LMFDB - Embedded newform 1156.1.o.a.47.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1156,1,Mod(47,1156)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1156, base_ring=CyclotomicField(68))

chi = DirichletCharacter(H, H._module([34, 25]))

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

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

chi := DirichletCharacter("1156.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$1156 = 2^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	1156.o (of order $68$ , degree $32$ , 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:	$0.576919154604$
Analytic rank:	$0$
Dimension:	$32$
Coefficient field:	$\Q(\zeta_{68})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{32} - x^{30} + x^{28} - x^{26} + x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} + \cdots + 1$ x^32 - x^30 + x^28 - x^26 + x^24 - x^22 + x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	yes
Projective image:	$D_{68}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{68} - \cdots)$

Embedding invariants

Embedding label			47.1
Root			$0.526432 + 0.850217i$ of defining polynomial
Character	$\chi$	$=$	1156.47
Dual form			1156.1.o.a.123.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.895163 + 0.445738i) q^{2} +(0.602635 - 0.798017i) q^{4} +(-0.932472 - 0.638758i) q^{5} +(-0.183750 + 0.982973i) q^{8} +(-0.673696 + 0.739009i) q^{9} +(1.11943 + 0.156154i) q^{10} +(-1.32445 - 0.247582i) q^{13} +(-0.273663 - 0.961826i) q^{16} +(0.273663 + 0.961826i) q^{17} +(0.273663 - 0.961826i) q^{18} +(-1.07168 + 0.359191i) q^{20} +(0.100251 + 0.258777i) q^{25} +(1.29596 - 0.368731i) q^{26} +(-1.97871 + 0.276018i) q^{29} +(0.673696 + 0.739009i) q^{32} +(-0.673696 - 0.739009i) q^{34} +(0.183750 + 0.982973i) q^{36} +(0.434164 + 1.84595i) q^{37} +(0.799224 - 0.799224i) q^{40} +(-1.50941 + 0.666468i) q^{41} +(1.10025 - 0.258777i) q^{45} +(0.995734 - 0.0922684i) q^{49} +(-0.205087 - 0.186962i) q^{50} +(-0.995734 + 0.907732i) q^{52} +(-1.34164 + 1.47171i) q^{53} +(1.64823 - 1.12907i) q^{58} +(-1.40756 - 1.16883i) q^{61} +(-0.932472 - 0.361242i) q^{64} +(1.07687 + 1.07687i) q^{65} +(0.932472 + 0.361242i) q^{68} +(-0.602635 - 0.798017i) q^{72} +(0.0449462 - 0.0806938i) q^{73} +(-1.21146 - 1.45890i) q^{74} +(-0.359191 + 1.07168i) q^{80} +(-0.0922684 - 0.995734i) q^{81} +(1.05409 - 1.26940i) q^{82} +(0.359191 - 1.07168i) q^{85} +(1.75984 - 0.328972i) q^{89} +(-0.869557 + 0.722071i) q^{90} +(-0.0127611 - 0.276018i) q^{97} +(-0.850217 + 0.526432i) q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$32 q + 2 q^{4} + 2 q^{5} - 2 q^{10} - 2 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{20} - 34 q^{25} - 2 q^{29} + 2 q^{37} + 2 q^{40} - 2 q^{41} - 2 q^{45} + 2 q^{50} + 2 q^{58} + 2 q^{61} + 2 q^{64} - 2 q^{68}+ \cdots - 2 q^{98}+O(q^{100})$ 32 * q + 2 * q^4 + 2 * q^5 - 2 * q^10 - 2 * q^16 + 2 * q^17 + 2 * q^18 - 2 * q^20 - 34 * q^25 - 2 * q^29 + 2 * q^37 + 2 * q^40 - 2 * q^41 - 2 * q^45 + 2 * q^50 + 2 * q^58 + 2 * q^61 + 2 * q^64 - 2 * q^68 - 2 * q^72 - 2 * q^73 - 2 * q^74 + 2 * q^80 + 2 * q^81 - 2 * q^82 - 2 * q^85 - 2 * q^90 + 2 * q^97 - 2 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1156\mathbb{Z}\right)^\times$ .

$n$	$579$	$581$
$\chi(n)$	$-1$	$e\left(\frac{25}{68}\right)$

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

)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$\alpha_n$			$\theta_n$
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$\alpha_p$			$\theta_p$

$2$	−0.895163	+	0.445738i	−0.895163	+	0.445738i
$2$	−0.895163	+	0.445738i	−0.895163	+	0.445738i
$3$		0			0		−0.403921	−	0.914794i	$-0.632353\pi$
$3$		0			0		0.403921	+	0.914794i	$0.367647\pi$
$4$	0.602635	−	0.798017i	0.602635	−	0.798017i
$5$	−0.932472	−	0.638758i	−0.932472	−	0.638758i		−	1.00000i	$-0.5\pi$
$5$	−0.932472	−	0.638758i	−0.932472	−	0.638758i	−0.932472	+	0.361242i	$0.882353\pi$
$6$		0			0
$7$		0			0		0.998933	−	0.0461835i	$-0.0147059\pi$
$7$		0			0		−0.998933	+	0.0461835i	$0.985294\pi$
$8$	−0.183750	+	0.982973i	−0.183750	+	0.982973i
$9$	−0.673696	+	0.739009i	−0.673696	+	0.739009i
$10$	1.11943	+	0.156154i	1.11943	+	0.156154i
$11$		0			0		−0.138156	−	0.990410i	$-0.544118\pi$
$11$		0			0		0.138156	+	0.990410i	$0.455882\pi$
$12$		0			0
$13$	−1.32445	−	0.247582i	−1.32445	−	0.247582i	−0.526432	−	0.850217i	$-0.676471\pi$
$13$	−1.32445	−	0.247582i	−1.32445	−	0.247582i	−0.798017	+	0.602635i	$0.794118\pi$
$14$		0			0
$15$		0			0
$16$	−0.273663	−	0.961826i	−0.273663	−	0.961826i
$17$	0.273663	+	0.961826i	0.273663	+	0.961826i
$17$	0.273663	+	0.961826i	0.273663	+	0.961826i
$18$	0.273663	−	0.961826i	0.273663	−	0.961826i
$19$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$19$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$20$	−1.07168	+	0.359191i	−1.07168	+	0.359191i
$21$		0			0
$22$		0			0
$23$		0			0		0.998933	−	0.0461835i	$-0.0147059\pi$
$23$		0			0		−0.998933	+	0.0461835i	$0.985294\pi$
$24$		0			0
$25$	0.100251	+	0.258777i	0.100251	+	0.258777i
$26$	1.29596	−	0.368731i	1.29596	−	0.368731i
$27$		0			0
$28$		0			0
$29$	−1.97871	+	0.276018i	−1.97871	+	0.276018i	−0.982973	+	0.183750i	$0.941176\pi$
$29$	−1.97871	+	0.276018i	−1.97871	+	0.276018i	−0.995734	+	0.0922684i	$0.970588\pi$
$30$		0			0
$31$		0			0		−0.565136	−	0.824997i	$-0.691176\pi$
$31$		0			0		0.565136	+	0.824997i	$0.308824\pi$
$32$	0.673696	+	0.739009i	0.673696	+	0.739009i
$33$		0			0
$34$	−0.673696	−	0.739009i	−0.673696	−	0.739009i
$35$		0			0
$36$	0.183750	+	0.982973i	0.183750	+	0.982973i
$37$	0.434164	+	1.84595i	0.434164	+	1.84595i	0.526432	+	0.850217i	$0.323529\pi$
