LMFDB - Embedded newform 1412.1.l.a.1195.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1412,1,Mod(295,1412)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1412, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 9]))

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

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

chi := DirichletCharacter("1412.295");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1412 = 2^{2} \cdot 353 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1412.l (of order $22$, degree $10$, 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:	$10$
Coefficient field:	$\Q(\zeta_{22})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 $ x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{22}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{22} - \cdots)$

Embedding invariants

Embedding label			1195.1
Root			$-0.841254 + 0.540641i$ of defining polynomial
Character	$\chi$	$=$	1412.1195
Dual form			1412.1.l.a.475.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.142315 - 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-1.49611 - 0.215109i) q^{5} +(0.415415 + 0.909632i) q^{8} +(0.959493 + 0.281733i) q^{9} +O(q^{10})$	\(q+(-0.142315 - 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-1.49611 - 0.215109i) q^{5} +(0.415415 + 0.909632i) q^{8} +(0.959493 + 0.281733i) q^{9} +1.51150i q^{10} +(1.07028 + 1.66538i) q^{13} +(0.841254 - 0.540641i) q^{16} +(-0.797176 - 1.74557i) q^{17} +(0.142315 - 0.989821i) q^{18} +(1.49611 - 0.215109i) q^{20} +(1.23259 + 0.361922i) q^{25} +(1.49611 - 1.29639i) q^{26} +(1.61435 + 0.474017i) q^{29} +(-0.654861 - 0.755750i) q^{32} +(-1.61435 + 1.03748i) q^{34} -1.00000 q^{36} +(1.65486 - 0.755750i) q^{37} +(-0.425839 - 1.45027i) q^{40} +(0.544078 - 0.627899i) q^{41} +(-1.37491 - 0.627899i) q^{45} -1.00000 q^{49} +(0.182822 - 1.27155i) q^{50} +(-1.49611 - 1.29639i) q^{52} +(0.557730 + 0.0801894i) q^{53} +(0.239446 - 1.66538i) q^{58} +(-1.41542 + 0.909632i) q^{61} +(-0.654861 + 0.755750i) q^{64} +(-1.24302 - 2.72183i) q^{65} +(1.25667 + 1.45027i) q^{68} +(0.142315 + 0.989821i) q^{72} +(1.91899 - 0.563465i) q^{73} +(-0.983568 - 1.53046i) q^{74} +(-1.37491 + 0.627899i) q^{80} +(0.841254 + 0.540641i) q^{81} +(-0.698939 - 0.449181i) q^{82} +(0.817178 + 2.78305i) q^{85} +(1.37491 + 0.627899i) q^{89} +(-0.425839 + 1.45027i) q^{90} +(-0.118239 + 0.258908i) q^{97} +(0.142315 + 0.989821i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - q^{2} - q^{4} - q^{8} + q^{9}+O(q^{10}) $ 10 * q - q^2 - q^4 - q^8 + q^9	$ 10 q - q^{2} - q^{4} - q^{8} + q^{9} - q^{16} - 2 q^{17} + q^{18} - q^{25} + 2 q^{29} - q^{32} - 2 q^{34} - 10 q^{36} + 11 q^{37} + 2 q^{41} - 10 q^{49} + 10 q^{50} + 2 q^{58} - 9 q^{61} - q^{64} - 2 q^{68} + q^{72} + 2 q^{73} - q^{81} + 2 q^{82} - 2 q^{97} + q^{98}+O(q^{100}) $ 10 * q - q^2 - q^4 - q^8 + q^9 - q^16 - 2 * q^17 + q^18 - q^25 + 2 * q^29 - q^32 - 2 * q^34 - 10 * q^36 + 11 * q^37 + 2 * q^41 - 10 * q^49 + 10 * q^50 + 2 * q^58 - 9 * q^61 - q^64 - 2 * q^68 + q^72 + 2 * q^73 - q^81 + 2 * q^82 - 2 * q^97 + q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$707$	$709$
$\chi(n)$	$-1$	$e\left(\frac{1}{22}\right)$

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		−0.989821	−	0.142315i	$-0.954545\pi$
$3$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$4$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$5$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$5$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	0.415415	+	0.909632i	0.415415	+	0.909632i
$9$	0.959493	+	0.281733i	0.959493	+	0.281733i
$10$			1.51150i			1.51150i
$11$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$11$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$12$		0			0
$13$	1.07028	+	1.66538i	1.07028	+	1.66538i	0.654861	+	0.755750i	$0.272727\pi$
$13$	1.07028	+	1.66538i	1.07028	+	1.66538i	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	$-0.727273\pi$
$17$	−0.797176	−	1.74557i	−0.797176	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$18$	0.142315	−	0.989821i	0.142315	−	0.989821i
$19$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$19$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$20$	1.49611	−	0.215109i	1.49611	−	0.215109i
$21$		0			0
$22$		0			0
$23$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$23$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$24$		0			0
$25$	1.23259	+	0.361922i	1.23259	+	0.361922i
$26$	1.49611	−	1.29639i	1.49611	−	1.29639i
$27$		0			0
$28$		0			0
$29$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$29$	1.61435	+	0.474017i	1.61435	+	0.474017i	0.654861	+	0.755750i	$0.272727\pi$
$30$		0			0
$31$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$31$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$32$	−0.654861	−	0.755750i	−0.654861	−	0.755750i
$33$		0			0
$34$	−1.61435	+	1.03748i	−1.61435	+	1.03748i
$35$		0			0
$36$	−1.00000			−1.00000
$37$	1.65486	−	0.755750i	1.65486	−	0.755750i	0.654861	−	0.755750i	$-0.272727\pi$
$37$	1.65486	−	0.755750i	1.65486	−	0.755750i	1.00000			$0$
$38$		0			0
$39$		0			0
$40$	−0.425839	−	1.45027i	−0.425839	−	1.45027i
$41$	0.544078	−	0.627899i	0.544078	−	0.627899i	−0.415415	−	0.909632i	$-0.636364\pi$
