LMFDB - Embedded newform 7644.2.e.m.4705.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7644,2,Mod(4705,7644)] 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("7644.4705"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7644, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7644.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,6,0,0,0,0,0,6,0,0,0,4,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$61.0376473051$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.50922496.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 8x^{4} + 13x^{2} + 4 $ x^6 + 8x^4 + 13x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4705.1
Root			$-2.43255i$ of defining polynomial
Character	$\chi$	$=$	7644.4705
Dual form			7644.2.e.m.4705.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -3.05352i q^{5} +1.00000 q^{9} +0.989386i q^{11} +(3.51056 + 0.822183i) q^{13} -3.05352i q^{15} +7.83457 q^{17} +4.04291i q^{19} +2.51056 q^{23} -4.32401 q^{25} +1.00000 q^{27} -5.32401 q^{29} -5.68728i q^{31} +0.989386i q^{33} +6.44146i q^{37} +(3.51056 + 0.822183i) q^{39} +9.07521i q^{41} +3.32401 q^{43} -3.05352i q^{45} +12.7837i q^{47} +7.83457 q^{51} +10.3451 q^{53} +3.02112 q^{55} +4.04291i q^{57} -8.74080i q^{59} +8.64803 q^{61} +(2.51056 - 10.7196i) q^{65} -9.73019i q^{67} +2.51056 q^{69} +16.4922i q^{71} +10.1500i q^{73} -4.32401 q^{75} -0.302899 q^{79} +1.00000 q^{81} +4.69789i q^{83} -23.9231i q^{85} -5.32401 q^{87} +0.235209i q^{89} -5.68728i q^{93} +12.3451 q^{95} +12.4631i q^{97} +0.989386i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 6 q^{9} + 4 q^{13} + 8 q^{17} - 2 q^{23} - 4 q^{25} + 6 q^{27} - 10 q^{29} + 4 q^{39} - 2 q^{43} + 8 q^{51} + 6 q^{53} - 16 q^{55} + 8 q^{61} - 2 q^{65} - 2 q^{69} - 4 q^{75} - 14 q^{79}+ \cdots + 18 q^{95}+O(q^{100}) $ 6 * q + 6 * q^3 + 6 * q^9 + 4 * q^13 + 8 * q^17 - 2 * q^23 - 4 * q^25 + 6 * q^27 - 10 * q^29 + 4 * q^39 - 2 * q^43 + 8 * q^51 + 6 * q^53 - 16 * q^55 + 8 * q^61 - 2 * q^65 - 2 * q^69 - 4 * q^75 - 14 * q^79 + 6 * q^81 - 10 * q^87 + 18 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/7644\mathbb{Z}\right)^\times$.

$n$	$2549$	$3433$	$3823$	$5293$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	1.00000			0.577350
$4$		0			0
$5$		−	3.05352i		−	1.36558i	−0.730616	−	0.682789i	$-0.760767\pi$
$5$		−	3.05352i		−	1.36558i	0.730616	−	0.682789i	$-0.239233\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			0.989386i			0.298311i	0.988814	+	0.149156i	$0.0476555\pi$
$11$			0.989386i			0.298311i	−0.988814	+	0.149156i	$0.952344\pi$
$12$		0			0
$13$	3.51056	+	0.822183i	0.973653	+	0.228033i
$13$	3.51056	+	0.822183i	0.973653	+	0.228033i
$14$		0			0
$15$		−	3.05352i		−	0.788417i
$16$		0			0
$17$	7.83457			1.90016			0.950081	−	0.312002i	$-0.101000\pi$
$17$	7.83457			1.90016			0.950081	+	0.312002i	$0.101000\pi$
$18$		0			0
$19$			4.04291i			0.927507i	0.885964	+	0.463754i	$0.153498\pi$
$19$			4.04291i			0.927507i	−0.885964	+	0.463754i	$0.846502\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	2.51056			0.523487			0.261744	−	0.965137i	$-0.415702\pi$
$23$	2.51056			0.523487			0.261744	+	0.965137i	$0.415702\pi$
$24$		0			0
$25$	−4.32401			−0.864803
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−5.32401			−0.988645			−0.494322	−	0.869279i	$-0.664584\pi$
$29$	−5.32401			−0.988645			−0.494322	+	0.869279i	$0.664584\pi$
$30$		0			0
$31$		−	5.68728i		−	1.02147i	−0.859740	−	0.510733i	$-0.829374\pi$
$31$		−	5.68728i		−	1.02147i	0.859740	−	0.510733i	$-0.170626\pi$
$32$		0			0
$33$			0.989386i			0.172230i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			6.44146i			1.05897i	0.848320	+	0.529484i	$0.177615\pi$
$37$			6.44146i			1.05897i	−0.848320	+	0.529484i	$0.822385\pi$
$38$		0			0
$39$	3.51056	+	0.822183i	0.562139	+	0.131655i
$40$		0			0
$41$			9.07521i			1.41731i	0.705555	+	0.708655i	$0.250698\pi$
$41$			9.07521i			1.41731i	−0.705555	+	0.708655i	$0.749302\pi$
$42$		0			0
$43$	3.32401			0.506907			0.253454	−	0.967348i	$-0.418433\pi$
$43$	3.32401			0.506907			0.253454	+	0.967348i	$0.418433\pi$
$44$		0			0
$45$		−	3.05352i		−	0.455193i
$46$		0			0
$47$			12.7837i			1.86470i	0.361562	+	0.932348i	$0.382244\pi$
$47$			12.7837i			1.86470i	−0.361562	+	0.932348i	$0.617756\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	7.83457			1.09706
$52$		0			0
$53$	10.3451			1.42101			0.710506	−	0.703691i	$-0.248466\pi$
$53$	10.3451			1.42101			0.710506	+	0.703691i	$0.248466\pi$
$54$		0			0
$55$	3.02112			0.407367
$56$		0			0
$57$			4.04291i			0.535497i
$58$		0			0
$59$		−	8.74080i		−	1.13796i	−0.822353	−	0.568978i	$-0.807339\pi$
$59$		−	8.74080i		−	1.13796i	0.822353	−	0.568978i	$-0.192661\pi$
$60$		0			0
$61$	8.64803			1.10727			0.553633	−	0.832761i	$-0.313241\pi$
$61$	8.64803			1.10727			0.553633	+	0.832761i	$0.313241\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.51056	−	10.7196i	0.311396	−	1.32960i
$66$		0			0
$67$		−	9.73019i		−	1.18873i	−0.804195	−	0.594366i	$-0.797403\pi$
$67$		−	9.73019i		−	1.18873i	0.804195	−	0.594366i	$-0.202597\pi$
$68$		0			0
$69$	2.51056			0.302236
$70$		0			0
$71$			16.4922i			1.95727i	0.205614	+	0.978633i	$0.434081\pi$