$37$	0.434164	+	1.84595i	0.434164	+	1.84595i	−0.0922684	+	0.995734i	$0.529412\pi$
$38$		0			0
$39$		0			0
$40$	0.799224	−	0.799224i	0.799224	−	0.799224i
$41$	−1.50941	+	0.666468i	−1.50941	+	0.666468i	−0.982973	−	0.183750i	$-0.941176\pi$
$41$	−1.50941	+	0.666468i	−1.50941	+	0.666468i	−0.526432	+	0.850217i	$0.676471\pi$
$42$		0			0
$43$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$43$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$44$		0			0
$45$	1.10025	−	0.258777i	1.10025	−	0.258777i
$46$		0			0
$47$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$47$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$48$		0			0
$49$	0.995734	−	0.0922684i	0.995734	−	0.0922684i
$50$	−0.205087	−	0.186962i	−0.205087	−	0.186962i
$51$		0			0
$52$	−0.995734	+	0.907732i	−0.995734	+	0.907732i
$53$	−1.34164	+	1.47171i	−1.34164	+	1.47171i	−0.602635	+	0.798017i	$0.705882\pi$
$53$	−1.34164	+	1.47171i	−1.34164	+	1.47171i	−0.739009	+	0.673696i	$0.764706\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.64823	−	1.12907i	1.64823	−	1.12907i
$59$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$59$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$60$		0			0
$61$	−1.40756	−	1.16883i	−1.40756	−	1.16883i	−0.961826	−	0.273663i	$-0.911765\pi$
$61$	−1.40756	−	1.16883i	−1.40756	−	1.16883i	−0.445738	−	0.895163i	$-0.647059\pi$
$62$		0			0
$63$		0			0
$64$	−0.932472	−	0.361242i	−0.932472	−	0.361242i
$65$	1.07687	+	1.07687i	1.07687	+	1.07687i
$66$		0			0
$67$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$67$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$68$	0.932472	+	0.361242i	0.932472	+	0.361242i
$69$		0			0
$70$		0			0
$71$		0			0		−0.0461835	−	0.998933i	$-0.514706\pi$
$71$		0			0		0.0461835	+	0.998933i	$0.485294\pi$
$72$	−0.602635	−	0.798017i	−0.602635	−	0.798017i
$73$	0.0449462	−	0.0806938i	0.0449462	−	0.0806938i	−0.850217	−	0.526432i	$-0.823529\pi$
$73$	0.0449462	−	0.0806938i	0.0449462	−	0.0806938i	0.895163	+	0.445738i	$0.147059\pi$
$74$	−1.21146	−	1.45890i	−1.21146	−	1.45890i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.317791	−	0.948161i	$-0.602941\pi$
$79$		0			0		0.317791	+	0.948161i	$0.397059\pi$
$80$	−0.359191	+	1.07168i	−0.359191	+	1.07168i
$81$	−0.0922684	−	0.995734i	−0.0922684	−	0.995734i
$82$	1.05409	−	1.26940i	1.05409	−	1.26940i
$83$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$83$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$84$		0			0
$85$	0.359191	−	1.07168i	0.359191	−	1.07168i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.75984	−	0.328972i	1.75984	−	0.328972i	0.798017	−	0.602635i	$-0.205882\pi$
$89$	1.75984	−	0.328972i	1.75984	−	0.328972i	0.961826	+	0.273663i	$0.0882353\pi$
$90$	−0.869557	+	0.722071i	−0.869557	+	0.722071i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.0127611	−	0.276018i	−0.0127611	−	0.276018i	−0.995734	−	0.0922684i	$-0.970588\pi$
$97$	−0.0127611	−	0.276018i	−0.0127611	−	0.276018i	0.982973	−	0.183750i	$-0.0588235\pi$
$98$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$99$		0			0
$100$	0.266923	+	0.0759460i	0.266923	+	0.0759460i
$101$	−0.177492	−	1.91545i	−0.177492	−	1.91545i	−0.361242	−	0.932472i	$-0.617647\pi$
$101$	−0.177492	−	1.91545i	−0.177492	−	1.91545i	0.183750	−	0.982973i	$-0.441176\pi$
$102$		0			0
$103$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$103$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$104$	0.486734	−	1.25640i	0.486734	−	1.25640i
$105$		0			0
$106$	0.544991	−	1.91545i	0.544991	−	1.91545i
$107$		0			0		0.948161	−	0.317791i	$-0.102941\pi$
$107$		0			0		−0.948161	+	0.317791i	$0.897059\pi$
$108$		0			0
$109$	0.256725	−	0.581427i	0.256725	−	0.581427i	−0.739009	−	0.673696i	$-0.764706\pi$
$109$	0.256725	−	0.581427i	0.256725	−	0.581427i	0.995734	+	0.0922684i	$0.0294118\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.29371	+	0.571231i	−1.29371	+	0.571231i	−0.932472	−	0.361242i	$-0.882353\pi$
$113$	−1.29371	+	0.571231i	−1.29371	+	0.571231i	−0.361242	+	0.932472i	$0.617647\pi$
$114$		0			0
$115$		0			0
$116$	−0.972171	+	1.74538i	−0.972171	+	1.74538i
$117$	1.07524	−	0.811985i	1.07524	−	0.811985i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.961826	+	0.273663i	−0.961826	+	0.273663i
$122$	1.78099	+	0.418885i	1.78099	+	0.418885i
$123$		0			0
$124$		0			0
$125$	−0.186962	+	0.794913i	−0.186962	+	0.794913i
$126$		0			0
$127$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$127$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$128$	0.995734	−	0.0922684i	0.995734	−	0.0922684i
$129$		0			0
$130$	−1.44397	−	0.483971i	−1.44397	−	0.483971i
$131$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$131$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	−0.995734	+	0.0922684i	−0.995734	+	0.0922684i
$137$	−0.221468	+	0.293271i	−0.221468	+	0.293271i	−0.895163	−	0.445738i	$-0.852941\pi$
$137$	−0.221468	+	0.293271i	−0.221468	+	0.293271i	0.673696	+	0.739009i	$0.264706\pi$
$138$		0			0
$139$		0			0		0.317791	−	0.948161i	$-0.397059\pi$
$139$		0			0		−0.317791	+	0.948161i	$0.602941\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	0.895163	+	0.445738i	0.895163	+	0.445738i
$145$	2.02140	+	1.00654i	2.02140	+	1.00654i
$146$	−0.00426582	+	0.0922684i	−0.00426582	+	0.0922684i
$147$		0			0
$148$	1.73474	+	0.765964i	1.73474	+	0.765964i
$149$	1.37821	−	0.533922i	1.37821	−	0.533922i	0.445738	−	0.895163i	$-0.352941\pi$
$149$	1.37821	−	0.533922i	1.37821	−	0.533922i	0.932472	+	0.361242i	$0.117647\pi$
$150$		0			0
$151$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$151$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$152$		0			0
