LMFDB - Embedded newform 7644.2.e.m.4705.4

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)

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")

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);

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			4705.4
Root			$0.634243i$ of defining polynomial
Character	$\chi$	$=$	7644.4705
Dual form			7644.2.e.m.4705.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.237279i q^{5} +1.00000 q^{9} +2.12216i q^{11} +(1.74823 - 3.15336i) q^{13} +0.237279i q^{15} -3.19547 q^{17} +1.88488i q^{19} +0.748228 q^{23} +4.94370 q^{25} +1.00000 q^{27} +3.94370 q^{29} +4.42185i q^{31} +2.12216i q^{33} +10.0765i q^{37} +(1.74823 - 3.15336i) q^{39} +5.89191i q^{41} -5.94370 q^{43} +0.237279i q^{45} -2.77425i q^{47} -3.19547 q^{51} -2.44724 q^{53} -0.503544 q^{55} +1.88488i q^{57} +4.65913i q^{59} -9.88740 q^{61} +(0.748228 + 0.414819i) q^{65} +2.53697i q^{67} +0.748228 q^{69} -11.4404i q^{71} +1.41032i q^{73} +4.94370 q^{75} +5.44015 q^{79} +1.00000 q^{81} -6.54401i q^{83} -0.758219i q^{85} +3.94370 q^{87} -12.3762i q^{89} +4.42185i q^{93} -0.447242 q^{95} +16.2057i q^{97} +2.12216i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$			0.237279i			0.106115i	0.998591	+	0.0530573i	$0.0168966\pi$
$5$			0.237279i			0.106115i	−0.998591	+	0.0530573i	$0.983103\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			2.12216i			0.639854i	0.947442	+	0.319927i	$0.103658\pi$
$11$			2.12216i			0.639854i	−0.947442	+	0.319927i	$0.896342\pi$
$12$		0			0
$13$	1.74823	−	3.15336i	0.484871	−	0.874586i
$13$	1.74823	−	3.15336i	0.484871	−	0.874586i
$14$		0			0
$15$			0.237279i			0.0612653i
$16$		0			0
$17$	−3.19547			−0.775015			−0.387508	−	0.921866i	$-0.626664\pi$
$17$	−3.19547			−0.775015			−0.387508	+	0.921866i	$0.626664\pi$
$18$		0			0
$19$			1.88488i			0.432420i	0.976347	+	0.216210i	$0.0693696\pi$
$19$			1.88488i			0.432420i	−0.976347	+	0.216210i	$0.930630\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.748228			0.156016			0.0780082	−	0.996953i	$-0.475144\pi$
$23$	0.748228			0.156016			0.0780082	+	0.996953i	$0.475144\pi$
$24$		0			0
$25$	4.94370			0.988740
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	3.94370			0.732326			0.366163	−	0.930551i	$-0.380671\pi$
$29$	3.94370			0.732326			0.366163	+	0.930551i	$0.380671\pi$
$30$		0			0
$31$			4.42185i			0.794188i	0.917778	+	0.397094i	$0.129981\pi$
$31$			4.42185i			0.794188i	−0.917778	+	0.397094i	$0.870019\pi$
$32$		0			0
$33$			2.12216i			0.369420i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			10.0765i			1.65656i	0.560312	+	0.828281i	$0.310681\pi$
$37$			10.0765i			1.65656i	−0.560312	+	0.828281i	$0.689319\pi$
$38$		0			0
$39$	1.74823	−	3.15336i	0.279941	−	0.504942i
$40$		0			0
$41$			5.89191i			0.920161i	0.887877	+	0.460081i	$0.152180\pi$
$41$			5.89191i			0.920161i	−0.887877	+	0.460081i	$0.847820\pi$
$42$		0			0
$43$	−5.94370			−0.906406			−0.453203	−	0.891407i	$-0.649719\pi$
$43$	−5.94370			−0.906406			−0.453203	+	0.891407i	$0.649719\pi$
$44$		0			0
$45$			0.237279i			0.0353715i
$46$		0			0
$47$		−	2.77425i		−	0.404666i	−0.979317	−	0.202333i	$-0.935148\pi$
$47$		−	2.77425i		−	0.404666i	0.979317	−	0.202333i	$-0.0648524\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−3.19547			−0.447455
$52$		0			0
$53$	−2.44724			−0.336155			−0.168077	−	0.985774i	$-0.553756\pi$
$53$	−2.44724			−0.336155			−0.168077	+	0.985774i	$0.553756\pi$
$54$		0			0
$55$	−0.503544			−0.0678978
$56$		0			0
$57$			1.88488i			0.249658i
$58$		0			0
$59$			4.65913i			0.606567i	0.952900	+	0.303283i	$0.0980829\pi$
$59$			4.65913i			0.606567i	−0.952900	+	0.303283i	$0.901917\pi$
$60$		0			0
$61$	−9.88740			−1.26595			−0.632976	−	0.774172i	$-0.718167\pi$
$61$	−9.88740			−1.26595			−0.632976	+	0.774172i	$0.718167\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.748228	+	0.414819i	0.0928063	+	0.0514519i
$66$		0			0
$67$			2.53697i			0.309941i	0.987919	+	0.154970i	$0.0495282\pi$
$67$			2.53697i			0.309941i	−0.987919	+	0.154970i	$0.950472\pi$
$68$		0			0
$69$	0.748228			0.0900761
$70$		0			0
$71$		−	11.4404i		−	1.35773i	−0.734264	−	0.678864i	$-0.762473\pi$
$71$		−	11.4404i		−	1.35773i	0.734264	−	0.678864i	$-0.237527\pi$
$72$		0			0
$73$			1.41032i			0.165065i	0.996588	+	0.0825326i	$0.0263009\pi$
$73$			1.41032i			0.165065i	−0.996588	+	0.0825326i	$0.973699\pi$
$74$		0			0
$75$	4.94370			0.570849
$76$		0			0
$77$		0			0
$78$		0			0
$79$	5.44015			0.612065			0.306033	−	0.952021i	$-0.400998\pi$
$79$	5.44015			0.612065			0.306033	+	0.952021i	$0.400998\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$		−	6.54401i		−	0.718298i	−0.933280	−	0.359149i	$-0.883067\pi$
$83$		−	6.54401i		−	0.718298i	0.933280	−	0.359149i	$-0.116933\pi$
$84$		0			0
$85$		−	0.758219i		−	0.0822404i
$86$		0			0
$87$	3.94370			0.422809
$88$		0			0
$89$		−	12.3762i		−	1.31187i	−0.754817	−	0.655936i	$-0.772274\pi$
$89$		−	12.3762i		−	1.31187i	0.754817	−	0.655936i	$-0.227726\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$			4.42185i			0.458524i
