LMFDB - Embedded newform 1912.1.x.a.1725.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1912,1,Mod(501,1912)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1912, base_ring=CyclotomicField(34))

chi = DirichletCharacter(H, H._module([0, 17, 9]))

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

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

chi := DirichletCharacter("1912.501");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 1912 = 2^{3} \cdot 239 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	1912.x (of order $34$, degree $16$, 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:	$0.954212304154$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{34})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + \cdots + 1 $ x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + 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:	$D_{34}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{34} - \cdots)$

Embedding invariants

Embedding label			1725.1
Root			$0.273663 - 0.961826i$ of defining polynomial
Character	$\chi$	$=$	1912.1725
Dual form			1912.1.x.a.501.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.0922684 - 0.995734i) q^{2} +(-0.982973 - 0.183750i) q^{4} +(-0.486734 + 0.533922i) q^{7} +(-0.273663 + 0.961826i) q^{8} +(-0.850217 + 0.526432i) q^{9} +O(q^{10})$	\(q+(0.0922684 - 0.995734i) q^{2} +(-0.982973 - 0.183750i) q^{4} +(-0.486734 + 0.533922i) q^{7} +(-0.273663 + 0.961826i) q^{8} +(-0.850217 + 0.526432i) q^{9} +(0.486734 + 0.533922i) q^{14} +(0.932472 + 0.361242i) q^{16} +(-0.136374 - 0.124322i) q^{17} +(0.445738 + 0.895163i) q^{18} +(1.01267 + 1.63552i) q^{23} +(-0.982973 + 0.183750i) q^{25} +(0.576554 - 0.435393i) q^{28} +(1.45285 + 1.32445i) q^{31} +(0.445738 - 0.895163i) q^{32} +(-0.136374 + 0.124322i) q^{34} +(0.932472 - 0.361242i) q^{36} +(-1.07524 + 0.811985i) q^{41} +(1.72198 - 0.857445i) q^{46} +(1.01267 + 0.288130i) q^{47} +(0.0441059 + 0.475979i) q^{49} +(0.0922684 + 0.995734i) q^{50} +(-0.380338 - 0.614268i) q^{56} +(1.45285 - 1.32445i) q^{62} +(0.132756 - 0.710182i) q^{63} +(-0.850217 - 0.526432i) q^{64} +(0.111208 + 0.147263i) q^{68} +(-0.658809 - 0.600584i) q^{71} +(-0.273663 - 0.961826i) q^{72} +(-0.380338 + 0.981767i) q^{73} +(-0.719401 + 1.85699i) q^{79} +(0.445738 - 0.895163i) q^{81} +(0.709310 + 1.14558i) q^{82} +(-0.694903 - 1.79375i) q^{89} +(-0.694903 - 1.79375i) q^{92} +(0.380338 - 0.981767i) q^{94} +(-0.486734 - 0.533922i) q^{97} +0.478018 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} - q^{4} - q^{8} - q^{9}+O(q^{10}) $ 16 * q - q^2 - q^4 - q^8 - q^9	$ 16 q - q^{2} - q^{4} - q^{8} - q^{9} - q^{16} + 2 q^{17} - q^{18} - q^{25} + 2 q^{31} - q^{32} + 2 q^{34} - q^{36} - q^{49} - q^{50} + 2 q^{62} - q^{64} + 2 q^{68} + 2 q^{71} - q^{72} - q^{81} - 18 q^{98}+O(q^{100}) $ 16 * q - q^2 - q^4 - q^8 - q^9 - q^16 + 2 * q^17 - q^18 - q^25 + 2 * q^31 - q^32 + 2 * q^34 - q^36 - q^49 - q^50 + 2 * q^62 - q^64 + 2 * q^68 + 2 * q^71 - q^72 - q^81 - 18 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1912\mathbb{Z}\right)^\times$.

$n$	$479$	$957$	$1441$
$\chi(n)$	$1$	$-1$	$e\left(\frac{25}{34}\right)$

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 $

$2$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$2$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$3$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$3$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$4$	−0.982973	−	0.183750i	−0.982973	−	0.183750i
$5$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$5$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$6$		0			0
$7$	−0.486734	+	0.533922i	−0.486734	+	0.533922i	−0.932472	−	0.361242i	$-0.882353\pi$
$7$	−0.486734	+	0.533922i	−0.486734	+	0.533922i	0.445738	+	0.895163i	$0.352941\pi$
$8$	−0.273663	+	0.961826i	−0.273663	+	0.961826i
$9$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$10$		0			0
$11$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$11$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$12$		0			0
$13$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$13$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$14$	0.486734	+	0.533922i	0.486734	+	0.533922i
$15$		0			0
$16$	0.932472	+	0.361242i	0.932472	+	0.361242i
$17$	−0.136374	−	0.124322i	−0.136374	−	0.124322i	0.602635	−	0.798017i	$-0.294118\pi$
$17$	−0.136374	−	0.124322i	−0.136374	−	0.124322i	−0.739009	+	0.673696i	$0.764706\pi$
$18$	0.445738	+	0.895163i	0.445738	+	0.895163i
$19$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$19$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.01267	+	1.63552i	1.01267	+	1.63552i	0.739009	+	0.673696i	$0.235294\pi$
$23$	1.01267	+	1.63552i	1.01267	+	1.63552i	0.273663	+	0.961826i	$0.411765\pi$
$24$		0			0
$25$	−0.982973	+	0.183750i	−0.982973	+	0.183750i
$26$		0			0
$27$		0			0
$28$	0.576554	−	0.435393i	0.576554	−	0.435393i
$29$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$29$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$30$		0			0
$31$	1.45285	+	1.32445i	1.45285	+	1.32445i	0.850217	+	0.526432i	$0.176471\pi$
$31$	1.45285	+	1.32445i	1.45285	+	1.32445i	0.602635	+	0.798017i	$0.294118\pi$
$32$	0.445738	−	0.895163i	0.445738	−	0.895163i
$33$		0			0
$34$	−0.136374	+	0.124322i	−0.136374	+	0.124322i
$35$		0			0
$36$	0.932472	−	0.361242i	0.932472	−	0.361242i