$41$	0.544078	−	0.627899i	0.544078	−	0.627899i	0.959493	+	0.281733i	$0.0909091\pi$
$42$		0			0
$43$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$43$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$44$		0			0
$45$	−1.37491	−	0.627899i	−1.37491	−	0.627899i
$46$		0			0
$47$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$47$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$48$		0			0
$49$	−1.00000			−1.00000
$50$	0.182822	−	1.27155i	0.182822	−	1.27155i
$51$		0			0
$52$	−1.49611	−	1.29639i	−1.49611	−	1.29639i
$53$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.415415	−	0.909632i	$-0.363636\pi$
$53$	0.557730	+	0.0801894i	0.557730	+	0.0801894i	0.142315	+	0.989821i	$0.454545\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	0.239446	−	1.66538i	0.239446	−	1.66538i
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$61$	−1.41542	+	0.909632i	−1.41542	+	0.909632i	−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−0.654861	+	0.755750i	−0.654861	+	0.755750i
$65$	−1.24302	−	2.72183i	−1.24302	−	2.72183i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$	1.25667	+	1.45027i	1.25667	+	1.45027i
$69$		0			0
$70$		0			0
$71$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$71$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$72$	0.142315	+	0.989821i	0.142315	+	0.989821i
$73$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$73$	1.91899	−	0.563465i	1.91899	−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$74$	−0.983568	−	1.53046i	−0.983568	−	1.53046i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$79$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$80$	−1.37491	+	0.627899i	−1.37491	+	0.627899i
$81$	0.841254	+	0.540641i	0.841254	+	0.540641i
$82$	−0.698939	−	0.449181i	−0.698939	−	0.449181i
$83$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$83$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$84$		0			0
$85$	0.817178	+	2.78305i	0.817178	+	2.78305i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.959493	−	0.281733i	$-0.0909091\pi$
$89$	1.37491	+	0.627899i	1.37491	+	0.627899i	0.415415	+	0.909632i	$0.363636\pi$
$90$	−0.425839	+	1.45027i	−0.425839	+	1.45027i
$91$		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	$0.181818\pi$
$98$	0.142315	+	0.989821i	0.142315	+	0.989821i
$99$		0			0
$100$	−1.28463			−1.28463
$101$			1.97964i			1.97964i	0.142315	+	0.989821i	$0.454545\pi$
$101$			1.97964i			1.97964i	−0.142315	+	0.989821i	$0.545455\pi$
$102$		0			0
$103$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$103$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$104$	−1.07028	+	1.66538i	−1.07028	+	1.66538i
$105$		0			0
$106$		−	0.563465i		−	0.563465i
$107$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$107$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$108$		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
$113$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−0.841254	+	0.540641i	$0.818182\pi$
$113$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−1.00000			$1.00000\pi$
$114$		0			0
$115$		0			0
$116$	−1.68251			−1.68251
$117$	0.557730	+	1.89945i	0.557730	+	1.89945i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.142315	−	0.989821i	−0.142315	−	0.989821i
$122$	1.10181	+	1.27155i	1.10181	+	1.27155i
$123$		0			0
$124$		0			0
$125$	−0.391340	−	0.178719i	−0.391340	−	0.178719i
$126$		0			0
$127$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$127$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$128$	0.841254	+	0.540641i	0.841254	+	0.540641i
$129$		0			0
$130$	−2.51722	+	1.61772i	−2.51722	+	1.61772i
$131$		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
$137$		−	1.08128i		−	1.08128i	−0.841254	−	0.540641i	$-0.818182\pi$
$137$		−	1.08128i		−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$138$		0			0
$139$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$139$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.959493	−	0.281733i	0.959493	−	0.281733i
$145$	−2.31329	−	1.05645i	−2.31329	−	1.05645i
$146$	−0.830830	−	1.81926i	−0.830830	−	1.81926i
$147$		0			0
$148$	−1.37491	+	1.19136i	−1.37491	+	1.19136i
$149$	0.584585	+	0.909632i	0.584585	+	0.909632i	1.00000			$0$
$149$	0.584585	+	0.909632i	0.584585	+	0.909632i	−0.415415	+	0.909632i	$0.636364\pi$
$150$		0			0
$151$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$151$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$152$		0			0
$153$	−0.273100	−	1.89945i	−0.273100	−	1.89945i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$157$	−1.25667	+	0.368991i	−1.25667	+	0.368991i	−0.415415	+	0.909632i	$0.636364\pi$
$158$		0			0
$159$		0			0
$160$	0.817178	+	1.27155i	0.817178	+	1.27155i
$161$		0			0
$162$	0.415415	−	0.909632i	0.415415	−	0.909632i
$163$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$163$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$164$	−0.345139	+	0.755750i	−0.345139	+	0.755750i
$165$		0			0
$166$		0			0
$167$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$167$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$168$		0			0
$169$	−1.21259	+	2.65520i	−1.21259	+	2.65520i
$170$	2.63843	−	1.20493i	2.63843	−	1.20493i
$171$		0			0
$172$		0			0
