LMFDB - Embedded newform 2496.2.m.a.2209.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2496,2,Mod(2209,2496)] 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("2496.2209"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2496, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2496 = 2^{6} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2496.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,0,0,-8,0,0,0,0,0,0,0,-16] 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:	$19.9306603445$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2209.6
Root			$-0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	2496.2209
Dual form			2496.2.m.a.2209.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.41421 q^{5} -1.00000 q^{9} +3.16228 q^{11} +(2.82843 - 2.23607i) q^{13} -1.41421i q^{15} -2.00000 q^{17} -6.32456 q^{19} +8.94427 q^{23} -3.00000 q^{25} -1.00000i q^{27} +4.47214i q^{29} -8.48528i q^{31} +3.16228i q^{33} +2.82843 q^{37} +(2.23607 + 2.82843i) q^{39} -9.48683i q^{41} +6.00000i q^{43} +1.41421 q^{45} -9.89949i q^{47} +7.00000 q^{49} -2.00000i q^{51} -4.47214i q^{53} -4.47214 q^{55} -6.32456i q^{57} -9.48683 q^{59} +13.4164i q^{61} +(-4.00000 + 3.16228i) q^{65} +12.6491 q^{67} +8.94427i q^{69} -15.5563i q^{71} +6.32456i q^{73} -3.00000i q^{75} +13.4164 q^{79} +1.00000 q^{81} +9.48683 q^{83} +2.82843 q^{85} -4.47214 q^{87} +9.48683i q^{89} +8.48528 q^{93} +8.94427 q^{95} +6.32456i q^{97} -3.16228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{9} - 16 q^{17} - 24 q^{25} + 56 q^{49} - 32 q^{65} + 8 q^{81}+O(q^{100}) $ 8 * q - 8 * q^9 - 16 * q^17 - 24 * q^25 + 56 * q^49 - 32 * q^65 + 8 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2496\mathbb{Z}\right)^\times$.

$n$	$703$	$769$	$833$	$1093$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$			1.00000i			0.577350i
$3$			1.00000i			0.577350i
$4$		0			0
$5$	−1.41421			−0.632456			−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421			−0.632456			−0.316228	+	0.948683i	$0.602416\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−1.00000			−0.333333
$10$		0			0
$11$	3.16228			0.953463			0.476731	−	0.879049i	$-0.341821\pi$
$11$	3.16228			0.953463			0.476731	+	0.879049i	$0.341821\pi$
$12$		0			0
$13$	2.82843	−	2.23607i	0.784465	−	0.620174i
$13$	2.82843	−	2.23607i	0.784465	−	0.620174i
$14$		0			0
$15$		−	1.41421i		−	0.365148i
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	−6.32456			−1.45095			−0.725476	−	0.688247i	$-0.758380\pi$
$19$	−6.32456			−1.45095			−0.725476	+	0.688247i	$0.758380\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	8.94427			1.86501			0.932505	−	0.361158i	$-0.117618\pi$
$23$	8.94427			1.86501			0.932505	+	0.361158i	$0.117618\pi$
$24$		0			0
$25$	−3.00000			−0.600000
$26$		0			0
$27$		−	1.00000i		−	0.192450i
$28$		0			0
$29$			4.47214i			0.830455i	0.909718	+	0.415227i	$0.136298\pi$
$29$			4.47214i			0.830455i	−0.909718	+	0.415227i	$0.863702\pi$
$30$		0			0
$31$		−	8.48528i		−	1.52400i	−0.647576	−	0.762001i	$-0.724217\pi$
$31$		−	8.48528i		−	1.52400i	0.647576	−	0.762001i	$-0.275783\pi$
$32$		0			0
$33$			3.16228i			0.550482i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.82843			0.464991			0.232495	−	0.972598i	$-0.425311\pi$
$37$	2.82843			0.464991			0.232495	+	0.972598i	$0.425311\pi$
$38$		0			0
$39$	2.23607	+	2.82843i	0.358057	+	0.452911i
$40$		0			0
$41$		−	9.48683i		−	1.48159i	−0.671729	−	0.740797i	$-0.734448\pi$
$41$		−	9.48683i		−	1.48159i	0.671729	−	0.740797i	$-0.265552\pi$
$42$		0			0
$43$			6.00000i			0.914991i	0.889212	+	0.457496i	$0.151253\pi$
$43$			6.00000i			0.914991i	−0.889212	+	0.457496i	$0.848747\pi$
$44$		0			0
$45$	1.41421			0.210819
$46$		0			0
$47$		−	9.89949i		−	1.44399i	−0.691898	−	0.721995i	$-0.743225\pi$
$47$		−	9.89949i		−	1.44399i	0.691898	−	0.721995i	$-0.256775\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$		−	2.00000i		−	0.280056i
$52$		0			0
$53$		−	4.47214i		−	0.614295i	−0.951662	−	0.307148i	$-0.900625\pi$
$53$		−	4.47214i		−	0.614295i	0.951662	−	0.307148i	$-0.0993745\pi$
$54$		0			0
$55$	−4.47214			−0.603023
$56$		0			0
$57$		−	6.32456i		−	0.837708i
$58$		0			0
$59$	−9.48683			−1.23508			−0.617540	−	0.786539i	$-0.711871\pi$
$59$	−9.48683			−1.23508			−0.617540	+	0.786539i	$0.711871\pi$
$60$		0			0
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	+	3.16228i	−0.496139	+	0.392232i
$66$		0			0
$67$	12.6491			1.54533			0.772667	−	0.634811i	$-0.218922\pi$
$67$	12.6491			1.54533			0.772667	+	0.634811i	$0.218922\pi$
$68$		0			0
$69$			8.94427i			1.07676i
$70$		0			0
$71$		−	15.5563i		−	1.84620i	−0.384561	−	0.923099i	$-0.625647\pi$
$71$		−	15.5563i		−	1.84620i	0.384561	−	0.923099i	$-0.374353\pi$
$72$		0			0
$73$			6.32456i			0.740233i	0.928985	+	0.370117i	$0.120682\pi$
$73$			6.32456i			0.740233i	−0.928985	+	0.370117i	$0.879318\pi$
$74$		0			0
$75$		−	3.00000i		−	0.346410i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	13.4164			1.50946			0.754732	−	0.656033i	$-0.227767\pi$