$153$	−0.895163	−	0.445738i	−0.895163	−	0.445738i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	−0.181395	+	1.95756i	−0.181395	+	1.95756i	0.0922684	+	0.995734i	$0.470588\pi$
$157$	−0.181395	+	1.95756i	−0.181395	+	1.95756i	−0.273663	+	0.961826i	$0.588235\pi$
$158$		0			0
$159$		0			0
$160$	−0.156154	−	1.11943i	−0.156154	−	1.11943i
$161$		0			0
$162$	0.526432	+	0.850217i	0.526432	+	0.850217i
$163$		0			0		0.228951	−	0.973438i	$-0.426471\pi$
$163$		0			0		−0.228951	+	0.973438i	$0.573529\pi$
$164$	−0.377767	+	1.60617i	−0.377767	+	1.60617i
$165$		0			0
$166$		0			0
$167$		0			0		−0.973438	−	0.228951i	$-0.926471\pi$
$167$		0			0		0.973438	+	0.228951i	$0.0735294\pi$
$168$		0			0
$169$	0.760397	+	0.294579i	0.760397	+	0.294579i
$170$	0.156154	+	1.11943i	0.156154	+	1.11943i
$171$		0			0
$172$		0			0
$173$	0.748723	−	1.34421i	0.748723	−	1.34421i	−0.183750	−	0.982973i	$-0.558824\pi$
$173$	0.748723	−	1.34421i	0.748723	−	1.34421i	0.932472	−	0.361242i	$-0.117647\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$	−1.42871	+	1.07891i	−1.42871	+	1.07891i
$179$		0			0			−	1.00000i	$-0.5\pi$
$179$		0			0				1.00000i	$0.5\pi$
$180$	0.456541	−	1.03397i	0.456541	−	1.03397i
$181$	0.666468	−	0.456541i	0.666468	−	0.456541i	−0.183750	−	0.982973i	$-0.558824\pi$
$181$	0.666468	−	0.456541i	0.666468	−	0.456541i	0.850217	+	0.526432i	$0.176471\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	0.774271	−	1.99862i	0.774271	−	1.99862i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$191$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$192$		0			0
$193$	−0.516087	+	1.16883i	−0.516087	+	1.16883i	0.445738	+	0.895163i	$0.352941\pi$
$193$	−0.516087	+	1.16883i	−0.516087	+	1.16883i	−0.961826	+	0.273663i	$0.911765\pi$
$194$	0.134455	+	0.241393i	0.134455	+	0.241393i
$195$		0			0
$196$	0.526432	−	0.850217i	0.526432	−	0.850217i
$197$	0.987432	−	1.44147i	0.987432	−	1.44147i	0.0922684	−	0.995734i	$-0.470588\pi$
$197$	0.987432	−	1.44147i	0.987432	−	1.44147i	0.895163	−	0.445738i	$-0.147059\pi$
$198$		0			0
$199$		0			0		0.769334	−	0.638847i	$-0.220588\pi$
$199$		0			0		−0.769334	+	0.638847i	$0.779412\pi$
$200$	−0.272791	+	0.0509936i	−0.272791	+	0.0509936i
$201$		0			0
$202$	1.01267	+	1.63552i	1.01267	+	1.63552i
$203$		0			0
$204$		0			0
$205$	1.83319	+	0.342683i	1.83319	+	0.342683i
$206$		0			0
$207$		0			0
$208$	0.124322	+	1.34164i	0.124322	+	1.34164i
$209$		0			0
$210$		0			0
$211$		0			0		−0.638847	−	0.769334i	$-0.720588\pi$
$211$		0			0		0.638847	+	0.769334i	$0.279412\pi$
$212$	0.365931	+	1.95756i	0.365931	+	1.95756i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	0.0293534	+	0.634905i	0.0293534	+	0.634905i
$219$		0			0
$220$		0			0
$221$	−0.124322	−	1.34164i	−0.124322	−	1.34164i
$222$		0			0
$223$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$223$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$224$		0			0
$225$	−0.258777	−	0.100251i	−0.258777	−	0.100251i
$226$	0.903466	−	1.08800i	0.903466	−	1.08800i
$227$		0			0		−0.873622	−	0.486604i	$-0.838235\pi$
$227$		0			0		0.873622	+	0.486604i	$0.161765\pi$
$228$		0			0
$229$	1.14558	+	1.25664i	1.14558	+	1.25664i	0.961826	+	0.273663i	$0.0882353\pi$
$229$	1.14558	+	1.25664i	1.14558	+	1.25664i	0.183750	+	0.982973i	$0.441176\pi$
$230$		0			0
$231$		0			0
$232$	0.0922684	−	1.99573i	0.0922684	−	1.99573i
$233$	0.241393	−	1.73049i	0.241393	−	1.73049i	−0.361242	−	0.932472i	$-0.617647\pi$
$233$	0.241393	−	1.73049i	0.241393	−	1.73049i	0.602635	−	0.798017i	$-0.294118\pi$
$234$	−0.600584	+	1.20614i	−0.600584	+	1.20614i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$239$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$240$		0			0
$241$	0.621500	−	0.516087i	0.621500	−	0.516087i	−0.273663	−	0.961826i	$-0.588235\pi$
$241$	0.621500	−	0.516087i	0.621500	−	0.516087i	0.895163	+	0.445738i	$0.147059\pi$
$242$	0.739009	−	0.673696i	0.739009	−	0.673696i
$243$		0			0
$244$	−1.78099	+	0.418885i	−1.78099	+	0.418885i
$245$	−0.987432	−	0.549996i	−0.987432	−	0.549996i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	−0.186962	−	0.794913i	−0.186962	−	0.794913i
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$257$	0.600584	+	0.658809i	0.600584	+	0.658809i	0.961826	−	0.273663i	$-0.0882353\pi$
$257$	0.600584	+	0.658809i	0.600584	+	0.658809i	−0.361242	+	0.932472i	$0.617647\pi$
$258$		0			0
$259$		0			0
$260$	1.50832	−	0.210401i	1.50832	−	0.210401i
$261$	1.12907	−	1.64823i	1.12907	−	1.64823i
$262$		0			0
$263$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$263$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$264$		0			0
$265$	2.19111	−	0.515345i	2.19111	−	0.515345i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.922758	+	0.309277i	−0.922758	+	0.309277i	−0.739009	−	0.673696i	$-0.764706\pi$
$269$	−0.922758	+	0.309277i	−0.922758	+	0.309277i	−0.183750	+	0.982973i	$0.558824\pi$
$270$		0			0
$271$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$271$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$272$	0.850217	−	0.526432i	0.850217	−	0.526432i
$273$		0			0
$274$	0.0675278	−	0.361242i	0.0675278	−	0.361242i
$275$		0			0
$276$		0			0
$277$	−1.49780	−	0.352279i	−1.49780	−	0.352279i	−0.602635	−	0.798017i	$-0.705882\pi$
$277$	−1.49780	−	0.352279i	−1.49780	−	0.352279i	−0.895163	+	0.445738i	$0.852941\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.293271	+	1.56886i	−0.293271	+	1.56886i	0.445738	+	0.895163i	$0.352941\pi$
$281$	−0.293271	+	1.56886i	−0.293271	+	1.56886i	−0.739009	+	0.673696i	$0.764706\pi$
$282$		0			0
