LMFDB - Embedded newform 4608.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$36.7950652514$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 512)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.00000 q^{5} -2.82843 q^{7} +4.24264 q^{11} -6.00000 q^{13} +4.24264 q^{19} -8.48528 q^{23} -1.00000 q^{25} +2.00000 q^{29} +5.65685 q^{31} -5.65685 q^{35} -6.00000 q^{37} -6.00000 q^{41} +4.24264 q^{43} +1.00000 q^{49} -2.00000 q^{53} +8.48528 q^{55} +1.41421 q^{59} -6.00000 q^{61} -12.0000 q^{65} -12.7279 q^{67} +8.48528 q^{71} -12.0000 q^{73} -12.0000 q^{77} -5.65685 q^{79} +4.24264 q^{83} +12.0000 q^{89} +16.9706 q^{91} +8.48528 q^{95} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{5} - 12 q^{13} - 2 q^{25} + 4 q^{29} - 12 q^{37} - 12 q^{41} + 2 q^{49} - 4 q^{53} - 12 q^{61} - 24 q^{65} - 24 q^{73} - 24 q^{77} + 24 q^{89} - 16 q^{97}+O(q^{100}) $ 2 * q + 4 * q^5 - 12 * q^13 - 2 * q^25 + 4 * q^29 - 12 * q^37 - 12 * q^41 + 2 * q^49 - 4 * q^53 - 12 * q^61 - 24 * q^65 - 24 * q^73 - 24 * q^77 + 24 * q^89 - 16 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.24264	1.27920	0.639602	−	0.768706i	$-0.279099\pi$
$11$	4.24264	1.27920	0.639602	+	0.768706i	$0.279099\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.48528	−1.76930	−0.884652	−	0.466252i	$-0.845604\pi$
$23$	−8.48528	−1.76930	−0.884652	+	0.466252i	$0.845604\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.65685	−0.956183
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	4.24264	0.646997	0.323498	−	0.946229i	$-0.395141\pi$
$43$	4.24264	0.646997	0.323498	+	0.946229i	$0.395141\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	8.48528	1.14416
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.41421	0.184115	0.0920575	−	0.995754i	$-0.470656\pi$
$59$	1.41421	0.184115	0.0920575	+	0.995754i	$0.470656\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−12.0000	−1.48842
$66$	0	0
$67$	−12.7279	−1.55496	−0.777482	−	0.628906i	$-0.783503\pi$
$67$	−12.7279	−1.55496	−0.777482	+	0.628906i	$0.783503\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	−12.0000	−1.40449	−0.702247	−	0.711934i	$-0.747820\pi$
$73$	−12.0000	−1.40449	−0.702247	+	0.711934i	$0.747820\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−5.65685	−0.636446	−0.318223	−	0.948016i	$-0.603086\pi$
$79$	−5.65685	−0.636446	−0.318223	+	0.948016i	$0.603086\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.24264	0.465690	0.232845	−	0.972514i	$-0.425196\pi$
$83$	4.24264	0.465690	0.232845	+	0.972514i	$0.425196\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	12.0000	1.27200	0.635999	−	0.771690i	$-0.280588\pi$
$89$	12.0000	1.27200	0.635999	+	0.771690i	$0.280588\pi$
$90$	0	0
$91$	16.9706	1.77900
$92$	0	0
$93$	0	0
$94$	0	0
$95$	8.48528	0.870572
$96$	0	0
$97$	−8.00000	−0.812277	−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000	−0.812277	−0.406138	+	0.913812i	$0.633125\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.0000	−0.995037	−0.497519	−	0.867453i	$-0.665755\pi$
$101$	−10.0000	−0.995037	−0.497519	+	0.867453i	$0.665755\pi$
$102$	0	0
$103$	2.82843	0.278693	0.139347	−	0.990244i	$-0.455500\pi$
$103$	2.82843	0.278693	0.139347	+	0.990244i	$0.455500\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.07107	−0.683586	−0.341793	−	0.939775i	$-0.611034\pi$
$107$	−7.07107	−0.683586	−0.341793	+	0.939775i	$0.611034\pi$
$108$	0	0
$109$	6.00000	0.574696	0.287348	−	0.957826i	$-0.407226\pi$
$109$	6.00000	0.574696	0.287348	+	0.957826i	$0.407226\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−16.9706	−1.58251
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	7.00000	0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.41421	0.123560	0.0617802	−	0.998090i	$-0.480322\pi$
