LMFDB - Embedded newform 1575.2.d.a.1324.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1575,2,Mod(1324,1575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1575, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1575.1324");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1575 = 3^{2} \cdot 5^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1575.d (of order $2$, degree $1$, not 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:	$12.5764383184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1324.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1575.1324
Dual form			1575.2.d.a.1324.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{7} +3.00000i q^{8} +O(q^{10})$	$q+1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{7} +3.00000i q^{8} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} +6.00000i q^{17} -4.00000 q^{19} -4.00000i q^{22} -2.00000 q^{26} -1.00000i q^{28} -2.00000 q^{29} +5.00000i q^{32} -6.00000 q^{34} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -1.00000 q^{49} +2.00000i q^{52} +6.00000i q^{53} +3.00000 q^{56} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -7.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +6.00000i q^{73} -6.00000 q^{74} -4.00000 q^{76} +4.00000i q^{77} +16.0000 q^{79} -2.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -12.0000i q^{88} -14.0000 q^{89} +2.00000 q^{91} +18.0000i q^{97} -1.00000i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{4}+O(q^{10}) $ 2 * q + 2 * q^4	$ 2 q + 2 q^{4} - 8 q^{11} + 2 q^{14} - 2 q^{16} - 8 q^{19} - 4 q^{26} - 4 q^{29} - 12 q^{34} - 4 q^{41} - 8 q^{44} - 2 q^{49} + 6 q^{56} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 12 q^{74} - 8 q^{76} + 32 q^{79} - 8 q^{86} - 28 q^{89} + 4 q^{91}+O(q^{100}) $ 2 * q + 2 * q^4 - 8 * q^11 + 2 * q^14 - 2 * q^16 - 8 * q^19 - 4 * q^26 - 4 * q^29 - 12 * q^34 - 4 * q^41 - 8 * q^44 - 2 * q^49 + 6 * q^56 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 12 * q^74 - 8 * q^76 + 32 * q^79 - 8 * q^86 - 28 * q^89 + 4 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times$.

$n$	$127$	$451$	$1226$
$\chi(n)$	$-1$	$1$	$1$

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 $

$2$	1.00000i	0.707107i	0.935414	+	0.353553i	$0.115027\pi$
$2$	1.00000i	0.707107i	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i
$7$	− 1.00000i	− 0.377964i
$8$	3.00000i	1.06066i
$9$	0	0
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	2.00000i	0.554700i	0.960769	+	0.277350i	$0.0894562\pi$
$13$	2.00000i	0.554700i	−0.960769	+	0.277350i	$0.910544\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	0	0
$22$	− 4.00000i	− 0.852803i
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0	0
$26$	−2.00000	−0.392232
$27$	0	0
$28$	− 1.00000i	− 0.188982i
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	5.00000i	0.883883i
$33$	0	0
$34$	−6.00000	−1.02899
$35$	0	0
$36$	0	0
$37$	6.00000i	0.986394i	0.869918	+	0.493197i	$0.164172\pi$
$37$	6.00000i	0.986394i	−0.869918	+	0.493197i	$0.835828\pi$
$38$	− 4.00000i	− 0.648886i
$39$	0	0
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	4.00000i	0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$	4.00000i	0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	−4.00000	−0.603023
$45$	0	0
$46$	0	0
$47$	0	0	1.00000			$0$
$47$	0	0	−1.00000			$\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	0	0
$52$	2.00000i	0.277350i
$53$	6.00000i	0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$	6.00000i	0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$	0	0
$55$	0	0
$56$	3.00000	0.400892
$57$	0	0
$58$	− 2.00000i	− 0.262613i
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	0	0
$63$	0	0
$64$	−7.00000	−0.875000
$65$	0	0
$66$	0	0
$67$	4.00000i	0.488678i	0.969690	+	0.244339i	$0.0785709\pi$
$67$	4.00000i	0.488678i	−0.969690	+	0.244339i	$0.921429\pi$
$68$	6.00000i	0.727607i
$69$	0	0
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	6.00000i	0.702247i	0.936329	+	0.351123i	$0.114200\pi$
$73$	6.00000i	0.702247i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	−4.00000	−0.458831
$77$	4.00000i	0.455842i
$78$	0	0
$79$	16.0000	1.80014	0.900070	−	0.435745i	$-0.143515\pi$