$37$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$37$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.07524	+	0.811985i	−1.07524	+	0.811985i	−0.982973	−	0.183750i	$-0.941176\pi$
$41$	−1.07524	+	0.811985i	−1.07524	+	0.811985i	−0.0922684	+	0.995734i	$0.529412\pi$
$42$		0			0
$43$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$43$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$44$		0			0
$45$		0			0
$46$	1.72198	−	0.857445i	1.72198	−	0.857445i
$47$	1.01267	+	0.288130i	1.01267	+	0.288130i	0.739009	−	0.673696i	$-0.235294\pi$
$47$	1.01267	+	0.288130i	1.01267	+	0.288130i	0.273663	+	0.961826i	$0.411765\pi$
$48$		0			0
$49$	0.0441059	+	0.475979i	0.0441059	+	0.475979i
$50$	0.0922684	+	0.995734i	0.0922684	+	0.995734i
$51$		0			0
$52$		0			0
$53$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$53$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$54$		0			0
$55$		0			0
$56$	−0.380338	−	0.614268i	−0.380338	−	0.614268i
$57$		0			0
$58$		0			0
$59$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$59$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$60$		0			0
$61$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$61$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$62$	1.45285	−	1.32445i	1.45285	−	1.32445i
$63$	0.132756	−	0.710182i	0.132756	−	0.710182i
$64$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$65$		0			0
$66$		0			0
$67$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$67$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$68$	0.111208	+	0.147263i	0.111208	+	0.147263i
$69$		0			0
$70$		0			0
$71$	−0.658809	−	0.600584i	−0.658809	−	0.600584i	0.273663	−	0.961826i	$-0.411765\pi$
$71$	−0.658809	−	0.600584i	−0.658809	−	0.600584i	−0.932472	+	0.361242i	$0.882353\pi$
$72$	−0.273663	−	0.961826i	−0.273663	−	0.961826i
$73$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	0.602635	+	0.798017i	$0.294118\pi$
$73$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	−0.982973	+	0.183750i	$0.941176\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.719401	+	1.85699i	−0.719401	+	1.85699i	−0.273663	+	0.961826i	$0.588235\pi$
$79$	−0.719401	+	1.85699i	−0.719401	+	1.85699i	−0.445738	+	0.895163i	$0.647059\pi$
$80$		0			0
$81$	0.445738	−	0.895163i	0.445738	−	0.895163i
$82$	0.709310	+	1.14558i	0.709310	+	1.14558i
$83$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$83$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−0.694903	−	1.79375i	−0.694903	−	1.79375i	−0.602635	−	0.798017i	$-0.705882\pi$
$89$	−0.694903	−	1.79375i	−0.694903	−	1.79375i	−0.0922684	−	0.995734i	$-0.529412\pi$
$90$		0			0
$91$		0			0
$92$	−0.694903	−	1.79375i	−0.694903	−	1.79375i
$93$		0			0
$94$	0.380338	−	0.981767i	0.380338	−	0.981767i
$95$		0			0
$96$		0			0
$97$	−0.486734	−	0.533922i	−0.486734	−	0.533922i	0.445738	−	0.895163i	$-0.352941\pi$
$97$	−0.486734	−	0.533922i	−0.486734	−	0.533922i	−0.932472	+	0.361242i	$0.882353\pi$
$98$	0.478018			0.478018
$99$		0			0
$100$	1.00000			1.00000
$101$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$101$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$102$		0			0
$103$	−0.942485	−	1.52217i	−0.942485	−	1.52217i	−0.850217	−	0.526432i	$-0.823529\pi$
$103$	−0.942485	−	1.52217i	−0.942485	−	1.52217i	−0.0922684	−	0.995734i	$-0.529412\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$107$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$108$		0			0
$109$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$109$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$110$		0			0
$111$		0			0
$112$	−0.646741	+	0.322039i	−0.646741	+	0.322039i
$113$	0.404479	−	1.42160i	0.404479	−	1.42160i	−0.445738	−	0.895163i	$-0.647059\pi$
$113$	0.404479	−	1.42160i	0.404479	−	1.42160i	0.850217	−	0.526432i	$-0.176471\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$	0.132756	−	0.0123017i	0.132756	−	0.0123017i
$120$		0			0
$121$	0.932472	−	0.361242i	0.932472	−	0.361242i
$122$		0			0
$123$		0			0
$124$	−1.18475	−	1.56886i	−1.18475	−	1.56886i
$125$		0			0
$126$	−0.694903	−	0.197717i	−0.694903	−	0.197717i
$127$	0.537235	+	1.07891i	0.537235	+	1.07891i	0.982973	+	0.183750i	$0.0588235\pi$
$127$	0.537235	+	1.07891i	0.537235	+	1.07891i	−0.445738	+	0.895163i	$0.647059\pi$
$128$	−0.602635	+	0.798017i	−0.602635	+	0.798017i
$129$		0			0
$130$		0			0
$131$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$131$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$	0.156896	−	0.0971461i	0.156896	−	0.0971461i
$137$	−0.840204	+	1.35698i	−0.840204	+	1.35698i	0.0922684	+	0.995734i	$0.470588\pi$
$137$	−0.840204	+	1.35698i	−0.840204	+	1.35698i	−0.932472	+	0.361242i	$0.882353\pi$
$138$		0			0
$139$		0			0			−	1.00000i	$-0.5\pi$
$139$		0			0				1.00000i	$0.5\pi$
$140$		0			0
$141$		0			0
$142$	−0.658809	+	0.600584i	−0.658809	+	0.600584i
$143$		0			0
$144$	−0.982973	+	0.183750i	−0.982973	+	0.183750i
$145$		0			0
$146$	0.942485	+	0.469302i	0.942485	+	0.469302i
$147$		0			0
$148$		0			0
$149$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$149$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$150$		0			0
$151$	0.365931	+	0.0339085i	0.365931	+	0.0339085i	0.273663	−	0.961826i	$-0.411765\pi$
$151$	0.365931	+	0.0339085i	0.365931	+	0.0339085i	0.0922684	+	0.995734i	$0.470588\pi$
$152$		0			0
$153$	0.181395	+	0.0339085i	0.181395	+	0.0339085i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$157$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$158$	1.78269	+	0.887674i	1.78269	+	0.887674i
$159$		0			0