$94$		0			0
$95$	−0.447242			−0.0458861
$96$		0			0
$97$			16.2057i			1.64544i	0.568450	+	0.822718i	$0.307543\pi$
$97$			16.2057i			1.64544i	−0.568450	+	0.822718i	$0.692457\pi$
$98$		0			0
$99$			2.12216i			0.213285i
$100$		0			0
$101$	16.5793			1.64970			0.824852	−	0.565348i	$-0.191258\pi$
$101$	16.5793			1.64970			0.824852	+	0.565348i	$0.191258\pi$
$102$		0			0
$103$	9.88740			0.974234			0.487117	−	0.873337i	$-0.338048\pi$
$103$	9.88740			0.974234			0.487117	+	0.873337i	$0.338048\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	10.6919			1.03363			0.516814	−	0.856098i	$-0.327118\pi$
$107$	10.6919			1.03363			0.516814	+	0.856098i	$0.327118\pi$
$108$		0			0
$109$			8.36914i			0.801618i	0.916162	+	0.400809i	$0.131271\pi$
$109$			8.36914i			0.801618i	−0.916162	+	0.400809i	$0.868729\pi$
$110$		0			0
$111$			10.0765i			0.956417i
$112$		0			0
$113$	−7.94370			−0.747280			−0.373640	−	0.927574i	$-0.621891\pi$
$113$	−7.94370			−0.747280			−0.373640	+	0.927574i	$0.621891\pi$
$114$		0			0
$115$			0.177539i			0.0165556i
$116$		0			0
$117$	1.74823	−	3.15336i	0.161624	−	0.291529i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	6.49646			0.590587
$122$		0			0
$123$			5.89191i			0.531255i
$124$		0			0
$125$			2.35943i			0.211034i
$126$		0			0
$127$	6.89448			0.611787			0.305893	−	0.952066i	$-0.401045\pi$
$127$	6.89448			0.611787			0.305893	+	0.952066i	$0.401045\pi$
$128$		0			0
$129$	−5.94370			−0.523313
$130$		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
$135$			0.237279i			0.0204218i
$136$		0			0
$137$			18.0308i			1.54048i	0.637757	+	0.770238i	$0.279862\pi$
$137$			18.0308i			1.54048i	−0.637757	+	0.770238i	$0.720138\pi$
$138$		0			0
$139$	13.8874			1.17791			0.588957	−	0.808165i	$-0.299539\pi$
$139$	13.8874			1.17791			0.588957	+	0.808165i	$0.299539\pi$
$140$		0			0
$141$		−	2.77425i		−	0.233634i
$142$		0			0
$143$	6.69193	+	3.71001i	0.559607	+	0.310247i
$144$		0			0
$145$			0.935758i			0.0777105i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			13.7865i			1.12943i	0.825285	+	0.564717i	$0.191015\pi$
$149$			13.7865i			1.12943i	−0.825285	+	0.564717i	$0.808985\pi$
$150$		0			0
$151$			10.9061i			0.887527i	0.896144	+	0.443764i	$0.146357\pi$
$151$			10.9061i			0.887527i	−0.896144	+	0.443764i	$0.853643\pi$
$152$		0			0
$153$	−3.19547			−0.258338
$154$		0			0
$155$	−1.04921			−0.0842749
$156$		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
$159$	−2.44724			−0.194079
$160$		0			0
$161$		0			0
$162$		0			0
$163$			17.8069i			1.39474i	0.716710	+	0.697371i	$0.245647\pi$
$163$			17.8069i			1.39474i	−0.716710	+	0.697371i	$0.754353\pi$
$164$		0			0
$165$	−0.503544			−0.0392008
$166$		0			0
$167$		−	18.3278i		−	1.41825i	−0.705083	−	0.709125i	$-0.749090\pi$
$167$		−	18.3278i		−	1.41825i	0.705083	−	0.709125i	$-0.250910\pi$
$168$		0			0
$169$	−6.88740	−	11.0256i	−0.529800	−	0.848123i
$170$		0			0
$171$			1.88488i			0.144140i
$172$		0			0
$173$	4.20256			0.319515			0.159757	−	0.987156i	$-0.448929\pi$
$173$	4.20256			0.319515			0.159757	+	0.987156i	$0.448929\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$			4.65913i			0.350202i
$178$		0			0
$179$	24.5230			1.83294			0.916468	−	0.400107i	$-0.131027\pi$
$179$	24.5230			1.83294			0.916468	+	0.400107i	$0.131027\pi$
$180$		0			0
$181$	−2.39094			−0.177717			−0.0888586	−	0.996044i	$-0.528322\pi$
$181$	−2.39094			−0.177717			−0.0888586	+	0.996044i	$0.528322\pi$
$182$		0			0
$183$	−9.88740			−0.730897
$184$		0			0
$185$	−2.39094			−0.175785
$186$		0			0
$187$		−	6.78128i		−	0.495897i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.300986			0.0217786			0.0108893	−	0.999941i	$-0.496534\pi$
$191$	0.300986			0.0217786			0.0108893	+	0.999941i	$0.496534\pi$
$192$		0			0
$193$			2.53697i			0.182615i	0.995823	+	0.0913077i	$0.0291047\pi$
$193$			2.53697i			0.182615i	−0.995823	+	0.0913077i	$0.970895\pi$
$194$		0			0
$195$	0.748228	+	0.414819i	0.0535817	+	0.0297058i
$196$		0			0
$197$			9.66166i			0.688365i	0.938903	+	0.344182i	$0.111844\pi$
$197$			9.66166i			0.688365i	−0.938903	+	0.344182i	$0.888156\pi$
$198$		0			0
$199$	−2.39094			−0.169489			−0.0847446	−	0.996403i	$-0.527007\pi$
$199$	−2.39094			−0.169489			−0.0847446	+	0.996403i	$0.527007\pi$
$200$		0			0
$201$			2.53697i			0.178944i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−1.39803			−0.0976425
$206$		0			0
$207$	0.748228			0.0520054
$208$		0			0
$209$	−4.00000			−0.276686
$210$		0			0
$211$	−14.9366			−1.02828			−0.514139	−	0.857707i	$-0.671889\pi$
$211$	−14.9366			−1.02828			−0.514139	+	0.857707i	$0.671889\pi$
$212$		0			0
$213$		−	11.4404i		−	0.783884i
$214$		0			0
$215$		−	1.41032i		−	0.0961828i
$216$		0			0
$217$		0			0
$218$		0			0
$219$			1.41032i			0.0953004i
$220$		0			0
$221$	−5.58641	+	10.0765i	−0.375783	+	0.677817i
$222$		0			0
$223$			11.6777i			0.781996i	0.920391	+	0.390998i	$0.127870\pi$
$223$			11.6777i			0.781996i	−0.920391	+	0.390998i	$0.872130\pi$
$224$		0			0
$225$	4.94370			0.329580