$173$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$173$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	0.425839	−	1.45027i	0.425839	−	1.45027i
$179$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$179$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$180$	1.49611	+	0.215109i	1.49611	+	0.215109i
$181$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	−0.959493	−	0.281733i	$-0.909091\pi$
$181$	−0.118239	+	0.258908i	−0.118239	+	0.258908i	0.841254	+	0.540641i	$0.181818\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.63843	+	0.774713i	−2.63843	+	0.774713i
$186$		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
$193$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$193$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$194$	0.273100	+	0.0801894i	0.273100	+	0.0801894i
$195$		0			0
$196$	0.959493	−	0.281733i	0.959493	−	0.281733i
$197$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	0.654861	−	0.755750i	$-0.272727\pi$
$197$	−0.186393	−	0.215109i	−0.186393	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$198$		0			0
$199$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$199$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$200$	0.182822	+	1.27155i	0.182822	+	1.27155i
$201$		0			0
$202$	1.95949	−	0.281733i	1.95949	−	0.281733i
$203$		0			0
$204$		0			0
$205$	−0.949069	+	0.822373i	−0.949069	+	0.822373i
$206$		0			0
$207$		0			0
$208$	1.80075	+	0.822373i	1.80075	+	0.822373i
$209$		0			0
$210$		0			0
$211$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$211$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$212$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i
$213$		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
$221$	2.05384	−	3.19584i	2.05384	−	3.19584i
$222$		0			0
$223$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$223$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$224$		0			0
$225$	1.08070	+	0.694523i	1.08070	+	0.694523i
$226$	0.797176	+	1.74557i	0.797176	+	1.74557i
$227$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$227$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$228$		0			0
$229$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$229$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$230$		0			0
$231$		0			0
$232$	0.239446	+	1.66538i	0.239446	+	1.66538i
$233$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	−0.959493	−	0.281733i	$-0.909091\pi$
$233$	−0.118239	−	0.822373i	−0.118239	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$234$	1.80075	−	0.822373i	1.80075	−	0.822373i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$239$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$240$		0			0
$241$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	0.654861	−	0.755750i	$-0.272727\pi$
$241$	−0.304632	−	0.474017i	−0.304632	−	0.474017i	−0.959493	+	0.281733i	$0.909091\pi$
$242$	−0.959493	+	0.281733i	−0.959493	+	0.281733i
$243$		0			0
$244$	1.10181	−	1.27155i	1.10181	−	1.27155i
$245$	1.49611	+	0.215109i	1.49611	+	0.215109i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.121206	+	0.412791i	−0.121206	+	0.412791i
$251$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$251$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.415415	−	0.909632i	0.415415	−	0.909632i
$257$	−0.817178	−	1.27155i	−0.817178	−	1.27155i	−0.959493	−	0.281733i	$-0.909091\pi$
$257$	−0.817178	−	1.27155i	−0.817178	−	1.27155i	0.142315	−	0.989821i	$-0.454545\pi$
$258$		0			0
$259$		0			0
$260$	1.95949	+	2.26138i	1.95949	+	2.26138i
$261$	1.41542	+	0.909632i	1.41542	+	0.909632i
$262$		0			0
$263$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$263$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$264$		0			0
$265$	−0.817178	−	0.239945i	−0.817178	−	0.239945i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.186393	−	1.29639i	−0.186393	−	1.29639i	−0.841254	−	0.540641i	$-0.818182\pi$
$269$	−0.186393	−	1.29639i	−0.186393	−	1.29639i	0.654861	−	0.755750i	$-0.272727\pi$
$270$		0			0
$271$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$271$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$272$	−1.61435	−	1.03748i	−1.61435	−	1.03748i
$273$		0			0
$274$	−1.07028	+	0.153882i	−1.07028	+	0.153882i
$275$		0			0
$276$		0			0
$277$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.142315	−	0.989821i	$-0.545455\pi$
$277$	−0.797176	−	0.234072i	−0.797176	−	0.234072i	−0.654861	+	0.755750i	$0.727273\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	0.186393	+	1.29639i	0.186393	+	1.29639i	0.841254	+	0.540641i	$0.181818\pi$
$281$	0.186393	+	1.29639i	0.186393	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$289$	−1.75667	+	2.02730i	−1.75667	+	2.02730i
$290$	−0.716476	+	2.44009i	−0.716476	+	2.44009i
$291$		0			0
$292$	−1.68251	+	1.08128i	−1.68251	+	1.08128i
$293$	−2.00000			−2.00000			−1.00000			$\pi$
$293$	−2.00000			−2.00000			−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$	1.37491	+	1.19136i	1.37491	+	1.19136i
$297$		0			0
$298$	0.817178	−	0.708089i	0.817178	−	0.708089i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.31329	−	1.05645i	2.31329	−	1.05645i
$306$	−1.84125	+	0.540641i	−1.84125	+	0.540641i
$307$		0			0		0.654861	−	0.755750i	$-0.272727\pi$
