LMFDB - Embedded newform 832.2.f.b.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,2,Mod(129,832)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("832.129"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	832.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-2,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.64355344817$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 26)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			129.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	832.129
Dual form			832.2.f.b.129.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.00000i q^{5} +3.00000i q^{7} -2.00000 q^{9} +(-2.00000 + 3.00000i) q^{13} +3.00000i q^{15} +3.00000 q^{17} +6.00000i q^{19} -3.00000i q^{21} +6.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +9.00000 q^{35} +3.00000i q^{37} +(2.00000 - 3.00000i) q^{39} -1.00000 q^{43} +6.00000i q^{45} +3.00000i q^{47} -2.00000 q^{49} -3.00000 q^{51} +6.00000 q^{53} -6.00000i q^{57} +6.00000i q^{59} +8.00000 q^{61} -6.00000i q^{63} +(9.00000 + 6.00000i) q^{65} +12.0000i q^{67} -6.00000 q^{69} +15.0000i q^{71} -6.00000i q^{73} +4.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -9.00000i q^{85} +6.00000i q^{89} +(-9.00000 - 6.00000i) q^{91} +18.0000 q^{95} +12.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{9} - 4 q^{13} + 6 q^{17} + 12 q^{23} - 8 q^{25} + 10 q^{27} + 18 q^{35} + 4 q^{39} - 2 q^{43} - 4 q^{49} - 6 q^{51} + 12 q^{53} + 16 q^{61} + 18 q^{65} - 12 q^{69} + 8 q^{75} - 20 q^{79}+ \cdots + 36 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^9 - 4 * q^13 + 6 * q^17 + 12 * q^23 - 8 * q^25 + 10 * q^27 + 18 * q^35 + 4 * q^39 - 2 * q^43 - 4 * q^49 - 6 * q^51 + 12 * q^53 + 16 * q^61 + 18 * q^65 - 12 * q^69 + 8 * q^75 - 20 * q^79 + 2 * q^81 - 18 * q^91 + 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/832\mathbb{Z}\right)^\times$.

$n$	$261$	$703$	$769$
$\chi(n)$	$1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.00000			−0.577350			−0.288675	−	0.957427i	$-0.593215\pi$
$3$	−1.00000			−0.577350			−0.288675	+	0.957427i	$0.593215\pi$
$4$		0			0
$5$		−	3.00000i		−	1.34164i	−0.741620	−	0.670820i	$-0.765942\pi$
$5$		−	3.00000i		−	1.34164i	0.741620	−	0.670820i	$-0.234058\pi$
$6$		0			0
$7$			3.00000i			1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$			3.00000i			1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$		0			0
$9$	−2.00000			−0.666667
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$14$		0			0
$15$			3.00000i			0.774597i
$16$		0			0
$17$	3.00000			0.727607			0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000			0.727607			0.363803	+	0.931476i	$0.381478\pi$
$18$		0			0
$19$			6.00000i			1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$			6.00000i			1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		−	3.00000i		−	0.654654i
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$	5.00000			0.962250
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.00000			1.52128
$36$		0			0
$37$			3.00000i			0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$			3.00000i			0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$		0			0
$39$	2.00000	−	3.00000i	0.320256	−	0.480384i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−1.00000			−0.152499			−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000			−0.152499			−0.0762493	+	0.997089i	$0.524294\pi$
$44$		0			0
$45$			6.00000i			0.894427i
$46$		0			0
$47$			3.00000i			0.437595i	0.975770	+	0.218797i	$0.0702134\pi$
$47$			3.00000i			0.437595i	−0.975770	+	0.218797i	$0.929787\pi$
$48$		0			0
$49$	−2.00000			−0.285714
$50$		0			0
$51$	−3.00000			−0.420084
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		−	6.00000i		−	0.794719i
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$	8.00000			1.02430			0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000			1.02430			0.512148	+	0.858898i	$0.328850\pi$
$62$		0			0
$63$		−	6.00000i		−	0.755929i
$64$		0			0
$65$	9.00000	+	6.00000i	1.11631	+	0.744208i
$66$		0			0
$67$			12.0000i			1.46603i	0.680211	+	0.733017i	$0.261888\pi$
$67$			12.0000i			1.46603i	−0.680211	+	0.733017i	$0.738112\pi$
$68$		0			0
$69$	−6.00000			−0.722315
$70$		0			0
$71$			15.0000i			1.78017i	0.455792	+	0.890086i	$0.349356\pi$
$71$			15.0000i			1.78017i	−0.455792	+	0.890086i	$0.650644\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$	4.00000			0.461880
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−10.0000			−1.12509			−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000			−1.12509			−0.562544	+	0.826767i	$0.690177\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$		−	9.00000i		−	0.976187i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	−9.00000	−	6.00000i	−0.943456	−	0.628971i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	18.0000			1.84676
$96$		0			0
$97$			12.0000i			1.21842i	0.793011	+	0.609208i	$0.208512\pi$