$226$		0			0
$227$			13.1478i			0.872647i	0.899790	+	0.436323i	$0.143720\pi$
$227$			13.1478i			0.872647i	−0.899790	+	0.436323i	$0.856280\pi$
$228$		0			0
$229$			1.30420i			0.0861837i	0.999071	+	0.0430918i	$0.0137208\pi$
$229$			1.30420i			0.0861837i	−0.999071	+	0.0430918i	$0.986279\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	12.4331			0.814517			0.407259	−	0.913313i	$-0.366485\pi$
$233$	12.4331			0.814517			0.407259	+	0.913313i	$0.366485\pi$
$234$		0			0
$235$	0.658273			0.0429410
$236$		0			0
$237$	5.44015			0.353376
$238$		0			0
$239$		−	21.3260i		−	1.37946i	−0.724065	−	0.689732i	$-0.757728\pi$
$239$		−	21.3260i		−	1.37946i	0.724065	−	0.689732i	$-0.242272\pi$
$240$		0			0
$241$		−	1.76540i		−	0.113719i	−0.998382	−	0.0568596i	$-0.981891\pi$
$241$		−	1.76540i		−	0.113719i	0.998382	−	0.0568596i	$-0.0181087\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		0			0
$246$		0			0
$247$	5.94370	+	3.29519i	0.378188	+	0.209668i
$248$		0			0
$249$		−	6.54401i		−	0.414709i
$250$		0			0
$251$	−16.3768			−1.03369			−0.516846	−	0.856078i	$-0.672894\pi$
$251$	−16.3768			−1.03369			−0.516846	+	0.856078i	$0.672894\pi$
$252$		0			0
$253$			1.58786i			0.0998277i
$254$		0			0
$255$		−	0.758219i		−	0.0474815i
$256$		0			0
$257$	−3.79744			−0.236878			−0.118439	−	0.992961i	$-0.537789\pi$
$257$	−3.79744			−0.236878			−0.118439	+	0.992961i	$0.537789\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	3.94370			0.244109
$262$		0			0
$263$	6.13208			0.378120			0.189060	−	0.981966i	$-0.439456\pi$
$263$	6.13208			0.378120			0.189060	+	0.981966i	$0.439456\pi$
$264$		0			0
$265$		−	0.580680i		−	0.0356709i
$266$		0			0
$267$		−	12.3762i		−	0.757409i
$268$		0			0
$269$	−7.19547			−0.438716			−0.219358	−	0.975644i	$-0.570396\pi$
$269$	−7.19547			−0.438716			−0.219358	+	0.975644i	$0.570396\pi$
$270$		0			0
$271$			24.5882i			1.49362i	0.665035	+	0.746812i	$0.268417\pi$
$271$			24.5882i			1.49362i	−0.665035	+	0.746812i	$0.731583\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			10.4913i			0.632649i
$276$		0			0
$277$	0.545670			0.0327861			0.0163931	−	0.999866i	$-0.494782\pi$
$277$	0.545670			0.0327861			0.0163931	+	0.999866i	$0.494782\pi$
$278$		0			0
$279$			4.42185i			0.264729i
$280$		0			0
$281$		−	0.223920i		−	0.0133580i	−0.999978	−	0.00667898i	$-0.997874\pi$
$281$		−	0.223920i		−	0.0133580i	0.999978	−	0.00667898i	$-0.00212600\pi$
$282$		0			0
$283$	−6.39094			−0.379902			−0.189951	−	0.981794i	$-0.560833\pi$
$283$	−6.39094			−0.379902			−0.189951	+	0.981794i	$0.560833\pi$
$284$		0			0
$285$	−0.447242			−0.0264923
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−6.78897			−0.399351
$290$		0			0
$291$			16.2057i			0.949993i
$292$		0			0
$293$		−	12.8507i		−	0.750748i	−0.926873	−	0.375374i	$-0.877514\pi$
$293$		−	12.8507i		−	0.750748i	0.926873	−	0.375374i	$-0.122486\pi$
$294$		0			0
$295$	−1.10552			−0.0643656
$296$		0			0
$297$			2.12216i			0.123140i
$298$		0			0
$299$	1.30807	−	2.35943i	0.0756478	−	0.136450i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	16.5793			0.952457
$304$		0			0
$305$		−	2.34608i		−	0.134336i
$306$		0			0
$307$		−	10.8481i		−	0.619131i	−0.950878	−	0.309566i	$-0.899816\pi$
$307$		−	10.8481i		−	0.619131i	0.950878	−	0.309566i	$-0.100184\pi$
$308$		0			0
$309$	9.88740			0.562474
$310$		0			0
$311$	−10.3909			−0.589216			−0.294608	−	0.955618i	$-0.595189\pi$
$311$	−10.3909			−0.589216			−0.294608	+	0.955618i	$0.595189\pi$
$312$		0			0
$313$	−2.11260			−0.119411			−0.0597057	−	0.998216i	$-0.519016\pi$
$313$	−2.11260			−0.119411			−0.0597057	+	0.998216i	$0.519016\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			28.2982i			1.58938i	0.607013	+	0.794692i	$0.292368\pi$
$317$			28.2982i			1.58938i	−0.607013	+	0.794692i	$0.707632\pi$
$318$		0			0
$319$			8.36914i			0.468582i
$320$		0			0
$321$	10.6919			0.596765
$322$		0			0
$323$		−	6.02307i		−	0.335132i
$324$		0			0
$325$	8.64271	−	15.5893i	0.479411	−	0.864737i
$326$		0			0
$327$			8.36914i			0.462815i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			3.17571i			0.174553i	0.996184	+	0.0872765i	$0.0278164\pi$
$331$			3.17571i			0.174553i	−0.996184	+	0.0872765i	$0.972184\pi$
$332$		0			0
$333$			10.0765i			0.552188i
$334$		0			0
$335$	−0.601972			−0.0328892
$336$		0			0
$337$	13.8311			0.753428			0.376714	−	0.926330i	$-0.377054\pi$
$337$	13.8311			0.753428			0.376714	+	0.926330i	$0.377054\pi$
$338$		0			0
$339$	−7.94370			−0.431442
$340$		0			0
$341$	−9.38385			−0.508164
$342$		0			0
$343$		0			0
$344$		0			0
$345$			0.177539i			0.00955838i
$346$		0			0
$347$	−4.70610			−0.252637			−0.126318	−	0.991990i	$-0.540316\pi$
$347$	−4.70610			−0.252637			−0.126318	+	0.991990i	$0.540316\pi$
$348$		0			0
$349$			6.29337i			0.336876i	0.985712	+	0.168438i	$0.0538723\pi$
$349$			6.29337i			0.336876i	−0.985712	+	0.168438i	$0.946128\pi$
$350$		0			0
$351$	1.74823	−	3.15336i	0.0933135	−	0.168314i
$352$		0			0
$353$			17.6757i			0.940784i	0.882458	+	0.470392i	$0.155887\pi$
