LMFDB - Embedded newform 1412.1.r.a.1123.1

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			1123.1
Root			$-0.540641 - 0.841254i$ of defining polynomial
Character	$\chi$	$=$	1412.1123
Dual form			1412.1.r.a.171.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.142315 + 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-0.114220 + 0.0855040i) q^{5} +(-0.415415 - 0.909632i) q^{8} +(0.281733 - 0.959493i) q^{9} +O(q^{10})$	\(q+(0.142315 + 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-0.114220 + 0.0855040i) q^{5} +(-0.415415 - 0.909632i) q^{8} +(0.281733 - 0.959493i) q^{9} +(-0.100889 - 0.100889i) q^{10} +(1.17116 + 0.254771i) q^{13} +(0.841254 - 0.540641i) q^{16} +(0.797176 + 1.74557i) q^{17} +(0.989821 + 0.142315i) q^{18} +(0.0855040 - 0.114220i) q^{20} +(-0.275997 + 0.939960i) q^{25} +(-0.0855040 + 1.19550i) q^{26} +(1.03748 + 0.304632i) q^{29} +(0.654861 + 0.755750i) q^{32} +(-1.61435 + 1.03748i) q^{34} +1.00000i q^{36} +(-1.75575 - 0.654861i) q^{37} +(0.125226 + 0.0683785i) q^{40} +(0.627899 + 0.544078i) q^{41} +(0.0498610 + 0.133682i) q^{45} -1.00000i q^{49} +(-0.969672 - 0.139418i) q^{50} +(-1.19550 + 0.0855040i) q^{52} +(-0.574406 - 0.767317i) q^{53} +(-0.153882 + 1.07028i) q^{58} +(0.909632 - 0.584585i) q^{61} +(-0.654861 + 0.755750i) q^{64} +(-0.155554 + 0.0710393i) q^{65} +(-1.25667 - 1.45027i) q^{68} +(-0.989821 + 0.142315i) q^{72} +(0.398326 - 1.83107i) q^{74} +(-0.0498610 + 0.133682i) q^{80} +(-0.841254 - 0.540641i) q^{81} +(-0.449181 + 0.698939i) q^{82} +(-0.240307 - 0.131217i) q^{85} +(0.133682 - 0.0498610i) q^{89} +(-0.125226 + 0.0683785i) q^{90} +(-0.118239 + 0.258908i) q^{97} +(0.989821 - 0.142315i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1412\mathbb{Z}\right)^\times$.

$n$	$707$	$709$
$\chi(n)$	$-1$	$e\left(\frac{35}{44}\right)$

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 $

$2$	0.142315	+	0.989821i	0.142315	+	0.989821i
$2$	0.142315	+	0.989821i	0.142315	+	0.989821i
$3$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$3$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$4$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$5$	−0.114220	+	0.0855040i	−0.114220	+	0.0855040i	−0.654861	−	0.755750i	$-0.727273\pi$
$5$	−0.114220	+	0.0855040i	−0.114220	+	0.0855040i	0.540641	+	0.841254i	$0.318182\pi$
$6$		0			0
$7$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$7$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$8$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$9$	0.281733	−	0.959493i	0.281733	−	0.959493i
$10$	−0.100889	−	0.100889i	−0.100889	−	0.100889i
$11$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$11$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$12$		0			0
$13$	1.17116	+	0.254771i	1.17116	+	0.254771i	0.755750	−	0.654861i	$-0.227273\pi$
$13$	1.17116	+	0.254771i	1.17116	+	0.254771i	0.415415	+	0.909632i	$0.363636\pi$
$14$		0			0
$15$		0			0
$16$	0.841254	−	0.540641i	0.841254	−	0.540641i
$17$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.654861	+	0.755750i	$0.272727\pi$
$17$	0.797176	+	1.74557i	0.797176	+	1.74557i	0.142315	+	0.989821i	$0.454545\pi$
$18$	0.989821	+	0.142315i	0.989821	+	0.142315i
$19$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$19$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$20$	0.0855040	−	0.114220i	0.0855040	−	0.114220i
$21$		0			0
$22$		0			0
$23$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$23$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$24$		0			0
$25$	−0.275997	+	0.939960i	−0.275997	+	0.939960i
$26$	−0.0855040	+	1.19550i	−0.0855040	+	1.19550i
$27$		0			0
$28$		0			0
$29$	1.03748	+	0.304632i	1.03748	+	0.304632i	0.755750	−	0.654861i	$-0.227273\pi$
$29$	1.03748	+	0.304632i	1.03748	+	0.304632i	0.281733	+	0.959493i	$0.409091\pi$
$30$		0			0
$31$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$31$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$32$	0.654861	+	0.755750i	0.654861	+	0.755750i
$33$		0			0
$34$	−1.61435	+	1.03748i	−1.61435	+	1.03748i
$35$		0			0
$36$			1.00000i			1.00000i
$37$	−1.75575	−	0.654861i	−1.75575	−	0.654861i	−0.755750	−	0.654861i	$-0.772727\pi$
$37$	−1.75575	−	0.654861i	−1.75575	−	0.654861i	−1.00000			$\pi$
$38$		0			0
$39$		0			0
$40$	0.125226	+	0.0683785i	0.125226	+	0.0683785i
$41$	0.627899	+	0.544078i	0.627899	+	0.544078i	0.909632	−	0.415415i	$-0.136364\pi$
$41$	0.627899	+	0.544078i	0.627899	+	0.544078i	−0.281733	+	0.959493i	$0.590909\pi$
$42$		0			0
$43$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$43$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$44$		0			0
$45$	0.0498610	+	0.133682i	0.0498610	+	0.133682i
$46$		0			0
$47$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$47$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$48$		0			0
$49$		−	1.00000i		−	1.00000i
$50$	−0.969672	−	0.139418i	−0.969672	−	0.139418i
$51$		0			0
$52$	−1.19550	+	0.0855040i	−1.19550	+	0.0855040i
$53$	−0.574406	−	0.767317i	−0.574406	−	0.767317i	0.415415	−	0.909632i	$-0.363636\pi$
$53$	−0.574406	−	0.767317i	−0.574406	−	0.767317i	−0.989821	+	0.142315i	$0.954545\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−0.153882	+	1.07028i	−0.153882	+	1.07028i
$59$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$59$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$60$		0			0
$61$	0.909632	−	0.584585i	0.909632	−	0.584585i		−	1.00000i	$-0.5\pi$
$61$	0.909632	−	0.584585i	0.909632	−	0.584585i	0.909632	+	0.415415i	$0.136364\pi$
