LMFDB - Embedded newform 927.1.v.a.145.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [927,1,Mod(10,927)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(927, base_ring=CyclotomicField(34))

chi = DirichletCharacter(H, H._module([0, 15]))

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

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

chi := DirichletCharacter("927.10");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 927 = 3^{2} \cdot 103 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	927.v (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:	$0.462633266711$
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:	$\Z[a_1, \ldots, a_{4}]$
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			145.1
Root			$0.850217 - 0.526432i$ of defining polynomial
Character	$\chi$	$=$	927.145
Dual form			927.1.v.a.748.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.982973 + 0.183750i) q^{4} +(-0.404479 - 1.42160i) q^{7} +(-0.538007 - 1.89090i) q^{13} +(0.932472 + 0.361242i) q^{16} +(0.658809 + 0.600584i) q^{19} +(-0.602635 + 0.798017i) q^{25} +(-0.136374 - 1.47171i) q^{28} +(-0.380338 + 0.981767i) q^{31} +(1.01267 + 1.63552i) q^{37} +(-0.840204 + 1.35698i) q^{43} +(-1.00711 + 0.623578i) q^{49} +(-0.181395 - 1.95756i) q^{52} +(1.12388 - 1.48826i) q^{61} +(0.850217 + 0.526432i) q^{64} +(-1.91545 - 0.544991i) q^{67} +(0.646741 + 0.322039i) q^{73} +(0.537235 + 0.711414i) q^{76} +(0.243964 + 0.489946i) q^{79} +(-2.47048 + 1.52966i) q^{91} +(-0.726337 - 0.961826i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{4} - 2 q^{7} + 2 q^{13} - q^{16} - 2 q^{19} - q^{25} + 2 q^{28} - 3 q^{49} - 2 q^{52} + 2 q^{61} + q^{64} + 2 q^{76} + 2 q^{79} - 13 q^{91} - 15 q^{97}+O(q^{100}) $ 16 * q + q^4 - 2 * q^7 + 2 * q^13 - q^16 - 2 * q^19 - q^25 + 2 * q^28 - 3 * q^49 - 2 * q^52 + 2 * q^61 + q^64 + 2 * q^76 + 2 * q^79 - 13 * q^91 - 15 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/927\mathbb{Z}\right)^\times$.

$n$	$722$	$829$
$\chi(n)$	$1$	$e\left(\frac{29}{34}\right)$

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			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$2$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$3$		0			0
$3$		0			0
$4$	0.982973	+	0.183750i	0.982973	+	0.183750i
$5$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$5$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$6$		0			0
$7$	−0.404479	−	1.42160i	−0.404479	−	1.42160i	−0.850217	−	0.526432i	$-0.823529\pi$
$7$	−0.404479	−	1.42160i	−0.404479	−	1.42160i	0.445738	−	0.895163i	$-0.352941\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$11$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$12$		0			0
$13$	−0.538007	−	1.89090i	−0.538007	−	1.89090i	−0.445738	−	0.895163i	$-0.647059\pi$
$13$	−0.538007	−	1.89090i	−0.538007	−	1.89090i	−0.0922684	−	0.995734i	$-0.529412\pi$
$14$		0			0
$15$		0			0
$16$	0.932472	+	0.361242i	0.932472	+	0.361242i
$17$		0			0		−0.798017	−	0.602635i	$-0.794118\pi$
$17$		0			0		0.798017	+	0.602635i	$0.205882\pi$
$18$		0			0
$19$	0.658809	+	0.600584i	0.658809	+	0.600584i	0.932472	−	0.361242i	$-0.117647\pi$
$19$	0.658809	+	0.600584i	0.658809	+	0.600584i	−0.273663	+	0.961826i	$0.588235\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$23$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$24$		0			0
$25$	−0.602635	+	0.798017i	−0.602635	+	0.798017i
$26$		0			0
$27$		0			0
$28$	−0.136374	−	1.47171i	−0.136374	−	1.47171i
$29$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$29$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$30$		0			0
$31$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	0.602635	+	0.798017i	$0.294118\pi$
$31$	−0.380338	+	0.981767i	−0.380338	+	0.981767i	−0.982973	+	0.183750i	$0.941176\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	1.01267	+	1.63552i	1.01267	+	1.63552i	0.739009	+	0.673696i	$0.235294\pi$
$37$	1.01267	+	1.63552i	1.01267	+	1.63552i	0.273663	+	0.961826i	$0.411765\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$41$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$42$		0			0
$43$	−0.840204	+	1.35698i	−0.840204	+	1.35698i	0.0922684	+	0.995734i	$0.470588\pi$
$43$	−0.840204	+	1.35698i	−0.840204	+	1.35698i	−0.932472	+	0.361242i	$0.882353\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$		0			0
$49$	−1.00711	+	0.623578i	−1.00711	+	0.623578i
$50$		0			0
$51$		0			0