$353$			17.6757i			0.940784i	−0.882458	+	0.470392i	$0.844113\pi$
$354$		0			0
$355$	2.71457			0.144075
$356$		0			0
$357$		0			0
$358$		0			0
$359$			24.1733i			1.27582i	0.770111	+	0.637910i	$0.220201\pi$
$359$			24.1733i			1.27582i	−0.770111	+	0.637910i	$0.779799\pi$
$360$		0			0
$361$	15.4472			0.813013
$362$		0			0
$363$	6.49646			0.340976
$364$		0			0
$365$	−0.334639			−0.0175158
$366$		0			0
$367$	21.2854			1.11109			0.555545	−	0.831486i	$-0.312510\pi$
$367$	21.2854			1.11109			0.555545	+	0.831486i	$0.312510\pi$
$368$		0			0
$369$			5.89191i			0.306720i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	32.2642			1.67058			0.835288	−	0.549813i	$-0.185301\pi$
$373$	32.2642			1.67058			0.835288	+	0.549813i	$0.185301\pi$
$374$		0			0
$375$			2.35943i			0.121841i
$376$		0			0
$377$	6.89448	−	12.4359i	0.355084	−	0.640482i
$378$		0			0
$379$		−	27.8834i		−	1.43227i	−0.697961	−	0.716136i	$-0.745909\pi$
$379$		−	27.8834i		−	1.43227i	0.697961	−	0.716136i	$-0.254091\pi$
$380$		0			0
$381$	6.89448			0.353215
$382$		0			0
$383$		−	16.4429i		−	0.840195i	−0.907479	−	0.420098i	$-0.861996\pi$
$383$		−	16.4429i		−	0.840195i	0.907479	−	0.420098i	$-0.138004\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.94370			−0.302135
$388$		0			0
$389$	25.7748			1.30683			0.653417	−	0.756998i	$-0.273335\pi$
$389$	25.7748			1.30683			0.653417	+	0.756998i	$0.273335\pi$
$390$		0			0
$391$	−2.39094			−0.120915
$392$		0			0
$393$	−8.00000			−0.403547
$394$		0			0
$395$			1.29084i			0.0649490i
$396$		0			0
$397$		−	9.42438i		−	0.472996i	−0.971632	−	0.236498i	$-0.924000\pi$
$397$		−	9.42438i		−	0.472996i	0.971632	−	0.236498i	$-0.0759997\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			22.1556i			1.10640i	0.833049	+	0.553200i	$0.186593\pi$
$401$			22.1556i			1.10640i	−0.833049	+	0.553200i	$0.813407\pi$
$402$		0			0
$403$	13.9437	+	7.73040i	0.694585	+	0.385079i
$404$		0			0
$405$			0.237279i			0.0117905i
$406$		0			0
$407$	−21.3839			−1.05996
$408$		0			0
$409$		−	0.532617i		−	0.0263362i	−0.999913	−	0.0131681i	$-0.995808\pi$
$409$		−	0.532617i		−	0.0263362i	0.999913	−	0.0131681i	$-0.00419166\pi$
$410$		0			0
$411$			18.0308i			0.889394i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1.55276			0.0762219
$416$		0			0
$417$	13.8874			0.680069
$418$		0			0
$419$	−25.9858			−1.26949			−0.634745	−	0.772721i	$-0.718895\pi$
$419$	−25.9858			−1.26949			−0.634745	+	0.772721i	$0.718895\pi$
$420$		0			0
$421$			7.73040i			0.376757i	0.982097	+	0.188378i	$0.0603231\pi$
$421$			7.73040i			0.376757i	−0.982097	+	0.188378i	$0.939677\pi$
$422$		0			0
$423$		−	2.77425i		−	0.134889i
$424$		0			0
$425$	−15.7974			−0.766288
$426$		0			0
$427$		0			0
$428$		0			0
$429$	6.69193	+	3.71001i	0.323089	+	0.179121i
$430$		0			0
$431$			30.6443i			1.47608i	0.674756	+	0.738041i	$0.264249\pi$
$431$			30.6443i			1.47608i	−0.674756	+	0.738041i	$0.735751\pi$
$432$		0			0
$433$	1.88740			0.0907025			0.0453513	−	0.998971i	$-0.485559\pi$
$433$	1.88740			0.0907025			0.0453513	+	0.998971i	$0.485559\pi$
$434$		0			0
$435$			0.935758i			0.0448662i
$436$		0			0
$437$			1.41032i			0.0674646i
$438$		0			0
$439$	−9.28543			−0.443169			−0.221585	−	0.975141i	$-0.571123\pi$
$439$	−9.28543			−0.443169			−0.221585	+	0.975141i	$0.571123\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−9.53011			−0.452789			−0.226395	−	0.974036i	$-0.572694\pi$
$443$	−9.53011			−0.452789			−0.226395	+	0.974036i	$0.572694\pi$
$444$		0			0
$445$	2.93661			0.139209
$446$		0			0
$447$			13.7865i			0.652079i
$448$		0			0
$449$			15.8042i			0.745847i	0.927862	+	0.372923i	$0.121645\pi$
$449$			15.8042i			0.745847i	−0.927862	+	0.372923i	$0.878355\pi$
$450$		0			0
$451$	−12.5035			−0.588769
$452$		0			0
$453$			10.9061i			0.512414i
$454$		0			0
$455$		0			0
$456$		0			0
$457$			31.9368i			1.49394i	0.664858	+	0.746970i	$0.268492\pi$
$457$			31.9368i			1.49394i	−0.664858	+	0.746970i	$0.731508\pi$
$458$		0			0
$459$	−3.19547			−0.149152
$460$		0			0
$461$		−	25.9254i		−	1.20747i	−0.797187	−	0.603733i	$-0.793679\pi$
$461$		−	25.9254i		−	1.20747i	0.797187	−	0.603733i	$-0.206321\pi$
$462$		0			0
$463$			0.310379i			0.0144245i	0.999974	+	0.00721226i	$0.00229576\pi$
$463$			0.310379i			0.0144245i	−0.999974	+	0.00721226i	$0.997704\pi$
$464$		0			0
$465$	−1.04921			−0.0486561
$466$		0			0
$467$	−17.3839			−0.804429			−0.402214	−	0.915546i	$-0.631759\pi$
$467$	−17.3839			−0.804429			−0.402214	+	0.915546i	$0.631759\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−4.00000			−0.184310
$472$		0			0
$473$		−	12.6135i		−	0.579967i
$474$		0			0
$475$			9.31826i			0.427551i
$476$		0			0
$477$	−2.44724			−0.112052
$478$		0			0
$479$		−	29.9922i		−	1.37038i	−0.728366	−	0.685188i	$-0.759720\pi$
$479$		−	29.9922i		−	1.37038i	0.728366	−	0.685188i	$-0.240280\pi$
$480$		0			0
$481$	31.7748	+	17.6160i	1.44881	+	0.803220i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.84527			−0.174605
$486$		0			0
$487$			23.5196i			1.06577i	0.846187	+	0.532887i	$0.178893\pi$