$79$	13.4164			1.50946			0.754732	+	0.656033i	$0.227767\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	9.48683			1.04132			0.520658	−	0.853766i	$-0.325687\pi$
$83$	9.48683			1.04132			0.520658	+	0.853766i	$0.325687\pi$
$84$		0			0
$85$	2.82843			0.306786
$86$		0			0
$87$	−4.47214			−0.479463
$88$		0			0
$89$			9.48683i			1.00560i	0.864402	+	0.502801i	$0.167697\pi$
$89$			9.48683i			1.00560i	−0.864402	+	0.502801i	$0.832303\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	8.48528			0.879883
$94$		0			0
$95$	8.94427			0.917663
$96$		0			0
$97$			6.32456i			0.642161i	0.947052	+	0.321081i	$0.104046\pi$
$97$			6.32456i			0.642161i	−0.947052	+	0.321081i	$0.895954\pi$
$98$		0			0
$99$	−3.16228			−0.317821
$100$		0			0
$101$		−	4.47214i		−	0.444994i	−0.974933	−	0.222497i	$-0.928579\pi$
$101$		−	4.47214i		−	0.444994i	0.974933	−	0.222497i	$-0.0714208\pi$
$102$		0			0
$103$	8.94427			0.881305			0.440653	−	0.897678i	$-0.354747\pi$
$103$	8.94427			0.881305			0.440653	+	0.897678i	$0.354747\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	12.0000i		−	1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$		−	12.0000i		−	1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$		0			0
$109$	16.9706			1.62549			0.812743	−	0.582623i	$-0.197974\pi$
$109$	16.9706			1.62549			0.812743	+	0.582623i	$0.197974\pi$
$110$		0			0
$111$			2.82843i			0.268462i
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$	−12.6491			−1.17954
$116$		0			0
$117$	−2.82843	+	2.23607i	−0.261488	+	0.206725i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.00000			−0.0909091
$122$		0			0
$123$	9.48683			0.855399
$124$		0			0
$125$	11.3137			1.01193
$126$		0			0
$127$	−4.47214			−0.396838			−0.198419	−	0.980117i	$-0.563581\pi$
$127$	−4.47214			−0.396838			−0.198419	+	0.980117i	$0.563581\pi$
$128$		0			0
$129$	−6.00000			−0.528271
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$			1.41421i			0.121716i
$136$		0			0
$137$			3.16228i			0.270172i	0.990834	+	0.135086i	$0.0431310\pi$
$137$			3.16228i			0.270172i	−0.990834	+	0.135086i	$0.956869\pi$
$138$		0			0
$139$		−	6.00000i		−	0.508913i	−0.967084	−	0.254457i	$-0.918103\pi$
$139$		−	6.00000i		−	0.508913i	0.967084	−	0.254457i	$-0.0818966\pi$
$140$		0			0
$141$	9.89949			0.833688
$142$		0			0
$143$	8.94427	−	7.07107i	0.747958	−	0.591312i
$144$		0			0
$145$		−	6.32456i		−	0.525226i
$146$		0			0
$147$			7.00000i			0.577350i
$148$		0			0
$149$	4.24264			0.347571			0.173785	−	0.984784i	$-0.444400\pi$
$149$	4.24264			0.347571			0.173785	+	0.984784i	$0.444400\pi$
$150$		0			0
$151$			8.48528i			0.690522i	0.938507	+	0.345261i	$0.112210\pi$
$151$			8.48528i			0.690522i	−0.938507	+	0.345261i	$0.887790\pi$
$152$		0			0
$153$	2.00000			0.161690
$154$		0			0
$155$			12.0000i			0.963863i
$156$		0			0
$157$			17.8885i			1.42766i	0.700318	+	0.713831i	$0.253041\pi$
$157$			17.8885i			1.42766i	−0.700318	+	0.713831i	$0.746959\pi$
$158$		0			0
$159$	4.47214			0.354663
$160$		0			0
$161$		0			0
$162$		0			0
$163$	6.32456			0.495377			0.247689	−	0.968840i	$-0.420329\pi$
$163$	6.32456			0.495377			0.247689	+	0.968840i	$0.420329\pi$
$164$		0			0
$165$		−	4.47214i		−	0.348155i
$166$		0			0
$167$		−	9.89949i		−	0.766046i	−0.923739	−	0.383023i	$-0.874883\pi$
$167$		−	9.89949i		−	0.766046i	0.923739	−	0.383023i	$-0.125117\pi$
$168$		0			0
$169$	3.00000	−	12.6491i	0.230769	−	0.973009i
$170$		0			0
$171$	6.32456			0.483651
$172$		0			0
$173$			4.47214i			0.340010i	0.985443	+	0.170005i	$0.0543784\pi$
$173$			4.47214i			0.340010i	−0.985443	+	0.170005i	$0.945622\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	9.48683i		−	0.713074i
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			8.94427i			0.664822i	0.943135	+	0.332411i	$0.107862\pi$
$181$			8.94427i			0.664822i	−0.943135	+	0.332411i	$0.892138\pi$
$182$		0			0
$183$	−13.4164			−0.991769
$184$		0			0
$185$	−4.00000			−0.294086
$186$		0			0
$187$	−6.32456			−0.462497
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.94427			0.647185			0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427			0.647185			0.323592	+	0.946197i	$0.395109\pi$
$192$		0			0
$193$		−	12.6491i		−	0.910503i	−0.890363	−	0.455251i	$-0.849549\pi$
$193$		−	12.6491i		−	0.910503i	0.890363	−	0.455251i	$-0.150451\pi$
$194$		0			0
$195$	−3.16228	−	4.00000i	−0.226455	−	0.286446i
$196$		0			0
$197$	−21.2132			−1.51138			−0.755689	−	0.654931i	$-0.772698\pi$
$197$	−21.2132			−1.51138			−0.755689	+	0.654931i	$0.772698\pi$
$198$		0			0
$199$	−13.4164			−0.951064			−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164			−0.951064			−0.475532	+	0.879698i	$0.657744\pi$
$200$		0			0
$201$			12.6491i			0.892199i
$202$		0			0
$203$		0			0
$204$		0			0
$205$			13.4164i			0.937043i
$206$		0			0
$207$	−8.94427			−0.621670
$208$		0			0
$209$	−20.0000			−1.38343
$210$		0			0
$211$		−	12.0000i		−	0.826114i	−0.910705	−	0.413057i	$-0.864461\pi$
$211$		−	12.0000i		−	0.826114i	0.910705	−	0.413057i	$-0.135539\pi$
$212$		0			0
$213$	15.5563			1.06590
$214$		0			0
$215$		−	8.48528i		−	0.578691i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−6.32456			−0.427374