$97$			12.0000i			1.21842i	−0.793011	+	0.609208i	$0.791488\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−12.0000			−1.19404			−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000			−1.19404			−0.597022	+	0.802225i	$0.703650\pi$
$102$		0			0
$103$	−14.0000			−1.37946			−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000			−1.37946			−0.689730	+	0.724066i	$0.742271\pi$
$104$		0			0
$105$	−9.00000			−0.878310
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$			9.00000i			0.862044i	0.902342	+	0.431022i	$0.141847\pi$
$109$			9.00000i			0.862044i	−0.902342	+	0.431022i	$0.858153\pi$
$110$		0			0
$111$		−	3.00000i		−	0.284747i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$		−	18.0000i		−	1.67851i
$116$		0			0
$117$	4.00000	−	6.00000i	0.369800	−	0.554700i
$118$		0			0
$119$			9.00000i			0.825029i
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$		0			0
$124$		0			0
$125$		−	3.00000i		−	0.268328i
$126$		0			0
$127$	2.00000			0.177471			0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000			0.177471			0.0887357	+	0.996055i	$0.471717\pi$
$128$		0			0
$129$	1.00000			0.0880451
$130$		0			0
$131$	−3.00000			−0.262111			−0.131056	−	0.991375i	$-0.541837\pi$
$131$	−3.00000			−0.262111			−0.131056	+	0.991375i	$0.541837\pi$
$132$		0			0
$133$	−18.0000			−1.56080
$134$		0			0
$135$		−	15.0000i		−	1.29099i
$136$		0			0
$137$		−	18.0000i		−	1.53784i	−0.639343	−	0.768922i	$-0.720793\pi$
$137$		−	18.0000i		−	1.53784i	0.639343	−	0.768922i	$-0.279207\pi$
$138$		0			0
$139$	5.00000			0.424094			0.212047	−	0.977259i	$-0.431987\pi$
$139$	5.00000			0.424094			0.212047	+	0.977259i	$0.431987\pi$
$140$		0			0
$141$		−	3.00000i		−	0.252646i
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	2.00000			0.164957
$148$		0			0
$149$		−	6.00000i		−	0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$		−	6.00000i		−	0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$		0			0
$151$		−	15.0000i		−	1.22068i	−0.792139	−	0.610341i	$-0.791032\pi$
$151$		−	15.0000i		−	1.22068i	0.792139	−	0.610341i	$-0.208968\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$		0			0
$156$		0			0
$157$	22.0000			1.75579			0.877896	−	0.478852i	$-0.158947\pi$
$157$	22.0000			1.75579			0.877896	+	0.478852i	$0.158947\pi$
$158$		0			0
$159$	−6.00000			−0.475831
$160$		0			0
$161$			18.0000i			1.41860i
$162$		0			0
$163$		−	6.00000i		−	0.469956i	−0.972001	−	0.234978i	$-0.924498\pi$
$163$		−	6.00000i		−	0.469956i	0.972001	−	0.234978i	$-0.0755019\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	12.0000i		−	0.928588i	−0.885681	−	0.464294i	$-0.846308\pi$
$167$		−	12.0000i		−	0.928588i	0.885681	−	0.464294i	$-0.153692\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$		−	12.0000i		−	0.917663i
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$		−	12.0000i		−	0.907115i
$176$		0			0
$177$		−	6.00000i		−	0.450988i
$178$		0			0
$179$	15.0000			1.12115			0.560576	−	0.828103i	$-0.310580\pi$
$179$	15.0000			1.12115			0.560576	+	0.828103i	$0.310580\pi$
$180$		0			0
$181$	−2.00000			−0.148659			−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000			−0.148659			−0.0743294	+	0.997234i	$0.523682\pi$
$182$		0			0
$183$	−8.00000			−0.591377
$184$		0			0
$185$	9.00000			0.661693
$186$		0			0
$187$		0			0
$188$		0			0
$189$			15.0000i			1.09109i
$190$		0			0
$191$	−12.0000			−0.868290			−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000			−0.868290			−0.434145	+	0.900843i	$0.642949\pi$
$192$		0			0
$193$		−	6.00000i		−	0.431889i	−0.976406	−	0.215945i	$-0.930717\pi$
$193$		−	6.00000i		−	0.431889i	0.976406	−	0.215945i	$-0.0692831\pi$
$194$		0			0
$195$	−9.00000	−	6.00000i	−0.644503	−	0.429669i
$196$		0			0
$197$			3.00000i			0.213741i	0.994273	+	0.106871i	$0.0340831\pi$
$197$			3.00000i			0.213741i	−0.994273	+	0.106871i	$0.965917\pi$
$198$		0			0
$199$	20.0000			1.41776			0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000			1.41776			0.708881	+	0.705328i	$0.249200\pi$
$200$		0			0
$201$		−	12.0000i		−	0.846415i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−12.0000			−0.834058
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−23.0000			−1.58339			−0.791693	−	0.610920i	$-0.790800\pi$
$211$	−23.0000			−1.58339			−0.791693	+	0.610920i	$0.790800\pi$
$212$		0			0
$213$		−	15.0000i		−	1.02778i
$214$		0			0
$215$			3.00000i			0.204598i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			6.00000i			0.405442i
$220$		0			0
$221$	−6.00000	+	9.00000i	−0.403604	+	0.605406i
$222$		0			0
$223$		−	9.00000i		−	0.602685i	−0.953516	−	0.301342i	$-0.902565\pi$
$223$		−	9.00000i		−	0.602685i	0.953516	−	0.301342i	$-0.0974347\pi$
$224$		0			0
$225$	8.00000			0.533333
$226$		0			0
$227$			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
$229$			9.00000i			0.594737i	0.954763	+	0.297368i	$0.0961089\pi$
$229$			9.00000i			0.594737i	−0.954763	+	0.297368i	$0.903891\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−21.0000			−1.37576			−0.687878	−	0.725826i	$-0.741458\pi$
$233$	−21.0000			−1.37576			−0.687878	+	0.725826i	$0.741458\pi$
$234$		0			0
$235$	9.00000			0.587095
$236$		0			0
$237$	10.0000			0.649570
