.841254i) q^{9} +O(q^{10})$	\(q+(0.959493 - 0.281733i) q^{2} +(0.841254 - 0.540641i) q^{4} +(-1.05195 + 0.574406i) q^{5} +(0.654861 - 0.755750i) q^{8} +(0.540641 - 0.841254i) q^{9} +(-0.847507 + 0.847507i) q^{10} +(0.334961 + 0.898064i) q^{13} +(0.415415 - 0.909632i) q^{16} +(1.10181 - 1.27155i) q^{17} +(0.281733 - 0.959493i) q^{18} +(-0.574406 + 1.05195i) q^{20} +(0.236009 - 0.367237i) q^{25} +(0.574406 + 0.767317i) q^{26} +(1.53046 + 0.983568i) q^{29} +(0.142315 - 0.989821i) q^{32} +(0.698939 - 1.53046i) q^{34} -1.00000i q^{36} +(-1.98982 + 0.142315i) q^{37} +(-0.254771 + 1.17116i) q^{40} +(-1.29639 + 0.186393i) q^{41} +(-0.0855040 + 1.19550i) q^{45} +1.00000i q^{49} +(0.122986 - 0.418852i) q^{50} +(0.767317 + 0.574406i) q^{52} +(-0.936593 - 1.71524i) q^{53} +(1.74557 + 0.512546i) q^{58} +(-0.755750 + 1.65486i) q^{61} +(-0.142315 - 0.989821i) q^{64} +(-0.868215 - 0.752312i) q^{65} +(0.239446 - 1.66538i) q^{68} +(-0.281733 - 0.959493i) q^{72} +(-1.86912 + 0.697148i) q^{74} +(0.0855040 + 1.19550i) q^{80} +(-0.415415 - 0.909632i) q^{81} +(-1.19136 + 0.544078i) q^{82} +(-0.428654 + 1.97049i) q^{85} +(-1.19550 - 0.0855040i) q^{89} +(0.254771 + 1.17116i) q^{90} +(1.25667 + 1.45027i) q^{97} +(0.281733 + 0.959493i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1412\mathbb{Z}\right)^\times$.

$n$	$707$	$709$
$\chi(n)$	$-1$	$e\left(\frac{37}{44}\right)$

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 $

$2$	0.959493	−	0.281733i	0.959493	−	0.281733i
$2$	0.959493	−	0.281733i	0.959493	−	0.281733i
$3$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$3$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$4$	0.841254	−	0.540641i	0.841254	−	0.540641i
$5$	−1.05195	+	0.574406i	−1.05195	+	0.574406i	−0.909632	−	0.415415i	$-0.863636\pi$
$5$	−1.05195	+	0.574406i	−1.05195	+	0.574406i	−0.142315	+	0.989821i	$0.545455\pi$
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$	0.654861	−	0.755750i	0.654861	−	0.755750i
$9$	0.540641	−	0.841254i	0.540641	−	0.841254i
$10$	−0.847507	+	0.847507i	−0.847507	+	0.847507i
$11$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$11$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$12$		0			0
$13$	0.334961	+	0.898064i	0.334961	+	0.898064i	0.989821	+	0.142315i	$0.0454545\pi$
$13$	0.334961	+	0.898064i	0.334961	+	0.898064i	−0.654861	+	0.755750i	$0.727273\pi$
$14$		0			0
$15$		0			0
$16$	0.415415	−	0.909632i	0.415415	−	0.909632i
$17$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$17$	1.10181	−	1.27155i	1.10181	−	1.27155i	0.959493	−	0.281733i	$-0.0909091\pi$
$18$	0.281733	−	0.959493i	0.281733	−	0.959493i
$19$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$19$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$20$	−0.574406	+	1.05195i	−0.574406	+	1.05195i
$21$		0			0
$22$		0			0
$23$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$23$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$24$		0			0
$25$	0.236009	−	0.367237i	0.236009	−	0.367237i
$26$	0.574406	+	0.767317i	0.574406	+	0.767317i
$27$		0			0
$28$		0			0
$29$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.989821	+	0.142315i	$0.0454545\pi$
$29$	1.53046	+	0.983568i	1.53046	+	0.983568i	0.540641	+	0.841254i	$0.318182\pi$
$30$		0			0
$31$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$31$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$32$	0.142315	−	0.989821i	0.142315	−	0.989821i
$33$		0			0
$34$	0.698939	−	1.53046i	0.698939	−	1.53046i
$35$		0			0
$36$		−	1.00000i		−	1.00000i
$37$	−1.98982	+	0.142315i	−1.98982	+	0.142315i	−0.989821	+	0.142315i	$0.954545\pi$
$37$	−1.98982	+	0.142315i	−1.98982	+	0.142315i	−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	−0.254771	+	1.17116i	−0.254771	+	1.17116i
$41$	−1.29639	+	0.186393i	−1.29639	+	0.186393i	−0.755750	−	0.654861i	$-0.772727\pi$
$41$	−1.29639	+	0.186393i	−1.29639	+	0.186393i	−0.540641	+	0.841254i	$0.681818\pi$
$42$		0			0
$43$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$43$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$44$		0			0
$45$	−0.0855040	+	1.19550i	−0.0855040	+	1.19550i
$46$		0			0
$47$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$47$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$48$		0			0
$49$			1.00000i			1.00000i
$50$	0.122986	−	0.418852i	0.122986	−	0.418852i
$51$		0			0
$52$	0.767317	+	0.574406i	0.767317	+	0.574406i
$53$	−0.936593	−	1.71524i	−0.936593	−	1.71524i	−0.654861	−	0.755750i	$-0.727273\pi$
$53$	−0.936593	−	1.71524i	−0.936593	−	1.71524i	−0.281733	−	0.959493i	$-0.590909\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.74557	+	0.512546i	1.74557	+	0.512546i
$59$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$59$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$60$		0			0
$61$	−0.755750	+	1.65486i	−0.755750	+	1.65486i			1.00000i	$0.5\pi$
$61$	−0.755750	+	1.65486i	−0.755750	+	1.65486i	−0.755750	+	0.654861i	$0.772727\pi$
$62$		0			0
$63$		0			0
$64$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$65$	−0.868215	−	0.752312i	−0.868215	−	0.752312i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$	0.239446	−	1.66538i	0.239446	−	1.66538i
$69$		0			0
$70$		0			0