$283$		0			0		−0.769334	−	0.638847i	$-0.779412\pi$
$283$		0			0		0.769334	+	0.638847i	$0.220588\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−1.00000			−1.00000
$289$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$290$	−2.25813			−2.25813
$291$		0			0
$292$	−0.0373089	−	0.0844967i	−0.0373089	−	0.0844967i
$293$	0.537235	−	0.711414i	0.537235	−	0.711414i	−0.445738	−	0.895163i	$-0.647059\pi$
$293$	0.537235	−	0.711414i	0.537235	−	0.711414i	0.982973	+	0.183750i	$0.0588235\pi$
$294$		0			0
$295$		0			0
$296$	−1.89430	+	0.0875787i	−1.89430	+	0.0875787i
$297$		0			0
$298$	−0.995734	+	1.09227i	−0.995734	+	1.09227i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	0.565917	+	1.98899i	0.565917	+	1.98899i
$306$	1.00000			1.00000
$307$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$307$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.565136	−	0.824997i	$-0.691176\pi$
$311$		0			0		0.565136	+	0.824997i	$0.308824\pi$
$312$		0			0
$313$	−0.947359	+	0.222817i	−0.947359	+	0.222817i	−0.673696	−	0.739009i	$-0.735294\pi$
$313$	−0.947359	+	0.222817i	−0.947359	+	0.222817i	−0.273663	+	0.961826i	$0.588235\pi$
$314$	−0.710182	−	1.83319i	−0.710182	−	1.83319i
$315$		0			0
$316$		0			0
$317$	−0.549996	+	0.802895i	−0.549996	+	0.802895i	−0.995734	−	0.0922684i	$-0.970588\pi$
$317$	−0.549996	+	0.802895i	−0.549996	+	0.802895i	0.445738	+	0.895163i	$0.352941\pi$
$318$		0			0
$319$		0			0
$320$	0.638758	+	0.932472i	0.638758	+	0.932472i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$325$	−0.0687083	−	0.367557i	−0.0687083	−	0.367557i
$326$		0			0
$327$		0			0
$328$	−0.377767	−	1.60617i	−0.377767	−	1.60617i
$329$		0			0
$330$		0			0
$331$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$331$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$332$		0			0
$333$	−1.65667	−	0.922758i	−1.65667	−	0.922758i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−1.49780	+	1.24376i	−1.49780	+	1.24376i	−0.602635	+	0.798017i	$0.705882\pi$
$337$	−1.49780	+	1.24376i	−1.49780	+	1.24376i	−0.895163	+	0.445738i	$0.852941\pi$
$338$	−0.811985	+	0.0752415i	−0.811985	+	0.0752415i
$339$		0			0
$340$	−0.638758	−	0.932472i	−0.638758	−	0.932472i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.0710610	+	1.53703i	−0.0710610	+	1.53703i
$347$		0			0		0.824997	−	0.565136i	$-0.191176\pi$
$347$		0			0		−0.824997	+	0.565136i	$0.808824\pi$
$348$		0			0
$349$	−0.709310	−	0.778076i	−0.709310	−	0.778076i	0.273663	−	0.961826i	$-0.411765\pi$
$349$	−0.709310	−	0.778076i	−0.709310	−	0.778076i	−0.982973	+	0.183750i	$0.941176\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	0.510366	+	0.197717i	0.510366	+	0.197717i	0.602635	−	0.798017i	$-0.294118\pi$
$353$	0.510366	+	0.197717i	0.510366	+	0.197717i	−0.0922684	+	0.995734i	$0.529412\pi$
$354$		0			0
$355$		0			0
$356$	0.798017	−	1.60263i	0.798017	−	1.60263i
$357$		0			0
$358$		0			0
$359$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$359$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$360$	0.0521999	+	1.12907i	0.0521999	+	1.12907i
$361$	0.602635	+	0.798017i	0.602635	+	0.798017i
$362$	−0.393100	+	0.705749i	−0.393100	+	0.705749i
$363$		0			0
$364$		0			0
$365$	−0.0934549	+	0.0465350i	−0.0934549	+	0.0465350i
$366$		0			0
$367$		0			0		−0.638847	−	0.769334i	$-0.720588\pi$
$367$		0			0		0.638847	+	0.769334i	$0.279412\pi$
$368$		0			0
$369$	0.524354	−	1.56446i	0.524354	−	1.56446i
$370$	0.197764	+	2.13422i	0.197764	+	2.13422i
$371$		0			0
$372$		0			0
$373$	1.03494	+	0.193463i	1.03494	+	0.193463i	0.673696	−	0.739009i	$-0.264706\pi$
$373$	1.03494	+	0.193463i	1.03494	+	0.193463i	0.361242	+	0.932472i	$0.382353\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.68903	+	0.124322i	2.68903	+	0.124322i
$378$		0			0
$379$		0			0		0.769334	−	0.638847i	$-0.220588\pi$
$379$		0			0		−0.769334	+	0.638847i	$0.779412\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$383$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$384$		0			0
$385$		0			0
$386$	−0.0590083	−	1.27633i	−0.0590083	−	1.27633i
$387$		0			0
$388$	−0.227957	−	0.156154i	−0.227957	−	0.156154i
$389$	0.177492	+	0.0505009i	0.177492	+	0.0505009i	0.361242	−	0.932472i	$-0.382353\pi$
$389$	0.177492	+	0.0505009i	0.177492	+	0.0505009i	−0.183750	+	0.982973i	$0.558824\pi$
$390$		0			0
$391$		0			0
$392$	−0.0922684	+	0.995734i	−0.0922684	+	0.995734i
$393$		0			0
$394$	−0.241393	+	1.73049i	−0.241393	+	1.73049i
$395$		0			0
$396$		0			0
$397$	−0.524354	+	0.359191i	−0.524354	+	0.359191i	−0.798017	−	0.602635i	$-0.794118\pi$
$397$	−0.524354	+	0.359191i	−0.524354	+	0.359191i	0.273663	+	0.961826i	$0.411765\pi$
$398$		0			0
$399$		0			0
$400$	0.221463	−	0.167241i	0.221463	−	0.167241i
$401$	−0.488975	−	1.45890i	−0.488975	−	1.45890i	−0.850217	−	0.526432i	$-0.823529\pi$
$401$	−0.488975	−	1.45890i	−0.488975	−	1.45890i	0.361242	−	0.932472i	$-0.382353\pi$
$402$		0			0
$403$		0			0
$404$	−1.63552	−	1.01267i	−1.63552	−	1.01267i
$405$	−0.549996	+	0.987432i	−0.549996	+	0.987432i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.73901	−	0.673696i	−1.73901	−	0.673696i	−0.739009	−	0.673696i	$-0.764706\pi$
$409$	−1.73901	−	0.673696i	−1.73901	−	0.673696i	−1.00000			$\pi$
$410$	−1.79375	+	0.510366i	−1.79375	+	0.510366i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	−0.709310	−	1.14558i	−0.709310	−	1.14558i
$417$		0			0
$418$		0			0
$419$		0			0		−0.948161	−	0.317791i	$-0.897059\pi$
$419$		0			0		0.948161	+	0.317791i	$0.102941\pi$
$420$		0			0
$421$	−0.177492	+	1.91545i	−0.177492	+	1.91545i	0.183750	+	0.982973i	$0.441176\pi$
$421$	−0.177492	+	1.91545i	−0.177492	+	1.91545i	−0.361242	+	0.932472i	$0.617647\pi$