$160$		0			0
$161$	−1.36614	−	0.255376i	−1.36614	−	0.255376i
$162$	−0.850217	−	0.526432i	−0.850217	−	0.526432i
$163$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$163$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$164$	1.20614	−	0.600584i	1.20614	−	0.600584i
$165$		0			0
$166$		0			0
$167$	1.53511	−	0.436776i	1.53511	−	0.436776i	0.602635	−	0.798017i	$-0.294118\pi$
$167$	1.53511	−	0.436776i	1.53511	−	0.436776i	0.932472	+	0.361242i	$0.117647\pi$
$168$		0			0
$169$	−0.739009	−	0.673696i	−0.739009	−	0.673696i
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$173$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$174$		0			0
$175$	0.380338	−	0.614268i	0.380338	−	0.614268i
$176$		0			0
$177$		0			0
$178$	−1.85022	+	0.526432i	−1.85022	+	0.526432i
$179$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$179$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$180$		0			0
$181$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$181$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$182$		0			0
$183$		0			0
$184$	−1.85022	+	0.526432i	−1.85022	+	0.526432i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	−0.942485	−	0.469302i	−0.942485	−	0.469302i
$189$		0			0
$190$		0			0
$191$	0.365931	+	0.0339085i	0.365931	+	0.0339085i	0.273663	−	0.961826i	$-0.411765\pi$
$191$	0.365931	+	0.0339085i	0.365931	+	0.0339085i	0.0922684	+	0.995734i	$0.470588\pi$
$192$		0			0
$193$	0.757949	+	1.52217i	0.757949	+	1.52217i	0.850217	+	0.526432i	$0.176471\pi$
$193$	0.757949	+	1.52217i	0.757949	+	1.52217i	−0.0922684	+	0.995734i	$0.529412\pi$
$194$	−0.576554	+	0.435393i	−0.576554	+	0.435393i
$195$		0			0
$196$	0.0441059	−	0.475979i	0.0441059	−	0.475979i
$197$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$197$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$198$		0			0
$199$	−0.328972	+	1.75984i	−0.328972	+	1.75984i	0.273663	+	0.961826i	$0.411765\pi$
$199$	−0.328972	+	1.75984i	−0.328972	+	1.75984i	−0.602635	+	0.798017i	$0.705882\pi$
$200$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$	−1.60263	+	0.798017i	−1.60263	+	0.798017i
$207$	−1.72198	−	0.857445i	−1.72198	−	0.857445i
$208$		0			0
$209$		0			0
$210$		0			0
$211$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$211$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.41430	+	0.131055i	−1.41430	+	0.131055i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	0.576554	−	1.48826i	0.576554	−	1.48826i	−0.273663	−	0.961826i	$-0.588235\pi$
$223$	0.576554	−	1.48826i	0.576554	−	1.48826i	0.850217	−	0.526432i	$-0.176471\pi$
$224$	0.260991	+	0.673696i	0.260991	+	0.673696i
$225$	0.739009	−	0.673696i	0.739009	−	0.673696i
$226$	−1.37821	−	0.533922i	−1.37821	−	0.533922i
$227$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$227$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.293271	−	1.56886i	−0.293271	−	1.56886i	−0.739009	−	0.673696i	$-0.764706\pi$
$233$	−0.293271	−	1.56886i	−0.293271	−	1.56886i	0.445738	−	0.895163i	$-0.352941\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		−	0.133325i		−	0.133325i
$239$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$239$	0.0922684	−	0.995734i	0.0922684	−	0.995734i
$240$		0			0
$241$	−0.111208	+	1.20013i	−0.111208	+	1.20013i	0.739009	+	0.673696i	$0.235294\pi$
$241$	−0.111208	+	1.20013i	−0.111208	+	1.20013i	−0.850217	+	0.526432i	$0.823529\pi$
$242$	−0.273663	−	0.961826i	−0.273663	−	0.961826i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$	−1.67148	+	1.03494i	−1.67148	+	1.03494i
$249$		0			0
$250$		0			0
$251$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$251$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$252$	−0.260991	+	0.673696i	−0.260991	+	0.673696i
$253$		0			0
$254$	1.12388	−	0.435393i	1.12388	−	0.435393i
$255$		0			0
$256$	0.739009	+	0.673696i	0.739009	+	0.673696i
$257$	−0.757949	−	1.52217i	−0.757949	−	1.52217i	−0.850217	−	0.526432i	$-0.823529\pi$
$257$	−0.757949	−	1.52217i	−0.757949	−	1.52217i	0.0922684	−	0.995734i	$-0.470588\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.96595			1.96595			0.982973	−	0.183750i	$-0.0588235\pi$
$263$	1.96595			1.96595			0.982973	+	0.183750i	$0.0588235\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$269$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$270$		0			0
$271$	−0.876298	+	1.75984i	−0.876298	+	1.75984i	−0.273663	+	0.961826i	$0.588235\pi$
$271$	−0.876298	+	1.75984i	−0.876298	+	1.75984i	−0.602635	+	0.798017i	$0.705882\pi$
$272$	−0.0822551	−	0.165190i	−0.0822551	−	0.165190i
$273$		0			0
$274$	1.27366	+	0.961826i	1.27366	+	0.961826i
$275$		0			0
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$		0			0
$279$	−1.93247	−	0.361242i	−1.93247	−	0.361242i
$280$		0			0
$281$	−0.554262	+	0.895163i	−0.554262	+	0.895163i	0.445738	+	0.895163i	$0.352941\pi$
$281$	−0.554262	+	0.895163i	−0.554262	+	0.895163i	−1.00000			$\pi$
$282$		0			0
$283$		0			0		1.00000			$0$
$283$		0			0		−1.00000			$\pi$
$284$	0.537235	+	0.711414i	0.537235	+	0.711414i
$285$		0			0
$286$		0			0
$287$	0.0898203	−	0.969315i	0.0898203	−	0.969315i
$288$	0.0922684	+	0.995734i	0.0922684	+	0.995734i
$289$	−0.0891263	−	0.961826i	−0.0891263	−	0.961826i
$290$		0			0
$291$		0			0
$292$	0.554262	−	0.895163i	0.554262	−	0.895163i
$293$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$293$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$	0.0675278	−	0.361242i	0.0675278	−	0.361242i
$303$		0			0
$304$		0			0