$487$			23.5196i			1.06577i	−0.846187	+	0.532887i	$0.821107\pi$
$488$		0			0
$489$			17.8069i			0.805255i
$490$		0			0
$491$	−1.68484			−0.0760357			−0.0380179	−	0.999277i	$-0.512104\pi$
$491$	−1.68484			−0.0760357			−0.0380179	+	0.999277i	$0.512104\pi$
$492$		0			0
$493$	−12.6020			−0.567564
$494$		0			0
$495$	−0.503544			−0.0226326
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	10.9542i		−	0.490376i	−0.969476	−	0.245188i	$-0.921150\pi$
$499$		−	10.9542i		−	0.490376i	0.969476	−	0.245188i	$-0.0788497\pi$
$500$		0			0
$501$		−	18.3278i		−	0.818827i
$502$		0			0
$503$	−12.3768			−0.551853			−0.275926	−	0.961179i	$-0.588985\pi$
$503$	−12.3768			−0.551853			−0.275926	+	0.961179i	$0.588985\pi$
$504$		0			0
$505$			3.93393i			0.175058i
$506$		0			0
$507$	−6.88740	−	11.0256i	−0.305880	−	0.489664i
$508$		0			0
$509$		−	41.9669i		−	1.86015i	−0.367371	−	0.930074i	$-0.619742\pi$
$509$		−	41.9669i		−	1.86015i	0.367371	−	0.930074i	$-0.380258\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			1.88488i			0.0832193i
$514$		0			0
$515$			2.34608i			0.103380i
$516$		0			0
$517$	5.88740			0.258927
$518$		0			0
$519$	4.20256			0.184472
$520$		0			0
$521$	−1.81162			−0.0793684			−0.0396842	−	0.999212i	$-0.512635\pi$
$521$	−1.81162			−0.0793684			−0.0396842	+	0.999212i	$0.512635\pi$
$522$		0			0
$523$	29.6622			1.29704			0.648519	−	0.761199i	$-0.275389\pi$
$523$	29.6622			1.29704			0.648519	+	0.761199i	$0.275389\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	14.1299i		−	0.615508i
$528$		0			0
$529$	−22.4402			−0.975659
$530$		0			0
$531$			4.65913i			0.202189i
$532$		0			0
$533$	18.5793	+	10.3004i	0.804760	+	0.446160i
$534$		0			0
$535$			2.53697i			0.109683i
$536$		0			0
$537$	24.5230			1.05825
$538$		0			0
$539$		0			0
$540$		0			0
$541$			21.2930i			0.915456i	0.889092	+	0.457728i	$0.151337\pi$
$541$			21.2930i			0.915456i	−0.889092	+	0.457728i	$0.848663\pi$
$542$		0			0
$543$	−2.39094			−0.102605
$544$		0			0
$545$	−1.98582			−0.0850634
$546$		0			0
$547$	−3.55276			−0.151905			−0.0759525	−	0.997111i	$-0.524200\pi$
$547$	−3.55276			−0.151905			−0.0759525	+	0.997111i	$0.524200\pi$
$548$		0			0
$549$	−9.88740			−0.421984
$550$		0			0
$551$			7.43338i			0.316673i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−2.39094			−0.101490
$556$		0			0
$557$			15.5652i			0.659521i	0.944065	+	0.329760i	$0.106968\pi$
$557$			15.5652i			0.659521i	−0.944065	+	0.329760i	$0.893032\pi$
$558$		0			0
$559$	−10.3909	+	18.7426i	−0.439490	+	0.792729i
$560$		0			0
$561$		−	6.78128i		−	0.286306i
$562$		0			0
$563$	42.1657			1.77707			0.888537	−	0.458805i	$-0.151722\pi$
$563$	42.1657			1.77707			0.888537	+	0.458805i	$0.151722\pi$
$564$		0			0
$565$		−	1.88488i		−	0.0792973i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	22.8240			0.956832			0.478416	−	0.878133i	$-0.341211\pi$
$569$	22.8240			0.956832			0.478416	+	0.878133i	$0.341211\pi$
$570$		0			0
$571$	0.545670			0.0228356			0.0114178	−	0.999935i	$-0.496366\pi$
$571$	0.545670			0.0228356			0.0114178	+	0.999935i	$0.496366\pi$
$572$		0			0
$573$	0.300986			0.0125739
$574$		0			0
$575$	3.69901			0.154260
$576$		0			0
$577$		−	30.3936i		−	1.26530i	−0.774437	−	0.632651i	$-0.781967\pi$
$577$		−	30.3936i		−	1.26530i	0.774437	−	0.632651i	$-0.218033\pi$
$578$		0			0
$579$			2.53697i			0.105433i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	5.19343i		−	0.215090i
$584$		0			0
$585$	0.748228	+	0.414819i	0.0309354	+	0.0171506i
$586$		0			0
$587$		−	39.1909i		−	1.61758i	−0.588096	−	0.808791i	$-0.700122\pi$
$587$		−	39.1909i		−	1.61758i	0.588096	−	0.808791i	$-0.299878\pi$
$588$		0			0
$589$	−8.33464			−0.343423
$590$		0			0
$591$			9.66166i			0.397428i
$592$		0			0
$593$		−	34.7824i		−	1.42834i	−0.699970	−	0.714172i	$-0.746803\pi$
$593$		−	34.7824i		−	1.42834i	0.699970	−	0.714172i	$-0.253197\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.39094			−0.0978547
$598$		0			0
$599$	−1.53011			−0.0625186			−0.0312593	−	0.999511i	$-0.509952\pi$
$599$	−1.53011			−0.0625186			−0.0312593	+	0.999511i	$0.509952\pi$
$600$		0			0
$601$	−10.8945			−0.444395			−0.222198	−	0.975002i	$-0.571323\pi$
$601$	−10.8945			−0.444395			−0.222198	+	0.975002i	$0.571323\pi$
$602$		0			0
$603$			2.53697i			0.103314i
$604$		0			0
$605$			1.54148i			0.0626699i
$606$		0			0
$607$	18.2642			0.741319			0.370660	−	0.928769i	$-0.379132\pi$
$607$	18.2642			0.741319			0.370660	+	0.928769i	$0.379132\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−8.74823	−	4.85003i	−0.353915	−	0.196211i
$612$		0			0
$613$			16.9772i			0.685704i	0.939389	+	0.342852i	$0.111393\pi$
$613$			16.9772i			0.685704i	−0.939389	+	0.342852i	$0.888607\pi$
$614$		0			0
$615$	−1.39803			−0.0563739
$616$		0			0
$617$		−	6.48595i		−	0.261114i	−0.991441	−	0.130557i	$-0.958323\pi$
$617$		−	6.48595i		−	0.261114i	0.991441	−	0.130557i	$-0.0416766\pi$
$618$		0			0
$619$		−	17.0487i		−	0.685244i	−0.939473	−	0.342622i	$-0.888685\pi$
$619$		−	17.0487i		−	0.685244i	0.939473	−	0.342622i	$-0.111315\pi$