$422$		0			0
$423$		0			0
$424$	−1.20013	−	1.58923i	−1.20013	−	1.58923i
$425$	−0.221463	+	0.167241i	−0.221463	+	0.167241i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.486604	−	0.873622i	$-0.661765\pi$
$431$		0			0		0.486604	+	0.873622i	$0.338235\pi$
$432$		0			0
$433$	−1.75984	−	0.876298i	−1.75984	−	0.876298i	−0.961826	−	0.273663i	$-0.911765\pi$
$433$	−1.75984	−	0.876298i	−1.75984	−	0.876298i	−0.798017	−	0.602635i	$-0.794118\pi$
$434$		0			0
$435$		0			0
$436$	−0.309277	−	0.555259i	−0.309277	−	0.555259i
$437$		0			0
$438$		0			0
$439$		0			0		0.317791	−	0.948161i	$-0.397059\pi$
$439$		0			0		−0.317791	+	0.948161i	$0.602941\pi$
$440$		0			0
$441$	−0.602635	+	0.798017i	−0.602635	+	0.798017i
$442$	0.709310	+	1.14558i	0.709310	+	1.14558i
$443$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$443$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$444$		0			0
$445$	−1.85114	−	0.817357i	−1.85114	−	0.817357i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	0.176521	+	1.26544i	0.176521	+	1.26544i	0.850217	+	0.526432i	$0.176471\pi$
$449$	0.176521	+	1.26544i	0.176521	+	1.26544i	−0.673696	+	0.739009i	$0.735294\pi$
$450$	0.276333	−	0.0256060i	0.276333	−	0.0256060i
$451$		0			0
$452$	−0.323785	+	1.37665i	−0.323785	+	1.37665i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.526432	+	0.149783i	−0.526432	+	0.149783i	−0.526432	−	0.850217i	$-0.676471\pi$
$457$	−0.526432	+	0.149783i	−0.526432	+	0.149783i			1.00000i	$0.5\pi$
$458$	−1.58561	−	0.614268i	−1.58561	−	0.614268i
$459$		0			0
$460$		0			0
$461$	−1.27366	+	0.961826i	−1.27366	+	0.961826i	−0.273663	+	0.961826i	$0.588235\pi$
$461$	−1.27366	+	0.961826i	−1.27366	+	0.961826i	−1.00000			$1.00000\pi$
$462$		0			0
$463$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$463$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$464$	0.806980	+	1.82764i	0.806980	+	1.82764i
$465$		0			0
$466$	0.555259	+	1.65667i	0.555259	+	1.65667i
$467$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$467$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$468$		−	1.34739i		−	1.34739i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.183750	−	1.98297i	−0.183750	−	1.98297i
$478$		0			0
$479$		0			0		−0.824997	−	0.565136i	$-0.808824\pi$
$479$		0			0		0.824997	+	0.565136i	$0.191176\pi$
$480$		0			0
$481$	−0.118003	−	2.55236i	−0.118003	−	2.55236i
$482$	−0.326304	+	0.739009i	−0.326304	+	0.739009i
$483$		0			0
$484$	−0.361242	+	0.932472i	−0.361242	+	0.932472i
$485$	−0.164409	+	0.265530i	−0.164409	+	0.265530i
$486$		0			0
$487$		0			0		−0.990410	−	0.138156i	$-0.955882\pi$
$487$		0			0		0.990410	+	0.138156i	$0.0441176\pi$
$488$	1.40756	−	1.16883i	1.40756	−	1.16883i
$489$		0			0
$490$	1.12907	+	0.0521999i	1.12907	+	0.0521999i
$491$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$491$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$492$		0			0
$493$	−0.806980	−	1.82764i	−0.806980	−	1.82764i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.317791	−	0.948161i	$-0.602941\pi$
$499$		0			0		0.317791	+	0.948161i	$0.397059\pi$
$500$	0.521684	+	0.628241i	0.521684	+	0.628241i
$501$		0			0
$502$		0			0
$503$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$503$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$504$		0			0
$505$	−1.05800	+	1.89947i	−1.05800	+	1.89947i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.0822551	−	0.165190i	−0.0822551	−	0.165190i	0.850217	−	0.526432i	$-0.176471\pi$
$509$	−0.0822551	−	0.165190i	−0.0822551	−	0.165190i	−0.932472	+	0.361242i	$0.882353\pi$
$510$		0			0
$511$		0			0
$512$	0.526432	−	0.850217i	0.526432	−	0.850217i
$513$		0			0
$514$	−0.831277	−	0.322039i	−0.831277	−	0.322039i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−1.25640	+	0.860657i	−1.25640	+	0.860657i
$521$	0.0653133	−	1.41270i	0.0653133	−	1.41270i	−0.673696	−	0.739009i	$-0.735294\pi$
$521$	0.0653133	−	1.41270i	0.0653133	−	1.41270i	0.739009	−	0.673696i	$-0.235294\pi$
$522$	−0.276018	+	1.97871i	−0.276018	+	1.97871i
$523$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$523$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.995734	−	0.0922684i	0.995734	−	0.0922684i
$530$	−1.73170	+	1.43798i	−1.73170	+	1.43798i
$531$		0			0
$532$		0			0
$533$	2.16414	−	0.509000i	2.16414	−	0.509000i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	0.688163	−	0.688163i	0.688163	−	0.688163i
$539$		0			0
$540$		0			0
$541$	0.184956	+	0.786384i	0.184956	+	0.786384i	0.982973	+	0.183750i	$0.0588235\pi$
$541$	0.184956	+	0.786384i	0.184956	+	0.786384i	−0.798017	+	0.602635i	$0.794118\pi$
$542$		0			0
$543$		0			0
$544$	−0.526432	+	0.850217i	−0.526432	+	0.850217i
$545$	−0.610781	+	0.378179i	−0.610781	+	0.378179i
$546$		0			0
$547$		0			0		−0.565136	−	0.824997i	$-0.691176\pi$
$547$		0			0		0.565136	+	0.824997i	$0.308824\pi$
$548$	0.100571	+	0.353470i	0.100571	+	0.353470i
$549$	1.81204	−	0.252769i	1.81204	−	0.252769i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	1.49780	−	0.352279i	1.49780	−	0.352279i
$555$		0			0
$556$		0			0
$557$	0.798017	+	1.60263i	0.798017	+	1.60263i	0.798017	+	0.602635i	$0.205882\pi$
$557$	0.798017	+	1.60263i	0.798017	+	1.60263i			1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	−0.436776	−	1.53511i	−0.436776	−	1.53511i
$563$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$563$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$564$		0			0
$565$	1.57123	+	0.293714i	1.57123	+	0.293714i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$569$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$570$		0			0