$71$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$71$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$72$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$73$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$73$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$74$	−1.86912	+	0.697148i	−1.86912	+	0.697148i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$79$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$80$	0.0855040	+	1.19550i	0.0855040	+	1.19550i
$81$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$82$	−1.19136	+	0.544078i	−1.19136	+	0.544078i
$83$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$83$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$84$		0			0
$85$	−0.428654	+	1.97049i	−0.428654	+	1.97049i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.19550	−	0.0855040i	−1.19550	−	0.0855040i	−0.540641	−	0.841254i	$-0.681818\pi$
$89$	−1.19550	−	0.0855040i	−1.19550	−	0.0855040i	−0.654861	+	0.755750i	$0.727273\pi$
$90$	0.254771	+	1.17116i	0.254771	+	1.17116i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.25667	+	1.45027i	1.25667	+	1.45027i	0.841254	+	0.540641i	$0.181818\pi$
$97$	1.25667	+	1.45027i	1.25667	+	1.45027i	0.415415	+	0.909632i	$0.363636\pi$
$98$	0.281733	+	0.959493i	0.281733	+	0.959493i
$99$		0			0
$100$		−	0.436535i		−	0.436535i
$101$	−0.677760	+	0.677760i	−0.677760	+	0.677760i	−0.959493	−	0.281733i	$-0.909091\pi$
$101$	−0.677760	+	0.677760i	−0.677760	+	0.677760i	0.281733	+	0.959493i	$0.409091\pi$
$102$		0			0
$103$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$103$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$104$	0.898064	+	0.334961i	0.898064	+	0.334961i
$105$		0			0
$106$	−1.38189	−	1.38189i	−1.38189	−	1.38189i
$107$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$107$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$108$		0			0
$109$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	1.00000			$0$
$109$	0.0405070	+	0.281733i	0.0405070	+	0.281733i	−0.959493	+	0.281733i	$0.909091\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.909632	−	1.41542i	−0.909632	−	1.41542i	−0.909632	−	0.415415i	$-0.863636\pi$
$113$	−0.909632	−	1.41542i	−0.909632	−	1.41542i		−	1.00000i	$-0.5\pi$
$114$		0			0
$115$		0			0
$116$	1.81926			1.81926
$117$	0.936593	+	0.203743i	0.936593	+	0.203743i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$122$	−0.258908	+	1.80075i	−0.258908	+	1.80075i
$123$		0			0
$124$		0			0
$125$	0.0481785	−	0.673623i	0.0481785	−	0.673623i
$126$		0			0
$127$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$127$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$128$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$129$		0			0
$130$	−1.04500	−	0.477234i	−1.04500	−	0.477234i
$131$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$131$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.239446	−	1.66538i	−0.239446	−	1.66538i
$137$	0.494217	+	0.494217i	0.494217	+	0.494217i	0.909632	−	0.415415i	$-0.136364\pi$
$137$	0.494217	+	0.494217i	0.494217	+	0.494217i	−0.415415	+	0.909632i	$0.636364\pi$
$138$		0			0
$139$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$139$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$145$	−2.17493	−	0.155554i	−2.17493	−	0.155554i
$146$		0			0
$147$		0			0
$148$	−1.59700	+	1.19550i	−1.59700	+	1.19550i
$149$	−0.244250	−	0.654861i	−0.244250	−	0.654861i	0.755750	−	0.654861i	$-0.227273\pi$
$149$	−0.244250	−	0.654861i	−0.244250	−	0.654861i	−1.00000			$\pi$
$150$		0			0
$151$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$151$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$152$		0			0
$153$	−0.474017	−	1.61435i	−0.474017	−	1.61435i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.153882	−	0.239446i	−0.153882	−	0.239446i	0.755750	−	0.654861i	$-0.227273\pi$
$157$	−0.153882	−	0.239446i	−0.153882	−	0.239446i	−0.909632	+	0.415415i	$0.863636\pi$
$158$		0			0
$159$		0			0
$160$	0.418852	+	1.12299i	0.418852	+	1.12299i
$161$		0			0
$162$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$163$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$163$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$164$	−0.989821	+	0.857685i	−0.989821	+	0.857685i
$165$		0			0
$166$		0			0
$167$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$167$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$168$		0			0
$169$	0.0614286	−	0.0532282i	0.0614286	−	0.0532282i
$170$	0.143861	+	2.01144i	0.143861	+	2.01144i
$171$		0			0
$172$		0			0
$173$	0.767317	+	1.40524i	0.767317	+	1.40524i	0.909632	+	0.415415i	$0.136364\pi$
$173$	0.767317	+	1.40524i	0.767317	+	1.40524i	−0.142315	+	0.989821i	$0.545455\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.17116	+	0.254771i	−1.17116	+	0.254771i
$179$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$179$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$180$	0.574406	+	1.05195i	0.574406	+	1.05195i
$181$	0.425839	−	0.368991i	0.425839	−	0.368991i	−0.415415	−	0.909632i	$-0.636364\pi$
$181$	0.425839	−	0.368991i	0.425839	−	0.368991i	0.841254	+	0.540641i	$0.181818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.01144	−	1.29267i	2.01144	−	1.29267i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$191$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$192$		0			0
$193$	0.494217	+	1.32505i	0.494217	+	1.32505i	0.909632	+	0.415415i	$0.136364\pi$
$193$	0.494217	+	1.32505i	0.494217	+	1.32505i	−0.415415	+	0.909632i	$0.636364\pi$
$194$	1.61435	+	1.03748i	1.61435	+	1.03748i
$195$		0			0