$620$		0			0
$621$	0.748228			0.0300254
$622$		0			0
$623$		0			0
$624$		0			0
$625$	24.1586			0.966346
$626$		0			0
$627$	−4.00000			−0.159745
$628$		0			0
$629$		−	32.1991i		−	1.28386i
$630$		0			0
$631$		−	32.6469i		−	1.29965i	−0.760082	−	0.649827i	$-0.774841\pi$
$631$		−	32.6469i		−	1.29965i	0.760082	−	0.649827i	$-0.225159\pi$
$632$		0			0
$633$	−14.9366			−0.593677
$634$		0			0
$635$			1.63592i			0.0649195i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		−	11.4404i		−	0.452576i
$640$		0			0
$641$	−23.8311			−0.941272			−0.470636	−	0.882328i	$-0.655975\pi$
$641$	−23.8311			−0.941272			−0.470636	+	0.882328i	$0.655975\pi$
$642$		0			0
$643$		−	4.12483i		−	0.162667i	−0.996687	−	0.0813337i	$-0.974082\pi$
$643$		−	4.12483i		−	0.162667i	0.996687	−	0.0813337i	$-0.0259180\pi$
$644$		0			0
$645$		−	1.41032i		−	0.0555312i
$646$		0			0
$647$	−29.1586			−1.14634			−0.573172	−	0.819435i	$-0.694287\pi$
$647$	−29.1586			−1.14634			−0.573172	+	0.819435i	$0.694287\pi$
$648$		0			0
$649$	−9.88740			−0.388114
$650$		0			0
$651$		0			0
$652$		0			0
$653$	40.7677			1.59536			0.797682	−	0.603079i	$-0.206059\pi$
$653$	40.7677			1.59536			0.797682	+	0.603079i	$0.206059\pi$
$654$		0			0
$655$		−	1.89824i		−	0.0741702i
$656$		0			0
$657$			1.41032i			0.0550217i
$658$		0			0
$659$	−30.2447			−1.17817			−0.589083	−	0.808073i	$-0.700511\pi$
$659$	−30.2447			−1.17817			−0.589083	+	0.808073i	$0.700511\pi$
$660$		0			0
$661$		−	23.2973i		−	0.906161i	−0.891470	−	0.453081i	$-0.850325\pi$
$661$		−	23.2973i		−	0.906161i	0.891470	−	0.453081i	$-0.149675\pi$
$662$		0			0
$663$	−5.58641	+	10.0765i	−0.216958	+	0.391338i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	2.95079			0.114255
$668$		0			0
$669$			11.6777i			0.451486i
$670$		0			0
$671$		−	20.9826i		−	0.810024i
$672$		0			0
$673$	9.55276			0.368232			0.184116	−	0.982905i	$-0.441058\pi$
$673$	9.55276			0.368232			0.184116	+	0.982905i	$0.441058\pi$
$674$		0			0
$675$	4.94370			0.190283
$676$		0			0
$677$	−41.1813			−1.58273			−0.791363	−	0.611347i	$-0.790628\pi$
$677$	−41.1813			−1.58273			−0.791363	+	0.611347i	$0.790628\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$			13.1478i			0.503823i
$682$		0			0
$683$		−	11.5599i		−	0.442327i	−0.975237	−	0.221164i	$-0.929015\pi$
$683$		−	11.5599i		−	0.442327i	0.975237	−	0.221164i	$-0.0709855\pi$
$684$		0			0
$685$	−4.27834			−0.163467
$686$		0			0
$687$			1.30420i			0.0497582i
$688$		0			0
$689$	−4.27834	+	7.71704i	−0.162992	+	0.293996i
$690$		0			0
$691$			5.84553i			0.222374i	0.993799	+	0.111187i	$0.0354653\pi$
$691$			5.84553i			0.222374i	−0.993799	+	0.111187i	$0.964535\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			3.29519i			0.124994i
$696$		0			0
$697$		−	18.8274i		−	0.713139i
$698$		0			0
$699$	12.4331			0.470262
$700$		0			0
$701$	−20.8382			−0.787047			−0.393524	−	0.919314i	$-0.628744\pi$
$701$	−20.8382			−0.787047			−0.393524	+	0.919314i	$0.628744\pi$
$702$		0			0
$703$	−18.9929			−0.716331
$704$		0			0
$705$	0.658273			0.0247920
$706$		0			0
$707$		0			0
$708$		0			0
$709$			13.6821i			0.513840i	0.966433	+	0.256920i	$0.0827077\pi$
$709$			13.6821i			0.513840i	−0.966433	+	0.256920i	$0.917292\pi$
$710$		0			0
$711$	5.44015			0.204022
$712$		0			0
$713$			3.30855i			0.123906i
$714$		0			0
$715$	−0.880309	+	1.58786i	−0.0329217	+	0.0593825i
$716$		0			0
$717$		−	21.3260i		−	0.796434i
$718$		0			0
$719$	32.7819			1.22256			0.611279	−	0.791415i	$-0.290655\pi$
$719$	32.7819			1.22256			0.611279	+	0.791415i	$0.290655\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	1.76540i		−	0.0656558i
$724$		0			0
$725$	19.4965			0.724080
$726$		0			0
$727$	34.7677			1.28946			0.644731	−	0.764409i	$-0.276969\pi$
$727$	34.7677			1.28946			0.644731	+	0.764409i	$0.276969\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	18.9929			0.702478
$732$		0			0
$733$			38.8956i			1.43664i	0.695712	+	0.718321i	$0.255089\pi$
$733$			38.8956i			1.43664i	−0.695712	+	0.718321i	$0.744911\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.38385			−0.198317
$738$		0			0
$739$			11.9747i			0.440497i	0.975444	+	0.220248i	$0.0706868\pi$
$739$			11.9747i			0.440497i	−0.975444	+	0.220248i	$0.929313\pi$
$740$		0			0
$741$	5.94370	+	3.29519i	0.218347	+	0.121052i
$742$		0			0
$743$		−	18.1503i		−	0.665869i	−0.942950	−	0.332935i	$-0.891961\pi$
$743$		−	18.1503i		−	0.665869i	0.942950	−	0.332935i	$-0.108039\pi$
$744$		0			0
$745$	−3.27125			−0.119849
$746$		0			0
$747$		−	6.54401i		−	0.239433i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−28.9366			−1.05591			−0.527956	−	0.849272i	$-0.677041\pi$
$751$	−28.9366			−1.05591			−0.527956	+	0.849272i	$0.677041\pi$
$752$		0			0
$753$	−16.3768			−0.596803
$754$		0			0
$755$	−2.58780			−0.0941796
$756$		0			0
$757$	−7.06339			−0.256723			−0.128362	−	0.991727i	$-0.540972\pi$
$757$	−7.06339			−0.256723			−0.128362	+	0.991727i	$0.540972\pi$
$758$		0			0
$759$			1.58786i			0.0576355i
$760$		0			0
$761$			7.42171i			0.269037i	0.990911	+	0.134518i	$0.0429487\pi$