$71$			16.4922i			1.95727i	−0.205614	+	0.978633i	$0.565919\pi$
$72$		0			0
$73$			10.1500i			1.18796i	0.804479	+	0.593982i	$0.202445\pi$
$73$			10.1500i			1.18796i	−0.804479	+	0.593982i	$0.797555\pi$
$74$		0			0
$75$	−4.32401			−0.499294
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.302899			−0.0340788			−0.0170394	−	0.999855i	$-0.505424\pi$
$79$	−0.302899			−0.0340788			−0.0170394	+	0.999855i	$0.505424\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			4.69789i			0.515661i	0.966190	+	0.257830i	$0.0830076\pi$
$83$			4.69789i			0.515661i	−0.966190	+	0.257830i	$0.916992\pi$
$84$		0			0
$85$		−	23.9231i		−	2.59482i
$86$		0			0
$87$	−5.32401			−0.570794
$88$		0			0
$89$			0.235209i			0.0249321i	0.999922	+	0.0124660i	$0.00396817\pi$
$89$			0.235209i			0.0249321i	−0.999922	+	0.0124660i	$0.996032\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	5.68728i		−	0.589743i
$94$		0			0
$95$	12.3451			1.26658
$96$		0			0
$97$			12.4631i			1.26544i	0.774381	+	0.632720i	$0.218062\pi$
$97$			12.4631i			1.26544i	−0.774381	+	0.632720i	$0.781938\pi$
$98$		0			0
$99$			0.989386i			0.0994370i
$100$		0			0
$101$	−9.46149			−0.941453			−0.470727	−	0.882279i	$-0.656008\pi$
$101$	−9.46149			−0.941453			−0.470727	+	0.882279i	$0.656008\pi$
$102$		0			0
$103$	−8.64803			−0.852116			−0.426058	−	0.904696i	$-0.640098\pi$
$103$	−8.64803			−0.852116			−0.426058	+	0.904696i	$0.640098\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	3.18654			0.308055			0.154027	−	0.988067i	$-0.450776\pi$
$107$	3.18654			0.308055			0.154027	+	0.988067i	$0.450776\pi$
$108$		0			0
$109$		−	5.26751i		−	0.504536i	−0.967657	−	0.252268i	$-0.918824\pi$
$109$		−	5.26751i		−	0.504536i	0.967657	−	0.252268i	$-0.0811764\pi$
$110$		0			0
$111$			6.44146i			0.611396i
$112$		0			0
$113$	1.32401			0.124553			0.0622764	−	0.998059i	$-0.480164\pi$
$113$	1.32401			0.124553			0.0622764	+	0.998059i	$0.480164\pi$
$114$		0			0
$115$		−	7.66605i		−	0.714863i
$116$		0			0
$117$	3.51056	+	0.822183i	0.324551	+	0.0760109i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	10.0211			0.911010
$122$		0			0
$123$			9.07521i			0.818284i
$124$		0			0
$125$		−	2.06414i		−	0.184622i
$126$		0			0
$127$	−18.6903			−1.65849			−0.829246	−	0.558884i	$-0.811230\pi$
$127$	−18.6903			−1.65849			−0.829246	+	0.558884i	$0.811230\pi$
$128$		0			0
$129$	3.32401			0.292663
$130$		0			0
$131$	−8.00000			−0.698963			−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000			−0.698963			−0.349482	+	0.936943i	$0.613642\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		−	3.05352i		−	0.262806i
$136$		0			0
$137$			11.8935i			1.01613i	0.861318	+	0.508066i	$0.169639\pi$
$137$			11.8935i			1.01613i	−0.861318	+	0.508066i	$0.830361\pi$
$138$		0			0
$139$	−4.64803			−0.394241			−0.197120	−	0.980379i	$-0.563159\pi$
$139$	−4.64803			−0.394241			−0.197120	+	0.980379i	$0.563159\pi$
$140$		0			0
$141$			12.7837i			1.07658i
$142$		0			0
$143$	−0.813457	+	3.47330i	−0.0680247	+	0.290452i
$144$		0			0
$145$			16.2570i			1.35007i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			9.91475i			0.812248i	0.913818	+	0.406124i	$0.133120\pi$
$149$			9.91475i			0.812248i	−0.913818	+	0.406124i	$0.866880\pi$
$150$		0			0
$151$		−	14.9977i		−	1.22050i	−0.792211	−	0.610248i	$-0.791070\pi$
$151$		−	14.9977i		−	1.22050i	0.792211	−	0.610248i	$-0.208930\pi$
$152$		0			0
$153$	7.83457			0.633388
$154$		0			0
$155$	−17.3662			−1.39489
$156$		0			0
$157$	−4.00000			−0.319235			−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000			−0.319235			−0.159617	+	0.987179i	$0.551026\pi$
$158$		0			0
$159$	10.3451			0.820422
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	13.5241i		−	1.05929i	−0.848221	−	0.529643i	$-0.822326\pi$
$163$		−	13.5241i		−	1.05929i	0.848221	−	0.529643i	$-0.177674\pi$
$164$		0			0
$165$	3.02112			0.235193
$166$		0			0
$167$		−	13.4525i		−	1.04099i	−0.853865	−	0.520494i	$-0.825748\pi$
$167$		−	13.4525i		−	1.04099i	0.853865	−	0.520494i	$-0.174252\pi$
$168$		0			0
$169$	11.6480	+	5.77264i	0.896002	+	0.444050i
$170$		0			0
$171$			4.04291i			0.309169i
$172$		0			0
$173$	−13.8768			−1.05503			−0.527517	−	0.849545i	$-0.676877\pi$
$173$	−13.8768			−1.05503			−0.527517	+	0.849545i	$0.676877\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	8.74080i		−	0.656999i
$178$		0			0
$179$	−10.7855			−0.806146			−0.403073	−	0.915168i	$-0.632058\pi$
$179$	−10.7855			−0.806146			−0.403073	+	0.915168i	$0.632058\pi$
$180$		0			0
$181$	19.6691			1.46200			0.730998	−	0.682380i	$-0.239055\pi$
$181$	19.6691			1.46200			0.730998	+	0.682380i	$0.239055\pi$
$182$		0			0
$183$	8.64803			0.639281
$184$		0			0
$185$	19.6691			1.44610
$186$		0			0
$187$			7.75142i			0.566840i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	14.8557			1.07492			0.537460	−	0.843289i	$-0.319384\pi$
$191$	14.8557			1.07492			0.537460	+	0.843289i	$0.319384\pi$
$192$		0			0
$193$		−	9.73019i		−	0.700394i	−0.936676	−	0.350197i	$-0.886115\pi$
$193$		−	9.73019i		−	0.700394i	0.936676	−	0.350197i	$-0.113885\pi$
$194$		0			0
$195$	2.51056	−	10.7196i	0.179785	−	0.767645i
$196$		0			0
$197$			17.1610i			1.22267i	0.791371	+	0.611336i	$0.209368\pi$
$197$			17.1610i			1.22267i	−0.791371	+	0.611336i	$0.790632\pi$