$196$	0.540641	+	0.841254i	0.540641	+	0.841254i
$197$	1.89945	+	0.273100i	1.89945	+	0.273100i	0.989821	−	0.142315i	$-0.0454545\pi$
$197$	1.89945	+	0.273100i	1.89945	+	0.273100i	0.909632	+	0.415415i	$0.136364\pi$
$198$		0			0
$199$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$199$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$200$	−0.122986	−	0.418852i	−0.122986	−	0.418852i
$201$		0			0
$202$	−0.459359	+	0.841254i	−0.459359	+	0.841254i
$203$		0			0
$204$		0			0
$205$	1.25667	−	0.940730i	1.25667	−	0.940730i
$206$		0			0
$207$		0			0
$208$	0.956056	+	0.0683785i	0.956056	+	0.0683785i
$209$		0			0
$210$		0			0
$211$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$211$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$212$	−1.71524	−	0.936593i	−1.71524	−	0.936593i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.118239	+	0.258908i	0.118239	+	0.258908i
$219$		0			0
$220$		0			0
$221$	1.51100	+	0.563574i	1.51100	+	0.563574i
$222$		0			0
$223$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$223$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$224$		0			0
$225$	−0.181343	−	0.397086i	−0.181343	−	0.397086i
$226$	−1.27155	−	1.10181i	−1.27155	−	1.10181i
$227$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$227$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$228$		0			0
$229$	1.41061	+	0.100889i	1.41061	+	0.100889i	0.755750	−	0.654861i	$-0.227273\pi$
$229$	1.41061	+	0.100889i	1.41061	+	0.100889i	0.654861	+	0.755750i	$0.272727\pi$
$230$		0			0
$231$		0			0
$232$	1.74557	−	0.512546i	1.74557	−	0.512546i
$233$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$233$	−0.425839	−	1.45027i	−0.425839	−	1.45027i	0.415415	−	0.909632i	$-0.363636\pi$
$234$	0.956056	−	0.0683785i	0.956056	−	0.0683785i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$239$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$240$		0			0
$241$	−1.83107	+	0.682956i	−1.83107	+	0.682956i	−0.841254	+	0.540641i	$0.818182\pi$
$241$	−1.83107	+	0.682956i	−1.83107	+	0.682956i	−0.989821	+	0.142315i	$0.954545\pi$
$242$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$243$		0			0
$244$	0.258908	+	1.80075i	0.258908	+	1.80075i
$245$	−0.574406	−	1.05195i	−0.574406	−	1.05195i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.143555	−	0.659910i	−0.143555	−	0.659910i
$251$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$251$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$257$	0.559521	+	1.50013i	0.559521	+	1.50013i	0.841254	+	0.540641i	$0.181818\pi$
$257$	0.559521	+	1.50013i	0.559521	+	1.50013i	−0.281733	+	0.959493i	$0.590909\pi$
$258$		0			0
$259$		0			0
$260$	−1.13712	−	0.163493i	−1.13712	−	0.163493i
$261$	1.65486	−	0.755750i	1.65486	−	0.755750i
$262$		0			0
$263$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$263$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$264$		0			0
$265$	1.97049	+	1.26636i	1.97049	+	1.26636i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.89945	+	0.557730i	−1.89945	+	0.557730i	−0.909632	+	0.415415i	$0.863636\pi$
$269$	−1.89945	+	0.557730i	−1.89945	+	0.557730i	−0.989821	+	0.142315i	$0.954545\pi$
$270$		0			0
$271$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$271$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$272$	−0.698939	−	1.53046i	−0.698939	−	1.53046i
$273$		0			0
$274$	0.613435	+	0.334961i	0.613435	+	0.334961i
$275$		0			0
$276$		0			0
$277$	−0.817178	+	1.27155i	−0.817178	+	1.27155i	0.142315	+	0.989821i	$0.454545\pi$
$277$	−0.817178	+	1.27155i	−0.817178	+	1.27155i	−0.959493	+	0.281733i	$0.909091\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.415415	+	0.909632i	$0.363636\pi$
$281$	0.557730	+	1.89945i	0.557730	+	1.89945i	0.142315	+	0.989821i	$0.454545\pi$
$282$		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
$288$	−0.755750	−	0.654861i	−0.755750	−	0.654861i
$289$	−0.260554	−	1.81219i	−0.260554	−	1.81219i
$290$	−2.13066	+	0.463496i	−2.13066	+	0.463496i
$291$		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
$296$	−1.19550	+	1.59700i	−1.19550	+	1.59700i
$297$		0			0
$298$	−0.418852	−	0.559521i	−0.418852	−	0.559521i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.155554	−	2.17493i	−0.155554	−	2.17493i
$306$	−0.909632	−	1.41542i	−0.909632	−	1.41542i
$307$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$307$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	0.424047	−	1.94931i	0.424047	−	1.94931i	0.142315	−	0.989821i	$-0.454545\pi$
$313$	0.424047	−	1.94931i	0.424047	−	1.94931i	0.281733	−	0.959493i	$-0.409091\pi$
$314$	−0.215109	−	0.186393i	−0.215109	−	0.186393i
$315$		0			0
$316$		0			0
$317$		−	1.91899i		−	1.91899i	−0.281733	−	0.959493i	$-0.590909\pi$
$317$		−	1.91899i		−	1.91899i	0.281733	−	0.959493i	$-0.409091\pi$
$318$		0			0
$319$		0			0
$320$	0.718267	+	0.959493i	0.718267	+	0.959493i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$325$	0.408856	+	0.0889411i	0.408856	+	0.0889411i
$326$		0			0
$327$		0			0
$328$	−0.708089	+	1.10181i	−0.708089	+	1.10181i
$329$		0			0
$330$		0			0
$331$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$331$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$332$		0			0
$333$	−0.956056	+	1.75089i	−0.956056	+	1.75089i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.544078	−	1.19136i	0.544078	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$337$	0.544078	−	1.19136i	0.544078	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$338$	0.0439442	−	0.0683785i	0.0439442	−	0.0683785i