$220$		0			0
$221$	−5.65685	+	4.47214i	−0.380521	+	0.300828i
$222$		0			0
$223$		−	8.48528i		−	0.568216i	−0.958792	−	0.284108i	$-0.908302\pi$
$223$		−	8.48528i		−	0.568216i	0.958792	−	0.284108i	$-0.0916975\pi$
$224$		0			0
$225$	3.00000			0.200000
$226$		0			0
$227$	−15.8114			−1.04944			−0.524719	−	0.851275i	$-0.675830\pi$
$227$	−15.8114			−1.04944			−0.524719	+	0.851275i	$0.675830\pi$
$228$		0			0
$229$	11.3137			0.747631			0.373815	−	0.927503i	$-0.378049\pi$
$229$	11.3137			0.747631			0.373815	+	0.927503i	$0.378049\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$			14.0000i			0.913259i
$236$		0			0
$237$			13.4164i			0.871489i
$238$		0			0
$239$		−	9.89949i		−	0.640345i	−0.947359	−	0.320173i	$-0.896259\pi$
$239$		−	9.89949i		−	0.640345i	0.947359	−	0.320173i	$-0.103741\pi$
$240$		0			0
$241$			18.9737i			1.22220i	0.791553	+	0.611101i	$0.209273\pi$
$241$			18.9737i			1.22220i	−0.791553	+	0.611101i	$0.790727\pi$
$242$		0			0
$243$			1.00000i			0.0641500i
$244$		0			0
$245$	−9.89949			−0.632456
$246$		0			0
$247$	−17.8885	+	14.1421i	−1.13822	+	0.899843i
$248$		0			0
$249$			9.48683i			0.601204i
$250$		0			0
$251$			20.0000i			1.26239i	0.775625	+	0.631194i	$0.217435\pi$
$251$			20.0000i			1.26239i	−0.775625	+	0.631194i	$0.782565\pi$
$252$		0			0
$253$	28.2843			1.77822
$254$		0			0
$255$			2.82843i			0.177123i
$256$		0			0
$257$	22.0000			1.37232			0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000			1.37232			0.686161	+	0.727450i	$0.259294\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		−	4.47214i		−	0.276818i
$262$		0			0
$263$	26.8328			1.65458			0.827291	−	0.561773i	$-0.189881\pi$
$263$	26.8328			1.65458			0.827291	+	0.561773i	$0.189881\pi$
$264$		0			0
$265$			6.32456i			0.388514i
$266$		0			0
$267$	−9.48683			−0.580585
$268$		0			0
$269$		−	22.3607i		−	1.36335i	−0.731653	−	0.681677i	$-0.761251\pi$
$269$		−	22.3607i		−	1.36335i	0.731653	−	0.681677i	$-0.238749\pi$
$270$		0			0
$271$			5.65685i			0.343629i	0.985129	+	0.171815i	$0.0549630\pi$
$271$			5.65685i			0.343629i	−0.985129	+	0.171815i	$0.945037\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−9.48683			−0.572078
$276$		0			0
$277$		−	26.8328i		−	1.61223i	−0.591761	−	0.806114i	$-0.701567\pi$
$277$		−	26.8328i		−	1.61223i	0.591761	−	0.806114i	$-0.298433\pi$
$278$		0			0
$279$			8.48528i			0.508001i
$280$		0			0
$281$		−	15.8114i		−	0.943228i	−0.881805	−	0.471614i	$-0.843672\pi$
$281$		−	15.8114i		−	0.943228i	0.881805	−	0.471614i	$-0.156328\pi$
$282$		0			0
$283$		−	4.00000i		−	0.237775i	−0.992908	−	0.118888i	$-0.962067\pi$
$283$		−	4.00000i		−	0.237775i	0.992908	−	0.118888i	$-0.0379328\pi$
$284$		0			0
$285$			8.94427i			0.529813i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$	−6.32456			−0.370752
$292$		0			0
$293$	−26.8701			−1.56977			−0.784883	−	0.619644i	$-0.787277\pi$
$293$	−26.8701			−1.56977			−0.784883	+	0.619644i	$0.787277\pi$
$294$		0			0
$295$	13.4164			0.781133
$296$		0			0
$297$		−	3.16228i		−	0.183494i
$298$		0			0
$299$	25.2982	−	20.0000i	1.46303	−	1.15663i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	4.47214			0.256917
$304$		0			0
$305$		−	18.9737i		−	1.08643i
$306$		0			0
$307$	25.2982			1.44385			0.721923	−	0.691974i	$-0.243259\pi$
$307$	25.2982			1.44385			0.721923	+	0.691974i	$0.243259\pi$
$308$		0			0
$309$			8.94427i			0.508822i
$310$		0			0
$311$	8.94427			0.507183			0.253592	−	0.967311i	$-0.418388\pi$
$311$	8.94427			0.507183			0.253592	+	0.967311i	$0.418388\pi$
$312$		0			0
$313$	−4.00000			−0.226093			−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000			−0.226093			−0.113047	+	0.993590i	$0.536061\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.24264			0.238290			0.119145	−	0.992877i	$-0.461985\pi$
$317$	4.24264			0.238290			0.119145	+	0.992877i	$0.461985\pi$
$318$		0			0
$319$			14.1421i			0.791808i
$320$		0			0
$321$	12.0000			0.669775
$322$		0			0
$323$	12.6491			0.703815
$324$		0			0
$325$	−8.48528	+	6.70820i	−0.470679	+	0.372104i
$326$		0			0
$327$			16.9706i			0.938474i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	12.6491			0.695258			0.347629	−	0.937632i	$-0.386987\pi$
$331$	12.6491			0.695258			0.347629	+	0.937632i	$0.386987\pi$
$332$		0			0
$333$	−2.82843			−0.154997
$334$		0			0
$335$	−17.8885			−0.977356
$336$		0			0
$337$	−32.0000			−1.74315			−0.871576	−	0.490261i	$-0.836901\pi$
$337$	−32.0000			−1.74315			−0.871576	+	0.490261i	$0.836901\pi$
$338$		0			0
$339$			14.0000i			0.760376i
$340$		0			0
$341$		−	26.8328i		−	1.45308i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		−	12.6491i		−	0.681005i
$346$		0			0
$347$		−	8.00000i		−	0.429463i	−0.976673	−	0.214731i	$-0.931112\pi$
$347$		−	8.00000i		−	0.429463i	0.976673	−	0.214731i	$-0.0688876\pi$
$348$		0			0
$349$	−2.82843			−0.151402			−0.0757011	−	0.997131i	$-0.524119\pi$
$349$	−2.82843			−0.151402			−0.0757011	+	0.997131i	$0.524119\pi$
$350$		0			0
$351$	−2.23607	−	2.82843i	−0.119352	−	0.150970i
$352$		0			0