$79$	16.0000	1.80014	0.900070	+	0.435745i	$0.143515\pi$
$80$	0	0
$81$	0	0
$82$	− 2.00000i	− 0.220863i
$83$	− 12.0000i	− 1.31717i	−0.752506	−	0.658586i	$-0.771155\pi$
$83$	− 12.0000i	− 1.31717i	0.752506	−	0.658586i	$-0.228845\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	− 12.0000i	− 1.27920i
$89$	−14.0000	−1.48400	−0.741999	−	0.670402i	$-0.766122\pi$
$89$	−14.0000	−1.48400	−0.741999	+	0.670402i	$0.766122\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.0000i	1.82762i	0.406138	+	0.913812i	$0.366875\pi$
$97$	18.0000i	1.82762i	−0.406138	+	0.913812i	$0.633125\pi$
$98$	− 1.00000i	− 0.101015i
$99$	0	0
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	− 8.00000i	− 0.788263i	−0.919054	−	0.394132i	$-0.871045\pi$
$103$	− 8.00000i	− 0.788263i	0.919054	−	0.394132i	$-0.128955\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	−6.00000	−0.582772
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	18.0000	1.72409	0.862044	−	0.506834i	$-0.169184\pi$
$109$	18.0000	1.72409	0.862044	+	0.506834i	$0.169184\pi$
$110$	0	0
$111$	0	0
$112$	1.00000i	0.0944911i
$113$	− 14.0000i	− 1.31701i	−0.752577	−	0.658505i	$-0.771189\pi$
$113$	− 14.0000i	− 1.31701i	0.752577	−	0.658505i	$-0.228811\pi$
$114$	0	0
$115$	0	0
$116$	−2.00000	−0.185695
$117$	0	0
$118$	12.0000i	1.10469i
$119$	6.00000	0.550019
$120$	0	0
$121$	5.00000	0.454545
$122$	− 2.00000i	− 0.181071i
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	3.00000i	0.265165i
$129$	0	0
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	4.00000i	0.346844i
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−18.0000	−1.54349
$137$	6.00000i	0.512615i	0.966595	+	0.256307i	$0.0825059\pi$
$137$	6.00000i	0.512615i	−0.966595	+	0.256307i	$0.917494\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8.00000i	− 0.668994i
$144$	0	0
$145$	0	0
$146$	−6.00000	−0.496564
$147$	0	0
$148$	6.00000i	0.493197i
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	− 12.0000i	− 0.973329i
$153$	0	0
$154$	−4.00000	−0.322329
$155$	0	0
$156$	0	0
$157$	− 2.00000i	− 0.159617i	−0.996810	−	0.0798087i	$-0.974569\pi$
$157$	− 2.00000i	− 0.159617i	0.996810	−	0.0798087i	$-0.0254309\pi$
$158$	16.0000i	1.27289i
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	−2.00000	−0.156174
$165$	0	0
$166$	12.0000	0.931381
$167$	8.00000i	0.619059i	0.950890	+	0.309529i	$0.100171\pi$
$167$	8.00000i	0.619059i	−0.950890	+	0.309529i	$0.899829\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	0	0
$172$	4.00000i	0.304997i
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	− 14.0000i	− 1.04934i
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	−26.0000	−1.93256	−0.966282	−	0.257485i	$-0.917106\pi$
$181$	−26.0000	−1.93256	−0.966282	+	0.257485i	$0.917106\pi$
$182$	2.00000i	0.148250i
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 24.0000i	− 1.75505i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.00000	0.578860	0.289430	−	0.957199i	$-0.406534\pi$
$191$	8.00000	0.578860	0.289430	+	0.957199i	$0.406534\pi$
$192$	0	0
$193$	− 2.00000i	− 0.143963i	−0.997406	−	0.0719816i	$-0.977068\pi$
$193$	− 2.00000i	− 0.143963i	0.997406	−	0.0719816i	$-0.0229323\pi$
$194$	−18.0000	−1.29232
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	− 22.0000i	− 1.56744i	−0.621117	−	0.783718i	$-0.713321\pi$
$197$	− 22.0000i	− 1.56744i	0.621117	−	0.783718i	$-0.286679\pi$
$198$	0	0
$199$	−24.0000	−1.70131	−0.850657	−	0.525720i	$-0.823796\pi$
$199$	−24.0000	−1.70131	−0.850657	+	0.525720i	$0.823796\pi$
$200$	0	0
$201$	0	0
$202$	− 14.0000i	− 0.985037i
$203$	2.00000i	0.140372i
$204$	0	0
$205$	0	0
$206$	8.00000	0.557386
$207$	0	0
$208$	− 2.00000i	− 0.138675i
$209$	16.0000	1.10674
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000i	0.412082i
$213$	0	0
$214$	4.00000	0.273434
$215$	0	0
$216$	0	0
$217$	0	0
$218$	18.0000i	1.21911i
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	− 16.0000i	− 1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$	− 16.0000i	− 1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$	5.00000	0.334077
$225$	0	0
$226$	14.0000	0.931266
$227$	12.0000i	0.796468i	0.917284	+	0.398234i	$0.130377\pi$
$227$	12.0000i	0.796468i	−0.917284	+	0.398234i	$0.869623\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	0	0