$339$		0			0
$340$	0.704722	+	1.88943i	0.704722	+	1.88943i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.13214	+	1.13214i	1.13214	+	1.13214i
$347$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$347$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$348$		0			0
$349$	−1.27155	+	0.817178i	−1.27155	+	0.817178i	−0.989821	−	0.142315i	$-0.954545\pi$
$349$	−1.27155	+	0.817178i	−1.27155	+	0.817178i	−0.281733	+	0.959493i	$0.590909\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$353$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$354$		0			0
$355$		0			0
$356$	−1.05195	+	0.574406i	−1.05195	+	0.574406i
$357$		0			0
$358$		0			0
$359$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$359$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$360$	0.847507	+	0.847507i	0.847507	+	0.847507i
$361$	0.654861	−	0.755750i	0.654861	−	0.755750i
$362$	0.304632	−	0.474017i	0.304632	−	0.474017i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.877679	−	0.479249i	$-0.840909\pi$
$367$		0			0		0.877679	+	0.479249i	$0.159091\pi$
$368$		0			0
$369$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$370$	1.56577	−	1.80700i	1.56577	−	1.80700i
$371$		0			0
$372$		0			0
$373$	0.677760	−	1.24123i	0.677760	−	1.24123i	−0.281733	−	0.959493i	$-0.590909\pi$
$373$	0.677760	−	1.24123i	0.677760	−	1.24123i	0.959493	−	0.281733i	$-0.0909091\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.370663	+	1.70391i	−0.370663	+	1.70391i
$378$		0			0
$379$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$379$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$383$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$384$		0			0
$385$		0			0
$386$	0.847507	+	1.13214i	0.847507	+	1.13214i
$387$		0			0
$388$	1.84125	+	0.540641i	1.84125	+	0.540641i
$389$		−	1.08128i		−	1.08128i	−0.841254	−	0.540641i	$-0.818182\pi$
$389$		−	1.08128i		−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$390$		0			0
$391$		0			0
$392$	0.755750	+	0.654861i	0.755750	+	0.654861i
$393$		0			0
$394$	1.89945	−	0.273100i	1.89945	−	0.273100i
$395$		0			0
$396$		0			0
$397$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$397$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$398$		0			0
$399$		0			0
$400$	−0.236009	−	0.367237i	−0.236009	−	0.367237i
$401$	0.100889	+	1.41061i	0.100889	+	1.41061i	0.755750	+	0.654861i	$0.227273\pi$
$401$	0.100889	+	1.41061i	0.100889	+	1.41061i	−0.654861	+	0.755750i	$0.727273\pi$
$402$		0			0
$403$		0			0
$404$	−0.203743	+	0.936593i	−0.203743	+	0.936593i
$405$	0.959493	+	0.718267i	0.959493	+	0.718267i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.142315	+	0.0101786i	−0.142315	+	0.0101786i	−0.142315	−	0.989821i	$-0.545455\pi$
$409$	−0.142315	+	0.0101786i	−0.142315	+	0.0101786i			1.00000i	$0.5\pi$
$410$	0.940730	−	1.25667i	0.940730	−	1.25667i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.936593	−	0.203743i	0.936593	−	0.203743i
$417$		0			0
$418$		0			0
$419$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$419$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$420$		0			0
$421$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	0.755750	+	0.654861i	$0.227273\pi$
$421$	−0.153882	+	1.07028i	−0.153882	+	1.07028i	−0.909632	+	0.415415i	$0.863636\pi$
$422$		0			0
$423$		0			0
$424$	−1.90963	−	0.415415i	−1.90963	−	0.415415i
$425$	−0.206925	−	0.704722i	−0.206925	−	0.704722i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$431$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$432$		0			0
$433$	−0.125226	−	1.75089i	−0.125226	−	1.75089i	−0.540641	−	0.841254i	$-0.681818\pi$
$433$	−0.125226	−	1.75089i	−0.125226	−	1.75089i	0.415415	−	0.909632i	$-0.363636\pi$
$434$		0			0
$435$		0			0
$436$	0.186393	+	0.215109i	0.186393	+	0.215109i
$437$		0			0
$438$		0			0
$439$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$439$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$440$		0			0
$441$	0.841254	+	0.540641i	0.841254	+	0.540641i
$442$	1.60857	+	0.115047i	1.60857	+	0.115047i
$443$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$443$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$444$		0			0
$445$	1.30672	−	0.596758i	1.30672	−	0.596758i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.682956	+	1.83107i	0.682956	+	1.83107i	0.540641	+	0.841254i	$0.318182\pi$
$449$	0.682956	+	1.83107i	0.682956	+	1.83107i	0.142315	+	0.989821i	$0.454545\pi$
$450$	−0.285870	−	0.329911i	−0.285870	−	0.329911i
$451$		0			0
$452$	−1.53046	−	0.698939i	−1.53046	−	0.698939i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.398326	−	0.148568i	−0.398326	−	0.148568i	0.142315	−	0.989821i	$-0.454545\pi$
$457$	−0.398326	−	0.148568i	−0.398326	−	0.148568i	−0.540641	+	0.841254i	$0.681818\pi$
$458$	1.38189	−	0.300613i	1.38189	−	0.300613i
$459$		0			0
$460$		0			0
$461$	−0.334961	−	0.613435i	−0.334961	−	0.613435i	0.654861	−	0.755750i	$-0.272727\pi$
$461$	−0.334961	−	0.613435i	−0.334961	−	0.613435i	−0.989821	+	0.142315i	$0.954545\pi$
$462$		0			0
$463$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$463$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$464$	1.53046	−	0.983568i	1.53046	−	0.983568i
$465$		0			0
$466$	−0.817178	−	1.27155i	−0.817178	−	1.27155i