$131$	1.41421	0.123560	0.0617802	+	0.998090i	$0.480322\pi$
$132$	0	0
$133$	−12.0000	−1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	4.24264	0.359856	0.179928	−	0.983680i	$-0.442414\pi$
$139$	4.24264	0.359856	0.179928	+	0.983680i	$0.442414\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−25.4558	−2.12872
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−22.0000	−1.80231	−0.901155	−	0.433497i	$-0.857280\pi$
$149$	−22.0000	−1.80231	−0.901155	+	0.433497i	$0.857280\pi$
$150$	0	0
$151$	−14.1421	−1.15087	−0.575435	−	0.817847i	$-0.695167\pi$
$151$	−14.1421	−1.15087	−0.575435	+	0.817847i	$0.695167\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.3137	0.908739
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	24.0000	1.89146
$162$	0	0
$163$	−21.2132	−1.66155	−0.830773	−	0.556611i	$-0.812101\pi$
$163$	−21.2132	−1.66155	−0.830773	+	0.556611i	$0.812101\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.48528	−0.656611	−0.328305	−	0.944572i	$-0.606478\pi$
$167$	−8.48528	−0.656611	−0.328305	+	0.944572i	$0.606478\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	0	0
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−18.3848	−1.37414	−0.687071	−	0.726590i	$-0.741104\pi$
$179$	−18.3848	−1.37414	−0.687071	+	0.726590i	$0.741104\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−12.0000	−0.882258
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.9706	1.22795	0.613973	−	0.789327i	$-0.289570\pi$
$191$	16.9706	1.22795	0.613973	+	0.789327i	$0.289570\pi$
$192$	0	0
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	19.7990	1.40351	0.701757	−	0.712417i	$-0.252399\pi$
$199$	19.7990	1.40351	0.701757	+	0.712417i	$0.252399\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−5.65685	−0.397033
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	18.0000	1.24509
$210$	0	0
$211$	−4.24264	−0.292075	−0.146038	−	0.989279i	$-0.546652\pi$
$211$	−4.24264	−0.292075	−0.146038	+	0.989279i	$0.546652\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.48528	0.578691
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−5.65685	−0.378811	−0.189405	−	0.981899i	$-0.560656\pi$
$223$	−5.65685	−0.378811	−0.189405	+	0.981899i	$0.560656\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−12.7279	−0.844782	−0.422391	−	0.906414i	$-0.638809\pi$
$227$	−12.7279	−0.844782	−0.422391	+	0.906414i	$0.638809\pi$
$228$	0	0
$229$	−18.0000	−1.18947	−0.594737	−	0.803921i	$-0.702744\pi$
$229$	−18.0000	−1.18947	−0.594737	+	0.803921i	$0.702744\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	12.0000	0.786146	0.393073	−	0.919507i	$-0.371412\pi$
$233$	12.0000	0.786146	0.393073	+	0.919507i	$0.371412\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.9706	−1.09773	−0.548867	−	0.835910i	$-0.684941\pi$
$239$	−16.9706	−1.09773	−0.548867	+	0.835910i	$0.684941\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.00000	0.127775
$246$	0	0
$247$	−25.4558	−1.61972
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.7279	0.803379	0.401690	−	0.915776i	$-0.368423\pi$
$251$	12.7279	0.803379	0.401690	+	0.915776i	$0.368423\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	0	0
$259$	16.9706	1.05450
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.48528	0.523225	0.261612	−	0.965173i	$-0.415746\pi$
$263$	8.48528	0.523225	0.261612	+	0.965173i	$0.415746\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−22.6274	−1.37452	−0.687259	−	0.726413i	$-0.741186\pi$
$271$	−22.6274	−1.37452	−0.687259	+	0.726413i	$0.741186\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.24264	−0.255841
$276$	0	0
$277$	−6.00000	−0.360505	−0.180253	−	0.983620i	$-0.557691\pi$