$305$		0			0
$306$	0.0505009	−	0.177492i	0.0505009	−	0.177492i
$307$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$307$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−0.243964	−	0.857445i	−0.243964	−	0.857445i	−0.982973	−	0.183750i	$-0.941176\pi$
$311$	−0.243964	−	0.857445i	−0.243964	−	0.857445i	0.739009	−	0.673696i	$-0.235294\pi$
$312$		0			0
$313$	0.840204	+	1.35698i	0.840204	+	1.35698i	0.932472	+	0.361242i	$0.117647\pi$
$313$	0.840204	+	1.35698i	0.840204	+	1.35698i	−0.0922684	+	0.995734i	$0.529412\pi$
$314$		0			0
$315$		0			0
$316$	1.04837	−	1.69318i	1.04837	−	1.69318i
$317$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$317$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$	−0.380338	+	1.33675i	−0.380338	+	1.33675i
$323$		0			0
$324$	−0.602635	+	0.798017i	−0.602635	+	0.798017i
$325$		0			0
$326$		0			0
$327$		0			0
$328$	−0.486734	−	1.25640i	−0.486734	−	1.25640i
$329$	−0.646741	+	0.400445i	−0.646741	+	0.400445i
$330$		0			0
$331$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$331$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$332$		0			0
$333$		0			0
$334$	−0.293271	−	1.56886i	−0.293271	−	1.56886i
$335$		0			0
$336$		0			0
$337$	−1.20527			−1.20527			−0.602635	−	0.798017i	$-0.705882\pi$
$337$	−1.20527			−1.20527			−0.602635	+	0.798017i	$0.705882\pi$
$338$	−0.739009	+	0.673696i	−0.739009	+	0.673696i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−0.852157	−	0.643519i	−0.852157	−	0.643519i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$347$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$348$		0			0
$349$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$349$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$350$	−0.576554	−	0.435393i	−0.576554	−	0.435393i
$351$		0			0
$352$		0			0
$353$	1.91545	−	0.544991i	1.91545	−	0.544991i	0.932472	−	0.361242i	$-0.117647\pi$
$353$	1.91545	−	0.544991i	1.91545	−	0.544991i	0.982973	−	0.183750i	$-0.0588235\pi$
$354$		0			0
$355$		0			0
$356$	0.353470	+	1.89090i	0.353470	+	1.89090i
$357$		0			0
$358$		0			0
$359$	0.111208	+	1.20013i	0.111208	+	1.20013i	0.850217	+	0.526432i	$0.176471\pi$
$359$	0.111208	+	1.20013i	0.111208	+	1.20013i	−0.739009	+	0.673696i	$0.764706\pi$
$360$		0			0
$361$	0.982973	−	0.183750i	0.982973	−	0.183750i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$	−0.537235	+	0.711414i	−0.537235	+	0.711414i	−0.982973	−	0.183750i	$-0.941176\pi$
$367$	−0.537235	+	0.711414i	−0.537235	+	0.711414i	0.445738	+	0.895163i	$0.352941\pi$
$368$	0.353470	+	1.89090i	0.353470	+	1.89090i
$369$	0.486734	−	1.25640i	0.486734	−	1.25640i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$373$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$374$		0			0
$375$		0			0
$376$	−0.554262	+	0.895163i	−0.554262	+	0.895163i
$377$		0			0
$378$		0			0
$379$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$379$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$380$		0			0
$381$		0			0
$382$	0.0675278	−	0.361242i	0.0675278	−	0.361242i
$383$	0.876298	−	0.163808i	0.876298	−	0.163808i	0.273663	−	0.961826i	$-0.411765\pi$
$383$	0.876298	−	0.163808i	0.876298	−	0.163808i	0.602635	+	0.798017i	$0.294118\pi$
$384$		0			0
$385$		0			0
$386$	1.58561	−	0.614268i	1.58561	−	0.614268i
$387$		0			0
$388$	0.380338	+	0.614268i	0.380338	+	0.614268i
$389$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$389$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$390$		0			0
$391$	0.0652282	−	0.348940i	0.0652282	−	0.348940i
$392$	−0.469879	−	0.0878355i	−0.469879	−	0.0878355i
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$397$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$398$	1.72198	+	0.489946i	1.72198	+	0.489946i
$399$		0			0
$400$	−0.982973	−	0.183750i	−0.982973	−	0.183750i
$401$	1.44574	+	0.895163i	1.44574	+	0.895163i	1.00000			$0$
$401$	1.44574	+	0.895163i	1.44574	+	0.895163i	0.445738	+	0.895163i	$0.352941\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.404479	+	0.368731i	0.404479	+	0.368731i	0.850217	−	0.526432i	$-0.176471\pi$
$409$	0.404479	+	0.368731i	0.404479	+	0.368731i	−0.445738	+	0.895163i	$0.647059\pi$
$410$		0			0
$411$		0			0
$412$	0.646741	+	1.66943i	0.646741	+	1.66943i
$413$		0			0
$414$	−1.01267	+	1.63552i	−1.01267	+	1.63552i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$419$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$420$		0			0
$421$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$421$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$422$		0			0
$423$	−1.01267	+	0.288130i	−1.01267	+	0.288130i
$424$		0			0
$425$	0.156896	+	0.0971461i	0.156896	+	0.0971461i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.149783	+	0.526432i	−0.149783	+	0.526432i	0.850217	+	0.526432i	$0.176471\pi$
$431$	−0.149783	+	0.526432i	−0.149783	+	0.526432i	−1.00000			$\pi$
$432$		0			0
$433$	1.07524	−	0.811985i	1.07524	−	0.811985i	0.0922684	−	0.995734i	$-0.470588\pi$
$433$	1.07524	−	0.811985i	1.07524	−	0.811985i	0.982973	+	0.183750i	$0.0588235\pi$
$434$			1.42036i			1.42036i
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	−0.0505009	+	0.544991i	−0.0505009	+	0.544991i	0.932472	+	0.361242i	$0.117647\pi$
$439$	−0.0505009	+	0.544991i	−0.0505009	+	0.544991i	−0.982973	+	0.183750i	$0.941176\pi$
$440$		0			0
$441$	−0.288070	−	0.381466i	−0.288070	−	0.381466i
$442$		0			0
$443$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$443$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$444$		0			0