$232$	− 6.00000i	− 0.393919i
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	12.0000	0.781133
$237$	0	0
$238$	6.00000i	0.388922i
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	5.00000i	0.321412i
$243$	0	0
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	− 8.00000i	− 0.509028i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.0000	1.26239	0.631194	−	0.775625i	$-0.282565\pi$
$251$	20.0000	1.26239	0.631194	+	0.775625i	$0.282565\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	−17.0000	−1.06250
$257$	− 26.0000i	− 1.62184i	−0.585160	−	0.810918i	$-0.698968\pi$
$257$	− 26.0000i	− 1.62184i	0.585160	−	0.810918i	$-0.301032\pi$
$258$	0	0
$259$	6.00000	0.372822
$260$	0	0
$261$	0	0
$262$	− 4.00000i	− 0.247121i
$263$	16.0000i	0.986602i	0.869859	+	0.493301i	$0.164210\pi$
$263$	16.0000i	0.986602i	−0.869859	+	0.493301i	$0.835790\pi$
$264$	0	0
$265$	0	0
$266$	−4.00000	−0.245256
$267$	0	0
$268$	4.00000i	0.244339i
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	− 6.00000i	− 0.363803i
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	22.0000i	1.32185i	0.750451	+	0.660926i	$0.229836\pi$
$277$	22.0000i	1.32185i	−0.750451	+	0.660926i	$0.770164\pi$
$278$	− 12.0000i	− 0.719712i
$279$	0	0
$280$	0	0
$281$	22.0000	1.31241	0.656205	−	0.754583i	$-0.272161\pi$
$281$	22.0000	1.31241	0.656205	+	0.754583i	$0.272161\pi$
$282$	0	0
$283$	20.0000i	1.18888i	0.804141	+	0.594438i	$0.202626\pi$
$283$	20.0000i	1.18888i	−0.804141	+	0.594438i	$0.797374\pi$
$284$	0	0
$285$	0	0
$286$	8.00000	0.473050
$287$	2.00000i	0.118056i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	6.00000i	0.351123i
$293$	14.0000i	0.817889i	0.912559	+	0.408944i	$0.134103\pi$
$293$	14.0000i	0.817889i	−0.912559	+	0.408944i	$0.865897\pi$
$294$	0	0
$295$	0	0
$296$	−18.0000	−1.04623
$297$	0	0
$298$	6.00000i	0.347571i
$299$	0	0
$300$	0	0
$301$	4.00000	0.230556
$302$	8.00000i	0.460348i
$303$	0	0
$304$	4.00000	0.229416
$305$	0	0
$306$	0	0
$307$	4.00000i	0.228292i	0.993464	+	0.114146i	$0.0364132\pi$
$307$	4.00000i	0.228292i	−0.993464	+	0.114146i	$0.963587\pi$
$308$	4.00000i	0.227921i
$309$	0	0
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	− 26.0000i	− 1.46961i	−0.678280	−	0.734803i	$-0.737274\pi$
$313$	− 26.0000i	− 1.46961i	0.678280	−	0.734803i	$-0.262726\pi$
$314$	2.00000	0.112867
$315$	0	0
$316$	16.0000	0.900070
$317$	18.0000i	1.01098i	0.862832	+	0.505490i	$0.168688\pi$
$317$	18.0000i	1.01098i	−0.862832	+	0.505490i	$0.831312\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	4.00000	0.221540
$327$	0	0
$328$	− 6.00000i	− 0.331295i
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	− 12.0000i	− 0.658586i
$333$	0	0
$334$	−8.00000	−0.437741
$335$	0	0
$336$	0	0
$337$	− 14.0000i	− 0.762629i	−0.924445	−	0.381314i	$-0.875472\pi$
$337$	− 14.0000i	− 0.762629i	0.924445	−	0.381314i	$-0.124528\pi$
$338$	9.00000i	0.489535i
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	1.00000i	0.0539949i
$344$	−12.0000	−0.646997
$345$	0	0
$346$	10.0000	0.537603
$347$	28.0000i	1.50312i	0.659665	+	0.751559i	$0.270698\pi$
$347$	28.0000i	1.50312i	−0.659665	+	0.751559i	$0.729302\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	− 20.0000i	− 1.06600i
$353$	10.0000i	0.532246i	0.963939	+	0.266123i	$0.0857428\pi$
$353$	10.0000i	0.532246i	−0.963939	+	0.266123i	$0.914257\pi$
$354$	0	0
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	− 4.00000i	− 0.211407i
$359$	32.0000	1.68890	0.844448	−	0.535638i	$-0.179929\pi$
$359$	32.0000	1.68890	0.844448	+	0.535638i	$0.179929\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	− 26.0000i	− 1.36653i
$363$	0	0
$364$	2.00000	0.104828
$365$	0	0
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.00000	0.311504
$372$	0	0
$373$	10.0000i	0.517780i	0.965907	+	0.258890i	$0.0833568\pi$
$373$	10.0000i	0.517780i	−0.965907	+	0.258890i	$0.916643\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	0	0
$377$	− 4.00000i	− 0.206010i
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	0	0
$382$	8.00000i	0.409316i
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	2.00000	0.101797
$387$	0	0
$388$	18.0000i	0.913812i
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	− 3.00000i	− 0.151523i