$277$	−6.00000	−0.360505	−0.180253	+	0.983620i	$0.557691\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.0000	−0.715860	−0.357930	−	0.933748i	$-0.616517\pi$
$281$	−12.0000	−0.715860	−0.357930	+	0.933748i	$0.616517\pi$
$282$	0	0
$283$	12.7279	0.756596	0.378298	−	0.925684i	$-0.376509\pi$
$283$	12.7279	0.756596	0.378298	+	0.925684i	$0.376509\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	16.9706	1.00174
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	2.82843	0.164677
$296$	0	0
$297$	0	0
$298$	0	0
$299$	50.9117	2.94430
$300$	0	0
$301$	−12.0000	−0.691669
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.0000	−0.687118
$306$	0	0
$307$	4.24264	0.242140	0.121070	−	0.992644i	$-0.461367\pi$
$307$	4.24264	0.242140	0.121070	+	0.992644i	$0.461367\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.0000	0.561656	0.280828	−	0.959758i	$-0.409391\pi$
$317$	10.0000	0.561656	0.280828	+	0.959758i	$0.409391\pi$
$318$	0	0
$319$	8.48528	0.475085
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.00000	0.332820
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	29.6985	1.63238	0.816188	−	0.577786i	$-0.196083\pi$
$331$	29.6985	1.63238	0.816188	+	0.577786i	$0.196083\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−25.4558	−1.39080
$336$	0	0
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.2132	−1.13878	−0.569392	−	0.822066i	$-0.692821\pi$
$347$	−21.2132	−1.13878	−0.569392	+	0.822066i	$0.692821\pi$
$348$	0	0
$349$	6.00000	0.321173	0.160586	−	0.987022i	$-0.448662\pi$
$349$	6.00000	0.321173	0.160586	+	0.987022i	$0.448662\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.00000	0.319348	0.159674	−	0.987170i	$-0.448956\pi$
$353$	6.00000	0.319348	0.159674	+	0.987170i	$0.448956\pi$
$354$	0	0
$355$	16.9706	0.900704
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.48528	0.447836	0.223918	−	0.974608i	$-0.428115\pi$
$359$	8.48528	0.447836	0.223918	+	0.974608i	$0.428115\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−24.0000	−1.25622
$366$	0	0
$367$	−11.3137	−0.590571	−0.295285	−	0.955409i	$-0.595415\pi$
$367$	−11.3137	−0.590571	−0.295285	+	0.955409i	$0.595415\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	5.65685	0.293689
$372$	0	0
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−12.0000	−0.618031
$378$	0	0
$379$	−12.7279	−0.653789	−0.326895	−	0.945061i	$-0.606002\pi$
$379$	−12.7279	−0.653789	−0.326895	+	0.945061i	$0.606002\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	33.9411	1.73431	0.867155	−	0.498038i	$-0.165946\pi$
$383$	33.9411	1.73431	0.867155	+	0.498038i	$0.165946\pi$
$384$	0	0
$385$	−24.0000	−1.22315
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−11.3137	−0.569254
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	−33.9411	−1.69073
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.4558	−1.26180
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	8.48528	0.416526
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.7279	0.621800	0.310900	−	0.950443i	$-0.399370\pi$
$419$	12.7279	0.621800	0.310900	+	0.950443i	$0.399370\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	16.9706	0.821263
$428$	0	0
$429$	0	0
$430$	0	0
$431$	16.9706	0.817443	0.408722	−	0.912659i	$-0.365975\pi$
$431$	16.9706	0.817443	0.408722	+	0.912659i	$0.365975\pi$
$432$	0	0
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−36.0000	−1.72211
$438$	0	0
$439$	2.82843	0.134993	0.0674967	−	0.997719i	$-0.478499\pi$
$439$	2.82843	0.134993	0.0674967	+	0.997719i	$0.478499\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.7279	−0.604722	−0.302361	−	0.953194i	$-0.597775\pi$
$443$	−12.7279	−0.604722	−0.302361	+	0.953194i	$0.597775\pi$
$444$	0	0
$445$	24.0000	1.13771
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−24.0000	−1.13263	−0.566315	−	0.824189i	$-0.691631\pi$