$62$		0			0
$63$		0			0
$64$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$65$	−0.155554	+	0.0710393i	−0.155554	+	0.0710393i
$66$		0			0
$67$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$67$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$68$	−1.25667	−	1.45027i	−1.25667	−	1.45027i
$69$		0			0
$70$		0			0
$71$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$71$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$72$	−0.989821	+	0.142315i	−0.989821	+	0.142315i
$73$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$73$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$74$	0.398326	−	1.83107i	0.398326	−	1.83107i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$79$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$80$	−0.0498610	+	0.133682i	−0.0498610	+	0.133682i
$81$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$82$	−0.449181	+	0.698939i	−0.449181	+	0.698939i
$83$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$83$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$84$		0			0
$85$	−0.240307	−	0.131217i	−0.240307	−	0.131217i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	0.133682	−	0.0498610i	0.133682	−	0.0498610i	−0.281733	−	0.959493i	$-0.590909\pi$
$89$	0.133682	−	0.0498610i	0.133682	−	0.0498610i	0.415415	+	0.909632i	$0.363636\pi$
$90$	−0.125226	+	0.0683785i	−0.125226	+	0.0683785i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$97$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$98$	0.989821	−	0.142315i	0.989821	−	0.142315i
$99$		0			0
$100$		−	0.979643i		−	0.979643i
$101$	0.847507	+	0.847507i	0.847507	+	0.847507i	0.989821	−	0.142315i	$-0.0454545\pi$
$101$	0.847507	+	0.847507i	0.847507	+	0.847507i	−0.142315	+	0.989821i	$0.545455\pi$
$102$		0			0
$103$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$103$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$104$	−0.254771	−	1.17116i	−0.254771	−	1.17116i
$105$		0			0
$106$	0.677760	−	0.677760i	0.677760	−	0.677760i
$107$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$107$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$108$		0			0
$109$	0.857685	−	0.989821i	0.857685	−	0.989821i	−0.142315	−	0.989821i	$-0.545455\pi$
$109$	0.857685	−	0.989821i	0.857685	−	0.989821i	1.00000			$0$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.540641	+	1.84125i	0.540641	+	1.84125i	0.540641	+	0.841254i	$0.318182\pi$
$113$	0.540641	+	1.84125i	0.540641	+	1.84125i			1.00000i	$0.5\pi$
$114$		0			0
$115$		0			0
$116$	−1.08128			−1.08128
$117$	0.574406	−	1.05195i	0.574406	−	1.05195i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$122$	0.708089	+	0.817178i	0.708089	+	0.817178i
$123$		0			0
$124$		0			0
$125$	−0.0987069	−	0.264644i	−0.0987069	−	0.264644i
$126$		0			0
$127$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$127$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$128$	−0.841254	−	0.540641i	−0.841254	−	0.540641i
$129$		0			0
$130$	−0.0924539	−	0.143861i	−0.0924539	−	0.143861i
$131$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$131$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	1.25667	−	1.45027i	1.25667	−	1.45027i
$137$	−1.38189	+	1.38189i	−1.38189	+	1.38189i	−0.540641	+	0.841254i	$0.681818\pi$
$137$	−1.38189	+	1.38189i	−1.38189	+	1.38189i	−0.841254	+	0.540641i	$0.818182\pi$
$138$		0			0
$139$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$139$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.281733	−	0.959493i	−0.281733	−	0.959493i
$145$	−0.144548	+	0.0539138i	−0.144548	+	0.0539138i
$146$		0			0
$147$		0			0
$148$	1.86912	+	0.133682i	1.86912	+	0.133682i
$149$	−1.90963	−	0.415415i	−1.90963	−	0.415415i	−0.909632	−	0.415415i	$-0.863636\pi$
$149$	−1.90963	−	0.415415i	−1.90963	−	0.415415i	−1.00000			$\pi$
$150$		0			0
$151$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$151$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$152$		0			0
$153$	1.89945	−	0.273100i	1.89945	−	0.273100i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.368991	−	1.25667i	−0.368991	−	1.25667i	−0.909632	−	0.415415i	$-0.863636\pi$
$157$	−0.368991	−	1.25667i	−0.368991	−	1.25667i	0.540641	−	0.841254i	$-0.318182\pi$
$158$		0			0
$159$		0			0
$160$	−0.139418	−	0.0303285i	−0.139418	−	0.0303285i
$161$		0			0
$162$	0.415415	−	0.909632i	0.415415	−	0.909632i
$163$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$163$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$164$	−0.755750	−	0.345139i	−0.755750	−	0.345139i
$165$		0			0
$166$		0			0
$167$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$167$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$168$		0			0
$169$	0.397086	+	0.181343i	0.397086	+	0.181343i
$170$	0.0956825	−	0.256535i	0.0956825	−	0.256535i
$171$		0			0
$172$		0			0
$173$	−1.19550	−	1.59700i	−1.19550	−	1.59700i	−0.654861	−	0.755750i	$-0.727273\pi$
$173$	−1.19550	−	1.59700i	−1.19550	−	1.59700i	−0.540641	−	0.841254i	$-0.681818\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.0683785	+	0.125226i	0.0683785	+	0.125226i
$179$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$179$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$180$	−0.0855040	−	0.114220i	−0.0855040	−	0.114220i
$181$	−1.80075	−	0.822373i	−1.80075	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$181$	−1.80075	−	0.822373i	−1.80075	−	0.822373i	−0.841254	−	0.540641i	$-0.818182\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.256535	−	0.0753254i	0.256535	−	0.0753254i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$191$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$192$		0			0