$52$	−0.181395	−	1.95756i	−0.181395	−	1.95756i
$53$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$53$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$59$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$60$		0			0
$61$	1.12388	−	1.48826i	1.12388	−	1.48826i	0.273663	−	0.961826i	$-0.411765\pi$
$61$	1.12388	−	1.48826i	1.12388	−	1.48826i	0.850217	−	0.526432i	$-0.176471\pi$
$62$		0			0
$63$		0			0
$64$	0.850217	+	0.526432i	0.850217	+	0.526432i
$65$		0			0
$66$		0			0
$67$	−1.91545	−	0.544991i	−1.91545	−	0.544991i	−0.982973	−	0.183750i	$-0.941176\pi$
$67$	−1.91545	−	0.544991i	−1.91545	−	0.544991i	−0.932472	−	0.361242i	$-0.882353\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.445738	−	0.895163i	$-0.352941\pi$
$71$		0			0		−0.445738	+	0.895163i	$0.647059\pi$
$72$		0			0
$73$	0.646741	+	0.322039i	0.646741	+	0.322039i	0.739009	−	0.673696i	$-0.235294\pi$
$73$	0.646741	+	0.322039i	0.646741	+	0.322039i	−0.0922684	+	0.995734i	$0.529412\pi$
$74$		0			0
$75$		0			0
$76$	0.537235	+	0.711414i	0.537235	+	0.711414i
$77$		0			0
$78$		0			0
$79$	0.243964	+	0.489946i	0.243964	+	0.489946i	0.982973	−	0.183750i	$-0.0588235\pi$
$79$	0.243964	+	0.489946i	0.243964	+	0.489946i	−0.739009	+	0.673696i	$0.764706\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$83$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$89$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$90$		0			0
$91$	−2.47048	+	1.52966i	−2.47048	+	1.52966i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.726337	−	0.961826i	−0.726337	−	0.961826i	0.273663	−	0.961826i	$-0.411765\pi$
$97$	−0.726337	−	0.961826i	−0.726337	−	0.961826i	−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$	−0.739009	+	0.673696i	−0.739009	+	0.673696i
$101$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$101$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$102$		0			0
$103$	0.850217	−	0.526432i	0.850217	−	0.526432i
$103$	0.850217	−	0.526432i	0.850217	−	0.526432i
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.183750	−	0.982973i	$-0.441176\pi$
$107$		0			0		−0.183750	+	0.982973i	$0.558824\pi$
$108$		0			0
$109$	0.576554	−	0.435393i	0.576554	−	0.435393i	−0.273663	−	0.961826i	$-0.588235\pi$
$109$	0.576554	−	0.435393i	0.576554	−	0.435393i	0.850217	+	0.526432i	$0.176471\pi$
$110$		0			0
$111$		0			0
$112$	0.136374	−	1.47171i	0.136374	−	1.47171i
$113$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$113$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.982973	−	0.183750i	−0.982973	−	0.183750i
$122$		0			0
$123$		0			0
$124$	−0.554262	+	0.895163i	−0.554262	+	0.895163i
$125$		0			0
$126$		0			0
$127$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.850217	+	0.526432i	$0.823529\pi$
$127$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.932472	+	0.361242i	$0.882353\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$131$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$132$		0			0
$133$	0.587313	−	1.17948i	0.587313	−	1.17948i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$137$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$138$		0			0
$139$	−0.465346	+	1.63552i	−0.465346	+	1.63552i	0.273663	+	0.961826i	$0.411765\pi$
$139$	−0.465346	+	1.63552i	−0.465346	+	1.63552i	−0.739009	+	0.673696i	$0.764706\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$	0.694903	+	1.79375i	0.694903	+	1.79375i
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$		0			0
$151$	−0.646741	−	1.66943i	−0.646741	−	1.66943i	−0.739009	−	0.673696i	$-0.764706\pi$
$151$	−0.646741	−	1.66943i	−0.646741	−	1.66943i	0.0922684	−	0.995734i	$-0.470588\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.942485	+	1.52217i	0.942485	+	1.52217i	0.850217	+	0.526432i	$0.176471\pi$
$157$	0.942485	+	1.52217i	0.942485	+	1.52217i	0.0922684	+	0.995734i	$0.470588\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	0.156896	+	0.0971461i	0.156896	+	0.0971461i	0.602635	−	0.798017i	$-0.294118\pi$
$163$	0.156896	+	0.0971461i	0.156896	+	0.0971461i	−0.445738	+	0.895163i	$0.647059\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.526432	−	0.850217i	$-0.323529\pi$
$167$		0			0		−0.526432	+	0.850217i	$0.676471\pi$
$168$		0			0
$169$	−2.43582	+	1.50820i	−2.43582	+	1.50820i
$170$		0			0
$171$		0			0
$172$	−1.07524	+	1.17948i	−1.07524	+	1.17948i
$173$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$173$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$174$		0			0