$761$			7.42171i			0.269037i	−0.990911	+	0.134518i	$0.957051\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		−	0.758219i		−	0.0274135i
$766$		0			0
$767$	14.6919	+	8.14522i	0.530495	+	0.294107i
$768$		0			0
$769$			2.16854i			0.0781994i	0.999235	+	0.0390997i	$0.0124490\pi$
$769$			2.16854i			0.0781994i	−0.999235	+	0.0390997i	$0.987551\pi$
$770$		0			0
$771$	−3.79744			−0.136762
$772$		0			0
$773$			19.3350i			0.695431i	0.937600	+	0.347716i	$0.113043\pi$
$773$			19.3350i			0.695431i	−0.937600	+	0.347716i	$0.886957\pi$
$774$		0			0
$775$			21.8603i			0.785245i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−11.1055			−0.397896
$780$		0			0
$781$	24.2783			0.868747
$782$		0			0
$783$	3.94370			0.140936
$784$		0			0
$785$		−	0.949118i		−	0.0338755i
$786$		0			0
$787$			25.9985i			0.926746i	0.886163	+	0.463373i	$0.153361\pi$
$787$			25.9985i			0.926746i	−0.886163	+	0.463373i	$0.846639\pi$
$788$		0			0
$789$	6.13208			0.218308
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−17.2854	+	31.1786i	−0.613823	+	1.10718i
$794$		0			0
$795$		−	0.580680i		−	0.0205946i
$796$		0			0
$797$	−11.4207			−0.404541			−0.202271	−	0.979330i	$-0.564832\pi$
$797$	−11.4207			−0.404541			−0.202271	+	0.979330i	$0.564832\pi$
$798$		0			0
$799$			8.86504i			0.313623i
$800$		0			0
$801$		−	12.3762i		−	0.437291i
$802$		0			0
$803$	−2.99291			−0.105618
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−7.19547			−0.253293
$808$		0			0
$809$	−46.7114			−1.64229			−0.821143	−	0.570723i	$-0.806663\pi$
$809$	−46.7114			−1.64229			−0.821143	+	0.570723i	$0.806663\pi$
$810$		0			0
$811$			24.7076i			0.867603i	0.901009	+	0.433801i	$0.142828\pi$
$811$			24.7076i			0.867603i	−0.901009	+	0.433801i	$0.857172\pi$
$812$		0			0
$813$			24.5882i			0.862345i
$814$		0			0
$815$	−4.22521			−0.148003
$816$		0			0
$817$		−	11.2031i		−	0.391948i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	27.4685i		−	0.958659i	−0.877635	−	0.479329i	$-0.840880\pi$
$821$		−	27.4685i		−	0.958659i	0.877635	−	0.479329i	$-0.159120\pi$
$822$		0			0
$823$	27.5496			0.960318			0.480159	−	0.877181i	$-0.340579\pi$
$823$	27.5496			0.960318			0.480159	+	0.877181i	$0.340579\pi$
$824$		0			0
$825$			10.4913i			0.365260i
$826$		0			0
$827$		−	29.5757i		−	1.02845i	−0.857656	−	0.514223i	$-0.828080\pi$
$827$		−	29.5757i		−	1.02845i	0.857656	−	0.514223i	$-0.171920\pi$
$828$		0			0
$829$	−11.2713			−0.391467			−0.195733	−	0.980657i	$-0.562709\pi$
$829$	−11.2713			−0.391467			−0.195733	+	0.980657i	$0.562709\pi$
$830$		0			0
$831$	0.545670			0.0189291
$832$		0			0
$833$		0			0
$834$		0			0
$835$	4.34881			0.150497
$836$		0			0
$837$			4.42185i			0.152841i
$838$		0			0
$839$		−	38.2552i		−	1.32072i	−0.750951	−	0.660358i	$-0.770405\pi$
$839$		−	38.2552i		−	1.32072i	0.750951	−	0.660358i	$-0.229595\pi$
$840$		0			0
$841$	−13.4472			−0.463698
$842$		0			0
$843$		−	0.223920i		−	0.00771222i
$844$		0			0
$845$	2.61615	−	1.63424i	0.0899982	−	0.0562195i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−6.39094			−0.219337
$850$		0			0
$851$			7.53950i			0.258451i
$852$		0			0
$853$			8.47526i			0.290187i	0.989418	+	0.145094i	$0.0463484\pi$
$853$			8.47526i			0.290187i	−0.989418	+	0.145094i	$0.953652\pi$
$854$		0			0
$855$	−0.447242			−0.0152954
$856$		0			0
$857$	11.5722			0.395300			0.197650	−	0.980273i	$-0.436669\pi$
$857$	11.5722			0.395300			0.197650	+	0.980273i	$0.436669\pi$
$858$		0			0
$859$	−51.1728			−1.74599			−0.872997	−	0.487725i	$-0.837827\pi$
$859$	−51.1728			−1.74599			−0.872997	+	0.487725i	$0.837827\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	12.1506i		−	0.413610i	−0.978382	−	0.206805i	$-0.933693\pi$
$863$		−	12.1506i		−	0.413610i	0.978382	−	0.206805i	$-0.0663066\pi$
$864$		0			0
$865$			0.997180i			0.0339052i
$866$		0			0
$867$	−6.78897			−0.230565
$868$		0			0
$869$			11.5449i			0.391632i
$870$		0			0
$871$	8.00000	+	4.43521i	0.271070	+	0.150281i
$872$		0			0
$873$			16.2057i			0.548479i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	24.8985i		−	0.840764i	−0.907347	−	0.420382i	$-0.861896\pi$
$877$		−	24.8985i		−	0.840764i	0.907347	−	0.420382i	$-0.138104\pi$
$878$		0			0
$879$		−	12.8507i		−	0.433444i
$880$		0			0
$881$	−54.7451			−1.84441			−0.922204	−	0.386704i	$-0.873614\pi$
$881$	−54.7451			−1.84441			−0.922204	+	0.386704i	$0.873614\pi$
$882$		0			0
$883$	−8.27834			−0.278588			−0.139294	−	0.990251i	$-0.544483\pi$
$883$	−8.27834			−0.278588			−0.139294	+	0.990251i	$0.544483\pi$
$884$		0			0
$885$	−1.10552			−0.0371615
$886$		0			0
$887$	1.00709			0.0338147			0.0169073	−	0.999857i	$-0.494618\pi$
$887$	1.00709			0.0338147			0.0169073	+	0.999857i	$0.494618\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$			2.12216i			0.0710949i
$892$		0			0
$893$	5.22912			0.174986
$894$		0			0
$895$			5.81881i			0.194501i
$896$		0			0
$897$	1.30807	−	2.35943i	0.0436753	−	0.0787792i
$898$		0			0
$899$			17.4384i			0.581605i
$900$		0			0
$901$	7.82009			0.260525