$193$	−1.38189	−	0.300613i	−1.38189	−	0.300613i	−0.540641	−	0.841254i	$-0.681818\pi$
$193$	−1.38189	−	0.300613i	−1.38189	−	0.300613i	−0.841254	+	0.540641i	$0.818182\pi$
$194$	−0.273100	−	0.0801894i	−0.273100	−	0.0801894i
$195$		0			0
$196$	0.281733	+	0.959493i	0.281733	+	0.959493i
$197$	0.215109	−	0.186393i	0.215109	−	0.186393i	−0.540641	−	0.841254i	$-0.681818\pi$
$197$	0.215109	−	0.186393i	0.215109	−	0.186393i	0.755750	+	0.654861i	$0.227273\pi$
$198$		0			0
$199$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$199$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$200$	0.969672	−	0.139418i	0.969672	−	0.139418i
$201$		0			0
$202$	−0.718267	+	0.959493i	−0.718267	+	0.959493i
$203$		0			0
$204$		0			0
$205$	−0.118239	−	0.00845665i	−0.118239	−	0.00845665i
$206$		0			0
$207$		0			0
$208$	1.12299	−	0.418852i	1.12299	−	0.418852i
$209$		0			0
$210$		0			0
$211$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$211$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$212$	0.767317	+	0.574406i	0.767317	+	0.574406i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	1.10181	+	0.708089i	1.10181	+	0.708089i
$219$		0			0
$220$		0			0
$221$	0.488902	+	2.24745i	0.488902	+	2.24745i
$222$		0			0
$223$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$223$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$224$		0			0
$225$	0.824128	+	0.529635i	0.824128	+	0.529635i
$226$	−1.74557	+	0.797176i	−1.74557	+	0.797176i
$227$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$227$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$228$		0			0
$229$	−1.32505	+	0.494217i	−1.32505	+	0.494217i	−0.909632	−	0.415415i	$-0.863636\pi$
$229$	−1.32505	+	0.494217i	−1.32505	+	0.494217i	−0.415415	+	0.909632i	$0.636364\pi$
$230$		0			0
$231$		0			0
$232$	−0.153882	−	1.07028i	−0.153882	−	1.07028i
$233$	1.80075	−	0.258908i	1.80075	−	0.258908i	0.841254	−	0.540641i	$-0.181818\pi$
$233$	1.80075	−	0.258908i	1.80075	−	0.258908i	0.959493	+	0.281733i	$0.0909091\pi$
$234$	1.12299	+	0.418852i	1.12299	+	0.418852i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$239$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$240$		0			0
$241$	0.203743	−	0.936593i	0.203743	−	0.936593i	−0.755750	−	0.654861i	$-0.772727\pi$
$241$	0.203743	−	0.936593i	0.203743	−	0.936593i	0.959493	−	0.281733i	$-0.0909091\pi$
$242$	0.959493	−	0.281733i	0.959493	−	0.281733i
$243$		0			0
$244$	−0.708089	+	0.817178i	−0.708089	+	0.817178i
$245$	0.0855040	+	0.114220i	0.0855040	+	0.114220i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.247902	−	0.135365i	0.247902	−	0.135365i
$251$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$251$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.415415	−	0.909632i	0.415415	−	0.909632i
$257$	−1.94931	−	0.424047i	−1.94931	−	0.424047i	−0.989821	−	0.142315i	$-0.954545\pi$
$257$	−1.94931	−	0.424047i	−1.94931	−	0.424047i	−0.959493	−	0.281733i	$-0.909091\pi$
$258$		0			0
$259$		0			0
$260$	0.129239	−	0.111986i	0.129239	−	0.111986i
$261$	0.584585	−	0.909632i	0.584585	−	0.909632i
$262$		0			0
$263$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$263$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$264$		0			0
$265$	0.131217	+	0.0385289i	0.131217	+	0.0385289i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.215109	−	1.49611i	−0.215109	−	1.49611i	−0.755750	−	0.654861i	$-0.772727\pi$
$269$	−0.215109	−	1.49611i	−0.215109	−	1.49611i	0.540641	−	0.841254i	$-0.318182\pi$
$270$		0			0
$271$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$271$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$272$	1.61435	+	1.03748i	1.61435	+	1.03748i
$273$		0			0
$274$	−1.56449	−	1.17116i	−1.56449	−	1.17116i
$275$		0			0
$276$		0			0
$277$	0.512546	−	1.74557i	0.512546	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$277$	0.512546	−	1.74557i	0.512546	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$281$	1.49611	−	0.215109i	1.49611	−	0.215109i	0.841254	+	0.540641i	$0.181818\pi$
$282$		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
$288$	0.909632	−	0.415415i	0.909632	−	0.415415i
$289$	−1.75667	+	2.02730i	−1.75667	+	2.02730i
$290$	−0.0739364	−	0.135404i	−0.0739364	−	0.135404i
$291$		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
$296$	0.133682	+	1.86912i	0.133682	+	1.86912i
$297$		0			0
$298$	0.139418	−	1.94931i	0.139418	−	1.94931i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−0.0539138	+	0.144548i	−0.0539138	+	0.144548i
$306$	0.540641	+	1.84125i	0.540641	+	1.84125i
$307$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$307$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	1.64468	+	0.898064i	1.64468	+	0.898064i	0.989821	+	0.142315i	$0.0454545\pi$
$313$	1.64468	+	0.898064i	1.64468	+	0.898064i	0.654861	+	0.755750i	$0.272727\pi$
$314$	1.19136	−	0.544078i	1.19136	−	0.544078i
$315$		0			0
$316$		0			0
$317$			0.284630i			0.284630i	0.989821	+	0.142315i	$0.0454545\pi$
$317$			0.284630i			0.284630i	−0.989821	+	0.142315i	$0.954545\pi$
$318$		0			0
$319$		0			0
$320$	0.0101786	−	0.142315i	0.0101786	−	0.142315i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.959493	+	0.281733i	0.959493	+	0.281733i
$325$	−0.562713	+	1.03053i	−0.562713	+	1.03053i
$326$		0			0
$327$		0			0
$328$	0.234072	−	0.797176i	0.234072	−	0.797176i