$445$		0			0
$446$	−1.42871	−	0.711414i	−1.42871	−	0.711414i
$447$		0			0
$448$	0.694903	−	0.197717i	0.694903	−	0.197717i
$449$	−1.42871	−	0.711414i	−1.42871	−	0.711414i	−0.445738	−	0.895163i	$-0.647059\pi$
$449$	−1.42871	−	0.711414i	−1.42871	−	0.711414i	−0.982973	+	0.183750i	$0.941176\pi$
$450$	−0.602635	−	0.798017i	−0.602635	−	0.798017i
$451$		0			0
$452$	−0.658809	+	1.32307i	−0.658809	+	1.32307i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.397365	+	0.798017i	−0.397365	+	0.798017i	0.602635	+	0.798017i	$0.294118\pi$
$457$	−0.397365	+	0.798017i	−0.397365	+	0.798017i	−1.00000			$\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$461$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$462$		0			0
$463$	−0.486734	−	1.25640i	−0.486734	−	1.25640i	−0.932472	−	0.361242i	$-0.882353\pi$
$463$	−0.486734	−	1.25640i	−0.486734	−	1.25640i	0.445738	−	0.895163i	$-0.352941\pi$
$464$		0			0
$465$		0			0
$466$	−1.58923	+	0.147263i	−1.58923	+	0.147263i
$467$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$467$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$	−0.132756	−	0.0123017i	−0.132756	−	0.0123017i
$477$		0			0
$478$	−0.982973	−	0.183750i	−0.982973	−	0.183750i
$479$	1.86494			1.86494			0.932472	−	0.361242i	$-0.117647\pi$
$479$	1.86494			1.86494			0.932472	+	0.361242i	$0.117647\pi$
$480$		0			0
$481$		0			0
$482$	1.18475	+	0.221468i	1.18475	+	0.221468i
$483$		0			0
$484$	−0.982973	+	0.183750i	−0.982973	+	0.183750i
$485$		0			0
$486$		0			0
$487$	0.465346	−	0.288130i	0.465346	−	0.288130i	−0.273663	−	0.961826i	$-0.588235\pi$
$487$	0.465346	−	0.288130i	0.465346	−	0.288130i	0.739009	+	0.673696i	$0.235294\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$491$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	0.876298	+	1.75984i	0.876298	+	1.75984i
$497$	0.641330	−	0.0594279i	0.641330	−	0.0594279i
$498$		0			0
$499$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$499$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	1.67148	−	0.312454i	1.67148	−	0.312454i	0.739009	−	0.673696i	$-0.235294\pi$
$503$	1.67148	−	0.312454i	1.67148	−	0.312454i	0.932472	+	0.361242i	$0.117647\pi$
$504$	0.646741	+	0.322039i	0.646741	+	0.322039i
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−0.329838	−	1.15926i	−0.329838	−	1.15926i
$509$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$509$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$510$		0			0
$511$	−0.339063	−	0.680930i	−0.339063	−	0.680930i
$512$	0.739009	−	0.673696i	0.739009	−	0.673696i
$513$		0			0
$514$	−1.58561	+	0.614268i	−1.58561	+	0.614268i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.04837	+	0.0971461i	1.04837	+	0.0971461i	0.602635	−	0.798017i	$-0.294118\pi$
$521$	1.04837	+	0.0971461i	1.04837	+	0.0971461i	0.445738	+	0.895163i	$0.352941\pi$
$522$		0			0
$523$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$523$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$524$		0			0
$525$		0			0
$526$	0.181395	−	1.95756i	0.181395	−	1.95756i
$527$	−0.0334740	−	0.361242i	−0.0334740	−	0.361242i
$528$		0			0
$529$	−1.20369	+	2.41733i	−1.20369	+	2.41733i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$541$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$542$	1.67148	+	1.03494i	1.67148	+	1.03494i
$543$		0			0
$544$	−0.172075	+	0.0666624i	−0.172075	+	0.0666624i
$545$		0			0
$546$		0			0
$547$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$547$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$548$	1.07524	−	1.17948i	1.07524	−	1.17948i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−0.641330	−	1.28796i	−0.641330	−	1.28796i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$557$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$558$	−0.538007	+	1.89090i	−0.538007	+	1.89090i
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.840204	+	0.634493i	0.840204	+	0.634493i
$563$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$563$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	0.260991	+	0.673696i	0.260991	+	0.673696i
$568$	0.757949	−	0.469302i	0.757949	−	0.469302i
$569$	−0.831277	−	1.66943i	−0.831277	−	1.66943i	−0.739009	−	0.673696i	$-0.764706\pi$
$569$	−0.831277	−	1.66943i	−0.831277	−	1.66943i	−0.0922684	−	0.995734i	$-0.529412\pi$
$570$		0			0
$571$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$571$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$572$		0			0
$573$		0			0
$574$	−0.956893	−	0.178874i	−0.956893	−	0.178874i
$575$	−1.29596	−	1.42160i	−1.29596	−	1.42160i
$576$	1.00000			1.00000
$577$	1.37821	−	1.25640i	1.37821	−	1.25640i	0.445738	−	0.895163i	$-0.352941\pi$
$577$	1.37821	−	1.25640i	1.37821	−	1.25640i	0.932472	−	0.361242i	$-0.117647\pi$
$578$	−0.965946			−0.965946
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$	−0.840204	−	0.634493i	−0.840204	−	0.634493i
$585$		0			0
$586$		0			0
$587$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$587$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	1.58923	−	1.20013i	1.58923	−	1.20013i	0.739009	−	0.673696i	$-0.235294\pi$
$593$	1.58923	−	1.20013i	1.58923	−	1.20013i	0.850217	−	0.526432i	$-0.176471\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	0.172075	−	0.0666624i	0.172075	−	0.0666624i	−0.273663	−	0.961826i	$-0.588235\pi$
$599$	0.172075	−	0.0666624i	0.172075	−	0.0666624i	0.445738	+	0.895163i	$0.352941\pi$
$600$		0			0