$198$		0			0
$199$	19.6691			1.39431			0.697154	−	0.716921i	$-0.254449\pi$
$199$	19.6691			1.39431			0.697154	+	0.716921i	$0.254449\pi$
$200$		0			0
$201$		−	9.73019i		−	0.686315i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	27.7114			1.93545
$206$		0			0
$207$	2.51056			0.174496
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−12.7182			−0.875558			−0.437779	−	0.899083i	$-0.644235\pi$
$211$	−12.7182			−0.875558			−0.437779	+	0.899083i	$0.644235\pi$
$212$		0			0
$213$			16.4922i			1.13003i
$214$		0			0
$215$		−	10.1500i		−	0.692222i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			10.1500i			0.685871i
$220$		0			0
$221$	27.5037	+	6.44146i	1.85010	+	0.433299i
$222$		0			0
$223$		−	19.5457i		−	1.30888i	−0.756114	−	0.654440i	$-0.772904\pi$
$223$		−	19.5457i		−	1.30888i	0.756114	−	0.654440i	$-0.227096\pi$
$224$		0			0
$225$	−4.32401			−0.288268
$226$		0			0
$227$		−	4.78326i		−	0.317476i	−0.987321	−	0.158738i	$-0.949257\pi$
$227$		−	4.78326i		−	0.317476i	0.987321	−	0.158738i	$-0.0507425\pi$
$228$		0			0
$229$		−	27.5462i		−	1.82030i	−0.414274	−	0.910152i	$-0.635965\pi$
$229$		−	27.5462i		−	1.82030i	0.414274	−	0.910152i	$-0.364035\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	13.7393			0.900093			0.450047	−	0.893005i	$-0.351407\pi$
$233$	13.7393			0.900093			0.450047	+	0.893005i	$0.351407\pi$
$234$		0			0
$235$	39.0354			2.54639
$236$		0			0
$237$	−0.302899			−0.0196754
$238$		0			0
$239$		−	26.0864i		−	1.68739i	−0.536824	−	0.843694i	$-0.680376\pi$
$239$		−	26.0864i		−	1.68739i	0.536824	−	0.843694i	$-0.319624\pi$
$240$		0			0
$241$			5.18214i			0.333811i	0.985973	+	0.166905i	$0.0533775\pi$
$241$			5.18214i			0.333811i	−0.985973	+	0.166905i	$0.946623\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−3.32401	+	14.1929i	−0.211502	+	0.903071i
$248$		0			0
$249$			4.69789i			0.297717i
$250$		0			0
$251$	−8.41532			−0.531170			−0.265585	−	0.964087i	$-0.585565\pi$
$251$	−8.41532			−0.531170			−0.265585	+	0.964087i	$0.585565\pi$
$252$		0			0
$253$			2.48391i			0.156162i
$254$		0			0
$255$		−	23.9231i		−	1.49812i
$256$		0			0
$257$	−21.8768			−1.36464			−0.682319	−	0.731055i	$-0.739028\pi$
$257$	−21.8768			−1.36464			−0.682319	+	0.731055i	$0.739028\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−5.32401			−0.329548
$262$		0			0
$263$	−7.11636			−0.438813			−0.219407	−	0.975633i	$-0.570412\pi$
$263$	−7.11636			−0.438813			−0.219407	+	0.975633i	$0.570412\pi$
$264$		0			0
$265$		−	31.5891i		−	1.94050i
$266$		0			0
$267$			0.235209i			0.0143945i
$268$		0			0
$269$	3.83457			0.233798			0.116899	−	0.993144i	$-0.462705\pi$
$269$	3.83457			0.233798			0.116899	+	0.993144i	$0.462705\pi$
$270$		0			0
$271$		−	21.2755i		−	1.29239i	−0.763171	−	0.646197i	$-0.776359\pi$
$271$		−	21.2755i		−	1.29239i	0.763171	−	0.646197i	$-0.223641\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	4.27812i		−	0.257980i
$276$		0			0
$277$	20.3874			1.22496			0.612479	−	0.790487i	$-0.290172\pi$
$277$	20.3874			1.22496			0.612479	+	0.790487i	$0.290172\pi$
$278$		0			0
$279$		−	5.68728i		−	0.340488i
$280$		0			0
$281$		−	25.4176i		−	1.51629i	−0.652088	−	0.758143i	$-0.726107\pi$
$281$		−	25.4176i		−	1.51629i	0.652088	−	0.758143i	$-0.273893\pi$
$282$		0			0
$283$	15.6691			0.931434			0.465717	−	0.884934i	$-0.345796\pi$
$283$	15.6691			0.931434			0.465717	+	0.884934i	$0.345796\pi$
$284$		0			0
$285$	12.3451			0.731262
$286$		0			0
$287$		0			0
$288$		0			0
$289$	44.3805			2.61062
$290$		0			0
$291$			12.4631i			0.730602i
$292$		0			0
$293$			6.34226i			0.370519i	0.982690	+	0.185259i	$0.0593125\pi$
$293$			6.34226i			0.370519i	−0.982690	+	0.185259i	$0.940687\pi$
$294$		0			0
$295$	−26.6903			−1.55397
$296$		0			0
$297$			0.989386i			0.0574100i
$298$		0			0
$299$	8.81346	+	2.06414i	0.509695	+	0.119372i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−9.46149			−0.543548
$304$		0			0
$305$		−	26.4070i		−	1.51206i
$306$		0			0
$307$		−	1.89341i		−	0.108062i	−0.998539	−	0.0540312i	$-0.982793\pi$
$307$		−	1.89341i		−	0.108062i	0.998539	−	0.0540312i	$-0.0172070\pi$
$308$		0			0
$309$	−8.64803			−0.491969
$310$		0			0
$311$	11.6691			0.661696			0.330848	−	0.943684i	$-0.392665\pi$
$311$	11.6691			0.661696			0.330848	+	0.943684i	$0.392665\pi$
$312$		0			0
$313$	−20.6480			−1.16710			−0.583548	−	0.812079i	$-0.698336\pi$
$313$	−20.6480			−1.16710			−0.583548	+	0.812079i	$0.698336\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	17.8022i		−	0.999870i	−0.866063	−	0.499935i	$-0.833357\pi$
$317$		−	17.8022i		−	0.999870i	0.866063	−	0.499935i	$-0.166643\pi$
$318$		0			0
$319$		−	5.26751i		−	0.294924i
$320$		0			0
$321$	3.18654			0.177855
$322$		0			0
$323$			31.6745i			1.76242i
$324$		0			0
$325$	−15.1797	−	3.55513i	−0.842018	−	0.197203i
$326$		0			0
$327$		−	5.26751i		−	0.291294i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			4.96782i			0.273056i	0.990636	+	0.136528i	$0.0435944\pi$
$331$			4.96782i			0.273056i	−0.990636	+	0.136528i	$0.956406\pi$
$332$		0			0
$333$			6.44146i			0.352990i
$334$		0			0
$335$	−29.7114			−1.62331
$336$		0			0
$337$	−13.9720			−0.761106			−0.380553	−	0.924759i	$-0.624266\pi$