$238$		0			0
$239$			9.00000i			0.582162i	0.956698	+	0.291081i	$0.0940149\pi$
$239$			9.00000i			0.582162i	−0.956698	+	0.291081i	$0.905985\pi$
$240$		0			0
$241$			30.0000i			1.93247i	0.257663	+	0.966235i	$0.417048\pi$
$241$			30.0000i			1.93247i	−0.257663	+	0.966235i	$0.582952\pi$
$242$		0			0
$243$	−16.0000			−1.02640
$244$		0			0
$245$			6.00000i			0.383326i
$246$		0			0
$247$	−18.0000	−	12.0000i	−1.14531	−	0.763542i
$248$		0			0
$249$			6.00000i			0.380235i
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$			9.00000i			0.563602i
$256$		0			0
$257$	3.00000			0.187135			0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000			0.187135			0.0935674	+	0.995613i	$0.470173\pi$
$258$		0			0
$259$	−9.00000			−0.559233
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−24.0000			−1.47990			−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000			−1.47990			−0.739952	+	0.672660i	$0.765152\pi$
$264$		0			0
$265$		−	18.0000i		−	1.10573i
$266$		0			0
$267$		−	6.00000i		−	0.367194i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			15.0000i			0.911185i	0.890188	+	0.455593i	$0.150573\pi$
$271$			15.0000i			0.911185i	−0.890188	+	0.455593i	$0.849427\pi$
$272$		0			0
$273$	9.00000	+	6.00000i	0.544705	+	0.363137i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−8.00000			−0.480673			−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000			−0.480673			−0.240337	+	0.970690i	$0.577258\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			30.0000i			1.78965i	0.446417	+	0.894825i	$0.352700\pi$
$281$			30.0000i			1.78965i	−0.446417	+	0.894825i	$0.647300\pi$
$282$		0			0
$283$	4.00000			0.237775			0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000			0.237775			0.118888	+	0.992908i	$0.462067\pi$
$284$		0			0
$285$	−18.0000			−1.06623
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.00000			−0.470588
$290$		0			0
$291$		−	12.0000i		−	0.703452i
$292$		0			0
$293$		−	9.00000i		−	0.525786i	−0.964825	−	0.262893i	$-0.915323\pi$
$293$		−	9.00000i		−	0.525786i	0.964825	−	0.262893i	$-0.0846766\pi$
$294$		0			0
$295$	18.0000			1.04800
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−12.0000	+	18.0000i	−0.693978	+	1.04097i
$300$		0			0
$301$		−	3.00000i		−	0.172917i
$302$		0			0
$303$	12.0000			0.689382
$304$		0			0
$305$		−	24.0000i		−	1.37424i
$306$		0			0
$307$		−	18.0000i		−	1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$		−	18.0000i		−	1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$		0			0
$309$	14.0000			0.796432
$310$		0			0
$311$	18.0000			1.02069			0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000			1.02069			0.510343	+	0.859971i	$0.329518\pi$
$312$		0			0
$313$	19.0000			1.07394			0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000			1.07394			0.536972	+	0.843600i	$0.319568\pi$
$314$		0			0
$315$	−18.0000			−1.01419
$316$		0			0
$317$			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
$319$		0			0
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$			18.0000i			1.00155i
$324$		0			0
$325$	8.00000	−	12.0000i	0.443760	−	0.665640i
$326$		0			0
$327$		−	9.00000i		−	0.497701i
$328$		0			0
$329$	−9.00000			−0.496186
$330$		0			0
$331$		−	30.0000i		−	1.64895i	−0.565899	−	0.824475i	$-0.691471\pi$
$331$		−	30.0000i		−	1.64895i	0.565899	−	0.824475i	$-0.308529\pi$
$332$		0			0
$333$		−	6.00000i		−	0.328798i
$334$		0			0
$335$	36.0000			1.96689
$336$		0			0
$337$	13.0000			0.708155			0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000			0.708155			0.354078	+	0.935216i	$0.384795\pi$
$338$		0			0
$339$	6.00000			0.325875
$340$		0			0
$341$		0			0
$342$		0			0
$343$			15.0000i			0.809924i
$344$		0			0
$345$			18.0000i			0.969087i
$346$		0			0
$347$	33.0000			1.77153			0.885766	−	0.464131i	$-0.153633\pi$
$347$	33.0000			1.77153			0.885766	+	0.464131i	$0.153633\pi$
$348$		0			0
$349$		−	21.0000i		−	1.12410i	−0.827102	−	0.562052i	$-0.810012\pi$
$349$		−	21.0000i		−	1.12410i	0.827102	−	0.562052i	$-0.189988\pi$
$350$		0			0
$351$	−10.0000	+	15.0000i	−0.533761	+	0.800641i
$352$		0			0
$353$		−	6.00000i		−	0.319348i	−0.987170	−	0.159674i	$-0.948956\pi$
$353$		−	6.00000i		−	0.319348i	0.987170	−	0.159674i	$-0.0510443\pi$
$354$		0			0
$355$	45.0000			2.38835
$356$		0			0
$357$		−	9.00000i		−	0.476331i
$358$		0			0
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$	−11.0000			−0.577350
$364$		0			0
$365$	−18.0000			−0.942163
$366$		0			0
$367$	−8.00000			−0.417597			−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000			−0.417597			−0.208798	+	0.977959i	$0.566955\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			18.0000i			0.934513i
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$		0			0
$375$			3.00000i			0.154919i
$376$		0			0
$377$		0			0
$378$		0			0
$379$			6.00000i			0.308199i	0.988055	+	0.154100i	$0.0492477\pi$
$379$			6.00000i			0.308199i	−0.988055	+	0.154100i	$0.950752\pi$
$380$		0			0
$381$	−2.00000			−0.102463
$382$		0			0
$383$		−	9.00000i		−	0.459879i	−0.973205	−	0.229939i	$-0.926147\pi$