$175$	1.37821	+	0.533922i	1.37821	+	0.533922i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$179$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$180$		0			0
$181$	0.576554	+	0.435393i	0.576554	+	0.435393i	0.850217	−	0.526432i	$-0.176471\pi$
$181$	0.576554	+	0.435393i	0.576554	+	0.435393i	−0.273663	+	0.961826i	$0.588235\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$191$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$192$		0			0
$193$	1.53511	−	0.436776i	1.53511	−	0.436776i	0.602635	−	0.798017i	$-0.294118\pi$
$193$	1.53511	−	0.436776i	1.53511	−	0.436776i	0.932472	+	0.361242i	$0.117647\pi$
$194$		0			0
$195$		0			0
$196$	−1.10455	+	0.427904i	−1.10455	+	0.427904i
$197$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$197$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$198$		0			0
$199$	−1.53511	+	0.436776i	−1.53511	+	0.436776i	−0.932472	−	0.361242i	$-0.882353\pi$
$199$	−1.53511	+	0.436776i	−1.53511	+	0.436776i	−0.602635	+	0.798017i	$0.705882\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	0.181395	−	1.95756i	0.181395	−	1.95756i
$209$		0			0
$210$		0			0
$211$	0.328972	−	0.163808i	0.328972	−	0.163808i	−0.273663	−	0.961826i	$-0.588235\pi$
$211$	0.328972	−	0.163808i	0.328972	−	0.163808i	0.602635	+	0.798017i	$0.294118\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	1.54951	+	0.143584i	1.54951	+	0.143584i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−0.111208	+	0.147263i	−0.111208	+	0.147263i	−0.850217	−	0.526432i	$-0.823529\pi$
$223$	−0.111208	+	0.147263i	−0.111208	+	0.147263i	0.739009	+	0.673696i	$0.235294\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$227$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$228$		0			0
$229$	−0.156896	−	1.69318i	−0.156896	−	1.69318i	−0.602635	−	0.798017i	$-0.705882\pi$
$229$	−0.156896	−	1.69318i	−0.156896	−	1.69318i	0.445738	−	0.895163i	$-0.352941\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$233$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$239$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$240$		0			0
$241$	−1.34164	−	1.47171i	−1.34164	−	1.47171i	−0.739009	−	0.673696i	$-0.764706\pi$
$241$	−1.34164	−	1.47171i	−1.34164	−	1.47171i	−0.602635	−	0.798017i	$-0.705882\pi$
$242$		0			0
$243$		0			0
$244$	1.37821	−	1.25640i	1.37821	−	1.25640i
$245$		0			0
$246$		0			0
$247$	0.781199	−	1.56886i	0.781199	−	1.56886i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$251$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.739009	+	0.673696i	0.739009	+	0.673696i
$257$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$257$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$258$		0			0
$259$	1.91545	−	2.10114i	1.91545	−	2.10114i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.78269	−	0.887674i	−1.78269	−	0.887674i
$269$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$269$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$270$		0			0
$271$	−1.29596	−	1.42160i	−1.29596	−	1.42160i	−0.850217	−	0.526432i	$-0.823529\pi$
$271$	−1.29596	−	1.42160i	−1.29596	−	1.42160i	−0.445738	−	0.895163i	$-0.647059\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.60263	+	0.798017i	1.60263	+	0.798017i	1.00000			$0$
$277$	1.60263	+	0.798017i	1.60263	+	0.798017i	0.602635	+	0.798017i	$0.294118\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$281$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$282$		0			0
$283$	0.0675278	+	0.361242i	0.0675278	+	0.361242i	1.00000			$0$
$283$	0.0675278	+	0.361242i	0.0675278	+	0.361242i	−0.932472	+	0.361242i	$0.882353\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	0.273663	+	0.961826i	0.273663	+	0.961826i
$290$		0			0
$291$		0			0
$292$	0.576554	+	0.435393i	0.576554	+	0.435393i
$293$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$293$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	2.26892	+	0.645562i	2.26892	+	0.645562i
$302$		0			0
$303$		0			0
$304$	0.397365	+	0.798017i	0.397365	+	0.798017i
$305$		0			0
$306$		0			0
$307$	1.04837	+	0.0971461i	1.04837	+	0.0971461i	0.602635	−	0.798017i	$-0.294118\pi$
$307$	1.04837	+	0.0971461i	1.04837	+	0.0971461i	0.445738	+	0.895163i	$0.352941\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$311$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$312$		0			0
$313$	−0.538007	−	0.100571i	−0.538007	−	0.100571i	−0.0922684	−	0.995734i	$-0.529412\pi$