$353$			9.48683i			0.504933i	0.967606	+	0.252467i	$0.0812418\pi$
$353$			9.48683i			0.504933i	−0.967606	+	0.252467i	$0.918758\pi$
$354$		0			0
$355$			22.0000i			1.16764i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	24.0416i		−	1.26887i	−0.772977	−	0.634434i	$-0.781233\pi$
$359$		−	24.0416i		−	1.26887i	0.772977	−	0.634434i	$-0.218767\pi$
$360$		0			0
$361$	21.0000			1.10526
$362$		0			0
$363$		−	1.00000i		−	0.0524864i
$364$		0			0
$365$		−	8.94427i		−	0.468165i
$366$		0			0
$367$	−8.94427			−0.466887			−0.233444	−	0.972370i	$-0.574999\pi$
$367$	−8.94427			−0.466887			−0.233444	+	0.972370i	$0.574999\pi$
$368$		0			0
$369$			9.48683i			0.493865i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	4.47214i		−	0.231558i	−0.993275	−	0.115779i	$-0.963063\pi$
$373$		−	4.47214i		−	0.231558i	0.993275	−	0.115779i	$-0.0369365\pi$
$374$		0			0
$375$			11.3137i			0.584237i
$376$		0			0
$377$	10.0000	+	12.6491i	0.515026	+	0.651462i
$378$		0			0
$379$	−31.6228			−1.62435			−0.812176	−	0.583412i	$-0.801717\pi$
$379$	−31.6228			−1.62435			−0.812176	+	0.583412i	$0.801717\pi$
$380$		0			0
$381$		−	4.47214i		−	0.229114i
$382$		0			0
$383$		−	7.07107i		−	0.361315i	−0.983546	−	0.180657i	$-0.942177\pi$
$383$		−	7.07107i		−	0.361315i	0.983546	−	0.180657i	$-0.0578225\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	6.00000i		−	0.304997i
$388$		0			0
$389$		−	22.3607i		−	1.13373i	−0.823810	−	0.566866i	$-0.808156\pi$
$389$		−	22.3607i		−	1.13373i	0.823810	−	0.566866i	$-0.191844\pi$
$390$		0			0
$391$	−17.8885			−0.904663
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−18.9737			−0.954669
$396$		0			0
$397$	−14.1421			−0.709773			−0.354887	−	0.934909i	$-0.615481\pi$
$397$	−14.1421			−0.709773			−0.354887	+	0.934909i	$0.615481\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			28.4605i			1.42125i	0.703571	+	0.710625i	$0.251588\pi$
$401$			28.4605i			1.42125i	−0.703571	+	0.710625i	$0.748412\pi$
$402$		0			0
$403$	−18.9737	−	24.0000i	−0.945146	−	1.19553i
$404$		0			0
$405$	−1.41421			−0.0702728
$406$		0			0
$407$	8.94427			0.443351
$408$		0			0
$409$		−	31.6228i		−	1.56365i	−0.623501	−	0.781823i	$-0.714290\pi$
$409$		−	31.6228i		−	1.56365i	0.623501	−	0.781823i	$-0.285710\pi$
$410$		0			0
$411$	−3.16228			−0.155984
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−13.4164			−0.658586
$416$		0			0
$417$	6.00000			0.293821
$418$		0			0
$419$			16.0000i			0.781651i	0.920465	+	0.390826i	$0.127810\pi$
$419$			16.0000i			0.781651i	−0.920465	+	0.390826i	$0.872190\pi$
$420$		0			0
$421$	−33.9411			−1.65419			−0.827095	−	0.562063i	$-0.810008\pi$
$421$	−33.9411			−1.65419			−0.827095	+	0.562063i	$0.810008\pi$
$422$		0			0
$423$			9.89949i			0.481330i
$424$		0			0
$425$	6.00000			0.291043
$426$		0			0
$427$		0			0
$428$		0			0
$429$	7.07107	+	8.94427i	0.341394	+	0.431834i
$430$		0			0
$431$			15.5563i			0.749323i	0.927162	+	0.374661i	$0.122241\pi$
$431$			15.5563i			0.749323i	−0.927162	+	0.374661i	$0.877759\pi$
$432$		0			0
$433$	−4.00000			−0.192228			−0.0961139	−	0.995370i	$-0.530641\pi$
$433$	−4.00000			−0.192228			−0.0961139	+	0.995370i	$0.530641\pi$
$434$		0			0
$435$	6.32456			0.303239
$436$		0			0
$437$	−56.5685			−2.70604
$438$		0			0
$439$	−8.94427			−0.426887			−0.213443	−	0.976955i	$-0.568468\pi$
$439$	−8.94427			−0.426887			−0.213443	+	0.976955i	$0.568468\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$		−	16.0000i		−	0.760183i	−0.924949	−	0.380091i	$-0.875893\pi$
$443$		−	16.0000i		−	0.760183i	0.924949	−	0.380091i	$-0.124107\pi$
$444$		0			0
$445$		−	13.4164i		−	0.635999i
$446$		0			0
$447$			4.24264i			0.200670i
$448$		0			0
$449$		−	34.7851i		−	1.64161i	−0.571210	−	0.820804i	$-0.693526\pi$
$449$		−	34.7851i		−	1.64161i	0.571210	−	0.820804i	$-0.306474\pi$
$450$		0			0
$451$		−	30.0000i		−	1.41264i
$452$		0			0
$453$	−8.48528			−0.398673
$454$		0			0
$455$		0			0
$456$		0			0
$457$			18.9737i			0.887551i	0.896138	+	0.443775i	$0.146361\pi$
$457$			18.9737i			0.887551i	−0.896138	+	0.443775i	$0.853639\pi$
$458$		0			0
$459$			2.00000i			0.0933520i
$460$		0			0
$461$	−29.6985			−1.38320			−0.691598	−	0.722282i	$-0.743093\pi$
$461$	−29.6985			−1.38320			−0.691598	+	0.722282i	$0.743093\pi$
$462$		0			0
$463$			5.65685i			0.262896i	0.991323	+	0.131448i	$0.0419627\pi$
$463$			5.65685i			0.262896i	−0.991323	+	0.131448i	$0.958037\pi$
$464$		0			0
$465$	−12.0000			−0.556487
$466$		0			0
$467$			12.0000i			0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$			12.0000i			0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−17.8885			−0.824261
$472$		0			0
$473$			18.9737i			0.872410i
$474$		0			0
$475$	18.9737			0.870572
$476$		0			0
$477$			4.47214i			0.204765i
$478$		0			0
$479$			9.89949i			0.452319i	0.974090	+	0.226160i	$0.0726171\pi$
$479$			9.89949i			0.452319i	−0.974090	+	0.226160i	$0.927383\pi$
$480$		0			0
$481$	8.00000	−	6.32456i	0.364769	−	0.288375i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	8.94427i		−	0.406138i
$486$		0			0
$487$			14.1421i			0.640841i	0.947275	+	0.320421i	$0.103824\pi$