$383$		−	9.00000i		−	0.459879i	0.973205	−	0.229939i	$-0.0738528\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.00000			0.101666
$388$		0			0
$389$	30.0000			1.52106			0.760530	−	0.649303i	$-0.224939\pi$
$389$	30.0000			1.52106			0.760530	+	0.649303i	$0.224939\pi$
$390$		0			0
$391$	18.0000			0.910299
$392$		0			0
$393$	3.00000			0.151330
$394$		0			0
$395$			30.0000i			1.50946i
$396$		0			0
$397$			18.0000i			0.903394i	0.892171	+	0.451697i	$0.149181\pi$
$397$			18.0000i			0.903394i	−0.892171	+	0.451697i	$0.850819\pi$
$398$		0			0
$399$	18.0000			0.901127
$400$		0			0
$401$		−	30.0000i		−	1.49813i	−0.662497	−	0.749064i	$-0.730503\pi$
$401$		−	30.0000i		−	1.49813i	0.662497	−	0.749064i	$-0.269497\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		−	3.00000i		−	0.149071i
$406$		0			0
$407$		0			0
$408$		0			0
$409$			6.00000i			0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$		0			0
$411$			18.0000i			0.887875i
$412$		0			0
$413$	−18.0000			−0.885722
$414$		0			0
$415$	−18.0000			−0.883585
$416$		0			0
$417$	−5.00000			−0.244851
$418$		0			0
$419$	15.0000			0.732798			0.366399	−	0.930458i	$-0.380591\pi$
$419$	15.0000			0.732798			0.366399	+	0.930458i	$0.380591\pi$
$420$		0			0
$421$		−	15.0000i		−	0.731055i	−0.930800	−	0.365528i	$-0.880889\pi$
$421$		−	15.0000i		−	0.731055i	0.930800	−	0.365528i	$-0.119111\pi$
$422$		0			0
$423$		−	6.00000i		−	0.291730i
$424$		0			0
$425$	−12.0000			−0.582086
$426$		0			0
$427$			24.0000i			1.16144i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	15.0000i		−	0.722525i	−0.932464	−	0.361262i	$-0.882346\pi$
$431$		−	15.0000i		−	0.722525i	0.932464	−	0.361262i	$-0.117654\pi$
$432$		0			0
$433$	−11.0000			−0.528626			−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000			−0.528626			−0.264313	+	0.964437i	$0.585145\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			36.0000i			1.72211i
$438$		0			0
$439$	10.0000			0.477274			0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000			0.477274			0.238637	+	0.971109i	$0.423299\pi$
$440$		0			0
$441$	4.00000			0.190476
$442$		0			0
$443$	−21.0000			−0.997740			−0.498870	−	0.866677i	$-0.666252\pi$
$443$	−21.0000			−0.997740			−0.498870	+	0.866677i	$0.666252\pi$
$444$		0			0
$445$	18.0000			0.853282
$446$		0			0
$447$			6.00000i			0.283790i
$448$		0			0
$449$		−	24.0000i		−	1.13263i	−0.824189	−	0.566315i	$-0.808369\pi$
$449$		−	24.0000i		−	1.13263i	0.824189	−	0.566315i	$-0.191631\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			15.0000i			0.704761i
$454$		0			0
$455$	−18.0000	+	27.0000i	−0.843853	+	1.26578i
$456$		0			0
$457$		−	18.0000i		−	0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$		−	18.0000i		−	0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$		0			0
$459$	15.0000			0.700140
$460$		0			0
$461$		−	15.0000i		−	0.698620i	−0.937007	−	0.349310i	$-0.886416\pi$
$461$		−	15.0000i		−	0.698620i	0.937007	−	0.349310i	$-0.113584\pi$
$462$		0			0
$463$		−	24.0000i		−	1.11537i	−0.830051	−	0.557687i	$-0.811689\pi$
$463$		−	24.0000i		−	1.11537i	0.830051	−	0.557687i	$-0.188311\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−36.0000			−1.66233
$470$		0			0
$471$	−22.0000			−1.01371
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	24.0000i		−	1.10120i
$476$		0			0
$477$	−12.0000			−0.549442
$478$		0			0
$479$			39.0000i			1.78196i	0.454047	+	0.890978i	$0.349980\pi$
$479$			39.0000i			1.78196i	−0.454047	+	0.890978i	$0.650020\pi$
$480$		0			0
$481$	−9.00000	−	6.00000i	−0.410365	−	0.273576i
$482$		0			0
$483$		−	18.0000i		−	0.819028i
$484$		0			0
$485$	36.0000			1.63468
$486$		0			0
$487$		−	12.0000i		−	0.543772i	−0.962329	−	0.271886i	$-0.912353\pi$
$487$		−	12.0000i		−	0.543772i	0.962329	−	0.271886i	$-0.0876473\pi$
$488$		0			0
$489$			6.00000i			0.271329i
$490$		0			0
$491$	27.0000			1.21849			0.609246	−	0.792981i	$-0.291472\pi$
$491$	27.0000			1.21849			0.609246	+	0.792981i	$0.291472\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−45.0000			−2.01853
$498$		0			0
$499$			36.0000i			1.61158i	0.592200	+	0.805791i	$0.298259\pi$
$499$			36.0000i			1.61158i	−0.592200	+	0.805791i	$0.701741\pi$
$500$		0			0
$501$			12.0000i			0.536120i
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$			36.0000i			1.60198i
$506$		0			0
$507$	5.00000	+	12.0000i	0.222058	+	0.532939i
$508$		0			0
$509$		−	6.00000i		−	0.265945i	−0.991120	−	0.132973i	$-0.957548\pi$
$509$		−	6.00000i		−	0.265945i	0.991120	−	0.132973i	$-0.0424523\pi$
$510$		0			0
$511$	18.0000			0.796273
$512$		0			0
$513$			30.0000i			1.32453i
$514$		0			0
$515$			42.0000i			1.85074i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−6.00000			−0.263371
$520$		0			0
$521$	27.0000			1.18289			0.591446	−	0.806345i	$-0.298557\pi$
$521$	27.0000			1.18289			0.591446	+	0.806345i	$0.298557\pi$
$522$		0			0
$523$	−16.0000			−0.699631			−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000			−0.699631			−0.349816	+	0.936819i	$0.613756\pi$
$524$		0			0
$525$			12.0000i			0.523723i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		−	12.0000i		−	0.520756i