$307$		0			0		−0.654861	+	0.755750i	$0.727273\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	−0.512546	−	1.74557i	−0.512546	−	1.74557i	−0.654861	−	0.755750i	$-0.727273\pi$
$313$	−0.512546	−	1.74557i	−0.512546	−	1.74557i	0.142315	−	0.989821i	$-0.454545\pi$
$314$	0.544078	+	1.19136i	0.544078	+	1.19136i
$315$		0			0
$316$		0			0
$317$	0.284630			0.284630			0.142315	−	0.989821i	$-0.454545\pi$
$317$	0.284630			0.284630			0.142315	+	0.989821i	$0.454545\pi$
$318$		0			0
$319$		0			0
$320$	1.14231	−	0.989821i	1.14231	−	0.989821i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$325$	0.716476	+	2.44009i	0.716476	+	2.44009i
$326$		0			0
$327$		0			0
$328$	0.797176	+	0.234072i	0.797176	+	0.234072i
$329$		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
$333$	1.80075	−	0.258908i	1.80075	−	0.258908i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$337$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$338$	2.80075	+	0.822373i	2.80075	+	0.822373i
$339$		0			0
$340$	−1.56815	−	2.44009i	−1.56815	−	2.44009i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$			1.51150i			1.51150i
$347$		0			0			−	1.00000i	$-0.5\pi$
$347$		0			0				1.00000i	$0.5\pi$
$348$		0			0
$349$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$349$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$350$		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
$356$	−1.49611	−	0.215109i	−1.49611	−	0.215109i
$357$		0			0
$358$		0			0
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		−	1.51150i		−	1.51150i
$361$	0.415415	+	0.909632i	0.415415	+	0.909632i
$362$	0.273100	+	0.0801894i	0.273100	+	0.0801894i
$363$		0			0
$364$		0			0
$365$	−2.99223	+	0.430218i	−2.99223	+	0.430218i
$366$		0			0
$367$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$367$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$368$		0			0
$369$	0.698939	−	0.449181i	0.698939	−	0.449181i
$370$	1.14231	+	2.50132i	1.14231	+	2.50132i
$371$		0			0
$372$		0			0
$373$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$373$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	0.938384	+	3.19584i	0.938384	+	3.19584i
$378$		0			0
$379$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$379$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$383$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	0.0405070	−	0.281733i	0.0405070	−	0.281733i
$389$	−1.91899			−1.91899			−0.959493	−	0.281733i	$-0.909091\pi$
$389$	−1.91899			−1.91899			−0.959493	+	0.281733i	$0.909091\pi$
$390$		0			0
$391$		0			0
$392$	−0.415415	−	0.909632i	−0.415415	−	0.909632i
$393$		0			0
$394$	−0.186393	+	0.215109i	−0.186393	+	0.215109i
$395$		0			0
$396$		0			0
$397$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	0.415415	+	0.909632i	$0.363636\pi$
$397$	−0.544078	+	1.19136i	−0.544078	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$398$		0			0
$399$		0			0
$400$	1.23259	−	0.361922i	1.23259	−	0.361922i
$401$		0			0		−0.415415	−	0.909632i	$-0.636364\pi$
$401$		0			0		0.415415	+	0.909632i	$0.363636\pi$
$402$		0			0
$403$		0			0
$404$	−0.557730	−	1.89945i	−0.557730	−	1.89945i
$405$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−0.654861	+	0.755750i	$0.727273\pi$
$409$	−1.65486	+	0.755750i	−1.65486	+	0.755750i	−1.00000			$\pi$
$410$	0.949069	+	0.822373i	0.949069	+	0.822373i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.557730	−	1.89945i	0.557730	−	1.89945i
$417$		0			0
$418$		0			0
$419$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$419$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$420$		0			0
$421$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.841254	−	0.540641i	$-0.818182\pi$
$421$	−1.25667	−	1.45027i	−1.25667	−	1.45027i	−0.415415	−	0.909632i	$-0.636364\pi$
$422$		0			0
$423$		0			0
$424$	0.158746	+	0.540641i	0.158746	+	0.540641i
$425$	−0.350833	−	2.44009i	−0.350833	−	2.44009i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$431$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$432$		0			0
$433$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.841254	−	0.540641i	$-0.181818\pi$
$433$	1.80075	−	0.822373i	1.80075	−	0.822373i	0.959493	−	0.281733i	$-0.0909091\pi$
$434$		0			0
$435$		0			0
$436$	−0.544078	+	1.19136i	−0.544078	+	1.19136i
$437$		0			0
$438$		0			0
$439$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$439$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$440$		0			0
$441$	−0.959493	−	0.281733i	−0.959493	−	0.281733i
$442$	−3.45561	−	1.57812i	−3.45561	−	1.57812i
$443$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$443$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$444$		0			0
$445$	−1.92195	−	1.23516i	−1.92195	−	1.23516i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.304632	+	0.474017i	0.304632	+	0.474017i	0.959493	−	0.281733i	$-0.0909091\pi$
$449$	0.304632	+	0.474017i	0.304632	+	0.474017i	−0.654861	+	0.755750i	$0.727273\pi$
$450$	0.533654	−	1.16854i	0.533654	−	1.16854i
$451$		0			0
$452$	1.61435	−	1.03748i	1.61435	−	1.03748i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.304632	−	0.474017i	0.304632	−	0.474017i	−0.654861	−	0.755750i	$-0.727273\pi$