$467$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$467$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$468$	0.898064	−	0.334961i	0.898064	−	0.334961i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−1.94931	−	0.139418i	−1.94931	−	0.139418i
$478$		0			0
$479$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$479$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$480$		0			0
$481$	−0.794320	−	1.73932i	−0.794320	−	1.73932i
$482$	−1.56449	+	1.17116i	−1.56449	+	1.17116i
$483$		0			0
$484$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$485$	−2.15499	−	0.803771i	−2.15499	−	0.803771i
$486$		0			0
$487$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$487$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$488$	0.755750	+	1.65486i	0.755750	+	1.65486i
$489$		0			0
$490$	−0.847507	−	0.847507i	−0.847507	−	0.847507i
$491$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$491$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$492$		0			0
$493$	2.93694	−	0.862362i	2.93694	−	0.862362i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$499$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$500$	−0.323658	−	0.592735i	−0.323658	−	0.592735i
$501$		0			0
$502$		0			0
$503$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$503$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$504$		0			0
$505$	0.323658	−	1.10228i	0.323658	−	1.10228i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	1.49611	+	0.215109i	1.49611	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$509$	1.49611	+	0.215109i	1.49611	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$510$		0			0
$511$		0			0
$512$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$513$		0			0
$514$	0.959493	+	1.28173i	0.959493	+	1.28173i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−1.13712	+	0.163493i	−1.13712	+	0.163493i
$521$	1.27155	−	0.817178i	1.27155	−	0.817178i	0.281733	−	0.959493i	$-0.409091\pi$
$521$	1.27155	−	0.817178i	1.27155	−	0.817178i	0.989821	+	0.142315i	$0.0454545\pi$
$522$	1.37491	−	1.19136i	1.37491	−	1.19136i
$523$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$523$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.959493	+	0.281733i	0.959493	+	0.281733i
$530$	2.24745	+	0.659910i	2.24745	+	0.659910i
$531$		0			0
$532$		0			0
$533$	−0.601633	−	1.10181i	−0.601633	−	1.10181i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.66538	+	1.07028i	−1.66538	+	1.07028i
$539$		0			0
$540$		0			0
$541$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$541$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$542$		0			0
$543$		0			0
$544$	−1.10181	−	1.27155i	−1.10181	−	1.27155i
$545$	−0.204440	−	0.273100i	−0.204440	−	0.273100i
$546$		0			0
$547$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$547$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$548$	0.682956	+	0.148568i	0.682956	+	0.148568i
$549$	0.983568	+	1.53046i	0.983568	+	1.53046i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.425839	+	1.45027i	−0.425839	+	1.45027i
$555$		0			0
$556$		0			0
$557$	0.654861	+	1.75575i	0.654861	+	1.75575i	0.654861	+	0.755750i	$0.272727\pi$
$557$	0.654861	+	1.75575i	0.654861	+	1.75575i			1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.07028	+	1.66538i	1.07028	+	1.66538i
$563$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$563$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$564$		0			0
$565$	1.76991	+	0.966443i	1.76991	+	0.966443i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.300613	+	0.300613i	0.300613	+	0.300613i	0.841254	−	0.540641i	$-0.181818\pi$
$569$	0.300613	+	0.300613i	0.300613	+	0.300613i	−0.540641	+	0.841254i	$0.681818\pi$
$570$		0			0
$571$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$571$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−0.909632	−	0.415415i	−0.909632	−	0.415415i
$577$	−0.559521	+	0.418852i	−0.559521	+	0.418852i	−0.841254	−	0.540641i	$-0.818182\pi$
$577$	−0.559521	+	0.418852i	−0.559521	+	0.418852i	0.281733	+	0.959493i	$0.409091\pi$
$578$	−0.760554	−	1.66538i	−0.760554	−	1.66538i
$579$		0			0
$580$	−1.91377	+	1.04500i	−1.91377	+	1.04500i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−1.10228	+	0.323658i	−1.10228	+	0.323658i
$586$	−0.563465	−	1.91899i	−0.563465	−	1.91899i
$587$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$587$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.697148	+	1.86912i	−0.697148	+	1.86912i
$593$	−0.909632	−	1.41542i	−0.909632	−	1.41542i	−0.909632	−	0.415415i	$-0.863636\pi$
$593$	−0.909632	−	1.41542i	−0.909632	−	1.41542i		−	1.00000i	$-0.5\pi$
$594$		0			0
$595$		0			0
$596$	−0.559521	−	0.418852i	−0.559521	−	0.418852i
$597$		0			0
$598$		0			0
$599$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$599$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$600$		0			0
$601$	1.94931	−	0.424047i	1.94931	−	0.424047i	0.959493	−	0.281733i	$-0.0909091\pi$
$601$	1.94931	−	0.424047i	1.94931	−	0.424047i	0.989821	−	0.142315i	$-0.0454545\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.847507	−	0.847507i	0.847507	−	0.847507i
$606$		0			0
$607$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$607$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$608$		0			0
$609$		0			0
$610$	−0.762003	−	2.04301i	−0.762003	−	2.04301i
$611$		0			0
$612$	−1.27155	−	1.10181i	−1.27155	−	1.10181i