$337$	−13.9720			−0.761106			−0.380553	+	0.924759i	$0.624266\pi$
$338$		0			0
$339$	1.32401			0.0719106
$340$		0			0
$341$	5.62691			0.304714
$342$		0			0
$343$		0			0
$344$		0			0
$345$		−	7.66605i		−	0.412726i
$346$		0			0
$347$	16.8979			0.907128			0.453564	−	0.891224i	$-0.350152\pi$
$347$	16.8979			0.907128			0.453564	+	0.891224i	$0.350152\pi$
$348$		0			0
$349$			26.8267i			1.43600i	0.696042	+	0.718001i	$0.254943\pi$
$349$			26.8267i			1.43600i	−0.696042	+	0.718001i	$0.745057\pi$
$350$		0			0
$351$	3.51056	+	0.822183i	0.187380	+	0.0438849i
$352$		0			0
$353$			27.2256i			1.44907i	0.689236	+	0.724537i	$0.257946\pi$
$353$			27.2256i			1.44907i	−0.689236	+	0.724537i	$0.742054\pi$
$354$		0			0
$355$	50.3594			2.67280
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	10.5559i		−	0.557119i	−0.960419	−	0.278560i	$-0.910143\pi$
$359$		−	10.5559i		−	0.557119i	0.960419	−	0.278560i	$-0.0898570\pi$
$360$		0			0
$361$	2.65487			0.139730
$362$		0			0
$363$	10.0211			0.525972
$364$		0			0
$365$	30.9932			1.62226
$366$		0			0
$367$	−26.3594			−1.37595			−0.687975	−	0.725735i	$-0.741500\pi$
$367$	−26.3594			−1.37595			−0.687975	+	0.725735i	$0.741500\pi$
$368$		0			0
$369$			9.07521i			0.472436i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.76729			0.298619			0.149309	−	0.988791i	$-0.452295\pi$
$373$	5.76729			0.298619			0.149309	+	0.988791i	$0.452295\pi$
$374$		0			0
$375$		−	2.06414i		−	0.106592i
$376$		0			0
$377$	−18.6903	−	4.37732i	−0.962597	−	0.225443i
$378$		0			0
$379$			7.08261i			0.363809i	0.983316	+	0.181905i	$0.0582262\pi$
$379$			7.08261i			0.363809i	−0.983316	+	0.181905i	$0.941774\pi$
$380$		0			0
$381$	−18.6903			−0.957531
$382$		0			0
$383$		−	9.40961i		−	0.480809i	−0.970673	−	0.240404i	$-0.922720\pi$
$383$		−	9.40961i		−	0.480809i	0.970673	−	0.240404i	$-0.0772801\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.32401			0.168969
$388$		0			0
$389$	−11.2961			−0.572733			−0.286366	−	0.958120i	$-0.592447\pi$
$389$	−11.2961			−0.572733			−0.286366	+	0.958120i	$0.592447\pi$
$390$		0			0
$391$	19.6691			0.994711
$392$		0			0
$393$	−8.00000			−0.403547
$394$		0			0
$395$			0.924911i			0.0465373i
$396$		0			0
$397$		−	20.2146i		−	1.01454i	−0.861787	−	0.507270i	$-0.830655\pi$
$397$		−	20.2146i		−	1.01454i	0.861787	−	0.507270i	$-0.169345\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			4.64725i			0.232072i	0.993245	+	0.116036i	$0.0370189\pi$
$401$			4.64725i			0.232072i	−0.993245	+	0.116036i	$0.962981\pi$
$402$		0			0
$403$	4.67599	−	19.9655i	0.232927	−	0.994553i
$404$		0			0
$405$		−	3.05352i		−	0.151731i
$406$		0			0
$407$	−6.37309			−0.315902
$408$		0			0
$409$			22.9982i			1.13719i	0.822619	+	0.568593i	$0.192512\pi$
$409$			22.9982i			1.13719i	−0.822619	+	0.568593i	$0.807488\pi$
$410$		0			0
$411$			11.8935i			0.586664i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	14.3451			0.704175
$416$		0			0
$417$	−4.64803			−0.227615
$418$		0			0
$419$	−40.0845			−1.95825			−0.979127	−	0.203249i	$-0.934850\pi$
$419$	−40.0845			−1.95825			−0.979127	+	0.203249i	$0.934850\pi$
$420$		0			0
$421$		−	19.9655i		−	0.973060i	−0.873664	−	0.486530i	$-0.838262\pi$
$421$		−	19.9655i		−	0.973060i	0.873664	−	0.486530i	$-0.161738\pi$
$422$		0			0
$423$			12.7837i			0.621565i
$424$		0			0
$425$	−33.8768			−1.64327
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−0.813457	+	3.47330i	−0.0392741	+	0.167692i
$430$		0			0
$431$			8.60479i			0.414478i	0.978290	+	0.207239i	$0.0664478\pi$
$431$			8.60479i			0.414478i	−0.978290	+	0.207239i	$0.933552\pi$
$432$		0			0
$433$	−16.6480			−0.800053			−0.400027	−	0.916504i	$-0.630999\pi$
$433$	−16.6480			−0.800053			−0.400027	+	0.916504i	$0.630999\pi$
$434$		0			0
$435$			16.2570i			0.779464i
$436$		0			0
$437$			10.1500i			0.485538i
$438$		0			0
$439$	38.3594			1.83079			0.915397	−	0.402552i	$-0.131877\pi$
$439$	38.3594			1.83079			0.915397	+	0.402552i	$0.131877\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	32.8277			1.55969			0.779846	−	0.625971i	$-0.215297\pi$
$443$	32.8277			1.55969			0.779846	+	0.625971i	$0.215297\pi$
$444$		0			0
$445$	0.718216			0.0340467
$446$		0			0
$447$			9.91475i			0.468952i
$448$		0			0
$449$		−	5.28840i		−	0.249575i	−0.992183	−	0.124787i	$-0.960175\pi$
$449$		−	5.28840i		−	0.249575i	0.992183	−	0.124787i	$-0.0398249\pi$
$450$		0			0
$451$	−8.97888			−0.422799
$452$		0			0
$453$		−	14.9977i		−	0.704653i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			31.0333i			1.45168i	0.687865	+	0.725839i	$0.258548\pi$
$457$			31.0333i			1.45168i	−0.687865	+	0.725839i	$0.741452\pi$
$458$		0			0
$459$	7.83457			0.365687
$460$		0			0
$461$		−	12.7331i		−	0.593038i	−0.955027	−	0.296519i	$-0.904174\pi$
$461$		−	12.7331i		−	0.593038i	0.955027	−	0.296519i	$-0.0958258\pi$
$462$		0			0
$463$		−	26.9121i		−	1.25071i	−0.780340	−	0.625356i	$-0.784954\pi$
$463$		−	26.9121i		−	1.25071i	0.780340	−	0.625356i	$-0.215046\pi$
$464$		0			0
$465$	−17.3662			−0.805340
$466$		0			0
$467$	−2.37309			−0.109813			−0.0549067	−	0.998491i	$-0.517486\pi$
$467$	−2.37309			−0.109813			−0.0549067	+	0.998491i	$0.517486\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.00000			−0.184310
$472$		0			0