$457$	0.304632	−	0.474017i	0.304632	−	0.474017i	0.959493	+	0.281733i	$0.0909091\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1.07028	+	0.153882i	1.07028	+	0.153882i	0.654861	−	0.755750i	$-0.272727\pi$
$461$	1.07028	+	0.153882i	1.07028	+	0.153882i	0.415415	+	0.909632i	$0.363636\pi$
$462$		0			0
$463$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$463$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$464$	1.61435	−	0.474017i	1.61435	−	0.474017i
$465$		0			0
$466$	−0.797176	+	0.234072i	−0.797176	+	0.234072i
$467$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$467$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$468$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.512546	+	0.234072i	0.512546	+	0.234072i
$478$		0			0
$479$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$479$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$480$		0			0
$481$	3.02977	+	1.94711i	3.02977	+	1.94711i
$482$	−0.425839	+	0.368991i	−0.425839	+	0.368991i
$483$		0			0
$484$	0.415415	+	0.909632i	0.415415	+	0.909632i
$485$	0.232593	−	0.361922i	0.232593	−	0.361922i
$486$		0			0
$487$		0			0		0.989821	−	0.142315i	$-0.0454545\pi$
$487$		0			0		−0.989821	+	0.142315i	$0.954545\pi$
$488$	−1.41542	−	0.909632i	−1.41542	−	0.909632i
$489$		0			0
$490$		−	1.51150i		−	1.51150i
$491$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$491$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$492$		0			0
$493$	−0.459493	−	3.19584i	−0.459493	−	3.19584i
$494$		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	$0.363636\pi$
$500$	0.425839	+	0.0612263i	0.425839	+	0.0612263i
$501$		0			0
$502$		0			0
$503$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$503$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$504$		0			0
$505$	0.425839	−	2.96177i	0.425839	−	2.96177i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$509$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−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
$514$	−1.14231	+	0.989821i	−1.14231	+	0.989821i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	1.95949	−	2.26138i	1.95949	−	2.26138i
$521$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.142315	−	0.989821i	$-0.454545\pi$
$521$	0.797176	−	0.234072i	0.797176	−	0.234072i	0.654861	+	0.755750i	$0.272727\pi$
$522$	0.698939	−	1.53046i	0.698939	−	1.53046i
$523$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$523$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$530$	−0.121206	+	0.843008i	−0.121206	+	0.843008i
$531$		0			0
$532$		0			0
$533$	1.62801	+	0.234072i	1.62801	+	0.234072i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−1.25667	+	0.368991i	−1.25667	+	0.368991i
$539$		0			0
$540$		0			0
$541$	0.698939	−	0.449181i	0.698939	−	0.449181i	−0.142315	−	0.989821i	$-0.545455\pi$
$541$	0.698939	−	0.449181i	0.698939	−	0.449181i	0.841254	+	0.540641i	$0.181818\pi$
$542$		0			0
$543$		0			0
$544$	−0.797176	+	1.74557i	−0.797176	+	1.74557i
$545$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$546$		0			0
$547$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$547$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$548$	0.304632	+	1.03748i	0.304632	+	1.03748i
$549$	−1.61435	+	0.474017i	−1.61435	+	0.474017i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	−0.118239	+	0.822373i	−0.118239	+	0.822373i
$555$		0			0
$556$		0			0
$557$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	0.415415	−	0.909632i	$-0.363636\pi$
$557$	−0.584585	−	0.909632i	−0.584585	−	0.909632i	−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	1.25667	−	0.368991i	1.25667	−	0.368991i
$563$		0			0		−0.989821	−	0.142315i	$-0.954545\pi$
$563$		0			0		0.989821	+	0.142315i	$0.0454545\pi$
$564$		0			0
$565$	2.87102	−	0.412791i	2.87102	−	0.412791i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			0.563465i			0.563465i	0.959493	+	0.281733i	$0.0909091\pi$
$569$			0.563465i			0.563465i	−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
$576$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$577$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	−0.959493	−	0.281733i	$-0.909091\pi$
$577$	−0.817178	+	0.708089i	−0.817178	+	0.708089i	0.142315	+	0.989821i	$0.454545\pi$
$578$	2.25667	+	1.45027i	2.25667	+	1.45027i
$579$		0			0
$580$	2.51722	+	0.361922i	2.51722	+	0.361922i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	1.30972	+	1.51150i	1.30972	+	1.51150i
$585$	−0.425839	−	2.96177i	−0.425839	−	2.96177i
$586$	0.284630	+	1.97964i	0.284630	+	1.97964i
$587$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$587$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.983568	−	1.53046i	0.983568	−	1.53046i
$593$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−0.841254	+	0.540641i	$0.818182\pi$
$593$	−1.84125	+	0.540641i	−1.84125	+	0.540641i	−1.00000			$1.00000\pi$
$594$		0			0
$595$		0			0
$596$	−0.817178	−	0.708089i	−0.817178	−	0.708089i
$597$		0			0
$598$		0			0
$599$		0			0		−0.755750	−	0.654861i	$-0.772727\pi$
$599$		0			0		0.755750	+	0.654861i	$0.227273\pi$
$600$		0			0
$601$	0.512546	−	1.74557i	0.512546	−	1.74557i	−0.142315	−	0.989821i	$-0.545455\pi$
$601$	0.512546	−	1.74557i	0.512546	−	1.74557i	0.654861	−	0.755750i	$-0.272727\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$			1.51150i			1.51150i