$532$		0			0
$533$		0			0
$534$		0			0
$535$			36.0000i			1.55642i
$536$		0			0
$537$	−15.0000			−0.647298
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	15.0000i		−	0.644900i	−0.946586	−	0.322450i	$-0.895494\pi$
$541$		−	15.0000i		−	0.644900i	0.946586	−	0.322450i	$-0.104506\pi$
$542$		0			0
$543$	2.00000			0.0858282
$544$		0			0
$545$	27.0000			1.15655
$546$		0			0
$547$	−37.0000			−1.58201			−0.791003	−	0.611812i	$-0.790441\pi$
$547$	−37.0000			−1.58201			−0.791003	+	0.611812i	$0.790441\pi$
$548$		0			0
$549$	−16.0000			−0.682863
$550$		0			0
$551$		0			0
$552$		0			0
$553$		−	30.0000i		−	1.27573i
$554$		0			0
$555$	−9.00000			−0.382029
$556$		0			0
$557$		−	27.0000i		−	1.14403i	−0.820244	−	0.572013i	$-0.806163\pi$
$557$		−	27.0000i		−	1.14403i	0.820244	−	0.572013i	$-0.193837\pi$
$558$		0			0
$559$	2.00000	−	3.00000i	0.0845910	−	0.126886i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	39.0000			1.64365			0.821827	−	0.569737i	$-0.192955\pi$
$563$	39.0000			1.64365			0.821827	+	0.569737i	$0.192955\pi$
$564$		0			0
$565$			18.0000i			0.757266i
$566$		0			0
$567$			3.00000i			0.125988i
$568$		0			0
$569$	45.0000			1.88650			0.943249	−	0.332086i	$-0.107752\pi$
$569$	45.0000			1.88650			0.943249	+	0.332086i	$0.107752\pi$
$570$		0			0
$571$	−23.0000			−0.962520			−0.481260	−	0.876578i	$-0.659821\pi$
$571$	−23.0000			−0.962520			−0.481260	+	0.876578i	$0.659821\pi$
$572$		0			0
$573$	12.0000			0.501307
$574$		0			0
$575$	−24.0000			−1.00087
$576$		0			0
$577$			42.0000i			1.74848i	0.485491	+	0.874241i	$0.338641\pi$
$577$			42.0000i			1.74848i	−0.485491	+	0.874241i	$0.661359\pi$
$578$		0			0
$579$			6.00000i			0.249351i
$580$		0			0
$581$	18.0000			0.746766
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−18.0000	−	12.0000i	−0.744208	−	0.496139i
$586$		0			0
$587$		−	18.0000i		−	0.742940i	−0.928445	−	0.371470i	$-0.878854\pi$
$587$		−	18.0000i		−	0.742940i	0.928445	−	0.371470i	$-0.121146\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		−	3.00000i		−	0.123404i
$592$		0			0
$593$		−	36.0000i		−	1.47834i	−0.673517	−	0.739171i	$-0.735217\pi$
$593$		−	36.0000i		−	1.47834i	0.673517	−	0.739171i	$-0.264783\pi$
$594$		0			0
$595$	27.0000			1.10689
$596$		0			0
$597$	−20.0000			−0.818546
$598$		0			0
$599$	30.0000			1.22577			0.612883	−	0.790173i	$-0.290010\pi$
$599$	30.0000			1.22577			0.612883	+	0.790173i	$0.290010\pi$
$600$		0			0
$601$	37.0000			1.50926			0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000			1.50926			0.754631	+	0.656150i	$0.227816\pi$
$602$		0			0
$603$		−	24.0000i		−	0.977356i
$604$		0			0
$605$		−	33.0000i		−	1.34164i
$606$		0			0
$607$	22.0000			0.892952			0.446476	−	0.894795i	$-0.352679\pi$
$607$	22.0000			0.892952			0.446476	+	0.894795i	$0.352679\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−9.00000	−	6.00000i	−0.364101	−	0.242734i
$612$		0			0
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			12.0000i			0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$			12.0000i			0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$		0			0
$619$		−	24.0000i		−	0.964641i	−0.875995	−	0.482321i	$-0.839794\pi$
$619$		−	24.0000i		−	0.964641i	0.875995	−	0.482321i	$-0.160206\pi$
$620$		0			0
$621$	30.0000			1.20386
$622$		0			0
$623$	−18.0000			−0.721155
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			9.00000i			0.358854i
$630$		0			0
$631$		−	15.0000i		−	0.597141i	−0.954388	−	0.298570i	$-0.903490\pi$
$631$		−	15.0000i		−	0.597141i	0.954388	−	0.298570i	$-0.0965097\pi$
$632$		0			0
$633$	23.0000			0.914168
$634$		0			0
$635$		−	6.00000i		−	0.238103i
$636$		0			0
$637$	4.00000	−	6.00000i	0.158486	−	0.237729i
$638$		0			0
$639$		−	30.0000i		−	1.18678i
$640$		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
$643$		−	36.0000i		−	1.41970i	−0.704352	−	0.709851i	$-0.748762\pi$
$643$		−	36.0000i		−	1.41970i	0.704352	−	0.709851i	$-0.251238\pi$
$644$		0			0
$645$		−	3.00000i		−	0.118125i
$646$		0			0
$647$	42.0000			1.65119			0.825595	−	0.564263i	$-0.190840\pi$
$647$	42.0000			1.65119			0.825595	+	0.564263i	$0.190840\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$	36.0000			1.40879			0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000			1.40879			0.704394	+	0.709809i	$0.251219\pi$
$654$		0			0
$655$			9.00000i			0.351659i
$656$		0			0
$657$			12.0000i			0.468165i
$658$		0			0
$659$		0			0			−	1.00000i	$-0.5\pi$
$659$		0			0				1.00000i	$0.5\pi$
$660$		0			0
$661$			30.0000i			1.16686i	0.812162	+	0.583432i	$0.198291\pi$
$661$			30.0000i			1.16686i	−0.812162	+	0.583432i	$0.801709\pi$
$662$		0			0
$663$	6.00000	−	9.00000i	0.233021	−	0.349531i
$664$		0			0
$665$			54.0000i			2.09403i
$666$		0			0
$667$		0			0
$668$		0			0
$669$			9.00000i			0.347960i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$		0			0
$675$	−20.0000			−0.769800
$676$		0			0
$677$	−18.0000			−0.691796			−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000			−0.691796			−0.345898	+	0.938272i	$0.612426\pi$