$313$	−0.538007	−	0.100571i	−0.538007	−	0.100571i	−0.445738	+	0.895163i	$0.647059\pi$
$314$		0			0
$315$		0			0
$316$	0.149783	+	0.526432i	0.149783	+	0.526432i
$317$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$317$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$	1.83319	+	0.710182i	1.83319	+	0.710182i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	0.353470	−	1.89090i	0.353470	−	1.89090i	−0.0922684	−	0.995734i	$-0.529412\pi$
$331$	0.353470	−	1.89090i	0.353470	−	1.89090i	0.445738	−	0.895163i	$-0.352941\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.0170269	+	0.183750i	0.0170269	+	0.183750i	1.00000			$0$
$337$	0.0170269	+	0.183750i	0.0170269	+	0.183750i	−0.982973	+	0.183750i	$0.941176\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	0.201564	+	0.183750i	0.201564	+	0.183750i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$347$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$348$		0			0
$349$	0.193463	+	1.03494i	0.193463	+	1.03494i	0.932472	+	0.361242i	$0.117647\pi$
$349$	0.193463	+	1.03494i	0.193463	+	1.03494i	−0.739009	+	0.673696i	$0.764706\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$353$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$359$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$360$		0			0
$361$	−0.0189399	−	0.204394i	−0.0189399	−	0.204394i
$362$		0			0
$363$		0			0
$364$	−2.70949	+	1.04966i	−2.70949	+	1.04966i
$365$		0			0
$366$		0			0
$367$	1.58561	−	0.614268i	1.58561	−	0.614268i	0.602635	−	0.798017i	$-0.294118\pi$
$367$	1.58561	−	0.614268i	1.58561	−	0.614268i	0.982973	+	0.183750i	$0.0588235\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.44574	−	0.895163i	−1.44574	−	0.895163i	−0.445738	−	0.895163i	$-0.647059\pi$
$373$	−1.44574	−	0.895163i	−1.44574	−	0.895163i	−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.07524	−	0.811985i	1.07524	−	0.811985i	0.0922684	−	0.995734i	$-0.470588\pi$
$379$	1.07524	−	0.811985i	1.07524	−	0.811985i	0.982973	+	0.183750i	$0.0588235\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.445738	−	0.895163i	$-0.647059\pi$
$383$		0			0		0.445738	+	0.895163i	$0.352941\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	−0.537235	−	1.07891i	−0.537235	−	1.07891i
$389$		0			0		−0.0922684	−	0.995734i	$-0.529412\pi$
$389$		0			0		0.0922684	+	0.995734i	$0.470588\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	0.486734	−	1.25640i	0.486734	−	1.25640i	−0.445738	−	0.895163i	$-0.647059\pi$
$397$	0.486734	−	1.25640i	0.486734	−	1.25640i	0.932472	−	0.361242i	$-0.117647\pi$
$398$		0			0
$399$		0			0
$400$	−0.850217	+	0.526432i	−0.850217	+	0.526432i
$401$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$401$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$402$		0			0
$403$	2.06104	+	0.190984i	2.06104	+	0.190984i
$404$		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	$-0.823529\pi$
$409$	−0.404479	+	0.368731i	−0.404479	+	0.368731i	0.445738	+	0.895163i	$0.352941\pi$
$410$		0			0
$411$		0			0
$412$	0.932472	−	0.361242i	0.932472	−	0.361242i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$419$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$420$		0			0
$421$	−0.0822551	+	0.887674i	−0.0822551	+	0.887674i	0.850217	+	0.526432i	$0.176471\pi$
$421$	−0.0822551	+	0.887674i	−0.0822551	+	0.887674i	−0.932472	+	0.361242i	$0.882353\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.57029	−	0.995734i	−2.57029	−	0.995734i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$431$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$432$		0			0
$433$	0.193463	−	0.312454i	0.193463	−	0.312454i	−0.739009	−	0.673696i	$-0.764706\pi$
$433$	0.193463	−	0.312454i	0.193463	−	0.312454i	0.932472	+	0.361242i	$0.117647\pi$
$434$		0			0
$435$		0			0
$436$	0.646741	−	0.322039i	0.646741	−	0.322039i
$437$		0			0
$438$		0			0
$439$	−1.27366	+	0.961826i	−1.27366	+	0.961826i	−0.273663	+	0.961826i	$0.588235\pi$
$439$	−1.27366	+	0.961826i	−1.27366	+	0.961826i	−1.00000			$1.00000\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$443$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$	0.404479	−	1.42160i	0.404479	−	1.42160i
$449$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$449$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	0.486734	+	1.25640i	0.486734	+	1.25640i	0.932472	+	0.361242i	$0.117647\pi$