$613$	−0.281733	−	0.0405070i	−0.281733	−	0.0405070i		−	1.00000i	$-0.5\pi$
$613$	−0.281733	−	0.0405070i	−0.281733	−	0.0405070i	−0.281733	+	0.959493i	$0.590909\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.956056	−	0.0683785i	−0.956056	−	0.0683785i	−0.415415	−	0.909632i	$-0.636364\pi$
$617$	−0.956056	−	0.0683785i	−0.956056	−	0.0683785i	−0.540641	+	0.841254i	$0.681818\pi$
$618$		0			0
$619$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$619$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.517596	+	1.13338i	0.517596	+	1.13338i
$626$	−0.142315	−	1.98982i	−0.142315	−	1.98982i
$627$		0			0
$628$	−0.258908	−	0.118239i	−0.258908	−	0.118239i
$629$	−2.01144	+	2.68697i	−2.01144	+	2.68697i
$630$		0			0
$631$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$631$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$632$		0			0
$633$		0			0
$634$	−0.540641	−	1.84125i	−0.540641	−	1.84125i
$635$		0			0
$636$		0			0
$637$	−0.898064	+	0.334961i	−0.898064	+	0.334961i
$638$		0			0
$639$		0			0
$640$	0.959493	+	0.718267i	0.959493	+	0.718267i
$641$	−0.215109	−	0.186393i	−0.215109	−	0.186393i	0.540641	−	0.841254i	$-0.318182\pi$
$641$	−0.215109	−	0.186393i	−0.215109	−	0.186393i	−0.755750	+	0.654861i	$0.772727\pi$
$642$		0			0
$643$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$643$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$647$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$648$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$649$		0			0
$650$	0.417352	−	0.0298496i	0.417352	−	0.0298496i
$651$		0			0
$652$		0			0
$653$	−0.677760	−	1.24123i	−0.677760	−	1.24123i	−0.959493	−	0.281733i	$-0.909091\pi$
$653$	−0.677760	−	1.24123i	−0.677760	−	1.24123i	0.281733	−	0.959493i	$-0.409091\pi$
$654$		0			0
$655$		0			0
$656$	−0.368991	+	1.25667i	−0.368991	+	1.25667i
$657$		0			0
$658$		0			0
$659$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$659$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$660$		0			0
$661$	0.125226	−	1.75089i	0.125226	−	1.75089i	−0.415415	−	0.909632i	$-0.636364\pi$
$661$	0.125226	−	1.75089i	0.125226	−	1.75089i	0.540641	−	0.841254i	$-0.318182\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.424047	+	1.94931i	−0.424047	+	1.94931i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.254771	+	0.340335i	0.254771	+	0.340335i	0.909632	−	0.415415i	$-0.136364\pi$
$673$	0.254771	+	0.340335i	0.254771	+	0.340335i	−0.654861	+	0.755750i	$0.727273\pi$
$674$	0.186393	−	1.29639i	0.186393	−	1.29639i
$675$		0			0
$676$	0.0228997	−	0.0779892i	0.0228997	−	0.0779892i
$677$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.959493	−	0.281733i	$-0.909091\pi$
$677$	−1.61435	−	1.03748i	−1.61435	−	1.03748i	−0.654861	−	0.755750i	$-0.727273\pi$
$678$		0			0
$679$		0			0
$680$	1.20849	+	1.61435i	1.20849	+	1.61435i
$681$		0			0
$682$		0			0
$683$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$683$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$684$		0			0
$685$	−0.803771	−	0.236009i	−0.803771	−	0.236009i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.22668	−	1.41566i	1.22668	−	1.41566i
$690$		0			0
$691$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$691$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$692$	1.40524	+	0.767317i	1.40524	+	0.767317i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.19136	+	1.85380i	−1.19136	+	1.85380i
$698$	−0.989821	+	1.14231i	−0.989821	+	1.14231i
$699$		0			0
$700$		0			0
$701$	1.05195	−	0.574406i	1.05195	−	0.574406i	0.142315	−	0.989821i	$-0.454545\pi$
$701$	1.05195	−	0.574406i	1.05195	−	0.574406i	0.909632	+	0.415415i	$0.136364\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$707$		0			0
$708$		0			0
$709$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.841254	+	0.540641i	$0.818182\pi$
$709$	−1.75089	+	0.956056i	−1.75089	+	0.956056i	−0.909632	+	0.415415i	$0.863636\pi$
$710$		0			0
$711$		0			0
$712$	−0.847507	+	0.847507i	−0.847507	+	0.847507i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$719$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$720$	1.05195	+	0.574406i	1.05195	+	0.574406i
$721$		0			0
$722$	0.415415	−	0.909632i	0.415415	−	0.909632i
$723$		0			0
$724$	0.158746	−	0.540641i	0.158746	−	0.540641i
$725$	0.722404	−	0.329911i	0.722404	−	0.329911i
$726$		0			0
$727$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$727$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$728$		0			0
$729$	−0.989821	−	0.142315i	−0.989821	−	0.142315i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	0.133682	−	1.86912i	0.133682	−	1.86912i	−0.281733	−	0.959493i	$-0.590909\pi$
$733$	0.133682	−	1.86912i	0.133682	−	1.86912i	0.415415	−	0.909632i	$-0.363636\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.186393	+	1.29639i	−0.186393	+	1.29639i
$739$		0			0		−0.599278	−	0.800541i	$-0.704545\pi$
$739$		0			0		0.599278	+	0.800541i	$0.295455\pi$
$740$	0.993259	−	2.17493i	0.993259	−	2.17493i
$741$		0			0
$742$		0			0
$743$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$743$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$744$		0			0
$745$	0.633095	+	0.548580i	0.633095	+	0.548580i
$746$	0.300613	−	1.38189i	0.300613	−	1.38189i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$751$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$752$		0			0
$753$		0			0
$754$	0.124398	+	1.73932i	0.124398	+	1.73932i
$755$		0			0