$449$	−24.0000	−1.13263	−0.566315	+	0.824189i	$0.691631\pi$
$450$	0	0
$451$	−25.4558	−1.19867
$452$	0	0
$453$	0	0
$454$	0	0
$455$	33.9411	1.59118
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.0000	1.21094	0.605470	−	0.795868i	$-0.292985\pi$
$461$	26.0000	1.21094	0.605470	+	0.795868i	$0.292985\pi$
$462$	0	0
$463$	5.65685	0.262896	0.131448	−	0.991323i	$-0.458037\pi$
$463$	5.65685	0.262896	0.131448	+	0.991323i	$0.458037\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4.24264	−0.196326	−0.0981630	−	0.995170i	$-0.531297\pi$
$467$	−4.24264	−0.196326	−0.0981630	+	0.995170i	$0.531297\pi$
$468$	0	0
$469$	36.0000	1.66233
$470$	0	0
$471$	0	0
$472$	0	0
$473$	18.0000	0.827641
$474$	0	0
$475$	−4.24264	−0.194666
$476$	0	0
$477$	0	0
$478$	0	0
$479$	16.9706	0.775405	0.387702	−	0.921785i	$-0.373269\pi$
$479$	16.9706	0.775405	0.387702	+	0.921785i	$0.373269\pi$
$480$	0	0
$481$	36.0000	1.64146
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.0000	−0.726523
$486$	0	0
$487$	−36.7696	−1.66619	−0.833094	−	0.553132i	$-0.813433\pi$
$487$	−36.7696	−1.66619	−0.833094	+	0.553132i	$0.813433\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−26.8701	−1.21263	−0.606314	−	0.795225i	$-0.707353\pi$
$491$	−26.8701	−1.21263	−0.606314	+	0.795225i	$0.707353\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−4.24264	−0.189927	−0.0949633	−	0.995481i	$-0.530273\pi$
$499$	−4.24264	−0.189927	−0.0949633	+	0.995481i	$0.530273\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−42.4264	−1.89170	−0.945850	−	0.324604i	$-0.894769\pi$
$503$	−42.4264	−1.89170	−0.945850	+	0.324604i	$0.894769\pi$
$504$	0	0
$505$	−20.0000	−0.889988
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−34.0000	−1.50702	−0.753512	−	0.657434i	$-0.771642\pi$
$509$	−34.0000	−1.50702	−0.753512	+	0.657434i	$0.771642\pi$
$510$	0	0
$511$	33.9411	1.50147
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.65685	0.249271
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	−4.24264	−0.185518	−0.0927589	−	0.995689i	$-0.529569\pi$
$523$	−4.24264	−0.185518	−0.0927589	+	0.995689i	$0.529569\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	49.0000	2.13043
$530$	0	0
$531$	0	0
$532$	0	0
$533$	36.0000	1.55933
$534$	0	0
$535$	−14.1421	−0.611418
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.24264	0.182743
$540$	0	0
$541$	42.0000	1.80572	0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000	1.80572	0.902861	+	0.429934i	$0.141463\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.0000	0.514024
$546$	0	0
$547$	21.2132	0.907011	0.453506	−	0.891253i	$-0.350173\pi$
$547$	21.2132	0.907011	0.453506	+	0.891253i	$0.350173\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	8.48528	0.361485
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	−25.4558	−1.07667
$560$	0	0
$561$	0	0
$562$	0	0
$563$	29.6985	1.25164	0.625821	−	0.779967i	$-0.284764\pi$
$563$	29.6985	1.25164	0.625821	+	0.779967i	$0.284764\pi$
$564$	0	0
$565$	−12.0000	−0.504844
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	12.7279	0.532647	0.266323	−	0.963884i	$-0.414191\pi$
$571$	12.7279	0.532647	0.266323	+	0.963884i	$0.414191\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.48528	0.353861
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−8.48528	−0.351424
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.5563	0.642079	0.321040	−	0.947066i	$-0.395968\pi$
$587$	15.5563	0.642079	0.321040	+	0.947066i	$0.395968\pi$
$588$	0	0
$589$	24.0000	0.988903
$590$	0	0
$591$	0	0
$592$	0	0
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	0	0
$601$	−12.0000	−0.489490	−0.244745	−	0.969587i	$-0.578704\pi$