$457$	0.486734	+	1.25640i	0.486734	+	1.25640i	−0.445738	+	0.895163i	$0.647059\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$461$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$462$		0			0
$463$	−0.365931	+	0.0339085i	−0.365931	+	0.0339085i	−0.273663	−	0.961826i	$-0.588235\pi$
$463$	−0.365931	+	0.0339085i	−0.365931	+	0.0339085i	−0.0922684	+	0.995734i	$0.529412\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.361242	−	0.932472i	$-0.617647\pi$
$467$		0			0		0.361242	+	0.932472i	$0.382353\pi$
$468$		0			0
$469$			2.94343i			2.94343i
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−0.876298	+	0.163808i	−0.876298	+	0.163808i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$479$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$480$		0			0
$481$	2.54778	−	2.79478i	2.54778	−	2.79478i
$482$		0			0
$483$		0			0
$484$	−0.932472	−	0.361242i	−0.932472	−	0.361242i
$485$		0			0
$486$		0			0
$487$	−1.04837	+	0.0971461i	−1.04837	+	0.0971461i	−0.602635	−	0.798017i	$-0.705882\pi$
$487$	−1.04837	+	0.0971461i	−1.04837	+	0.0971461i	−0.445738	+	0.895163i	$0.647059\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$491$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−0.709310	+	0.778076i	−0.709310	+	0.778076i
$497$		0			0
$498$		0			0
$499$	−0.694903	+	1.79375i	−0.694903	+	1.79375i	−0.0922684	+	0.995734i	$0.529412\pi$
$499$	−0.694903	+	1.79375i	−0.694903	+	1.79375i	−0.602635	+	0.798017i	$0.705882\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.526432	−	0.850217i	$-0.676471\pi$
$503$		0			0		0.526432	+	0.850217i	$0.323529\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$	−1.91545	+	0.544991i	−1.91545	+	0.544991i
$509$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$509$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$510$		0			0
$511$	0.196216	−	1.04966i	0.196216	−	1.04966i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.602635	−	0.798017i	$-0.705882\pi$
$521$		0			0		0.602635	+	0.798017i	$0.294118\pi$
$522$		0			0
$523$	0.243964	−	0.857445i	0.243964	−	0.857445i	−0.739009	−	0.673696i	$-0.764706\pi$
$523$	0.243964	−	0.857445i	0.243964	−	0.857445i	0.982973	−	0.183750i	$-0.0588235\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.982973	−	0.183750i	0.982973	−	0.183750i
$530$		0			0
$531$		0			0
$532$	0.794043	−	1.05148i	0.794043	−	1.05148i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	0.538007	−	0.100571i	0.538007	−	0.100571i	0.0922684	−	0.995734i	$-0.470588\pi$
$541$	0.538007	−	0.100571i	0.538007	−	0.100571i	0.445738	+	0.895163i	$0.352941\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.757949	+	1.52217i	−0.757949	+	1.52217i	0.0922684	+	0.995734i	$0.470588\pi$
$547$	−0.757949	+	1.52217i	−0.757949	+	1.52217i	−0.850217	+	0.526432i	$0.823529\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	0.597827	−	0.544991i	0.597827	−	0.544991i
$554$		0			0
$555$		0			0
$556$	−0.757949	+	1.52217i	−0.757949	+	1.52217i
$557$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$557$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$558$		0			0
$559$	3.01794	+	0.858677i	3.01794	+	0.858677i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.932472	−	0.361242i	$-0.117647\pi$
$563$		0			0		−0.932472	+	0.361242i	$0.882353\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		0.850217	−	0.526432i	$-0.176471\pi$
$569$		0			0		−0.850217	+	0.526432i	$0.823529\pi$
$570$		0			0
$571$	0.184537			0.184537			0.0922684	−	0.995734i	$-0.470588\pi$
$571$	0.184537			0.184537			0.0922684	+	0.995734i	$0.470588\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.20614	−	0.600584i	−1.20614	−	0.600584i	−0.273663	−	0.961826i	$-0.588235\pi$
$577$	−1.20614	−	0.600584i	−1.20614	−	0.600584i	−0.932472	+	0.361242i	$0.882353\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$587$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$588$		0			0
$589$	−0.840204	+	0.418372i	−0.840204	+	0.418372i
$590$		0			0
$591$		0			0
$592$	0.353470	+	1.89090i	0.353470	+	1.89090i
$593$		0			0		0.602635	−	0.798017i	$-0.294118\pi$
$593$		0			0		−0.602635	+	0.798017i	$0.705882\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.739009	−	0.673696i	$-0.764706\pi$
$599$		0			0		0.739009	+	0.673696i	$0.235294\pi$
$600$		0			0
$601$	−1.53511	−	1.15926i	−1.53511	−	1.15926i	−0.932472	−	0.361242i	$-0.882353\pi$
$601$	−1.53511	−	1.15926i	−1.53511	−	1.15926i	−0.602635	−	0.798017i	$-0.705882\pi$
$602$		0			0
$603$		0			0