$487$			14.1421i			0.640841i	−0.947275	+	0.320421i	$0.896176\pi$
$488$		0			0
$489$			6.32456i			0.286006i
$490$		0			0
$491$			8.00000i			0.361035i	0.983572	+	0.180517i	$0.0577772\pi$
$491$			8.00000i			0.361035i	−0.983572	+	0.180517i	$0.942223\pi$
$492$		0			0
$493$		−	8.94427i		−	0.402830i
$494$		0			0
$495$	4.47214			0.201008
$496$		0			0
$497$		0			0
$498$		0			0
$499$	25.2982			1.13250			0.566252	−	0.824232i	$-0.308393\pi$
$499$	25.2982			1.13250			0.566252	+	0.824232i	$0.308393\pi$
$500$		0			0
$501$	9.89949			0.442277
$502$		0			0
$503$	−8.94427			−0.398805			−0.199403	−	0.979918i	$-0.563900\pi$
$503$	−8.94427			−0.398805			−0.199403	+	0.979918i	$0.563900\pi$
$504$		0			0
$505$			6.32456i			0.281439i
$506$		0			0
$507$	12.6491	+	3.00000i	0.561767	+	0.133235i
$508$		0			0
$509$	18.3848			0.814891			0.407445	−	0.913230i	$-0.366420\pi$
$509$	18.3848			0.814891			0.407445	+	0.913230i	$0.366420\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			6.32456i			0.279236i
$514$		0			0
$515$	−12.6491			−0.557386
$516$		0			0
$517$		−	31.3050i		−	1.37679i
$518$		0			0
$519$	−4.47214			−0.196305
$520$		0			0
$521$	−18.0000			−0.788594			−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000			−0.788594			−0.394297	+	0.918983i	$0.629012\pi$
$522$		0			0
$523$		−	26.0000i		−	1.13690i	−0.822718	−	0.568450i	$-0.807543\pi$
$523$		−	26.0000i		−	1.13690i	0.822718	−	0.568450i	$-0.192457\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			16.9706i			0.739249i
$528$		0			0
$529$	57.0000			2.47826
$530$		0			0
$531$	9.48683			0.411693
$532$		0			0
$533$	−21.2132	−	26.8328i	−0.918846	−	1.16226i
$534$		0			0
$535$			16.9706i			0.733701i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	22.1359			0.953463
$540$		0			0
$541$	33.9411			1.45924			0.729621	−	0.683851i	$-0.239696\pi$
$541$	33.9411			1.45924			0.729621	+	0.683851i	$0.239696\pi$
$542$		0			0
$543$	−8.94427			−0.383835
$544$		0			0
$545$	−24.0000			−1.02805
$546$		0			0
$547$			22.0000i			0.940652i	0.882493	+	0.470326i	$0.155864\pi$
$547$			22.0000i			0.940652i	−0.882493	+	0.470326i	$0.844136\pi$
$548$		0			0
$549$		−	13.4164i		−	0.572598i
$550$		0			0
$551$		−	28.2843i		−	1.20495i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		−	4.00000i		−	0.169791i
$556$		0			0
$557$	−21.2132			−0.898832			−0.449416	−	0.893323i	$-0.648368\pi$
$557$	−21.2132			−0.898832			−0.449416	+	0.893323i	$0.648368\pi$
$558$		0			0
$559$	13.4164	+	16.9706i	0.567454	+	0.717778i
$560$		0			0
$561$		−	6.32456i		−	0.267023i
$562$		0			0
$563$			24.0000i			1.01148i	0.862686	+	0.505740i	$0.168780\pi$
$563$			24.0000i			1.01148i	−0.862686	+	0.505740i	$0.831220\pi$
$564$		0			0
$565$	−19.7990			−0.832950
$566$		0			0
$567$		0			0
$568$		0			0
$569$	46.0000			1.92842			0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000			1.92842			0.964210	+	0.265139i	$0.0854179\pi$
$570$		0			0
$571$			20.0000i			0.836974i	0.908223	+	0.418487i	$0.137439\pi$
$571$			20.0000i			0.836974i	−0.908223	+	0.418487i	$0.862561\pi$
$572$		0			0
$573$			8.94427i			0.373652i
$574$		0			0
$575$	−26.8328			−1.11901
$576$		0			0
$577$		−	12.6491i		−	0.526589i	−0.964715	−	0.263295i	$-0.915191\pi$
$577$		−	12.6491i		−	0.526589i	0.964715	−	0.263295i	$-0.0848092\pi$
$578$		0			0
$579$	12.6491			0.525679
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	14.1421i		−	0.585707i
$584$		0			0
$585$	4.00000	−	3.16228i	0.165380	−	0.130744i
$586$		0			0
$587$	−9.48683			−0.391564			−0.195782	−	0.980647i	$-0.562724\pi$
$587$	−9.48683			−0.391564			−0.195782	+	0.980647i	$0.562724\pi$
$588$		0			0
$589$			53.6656i			2.21125i
$590$		0			0
$591$		−	21.2132i		−	0.872595i
$592$		0			0
$593$			22.1359i			0.909014i	0.890743	+	0.454507i	$0.150185\pi$
$593$			22.1359i			0.909014i	−0.890743	+	0.454507i	$0.849815\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	13.4164i		−	0.549097i
$598$		0			0
$599$	17.8885			0.730906			0.365453	−	0.930830i	$-0.380914\pi$
$599$	17.8885			0.730906			0.365453	+	0.930830i	$0.380914\pi$
$600$		0			0
$601$	−20.0000			−0.815817			−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000			−0.815817			−0.407909	+	0.913023i	$0.633742\pi$
$602$		0			0
$603$	−12.6491			−0.515112
$604$		0			0
$605$	1.41421			0.0574960
$606$		0			0
$607$	−26.8328			−1.08911			−0.544555	−	0.838725i	$-0.683302\pi$
$607$	−26.8328			−1.08911			−0.544555	+	0.838725i	$0.683302\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−22.1359	−	28.0000i	−0.895524	−	1.13276i
$612$		0			0
$613$	−42.4264			−1.71359			−0.856793	−	0.515660i	$-0.827547\pi$
$613$	−42.4264			−1.71359			−0.856793	+	0.515660i	$0.827547\pi$
$614$		0			0
$615$	−13.4164			−0.541002
$616$		0			0
$617$			28.4605i			1.14578i	0.819633	+	0.572888i	$0.194177\pi$
$617$			28.4605i			1.14578i	−0.819633	+	0.572888i	$0.805823\pi$
$618$		0			0
$619$		0			0			−	1.00000i	$-0.5\pi$
$619$		0			0				1.00000i	$0.5\pi$
$620$		0			0
$621$		−	8.94427i		−	0.358921i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		−	20.0000i		−	0.798723i
$628$		0			0