$601$	1.42871	+	1.07891i	1.42871	+	1.07891i	0.982973	+	0.183750i	$0.0588235\pi$
$601$	1.42871	+	1.07891i	1.42871	+	1.07891i	0.445738	+	0.895163i	$0.352941\pi$
$602$		0			0
$603$		0			0
$604$	−0.353470	−	0.100571i	−0.353470	−	0.100571i
$605$		0			0
$606$		0			0
$607$	−0.247582	−	1.32445i	−0.247582	−	1.32445i	−0.850217	−	0.526432i	$-0.823529\pi$
$607$	−0.247582	−	1.32445i	−0.247582	−	1.32445i	0.602635	−	0.798017i	$-0.294118\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−0.172075	−	0.0666624i	−0.172075	−	0.0666624i
$613$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$613$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.34739i		−	1.34739i	−0.739009	−	0.673696i	$-0.764706\pi$
$617$		−	1.34739i		−	1.34739i	0.739009	−	0.673696i	$-0.235294\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		0			0
$622$	−0.876298	+	0.163808i	−0.876298	+	0.163808i
$623$	1.29596	+	0.502056i	1.29596	+	0.502056i
$624$		0			0
$625$	0.932472	−	0.361242i	0.932472	−	0.361242i
$626$	1.42871	−	0.711414i	1.42871	−	0.711414i
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−1.45285	−	0.271585i	−1.45285	−	0.271585i	−0.602635	−	0.798017i	$-0.705882\pi$
$631$	−1.45285	−	0.271585i	−1.45285	−	0.271585i	−0.850217	+	0.526432i	$0.823529\pi$
$632$	−1.58923	−	1.20013i	−1.58923	−	1.20013i
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	0.876298	+	0.163808i	0.876298	+	0.163808i
$640$		0			0
$641$	0.0505009	−	0.544991i	0.0505009	−	0.544991i	−0.932472	−	0.361242i	$-0.882353\pi$
$641$	0.0505009	−	0.544991i	0.0505009	−	0.544991i	0.982973	−	0.183750i	$-0.0588235\pi$
$642$		0			0
$643$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$643$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$644$	1.29596	+	0.502056i	1.29596	+	0.502056i
$645$		0			0
$646$		0			0
$647$	0.658809	+	0.600584i	0.658809	+	0.600584i	0.932472	−	0.361242i	$-0.117647\pi$
$647$	0.658809	+	0.600584i	0.658809	+	0.600584i	−0.273663	+	0.961826i	$0.588235\pi$
$648$	0.739009	+	0.673696i	0.739009	+	0.673696i
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$653$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$654$		0			0
$655$		0			0
$656$	−1.29596	+	0.368731i	−1.29596	+	0.368731i
$657$	−0.193463	−	1.03494i	−0.193463	−	1.03494i
$658$	0.339063	+	0.680930i	0.339063	+	0.680930i
$659$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$659$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$660$		0			0
$661$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$661$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−1.58923	+	0.147263i	−1.58923	+	0.147263i
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$	−0.111208	+	1.20013i	−0.111208	+	1.20013i
$675$		0			0
$676$	0.602635	+	0.798017i	0.602635	+	0.798017i
$677$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$677$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$678$		0			0
$679$	0.521982			0.521982
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$683$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$684$		0			0
$685$		0			0
$686$	−0.719401	+	0.789146i	−0.719401	+	0.789146i
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$691$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.247582	+	0.0229419i	0.247582	+	0.0229419i
$698$		0			0
$699$		0			0
$700$	−0.486734	+	0.533922i	−0.486734	+	0.533922i
$701$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$701$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	−0.365931	−	1.95756i	−0.365931	−	1.95756i
$707$		0			0
$708$		0			0
$709$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$709$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$710$		0			0
$711$	−0.365931	−	1.95756i	−0.365931	−	1.95756i
$712$	1.91545	−	0.177492i	1.91545	−	0.177492i
$713$	−0.694903	+	3.71740i	−0.694903	+	3.71740i
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$	1.20527			1.20527
$719$	0.156896	−	1.69318i	0.156896	−	1.69318i	−0.445738	−	0.895163i	$-0.647059\pi$
$719$	0.156896	−	1.69318i	0.156896	−	1.69318i	0.602635	−	0.798017i	$-0.294118\pi$
$720$		0			0
$721$	1.27146	+	0.237677i	1.27146	+	0.237677i
$722$	−0.0922684	−	0.995734i	−0.0922684	−	0.995734i
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$	1.70043			1.70043			0.850217	−	0.526432i	$-0.176471\pi$
$727$	1.70043			1.70043			0.850217	+	0.526432i	$0.176471\pi$
$728$		0			0
$729$	0.0922684	+	0.995734i	0.0922684	+	0.995734i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$733$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$734$	0.658809	+	0.600584i	0.658809	+	0.600584i
$735$		0			0
$736$	1.91545	−	0.177492i	1.91545	−	0.177492i
$737$		0			0
$738$	−1.20614	−	0.600584i	−1.20614	−	0.600584i
$739$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$739$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.42871	−	0.711414i	−1.42871	−	0.711414i	−0.445738	−	0.895163i	$-0.647059\pi$
$743$	−1.42871	−	0.711414i	−1.42871	−	0.711414i	−0.982973	+	0.183750i	$0.941176\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−1.45285	+	1.32445i	−1.45285	+	1.32445i	−0.602635	+	0.798017i	$0.705882\pi$
$751$	−1.45285	+	1.32445i	−1.45285	+	1.32445i	−0.850217	+	0.526432i	$0.823529\pi$
$752$	0.840204	+	0.634493i	0.840204	+	0.634493i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$757$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.70043			1.70043			0.850217	−	0.526432i	$-0.176471\pi$
$761$	1.70043			1.70043			0.850217	+	0.526432i	$0.176471\pi$