$678$		0			0
$679$	−36.0000			−1.38155
$680$		0			0
$681$		−	12.0000i		−	0.459841i
$682$		0			0
$683$		−	6.00000i		−	0.229584i	−0.993390	−	0.114792i	$-0.963380\pi$
$683$		−	6.00000i		−	0.229584i	0.993390	−	0.114792i	$-0.0366201\pi$
$684$		0			0
$685$	−54.0000			−2.06323
$686$		0			0
$687$		−	9.00000i		−	0.343371i
$688$		0			0
$689$	−12.0000	+	18.0000i	−0.457164	+	0.685745i
$690$		0			0
$691$		0			0		1.00000			$0$
$691$		0			0		−1.00000			$\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	15.0000i		−	0.568982i
$696$		0			0
$697$		0			0
$698$		0			0
$699$	21.0000			0.794293
$700$		0			0
$701$	−42.0000			−1.58632			−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000			−1.58632			−0.793159	+	0.609015i	$0.791565\pi$
$702$		0			0
$703$	−18.0000			−0.678883
$704$		0			0
$705$	−9.00000			−0.338960
$706$		0			0
$707$		−	36.0000i		−	1.35392i
$708$		0			0
$709$		−	6.00000i		−	0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$		−	6.00000i		−	0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$		0			0
$711$	20.0000			0.750059
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		−	9.00000i		−	0.336111i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		−	42.0000i		−	1.56416i
$722$		0			0
$723$		−	30.0000i		−	1.11571i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−28.0000			−1.03846			−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000			−1.03846			−0.519231	+	0.854634i	$0.673782\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−3.00000			−0.110959
$732$		0			0
$733$		−	9.00000i		−	0.332423i	−0.986090	−	0.166211i	$-0.946847\pi$
$733$		−	9.00000i		−	0.332423i	0.986090	−	0.166211i	$-0.0531534\pi$
$734$		0			0
$735$		−	6.00000i		−	0.221313i
$736$		0			0
$737$		0			0
$738$		0			0
$739$			36.0000i			1.32428i	0.749380	+	0.662141i	$0.230352\pi$
$739$			36.0000i			1.32428i	−0.749380	+	0.662141i	$0.769648\pi$
$740$		0			0
$741$	18.0000	+	12.0000i	0.661247	+	0.440831i
$742$		0			0
$743$		−	39.0000i		−	1.43077i	−0.698730	−	0.715386i	$-0.746251\pi$
$743$		−	39.0000i		−	1.43077i	0.698730	−	0.715386i	$-0.253749\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$		0			0
$747$			12.0000i			0.439057i
$748$		0			0
$749$		−	36.0000i		−	1.31541i
$750$		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
$753$	−12.0000			−0.437304
$754$		0			0
$755$	−45.0000			−1.63772
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			30.0000i			1.08750i	0.839248	+	0.543750i	$0.182996\pi$
$761$			30.0000i			1.08750i	−0.839248	+	0.543750i	$0.817004\pi$
$762$		0			0
$763$	−27.0000			−0.977466
$764$		0			0
$765$			18.0000i			0.650791i
$766$		0			0
$767$	−18.0000	−	12.0000i	−0.649942	−	0.433295i
$768$		0			0
$769$		−	24.0000i		−	0.865462i	−0.901523	−	0.432731i	$-0.857550\pi$
$769$		−	24.0000i		−	0.865462i	0.901523	−	0.432731i	$-0.142450\pi$
$770$		0			0
$771$	−3.00000			−0.108042
$772$		0			0
$773$			21.0000i			0.755318i	0.925945	+	0.377659i	$0.123271\pi$
$773$			21.0000i			0.755318i	−0.925945	+	0.377659i	$0.876729\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	9.00000			0.322873
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	66.0000i		−	2.35564i
$786$		0			0
$787$			12.0000i			0.427754i	0.976861	+	0.213877i	$0.0686091\pi$
$787$			12.0000i			0.427754i	−0.976861	+	0.213877i	$0.931391\pi$
$788$		0			0
$789$	24.0000			0.854423
$790$		0			0
$791$		−	18.0000i		−	0.640006i
$792$		0			0
$793$	−16.0000	+	24.0000i	−0.568177	+	0.852265i
$794$		0			0
$795$			18.0000i			0.638394i
$796$		0			0
$797$	−18.0000			−0.637593			−0.318796	−	0.947823i	$-0.603279\pi$
$797$	−18.0000			−0.637593			−0.318796	+	0.947823i	$0.603279\pi$
$798$		0			0
$799$			9.00000i			0.318397i
$800$		0			0
$801$		−	12.0000i		−	0.423999i
$802$		0			0
$803$		0			0
$804$		0			0
$805$	54.0000			1.90325
$806$		0			0
$807$		0			0
$808$		0			0
$809$	15.0000			0.527372			0.263686	−	0.964609i	$-0.415062\pi$
$809$	15.0000			0.527372			0.263686	+	0.964609i	$0.415062\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		−	15.0000i		−	0.526073i
$814$		0			0
$815$	−18.0000			−0.630512
$816$		0			0
$817$		−	6.00000i		−	0.209913i
$818$		0			0
$819$	18.0000	+	12.0000i	0.628971	+	0.419314i
$820$		0			0
$821$			45.0000i			1.57051i	0.619172	+	0.785255i	$0.287468\pi$
$821$			45.0000i			1.57051i	−0.619172	+	0.785255i	$0.712532\pi$
$822$		0			0
$823$	−14.0000			−0.488009			−0.244005	−	0.969774i	$-0.578461\pi$
$823$	−14.0000			−0.488009			−0.244005	+	0.969774i	$0.578461\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	18.0000i		−	0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$		−	18.0000i		−	0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$		0			0
$829$	20.0000			0.694629			0.347314	−	0.937749i	$-0.387094\pi$
$829$	20.0000			0.694629			0.347314	+	0.937749i	$0.387094\pi$
$830$		0			0
$831$	8.00000			0.277517
$832$		0			0
$833$	−6.00000			−0.207888
$834$		0			0
$835$	−36.0000			−1.24583
$836$		0			0
$837$		0			0
$838$		0			0
$839$			24.0000i			0.828572i	0.910147	+	0.414286i	$0.135969\pi$