$629$	−5.65685			−0.225554
$630$		0			0
$631$			48.0833i			1.91416i	0.289817	+	0.957082i	$0.406406\pi$
$631$			48.0833i			1.91416i	−0.289817	+	0.957082i	$0.593594\pi$
$632$		0			0
$633$	12.0000			0.476957
$634$		0			0
$635$	6.32456			0.250982
$636$		0			0
$637$	19.7990	−	15.6525i	0.784465	−	0.620174i
$638$		0			0
$639$			15.5563i			0.615400i
$640$		0			0
$641$	−30.0000			−1.18493			−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000			−1.18493			−0.592464	+	0.805597i	$0.701845\pi$
$642$		0			0
$643$	−25.2982			−0.997664			−0.498832	−	0.866699i	$-0.666238\pi$
$643$	−25.2982			−0.997664			−0.498832	+	0.866699i	$0.666238\pi$
$644$		0			0
$645$	8.48528			0.334108
$646$		0			0
$647$	35.7771			1.40654			0.703271	−	0.710922i	$-0.251722\pi$
$647$	35.7771			1.40654			0.703271	+	0.710922i	$0.251722\pi$
$648$		0			0
$649$	−30.0000			−1.17760
$650$		0			0
$651$		0			0
$652$		0			0
$653$			13.4164i			0.525025i	0.964929	+	0.262512i	$0.0845510\pi$
$653$			13.4164i			0.525025i	−0.964929	+	0.262512i	$0.915449\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	6.32456i		−	0.246744i
$658$		0			0
$659$		−	36.0000i		−	1.40236i	−0.712984	−	0.701180i	$-0.752657\pi$
$659$		−	36.0000i		−	1.40236i	0.712984	−	0.701180i	$-0.247343\pi$
$660$		0			0
$661$	−8.48528			−0.330039			−0.165020	−	0.986290i	$-0.552769\pi$
$661$	−8.48528			−0.330039			−0.165020	+	0.986290i	$0.552769\pi$
$662$		0			0
$663$	−4.47214	−	5.65685i	−0.173683	−	0.219694i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			40.0000i			1.54881i
$668$		0			0
$669$	8.48528			0.328060
$670$		0			0
$671$			42.4264i			1.63785i
$672$		0			0
$673$	−14.0000			−0.539660			−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000			−0.539660			−0.269830	+	0.962908i	$0.586968\pi$
$674$		0			0
$675$			3.00000i			0.115470i
$676$		0			0
$677$		−	22.3607i		−	0.859391i	−0.902974	−	0.429695i	$-0.858621\pi$
$677$		−	22.3607i		−	0.859391i	0.902974	−	0.429695i	$-0.141379\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		−	15.8114i		−	0.605894i
$682$		0			0
$683$	9.48683			0.363004			0.181502	−	0.983391i	$-0.441904\pi$
$683$	9.48683			0.363004			0.181502	+	0.983391i	$0.441904\pi$
$684$		0			0
$685$		−	4.47214i		−	0.170872i
$686$		0			0
$687$			11.3137i			0.431645i
$688$		0			0
$689$	−10.0000	−	12.6491i	−0.380970	−	0.481893i
$690$		0			0
$691$	12.6491			0.481195			0.240597	−	0.970625i	$-0.422657\pi$
$691$	12.6491			0.481195			0.240597	+	0.970625i	$0.422657\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			8.48528i			0.321865i
$696$		0			0
$697$			18.9737i			0.718679i
$698$		0			0
$699$		−	6.00000i		−	0.226941i
$700$		0			0
$701$			13.4164i			0.506731i	0.967371	+	0.253365i	$0.0815375\pi$
$701$			13.4164i			0.506731i	−0.967371	+	0.253365i	$0.918463\pi$
$702$		0			0
$703$	−17.8885			−0.674679
$704$		0			0
$705$	−14.0000			−0.527271
$706$		0			0
$707$		0			0
$708$		0			0
$709$	16.9706			0.637343			0.318671	−	0.947865i	$-0.396763\pi$
$709$	16.9706			0.637343			0.318671	+	0.947865i	$0.396763\pi$
$710$		0			0
$711$	−13.4164			−0.503155
$712$		0			0
$713$		−	75.8947i		−	2.84228i
$714$		0			0
$715$	−12.6491	+	10.0000i	−0.473050	+	0.373979i
$716$		0			0
$717$	9.89949			0.369703
$718$		0			0
$719$	−26.8328			−1.00070			−0.500348	−	0.865825i	$-0.666794\pi$
$719$	−26.8328			−1.00070			−0.500348	+	0.865825i	$0.666794\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−18.9737			−0.705638
$724$		0			0
$725$		−	13.4164i		−	0.498273i
$726$		0			0
$727$	−31.3050			−1.16104			−0.580518	−	0.814247i	$-0.697150\pi$
$727$	−31.3050			−1.16104			−0.580518	+	0.814247i	$0.697150\pi$
$728$		0			0
$729$	−1.00000			−0.0370370
$730$		0			0
$731$		−	12.0000i		−	0.443836i
$732$		0			0
$733$	−5.65685			−0.208941			−0.104470	−	0.994528i	$-0.533315\pi$
$733$	−5.65685			−0.208941			−0.104470	+	0.994528i	$0.533315\pi$
$734$		0			0
$735$		−	9.89949i		−	0.365148i
$736$		0			0
$737$	40.0000			1.47342
$738$		0			0
$739$		0			0			−	1.00000i	$-0.5\pi$
$739$		0			0				1.00000i	$0.5\pi$
$740$		0			0
$741$	−14.1421	−	17.8885i	−0.519524	−	0.657152i
$742$		0			0
$743$		−	35.3553i		−	1.29706i	−0.761188	−	0.648531i	$-0.775384\pi$
$743$		−	35.3553i		−	1.29706i	0.761188	−	0.648531i	$-0.224616\pi$
$744$		0			0
$745$	−6.00000			−0.219823
$746$		0			0
$747$	−9.48683			−0.347105
$748$		0			0
$749$		0			0
$750$		0			0
$751$	26.8328			0.979143			0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328			0.979143			0.489572	+	0.871963i	$0.337153\pi$
$752$		0			0
$753$	−20.0000			−0.728841
$754$		0			0
$755$		−	12.0000i		−	0.436725i
$756$		0			0
$757$			40.2492i			1.46288i	0.681904	+	0.731441i	$0.261152\pi$
$757$			40.2492i			1.46288i	−0.681904	+	0.731441i	$0.738848\pi$
$758$		0			0
$759$			28.2843i			1.02665i
$760$		0			0
$761$			15.8114i			0.573162i	0.958056	+	0.286581i	$0.0925188\pi$
$761$			15.8114i			0.573162i	−0.958056	+	0.286581i	$0.907481\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−2.82843			−0.102262
$766$		0			0
$767$	−26.8328	+	21.2132i	−0.968877	+	0.765964i
$768$		0			0