$756$		0			0
$757$	0.0903680	−	0.415415i	0.0903680	−	0.415415i	−0.909632	−	0.415415i	$-0.863636\pi$
$757$	0.0903680	−	0.415415i	0.0903680	−	0.415415i	1.00000			$0$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.0855040	−	0.114220i	−0.0855040	−	0.114220i	0.755750	−	0.654861i	$-0.227273\pi$
$761$	−0.0855040	−	0.114220i	−0.0855040	−	0.114220i	−0.841254	+	0.540641i	$0.818182\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	1.42594	+	1.42594i	1.42594	+	1.42594i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.38189	−	0.300613i	1.38189	−	0.300613i	0.540641	−	0.841254i	$-0.318182\pi$
$769$	1.38189	−	0.300613i	1.38189	−	0.300613i	0.841254	+	0.540641i	$0.181818\pi$
$770$		0			0
$771$		0			0
$772$	1.13214	+	0.847507i	1.13214	+	0.847507i
$773$	−0.300613	+	0.300613i	−0.300613	+	0.300613i	−0.841254	−	0.540641i	$-0.818182\pi$
$773$	−0.300613	+	0.300613i	−0.300613	+	0.300613i	0.540641	+	0.841254i	$0.318182\pi$
$774$		0			0
$775$		0			0
$776$	1.91899			1.91899
$777$		0			0
$778$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	0.909632	+	0.415415i	0.909632	+	0.415415i
$785$	0.299415	+	0.163493i	0.299415	+	0.163493i
$786$		0			0
$787$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$787$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$788$	1.74557	−	0.797176i	1.74557	−	0.797176i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.73932	−	0.124398i	−1.73932	−	0.124398i
$794$	−0.239446	−	0.153882i	−0.239446	−	0.153882i
$795$		0			0
$796$		0			0
$797$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.142315	+	0.989821i	$0.545455\pi$
$797$	−0.797176	+	1.74557i	−0.797176	+	1.74557i	−0.654861	+	0.755750i	$0.727273\pi$
$798$		0			0
$799$		0			0
$800$	−0.329911	−	0.285870i	−0.329911	−	0.285870i
$801$	−0.718267	+	0.959493i	−0.718267	+	0.959493i
$802$	0.494217	+	1.32505i	0.494217	+	1.32505i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.0683785	+	0.956056i	0.0683785	+	0.956056i
$809$	−0.203743	−	0.936593i	−0.203743	−	0.936593i	−0.959493	−	0.281733i	$-0.909091\pi$
$809$	−0.203743	−	0.936593i	−0.203743	−	0.936593i	0.755750	−	0.654861i	$-0.227273\pi$
$810$	1.12299	+	0.418852i	1.12299	+	0.418852i
$811$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$811$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	−0.133682	+	0.0498610i	−0.133682	+	0.0498610i
$819$		0			0
$820$	0.548580	−	1.47080i	0.548580	−	1.47080i
$821$	1.50013	−	0.559521i	1.50013	−	0.559521i	0.540641	−	0.841254i	$-0.318182\pi$
$821$	1.50013	−	0.559521i	1.50013	−	0.559521i	0.959493	+	0.281733i	$0.0909091\pi$
$822$		0			0
$823$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$823$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$827$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$828$		0			0
$829$	−0.767317	+	0.574406i	−0.767317	+	0.574406i	−0.909632	−	0.415415i	$-0.863636\pi$
$829$	−0.767317	+	0.574406i	−0.767317	+	0.574406i	0.142315	+	0.989821i	$0.454545\pi$
$830$		0			0
$831$		0			0
$832$	0.841254	−	0.459359i	0.841254	−	0.459359i
$833$	1.27155	+	1.10181i	1.27155	+	1.10181i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$839$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$840$		0			0
$841$	0.959493	+	2.10100i	0.959493	+	2.10100i
$842$	0.153882	+	1.07028i	0.153882	+	1.07028i
$843$		0			0
$844$		0			0
$845$	−0.0340450	+	0.0912781i	−0.0340450	+	0.0912781i
$846$		0			0
$847$		0			0
$848$	−1.94931	+	0.139418i	−1.94931	+	0.139418i
$849$		0			0
$850$	−0.397086	−	0.617878i	−0.397086	−	0.617878i
$851$		0			0
$852$		0			0
$853$	−0.574406	−	1.05195i	−0.574406	−	1.05195i	−0.989821	−	0.142315i	$-0.954545\pi$
$853$	−0.574406	−	1.05195i	−0.574406	−	1.05195i	0.415415	−	0.909632i	$-0.363636\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.203743	−	0.373128i	0.203743	−	0.373128i	−0.755750	−	0.654861i	$-0.772727\pi$
$857$	0.203743	−	0.373128i	0.203743	−	0.373128i	0.959493	+	0.281733i	$0.0909091\pi$
$858$		0			0
$859$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$859$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$863$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$864$		0			0
$865$	−1.61435	−	1.03748i	−1.61435	−	1.03748i
$866$	−0.613435	−	1.64468i	−0.613435	−	1.64468i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.239446	+	0.153882i	0.239446	+	0.153882i
$873$	1.89945	−	0.273100i	1.89945	−	0.273100i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	0.654861	+	0.755750i	$0.272727\pi$
$877$	−0.186393	+	1.29639i	−0.186393	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.415415	+	0.0903680i	−0.415415	+	0.0903680i	−0.415415	−	0.909632i	$-0.636364\pi$
$881$	−0.415415	+	0.0903680i	−0.415415	+	0.0903680i			1.00000i	$0.5\pi$
$882$	0.959493	+	0.281733i	0.959493	+	0.281733i
$883$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$883$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$884$	1.57582	−	0.342800i	1.57582	−	0.342800i
$885$		0			0
$886$		0			0
$887$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$887$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$888$		0			0
$889$		0			0
$890$	1.08566	−	0.940730i	1.08566	−	0.940730i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	1.17116	+	1.56449i	1.17116	+	1.56449i
$899$		0			0
$900$	−0.367237	−	0.236009i	−0.367237	−	0.236009i
$901$	−3.21297	−	0.698939i	−3.21297	−	0.698939i
$902$		0			0
$903$		0			0