$329$		0			0
$330$		0			0
$331$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$331$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$332$		0			0
$333$	−1.12299	+	1.50013i	−1.12299	+	1.50013i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	−0.841254	−	0.540641i	$-0.818182\pi$
$337$	−0.698939	+	0.449181i	−0.698939	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$338$	−0.122986	+	0.418852i	−0.122986	+	0.418852i
$339$		0			0
$340$	0.267541	+	0.0581999i	0.267541	+	0.0581999i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	1.41061	−	1.41061i	1.41061	−	1.41061i
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$	−1.74557	+	0.512546i	−1.74557	+	0.512546i	−0.989821	−	0.142315i	$-0.954545\pi$
$349$	−1.74557	+	0.512546i	−1.74557	+	0.512546i	−0.755750	+	0.654861i	$0.772727\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.415415	−	0.909632i	0.415415	−	0.909632i
$353$	0.415415	−	0.909632i	0.415415	−	0.909632i
$354$		0			0
$355$		0			0
$356$	−0.114220	+	0.0855040i	−0.114220	+	0.0855040i
$357$		0			0
$358$		0			0
$359$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$359$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$360$	0.100889	−	0.100889i	0.100889	−	0.100889i
$361$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$362$	0.557730	−	1.89945i	0.557730	−	1.89945i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.800541	−	0.599278i	$-0.795455\pi$
$367$		0			0		0.800541	+	0.599278i	$0.204545\pi$
$368$		0			0
$369$	0.698939	−	0.449181i	0.698939	−	0.449181i
$370$	0.111067	+	0.243204i	0.111067	+	0.243204i
$371$		0			0
$372$		0			0
$373$	−0.847507	+	1.13214i	−0.847507	+	1.13214i	0.142315	+	0.989821i	$0.454545\pi$
$373$	−0.847507	+	1.13214i	−0.847507	+	1.13214i	−0.989821	+	0.142315i	$0.954545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	1.13745	+	0.621095i	1.13745	+	0.621095i
$378$		0			0
$379$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$379$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$383$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$384$		0			0
$385$		0			0
$386$	0.100889	−	1.41061i	0.100889	−	1.41061i
$387$		0			0
$388$	0.0405070	−	0.281733i	0.0405070	−	0.281733i
$389$			0.563465i			0.563465i	0.959493	+	0.281733i	$0.0909091\pi$
$389$			0.563465i			0.563465i	−0.959493	+	0.281733i	$0.909091\pi$
$390$		0			0
$391$		0			0
$392$	−0.909632	+	0.415415i	−0.909632	+	0.415415i
$393$		0			0
$394$	0.215109	+	0.186393i	0.215109	+	0.186393i
$395$		0			0
$396$		0			0
$397$	0.544078	−	1.19136i	0.544078	−	1.19136i	−0.415415	−	0.909632i	$-0.636364\pi$
$397$	0.544078	−	1.19136i	0.544078	−	1.19136i	0.959493	−	0.281733i	$-0.0909091\pi$
$398$		0			0
$399$		0			0
$400$	0.275997	+	0.939960i	0.275997	+	0.939960i
$401$	−0.494217	+	1.32505i	−0.494217	+	1.32505i	0.415415	+	0.909632i	$0.363636\pi$
$401$	−0.494217	+	1.32505i	−0.494217	+	1.32505i	−0.909632	+	0.415415i	$0.863636\pi$
$402$		0			0
$403$		0			0
$404$	−1.05195	−	0.574406i	−1.05195	−	0.574406i
$405$	0.142315	−	0.0101786i	0.142315	−	0.0101786i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−0.654861	−	0.244250i	−0.654861	−	0.244250i		−	1.00000i	$-0.5\pi$
$409$	−0.654861	−	0.244250i	−0.654861	−	0.244250i	−0.654861	+	0.755750i	$0.727273\pi$
$410$	−0.00845665	−	0.118239i	−0.00845665	−	0.118239i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.574406	+	1.05195i	0.574406	+	1.05195i
$417$		0			0
$418$		0			0
$419$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$419$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$420$		0			0
$421$	−0.368991	−	0.425839i	−0.368991	−	0.425839i	0.540641	−	0.841254i	$-0.318182\pi$
$421$	−0.368991	−	0.425839i	−0.368991	−	0.425839i	−0.909632	+	0.415415i	$0.863636\pi$
$422$		0			0
$423$		0			0
$424$	−0.459359	+	0.841254i	−0.459359	+	0.841254i
$425$	−1.86079	+	0.267541i	−1.86079	+	0.267541i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$431$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$432$		0			0
$433$	0.559521	−	1.50013i	0.559521	−	1.50013i	−0.281733	−	0.959493i	$-0.590909\pi$
$433$	0.559521	−	1.50013i	0.559521	−	1.50013i	0.841254	−	0.540641i	$-0.181818\pi$
$434$		0			0
$435$		0			0
$436$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$437$		0			0
$438$		0			0
$439$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$439$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$440$		0			0
$441$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$442$	−2.15499	+	0.803771i	−2.15499	+	0.803771i
$443$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$443$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$444$		0			0
$445$	−0.0110059	+	0.0171255i	−0.0110059	+	0.0171255i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.936593	+	0.203743i	0.936593	+	0.203743i	0.654861	−	0.755750i	$-0.272727\pi$
$449$	0.936593	+	0.203743i	0.936593	+	0.203743i	0.281733	+	0.959493i	$0.409091\pi$
$450$	−0.406958	+	0.891115i	−0.406958	+	0.891115i
$451$		0			0
$452$	−1.03748	−	1.61435i	−1.03748	−	1.61435i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.373128	+	1.71524i	0.373128	+	1.71524i	0.654861	+	0.755750i	$0.272727\pi$
$457$	0.373128	+	1.71524i	0.373128	+	1.71524i	−0.281733	+	0.959493i	$0.590909\pi$
$458$	−0.677760	−	1.24123i	−0.677760	−	1.24123i
$459$		0			0
$460$		0			0
$461$	−1.17116	−	1.56449i	−1.17116	−	1.56449i	−0.755750	−	0.654861i	$-0.772727\pi$
$461$	−1.17116	−	1.56449i	−1.17116	−	1.56449i	−0.415415	−	0.909632i	$-0.636364\pi$
$462$		0			0