$762$		0			0
$763$		0			0
$764$	−0.353470	−	0.100571i	−0.353470	−	0.100571i
$765$		0			0
$766$	−0.0822551	−	0.887674i	−0.0822551	−	0.887674i
$767$		0			0
$768$		0			0
$769$	−0.193463	+	0.312454i	−0.193463	+	0.312454i	−0.932472	−	0.361242i	$-0.882353\pi$
$769$	−0.193463	+	0.312454i	−0.193463	+	0.312454i	0.739009	+	0.673696i	$0.235294\pi$
$770$		0			0
$771$		0			0
$772$	−0.465346	−	1.63552i	−0.465346	−	1.63552i
$773$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$773$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$774$		0			0
$775$	−1.67148	−	1.03494i	−1.67148	−	1.03494i
$776$	0.646741	−	0.322039i	0.646741	−	0.322039i
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$	−0.341433	−	0.0971461i	−0.341433	−	0.0971461i
$783$		0			0
$784$	−0.130816	+	0.459770i	−0.130816	+	0.459770i
$785$		0			0
$786$		0			0
$787$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$787$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	0.562147	+	0.907899i	0.562147	+	0.907899i
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	0.646741	−	1.66943i	0.646741	−	1.66943i
$797$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$797$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$798$		0			0
$799$	−0.102282	−	0.165190i	−0.102282	−	0.165190i
$800$	−0.273663	+	0.961826i	−0.273663	+	0.961826i
$801$	1.53511	+	1.15926i	1.53511	+	1.15926i
$802$	1.02474	−	1.35698i	1.02474	−	1.35698i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.719401	+	1.85699i	0.719401	+	1.85699i	0.445738	+	0.895163i	$0.352941\pi$
$809$	0.719401	+	1.85699i	0.719401	+	1.85699i	0.273663	+	0.961826i	$0.411765\pi$
$810$		0			0
$811$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$811$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$		0			0
$818$	0.404479	−	0.368731i	0.404479	−	0.368731i
$819$		0			0
$820$		0			0
$821$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$821$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$822$		0			0
$823$	1.53511	+	1.15926i	1.53511	+	1.15926i	0.932472	+	0.361242i	$0.117647\pi$
$823$	1.53511	+	1.15926i	1.53511	+	1.15926i	0.602635	+	0.798017i	$0.294118\pi$
$824$	1.72198	−	0.489946i	1.72198	−	0.489946i
$825$		0			0
$826$		0			0
$827$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$827$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$828$	1.53511	+	1.15926i	1.53511	+	1.15926i
$829$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$829$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	0.0531595	−	0.0703946i	0.0531595	−	0.0703946i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−0.181395	+	0.0339085i	−0.181395	+	0.0339085i	−0.273663	−	0.961826i	$-0.588235\pi$
$839$	−0.181395	+	0.0339085i	−0.181395	+	0.0339085i	0.0922684	+	0.995734i	$0.470588\pi$
$840$		0			0
$841$	−0.602635	−	0.798017i	−0.602635	−	0.798017i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		0			0
$846$	0.193463	+	1.03494i	0.193463	+	1.03494i
$847$	−0.260991	+	0.673696i	−0.260991	+	0.673696i
$848$		0			0
$849$		0			0
$850$	0.111208	−	0.147263i	0.111208	−	0.147263i
$851$		0			0
$852$		0			0
$853$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$853$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.29596	+	1.42160i	1.29596	+	1.42160i	0.850217	+	0.526432i	$0.176471\pi$
$857$	1.29596	+	1.42160i	1.29596	+	1.42160i	0.445738	+	0.895163i	$0.352941\pi$
$858$		0			0
$859$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$859$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$860$		0			0
$861$		0			0
$862$	0.510366	+	0.197717i	0.510366	+	0.197717i
$863$	−0.942485	−	0.469302i	−0.942485	−	0.469302i	−0.0922684	−	0.995734i	$-0.529412\pi$
$863$	−0.942485	−	0.469302i	−0.942485	−	0.469302i	−0.850217	+	0.526432i	$0.823529\pi$
$864$		0			0
$865$		0			0
$866$	−0.709310	−	1.14558i	−0.709310	−	1.14558i
$867$		0			0
$868$	1.41430	+	0.131055i	1.41430	+	0.131055i
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.694903	+	0.197717i	0.694903	+	0.197717i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$877$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$878$	0.538007	+	0.100571i	0.538007	+	0.100571i
$879$		0			0
$880$		0			0
$881$	0.646741	−	0.322039i	0.646741	−	0.322039i	−0.0922684	−	0.995734i	$-0.529412\pi$
$881$	0.646741	−	0.322039i	0.646741	−	0.322039i	0.739009	+	0.673696i	$0.235294\pi$
$882$	−0.406419	+	0.251644i	−0.406419	+	0.251644i
$883$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$883$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.136374	+	0.124322i	0.136374	+	0.124322i	0.739009	−	0.673696i	$-0.235294\pi$
$887$	0.136374	+	0.124322i	0.136374	+	0.124322i	−0.602635	+	0.798017i	$0.705882\pi$
$888$		0			0
$889$	−0.837545	−	0.238302i	−0.837545	−	0.238302i
$890$		0			0
$891$		0			0
$892$	−0.840204	+	1.35698i	−0.840204	+	1.35698i
$893$		0			0
$894$		0			0
$895$		0			0
$896$	−0.132756	−	0.710182i	−0.132756	−	0.710182i
$897$		0			0
$898$	−0.840204	+	1.35698i	−0.840204	+	1.35698i
$899$		0			0
$900$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	1.25664	+	0.778076i	1.25664	+	0.778076i
$905$		0			0
$906$		0			0
$907$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$907$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.07524	+	0.811985i	−1.07524	+	0.811985i	−0.982973	−	0.183750i	$-0.941176\pi$
$911$	−1.07524	+	0.811985i	−1.07524	+	0.811985i	−0.0922684	+	0.995734i	$0.529412\pi$
$912$		0			0
$913$		0			0