$606$		0			0
$607$		0			0		0.841254	−	0.540641i	$-0.181818\pi$
$607$		0			0		−0.841254	+	0.540641i	$0.818182\pi$
$608$		0			0
$609$		0			0
$610$	−1.37491	−	2.13940i	−1.37491	−	2.13940i
$611$		0			0
$612$	0.797176	+	1.74557i	0.797176	+	1.74557i
$613$	−0.857685	−	0.989821i	−0.857685	−	0.989821i	0.142315	−	0.989821i	$-0.454545\pi$
$613$	−0.857685	−	0.989821i	−0.857685	−	0.989821i	−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$617$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$618$		0			0
$619$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$619$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.533654	−	0.342959i	−0.533654	−	0.342959i
$626$	−1.65486	+	0.755750i	−1.65486	+	0.755750i
$627$		0			0
$628$	1.10181	−	0.708089i	1.10181	−	0.708089i
$629$	−2.63843	−	2.28621i	−2.63843	−	2.28621i
$630$		0			0
$631$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$631$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$632$		0			0
$633$		0			0
$634$	−0.0405070	−	0.281733i	−0.0405070	−	0.281733i
$635$		0			0
$636$		0			0
$637$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$638$		0			0
$639$		0			0
$640$	−1.14231	−	0.989821i	−1.14231	−	0.989821i
$641$	0.544078	+	1.19136i	0.544078	+	1.19136i	0.959493	+	0.281733i	$0.0909091\pi$
$641$	0.544078	+	1.19136i	0.544078	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$642$		0			0
$643$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$643$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−0.142315	+	0.989821i	−0.142315	+	0.989821i
$649$		0			0
$650$	2.31329	−	1.05645i	2.31329	−	1.05645i
$651$		0			0
$652$		0			0
$653$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$653$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$654$		0			0
$655$		0			0
$656$	0.118239	−	0.822373i	0.118239	−	0.822373i
$657$	2.00000			2.00000
$658$		0			0
$659$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$659$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$660$		0			0
$661$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.959493	+	0.281733i	$0.0909091\pi$
$661$	1.80075	+	0.822373i	1.80075	+	0.822373i	0.841254	+	0.540641i	$0.181818\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.512546	−	1.74557i	−0.512546	−	1.74557i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.425839	+	0.368991i	−0.425839	+	0.368991i	−0.841254	−	0.540641i	$-0.818182\pi$
$673$	−0.425839	+	0.368991i	−0.425839	+	0.368991i	0.415415	+	0.909632i	$0.363636\pi$
$674$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$675$		0			0
$676$	0.415415	−	2.88927i	0.415415	−	2.88927i
$677$	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
$680$	−2.19209	+	1.89945i	−2.19209	+	1.89945i
$681$		0			0
$682$		0			0
$683$		0			0		−0.654861	−	0.755750i	$-0.727273\pi$
$683$		0			0		0.654861	+	0.755750i	$0.272727\pi$
$684$		0			0
$685$	−0.232593	+	1.61772i	−0.232593	+	1.61772i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	0.463379	+	1.01466i	0.463379	+	1.01466i
$690$		0			0
$691$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$691$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$692$	1.49611	−	0.215109i	1.49611	−	0.215109i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.52977	−	0.449181i	−1.52977	−	0.449181i
$698$	−0.345139	−	0.755750i	−0.345139	−	0.755750i
$699$		0			0
$700$		0			0
$701$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.654861	−	0.755750i	$-0.727273\pi$
$701$	−1.49611	−	0.215109i	−1.49611	−	0.215109i	−0.841254	+	0.540641i	$0.818182\pi$
$702$		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
$709$	−1.80075	−	0.258908i	−1.80075	−	0.258908i	−0.841254	−	0.540641i	$-0.818182\pi$
$709$	−1.80075	−	0.258908i	−1.80075	−	0.258908i	−0.959493	+	0.281733i	$0.909091\pi$
$710$		0			0
$711$		0			0
$712$			1.51150i			1.51150i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$719$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$720$	−1.49611	+	0.215109i	−1.49611	+	0.215109i
$721$		0			0
$722$	0.841254	−	0.540641i	0.841254	−	0.540641i
$723$		0			0
$724$	0.0405070	−	0.281733i	0.0405070	−	0.281733i
$725$	1.81828	+	1.16854i	1.81828	+	1.16854i
$726$		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
$729$	0.654861	+	0.755750i	0.654861	+	0.755750i
$730$	0.851677	+	2.90055i	0.851677	+	2.90055i
$731$		0			0
$732$		0			0
$733$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.841254	−	0.540641i	$-0.181818\pi$
$733$	0.983568	+	0.449181i	0.983568	+	0.449181i	0.142315	+	0.989821i	$0.454545\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	−0.544078	−	0.627899i	−0.544078	−	0.627899i
$739$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$739$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$740$	2.31329	−	1.48666i	2.31329	−	1.48666i
$741$		0			0
$742$		0			0
$743$		0			0		0.909632	−	0.415415i	$-0.136364\pi$
$743$		0			0		−0.909632	+	0.415415i	$0.863636\pi$
$744$		0			0
$745$	−0.678936	−	1.48666i	−0.678936	−	1.48666i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.909632	−	0.415415i	$-0.863636\pi$
$751$		0			0		0.909632	+	0.415415i	$0.136364\pi$
$752$		0			0
$753$		0			0
$754$	3.02977	−	1.38365i	3.02977	−	1.38365i
$755$		0			0
$756$		0			0
$757$	0.158746	+	0.540641i	0.158746	+	0.540641i	1.00000			$0$