$769$			25.2982i			0.912277i	0.889909	+	0.456139i	$0.150768\pi$
$769$			25.2982i			0.912277i	−0.889909	+	0.456139i	$0.849232\pi$
$770$		0			0
$771$			22.0000i			0.792311i
$772$		0			0
$773$	7.07107			0.254329			0.127164	−	0.991882i	$-0.459412\pi$
$773$	7.07107			0.254329			0.127164	+	0.991882i	$0.459412\pi$
$774$		0			0
$775$			25.4558i			0.914401i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			60.0000i			2.14972i
$780$		0			0
$781$		−	49.1935i		−	1.76028i
$782$		0			0
$783$	4.47214			0.159821
$784$		0			0
$785$		−	25.2982i		−	0.902932i
$786$		0			0
$787$	−12.6491			−0.450892			−0.225446	−	0.974256i	$-0.572384\pi$
$787$	−12.6491			−0.450892			−0.225446	+	0.974256i	$0.572384\pi$
$788$		0			0
$789$			26.8328i			0.955274i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	30.0000	+	37.9473i	1.06533	+	1.34755i
$794$		0			0
$795$	−6.32456			−0.224309
$796$		0			0
$797$		−	49.1935i		−	1.74252i	−0.490819	−	0.871262i	$-0.663302\pi$
$797$		−	49.1935i		−	1.74252i	0.490819	−	0.871262i	$-0.336698\pi$
$798$		0			0
$799$			19.7990i			0.700438i
$800$		0			0
$801$		−	9.48683i		−	0.335201i
$802$		0			0
$803$			20.0000i			0.705785i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	22.3607			0.787133
$808$		0			0
$809$	−6.00000			−0.210949			−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000			−0.210949			−0.105474	+	0.994422i	$0.533636\pi$
$810$		0			0
$811$	−6.32456			−0.222085			−0.111043	−	0.993816i	$-0.535419\pi$
$811$	−6.32456			−0.222085			−0.111043	+	0.993816i	$0.535419\pi$
$812$		0			0
$813$	−5.65685			−0.198395
$814$		0			0
$815$	−8.94427			−0.313304
$816$		0			0
$817$		−	37.9473i		−	1.32761i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	15.5563			0.542920			0.271460	−	0.962450i	$-0.412493\pi$
$821$	15.5563			0.542920			0.271460	+	0.962450i	$0.412493\pi$
$822$		0			0
$823$	8.94427			0.311778			0.155889	−	0.987775i	$-0.450176\pi$
$823$	8.94427			0.311778			0.155889	+	0.987775i	$0.450176\pi$
$824$		0			0
$825$		−	9.48683i		−	0.330289i
$826$		0			0
$827$	−3.16228			−0.109963			−0.0549816	−	0.998487i	$-0.517510\pi$
$827$	−3.16228			−0.109963			−0.0549816	+	0.998487i	$0.517510\pi$
$828$		0			0
$829$			22.3607i			0.776619i	0.921529	+	0.388309i	$0.126941\pi$
$829$			22.3607i			0.776619i	−0.921529	+	0.388309i	$0.873059\pi$
$830$		0			0
$831$	26.8328			0.930820
$832$		0			0
$833$	−14.0000			−0.485071
$834$		0			0
$835$			14.0000i			0.484490i
$836$		0			0
$837$	−8.48528			−0.293294
$838$		0			0
$839$		−	24.0416i		−	0.830009i	−0.909819	−	0.415005i	$-0.863780\pi$
$839$		−	24.0416i		−	0.830009i	0.909819	−	0.415005i	$-0.136220\pi$
$840$		0			0
$841$	9.00000			0.310345
$842$		0			0
$843$	15.8114			0.544573
$844$		0			0
$845$	−4.24264	+	17.8885i	−0.145951	+	0.615385i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	4.00000			0.137280
$850$		0			0
$851$	25.2982			0.867212
$852$		0			0
$853$	−14.1421			−0.484218			−0.242109	−	0.970249i	$-0.577839\pi$
$853$	−14.1421			−0.484218			−0.242109	+	0.970249i	$0.577839\pi$
$854$		0			0
$855$	−8.94427			−0.305888
$856$		0			0
$857$	−38.0000			−1.29806			−0.649028	−	0.760765i	$-0.724824\pi$
$857$	−38.0000			−1.29806			−0.649028	+	0.760765i	$0.724824\pi$
$858$		0			0
$859$		−	20.0000i		−	0.682391i	−0.939992	−	0.341196i	$-0.889168\pi$
$859$		−	20.0000i		−	0.682391i	0.939992	−	0.341196i	$-0.110832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			41.0122i			1.39607i	0.716063	+	0.698036i	$0.245942\pi$
$863$			41.0122i			1.39607i	−0.716063	+	0.698036i	$0.754058\pi$
$864$		0			0
$865$		−	6.32456i		−	0.215041i
$866$		0			0
$867$		−	13.0000i		−	0.441503i
$868$		0			0
$869$	42.4264			1.43922
$870$		0			0
$871$	35.7771	−	28.2843i	1.21226	−	0.958376i
$872$		0			0
$873$		−	6.32456i		−	0.214054i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	14.1421			0.477546			0.238773	−	0.971075i	$-0.423255\pi$
$877$	14.1421			0.477546			0.238773	+	0.971075i	$0.423255\pi$
$878$		0			0
$879$		−	26.8701i		−	0.906305i
$880$		0			0
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$			36.0000i			1.21150i	0.795656	+	0.605748i	$0.207126\pi$
$883$			36.0000i			1.21150i	−0.795656	+	0.605748i	$0.792874\pi$
$884$		0			0
$885$			13.4164i			0.450988i
$886$		0			0
$887$	−35.7771			−1.20128			−0.600639	−	0.799521i	$-0.705087\pi$
$887$	−35.7771			−1.20128			−0.600639	+	0.799521i	$0.705087\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	3.16228			0.105940
$892$		0			0
$893$			62.6099i			2.09516i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	20.0000	+	25.2982i	0.667781	+	0.844683i
$898$		0			0
$899$	37.9473			1.26561
$900$		0			0
$901$			8.94427i			0.297977i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	12.6491i		−	0.420471i
$906$		0			0
$907$		−	52.0000i		−	1.72663i	−0.504664	−	0.863316i	$-0.668384\pi$
$907$		−	52.0000i		−	1.72663i	0.504664	−	0.863316i	$-0.331616\pi$
$908$		0			0
$909$			4.47214i			0.148331i
$910$		0			0
$911$	−17.8885			−0.592674			−0.296337	−	0.955083i	$-0.595765\pi$
$911$	−17.8885			−0.592674			−0.296337	+	0.955083i	$0.595765\pi$
$912$		0			0