$839$			24.0000i			0.828572i	−0.910147	+	0.414286i	$0.864031\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		−	30.0000i		−	1.03325i
$844$		0			0
$845$	−36.0000	+	15.0000i	−1.23844	+	0.516016i
$846$		0			0
$847$			33.0000i			1.13389i
$848$		0			0
$849$	−4.00000			−0.137280
$850$		0			0
$851$			18.0000i			0.617032i
$852$		0			0
$853$		−	39.0000i		−	1.33533i	−0.744460	−	0.667667i	$-0.767293\pi$
$853$		−	39.0000i		−	1.33533i	0.744460	−	0.667667i	$-0.232707\pi$
$854$		0			0
$855$	−36.0000			−1.23117
$856$		0			0
$857$	18.0000			0.614868			0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000			0.614868			0.307434	+	0.951569i	$0.400530\pi$
$858$		0			0
$859$	40.0000			1.36478			0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000			1.36478			0.682391	+	0.730987i	$0.260940\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	39.0000i		−	1.32758i	−0.747921	−	0.663788i	$-0.768948\pi$
$863$		−	39.0000i		−	1.32758i	0.747921	−	0.663788i	$-0.231052\pi$
$864$		0			0
$865$		−	18.0000i		−	0.612018i
$866$		0			0
$867$	8.00000			0.271694
$868$		0			0
$869$		0			0
$870$		0			0
$871$	−36.0000	−	24.0000i	−1.21981	−	0.813209i
$872$		0			0
$873$		−	24.0000i		−	0.812277i
$874$		0			0
$875$	9.00000			0.304256
$876$		0			0
$877$			3.00000i			0.101303i	0.998716	+	0.0506514i	$0.0161297\pi$
$877$			3.00000i			0.101303i	−0.998716	+	0.0506514i	$0.983870\pi$
$878$		0			0
$879$			9.00000i			0.303562i
$880$		0			0
$881$	−33.0000			−1.11180			−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000			−1.11180			−0.555899	+	0.831250i	$0.687626\pi$
$882$		0			0
$883$	−11.0000			−0.370179			−0.185090	−	0.982722i	$-0.559258\pi$
$883$	−11.0000			−0.370179			−0.185090	+	0.982722i	$0.559258\pi$
$884$		0			0
$885$	−18.0000			−0.605063
$886$		0			0
$887$	12.0000			0.402921			0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000			0.402921			0.201460	+	0.979497i	$0.435431\pi$
$888$		0			0
$889$			6.00000i			0.201234i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−18.0000			−0.602347
$894$		0			0
$895$		−	45.0000i		−	1.50418i
$896$		0			0
$897$	12.0000	−	18.0000i	0.400668	−	0.601003i
$898$		0			0
$899$		0			0
$900$		0			0
$901$	18.0000			0.599667
$902$		0			0
$903$			3.00000i			0.0998337i
$904$		0			0
$905$			6.00000i			0.199447i
$906$		0			0
$907$	−17.0000			−0.564476			−0.282238	−	0.959344i	$-0.591077\pi$
$907$	−17.0000			−0.564476			−0.282238	+	0.959344i	$0.591077\pi$
$908$		0			0
$909$	24.0000			0.796030
$910$		0			0
$911$	−12.0000			−0.397578			−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000			−0.397578			−0.198789	+	0.980042i	$0.563701\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$			24.0000i			0.793416i
$916$		0			0
$917$		−	9.00000i		−	0.297206i
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$		0			0
$921$			18.0000i			0.593120i
$922$		0			0
$923$	−45.0000	−	30.0000i	−1.48119	−	0.987462i
$924$		0			0
$925$		−	12.0000i		−	0.394558i
$926$		0			0
$927$	28.0000			0.919641
$928$		0			0
$929$			36.0000i			1.18112i	0.806993	+	0.590561i	$0.201093\pi$
$929$			36.0000i			1.18112i	−0.806993	+	0.590561i	$0.798907\pi$
$930$		0			0
$931$		−	12.0000i		−	0.393284i
$932$		0			0
$933$	−18.0000			−0.589294
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−2.00000			−0.0653372			−0.0326686	−	0.999466i	$-0.510401\pi$
$937$	−2.00000			−0.0653372			−0.0326686	+	0.999466i	$0.510401\pi$
$938$		0			0
$939$	−19.0000			−0.620042
$940$		0			0
$941$		−	45.0000i		−	1.46696i	−0.679712	−	0.733479i	$-0.737895\pi$
$941$		−	45.0000i		−	1.46696i	0.679712	−	0.733479i	$-0.262105\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	45.0000			1.46385
$946$		0			0
$947$		−	48.0000i		−	1.55979i	−0.625910	−	0.779895i	$-0.715272\pi$
$947$		−	48.0000i		−	1.55979i	0.625910	−	0.779895i	$-0.284728\pi$
$948$		0			0
$949$	18.0000	+	12.0000i	0.584305	+	0.389536i
$950$		0			0
$951$		−	18.0000i		−	0.583690i
$952$		0			0
$953$	9.00000			0.291539			0.145769	−	0.989319i	$-0.453434\pi$
$953$	9.00000			0.291539			0.145769	+	0.989319i	$0.453434\pi$
$954$		0			0
$955$			36.0000i			1.16493i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	54.0000			1.74375
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	24.0000			0.773389
$964$		0			0
$965$	−18.0000			−0.579441
$966$		0			0
$967$			3.00000i			0.0964735i	0.998836	+	0.0482367i	$0.0153602\pi$
$967$			3.00000i			0.0964735i	−0.998836	+	0.0482367i	$0.984640\pi$
$968$		0			0
$969$		−	18.0000i		−	0.578243i
$970$		0			0
$971$	27.0000			0.866471			0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000			0.866471			0.433236	+	0.901281i	$0.357372\pi$
$972$		0			0
$973$			15.0000i			0.480878i
$974$		0			0
$975$	−8.00000	+	12.0000i	−0.256205	+	0.384308i
$976$		0			0
$977$			12.0000i			0.383914i	0.981403	+	0.191957i	$0.0614834\pi$
$977$			12.0000i			0.383914i	−0.981403	+	0.191957i	$0.938517\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		−	18.0000i		−	0.574696i
$982$		0			0
$983$		−	9.00000i		−	0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$983$		−	9.00000i		−	0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$984$		0			0