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	0.567321i		−	0.0188584i
$906$		0			0
$907$	3.16182			0.104986			0.0524932	−	0.998621i	$-0.483283\pi$
$907$	3.16182			0.104986			0.0524932	+	0.998621i	$0.483283\pi$
$908$		0			0
$909$	16.5793			0.549901
$910$		0			0
$911$	47.3191			1.56775			0.783876	−	0.620918i	$-0.213240\pi$
$911$	47.3191			1.56775			0.783876	+	0.620918i	$0.213240\pi$
$912$		0			0
$913$	13.8874			0.459606
$914$		0			0
$915$		−	2.34608i		−	0.0775589i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−10.2642			−0.338583			−0.169292	−	0.985566i	$-0.554148\pi$
$919$	−10.2642			−0.338583			−0.169292	+	0.985566i	$0.554148\pi$
$920$		0			0
$921$		−	10.8481i		−	0.357456i
$922$		0			0
$923$	−36.0758	−	20.0005i	−1.18745	−	0.658323i
$924$		0			0
$925$			49.8151i			1.63791i
$926$		0			0
$927$	9.88740			0.324745
$928$		0			0
$929$		−	29.1145i		−	0.955214i	−0.878574	−	0.477607i	$-0.841504\pi$
$929$		−	29.1145i		−	0.955214i	0.878574	−	0.477607i	$-0.158496\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−10.3909			−0.340184
$934$		0			0
$935$	1.60906			0.0526219
$936$		0			0
$937$	40.6551			1.32814			0.664072	−	0.747668i	$-0.268827\pi$
$937$	40.6551			1.32814			0.664072	+	0.747668i	$0.268827\pi$
$938$		0			0
$939$	−2.11260			−0.0689422
$940$		0			0
$941$			40.8983i			1.33325i	0.745395	+	0.666623i	$0.232261\pi$
$941$			40.8983i			1.33325i	−0.745395	+	0.666623i	$0.767739\pi$
$942$		0			0
$943$			4.40849i			0.143560i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	51.1790i		−	1.66309i	−0.555454	−	0.831547i	$-0.687456\pi$
$947$		−	51.1790i		−	1.66309i	0.555454	−	0.831547i	$-0.312544\pi$
$948$		0			0
$949$	4.44724	+	2.46556i	0.144364	+	0.0800353i
$950$		0			0
$951$			28.2982i			0.917631i
$952$		0			0
$953$	39.4933			1.27931			0.639657	−	0.768661i	$-0.279077\pi$
$953$	39.4933			1.27931			0.639657	+	0.768661i	$0.279077\pi$
$954$		0			0
$955$			0.0714177i			0.00231102i
$956$		0			0
$957$			8.36914i			0.270536i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	11.4472			0.369266
$962$		0			0
$963$	10.6919			0.344543
$964$		0			0
$965$	−0.601972			−0.0193782
$966$		0			0
$967$			3.53415i			0.113651i	0.998384	+	0.0568254i	$0.0180978\pi$
$967$			3.53415i			0.113651i	−0.998384	+	0.0568254i	$0.981902\pi$
$968$		0			0
$969$		−	6.02307i		−	0.193489i
$970$		0			0
$971$	−16.6020			−0.532783			−0.266391	−	0.963865i	$-0.585831\pi$
$971$	−16.6020			−0.532783			−0.266391	+	0.963865i	$0.585831\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	8.64271	−	15.5893i	0.276788	−	0.499256i
$976$		0			0
$977$		−	38.1838i		−	1.22161i	−0.791782	−	0.610803i	$-0.790847\pi$
$977$		−	38.1838i		−	1.22161i	0.791782	−	0.610803i	$-0.209153\pi$
$978$		0			0
$979$	26.2642			0.839406
$980$		0			0
$981$			8.36914i			0.267206i
$982$		0			0
$983$			21.6497i			0.690519i	0.938507	+	0.345260i	$0.112209\pi$
$983$			21.6497i			0.690519i	−0.938507	+	0.345260i	$0.887791\pi$
$984$		0			0
$985$	−2.29251			−0.0730455
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−4.44724			−0.141414
$990$		0			0
$991$	−25.3839			−0.806345			−0.403172	−	0.915124i	$-0.632092\pi$
$991$	−25.3839			−0.806345			−0.403172	+	0.915124i	$0.632092\pi$
$992$		0			0
$993$			3.17571i			0.100778i
$994$		0			0
$995$		−	0.567321i		−	0.0179853i
$996$		0			0
$997$	−41.7606			−1.32257			−0.661286	−	0.750134i	$-0.729989\pi$
$997$	−41.7606			−1.32257			−0.661286	+	0.750134i	$0.729989\pi$
$998$		0			0
$999$			10.0765i			0.318806i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7644.2.e.m.4705.4	yes	6
7.6	odd	2		7644.2.e.j.4705.3	✓	6
13.12	even	2	inner	7644.2.e.m.4705.3	yes	6
91.90	odd	2		7644.2.e.j.4705.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7644.2.e.j.4705.3	✓	6	7.6	odd	2
7644.2.e.j.4705.4	yes	6	91.90	odd	2
7644.2.e.m.4705.3	yes	6	13.12	even	2	inner
7644.2.e.m.4705.4	yes	6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.237279i q^{5} +1.00000 q^{9} +2.12216i q^{11} +(1.74823 - 3.15336i) q^{13} +0.237279i q^{15} -3.19547 q^{17} +1.88488i q^{19} +0.748228 q^{23} +4.94370 q^{25} +1.00000 q^{27} +3.94370 q^{29} +4.42185i q^{31} +2.12216i q^{33} +10.0765i q^{37} +(1.74823 - 3.15336i) q^{39} +5.89191i q^{41} -5.94370 q^{43} +0.237279i q^{45} -2.77425i q^{47} -3.19547 q^{51} -2.44724 q^{53} -0.503544 q^{55} +1.88488i q^{57} +4.65913i q^{59} -9.88740 q^{61} +(0.748228 + 0.414819i) q^{65} +2.53697i q^{67} +0.748228 q^{69} -11.4404i q^{71} +1.41032i q^{73} +4.94370 q^{75} +5.44015 q^{79} +1.00000 q^{81} -6.54401i q^{83} -0.758219i q^{85} +3.94370 q^{87} -12.3762i q^{89} +4.42185i q^{93} -0.447242 q^{95} +16.2057i q^{97} +2.12216i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 6 q^{9} + 4 q^{13} + 8 q^{17} - 2 q^{23} - 4 q^{25} + 6 q^{27} - 10 q^{29} + 4 q^{39} - 2 q^{43} + 8 q^{51} + 6 q^{53} - 16 q^{55} + 8 q^{61} - 2 q^{65} - 2 q^{69} - 4 q^{75} - 14 q^{79}+ \cdots + 18 q^{95}+O(q^{100}) \) 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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