$463$		0			0		0.997452	−	0.0713392i	$-0.0227273\pi$
$463$		0			0		−0.997452	+	0.0713392i	$0.977273\pi$
$464$	1.03748	−	0.304632i	1.03748	−	0.304632i
$465$		0			0
$466$	0.512546	+	1.74557i	0.512546	+	1.74557i
$467$		0			0		0.977147	−	0.212565i	$-0.0681818\pi$
$467$		0			0		−0.977147	+	0.212565i	$0.931818\pi$
$468$	−0.254771	+	1.17116i	−0.254771	+	1.17116i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.898064	+	0.334961i	−0.898064	+	0.334961i
$478$		0			0
$479$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$479$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$480$		0			0
$481$	−1.88943	−	1.21426i	−1.88943	−	1.21426i
$482$	0.956056	+	0.0683785i	0.956056	+	0.0683785i
$483$		0			0
$484$	0.415415	+	0.909632i	0.415415	+	0.909632i
$485$	−0.00863238	−	0.0396824i	−0.00863238	−	0.0396824i
$486$		0			0
$487$		0			0		0.599278	−	0.800541i	$-0.295455\pi$
$487$		0			0		−0.599278	+	0.800541i	$0.704545\pi$
$488$	−0.909632	−	0.584585i	−0.909632	−	0.584585i
$489$		0			0
$490$	−0.100889	+	0.100889i	−0.100889	+	0.100889i
$491$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$491$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$492$		0			0
$493$	0.295298	+	2.05384i	0.295298	+	2.05384i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$499$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$500$	0.169267	+	0.226115i	0.169267	+	0.226115i
$501$		0			0
$502$		0			0
$503$		0			0		0.877679	−	0.479249i	$-0.159091\pi$
$503$		0			0		−0.877679	+	0.479249i	$0.840909\pi$
$504$		0			0
$505$	−0.169267	−	0.0243370i	−0.169267	−	0.0243370i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$509$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$510$		0			0
$511$		0			0
$512$	0.959493	+	0.281733i	0.959493	+	0.281733i
$513$		0			0
$514$	0.142315	−	1.98982i	0.142315	−	1.98982i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	0.129239	+	0.111986i	0.129239	+	0.111986i
$521$	1.74557	−	0.512546i	1.74557	−	0.512546i	0.755750	−	0.654861i	$-0.227273\pi$
$521$	1.74557	−	0.512546i	1.74557	−	0.512546i	0.989821	+	0.142315i	$0.0454545\pi$
$522$	0.983568	+	0.449181i	0.983568	+	0.449181i
$523$		0			0		0.349464	−	0.936950i	$-0.386364\pi$
$523$		0			0		−0.349464	+	0.936950i	$0.613636\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.142315	−	0.989821i	0.142315	−	0.989821i
$530$	−0.0194625	+	0.135365i	−0.0194625	+	0.135365i
$531$		0			0
$532$		0			0
$533$	0.596758	+	0.797176i	0.596758	+	0.797176i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	1.45027	−	0.425839i	1.45027	−	0.425839i
$539$		0			0
$540$		0			0
$541$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.841254	+	0.540641i	$0.181818\pi$
$541$	0.983568	+	1.53046i	0.983568	+	1.53046i	0.142315	+	0.989821i	$0.454545\pi$
$542$		0			0
$543$		0			0
$544$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$545$	−0.0133311	+	0.186393i	−0.0133311	+	0.186393i
$546$		0			0
$547$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$547$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$548$	0.936593	−	1.71524i	0.936593	−	1.71524i
$549$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.80075	+	0.258908i	1.80075	+	0.258908i
$555$		0			0
$556$		0			0
$557$	−0.415415	−	0.0903680i	−0.415415	−	0.0903680i		−	1.00000i	$-0.5\pi$
$557$	−0.415415	−	0.0903680i	−0.415415	−	0.0903680i	−0.415415	+	0.909632i	$0.636364\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.425839	+	1.45027i	0.425839	+	1.45027i
$563$		0			0		0.800541	−	0.599278i	$-0.204545\pi$
$563$		0			0		−0.800541	+	0.599278i	$0.795455\pi$
$564$		0			0
$565$	−0.219186	−	0.164081i	−0.219186	−	0.164081i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.24123	+	1.24123i	−1.24123	+	1.24123i	−0.281733	+	0.959493i	$0.590909\pi$
$569$	−1.24123	+	1.24123i	−1.24123	+	1.24123i	−0.959493	+	0.281733i	$0.909091\pi$
$570$		0			0
$571$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$571$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.540641	+	0.841254i	0.540641	+	0.841254i
$577$	1.94931	+	0.139418i	1.94931	+	0.139418i	0.989821	−	0.142315i	$-0.0454545\pi$
$577$	1.94931	+	0.139418i	1.94931	+	0.139418i	0.959493	+	0.281733i	$0.0909091\pi$
$578$	−2.25667	−	1.45027i	−2.25667	−	1.45027i
$579$		0			0
$580$	0.123504	−	0.0924539i	0.123504	−	0.0924539i
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	0.0243370	+	0.169267i	0.0243370	+	0.169267i
$586$	−1.97964	+	0.284630i	−1.97964	+	0.284630i
$587$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$587$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−1.83107	+	0.398326i	−1.83107	+	0.398326i
$593$	0.540641	+	1.84125i	0.540641	+	1.84125i	0.540641	+	0.841254i	$0.318182\pi$
$593$	0.540641	+	1.84125i	0.540641	+	1.84125i			1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$	1.94931	−	0.139418i	1.94931	−	0.139418i
$597$		0			0
$598$		0			0
$599$		0			0		−0.0713392	−	0.997452i	$-0.522727\pi$
$599$		0			0		0.0713392	+	0.997452i	$0.477273\pi$
$600$		0			0
$601$	0.898064	+	1.64468i	0.898064	+	1.64468i	0.755750	+	0.654861i	$0.227273\pi$
$601$	0.898064	+	1.64468i	0.898064	+	1.64468i	0.142315	+	0.989821i	$0.454545\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.100889	+	0.100889i	0.100889	+	0.100889i
$606$		0			0
$607$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$607$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$608$		0			0
$609$		0			0
$610$	−0.150750	−	0.0327936i	−0.150750	−	0.0327936i
$611$		0			0