$393$	0	0
$394$	22.0000	1.10834
$395$	0	0
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	− 24.0000i	− 1.20301i
$399$	0	0
$400$	0	0
$401$	30.0000	1.49813	0.749064	−	0.662497i	$-0.230503\pi$
$401$	30.0000	1.49813	0.749064	+	0.662497i	$0.230503\pi$
$402$	0	0
$403$	0	0
$404$	−14.0000	−0.696526
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	− 24.0000i	− 1.18964i
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	− 8.00000i	− 0.394132i
$413$	− 12.0000i	− 0.590481i
$414$	0	0
$415$	0	0
$416$	−10.0000	−0.490290
$417$	0	0
$418$	16.0000i	0.782586i
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	4.00000i	0.194717i
$423$	0	0
$424$	−18.0000	−0.874157
$425$	0	0
$426$	0	0
$427$	2.00000i	0.0967868i
$428$	− 4.00000i	− 0.193347i
$429$	0	0
$430$	0	0
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	0	0
$433$	14.0000i	0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$	14.0000i	0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$	0	0
$435$	0	0
$436$	18.0000	0.862044
$437$	0	0
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	0	0
$441$	0	0
$442$	− 12.0000i	− 0.570782i
$443$	36.0000i	1.71041i	0.518289	+	0.855206i	$0.326569\pi$
$443$	36.0000i	1.71041i	−0.518289	+	0.855206i	$0.673431\pi$
$444$	0	0
$445$	0	0
$446$	16.0000	0.757622
$447$	0	0
$448$	7.00000i	0.330719i
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	8.00000	0.376705
$452$	− 14.0000i	− 0.658505i
$453$	0	0
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	10.0000i	0.467780i	0.972263	+	0.233890i	$0.0751456\pi$
$457$	10.0000i	0.467780i	−0.972263	+	0.233890i	$0.924854\pi$
$458$	10.0000i	0.467269i
$459$	0	0
$460$	0	0
$461$	10.0000	0.465746	0.232873	−	0.972507i	$-0.425187\pi$
$461$	10.0000	0.465746	0.232873	+	0.972507i	$0.425187\pi$
$462$	0	0
$463$	− 16.0000i	− 0.743583i	−0.928316	−	0.371792i	$-0.878744\pi$
$463$	− 16.0000i	− 0.743583i	0.928316	−	0.371792i	$-0.121256\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	6.00000	0.277945
$467$	− 36.0000i	− 1.66588i	−0.553362	−	0.832941i	$-0.686655\pi$
$467$	− 36.0000i	− 1.66588i	0.553362	−	0.832941i	$-0.313345\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	36.0000i	1.65703i
$473$	− 16.0000i	− 0.735681i
$474$	0	0
$475$	0	0
$476$	6.00000	0.275010
$477$	0	0
$478$	24.0000i	1.09773i
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	−12.0000	−0.547153
$482$	2.00000i	0.0910975i
$483$	0	0
$484$	5.00000	0.227273
$485$	0	0
$486$	0	0
$487$	− 8.00000i	− 0.362515i	−0.983436	−	0.181257i	$-0.941983\pi$
$487$	− 8.00000i	− 0.362515i	0.983436	−	0.181257i	$-0.0580167\pi$
$488$	− 6.00000i	− 0.271607i
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	− 12.0000i	− 0.540453i
$494$	8.00000	0.359937
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	0	0
$502$	20.0000i	0.892644i
$503$	24.0000i	1.07011i	0.844818	+	0.535054i	$0.179709\pi$
$503$	24.0000i	1.07011i	−0.844818	+	0.535054i	$0.820291\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.0000	−0.443242	−0.221621	−	0.975133i	$-0.571135\pi$
$509$	−10.0000	−0.443242	−0.221621	+	0.975133i	$0.571135\pi$
$510$	0	0
$511$	6.00000	0.265424
$512$	− 11.0000i	− 0.486136i
$513$	0	0
$514$	26.0000	1.14681
$515$	0	0
$516$	0	0
$517$	0	0
$518$	6.00000i	0.263625i
$519$	0	0
$520$	0	0
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−16.0000	−0.697633
$527$	0	0
$528$	0	0
$529$	23.0000	1.00000
$530$	0	0
$531$	0	0
$532$	4.00000i	0.173422i
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	−12.0000	−0.518321
$537$	0	0
$538$	6.00000i	0.258678i
$539$	4.00000	0.172292
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	16.0000i	0.687259i
$543$	0	0
$544$	−30.0000	−1.28624
$545$	0	0
$546$	0	0
$547$	4.00000i	0.171028i	0.996337	+	0.0855138i	$0.0272532\pi$
$547$	4.00000i	0.171028i	−0.996337	+	0.0855138i	$0.972747\pi$
$548$	6.00000i	0.256307i
$549$	0	0
$550$	0	0
$551$	8.00000	0.340811
$552$	0	0
$553$	− 16.0000i	− 0.680389i
$554$	−22.0000	−0.934690
$555$	0	0
$556$	−12.0000	−0.508913
$557$	2.00000i	0.0847427i	0.999102	+	0.0423714i	$0.0134913\pi$
$557$	2.00000i	0.0847427i	−0.999102	+	0.0423714i	$0.986509\pi$
$558$	0	0
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	22.0000i	0.928014i