$914$	0.757949	+	0.469302i	0.757949	+	0.469302i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−0.890705	−	1.17948i	−0.890705	−	1.17948i	−0.982973	−	0.183750i	$-0.941176\pi$
$919$	−0.890705	−	1.17948i	−0.890705	−	1.17948i	0.0922684	−	0.995734i	$-0.470588\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$	−1.29596	+	0.368731i	−1.29596	+	0.368731i
$927$	1.60263	+	0.798017i	1.60263	+	0.798017i
$928$		0			0
$929$	−0.247582	+	0.271585i	−0.247582	+	0.271585i	−0.850217	−	0.526432i	$-0.823529\pi$
$929$	−0.247582	+	0.271585i	−0.247582	+	0.271585i	0.602635	+	0.798017i	$0.294118\pi$
$930$		0			0
$931$		0			0
$932$			1.59603i			1.59603i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.172075	−	1.85699i	−0.172075	−	1.85699i	−0.445738	−	0.895163i	$-0.647059\pi$
$937$	−0.172075	−	1.85699i	−0.172075	−	1.85699i	0.273663	−	0.961826i	$-0.411765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$941$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$942$		0			0
$943$	−2.41689	−	0.936306i	−2.41689	−	0.936306i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$947$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$	−0.0244984	+	0.131055i	−0.0244984	+	0.131055i
$953$	−1.85022	+	0.526432i	−1.85022	+	0.526432i	−0.850217	+	0.526432i	$0.823529\pi$
$953$	−1.85022	+	0.526432i	−1.85022	+	0.526432i	−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$	−0.273663	+	0.961826i	−0.273663	+	0.961826i
$957$		0			0
$958$	0.172075	−	1.85699i	0.172075	−	1.85699i
$959$	−0.315563	−	1.10909i	−0.315563	−	1.10909i
$960$		0			0
$961$	0.264344	+	2.85272i	0.264344	+	2.85272i
$962$		0			0
$963$		0			0
$964$	0.329838	−	1.15926i	0.329838	−	1.15926i
$965$		0			0
$966$		0			0
$967$	0.876298	−	0.163808i	0.876298	−	0.163808i	0.273663	−	0.961826i	$-0.411765\pi$
$967$	0.876298	−	0.163808i	0.876298	−	0.163808i	0.602635	+	0.798017i	$0.294118\pi$
$968$	0.0922684	+	0.995734i	0.0922684	+	0.995734i
$969$		0			0
$970$		0			0
$971$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$971$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$972$		0			0
$973$		0			0
$974$	−0.243964	−	0.489946i	−0.243964	−	0.489946i
$975$		0			0
$976$		0			0
$977$	−1.20614	−	0.600584i	−1.20614	−	0.600584i	−0.273663	−	0.961826i	$-0.588235\pi$
$977$	−1.20614	−	0.600584i	−1.20614	−	0.600584i	−0.932472	+	0.361242i	$0.882353\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.09227	−	0.995734i	−1.09227	−	0.995734i	−0.0922684	−	0.995734i	$-0.529412\pi$
$983$	−1.09227	−	0.995734i	−1.09227	−	0.995734i	−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−0.293271	−	0.221468i	−0.293271	−	0.221468i	0.445738	−	0.895163i	$-0.352941\pi$
$991$	−0.293271	−	0.221468i	−0.293271	−	0.221468i	−0.739009	+	0.673696i	$0.764706\pi$
$992$	1.83319	−	0.710182i	1.83319	−	0.710182i
$993$		0			0
$994$		−	0.644077i		−	0.644077i
$995$		0			0
$996$		0			0
$997$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$997$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1912.1.x.a.1725.1	yes	16
8.5	even	2	RM	1912.1.x.a.1725.1	yes	16
239.23	odd	34	inner	1912.1.x.a.501.1	✓	16
1912.501	odd	34	inner	1912.1.x.a.501.1	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1912.1.x.a.501.1	✓	16	239.23	odd	34	inner
1912.1.x.a.501.1	✓	16	1912.501	odd	34	inner
1912.1.x.a.1725.1	yes	16	1.1	even	1	trivial
1912.1.x.a.1725.1	yes	16	8.5	even	2	RM

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:	\( N \)	\(=\)	\( 1912 = 2^{3} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	1912.x (of order \(34\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.0922684 - 0.995734i) q^{2} +(-0.982973 - 0.183750i) q^{4} +(-0.486734 + 0.533922i) q^{7} +(-0.273663 + 0.961826i) q^{8} +(-0.850217 + 0.526432i) q^{9} +O(q^{10})\)	\(q+(0.0922684 - 0.995734i) q^{2} +(-0.982973 - 0.183750i) q^{4} +(-0.486734 + 0.533922i) q^{7} +(-0.273663 + 0.961826i) q^{8} +(-0.850217 + 0.526432i) q^{9} +(0.486734 + 0.533922i) q^{14} +(0.932472 + 0.361242i) q^{16} +(-0.136374 - 0.124322i) q^{17} +(0.445738 + 0.895163i) q^{18} +(1.01267 + 1.63552i) q^{23} +(-0.982973 + 0.183750i) q^{25} +(0.576554 - 0.435393i) q^{28} +(1.45285 + 1.32445i) q^{31} +(0.445738 - 0.895163i) q^{32} +(-0.136374 + 0.124322i) q^{34} +(0.932472 - 0.361242i) q^{36} +(-1.07524 + 0.811985i) q^{41} +(1.72198 - 0.857445i) q^{46} +(1.01267 + 0.288130i) q^{47} +(0.0441059 + 0.475979i) q^{49} +(0.0922684 + 0.995734i) q^{50} +(-0.380338 - 0.614268i) q^{56} +(1.45285 - 1.32445i) q^{62} +(0.132756 - 0.710182i) q^{63} +(-0.850217 - 0.526432i) q^{64} +(0.111208 + 0.147263i) q^{68} +(-0.658809 - 0.600584i) q^{71} +(-0.273663 - 0.961826i) q^{72} +(-0.380338 + 0.981767i) q^{73} +(-0.719401 + 1.85699i) q^{79} +(0.445738 - 0.895163i) q^{81} +(0.709310 + 1.14558i) q^{82} +(-0.694903 - 1.79375i) q^{89} +(-0.694903 - 1.79375i) q^{92} +(0.380338 - 0.981767i) q^{94} +(-0.486734 - 0.533922i) q^{97} +0.478018 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} - q^{4} - q^{8} - q^{9}+O(q^{10}) \) 16 * q - q^2 - q^4 - q^8 - q^9	\( 16 q - q^{2} - q^{4} - q^{8} - q^{9} - q^{16} + 2 q^{17} - q^{18} - q^{25} + 2 q^{31} - q^{32} + 2 q^{34} - q^{36} - q^{49} - q^{50} + 2 q^{62} - q^{64} + 2 q^{68} + 2 q^{71} - q^{72} - q^{81} - 18 q^{98}+O(q^{100}) \) 16 * q - q^2 - q^4 - q^8 - q^9 - q^16 + 2 * q^17 - q^18 - q^25 + 2 * q^31 - q^32 + 2 * q^34 - q^36 - q^49 - q^50 + 2 * q^62 - q^64 + 2 * q^68 + 2 * q^71 - q^72 - q^81 - 18 * q^98

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