$571$		0			0		0.998933	−	0.0461835i	$-0.0147059\pi$
$571$		0			0		−0.998933	+	0.0461835i	$0.985294\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.895163	−	0.445738i	0.895163	−	0.445738i
$577$	1.05286			1.05286			0.526432	−	0.850217i	$-0.323529\pi$
$577$	1.05286			1.05286			0.526432	+	0.850217i	$0.323529\pi$
$578$	0.526432	−	0.850217i	0.526432	−	0.850217i
$579$		0			0
$580$	2.02140	−	1.00654i	2.02140	−	1.00654i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.0710610	+	0.0590083i	0.0710610	+	0.0590083i
$585$	−1.52129	+	0.0703337i	−1.52129	+	0.0703337i
$586$	−0.163808	+	0.876298i	−0.163808	+	0.876298i
$587$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$587$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	1.65667	−	0.922758i	1.65667	−	0.922758i
$593$	0.0675278	−	0.361242i	0.0675278	−	0.361242i	−0.932472	−	0.361242i	$-0.882353\pi$
$593$	0.0675278	−	0.361242i	0.0675278	−	0.361242i	1.00000			$0$
$594$		0			0
$595$		0			0
$596$	0.404479	−	1.42160i	0.404479	−	1.42160i
$597$		0			0
$598$		0			0
$599$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$599$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$600$		0			0
$601$	0.0922684	−	0.00426582i	0.0922684	−	0.00426582i		−	1.00000i	$-0.5\pi$
$601$	0.0922684	−	0.00426582i	0.0922684	−	0.00426582i	0.0922684	+	0.995734i	$0.470588\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.07168	+	0.359191i	1.07168	+	0.359191i
$606$		0			0
$607$		0			0		0.990410	−	0.138156i	$-0.0441176\pi$
$607$		0			0		−0.990410	+	0.138156i	$0.955882\pi$
$608$		0			0
$609$		0			0
$610$	−1.39316	−	1.52822i	−1.39316	−	1.52822i
$611$		0			0
$612$	−0.895163	+	0.445738i	−0.895163	+	0.445738i
$613$	1.02474	+	0.634493i	1.02474	+	0.634493i	0.932472	−	0.361242i	$-0.117647\pi$
$613$	1.02474	+	0.634493i	1.02474	+	0.634493i	0.0922684	+	0.995734i	$0.470588\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.0211475	+	0.0899135i	0.0211475	+	0.0899135i	0.982973	−	0.183750i	$-0.0588235\pi$
$617$	0.0211475	+	0.0899135i	0.0211475	+	0.0899135i	−0.961826	+	0.273663i	$0.911765\pi$
$618$		0			0
$619$		0			0		0.914794	−	0.403921i	$-0.132353\pi$
$619$		0			0		−0.914794	+	0.403921i	$0.867647\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.887181	−	0.808772i	0.887181	−	0.808772i
$626$	0.748723	−	0.621731i	0.748723	−	0.621731i
$627$		0			0
$628$	1.45285	+	1.32445i	1.45285	+	1.32445i
$629$	−1.65667	+	0.922758i	−1.65667	+	0.922758i
$630$		0			0
$631$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$631$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$632$		0			0
$633$		0			0
$634$	0.134455	−	0.963876i	0.134455	−	0.963876i
$635$		0			0
$636$		0			0
$637$	−1.34164	−	0.124322i	−1.34164	−	0.124322i
$638$		0			0
$639$		0			0
$640$	−0.987432	−	0.549996i	−0.987432	−	0.549996i
$641$	−1.24376	+	1.49780i	−1.24376	+	1.49780i	−0.445738	+	0.895163i	$0.647059\pi$
$641$	−1.24376	+	1.49780i	−1.24376	+	1.49780i	−0.798017	+	0.602635i	$0.794118\pi$
$642$		0			0
$643$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$643$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$647$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$648$	0.995734	+	0.0922684i	0.995734	+	0.0922684i
$649$		0			0
$650$	0.225339	+	0.298397i	0.225339	+	0.298397i
$651$		0			0
$652$		0			0
$653$	−0.323785	+	0.323785i	−0.323785	+	0.323785i	−0.850217	−	0.526432i	$-0.823529\pi$
$653$	−0.323785	+	0.323785i	−0.323785	+	0.323785i	0.526432	+	0.850217i	$0.323529\pi$
$654$		0			0
$655$		0			0
$656$	1.05409	+	1.26940i	1.05409	+	1.26940i
$657$	0.0293534	+	0.0875787i	0.0293534	+	0.0875787i
$658$		0			0
$659$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$659$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$660$		0			0
$661$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$661$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.89430	+	0.0875787i	1.89430	+	0.0875787i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	0.963876	+	1.73049i	0.963876	+	1.73049i	0.602635	+	0.798017i	$0.294118\pi$
$673$	0.963876	+	1.73049i	0.963876	+	1.73049i	0.361242	+	0.932472i	$0.382353\pi$
$674$	0.786384	−	1.78099i	0.786384	−	1.78099i
$675$		0			0
$676$	0.693321	−	0.429286i	0.693321	−	0.429286i
$677$	1.63417	+	1.11943i	1.63417	+	1.11943i	0.895163	+	0.445738i	$0.147059\pi$
$677$	1.63417	+	1.11943i	1.63417	+	1.11943i	0.739009	+	0.673696i	$0.235294\pi$
$678$		0			0
$679$		0			0
$680$	0.987432	+	0.549996i	0.987432	+	0.549996i
$681$		0			0
$682$		0			0
$683$		0			0		0.138156	−	0.990410i	$-0.455882\pi$
$683$		0			0		−0.138156	+	0.990410i	$0.544118\pi$
$684$		0			0
$685$	0.393841	−	0.132002i	0.393841	−	0.132002i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.14131	−	1.61704i	2.14131	−	1.61704i
$690$		0			0
$691$		0			0		0.914794	−	0.403921i	$-0.132353\pi$
$691$		0			0		−0.914794	+	0.403921i	$0.867647\pi$
$692$	−0.621500	−	1.40756i	−0.621500	−	1.40756i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−1.05409	−	1.26940i	−1.05409	−	1.26940i
$698$	0.981767	+	0.380338i	0.981767	+	0.380338i
$699$		0			0
$700$		0			0
$701$	−1.42160	−	1.29596i	−1.42160	−	1.29596i	−0.895163	−	0.445738i	$-0.852941\pi$
$701$	−1.42160	−	1.29596i	−1.42160	−	1.29596i	−0.526432	−	0.850217i	$-0.676471\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.544991	+	0.0505009i	−0.544991	+	0.0505009i
$707$		0			0
$708$		0			0
$709$	−0.323785	−	0.323785i	−0.323785	−	0.323785i	0.526432	−	0.850217i	$-0.323529\pi$
$709$	−0.323785	−	0.323785i	−0.323785	−	0.323785i	−0.850217	+	0.526432i	$0.823529\pi$
$710$		0			0
$711$		0			0
$712$			1.79033i			1.79033i