$563$	4.00000i	0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$	4.00000i	0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$	0	0
$565$	0	0
$566$	−20.0000	−0.840663
$567$	0	0
$568$	0	0
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−4.00000	−0.167395	−0.0836974	−	0.996491i	$-0.526673\pi$
$571$	−4.00000	−0.167395	−0.0836974	+	0.996491i	$0.526673\pi$
$572$	− 8.00000i	− 0.334497i
$573$	0	0
$574$	−2.00000	−0.0834784
$575$	0	0
$576$	0	0
$577$	34.0000i	1.41544i	0.706494	+	0.707719i	$0.250276\pi$
$577$	34.0000i	1.41544i	−0.706494	+	0.707719i	$0.749724\pi$
$578$	− 19.0000i	− 0.790296i
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	− 24.0000i	− 0.993978i
$584$	−18.0000	−0.744845
$585$	0	0
$586$	−14.0000	−0.578335
$587$	− 28.0000i	− 1.15568i	−0.816149	−	0.577842i	$-0.803895\pi$
$587$	− 28.0000i	− 1.15568i	0.816149	−	0.577842i	$-0.196105\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	− 6.00000i	− 0.246598i
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	0	0
$599$	48.0000	1.96123	0.980613	−	0.195952i	$-0.0627798\pi$
$599$	48.0000	1.96123	0.980613	+	0.195952i	$0.0627798\pi$
$600$	0	0
$601$	−6.00000	−0.244745	−0.122373	−	0.992484i	$-0.539050\pi$
$601$	−6.00000	−0.244745	−0.122373	+	0.992484i	$0.539050\pi$
$602$	4.00000i	0.163028i
$603$	0	0
$604$	8.00000	0.325515
$605$	0	0
$606$	0	0
$607$	− 16.0000i	− 0.649420i	−0.945814	−	0.324710i	$-0.894733\pi$
$607$	− 16.0000i	− 0.649420i	0.945814	−	0.324710i	$-0.105267\pi$
$608$	− 20.0000i	− 0.811107i
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	26.0000i	1.05013i	0.851062	+	0.525065i	$0.175959\pi$
$613$	26.0000i	1.05013i	−0.851062	+	0.525065i	$0.824041\pi$
$614$	−4.00000	−0.161427
$615$	0	0
$616$	−12.0000	−0.483494
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	0	0
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	0	0
$622$	24.0000i	0.962312i
$623$	14.0000i	0.560898i
$624$	0	0
$625$	0	0
$626$	26.0000	1.03917
$627$	0	0
$628$	− 2.00000i	− 0.0798087i
$629$	−36.0000	−1.43541
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	48.0000i	1.90934i
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	8.00000i	0.316723i
$639$	0	0
$640$	0	0
$641$	−18.0000	−0.710957	−0.355479	−	0.934684i	$-0.615682\pi$
$641$	−18.0000	−0.710957	−0.355479	+	0.934684i	$0.615682\pi$
$642$	0	0
$643$	− 20.0000i	− 0.788723i	−0.918955	−	0.394362i	$-0.870966\pi$
$643$	− 20.0000i	− 0.788723i	0.918955	−	0.394362i	$-0.129034\pi$
$644$	0	0
$645$	0	0
$646$	24.0000	0.944267
$647$	40.0000i	1.57256i	0.617869	+	0.786281i	$0.287996\pi$
$647$	40.0000i	1.57256i	−0.617869	+	0.786281i	$0.712004\pi$
$648$	0	0
$649$	−48.0000	−1.88416
$650$	0	0
$651$	0	0
$652$	− 4.00000i	− 0.156652i
$653$	− 18.0000i	− 0.704394i	−0.935926	−	0.352197i	$-0.885435\pi$
$653$	− 18.0000i	− 0.704394i	0.935926	−	0.352197i	$-0.114565\pi$
$654$	0	0
$655$	0	0
$656$	2.00000	0.0780869
$657$	0	0
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	− 4.00000i	− 0.155464i
$663$	0	0
$664$	36.0000	1.39707
$665$	0	0
$666$	0	0
$667$	0	0
$668$	8.00000i	0.309529i
$669$	0	0
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	− 34.0000i	− 1.31060i	−0.755367	−	0.655302i	$-0.772541\pi$
$673$	− 34.0000i	− 1.31060i	0.755367	−	0.655302i	$-0.227459\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	9.00000	0.346154
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	18.0000	0.690777
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 12.0000i	− 0.459167i	−0.973289	−	0.229584i	$-0.926264\pi$
$683$	− 12.0000i	− 0.459167i	0.973289	−	0.229584i	$-0.0737364\pi$
$684$	0	0
$685$	0	0
$686$	−1.00000	−0.0381802
$687$	0	0
$688$	− 4.00000i	− 0.152499i
$689$	−12.0000	−0.457164
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	− 10.0000i	− 0.380143i
$693$	0	0
$694$	−28.0000	−1.06287
$695$	0	0
$696$	0	0
$697$	− 12.0000i	− 0.454532i
$698$	2.00000i	0.0757011i
$699$	0	0
$700$	0	0
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	− 24.0000i	− 0.905177i
$704$	28.0000	1.05529
$705$	0	0
$706$	−10.0000	−0.376355
$707$	14.0000i	0.526524i
$708$	0	0
$709$	−6.00000	−0.225335	−0.112667	−	0.993633i	$-0.535939\pi$
$709$	−6.00000	−0.225335	−0.112667	+	0.993633i	$0.535939\pi$
$710$	0	0
$711$	0	0
$712$	− 42.0000i	− 1.57402i