$757$	0.158746	+	0.540641i	0.158746	+	0.540641i	−0.841254	+	0.540641i	$0.818182\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.415415	+	0.909632i	$0.636364\pi$
$761$	−1.37491	+	1.19136i	−1.37491	+	1.19136i	−0.959493	+	0.281733i	$0.909091\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$			2.90055i			2.90055i
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$769$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	0.563465i		−	0.563465i	−0.959493	−	0.281733i	$-0.909091\pi$
$773$		−	0.563465i		−	0.563465i	0.959493	−	0.281733i	$-0.0909091\pi$
$774$		0			0
$775$		0			0
$776$	−0.284630			−0.284630
$777$		0			0
$778$	0.273100	+	1.89945i	0.273100	+	1.89945i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.841254	+	0.540641i	−0.841254	+	0.540641i
$785$	1.95949	−	0.281733i	1.95949	−	0.281733i
$786$		0			0
$787$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$787$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$788$	0.239446	+	0.153882i	0.239446	+	0.153882i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.02977	−	1.38365i	−3.02977	−	1.38365i
$794$	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
$800$	−0.533654	−	1.16854i	−0.533654	−	1.16854i
$801$	1.14231	+	0.989821i	1.14231	+	0.989821i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−1.80075	+	0.822373i	−1.80075	+	0.822373i
$809$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.142315	+	0.989821i	$0.545455\pi$
$809$	−0.557730	+	1.89945i	−0.557730	+	1.89945i	−0.415415	+	0.909632i	$0.636364\pi$
$810$	−0.817178	+	1.27155i	−0.817178	+	1.27155i
$811$		0			0		0.281733	−	0.959493i	$-0.409091\pi$
$811$		0			0		−0.281733	+	0.959493i	$0.590909\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.983568	+	1.53046i	0.983568	+	1.53046i
$819$		0			0
$820$	0.678936	−	1.05645i	0.678936	−	1.05645i
$821$	0.817178	+	1.27155i	0.817178	+	1.27155i	0.959493	+	0.281733i	$0.0909091\pi$
$821$	0.817178	+	1.27155i	0.817178	+	1.27155i	−0.142315	+	0.989821i	$0.545455\pi$
$822$		0			0
$823$		0			0		−0.281733	−	0.959493i	$-0.590909\pi$
$823$		0			0		0.281733	+	0.959493i	$0.409091\pi$
$824$		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
$829$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.654861	+	0.755750i	$0.727273\pi$
$829$	−1.49611	+	1.29639i	−1.49611	+	1.29639i	−0.841254	+	0.540641i	$0.818182\pi$
$830$		0			0
$831$		0			0
$832$	−1.95949	−	0.281733i	−1.95949	−	0.281733i
$833$	0.797176	+	1.74557i	0.797176	+	1.74557i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.540641	−	0.841254i	$-0.318182\pi$
$839$		0			0		−0.540641	+	0.841254i	$0.681818\pi$
$840$		0			0
$841$	1.54019	+	0.989821i	1.54019	+	0.989821i
$842$	−1.25667	+	1.45027i	−1.25667	+	1.45027i
$843$		0			0
$844$		0			0
$845$	2.38533	−	3.71165i	2.38533	−	3.71165i
$846$		0			0
$847$		0			0
$848$	0.512546	−	0.234072i	0.512546	−	0.234072i
$849$		0			0
$850$	−2.36533	+	0.694523i	−2.36533	+	0.694523i
$851$		0			0
$852$		0			0
$853$	1.49611	+	0.215109i	1.49611	+	0.215109i	0.841254	−	0.540641i	$-0.181818\pi$
$853$	1.49611	+	0.215109i	1.49611	+	0.215109i	0.654861	+	0.755750i	$0.272727\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.415415	−	0.909632i	$-0.636364\pi$
$857$	−0.557730	+	0.0801894i	−0.557730	+	0.0801894i	−0.142315	+	0.989821i	$0.545455\pi$
$858$		0			0
$859$		0			0		−0.142315	−	0.989821i	$-0.545455\pi$
$859$		0			0		0.142315	+	0.989821i	$0.454545\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.959493	−	0.281733i	$-0.0909091\pi$
$863$		0			0		−0.959493	+	0.281733i	$0.909091\pi$
$864$		0			0
$865$	2.19209	+	0.643655i	2.19209	+	0.643655i
$866$	−1.07028	−	1.66538i	−1.07028	−	1.66538i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	1.25667	+	0.368991i	1.25667	+	0.368991i
$873$	−0.186393	+	0.215109i	−0.186393	+	0.215109i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	0.415415	−	0.909632i	$-0.363636\pi$
$877$	−0.544078	−	0.627899i	−0.544078	−	0.627899i	−0.959493	+	0.281733i	$0.909091\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.158746	+	0.540641i	−0.158746	+	0.540641i	0.841254	+	0.540641i	$0.181818\pi$
$881$	−0.158746	+	0.540641i	−0.158746	+	0.540641i	−1.00000			$\pi$
$882$	−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	$0.545455\pi$
$884$	−1.07028	+	3.64502i	−1.07028	+	3.64502i
$885$		0			0
$886$		0			0
$887$		0			0		0.415415	−	0.909632i	$-0.363636\pi$
$887$		0			0		−0.415415	+	0.909632i	$0.636364\pi$
$888$		0			0
$889$		0			0
$890$	−0.949069	+	2.07817i	−0.949069	+	2.07817i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.425839	−	0.368991i	0.425839	−	0.368991i
$899$		0			0
$900$	−1.23259	−	0.361922i	−1.23259	−	0.361922i
$901$	−0.304632	−	1.03748i	−0.304632	−	1.03748i
$902$		0			0
$903$		0			0
$904$	−1.25667	−	1.45027i	−1.25667	−	1.45027i
$905$	0.232593	−	0.361922i	0.232593	−	0.361922i
$906$		0			0
$907$		0			0		0.142315	−	0.989821i	$-0.454545\pi$
$907$		0			0		−0.142315	+	0.989821i	$0.545455\pi$
$908$		0			0
$909$	−0.557730	+	1.89945i	−0.557730	+	1.89945i
$910$		0			0
$911$		0			0		0.755750	−	0.654861i	$-0.227273\pi$
$911$		0			0		−0.755750	+	0.654861i	$0.772727\pi$
$912$		0			0
$913$		0			0
$914$	−0.512546	−	0.234072i	−0.512546	−	0.234072i
$915$		0			0