$904$	−1.66538	−	0.239446i	−1.66538	−	0.239446i
$905$	−0.236009	+	0.632763i	−0.236009	+	0.632763i
$906$		0			0
$907$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$907$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$908$		0			0
$909$	0.203743	+	0.936593i	0.203743	+	0.936593i
$910$		0			0
$911$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$911$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$912$		0			0
$913$		0			0
$914$	−0.424047	−	0.0303285i	−0.424047	−	0.0303285i
$915$		0			0
$916$	1.24123	−	0.677760i	1.24123	−	0.677760i
$917$		0			0
$918$		0			0
$919$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$919$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$920$		0			0
$921$		0			0
$922$	−0.494217	−	0.494217i	−0.494217	−	0.494217i
$923$		0			0
$924$		0			0
$925$	−0.417352	+	0.764323i	−0.417352	+	0.764323i
$926$		0			0
$927$		0			0
$928$	1.19136	−	1.37491i	1.19136	−	1.37491i
$929$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$929$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$930$		0			0
$931$		0			0
$932$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.767317	−	0.574406i	0.767317	−	0.574406i
$937$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	−0.959493	−	0.281733i	$-0.909091\pi$
$937$	0.0405070	−	0.281733i	0.0405070	−	0.281733i	1.00000			$0$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.613435	−	0.334961i	−0.613435	−	0.334961i	0.142315	−	0.989821i	$-0.454545\pi$
$941$	−0.613435	−	0.334961i	−0.613435	−	0.334961i	−0.755750	+	0.654861i	$0.772727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$947$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1.41061	+	1.41061i	1.41061	+	1.41061i	0.755750	+	0.654861i	$0.227273\pi$
$953$	1.41061	+	1.41061i	1.41061	+	1.41061i	0.654861	+	0.755750i	$0.272727\pi$
$954$	−1.90963	+	0.415415i	−1.90963	+	0.415415i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.281733	+	0.959493i	0.281733	+	0.959493i
$962$	−1.25217	−	1.44508i	−1.25217	−	1.44508i
$963$		0			0
$964$	−1.17116	+	1.56449i	−1.17116	+	1.56449i
$965$	−1.28101	−	1.11000i	−1.28101	−	1.11000i
$966$		0			0
$967$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$967$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$968$	−0.415415	+	0.909632i	−0.415415	+	0.909632i
$969$		0			0
$970$	−2.29415	−	0.164081i	−2.29415	−	0.164081i
$971$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$971$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	1.19136	+	1.37491i	1.19136	+	1.37491i
$977$	−0.755750	+	0.345139i	−0.755750	+	0.345139i	−0.755750	−	0.654861i	$-0.772727\pi$
$977$	−0.755750	+	0.345139i	−0.755750	+	0.345139i			1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$	−1.05195	−	0.574406i	−1.05195	−	0.574406i
$981$	0.258908	+	0.118239i	0.258908	+	0.118239i
$982$		0			0
$983$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$983$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$984$		0			0
$985$	−2.15499	+	0.803771i	−2.15499	+	0.803771i
$986$	2.57501	−	1.65486i	2.57501	−	1.65486i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$991$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	1.75575	+	0.654861i	1.75575	+	0.654861i	1.00000			$0$
$997$	1.75575	+	0.654861i	1.75575	+	0.654861i	0.755750	+	0.654861i	$0.227273\pi$
$998$		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.1291.1	yes	20
4.3	odd	2	CM	1412.1.r.a.1291.1	yes	20
353.35	even	44	inner	1412.1.r.a.35.1	✓	20
1412.35	odd	44	inner	1412.1.r.a.35.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1412.1.r.a.35.1	✓	20	353.35	even	44	inner
1412.1.r.a.35.1	✓	20	1412.35	odd	44	inner
1412.1.r.a.1291.1	yes	20	1.1	even	1	trivial
1412.1.r.a.1291.1	yes	20	4.3	odd	2	CM

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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)

\(f(q)\)	\(=\)	\(q+(0.959493 - 0.281733i) q^{2} +(0.841254 - 0.540641i) q^{4} +(-1.05195 + 0.574406i) q^{5} +(0.654861 - 0.755750i) q^{8} +(0.540641 - 0.841254i) q^{9} +O(q^{10})\)	\(q+(0.959493 - 0.281733i) q^{2} +(0.841254 - 0.540641i) q^{4} +(-1.05195 + 0.574406i) q^{5} +(0.654861 - 0.755750i) q^{8} +(0.540641 - 0.841254i) q^{9} +(-0.847507 + 0.847507i) q^{10} +(0.334961 + 0.898064i) q^{13} +(0.415415 - 0.909632i) q^{16} +(1.10181 - 1.27155i) q^{17} +(0.281733 - 0.959493i) q^{18} +(-0.574406 + 1.05195i) q^{20} +(0.236009 - 0.367237i) q^{25} +(0.574406 + 0.767317i) q^{26} +(1.53046 + 0.983568i) q^{29} +(0.142315 - 0.989821i) q^{32} +(0.698939 - 1.53046i) q^{34} -1.00000i q^{36} +(-1.98982 + 0.142315i) q^{37} +(-0.254771 + 1.17116i) q^{40} +(-1.29639 + 0.186393i) q^{41} +(-0.0855040 + 1.19550i) q^{45} +1.00000i q^{49} +(0.122986 - 0.418852i) q^{50} +(0.767317 + 0.574406i) q^{52} +(-0.936593 - 1.71524i) q^{53} +(1.74557 + 0.512546i) q^{58} +(-0.755750 + 1.65486i) q^{61} +(-0.142315 - 0.989821i) q^{64} +(-0.868215 - 0.752312i) q^{65} +(0.239446 - 1.66538i) q^{68} +(-0.281733 - 0.959493i) q^{72} +(-1.86912 + 0.697148i) q^{74} +(0.0855040 + 1.19550i) q^{80} +(-0.415415 - 0.909632i) q^{81} +(-1.19136 + 0.544078i) q^{82} +(-0.428654 + 1.97049i) q^{85} +(-1.19550 - 0.0855040i) q^{89} +(0.254771 + 1.17116i) q^{90} +(1.25667 + 1.45027i) q^{97} +(0.281733 + 0.959493i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) 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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