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−4.00000	−0.149487
$717$	0	0
$718$	32.0000i	1.19423i
$719$	−48.0000	−1.79010	−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000	−1.79010	−0.895049	+	0.445968i	$0.852860\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	− 3.00000i	− 0.111648i
$723$	0	0
$724$	−26.0000	−0.966282
$725$	0	0
$726$	0	0
$727$	− 40.0000i	− 1.48352i	−0.670667	−	0.741759i	$-0.733992\pi$
$727$	− 40.0000i	− 1.48352i	0.670667	−	0.741759i	$-0.266008\pi$
$728$	6.00000i	0.222375i
$729$	0	0
$730$	0	0
$731$	−24.0000	−0.887672
$732$	0	0
$733$	18.0000i	0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$	18.0000i	0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 16.0000i	− 0.589368i
$738$	0	0
$739$	−36.0000	−1.32428	−0.662141	−	0.749380i	$-0.730352\pi$
$739$	−36.0000	−1.32428	−0.662141	+	0.749380i	$0.730352\pi$
$740$	0	0
$741$	0	0
$742$	6.00000i	0.220267i
$743$	0	0	1.00000			$0$
$743$	0	0	−1.00000			$\pi$
$744$	0	0
$745$	0	0
$746$	−10.0000	−0.366126
$747$	0	0
$748$	− 24.0000i	− 0.877527i
$749$	−4.00000	−0.146157
$750$	0	0
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	0	0
$754$	4.00000	0.145671
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	− 12.0000i	− 0.435860i
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	− 18.0000i	− 0.651644i
$764$	8.00000	0.289430
$765$	0	0
$766$	0	0
$767$	24.0000i	0.866590i
$768$	0	0
$769$	−2.00000	−0.0721218	−0.0360609	−	0.999350i	$-0.511481\pi$
$769$	−2.00000	−0.0721218	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	0	0
$771$	0	0
$772$	− 2.00000i	− 0.0719816i
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	0	0
$776$	−54.0000	−1.93849
$777$	0	0
$778$	6.00000i	0.215110i
$779$	8.00000	0.286630
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	1.00000	0.0357143
$785$	0	0
$786$	0	0
$787$	− 44.0000i	− 1.56843i	−0.620489	−	0.784215i	$-0.713066\pi$
$787$	− 44.0000i	− 1.56843i	0.620489	−	0.784215i	$-0.286934\pi$
$788$	− 22.0000i	− 0.783718i
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	− 4.00000i	− 0.142044i
$794$	18.0000	0.638796
$795$	0	0
$796$	−24.0000	−0.850657
$797$	26.0000i	0.920967i	0.887668	+	0.460484i	$0.152324\pi$
$797$	26.0000i	0.920967i	−0.887668	+	0.460484i	$0.847676\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	30.0000i	1.05934i
$803$	− 24.0000i	− 0.846942i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	− 42.0000i	− 1.47755i
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	44.0000	1.54505	0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000	1.54505	0.772524	+	0.634985i	$0.218994\pi$
$812$	2.00000i	0.0701862i
$813$	0	0
$814$	24.0000	0.841200
$815$	0	0
$816$	0	0
$817$	− 16.0000i	− 0.559769i
$818$	22.0000i	0.769212i
$819$	0	0
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	− 24.0000i	− 0.836587i	−0.908312	−	0.418294i	$-0.862628\pi$
$823$	− 24.0000i	− 0.836587i	0.908312	−	0.418294i	$-0.137372\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	12.0000	0.417533
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−14.0000	−0.486240	−0.243120	−	0.969996i	$-0.578171\pi$
$829$	−14.0000	−0.486240	−0.243120	+	0.969996i	$0.578171\pi$
$830$	0	0
$831$	0	0
$832$	− 14.0000i	− 0.485363i
$833$	− 6.00000i	− 0.207888i
$834$	0	0
$835$	0	0
$836$	16.0000	0.553372
$837$	0	0
$838$	− 12.0000i	− 0.414533i
$839$	−8.00000	−0.276191	−0.138095	−	0.990419i	$-0.544098\pi$
$839$	−8.00000	−0.276191	−0.138095	+	0.990419i	$0.544098\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	38.0000i	1.30957i
$843$	0	0
$844$	4.00000	0.137686
$845$	0	0
$846$	0	0
$847$	− 5.00000i	− 0.171802i
$848$	− 6.00000i	− 0.206041i
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	10.0000i	0.342393i	0.985237	+	0.171197i	$0.0547634\pi$
$853$	10.0000i	0.342393i	−0.985237	+	0.171197i	$0.945237\pi$
$854$	−2.00000	−0.0684386
$855$	0	0
$856$	12.0000	0.410152
$857$	14.0000i	0.478231i	0.970991	+	0.239115i	$0.0768574\pi$
$857$	14.0000i	0.478231i	−0.970991	+	0.239115i	$0.923143\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	24.0000i	0.817443i
$863$	− 24.0000i	− 0.816970i	−0.912765	−	0.408485i	$-0.866057\pi$
$863$	− 24.0000i	− 0.816970i	0.912765	−	0.408485i	$-0.133943\pi$
$864$	0	0
$865$	0	0
$866$	−14.0000	−0.475739