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.914794	−	0.403921i	$-0.867647\pi$
$719$		0			0		0.914794	+	0.403921i	$0.132353\pi$
$720$	−0.549996	−	0.987432i	−0.549996	−	0.987432i
$721$		0			0
$722$	−0.895163	−	0.445738i	−0.895163	−	0.445738i
$723$		0			0
$724$	0.0373089	−	0.806980i	0.0373089	−	0.806980i
$725$	−0.269794	−	0.484372i	−0.269794	−	0.484372i
$726$		0			0
$727$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$727$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$728$		0			0
$729$	0.798017	+	0.602635i	0.798017	+	0.602635i
$730$	0.0629149	−	0.0833129i	0.0629149	−	0.0833129i
$731$		0			0
$732$		0			0
$733$		−	0.367499i		−	0.367499i	−0.982973	−	0.183750i	$-0.941176\pi$
$733$		−	0.367499i		−	0.367499i	0.982973	−	0.183750i	$-0.0588235\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	0.227957	+	1.63417i	0.227957	+	1.63417i
$739$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$739$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$740$	−1.12833	−	1.82232i	−1.12833	−	1.82232i
$741$		0			0
$742$		0			0
$743$		0			0		−0.998933	−	0.0461835i	$-0.985294\pi$
$743$		0			0		0.998933	+	0.0461835i	$0.0147059\pi$
$744$		0			0
$745$	−1.62619	−	0.382476i	−1.62619	−	0.382476i
$746$	−1.01267	+	0.288130i	−1.01267	+	0.288130i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.486604	−	0.873622i	$-0.338235\pi$
$751$		0			0		−0.486604	+	0.873622i	$0.661765\pi$
$752$		0			0
$753$		0			0
$754$	−2.46254	+	1.08732i	−2.46254	+	1.08732i
$755$		0			0
$756$		0			0
$757$			1.20527i			1.20527i	0.798017	+	0.602635i	$0.205882\pi$
$757$			1.20527i			1.20527i	−0.798017	+	0.602635i	$0.794118\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.100571	−	0.353470i	0.100571	−	0.353470i	−0.895163	−	0.445738i	$-0.852941\pi$
$761$	0.100571	−	0.353470i	0.100571	−	0.353470i	0.995734	+	0.0922684i	$0.0294118\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.549996	+	0.987432i	0.549996	+	0.987432i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1.52217	+	0.942485i	−1.52217	+	0.942485i	−0.526432	+	0.850217i	$0.676471\pi$
$769$	−1.52217	+	0.942485i	−1.52217	+	0.942485i	−0.995734	+	0.0922684i	$0.970588\pi$
$770$		0			0
$771$		0			0
$772$	0.621731	+	1.11622i	0.621731	+	1.11622i
$773$	0.710182	−	1.83319i	0.710182	−	1.83319i	0.183750	−	0.982973i	$-0.441176\pi$
$773$	0.710182	−	1.83319i	0.710182	−	1.83319i	0.526432	−	0.850217i	$-0.323529\pi$
$774$		0			0
$775$		0			0
$776$	0.273663	+	0.0381744i	0.273663	+	0.0381744i
$777$		0			0
$778$	−0.181395	+	0.0339085i	−0.181395	+	0.0339085i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−0.361242	−	0.932472i	−0.361242	−	0.932472i
$785$	1.41955	−	1.70950i	1.41955	−	1.70950i
$786$		0			0
$787$		0			0		0.317791	−	0.948161i	$-0.397059\pi$
$787$		0			0		−0.317791	+	0.948161i	$0.602941\pi$
$788$	−0.555259	−	1.65667i	−0.555259	−	1.65667i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.57487	+	1.89654i	1.57487	+	1.89654i
$794$	0.309277	−	0.555259i	0.309277	−	0.555259i
$795$		0			0
$796$		0			0
$797$	−1.98297	−	0.183750i	−1.98297	−	0.183750i	−0.982973	−	0.183750i	$-0.941176\pi$
$797$	−1.98297	−	0.183750i	−1.98297	−	0.183750i	−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$	−0.123700	+	0.248423i	−0.123700	+	0.248423i
$801$	−0.942485	+	1.52217i	−0.942485	+	1.52217i
$802$	1.08800	+	1.08800i	1.08800	+	1.08800i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	1.91545	+	0.177492i	1.91545	+	0.177492i
$809$	−1.44147	+	0.987432i	−1.44147	+	0.987432i	−0.445738	+	0.895163i	$0.647059\pi$
$809$	−1.44147	+	0.987432i	−1.44147	+	0.987432i	−0.995734	+	0.0922684i	$0.970588\pi$
$810$	0.0521999	−	1.12907i	0.0521999	−	1.12907i
$811$		0			0		0.138156	−	0.990410i	$-0.455882\pi$
$811$		0			0		−0.138156	+	0.990410i	$0.544118\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	1.85699	−	0.172075i	1.85699	−	0.172075i
$819$		0			0
$820$	1.37821	−	1.25640i	1.37821	−	1.25640i
$821$	−1.59837	+	0.890286i	−1.59837	+	0.890286i	−0.602635	+	0.798017i	$0.705882\pi$
$821$	−1.59837	+	0.890286i	−1.59837	+	0.890286i	−0.995734	+	0.0922684i	$0.970588\pi$
$822$		0			0
$823$		0			0		−0.873622	−	0.486604i	$-0.838235\pi$
$823$		0			0		0.873622	+	0.486604i	$0.161765\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$827$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$828$		0			0
$829$	1.47802			1.47802			0.739009	−	0.673696i	$-0.235294\pi$
$829$	1.47802			1.47802			0.739009	+	0.673696i	$0.235294\pi$
$830$		0			0
$831$		0			0
$832$	1.14558	+	0.709310i	1.14558	+	0.709310i
$833$	0.361242	+	0.932472i	0.361242	+	0.932472i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.565136	−	0.824997i	$-0.308824\pi$
$839$		0			0		−0.565136	+	0.824997i	$0.691176\pi$
$840$		0			0
$841$	2.87727	−	0.818654i	2.87727	−	0.818654i
$842$	−0.694903	−	1.79375i	−0.694903	−	1.79375i
$843$		0			0
$844$		0			0
$845$	−0.520884	−	0.760397i	−0.520884	−	0.760397i
$846$		0			0
$847$		0			0
$848$	1.78269	+	0.887674i	1.78269	+	0.887674i
$849$		0			0
$850$	0.123700	−	0.248423i	0.123700	−	0.248423i
$851$		0			0
$852$		0			0
$853$	−1.70083	+	0.947359i	−1.70083	+	0.947359i	−0.739009	+	0.673696i	$0.764706\pi$
$853$	−1.70083	+	0.947359i	−1.70083	+	0.947359i	−0.961826	+	0.273663i	$0.911765\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	0.453510	+	0.0632619i	0.453510	+	0.0632619i	0.361242	−	0.932472i	$-0.382353\pi$
$857$	0.453510	+	0.0632619i	0.453510	+	0.0632619i	0.0922684	+	0.995734i	$0.470588\pi$
$858$		0			0
$859$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$859$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$863$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$864$		0			0