$916$		0			0
$917$		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
$922$		−	1.08128i		−	1.08128i
$923$		0			0
$924$		0			0
$925$	2.31329	−	0.332601i	2.31329	−	0.332601i
$926$		0			0
$927$		0			0
$928$	−0.698939	−	1.53046i	−0.698939	−	1.53046i
$929$	1.68251	−	1.08128i	1.68251	−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$929$	1.68251	−	1.08128i	1.68251	−	1.08128i	0.841254	−	0.540641i	$-0.181818\pi$
$930$		0			0
$931$		0			0
$932$	0.345139	+	0.755750i	0.345139	+	0.755750i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	−1.49611	+	1.29639i	−1.49611	+	1.29639i
$937$	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
$941$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.654861	−	0.755750i	$-0.727273\pi$
$941$	−1.07028	+	0.153882i	−1.07028	+	0.153882i	−0.415415	+	0.909632i	$0.636364\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.540641	−	0.841254i	$-0.681818\pi$
$947$		0			0		0.540641	+	0.841254i	$0.318182\pi$
$948$		0			0
$949$	2.99223	+	2.59278i	2.99223	+	2.59278i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1.81926i		−	1.81926i	−0.415415	−	0.909632i	$-0.636364\pi$
$953$		−	1.81926i		−	1.81926i	0.415415	−	0.909632i	$-0.363636\pi$
$954$	0.158746	−	0.540641i	0.158746	−	0.540641i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.142315	+	0.989821i	0.142315	+	0.989821i
$962$	1.49611	−	3.27603i	1.49611	−	3.27603i
$963$		0			0
$964$	0.425839	+	0.368991i	0.425839	+	0.368991i
$965$		0			0
$966$		0			0
$967$		0			0		−0.841254	−	0.540641i	$-0.818182\pi$
$967$		0			0		0.841254	+	0.540641i	$0.181818\pi$
$968$	0.841254	−	0.540641i	0.841254	−	0.540641i
$969$		0			0
$970$	−0.391340	−	0.178719i	−0.391340	−	0.178719i
$971$		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
$976$	−0.698939	+	1.53046i	−0.698939	+	1.53046i
$977$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−0.415415	−	0.909632i	$-0.636364\pi$
$977$	−1.41542	−	0.909632i	−1.41542	−	0.909632i	−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$	−1.49611	+	0.215109i	−1.49611	+	0.215109i
$981$	1.10181	−	0.708089i	1.10181	−	0.708089i
$982$		0			0
$983$		0			0		−0.959493	−	0.281733i	$-0.909091\pi$
$983$		0			0		0.959493	+	0.281733i	$0.0909091\pi$
$984$		0			0
$985$	0.232593	+	0.361922i	0.232593	+	0.361922i
$986$	−3.09792	+	0.909632i	−3.09792	+	0.909632i
$987$		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
$997$	0.584585	−	0.909632i	0.584585	−	0.909632i	−0.415415	−	0.909632i	$-0.636364\pi$
$997$	0.584585	−	0.909632i	0.584585	−	0.909632i	1.00000			$0$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1412.1.l.a.1195.1	yes	10
4.3	odd	2	CM	1412.1.l.a.1195.1	yes	10
353.122	even	22	inner	1412.1.l.a.475.1	✓	10
1412.475	odd	22	inner	1412.1.l.a.475.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1412.1.l.a.475.1	✓	10	353.122	even	22	inner
1412.1.l.a.475.1	✓	10	1412.475	odd	22	inner
1412.1.l.a.1195.1	yes	10	1.1	even	1	trivial
1412.1.l.a.1195.1	yes	10	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.l (of order \(22\), degree \(10\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.142315 - 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-1.49611 - 0.215109i) q^{5} +(0.415415 + 0.909632i) q^{8} +(0.959493 + 0.281733i) q^{9} +O(q^{10})\)	\(q+(-0.142315 - 0.989821i) q^{2} +(-0.959493 + 0.281733i) q^{4} +(-1.49611 - 0.215109i) q^{5} +(0.415415 + 0.909632i) q^{8} +(0.959493 + 0.281733i) q^{9} +1.51150i q^{10} +(1.07028 + 1.66538i) q^{13} +(0.841254 - 0.540641i) q^{16} +(-0.797176 - 1.74557i) q^{17} +(0.142315 - 0.989821i) q^{18} +(1.49611 - 0.215109i) q^{20} +(1.23259 + 0.361922i) q^{25} +(1.49611 - 1.29639i) q^{26} +(1.61435 + 0.474017i) q^{29} +(-0.654861 - 0.755750i) q^{32} +(-1.61435 + 1.03748i) q^{34} -1.00000 q^{36} +(1.65486 - 0.755750i) q^{37} +(-0.425839 - 1.45027i) q^{40} +(0.544078 - 0.627899i) q^{41} +(-1.37491 - 0.627899i) q^{45} -1.00000 q^{49} +(0.182822 - 1.27155i) q^{50} +(-1.49611 - 1.29639i) q^{52} +(0.557730 + 0.0801894i) q^{53} +(0.239446 - 1.66538i) q^{58} +(-1.41542 + 0.909632i) q^{61} +(-0.654861 + 0.755750i) q^{64} +(-1.24302 - 2.72183i) q^{65} +(1.25667 + 1.45027i) q^{68} +(0.142315 + 0.989821i) q^{72} +(1.91899 - 0.563465i) q^{73} +(-0.983568 - 1.53046i) q^{74} +(-1.37491 + 0.627899i) q^{80} +(0.841254 + 0.540641i) q^{81} +(-0.698939 - 0.449181i) q^{82} +(0.817178 + 2.78305i) q^{85} +(1.37491 + 0.627899i) q^{89} +(-0.425839 + 1.45027i) q^{90} +(-0.118239 + 0.258908i) q^{97} +(0.142315 + 0.989821i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - q^{2} - q^{4} - q^{8} + q^{9}+O(q^{10}) \) 10 * q - q^2 - q^4 - q^8 + q^9	\( 10 q - q^{2} - q^{4} - q^{8} + q^{9} - q^{16} - 2 q^{17} + q^{18} - q^{25} + 2 q^{29} - q^{32} - 2 q^{34} - 10 q^{36} + 11 q^{37} + 2 q^{41} - 10 q^{49} + 10 q^{50} + 2 q^{58} - 9 q^{61} - q^{64} - 2 q^{68} + q^{72} + 2 q^{73} - q^{81} + 2 q^{82} - 2 q^{97} + q^{98}+O(q^{100}) \) 10 * q - q^2 - q^4 - q^8 + q^9 - q^16 - 2 * q^17 + q^18 - q^25 + 2 * q^29 - q^32 - 2 * q^34 - 10 * q^36 + 11 * q^37 + 2 * q^41 - 10 * q^49 + 10 * q^50 + 2 * q^58 - 9 * q^61 - q^64 - 2 * q^68 + q^72 + 2 * q^73 - q^81 + 2 * q^82 - 2 * q^97 + q^98

Introduction

L-functions

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