$612$	−1.74557	+	0.797176i	−1.74557	+	0.797176i
$613$	−0.989821	+	0.857685i	−0.989821	+	0.857685i	−0.989821	−	0.142315i	$-0.954545\pi$
$613$	−0.989821	+	0.857685i	−0.989821	+	0.857685i			1.00000i	$0.5\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.12299	+	0.418852i	−1.12299	+	0.418852i	−0.841254	−	0.540641i	$-0.818182\pi$
$617$	−1.12299	+	0.418852i	−1.12299	+	0.418852i	−0.281733	+	0.959493i	$0.590909\pi$
$618$		0			0
$619$		0			0		0.936950	−	0.349464i	$-0.113636\pi$
$619$		0			0		−0.936950	+	0.349464i	$0.886364\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.790226	−	0.507847i	−0.790226	−	0.507847i
$626$	−0.654861	+	1.75575i	−0.654861	+	1.75575i
$627$		0			0
$628$	0.708089	+	1.10181i	0.708089	+	1.10181i
$629$	−0.256535	−	3.58682i	−0.256535	−	3.58682i
$630$		0			0
$631$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$631$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$632$		0			0
$633$		0			0
$634$	−0.281733	+	0.0405070i	−0.281733	+	0.0405070i
$635$		0			0
$636$		0			0
$637$	0.254771	−	1.17116i	0.254771	−	1.17116i
$638$		0			0
$639$		0			0
$640$	0.142315	−	0.0101786i	0.142315	−	0.0101786i
$641$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.281733	−	0.959493i	$-0.409091\pi$
$641$	1.19136	−	0.544078i	1.19136	−	0.544078i	0.909632	+	0.415415i	$0.136364\pi$
$642$		0			0
$643$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$643$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$644$		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	$0.750000\pi$
$648$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$649$		0			0
$650$	−1.10013	−	0.410326i	−1.10013	−	0.410326i
$651$		0			0
$652$		0			0
$653$	0.847507	+	1.13214i	0.847507	+	1.13214i	0.989821	+	0.142315i	$0.0454545\pi$
$653$	0.847507	+	1.13214i	0.847507	+	1.13214i	−0.142315	+	0.989821i	$0.545455\pi$
$654$		0			0
$655$		0			0
$656$	0.822373	+	0.118239i	0.822373	+	0.118239i
$657$		0			0
$658$		0			0
$659$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$659$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$660$		0			0
$661$	−0.559521	−	1.50013i	−0.559521	−	1.50013i	−0.841254	−	0.540641i	$-0.818182\pi$
$661$	−0.559521	−	1.50013i	−0.559521	−	1.50013i	0.281733	−	0.959493i	$-0.409091\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−1.64468	−	0.898064i	−1.64468	−	0.898064i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.125226	+	1.75089i	−0.125226	+	1.75089i	0.415415	+	0.909632i	$0.363636\pi$
$673$	−0.125226	+	1.75089i	−0.125226	+	1.75089i	−0.540641	+	0.841254i	$0.681818\pi$
$674$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$675$		0			0
$676$	−0.432092	−	0.0621254i	−0.432092	−	0.0621254i
$677$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$677$	0.273100	+	0.0801894i	0.273100	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$678$		0			0
$679$		0			0
$680$	−0.0195325	+	0.273100i	−0.0195325	+	0.273100i
$681$		0			0
$682$		0			0
$683$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$683$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$684$		0			0
$685$	0.0396824	−	0.275997i	0.0396824	−	0.275997i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.477234	−	1.04500i	−0.477234	−	1.04500i
$690$		0			0
$691$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$691$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$692$	1.59700	+	1.19550i	1.59700	+	1.19550i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.449181	+	1.52977i	−0.449181	+	1.52977i
$698$	−0.755750	−	1.65486i	−0.755750	−	1.65486i
$699$		0			0
$700$		0			0
$701$	0.114220	−	0.0855040i	0.114220	−	0.0855040i	−0.540641	−	0.841254i	$-0.681818\pi$
$701$	0.114220	−	0.0855040i	0.114220	−	0.0855040i	0.654861	+	0.755750i	$0.272727\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.959493	+	0.281733i	0.959493	+	0.281733i
$707$		0			0
$708$		0			0
$709$	1.50013	−	1.12299i	1.50013	−	1.12299i	0.540641	−	0.841254i	$-0.318182\pi$
$709$	1.50013	−	1.12299i	1.50013	−	1.12299i	0.959493	−	0.281733i	$-0.0909091\pi$
$710$		0			0
$711$		0			0
$712$	−0.100889	−	0.100889i	−0.100889	−	0.100889i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.977147	−	0.212565i	$-0.931818\pi$
$719$		0			0		0.977147	+	0.212565i	$0.0681818\pi$
$720$	0.114220	+	0.0855040i	0.114220	+	0.0855040i
$721$		0			0
$722$	0.841254	−	0.540641i	0.841254	−	0.540641i
$723$		0			0
$724$	1.95949	+	0.281733i	1.95949	+	0.281733i
$725$	−0.572685	+	0.891115i	−0.572685	+	0.891115i
$726$		0			0
$727$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$727$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$728$		0			0
$729$	−0.755750	+	0.654861i	−0.755750	+	0.654861i
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	0.841254	−	0.540641i	$-0.181818\pi$
$733$	−0.148568	−	0.398326i	−0.148568	−	0.398326i	−0.989821	+	0.142315i	$0.954545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.544078	+	0.627899i	0.544078	+	0.627899i
$739$		0			0		0.0713392	−	0.997452i	$-0.477273\pi$
$739$		0			0		−0.0713392	+	0.997452i	$0.522727\pi$
$740$	−0.224922	+	0.144548i	−0.224922	+	0.144548i
$741$		0			0
$742$		0			0
$743$		0			0		−0.936950	−	0.349464i	$-0.886364\pi$
$743$		0			0		0.936950	+	0.349464i	$0.113636\pi$
$744$		0			0
$745$	0.253638	−	0.115832i	0.253638	−	0.115832i
$746$	−1.24123	−	0.677760i	−1.24123	−	0.677760i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.349464	−	0.936950i	$-0.613636\pi$
$751$		0			0		0.349464	+	0.936950i	$0.386364\pi$
$752$		0			0
$753$		0			0