$604$	−0.328972	−	1.75984i	−0.328972	−	1.75984i
$605$		0			0
$606$		0			0
$607$	−0.172075	+	1.85699i	−0.172075	+	1.85699i	0.273663	+	0.961826i	$0.411765\pi$
$607$	−0.172075	+	1.85699i	−0.172075	+	1.85699i	−0.445738	+	0.895163i	$0.647059\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−0.876298	−	1.75984i	−0.876298	−	1.75984i	−0.602635	−	0.798017i	$-0.705882\pi$
$613$	−0.876298	−	1.75984i	−0.876298	−	1.75984i	−0.273663	−	0.961826i	$-0.588235\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−1.96595			−1.96595			−0.982973	−	0.183750i	$-0.941176\pi$
$619$	−1.96595			−1.96595			−0.982973	+	0.183750i	$0.941176\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.273663	−	0.961826i	−0.273663	−	0.961826i
$626$		0			0
$627$		0			0
$628$	0.646741	+	1.66943i	0.646741	+	1.66943i
$629$		0			0
$630$		0			0
$631$	−0.243964	−	0.857445i	−0.243964	−	0.857445i	−0.982973	−	0.183750i	$-0.941176\pi$
$631$	−0.243964	−	0.857445i	−0.243964	−	0.857445i	0.739009	−	0.673696i	$-0.235294\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.72096	+	1.56886i	1.72096	+	1.56886i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$641$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$642$		0			0
$643$	1.12388	−	1.48826i	1.12388	−	1.48826i	0.273663	−	0.961826i	$-0.411765\pi$
$643$	1.12388	−	1.48826i	1.12388	−	1.48826i	0.850217	−	0.526432i	$-0.176471\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.895163	−	0.445738i	$-0.147059\pi$
$647$		0			0		−0.895163	+	0.445738i	$0.852941\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$	0.136374	+	0.124322i	0.136374	+	0.124322i
$653$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$653$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		−0.895163	−	0.445738i	$-0.852941\pi$
$659$		0			0		0.895163	+	0.445738i	$0.147059\pi$
$660$		0			0
$661$	−0.193463	+	0.312454i	−0.193463	+	0.312454i	−0.932472	−	0.361242i	$-0.882353\pi$
$661$	−0.193463	+	0.312454i	−0.193463	+	0.312454i	0.739009	+	0.673696i	$0.235294\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.831277	+	0.322039i	−0.831277	+	0.322039i	−0.739009	−	0.673696i	$-0.764706\pi$
$673$	−0.831277	+	0.322039i	−0.831277	+	0.322039i	−0.0922684	+	0.995734i	$0.529412\pi$
$674$		0			0
$675$		0			0
$676$	−2.67148	+	1.03494i	−2.67148	+	1.03494i
$677$		0			0		−0.961826	−	0.273663i	$-0.911765\pi$
$677$		0			0		0.961826	+	0.273663i	$0.0882353\pi$
$678$		0			0
$679$	−1.07354	+	1.42160i	−1.07354	+	1.42160i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$683$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−1.27366	+	0.961826i	−1.27366	+	0.961826i
$689$		0			0
$690$		0			0
$691$	1.42871	+	0.711414i	1.42871	+	0.711414i	0.982973	−	0.183750i	$-0.0588235\pi$
$691$	1.42871	+	0.711414i	1.42871	+	0.711414i	0.445738	+	0.895163i	$0.352941\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.25664	+	0.778076i	1.25664	+	0.778076i
$701$		0			0		0.961826	−	0.273663i	$-0.0882353\pi$
$701$		0			0		−0.961826	+	0.273663i	$0.911765\pi$
$702$		0			0
$703$	−0.315110	+	1.68569i	−0.315110	+	1.68569i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.67148	−	1.03494i	1.67148	−	1.03494i	0.739009	−	0.673696i	$-0.235294\pi$
$709$	1.67148	−	1.03494i	1.67148	−	1.03494i	0.932472	−	0.361242i	$-0.117647\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.0922684	−	0.995734i	$-0.470588\pi$
$719$		0			0		−0.0922684	+	0.995734i	$0.529412\pi$
$720$		0			0
$721$	−1.09227	−	0.995734i	−1.09227	−	0.995734i
$722$		0			0
$723$		0			0
$724$	0.486734	+	0.533922i	0.486734	+	0.533922i
$725$		0			0
$726$		0			0
$727$	−1.07524	+	0.811985i	−1.07524	+	0.811985i	−0.982973	−	0.183750i	$-0.941176\pi$
$727$	−1.07524	+	0.811985i	−1.07524	+	0.811985i	−0.0922684	+	0.995734i	$0.529412\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	−0.840204	−	1.35698i	−0.840204	−	1.35698i	−0.932472	−	0.361242i	$-0.882353\pi$
$733$	−0.840204	−	1.35698i	−0.840204	−	1.35698i	0.0922684	−	0.995734i	$-0.470588\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.18475	−	0.221468i	−1.18475	−	0.221468i	−0.445738	−	0.895163i	$-0.647059\pi$
$739$	−1.18475	−	0.221468i	−1.18475	−	0.221468i	−0.739009	+	0.673696i	$0.764706\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		−0.982973	−	0.183750i	$-0.941176\pi$
$743$		0			0		0.982973	+	0.183750i	$0.0588235\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	0.876298	−	1.75984i	0.876298	−	1.75984i	0.273663	−	0.961826i	$-0.411765\pi$