$867$	0	0
$868$	0	0
$869$	−64.0000	−2.17105
$870$	0	0
$871$	−8.00000	−0.271070
$872$	54.0000i	1.82867i
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.0000i	1.55331i	0.629926	+	0.776655i	$0.283085\pi$
$877$	46.0000i	1.55331i	−0.629926	+	0.776655i	$0.716915\pi$
$878$	24.0000i	0.809961i
$879$	0	0
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	28.0000i	0.942275i	0.882060	+	0.471138i	$0.156156\pi$
$883$	28.0000i	0.942275i	−0.882060	+	0.471138i	$0.843844\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−36.0000	−1.20944
$887$	− 8.00000i	− 0.268614i	−0.990940	−	0.134307i	$-0.957119\pi$
$887$	− 8.00000i	− 0.268614i	0.990940	−	0.134307i	$-0.0428808\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	− 16.0000i	− 0.535720i
$893$	0	0
$894$	0	0
$895$	0	0
$896$	3.00000	0.100223
$897$	0	0
$898$	− 30.0000i	− 1.00111i
$899$	0	0
$900$	0	0
$901$	−36.0000	−1.19933
$902$	8.00000i	0.266371i
$903$	0	0
$904$	42.0000	1.39690
$905$	0	0
$906$	0	0
$907$	− 4.00000i	− 0.132818i	−0.997792	−	0.0664089i	$-0.978846\pi$
$907$	− 4.00000i	− 0.132818i	0.997792	−	0.0664089i	$-0.0211542\pi$
$908$	12.0000i	0.398234i
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	48.0000i	1.58857i
$914$	−10.0000	−0.330771
$915$	0	0
$916$	10.0000	0.330409
$917$	4.00000i	0.132092i
$918$	0	0
$919$	−8.00000	−0.263896	−0.131948	−	0.991257i	$-0.542123\pi$
$919$	−8.00000	−0.263896	−0.131948	+	0.991257i	$0.542123\pi$
$920$	0	0
$921$	0	0
$922$	10.0000i	0.329332i
$923$	0	0
$924$	0	0
$925$	0	0
$926$	16.0000	0.525793
$927$	0	0
$928$	− 10.0000i	− 0.328266i
$929$	26.0000	0.853032	0.426516	−	0.904480i	$-0.359741\pi$
$929$	26.0000	0.853032	0.426516	+	0.904480i	$0.359741\pi$
$930$	0	0
$931$	4.00000	0.131095
$932$	− 6.00000i	− 0.196537i
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	0	0
$937$	42.0000i	1.37208i	0.727564	+	0.686040i	$0.240653\pi$
$937$	42.0000i	1.37208i	−0.727564	+	0.686040i	$0.759347\pi$
$938$	4.00000i	0.130605i
$939$	0	0
$940$	0	0
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	0	0
$943$	0	0
$944$	−12.0000	−0.390567
$945$	0	0
$946$	16.0000	0.520205
$947$	− 44.0000i	− 1.42981i	−0.699223	−	0.714904i	$-0.746470\pi$
$947$	− 44.0000i	− 1.42981i	0.699223	−	0.714904i	$-0.253530\pi$
$948$	0	0
$949$	−12.0000	−0.389536
$950$	0	0
$951$	0	0
$952$	18.0000i	0.583383i
$953$	26.0000i	0.842223i	0.907009	+	0.421111i	$0.138360\pi$
$953$	26.0000i	0.842223i	−0.907009	+	0.421111i	$0.861640\pi$
$954$	0	0
$955$	0	0
$956$	24.0000	0.776215
$957$	0	0
$958$	− 16.0000i	− 0.516937i
$959$	6.00000	0.193750
$960$	0	0
$961$	−31.0000	−1.00000
$962$	− 12.0000i	− 0.386896i
$963$	0	0
$964$	2.00000	0.0644157
$965$	0	0
$966$	0	0
$967$	40.0000i	1.28631i	0.765735	+	0.643157i	$0.222376\pi$
$967$	40.0000i	1.28631i	−0.765735	+	0.643157i	$0.777624\pi$
$968$	15.0000i	0.482118i
$969$	0	0
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	0	0
$973$	12.0000i	0.384702i
$974$	8.00000	0.256337
$975$	0	0
$976$	2.00000	0.0640184
$977$	30.0000i	0.959785i	0.877327	+	0.479893i	$0.159324\pi$
$977$	30.0000i	0.959785i	−0.877327	+	0.479893i	$0.840676\pi$
$978$	0	0
$979$	56.0000	1.78977
$980$	0	0
$981$	0	0
$982$	− 20.0000i	− 0.638226i
$983$	24.0000i	0.765481i	0.923856	+	0.382741i	$0.125020\pi$
$983$	24.0000i	0.765481i	−0.923856	+	0.382741i	$0.874980\pi$
$984$	0	0
$985$	0	0
$986$	12.0000	0.382158
$987$	0	0
$988$	− 8.00000i	− 0.254514i
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 26.0000i	− 0.823428i	−0.911313	−	0.411714i	$-0.864930\pi$
$997$	− 26.0000i	− 0.823428i	0.911313	−	0.411714i	$-0.135070\pi$
$998$	− 4.00000i	− 0.126618i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1575.2.d.a.1324.2		2
3.2	odd	2		525.2.d.a.274.1		2
5.2	odd	4		1575.2.a.c.1.1		1
5.3	odd	4		63.2.a.a.1.1		1
5.4	even	2	inner	1575.2.d.a.1324.1		2
15.2	even	4		525.2.a.d.1.1		1
15.8	even	4		21.2.a.a.1.1	✓	1
15.14	odd	2		525.2.d.a.274.2		2
20.3	even	4		1008.2.a.l.1.1		1
35.3	even	12		441.2.e.b.226.1		2
35.13	even	4		441.2.a.f.1.1		1
35.18	odd	12		441.2.e.a.226.1		2
35.23	odd	12		441.2.e.a.361.1		2
35.33	even	12		441.2.e.b.361.1		2
40.3	even	4		4032.2.a.k.1.1		1
40.13	odd	4		4032.2.a.h.1.1		1
45.13	odd	12		567.2.f.b.379.1		2
45.23	even	12		567.2.f.g.379.1		2
45.38	even	12		567.2.f.g.190.1		2