$754$	−0.452897	+	1.21426i	−0.452897	+	1.21426i
$755$		0			0
$756$		0			0
$757$	1.54064	+	0.841254i	1.54064	+	0.841254i	1.00000			$0$
$757$	1.54064	+	0.841254i	1.54064	+	0.841254i	0.540641	+	0.841254i	$0.318182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	−0.909632	−	0.415415i	$-0.863636\pi$
$761$	0.0498610	−	0.697148i	0.0498610	−	0.697148i	0.959493	−	0.281733i	$-0.0909091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−0.193604	+	0.193604i	−0.193604	+	0.193604i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.677760	−	1.24123i	−0.677760	−	1.24123i	−0.959493	−	0.281733i	$-0.909091\pi$
$769$	−0.677760	−	1.24123i	−0.677760	−	1.24123i	0.281733	−	0.959493i	$-0.409091\pi$
$770$		0			0
$771$		0			0
$772$	1.41061	−	0.100889i	1.41061	−	0.100889i
$773$	1.24123	+	1.24123i	1.24123	+	1.24123i	0.959493	+	0.281733i	$0.0909091\pi$
$773$	1.24123	+	1.24123i	1.24123	+	1.24123i	0.281733	+	0.959493i	$0.409091\pi$
$774$		0			0
$775$		0			0
$776$	0.284630			0.284630
$777$		0			0
$778$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.540641	−	0.841254i	−0.540641	−	0.841254i
$785$	0.149596	+	0.111986i	0.149596	+	0.111986i
$786$		0			0
$787$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$787$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$788$	−0.153882	+	0.239446i	−0.153882	+	0.239446i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.21426	−	0.452897i	1.21426	−	0.452897i
$794$	1.25667	+	0.368991i	1.25667	+	0.368991i
$795$		0			0
$796$		0			0
$797$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$797$	−0.239446	+	0.153882i	−0.239446	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$798$		0			0
$799$		0			0
$800$	−0.891115	+	0.406958i	−0.891115	+	0.406958i
$801$	−0.0101786	−	0.142315i	−0.0101786	−	0.142315i
$802$	−1.38189	−	0.300613i	−1.38189	−	0.300613i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	0.418852	−	1.12299i	0.418852	−	1.12299i
$809$	−1.05195	+	0.574406i	−1.05195	+	0.574406i	−0.909632	−	0.415415i	$-0.863636\pi$
$809$	−1.05195	+	0.574406i	−1.05195	+	0.574406i	−0.142315	+	0.989821i	$0.545455\pi$
$810$	0.0303285	+	0.139418i	0.0303285	+	0.139418i
$811$		0			0		−0.479249	−	0.877679i	$-0.659091\pi$
$811$		0			0		0.479249	+	0.877679i	$0.340909\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.148568	−	0.682956i	0.148568	−	0.682956i
$819$		0			0
$820$	0.115832	−	0.0251978i	0.115832	−	0.0251978i
$821$	0.424047	−	1.94931i	0.424047	−	1.94931i	0.142315	−	0.989821i	$-0.454545\pi$
$821$	0.424047	−	1.94931i	0.424047	−	1.94931i	0.281733	−	0.959493i	$-0.409091\pi$
$822$		0			0
$823$		0			0		0.479249	−	0.877679i	$-0.340909\pi$
$823$		0			0		−0.479249	+	0.877679i	$0.659091\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$827$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$828$		0			0
$829$	1.19550	+	0.0855040i	1.19550	+	0.0855040i	0.654861	−	0.755750i	$-0.272727\pi$
$829$	1.19550	+	0.0855040i	1.19550	+	0.0855040i	0.540641	+	0.841254i	$0.318182\pi$
$830$		0			0
$831$		0			0
$832$	−0.959493	+	0.718267i	−0.959493	+	0.718267i
$833$	1.74557	−	0.797176i	1.74557	−	0.797176i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.212565	−	0.977147i	$-0.568182\pi$
$839$		0			0		0.212565	+	0.977147i	$0.431818\pi$
$840$		0			0
$841$	0.142315	+	0.0914602i	0.142315	+	0.0914602i
$842$	0.368991	−	0.425839i	0.368991	−	0.425839i
$843$		0			0
$844$		0			0
$845$	−0.0608607	+	0.0132394i	−0.0608607	+	0.0132394i
$846$		0			0
$847$		0			0
$848$	−0.898064	−	0.334961i	−0.898064	−	0.334961i
$849$		0			0
$850$	−0.529635	−	1.80377i	−0.529635	−	1.80377i
$851$		0			0
$852$		0			0
$853$	0.0855040	+	0.114220i	0.0855040	+	0.114220i	0.841254	−	0.540641i	$-0.181818\pi$
$853$	0.0855040	+	0.114220i	0.0855040	+	0.114220i	−0.755750	+	0.654861i	$0.772727\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.05195	−	1.40524i	1.05195	−	1.40524i	0.142315	−	0.989821i	$-0.454545\pi$
$857$	1.05195	−	1.40524i	1.05195	−	1.40524i	0.909632	−	0.415415i	$-0.136364\pi$
$858$		0			0
$859$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$859$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$863$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$864$		0			0
$865$	0.273100	+	0.0801894i	0.273100	+	0.0801894i
$866$	1.56449	+	0.340335i	1.56449	+	0.340335i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	−1.25667	−	0.368991i	−1.25667	−	0.368991i
$873$	0.215109	+	0.186393i	0.215109	+	0.186393i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.544078	+	0.627899i	0.544078	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$877$	0.544078	+	0.627899i	0.544078	+	0.627899i	−0.415415	+	0.909632i	$0.636364\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.841254	−	1.54064i	−0.841254	−	1.54064i	−0.841254	−	0.540641i	$-0.818182\pi$
$881$	−0.841254	−	1.54064i	−0.841254	−	1.54064i		−	1.00000i	$-0.5\pi$
$882$	0.142315	−	0.989821i	0.142315	−	0.989821i
$883$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$883$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$884$	−1.10228	−	2.01867i	−1.10228	−	2.01867i
$885$		0			0
$886$		0			0
$887$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$887$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$888$		0			0
$889$		0			0
$890$	−0.0185175	−	0.00845665i	−0.0185175	−	0.00845665i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.0683785	+	0.956056i	−0.0683785	+	0.956056i
$899$		0			0
$900$	−0.939960	−	0.275997i	−0.939960	−	0.275997i
$901$	0.881504	−	1.61435i	0.881504	−	1.61435i