$601$	−12.0000	−0.489490	−0.244745	+	0.969587i	$0.578704\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	14.0000	0.569181
$606$	0	0
$607$	22.6274	0.918419	0.459209	−	0.888328i	$-0.348133\pi$
$607$	22.6274	0.918419	0.459209	+	0.888328i	$0.348133\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	36.0000	1.44931	0.724653	−	0.689114i	$-0.242000\pi$
$617$	36.0000	1.44931	0.724653	+	0.689114i	$0.242000\pi$
$618$	0	0
$619$	−38.1838	−1.53474	−0.767368	−	0.641207i	$-0.778434\pi$
$619$	−38.1838	−1.53474	−0.767368	+	0.641207i	$0.778434\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−33.9411	−1.35982
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0	0
$630$	0	0
$631$	31.1127	1.23858	0.619288	−	0.785164i	$-0.287421\pi$
$631$	31.1127	1.23858	0.619288	+	0.785164i	$0.287421\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.3137	0.448971
$636$	0	0
$637$	−6.00000	−0.237729
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.0000	0.947943	0.473972	−	0.880540i	$-0.342820\pi$
$641$	24.0000	0.947943	0.473972	+	0.880540i	$0.342820\pi$
$642$	0	0
$643$	−12.7279	−0.501940	−0.250970	−	0.967995i	$-0.580750\pi$
$643$	−12.7279	−0.501940	−0.250970	+	0.967995i	$0.580750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.4558	−1.00077	−0.500386	−	0.865802i	$-0.666809\pi$
$647$	−25.4558	−1.00077	−0.500386	+	0.865802i	$0.666809\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.00000	0.0782660	0.0391330	−	0.999234i	$-0.487540\pi$
$653$	2.00000	0.0782660	0.0391330	+	0.999234i	$0.487540\pi$
$654$	0	0
$655$	2.82843	0.110516
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−41.0122	−1.59761	−0.798804	−	0.601591i	$-0.794534\pi$
$659$	−41.0122	−1.59761	−0.798804	+	0.601591i	$0.794534\pi$
$660$	0	0
$661$	−18.0000	−0.700119	−0.350059	−	0.936727i	$-0.613839\pi$
$661$	−18.0000	−0.700119	−0.350059	+	0.936727i	$0.613839\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−24.0000	−0.930680
$666$	0	0
$667$	−16.9706	−0.657103
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−25.4558	−0.982712
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	0	0
$679$	22.6274	0.868361
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−38.1838	−1.46106	−0.730531	−	0.682880i	$-0.760727\pi$
$683$	−38.1838	−1.46106	−0.730531	+	0.682880i	$0.760727\pi$
$684$	0	0
$685$	12.0000	0.458496
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	29.6985	1.12978	0.564892	−	0.825165i	$-0.308918\pi$
$691$	29.6985	1.12978	0.564892	+	0.825165i	$0.308918\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.48528	0.321865
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	46.0000	1.73740	0.868698	−	0.495342i	$-0.164957\pi$
$701$	46.0000	1.73740	0.868698	+	0.495342i	$0.164957\pi$
$702$	0	0
$703$	−25.4558	−0.960085
$704$	0	0
$705$	0	0
$706$	0	0
$707$	28.2843	1.06374
$708$	0	0
$709$	18.0000	0.676004	0.338002	−	0.941145i	$-0.390249\pi$
$709$	18.0000	0.676004	0.338002	+	0.941145i	$0.390249\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−48.0000	−1.79761
$714$	0	0
$715$	−50.9117	−1.90399
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.9706	−0.632895	−0.316448	−	0.948610i	$-0.602490\pi$
$719$	−16.9706	−0.632895	−0.316448	+	0.948610i	$0.602490\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	2.82843	0.104901	0.0524503	−	0.998624i	$-0.483297\pi$
$727$	2.82843	0.104901	0.0524503	+	0.998624i	$0.483297\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−54.0000	−1.98912
$738$	0	0
$739$	46.6690	1.71675	0.858374	−	0.513024i	$-0.171475\pi$
$739$	46.6690	1.71675	0.858374	+	0.513024i	$0.171475\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.4558	0.933884	0.466942	−	0.884288i	$-0.345356\pi$
$743$	25.4558	0.933884	0.466942	+	0.884288i	$0.345356\pi$
$744$	0	0
$745$	−44.0000	−1.61204