45.43	odd	12		567.2.f.b.190.1		2
55.43	even	4		7623.2.a.g.1.1		1
60.23	odd	4		336.2.a.a.1.1		1
60.47	odd	4		8400.2.a.bn.1.1		1
105.23	even	12		147.2.e.b.67.1		2
105.38	odd	12		147.2.e.c.79.1		2
105.53	even	12		147.2.e.b.79.1		2
105.62	odd	4		3675.2.a.n.1.1		1
105.68	odd	12		147.2.e.c.67.1		2
105.83	odd	4		147.2.a.a.1.1		1
120.53	even	4		1344.2.a.g.1.1		1
120.83	odd	4		1344.2.a.s.1.1		1
140.83	odd	4		7056.2.a.p.1.1		1
165.98	odd	4		2541.2.a.j.1.1		1
195.38	even	4		3549.2.a.c.1.1		1
240.53	even	4		5376.2.c.r.2689.1		2
240.83	odd	4		5376.2.c.l.2689.1		2
240.173	even	4		5376.2.c.r.2689.2		2
240.203	odd	4		5376.2.c.l.2689.2		2
255.203	even	4		6069.2.a.b.1.1		1
285.113	odd	4		7581.2.a.d.1.1		1
420.23	odd	12		2352.2.q.x.1537.1		2
420.83	even	4		2352.2.a.v.1.1		1
420.143	even	12		2352.2.q.e.961.1		2
420.263	odd	12		2352.2.q.x.961.1		2
420.383	even	12		2352.2.q.e.1537.1		2
840.83	even	4		9408.2.a.m.1.1		1
840.293	odd	4		9408.2.a.bv.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.2.a.a.1.1	✓	1	15.8	even	4
63.2.a.a.1.1		1	5.3	odd	4
147.2.a.a.1.1		1	105.83	odd	4
147.2.e.b.67.1		2	105.23	even	12
147.2.e.b.79.1		2	105.53	even	12
147.2.e.c.67.1		2	105.68	odd	12
147.2.e.c.79.1		2	105.38	odd	12
336.2.a.a.1.1		1	60.23	odd	4
441.2.a.f.1.1		1	35.13	even	4
441.2.e.a.226.1		2	35.18	odd	12
441.2.e.a.361.1		2	35.23	odd	12
441.2.e.b.226.1		2	35.3	even	12
441.2.e.b.361.1		2	35.33	even	12
525.2.a.d.1.1		1	15.2	even	4
525.2.d.a.274.1		2	3.2	odd	2
525.2.d.a.274.2		2	15.14	odd	2
567.2.f.b.190.1		2	45.43	odd	12
567.2.f.b.379.1		2	45.13	odd	12
567.2.f.g.190.1		2	45.38	even	12
567.2.f.g.379.1		2	45.23	even	12
1008.2.a.l.1.1		1	20.3	even	4
1344.2.a.g.1.1		1	120.53	even	4
1344.2.a.s.1.1		1	120.83	odd	4
1575.2.a.c.1.1		1	5.2	odd	4
1575.2.d.a.1324.1		2	5.4	even	2	inner
1575.2.d.a.1324.2		2	1.1	even	1	trivial
2352.2.a.v.1.1		1	420.83	even	4
2352.2.q.e.961.1		2	420.143	even	12
2352.2.q.e.1537.1		2	420.383	even	12
2352.2.q.x.961.1		2	420.263	odd	12
2352.2.q.x.1537.1		2	420.23	odd	12
2541.2.a.j.1.1		1	165.98	odd	4
3549.2.a.c.1.1		1	195.38	even	4
3675.2.a.n.1.1		1	105.62	odd	4
4032.2.a.h.1.1		1	40.13	odd	4
4032.2.a.k.1.1		1	40.3	even	4
5376.2.c.l.2689.1		2	240.83	odd	4
5376.2.c.l.2689.2		2	240.203	odd	4
5376.2.c.r.2689.1		2	240.53	even	4
5376.2.c.r.2689.2		2	240.173	even	4
6069.2.a.b.1.1		1	255.203	even	4
7056.2.a.p.1.1		1	140.83	odd	4
7581.2.a.d.1.1		1	285.113	odd	4
7623.2.a.g.1.1		1	55.43	even	4
8400.2.a.bn.1.1		1	60.47	odd	4
9408.2.a.m.1.1		1	840.83	even	4
9408.2.a.bv.1.1		1	840.293	odd	4

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:	\( N \)	\(=\)	\( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1575.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{7} +3.00000i q^{8} +O(q^{10})\)	\(q+1.00000i q^{2} +1.00000 q^{4} -1.00000i q^{7} +3.00000i q^{8} -4.00000 q^{11} +2.00000i q^{13} +1.00000 q^{14} -1.00000 q^{16} +6.00000i q^{17} -4.00000 q^{19} -4.00000i q^{22} -2.00000 q^{26} -1.00000i q^{28} -2.00000 q^{29} +5.00000i q^{32} -6.00000 q^{34} +6.00000i q^{37} -4.00000i q^{38} -2.00000 q^{41} +4.00000i q^{43} -4.00000 q^{44} -1.00000 q^{49} +2.00000i q^{52} +6.00000i q^{53} +3.00000 q^{56} -2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -7.00000 q^{64} +4.00000i q^{67} +6.00000i q^{68} +6.00000i q^{73} -6.00000 q^{74} -4.00000 q^{76} +4.00000i q^{77} +16.0000 q^{79} -2.00000i q^{82} -12.0000i q^{83} -4.00000 q^{86} -12.0000i q^{88} -14.0000 q^{89} +2.00000 q^{91} +18.0000i q^{97} -1.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{4}+O(q^{10}) \) 2 * q + 2 * q^4	\( 2 q + 2 q^{4} - 8 q^{11} + 2 q^{14} - 2 q^{16} - 8 q^{19} - 4 q^{26} - 4 q^{29} - 12 q^{34} - 4 q^{41} - 8 q^{44} - 2 q^{49} + 6 q^{56} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 12 q^{74} - 8 q^{76} + 32 q^{79} - 8 q^{86} - 28 q^{89} + 4 q^{91}+O(q^{100}) \) 2 * q + 2 * q^4 - 8 * q^11 + 2 * q^14 - 2 * q^16 - 8 * q^19 - 4 * q^26 - 4 * q^29 - 12 * q^34 - 4 * q^41 - 8 * q^44 - 2 * q^49 + 6 * q^56 + 24 * q^59 - 4 * q^61 - 14 * q^64 - 12 * q^74 - 8 * q^76 + 32 * q^79 - 8 * q^86 - 28 * q^89 + 4 * q^91

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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