$473$			3.28873i			0.151216i
$474$		0			0
$475$		−	17.4816i		−	0.802111i
$476$		0			0
$477$	10.3451			0.473671
$478$		0			0
$479$		−	22.3779i		−	1.02247i	−0.859440	−	0.511236i	$-0.829188\pi$
$479$		−	22.3779i		−	1.02247i	0.859440	−	0.511236i	$-0.170812\pi$
$480$		0			0
$481$	−5.29606	+	22.6131i	−0.241479	+	1.03107i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	38.0565			1.72806
$486$		0			0
$487$		−	18.2864i		−	0.828637i	−0.910132	−	0.414319i	$-0.864020\pi$
$487$		−	18.2864i		−	0.828637i	0.910132	−	0.414319i	$-0.135980\pi$
$488$		0			0
$489$		−	13.5241i		−	0.611579i
$490$		0			0
$491$	−1.22877			−0.0554538			−0.0277269	−	0.999616i	$-0.508827\pi$
$491$	−1.22877			−0.0554538			−0.0277269	+	0.999616i	$0.508827\pi$
$492$		0			0
$493$	−41.7114			−1.87859
$494$		0			0
$495$	3.02112			0.135789
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	39.5896i		−	1.77227i	−0.463425	−	0.886136i	$-0.653380\pi$
$499$		−	39.5896i		−	1.77227i	0.463425	−	0.886136i	$-0.346620\pi$
$500$		0			0
$501$		−	13.4525i		−	0.601014i
$502$		0			0
$503$	−4.41532			−0.196869			−0.0984346	−	0.995144i	$-0.531384\pi$
$503$	−4.41532			−0.196869			−0.0984346	+	0.995144i	$0.531384\pi$
$504$		0			0
$505$			28.8909i			1.28563i
$506$		0			0
$507$	11.6480	+	5.77264i	0.517307	+	0.256372i
$508$		0			0
$509$		−	4.39115i		−	0.194634i	−0.995253	−	0.0973171i	$-0.968974\pi$
$509$		−	4.39115i		−	0.194634i	0.995253	−	0.0973171i	$-0.0310261\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			4.04291i			0.178499i
$514$		0			0
$515$			26.4070i			1.16363i
$516$		0			0
$517$	−12.6480			−0.556260
$518$		0			0
$519$	−13.8768			−0.609124
$520$		0			0
$521$	−5.79234			−0.253767			−0.126884	−	0.991918i	$-0.540497\pi$
$521$	−5.79234			−0.253767			−0.126884	+	0.991918i	$0.540497\pi$
$522$		0			0
$523$	−25.9441			−1.13446			−0.567228	−	0.823561i	$-0.691984\pi$
$523$	−25.9441			−1.13446			−0.567228	+	0.823561i	$0.691984\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	44.5574i		−	1.94095i
$528$		0			0
$529$	−16.6971			−0.725961
$530$		0			0
$531$		−	8.74080i		−	0.379319i
$532$		0			0
$533$	−7.46149	+	31.8590i	−0.323193	+	1.37997i
$534$		0			0
$535$		−	9.73019i		−	0.420673i
$536$		0			0
$537$	−10.7855			−0.465429
$538$		0			0
$539$		0			0
$540$		0			0
$541$		−	35.4683i		−	1.52490i	−0.647045	−	0.762452i	$-0.723995\pi$
$541$		−	35.4683i		−	1.52490i	0.647045	−	0.762452i	$-0.276005\pi$
$542$		0			0
$543$	19.6691			0.844084
$544$		0			0
$545$	−16.0845			−0.688983
$546$		0			0
$547$	−16.3451			−0.698867			−0.349434	−	0.936961i	$-0.613626\pi$
$547$	−16.3451			−0.698867			−0.349434	+	0.936961i	$0.613626\pi$
$548$		0			0
$549$	8.64803			0.369089
$550$		0			0
$551$		−	21.5245i		−	0.916975i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	19.6691			0.834909
$556$		0			0
$557$		−	23.7385i		−	1.00583i	−0.864335	−	0.502916i	$-0.832260\pi$
$557$		−	23.7385i		−	1.00583i	0.864335	−	0.502916i	$-0.167740\pi$
$558$		0			0
$559$	11.6691	+	2.73295i	0.493552	+	0.115591i
$560$		0			0
$561$			7.75142i			0.327265i
$562$		0			0
$563$	−16.9652			−0.714998			−0.357499	−	0.933914i	$-0.616370\pi$
$563$	−16.9652			−0.714998			−0.357499	+	0.933914i	$0.616370\pi$
$564$		0			0
$565$		−	4.04291i		−	0.170087i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	2.07019			0.0867868			0.0433934	−	0.999058i	$-0.486183\pi$
$569$	2.07019			0.0867868			0.0433934	+	0.999058i	$0.486183\pi$
$570$		0			0
$571$	20.3874			0.853184			0.426592	−	0.904444i	$-0.359714\pi$
$571$	20.3874			0.853184			0.426592	+	0.904444i	$0.359714\pi$
$572$		0			0
$573$	14.8557			0.620605
$574$		0			0
$575$	−10.8557			−0.452713
$576$		0			0
$577$		−	40.1294i		−	1.67061i	−0.549787	−	0.835305i	$-0.685291\pi$
$577$		−	40.1294i		−	1.67061i	0.549787	−	0.835305i	$-0.314709\pi$
$578$		0			0
$579$		−	9.73019i		−	0.404373i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			10.2353i			0.423904i
$584$		0			0
$585$	2.51056	−	10.7196i	0.103799	−	0.443200i
$586$		0			0
$587$			4.32877i			0.178667i	0.996002	+	0.0893336i	$0.0284737\pi$
$587$			4.32877i			0.178667i	−0.996002	+	0.0893336i	$0.971526\pi$
$588$		0			0
$589$	22.9932			0.947417
$590$		0			0
$591$			17.1610i			0.705910i
$592$		0			0
$593$			27.1126i			1.11338i	0.830720	+	0.556690i	$0.187929\pi$
$593$			27.1126i			1.11338i	−0.830720	+	0.556690i	$0.812071\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	19.6691			0.805004
$598$		0			0
$599$	40.8277			1.66818			0.834088	−	0.551632i	$-0.185995\pi$
$599$	40.8277			1.66818			0.834088	+	0.551632i	$0.185995\pi$
$600$		0			0
$601$	14.6903			0.599228			0.299614	−	0.954060i	$-0.403142\pi$
$601$	14.6903			0.599228			0.299614	+	0.954060i	$0.403142\pi$
$602$		0			0
$603$		−	9.73019i		−	0.396244i
$604$		0			0
$605$		−	30.5997i		−	1.24406i
$606$		0			0
$607$	−8.23271			−0.334155			−0.167078	−	0.985944i	$-0.553433\pi$
$607$	−8.23271			−0.334155			−0.167078	+	0.985944i	$0.553433\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−10.5106	+	44.8780i	−0.425212	+	1.81557i
$612$		0			0
$613$			7.91509i			0.319687i	0.987142	+	0.159844i	$0.0510990\pi$
$613$			7.91509i			0.319687i	−0.987142	+	0.159844i	$0.948901\pi$
$614$		0			0
$615$	27.7114			1.11743
$616$		0			0