$865$	−1.55679	+	0.775190i	−1.55679	+	0.775190i
$866$	1.96595			1.96595
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$	0.524354	+	0.359191i	0.524354	+	0.359191i
$873$	0.212577	+	0.176521i	0.212577	+	0.176521i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	0.0914812	+	0.0127611i	0.0914812	+	0.0127611i	0.183750	−	0.982973i	$-0.441176\pi$
$877$	0.0914812	+	0.0127611i	0.0914812	+	0.0127611i	−0.0922684	+	0.995734i	$0.529412\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−0.241393	+	0.134455i	−0.241393	+	0.134455i	−0.602635	−	0.798017i	$-0.705882\pi$
$881$	−0.241393	+	0.134455i	−0.241393	+	0.134455i	0.361242	+	0.932472i	$0.382353\pi$
$882$	0.183750	−	0.982973i	0.183750	−	0.982973i
$883$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$883$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$884$	−1.14558	−	0.709310i	−1.14558	−	0.709310i
$885$		0			0
$886$		0			0
$887$		0			0		0.948161	−	0.317791i	$-0.102941\pi$
$887$		0			0		−0.948161	+	0.317791i	$0.897059\pi$
$888$		0			0
$889$		0			0
$890$	2.02140	−	0.0934549i	2.02140	−	0.0934549i
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−0.722071	−	1.05409i	−0.722071	−	1.05409i
$899$		0			0
$900$	−0.235949	+	0.146094i	−0.235949	+	0.146094i
$901$	−1.78269	−	0.887674i	−1.78269	−	0.887674i
$902$		0			0
$903$		0			0
$904$	−0.323785	−	1.37665i	−0.323785	−	1.37665i
$905$	−0.913082			−0.913082
$906$		0			0
$907$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$907$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$908$		0			0
$909$	1.53511	+	1.15926i	1.53511	+	1.15926i
$910$		0			0
$911$		0			0		−0.873622	−	0.486604i	$-0.838235\pi$
$911$		0			0		0.873622	+	0.486604i	$0.161765\pi$
$912$		0			0
$913$		0			0
$914$	0.404479	−	0.368731i	0.404479	−	0.368731i
$915$		0			0
$916$	1.69318	−	0.156896i	1.69318	−	0.156896i
$917$		0			0
$918$		0			0
$919$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$919$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$920$		0			0
$921$		0			0
$922$	0.711414	−	1.42871i	0.711414	−	1.42871i
$923$		0			0
$924$		0			0
$925$	−0.434164	+	0.297409i	−0.434164	+	0.297409i
$926$		0			0
$927$		0			0
$928$	−1.53703	−	1.27633i	−1.53703	−	1.27633i
$929$	1.73049	+	0.963876i	1.73049	+	0.963876i	0.932472	+	0.361242i	$0.117647\pi$
$929$	1.73049	+	0.963876i	1.73049	+	0.963876i	0.798017	+	0.602635i	$0.205882\pi$
$930$		0			0
$931$		0			0
$932$	−1.23549	−	1.23549i	−1.23549	−	1.23549i
$933$		0			0
$934$		0			0
$935$		0			0
$936$	0.600584	+	1.20614i	0.600584	+	1.20614i
$937$	1.58923	+	0.147263i	1.58923	+	0.147263i	0.850217	−	0.526432i	$-0.176471\pi$
$937$	1.58923	+	0.147263i	1.58923	+	0.147263i	0.739009	+	0.673696i	$0.235294\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.26544	+	1.52391i	1.26544	+	1.52391i	0.739009	+	0.673696i	$0.235294\pi$
$941$	1.26544	+	1.52391i	1.26544	+	1.52391i	0.526432	+	0.850217i	$0.323529\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.317791	−	0.948161i	$-0.397059\pi$
$947$		0			0		−0.317791	+	0.948161i	$0.602941\pi$
$948$		0			0
$949$	−0.0795073	+	0.0957470i	−0.0795073	+	0.0957470i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.75984	+	0.328972i	−1.75984	+	0.328972i	−0.961826	−	0.273663i	$-0.911765\pi$
$953$	−1.75984	+	0.328972i	−1.75984	+	0.328972i	−0.798017	+	0.602635i	$0.794118\pi$
$954$	1.04837	+	1.69318i	1.04837	+	1.69318i
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.361242	+	0.932472i	−0.361242	+	0.932472i
$962$	1.24332	+	2.23218i	1.24332	+	2.23218i
$963$		0			0
$964$	−0.0373089	−	0.806980i	−0.0373089	−	0.806980i
$965$	1.22783	−	0.760243i	1.22783	−	0.760243i
$966$		0			0
$967$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$967$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$968$	−0.0922684	−	0.995734i	−0.0922684	−	0.995734i
$969$		0			0
$970$	0.0288162	−	0.310976i	0.0288162	−	0.310976i
$971$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$971$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$	−0.739009	+	1.67370i	−0.739009	+	1.67370i
$977$		−	1.05286i		−	1.05286i	−0.850217	−	0.526432i	$-0.823529\pi$
$977$		−	1.05286i		−	1.05286i	0.850217	−	0.526432i	$-0.176471\pi$
$978$		0			0
$979$		0			0
$980$	−1.03397	+	0.456541i	−1.03397	+	0.456541i
$981$	0.256725	+	0.581427i	0.256725	+	0.581427i
$982$		0			0
$983$		0			0		0.486604	−	0.873622i	$-0.338235\pi$
$983$		0			0		−0.486604	+	0.873622i	$0.661765\pi$
$984$		0			0
$985$	−1.84151	+	0.713403i	−1.84151	+	0.713403i
$986$	1.53703	+	1.27633i	1.53703	+	1.27633i
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		−0.998933	−	0.0461835i	$-0.985294\pi$
$991$		0			0		0.998933	+	0.0461835i	$0.0147059\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−1.21146	−	0.406040i	−1.21146	−	0.406040i	−0.361242	−	0.932472i	$-0.617647\pi$
$997$	−1.21146	−	0.406040i	−1.21146	−	0.406040i	−0.850217	+	0.526432i	$0.823529\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1156.1.o.a.47.1	✓	32
4.3	odd	2	CM	1156.1.o.a.47.1	✓	32
289.123	even	68	inner	1156.1.o.a.123.1	yes	32
1156.123	odd	68	inner	1156.1.o.a.123.1	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1156.1.o.a.47.1	✓	32	1.1	even	1	trivial
1156.1.o.a.47.1	✓	32	4.3	odd	2	CM
1156.1.o.a.123.1	yes	32	289.123	even	68	inner
1156.1.o.a.123.1	yes	32	1156.123	odd	68	inner

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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1156.1.o.a.47.1

Level	$1156$
Weight	$1$
Character	1156.47
Analytic conductor	$0.577$
Analytic rank	$0$
Dimension	$32$
Projective image	$D_{68}$
CM discriminant	-4
Inner twists	$4$