$751$	0.876298	−	1.75984i	0.876298	−	1.75984i	0.602635	−	0.798017i	$-0.294118\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−0.510366	+	1.79375i	−0.510366	+	1.79375i	0.0922684	+	0.995734i	$0.470588\pi$
$757$	−0.510366	+	1.79375i	−0.510366	+	1.79375i	−0.602635	+	0.798017i	$0.705882\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.982973	−	0.183750i	$-0.0588235\pi$
$761$		0			0		−0.982973	+	0.183750i	$0.941176\pi$
$762$		0			0
$763$	−0.852157	−	0.643519i	−0.852157	−	0.643519i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.260991	−	0.673696i	−0.260991	−	0.673696i	0.739009	−	0.673696i	$-0.235294\pi$
$769$	−0.260991	−	0.673696i	−0.260991	−	0.673696i	−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	1.58923	−	0.147263i	1.58923	−	0.147263i
$773$		0			0		0.995734	−	0.0922684i	$-0.0294118\pi$
$773$		0			0		−0.995734	+	0.0922684i	$0.970588\pi$
$774$		0			0
$775$	−0.554262	−	0.895163i	−0.554262	−	0.895163i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.16437	+	0.217658i	−1.16437	+	0.217658i
$785$		0			0
$786$		0			0
$787$	1.02474	−	0.634493i	1.02474	−	0.634493i	0.0922684	−	0.995734i	$-0.470588\pi$
$787$	1.02474	−	0.634493i	1.02474	−	0.634493i	0.932472	+	0.361242i	$0.117647\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.41880	−	1.32445i	−3.41880	−	1.32445i
$794$		0			0
$795$		0			0
$796$	−1.58923	+	0.147263i	−1.58923	+	0.147263i
$797$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$797$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$809$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$810$		0			0
$811$	1.72198	−	0.489946i	1.72198	−	0.489946i	0.739009	−	0.673696i	$-0.235294\pi$
$811$	1.72198	−	0.489946i	1.72198	−	0.489946i	0.982973	+	0.183750i	$0.0588235\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.36851	+	0.389375i	−1.36851	+	0.389375i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.673696	−	0.739009i	$-0.735294\pi$
$821$		0			0		0.673696	+	0.739009i	$0.264706\pi$
$822$		0			0
$823$			1.05286i			1.05286i	0.850217	+	0.526432i	$0.176471\pi$
$823$			1.05286i			1.05286i	−0.850217	+	0.526432i	$0.823529\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		0.739009	−	0.673696i	$-0.235294\pi$
$827$		0			0		−0.739009	+	0.673696i	$0.764706\pi$
$828$		0			0
$829$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.850217	+	0.526432i	$0.823529\pi$
$829$	−1.78269	+	0.887674i	−1.78269	+	0.887674i	−0.932472	+	0.361242i	$0.882353\pi$
$830$		0			0
$831$		0			0
$832$	0.538007	−	1.89090i	0.538007	−	1.89090i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		0.361242	−	0.932472i	$-0.382353\pi$
$839$		0			0		−0.361242	+	0.932472i	$0.617647\pi$
$840$		0			0
$841$	0.602635	−	0.798017i	0.602635	−	0.798017i
$842$		0			0
$843$		0			0
$844$	0.353470	−	0.100571i	0.353470	−	0.100571i
$845$		0			0
$846$		0			0
$847$	0.136374	+	1.47171i	0.136374	+	1.47171i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	0.537235	+	1.07891i	0.537235	+	1.07891i	0.982973	+	0.183750i	$0.0588235\pi$
$853$	0.537235	+	1.07891i	0.537235	+	1.07891i	−0.445738	+	0.895163i	$0.647059\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.798017	−	0.602635i	$-0.205882\pi$
$857$		0			0		−0.798017	+	0.602635i	$0.794118\pi$
$858$		0			0
$859$	1.20614	+	1.32307i	1.20614	+	1.32307i	0.932472	+	0.361242i	$0.117647\pi$
$859$	1.20614	+	1.32307i	1.20614	+	1.32307i	0.273663	+	0.961826i	$0.411765\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.850217	−	0.526432i	$-0.823529\pi$
$863$		0			0		0.850217	+	0.526432i	$0.176471\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$	1.49675	+	0.425861i	1.49675	+	0.425861i
$869$		0			0
$870$		0			0
$871$			3.91512i			3.91512i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	1.29596	−	1.42160i	1.29596	−	1.42160i	0.445738	−	0.895163i	$-0.352941\pi$
$877$	1.29596	−	1.42160i	1.29596	−	1.42160i	0.850217	−	0.526432i	$-0.176471\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−0.329838	+	1.15926i	−0.329838	+	1.15926i	0.602635	+	0.798017i	$0.294118\pi$
$883$	−0.329838	+	1.15926i	−0.329838	+	1.15926i	−0.932472	+	0.361242i	$0.882353\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$887$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$888$		0			0
$889$	1.98297	+	2.17522i	1.98297	+	2.17522i
$890$		0			0
$891$		0			0