$617$		−	12.1932i		−	0.490880i	−0.969412	−	0.245440i	$-0.921068\pi$
$617$		−	12.1932i		−	0.490880i	0.969412	−	0.245440i	$-0.0789324\pi$
$618$		0			0
$619$			37.4471i			1.50513i	0.658520	+	0.752563i	$0.271183\pi$
$619$			37.4471i			1.50513i	−0.658520	+	0.752563i	$0.728817\pi$
$620$		0			0
$621$	2.51056			0.100745
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−27.9230			−1.11692
$626$		0			0
$627$	−4.00000			−0.159745
$628$		0			0
$629$			50.4660i			2.01221i
$630$		0			0
$631$		−	0.369126i		−	0.0146947i	−0.999973	−	0.00734734i	$-0.997661\pi$
$631$		−	0.369126i		−	0.0146947i	0.999973	−	0.00734734i	$-0.00233875\pi$
$632$		0			0
$633$	−12.7182			−0.505504
$634$		0			0
$635$			57.0712i			2.26480i
$636$		0			0
$637$		0			0
$638$		0			0
$639$			16.4922i			0.652422i
$640$		0			0
$641$	3.97204			0.156886			0.0784431	−	0.996919i	$-0.475005\pi$
$641$	3.97204			0.156886			0.0784431	+	0.996919i	$0.475005\pi$
$642$		0			0
$643$			7.24628i			0.285765i	0.989740	+	0.142883i	$0.0456372\pi$
$643$			7.24628i			0.285765i	−0.989740	+	0.142883i	$0.954363\pi$
$644$		0			0
$645$		−	10.1500i		−	0.399654i
$646$		0			0
$647$	22.9230			0.901195			0.450597	−	0.892727i	$-0.351211\pi$
$647$	22.9230			0.901195			0.450597	+	0.892727i	$0.351211\pi$
$648$		0			0
$649$	8.64803			0.339465
$650$		0			0
$651$		0			0
$652$		0			0
$653$	10.7462			0.420530			0.210265	−	0.977644i	$-0.432567\pi$
$653$	10.7462			0.420530			0.210265	+	0.977644i	$0.432567\pi$
$654$		0			0
$655$			24.4282i			0.954489i
$656$		0			0
$657$			10.1500i			0.395988i
$658$		0			0
$659$	−35.5317			−1.38412			−0.692059	−	0.721841i	$-0.743296\pi$
$659$	−35.5317			−1.38412			−0.692059	+	0.721841i	$0.743296\pi$
$660$		0			0
$661$			22.2004i			0.863495i	0.901995	+	0.431748i	$0.142103\pi$
$661$			22.2004i			0.863495i	−0.901995	+	0.431748i	$0.857897\pi$
$662$		0			0
$663$	27.5037	+	6.44146i	1.06816	+	0.250165i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−13.3662			−0.517543
$668$		0			0
$669$		−	19.5457i		−	0.755682i
$670$		0			0
$671$			8.55624i			0.330310i
$672$		0			0
$673$	22.3451			0.861341			0.430671	−	0.902509i	$-0.358277\pi$
$673$	22.3451			0.861341			0.430671	+	0.902509i	$0.358277\pi$
$674$		0			0
$675$	−4.32401			−0.166431
$676$		0			0
$677$	−44.2499			−1.70066			−0.850331	−	0.526249i	$-0.823598\pi$
$677$	−44.2499			−1.70066			−0.850331	+	0.526249i	$0.823598\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	4.78326i		−	0.183295i
$682$		0			0
$683$			7.26717i			0.278070i	0.990287	+	0.139035i	$0.0444001\pi$
$683$			7.26717i			0.278070i	−0.990287	+	0.139035i	$0.955600\pi$
$684$		0			0
$685$	36.3172			1.38761
$686$		0			0
$687$		−	27.5462i		−	1.05095i
$688$		0			0
$689$	36.3172	+	8.50559i	1.38357	+	0.324037i
$690$		0			0
$691$		−	24.0084i		−	0.913324i	−0.889640	−	0.456662i	$-0.849045\pi$
$691$		−	24.0084i		−	0.913324i	0.889640	−	0.456662i	$-0.150955\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			14.1929i			0.538366i
$696$		0			0
$697$			71.1004i			2.69312i
$698$		0			0
$699$	13.7393			0.519669
$700$		0			0
$701$	14.0143			0.529312			0.264656	−	0.964343i	$-0.414742\pi$
$701$	14.0143			0.529312			0.264656	+	0.964343i	$0.414742\pi$
$702$		0			0
$703$	−26.0422			−0.982201
$704$		0			0
$705$	39.0354			1.47016
$706$		0			0
$707$		0			0
$708$		0			0
$709$		−	6.27778i		−	0.235767i	−0.993027	−	0.117884i	$-0.962389\pi$
$709$		−	6.27778i		−	0.235767i	0.993027	−	0.117884i	$-0.0376110\pi$
$710$		0			0
$711$	−0.302899			−0.0113596
$712$		0			0
$713$		−	14.2782i		−	0.534724i
$714$		0			0
$715$	10.6058	+	2.48391i	0.396634	+	0.0928930i
$716$		0			0
$717$		−	26.0864i		−	0.974214i
$718$		0			0
$719$	−11.3383			−0.422847			−0.211423	−	0.977395i	$-0.567810\pi$
$719$	−11.3383			−0.422847			−0.211423	+	0.977395i	$0.567810\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$			5.18214i			0.192726i
$724$		0			0
$725$	23.0211			0.854983
$726$		0			0
$727$	4.74617			0.176026			0.0880129	−	0.996119i	$-0.471948\pi$
$727$	4.74617			0.176026			0.0880129	+	0.996119i	$0.471948\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	26.0422			0.963207
$732$		0			0
$733$			15.6159i			0.576785i	0.957512	+	0.288392i	$0.0931208\pi$
$733$			15.6159i			0.576785i	−0.957512	+	0.288392i	$0.906879\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	9.62691			0.354612
$738$		0			0
$739$		−	17.9867i		−	0.661653i	−0.943692	−	0.330827i	$-0.892672\pi$
$739$		−	17.9867i		−	0.661653i	0.943692	−	0.330827i	$-0.107328\pi$
$740$		0			0
$741$	−3.32401	+	14.1929i	−0.122111	+	0.521388i
$742$		0			0
$743$		−	21.1186i		−	0.774765i	−0.921919	−	0.387383i	$-0.873379\pi$
$743$		−	21.1186i		−	0.774765i	0.921919	−	0.387383i	$-0.126621\pi$
$744$		0			0
$745$	30.2749			1.10919
$746$		0			0
$747$			4.69789i			0.171887i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−26.7182			−0.974962			−0.487481	−	0.873134i	$-0.662084\pi$
$751$	−26.7182			−0.974962			−0.487481	+	0.873134i	$0.662084\pi$
$752$		0			0
$753$	−8.41532			−0.306671
$754$		0			0
$755$	−45.7958			−1.66668
$756$		0			0
$757$	−9.28178			−0.337352			−0.168676	−	0.985672i	$-0.553949\pi$
$757$	−9.28178			−0.337352			−0.168676	+	0.985672i	$0.553949\pi$
$758$		0			0