$985$	9.00000			0.286764
$986$		0			0
$987$	9.00000			0.286473
$988$		0			0
$989$	−6.00000			−0.190789
$990$		0			0
$991$	−2.00000			−0.0635321			−0.0317660	−	0.999495i	$-0.510113\pi$
$991$	−2.00000			−0.0635321			−0.0317660	+	0.999495i	$0.510113\pi$
$992$		0			0
$993$			30.0000i			0.952021i
$994$		0			0
$995$		−	60.0000i		−	1.90213i
$996$		0			0
$997$	−8.00000			−0.253363			−0.126681	−	0.991943i	$-0.540433\pi$
$997$	−8.00000			−0.253363			−0.126681	+	0.991943i	$0.540433\pi$
$998$		0			0
$999$			15.0000i			0.474579i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	832.2.f.b.129.1		2
4.3	odd	2		832.2.f.d.129.1		2
8.3	odd	2		26.2.b.a.25.1	✓	2
8.5	even	2		208.2.f.a.129.2		2
13.12	even	2	inner	832.2.f.b.129.2		2
24.5	odd	2		1872.2.c.f.1585.1		2
24.11	even	2		234.2.b.b.181.2		2
40.3	even	4		650.2.c.a.649.1		2
40.19	odd	2		650.2.d.b.51.2		2
40.27	even	4		650.2.c.d.649.2		2
52.51	odd	2		832.2.f.d.129.2		2
56.3	even	6		1274.2.n.c.961.2		4
56.11	odd	6		1274.2.n.d.961.2		4
56.19	even	6		1274.2.n.c.753.1		4
56.27	even	2		1274.2.d.c.883.1		2
56.51	odd	6		1274.2.n.d.753.1		4
104.3	odd	6		338.2.e.c.147.2		4
104.5	odd	4		2704.2.a.k.1.1		1
104.11	even	12		338.2.c.b.191.1		2
104.19	even	12		338.2.c.f.315.1		2
104.21	odd	4		2704.2.a.j.1.1		1
104.35	odd	6		338.2.e.c.23.1		4
104.43	odd	6		338.2.e.c.23.2		4
104.51	odd	2		26.2.b.a.25.2	yes	2
104.59	even	12		338.2.c.b.315.1		2
104.67	even	12		338.2.c.f.191.1		2
104.75	odd	6		338.2.e.c.147.1		4
104.77	even	2		208.2.f.a.129.1		2
104.83	even	4		338.2.a.b.1.1		1
104.99	even	4		338.2.a.d.1.1		1
312.77	odd	2		1872.2.c.f.1585.2		2
312.83	odd	4		3042.2.a.j.1.1		1
312.155	even	2		234.2.b.b.181.1		2
312.203	odd	4		3042.2.a.g.1.1		1
520.99	even	4		8450.2.a.h.1.1		1
520.259	odd	2		650.2.d.b.51.1		2
520.363	even	4		650.2.c.d.649.1		2
520.467	even	4		650.2.c.a.649.2		2
520.499	even	4		8450.2.a.u.1.1		1
728.51	odd	6		1274.2.n.d.753.2		4
728.363	even	2		1274.2.d.c.883.2		2
728.467	even	6		1274.2.n.c.753.2		4
728.571	odd	6		1274.2.n.d.961.1		4
728.675	even	6		1274.2.n.c.961.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.b.a.25.1	✓	2	8.3	odd	2
26.2.b.a.25.2	yes	2	104.51	odd	2
208.2.f.a.129.1		2	104.77	even	2
208.2.f.a.129.2		2	8.5	even	2
234.2.b.b.181.1		2	312.155	even	2
234.2.b.b.181.2		2	24.11	even	2
338.2.a.b.1.1		1	104.83	even	4
338.2.a.d.1.1		1	104.99	even	4
338.2.c.b.191.1		2	104.11	even	12
338.2.c.b.315.1		2	104.59	even	12
338.2.c.f.191.1		2	104.67	even	12
338.2.c.f.315.1		2	104.19	even	12
338.2.e.c.23.1		4	104.35	odd	6
338.2.e.c.23.2		4	104.43	odd	6
338.2.e.c.147.1		4	104.75	odd	6
338.2.e.c.147.2		4	104.3	odd	6
650.2.c.a.649.1		2	40.3	even	4
650.2.c.a.649.2		2	520.467	even	4
650.2.c.d.649.1		2	520.363	even	4
650.2.c.d.649.2		2	40.27	even	4
650.2.d.b.51.1		2	520.259	odd	2
650.2.d.b.51.2		2	40.19	odd	2
832.2.f.b.129.1		2	1.1	even	1	trivial
832.2.f.b.129.2		2	13.12	even	2	inner
832.2.f.d.129.1		2	4.3	odd	2
832.2.f.d.129.2		2	52.51	odd	2
1274.2.d.c.883.1		2	56.27	even	2
1274.2.d.c.883.2		2	728.363	even	2
1274.2.n.c.753.1		4	56.19	even	6
1274.2.n.c.753.2		4	728.467	even	6
1274.2.n.c.961.1		4	728.675	even	6
1274.2.n.c.961.2		4	56.3	even	6
1274.2.n.d.753.1		4	56.51	odd	6
1274.2.n.d.753.2		4	728.51	odd	6
1274.2.n.d.961.1		4	728.571	odd	6
1274.2.n.d.961.2		4	56.11	odd	6
1872.2.c.f.1585.1		2	24.5	odd	2
1872.2.c.f.1585.2		2	312.77	odd	2
2704.2.a.j.1.1		1	104.21	odd	4
2704.2.a.k.1.1		1	104.5	odd	4
3042.2.a.g.1.1		1	312.203	odd	4
3042.2.a.j.1.1		1	312.83	odd	4
8450.2.a.h.1.1		1	520.99	even	4
8450.2.a.u.1.1		1	520.499	even	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	832.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.00000i q^{5} +3.00000i q^{7} -2.00000 q^{9} +(-2.00000 + 3.00000i) q^{13} +3.00000i q^{15} +3.00000 q^{17} +6.00000i q^{19} -3.00000i q^{21} +6.00000 q^{23} -4.00000 q^{25} +5.00000 q^{27} +9.00000 q^{35} +3.00000i q^{37} +(2.00000 - 3.00000i) q^{39} -1.00000 q^{43} +6.00000i q^{45} +3.00000i q^{47} -2.00000 q^{49} -3.00000 q^{51} +6.00000 q^{53} -6.00000i q^{57} +6.00000i q^{59} +8.00000 q^{61} -6.00000i q^{63} +(9.00000 + 6.00000i) q^{65} +12.0000i q^{67} -6.00000 q^{69} +15.0000i q^{71} -6.00000i q^{73} +4.00000 q^{75} -10.0000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -9.00000i q^{85} +6.00000i q^{89} +(-9.00000 - 6.00000i) q^{91} +18.0000 q^{95} +12.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{9} - 4 q^{13} + 6 q^{17} + 12 q^{23} - 8 q^{25} + 10 q^{27} + 18 q^{35} + 4 q^{39} - 2 q^{43} - 4 q^{49} - 6 q^{51} + 12 q^{53} + 16 q^{61} + 18 q^{65} - 12 q^{69} + 8 q^{75} - 20 q^{79}+ \cdots + 36 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^9 - 4 * q^13 + 6 * q^17 + 12 * q^23 - 8 * q^25 + 10 * q^27 + 18 * q^35 + 4 * q^39 - 2 * q^43 - 4 * q^49 - 6 * q^51 + 12 * q^53 + 16 * q^61 + 18 * q^65 - 12 * q^69 + 8 * q^75 - 20 * q^79 + 2 * q^81 - 18 * q^91 + 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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