$892$	−0.136374	+	0.124322i	−0.136374	+	0.124322i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	0.149783	+	0.526432i	0.149783	+	0.526432i	1.00000			$0$
$907$	0.149783	+	0.526432i	0.149783	+	0.526432i	−0.850217	+	0.526432i	$0.823529\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.932472	−	0.361242i	$-0.882353\pi$
$911$		0			0		0.932472	+	0.361242i	$0.117647\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$		0			0
$916$	0.156896	−	1.69318i	0.156896	−	1.69318i
$917$		0			0
$918$		0			0
$919$	1.29596	+	0.368731i	1.29596	+	0.368731i	0.850217	−	0.526432i	$-0.176471\pi$
$919$	1.29596	+	0.368731i	1.29596	+	0.368731i	0.445738	+	0.895163i	$0.352941\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−1.91545	−	0.177492i	−1.91545	−	0.177492i
$926$		0			0
$927$		0			0
$928$		0			0
$929$		0			0		−0.995734	−	0.0922684i	$-0.970588\pi$
$929$		0			0		0.995734	+	0.0922684i	$0.0294118\pi$
$930$		0			0
$931$	−1.03801	−	0.194037i	−1.03801	−	0.194037i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−0.576554	−	1.48826i	−0.576554	−	1.48826i	−0.850217	−	0.526432i	$-0.823529\pi$
$937$	−0.576554	−	1.48826i	−0.576554	−	1.48826i	0.273663	−	0.961826i	$-0.411765\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$941$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.273663	−	0.961826i	$-0.588235\pi$
$947$		0			0		0.273663	+	0.961826i	$0.411765\pi$
$948$		0			0
$949$	0.260991	−	1.39618i	0.260991	−	1.39618i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.183750	−	0.982973i	$-0.558824\pi$
$953$		0			0		0.183750	+	0.982973i	$0.441176\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−0.0801997	−	0.0731117i	−0.0801997	−	0.0731117i
$962$		0			0
$963$		0			0
$964$	−1.04837	−	1.69318i	−1.04837	−	1.69318i
$965$		0			0
$966$		0			0
$967$	0.0675278	+	0.361242i	0.0675278	+	0.361242i	1.00000			$0$
$967$	0.0675278	+	0.361242i	0.0675278	+	0.361242i	−0.932472	+	0.361242i	$0.882353\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.273663	−	0.961826i	$-0.411765\pi$
$971$		0			0		−0.273663	+	0.961826i	$0.588235\pi$
$972$		0			0
$973$	2.51327			2.51327
$974$		0			0
$975$		0			0
$976$	1.58561	−	0.981767i	1.58561	−	0.981767i
$977$		0			0		0.673696	−	0.739009i	$-0.264706\pi$
$977$		0			0		−0.673696	+	0.739009i	$0.735294\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.05617	−	1.39860i	1.05617	−	1.39860i
$989$		0			0
$990$		0			0
$991$	−0.156896	−	0.0971461i	−0.156896	−	0.0971461i	0.445738	−	0.895163i	$-0.352941\pi$
$991$	−0.156896	−	0.0971461i	−0.156896	−	0.0971461i	−0.602635	+	0.798017i	$0.705882\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	0.293271	−	0.221468i	0.293271	−	0.221468i	−0.445738	−	0.895163i	$-0.647059\pi$
$997$	0.293271	−	0.221468i	0.293271	−	0.221468i	0.739009	+	0.673696i	$0.235294\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	927.1.v.a.145.1	✓	16
3.2	odd	2	CM	927.1.v.a.145.1	✓	16
103.27	odd	34	inner	927.1.v.a.748.1	yes	16
309.233	even	34	inner	927.1.v.a.748.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
927.1.v.a.145.1	✓	16	1.1	even	1	trivial
927.1.v.a.145.1	✓	16	3.2	odd	2	CM
927.1.v.a.748.1	yes	16	103.27	odd	34	inner
927.1.v.a.748.1	yes	16	309.233	even	34	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 927 = 3^{2} \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	927.v (of order \(34\), degree \(16\), minimal)

\(f(q)\)	\(=\)	\(q+(0.982973 + 0.183750i) q^{4} +(-0.404479 - 1.42160i) q^{7} +(-0.538007 - 1.89090i) q^{13} +(0.932472 + 0.361242i) q^{16} +(0.658809 + 0.600584i) q^{19} +(-0.602635 + 0.798017i) q^{25} +(-0.136374 - 1.47171i) q^{28} +(-0.380338 + 0.981767i) q^{31} +(1.01267 + 1.63552i) q^{37} +(-0.840204 + 1.35698i) q^{43} +(-1.00711 + 0.623578i) q^{49} +(-0.181395 - 1.95756i) q^{52} +(1.12388 - 1.48826i) q^{61} +(0.850217 + 0.526432i) q^{64} +(-1.91545 - 0.544991i) q^{67} +(0.646741 + 0.322039i) q^{73} +(0.537235 + 0.711414i) q^{76} +(0.243964 + 0.489946i) q^{79} +(-2.47048 + 1.52966i) q^{91} +(-0.726337 - 0.961826i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{4} - 2 q^{7} + 2 q^{13} - q^{16} - 2 q^{19} - q^{25} + 2 q^{28} - 3 q^{49} - 2 q^{52} + 2 q^{61} + q^{64} + 2 q^{76} + 2 q^{79} - 13 q^{91} - 15 q^{97}+O(q^{100}) \) 16 * q + q^4 - 2 * q^7 + 2 * q^13 - q^16 - 2 * q^19 - q^25 + 2 * q^28 - 3 * q^49 - 2 * q^52 + 2 * q^61 + q^64 + 2 * q^76 + 2 * q^79 - 13 * q^91 - 15 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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