$759$			2.48391i			0.0901602i
$760$		0			0
$761$			28.4502i			1.03132i	0.856793	+	0.515660i	$0.172453\pi$
$761$			28.4502i			1.03132i	−0.856793	+	0.515660i	$0.827547\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		−	23.9231i		−	0.864940i
$766$		0			0
$767$	7.18654	−	30.6851i	0.259491	−	1.10797i
$768$		0			0
$769$			34.0730i			1.22870i	0.789032	+	0.614352i	$0.210583\pi$
$769$			34.0730i			1.22870i	−0.789032	+	0.614352i	$0.789417\pi$
$770$		0			0
$771$	−21.8768			−0.787874
$772$		0			0
$773$		−	15.6527i		−	0.562988i	−0.959563	−	0.281494i	$-0.909170\pi$
$773$		−	15.6527i		−	0.562988i	0.959563	−	0.281494i	$-0.0908299\pi$
$774$		0			0
$775$			24.5919i			0.883366i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−36.6903			−1.31456
$780$		0			0
$781$	−16.3172			−0.583874
$782$		0			0
$783$	−5.32401			−0.190265
$784$		0			0
$785$			12.2141i			0.435940i
$786$		0			0
$787$		−	11.1255i		−	0.396582i	−0.980143	−	0.198291i	$-0.936461\pi$
$787$		−	11.1255i		−	0.396582i	0.980143	−	0.198291i	$-0.0635391\pi$
$788$		0			0
$789$	−7.11636			−0.253349
$790$		0			0
$791$		0			0
$792$		0			0
$793$	30.3594	+	7.11027i	1.07809	+	0.252493i
$794$		0			0
$795$		−	31.5891i		−	1.12035i
$796$		0			0
$797$	−37.4615			−1.32695			−0.663477	−	0.748197i	$-0.730920\pi$
$797$	−37.4615			−1.32695			−0.663477	+	0.748197i	$0.730920\pi$
$798$		0			0
$799$			100.155i			3.54323i
$800$		0			0
$801$			0.235209i			0.00831069i
$802$		0			0
$803$	−10.0422			−0.354383
$804$		0			0
$805$		0			0
$806$		0			0
$807$	3.83457			0.134983
$808$		0			0
$809$	−7.42216			−0.260949			−0.130475	−	0.991452i	$-0.541650\pi$
$809$	−7.42216			−0.260949			−0.130475	+	0.991452i	$0.541650\pi$
$810$		0			0
$811$		−	12.0504i		−	0.423148i	−0.977362	−	0.211574i	$-0.932141\pi$
$811$		−	12.0504i		−	0.423148i	0.977362	−	0.211574i	$-0.0678589\pi$
$812$		0			0
$813$		−	21.2755i		−	0.746164i
$814$		0			0
$815$	−41.2961			−1.44654
$816$		0			0
$817$			13.4387i			0.470160i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	3.63697i		−	0.126931i	−0.997984	−	0.0634655i	$-0.979785\pi$
$821$		−	3.63697i		−	0.126931i	0.997984	−	0.0634655i	$-0.0202153\pi$
$822$		0			0
$823$	−46.5921			−1.62410			−0.812050	−	0.583589i	$-0.801648\pi$
$823$	−46.5921			−1.62410			−0.812050	+	0.583589i	$0.801648\pi$
$824$		0			0
$825$		−	4.27812i		−	0.148945i
$826$		0			0
$827$		−	11.5938i		−	0.403157i	−0.979472	−	0.201579i	$-0.935393\pi$
$827$		−	11.5938i		−	0.403157i	0.979472	−	0.201579i	$-0.0646072\pi$
$828$		0			0
$829$	22.2749			0.773641			0.386820	−	0.922155i	$-0.373573\pi$
$829$	22.2749			0.773641			0.386820	+	0.922155i	$0.373573\pi$
$830$		0			0
$831$	20.3874			0.707230
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−41.0776			−1.42155
$836$		0			0
$837$		−	5.68728i		−	0.196581i
$838$		0			0
$839$			20.5858i			0.710700i	0.934733	+	0.355350i	$0.115638\pi$
$839$			20.5858i			0.710700i	−0.934733	+	0.355350i	$0.884362\pi$
$840$		0			0
$841$	−0.654870			−0.0225817
$842$		0			0
$843$		−	25.4176i		−	0.875428i
$844$		0			0
$845$	17.6269	−	35.5675i	0.606384	−	1.22356i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	15.6691			0.537764
$850$		0			0
$851$			16.1716i			0.554357i
$852$		0			0
$853$			32.4287i			1.11034i	0.831738	+	0.555168i	$0.187346\pi$
$853$			32.4287i			1.11034i	−0.831738	+	0.555168i	$0.812654\pi$
$854$		0			0
$855$	12.3451			0.422195
$856$		0			0
$857$	−7.41926			−0.253437			−0.126718	−	0.991939i	$-0.540444\pi$
$857$	−7.41926			−0.253437			−0.126718	+	0.991939i	$0.540444\pi$
$858$		0			0
$859$	15.0074			0.512047			0.256023	−	0.966671i	$-0.417588\pi$
$859$	15.0074			0.512047			0.256023	+	0.966671i	$0.417588\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			47.1564i			1.60522i	0.596502	+	0.802612i	$0.296557\pi$
$863$			47.1564i			1.60522i	−0.596502	+	0.802612i	$0.703443\pi$
$864$		0			0
$865$			42.3732i			1.44073i
$866$		0			0
$867$	44.3805			1.50724
$868$		0			0
$869$		−	0.299684i		−	0.0101661i
$870$		0			0
$871$	8.00000	−	34.1584i	0.271070	−	1.15741i
$872$		0			0
$873$			12.4631i			0.421813i
$874$		0			0
$875$		0			0
$876$		0			0
$877$			48.1876i			1.62718i	0.581440	+	0.813590i	$0.302490\pi$
$877$			48.1876i			1.62718i	−0.581440	+	0.813590i	$0.697510\pi$
$878$		0			0
$879$			6.34226i			0.213919i
$880$		0			0
$881$	30.4267			1.02510			0.512551	−	0.858657i	$-0.328701\pi$
$881$	30.4267			1.02510			0.512551	+	0.858657i	$0.328701\pi$
$882$		0			0
$883$	32.3172			1.08756			0.543780	−	0.839228i	$-0.316993\pi$
$883$	32.3172			1.08756			0.543780	+	0.839228i	$0.316993\pi$
$884$		0			0
$885$	−26.6903			−0.897183
$886$		0			0
$887$	−6.04223			−0.202878			−0.101439	−	0.994842i	$-0.532345\pi$
$887$	−6.04223			−0.202878			−0.101439	+	0.994842i	$0.532345\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			0.989386i			0.0331457i
$892$		0			0
$893$	−51.6834			−1.72952
$894$		0			0
$895$			32.9338i			1.10086i
$896$		0			0
$897$	8.81346	+	2.06414i	0.294273	+	0.0689196i
$898$		0			0
$899$			30.2791i			1.00987i
$900$		0			0
$901$	81.0497			2.70016
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	60.0602i		−	1.99647i
$906$		0			0
$907$	38.0143			1.26224			0.631122	−	0.775684i	$-0.282595\pi$