$913$	30.0000			0.992855
$914$		0			0
$915$	18.9737			0.627250
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−31.3050			−1.03266			−0.516328	−	0.856391i	$-0.672701\pi$
$919$	−31.3050			−1.03266			−0.516328	+	0.856391i	$0.672701\pi$
$920$		0			0
$921$			25.2982i			0.833605i
$922$		0			0
$923$	−34.7851	−	44.0000i	−1.14496	−	1.44828i
$924$		0			0
$925$	−8.48528			−0.278994
$926$		0			0
$927$	−8.94427			−0.293768
$928$		0			0
$929$			34.7851i			1.14126i	0.821207	+	0.570630i	$0.193301\pi$
$929$			34.7851i			1.14126i	−0.821207	+	0.570630i	$0.806699\pi$
$930$		0			0
$931$	−44.2719			−1.45095
$932$		0			0
$933$			8.94427i			0.292822i
$934$		0			0
$935$	8.94427			0.292509
$936$		0			0
$937$	12.0000			0.392023			0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000			0.392023			0.196011	+	0.980602i	$0.437201\pi$
$938$		0			0
$939$		−	4.00000i		−	0.130535i
$940$		0			0
$941$	26.8701			0.875939			0.437969	−	0.898990i	$-0.355698\pi$
$941$	26.8701			0.875939			0.437969	+	0.898990i	$0.355698\pi$
$942$		0			0
$943$		−	84.8528i		−	2.76319i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−9.48683			−0.308281			−0.154140	−	0.988049i	$-0.549261\pi$
$947$	−9.48683			−0.308281			−0.154140	+	0.988049i	$0.549261\pi$
$948$		0			0
$949$	14.1421	+	17.8885i	0.459073	+	0.580687i
$950$		0			0
$951$			4.24264i			0.137577i
$952$		0			0
$953$	−46.0000			−1.49009			−0.745043	−	0.667016i	$-0.767571\pi$
$953$	−46.0000			−1.49009			−0.745043	+	0.667016i	$0.767571\pi$
$954$		0			0
$955$	−12.6491			−0.409316
$956$		0			0
$957$	−14.1421			−0.457150
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−41.0000			−1.32258
$962$		0			0
$963$			12.0000i			0.386695i
$964$		0			0
$965$			17.8885i			0.575853i
$966$		0			0
$967$		−	2.82843i		−	0.0909561i	−0.998965	−	0.0454780i	$-0.985519\pi$
$967$		−	2.82843i		−	0.0909561i	0.998965	−	0.0454780i	$-0.0144811\pi$
$968$		0			0
$969$			12.6491i			0.406348i
$970$		0			0
$971$		−	28.0000i		−	0.898563i	−0.893390	−	0.449281i	$-0.851680\pi$
$971$		−	28.0000i		−	0.898563i	0.893390	−	0.449281i	$-0.148320\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−6.70820	−	8.48528i	−0.214834	−	0.271746i
$976$		0			0
$977$		−	9.48683i		−	0.303511i	−0.988418	−	0.151755i	$-0.951507\pi$
$977$		−	9.48683i		−	0.303511i	0.988418	−	0.151755i	$-0.0484926\pi$
$978$		0			0
$979$			30.0000i			0.958804i
$980$		0			0
$981$	−16.9706			−0.541828
$982$		0			0
$983$		−	29.6985i		−	0.947235i	−0.880731	−	0.473617i	$-0.842948\pi$
$983$		−	29.6985i		−	0.947235i	0.880731	−	0.473617i	$-0.157052\pi$
$984$		0			0
$985$	30.0000			0.955879
$986$		0			0
$987$		0			0
$988$		0			0
$989$			53.6656i			1.70647i
$990$		0			0
$991$	26.8328			0.852372			0.426186	−	0.904635i	$-0.359857\pi$
$991$	26.8328			0.852372			0.426186	+	0.904635i	$0.359857\pi$
$992$		0			0
$993$			12.6491i			0.401407i
$994$		0			0
$995$	18.9737			0.601506
$996$		0			0
$997$		−	53.6656i		−	1.69961i	−0.527099	−	0.849804i	$-0.676720\pi$
$997$		−	53.6656i		−	1.69961i	0.527099	−	0.849804i	$-0.323280\pi$
$998$		0			0
$999$		−	2.82843i		−	0.0894875i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2496.2.m.a.2209.6	yes	8
4.3	odd	2	inner	2496.2.m.a.2209.1	✓	8
8.3	odd	2	inner	2496.2.m.a.2209.8	yes	8
8.5	even	2	inner	2496.2.m.a.2209.3	yes	8
13.12	even	2	inner	2496.2.m.a.2209.7	yes	8
52.51	odd	2	inner	2496.2.m.a.2209.4	yes	8
104.51	odd	2	inner	2496.2.m.a.2209.5	yes	8
104.77	even	2	inner	2496.2.m.a.2209.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2496.2.m.a.2209.1	✓	8	4.3	odd	2	inner
2496.2.m.a.2209.2	yes	8	104.77	even	2	inner
2496.2.m.a.2209.3	yes	8	8.5	even	2	inner
2496.2.m.a.2209.4	yes	8	52.51	odd	2	inner
2496.2.m.a.2209.5	yes	8	104.51	odd	2	inner
2496.2.m.a.2209.6	yes	8	1.1	even	1	trivial
2496.2.m.a.2209.7	yes	8	13.12	even	2	inner
2496.2.m.a.2209.8	yes	8	8.3	odd	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:	\( N \)	\(=\)	\( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2496.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.41421 q^{5} -1.00000 q^{9} +3.16228 q^{11} +(2.82843 - 2.23607i) q^{13} -1.41421i q^{15} -2.00000 q^{17} -6.32456 q^{19} +8.94427 q^{23} -3.00000 q^{25} -1.00000i q^{27} +4.47214i q^{29} -8.48528i q^{31} +3.16228i q^{33} +2.82843 q^{37} +(2.23607 + 2.82843i) q^{39} -9.48683i q^{41} +6.00000i q^{43} +1.41421 q^{45} -9.89949i q^{47} +7.00000 q^{49} -2.00000i q^{51} -4.47214i q^{53} -4.47214 q^{55} -6.32456i q^{57} -9.48683 q^{59} +13.4164i q^{61} +(-4.00000 + 3.16228i) q^{65} +12.6491 q^{67} +8.94427i q^{69} -15.5563i q^{71} +6.32456i q^{73} -3.00000i q^{75} +13.4164 q^{79} +1.00000 q^{81} +9.48683 q^{83} +2.82843 q^{85} -4.47214 q^{87} +9.48683i q^{89} +8.48528 q^{93} +8.94427 q^{95} +6.32456i q^{97} -3.16228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{9} - 16 q^{17} - 24 q^{25} + 56 q^{49} - 32 q^{65} + 8 q^{81}+O(q^{100}) \) 8 * q - 8 * q^9 - 16 * q^17 - 24 * q^25 + 56 * q^49 - 32 * q^65 + 8 * q^81

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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