$902$		0			0
$903$		0			0
$904$	1.45027	−	1.25667i	1.45027	−	1.25667i
$905$	0.275997	−	0.0600395i	0.275997	−	0.0600395i
$906$		0			0
$907$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$907$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$908$		0			0
$909$	1.05195	−	0.574406i	1.05195	−	0.574406i
$910$		0			0
$911$		0			0		−0.997452	−	0.0713392i	$-0.977273\pi$
$911$		0			0		0.997452	+	0.0713392i	$0.0227273\pi$
$912$		0			0
$913$		0			0
$914$	−1.64468	+	0.613435i	−1.64468	+	0.613435i
$915$		0			0
$916$	1.13214	−	0.847507i	1.13214	−	0.847507i
$917$		0			0
$918$		0			0
$919$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$919$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$920$		0			0
$921$		0			0
$922$	1.38189	−	1.38189i	1.38189	−	1.38189i
$923$		0			0
$924$		0			0
$925$	1.10013	−	1.46960i	1.10013	−	1.46960i
$926$		0			0
$927$		0			0
$928$	0.449181	+	0.983568i	0.449181	+	0.983568i
$929$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$929$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$930$		0			0
$931$		0			0
$932$	−1.65486	+	0.755750i	−1.65486	+	0.755750i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.19550	−	0.0855040i	−1.19550	−	0.0855040i
$937$	0.857685	+	0.989821i	0.857685	+	0.989821i	1.00000			$0$
$937$	0.857685	+	0.989821i	0.857685	+	0.989821i	−0.142315	+	0.989821i	$0.545455\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.56449	+	1.17116i	1.56449	+	1.17116i	0.909632	+	0.415415i	$0.136364\pi$
$941$	1.56449	+	1.17116i	1.56449	+	1.17116i	0.654861	+	0.755750i	$0.272727\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.212565	−	0.977147i	$-0.431818\pi$
$947$		0			0		−0.212565	+	0.977147i	$0.568182\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.32505	+	1.32505i	−1.32505	+	1.32505i	−0.415415	+	0.909632i	$0.636364\pi$
$953$	−1.32505	+	1.32505i	−1.32505	+	1.32505i	−0.909632	+	0.415415i	$0.863636\pi$
$954$	−0.459359	−	0.841254i	−0.459359	−	0.841254i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.989821	−	0.142315i	0.989821	−	0.142315i
$962$	0.933011	−	2.04301i	0.933011	−	2.04301i
$963$		0			0
$964$	0.0683785	+	0.956056i	0.0683785	+	0.956056i
$965$	0.183543	−	0.0838215i	0.183543	−	0.0838215i
$966$		0			0
$967$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$967$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$968$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$969$		0			0
$970$	0.0380500	−	0.0141919i	0.0380500	−	0.0141919i
$971$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$971$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	0.449181	−	0.983568i	0.449181	−	0.983568i
$977$	0.909632	−	1.41542i	0.909632	−	1.41542i		−	1.00000i	$-0.5\pi$
$977$	0.909632	−	1.41542i	0.909632	−	1.41542i	0.909632	−	0.415415i	$-0.136364\pi$
$978$		0			0
$979$		0			0
$980$	−0.114220	−	0.0855040i	−0.114220	−	0.0855040i
$981$	−0.708089	−	1.10181i	−0.708089	−	1.10181i
$982$		0			0
$983$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$983$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$984$		0			0
$985$	−0.00863238	+	0.0396824i	−0.00863238	+	0.0396824i
$986$	−1.99091	+	0.584585i	−1.99091	+	0.584585i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$991$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.0903680	+	0.415415i	0.0903680	+	0.415415i	1.00000			$0$
$997$	0.0903680	+	0.415415i	0.0903680	+	0.415415i	−0.909632	+	0.415415i	$0.863636\pi$
$998$		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.1123.1	yes	20
4.3	odd	2	CM	1412.1.r.a.1123.1	yes	20
353.171	even	44	inner	1412.1.r.a.171.1	✓	20
1412.171	odd	44	inner	1412.1.r.a.171.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1412.1.r.a.171.1	✓	20	353.171	even	44	inner
1412.1.r.a.171.1	✓	20	1412.171	odd	44	inner
1412.1.r.a.1123.1	yes	20	1.1	even	1	trivial
1412.1.r.a.1123.1	yes	20	4.3	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(0.142315 + 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-0.114220 + 0.0855040i) q^{5} +(-0.415415 - 0.909632i) q^{8} +(0.281733 - 0.959493i) q^{9} +O(q^{10})\)	\(q+(0.142315 + 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-0.114220 + 0.0855040i) q^{5} +(-0.415415 - 0.909632i) q^{8} +(0.281733 - 0.959493i) q^{9} +(-0.100889 - 0.100889i) q^{10} +(1.17116 + 0.254771i) q^{13} +(0.841254 - 0.540641i) q^{16} +(0.797176 + 1.74557i) q^{17} +(0.989821 + 0.142315i) q^{18} +(0.0855040 - 0.114220i) q^{20} +(-0.275997 + 0.939960i) q^{25} +(-0.0855040 + 1.19550i) q^{26} +(1.03748 + 0.304632i) q^{29} +(0.654861 + 0.755750i) q^{32} +(-1.61435 + 1.03748i) q^{34} +1.00000i q^{36} +(-1.75575 - 0.654861i) q^{37} +(0.125226 + 0.0683785i) q^{40} +(0.627899 + 0.544078i) q^{41} +(0.0498610 + 0.133682i) q^{45} -1.00000i q^{49} +(-0.969672 - 0.139418i) q^{50} +(-1.19550 + 0.0855040i) q^{52} +(-0.574406 - 0.767317i) q^{53} +(-0.153882 + 1.07028i) q^{58} +(0.909632 - 0.584585i) q^{61} +(-0.654861 + 0.755750i) q^{64} +(-0.155554 + 0.0710393i) q^{65} +(-1.25667 - 1.45027i) q^{68} +(-0.989821 + 0.142315i) q^{72} +(0.398326 - 1.83107i) q^{74} +(-0.0498610 + 0.133682i) q^{80} +(-0.841254 - 0.540641i) q^{81} +(-0.449181 + 0.698939i) q^{82} +(-0.240307 - 0.131217i) q^{85} +(0.133682 - 0.0498610i) q^{89} +(-0.125226 + 0.0683785i) q^{90} +(-0.118239 + 0.258908i) q^{97} +(0.989821 - 0.142315i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{2} - 2 q^{4} - 2 q^{5} + 2 q^{8}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more