$907$	38.0143			1.26224			0.631122	+	0.775684i	$0.282595\pi$
$908$		0			0
$909$	−9.46149			−0.313818
$910$		0			0
$911$	−46.2082			−1.53095			−0.765474	−	0.643467i	$-0.777495\pi$
$911$	−46.2082			−1.53095			−0.765474	+	0.643467i	$0.777495\pi$
$912$		0			0
$913$	−4.64803			−0.153827
$914$		0			0
$915$		−	26.4070i		−	0.872988i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	16.2327			0.535468			0.267734	−	0.963493i	$-0.413725\pi$
$919$	16.2327			0.535468			0.267734	+	0.963493i	$0.413725\pi$
$920$		0			0
$921$		−	1.89341i		−	0.0623898i
$922$		0			0
$923$	−13.5596	+	57.8969i	−0.446321	+	1.90570i
$924$		0			0
$925$		−	27.8529i		−	0.915799i
$926$		0			0
$927$	−8.64803			−0.284039
$928$		0			0
$929$			10.7702i			0.353359i	0.984268	+	0.176680i	$0.0565357\pi$
$929$			10.7702i			0.353359i	−0.984268	+	0.176680i	$0.943464\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	11.6691			0.382031
$934$		0			0
$935$	23.6691			0.774064
$936$		0			0
$937$	−7.90186			−0.258142			−0.129071	−	0.991635i	$-0.541200\pi$
$937$	−7.90186			−0.258142			−0.129071	+	0.991635i	$0.541200\pi$
$938$		0			0
$939$	−20.6480			−0.673823
$940$		0			0
$941$			7.38020i			0.240588i	0.992738	+	0.120294i	$0.0383836\pi$
$941$			7.38020i			0.240588i	−0.992738	+	0.120294i	$0.961616\pi$
$942$		0			0
$943$			22.7838i			0.741944i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			50.7866i			1.65034i	0.564882	+	0.825172i	$0.308922\pi$
$947$			50.7866i			1.65034i	−0.564882	+	0.825172i	$0.691078\pi$
$948$		0			0
$949$	−8.34513	+	35.6320i	−0.270894	+	1.15666i
$950$		0			0
$951$		−	17.8022i		−	0.577275i
$952$		0			0
$953$	−43.9161			−1.42258			−0.711291	−	0.702897i	$-0.751889\pi$
$953$	−43.9161			−1.42258			−0.711291	+	0.702897i	$0.751889\pi$
$954$		0			0
$955$		−	45.3622i		−	1.46789i
$956$		0			0
$957$		−	5.26751i		−	0.170274i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.34513			−0.0433913
$962$		0			0
$963$	3.18654			0.102685
$964$		0			0
$965$	−29.7114			−0.956443
$966$		0			0
$967$			32.6430i			1.04973i	0.851186	+	0.524864i	$0.175884\pi$
$967$			32.6430i			1.04973i	−0.851186	+	0.524864i	$0.824116\pi$
$968$		0			0
$969$			31.6745i			1.01753i
$970$		0			0
$971$	−45.7114			−1.46695			−0.733474	−	0.679718i	$-0.762102\pi$
$971$	−45.7114			−1.46695			−0.733474	+	0.679718i	$0.762102\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−15.1797	−	3.55513i	−0.486140	−	0.113855i
$976$		0			0
$977$		−	24.7764i		−	0.792668i	−0.918106	−	0.396334i	$-0.870282\pi$
$977$		−	24.7764i		−	0.792668i	0.918106	−	0.396334i	$-0.129718\pi$
$978$		0			0
$979$	−0.232712			−0.00743752
$980$		0			0
$981$		−	5.26751i		−	0.168179i
$982$		0			0
$983$		−	29.2968i		−	0.934424i	−0.884145	−	0.467212i	$-0.845259\pi$
$983$		−	29.2968i		−	0.934424i	0.884145	−	0.467212i	$-0.154741\pi$
$984$		0			0
$985$	52.4016			1.66965
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.34513			0.265360
$990$		0			0
$991$	−10.3731			−0.329512			−0.164756	−	0.986334i	$-0.552684\pi$
$991$	−10.3731			−0.329512			−0.164756	+	0.986334i	$0.552684\pi$
$992$		0			0
$993$			4.96782i			0.157649i
$994$		0			0
$995$		−	60.0602i		−	1.90404i
$996$		0			0
$997$	−18.7884			−0.595035			−0.297517	−	0.954716i	$-0.596159\pi$
$997$	−18.7884			−0.595035			−0.297517	+	0.954716i	$0.596159\pi$
$998$		0			0
$999$			6.44146i			0.203799i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.e.m.4705.1	yes	6
7.6	odd	2		7644.2.e.j.4705.6	yes	6
13.12	even	2	inner	7644.2.e.m.4705.6	yes	6
91.90	odd	2		7644.2.e.j.4705.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7644.2.e.j.4705.1	✓	6	91.90	odd	2
7644.2.e.j.4705.6	yes	6	7.6	odd	2
7644.2.e.m.4705.1	yes	6	1.1	even	1	trivial
7644.2.e.m.4705.6	yes	6	13.12	even	2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 7644 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7644.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.05352i q^{5} +1.00000 q^{9} +0.989386i q^{11} +(3.51056 + 0.822183i) q^{13} -3.05352i q^{15} +7.83457 q^{17} +4.04291i q^{19} +2.51056 q^{23} -4.32401 q^{25} +1.00000 q^{27} -5.32401 q^{29} -5.68728i q^{31} +0.989386i q^{33} +6.44146i q^{37} +(3.51056 + 0.822183i) q^{39} +9.07521i q^{41} +3.32401 q^{43} -3.05352i q^{45} +12.7837i q^{47} +7.83457 q^{51} +10.3451 q^{53} +3.02112 q^{55} +4.04291i q^{57} -8.74080i q^{59} +8.64803 q^{61} +(2.51056 - 10.7196i) q^{65} -9.73019i q^{67} +2.51056 q^{69} +16.4922i q^{71} +10.1500i q^{73} -4.32401 q^{75} -0.302899 q^{79} +1.00000 q^{81} +4.69789i q^{83} -23.9231i q^{85} -5.32401 q^{87} +0.235209i q^{89} -5.68728i q^{93} +12.3451 q^{95} +12.4631i q^{97} +0.989386i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{9} + 4 q^{13} + 8 q^{17} - 2 q^{23} - 4 q^{25} + 6 q^{27} - 10 q^{29} + 4 q^{39} - 2 q^{43} + 8 q^{51} + 6 q^{53} - 16 q^{55} + 8 q^{61} - 2 q^{65} - 2 q^{69} - 4 q^{75} - 14 q^{79}+ \cdots + 18 q^{95}+O(q^{100}) \) 6 * q + 6 * q^3 + 6 * q^9 + 4 * q^13 + 8 * q^17 - 2 * q^23 - 4 * q^25 + 6 * q^27 - 10 * q^29 + 4 * q^39 - 2 * q^43 + 8 * q^51 + 6 * q^53 - 16 * q^55 + 8 * q^61 - 2 * q^65 - 2 * q^69 - 4 * q^75 - 14 * q^79 + 6 * q^81 - 10 * q^87 + 18 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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