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.0000	0.730784
$750$	0	0
$751$	−5.65685	−0.206422	−0.103211	−	0.994660i	$-0.532912\pi$
$751$	−5.65685	−0.206422	−0.103211	+	0.994660i	$0.532912\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−28.2843	−1.02937
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−16.9706	−0.614376
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.48528	−0.306386
$768$	0	0
$769$	−24.0000	−0.865462	−0.432731	−	0.901523i	$-0.642450\pi$
$769$	−24.0000	−0.865462	−0.432731	+	0.901523i	$0.642450\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.0000	0.503545	0.251773	−	0.967786i	$-0.418987\pi$
$773$	14.0000	0.503545	0.251773	+	0.967786i	$0.418987\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−25.4558	−0.912050
$780$	0	0
$781$	36.0000	1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−12.0000	−0.428298
$786$	0	0
$787$	29.6985	1.05864	0.529318	−	0.848423i	$-0.322448\pi$
$787$	29.6985	1.05864	0.529318	+	0.848423i	$0.322448\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	16.9706	0.603404
$792$	0	0
$793$	36.0000	1.27840
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.00000	−0.0708436	−0.0354218	−	0.999372i	$-0.511277\pi$
$797$	−2.00000	−0.0708436	−0.0354218	+	0.999372i	$0.511277\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−50.9117	−1.79663
$804$	0	0
$805$	48.0000	1.69178
$806$	0	0
$807$	0	0
$808$	0	0
$809$	42.0000	1.47664	0.738321	−	0.674450i	$-0.235619\pi$
$809$	42.0000	1.47664	0.738321	+	0.674450i	$0.235619\pi$
$810$	0	0
$811$	−4.24264	−0.148979	−0.0744896	−	0.997222i	$-0.523733\pi$
$811$	−4.24264	−0.148979	−0.0744896	+	0.997222i	$0.523733\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−42.4264	−1.48613
$816$	0	0
$817$	18.0000	0.629740
$818$	0	0
$819$	0	0
$820$	0	0
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	−19.7990	−0.690149	−0.345075	−	0.938575i	$-0.612146\pi$
$823$	−19.7990	−0.690149	−0.345075	+	0.938575i	$0.612146\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	43.8406	1.52449	0.762244	−	0.647290i	$-0.224098\pi$
$827$	43.8406	1.52449	0.762244	+	0.647290i	$0.224098\pi$
$828$	0	0
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−16.9706	−0.587291
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−8.48528	−0.292944	−0.146472	−	0.989215i	$-0.546792\pi$
$839$	−8.48528	−0.292944	−0.146472	+	0.989215i	$0.546792\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	46.0000	1.58245
$846$	0	0
$847$	−19.7990	−0.680301
$848$	0	0
$849$	0	0
$850$	0	0
$851$	50.9117	1.74523
$852$	0	0
$853$	18.0000	0.616308	0.308154	−	0.951336i	$-0.400289\pi$
$853$	18.0000	0.616308	0.308154	+	0.951336i	$0.400289\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	−46.6690	−1.59233	−0.796164	−	0.605081i	$-0.793141\pi$
$859$	−46.6690	−1.59233	−0.796164	+	0.605081i	$0.793141\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	50.9117	1.73305	0.866527	−	0.499130i	$-0.166347\pi$
$863$	50.9117	1.73305	0.866527	+	0.499130i	$0.166347\pi$
$864$	0	0
$865$	28.0000	0.952029
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	76.3675	2.58762
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33.9411	1.14742
$876$	0	0
$877$	6.00000	0.202606	0.101303	−	0.994856i	$-0.467699\pi$
$877$	6.00000	0.202606	0.101303	+	0.994856i	$0.467699\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	38.1838	1.28499	0.642493	−	0.766292i	$-0.277900\pi$
$883$	38.1838	1.28499	0.642493	+	0.766292i	$0.277900\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.4264	1.42454	0.712270	−	0.701906i	$-0.247667\pi$
$887$	42.4264	1.42454	0.712270	+	0.701906i	$0.247667\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−36.7696	−1.22907
$896$	0	0
$897$	0	0
$898$	0	0
$899$	11.3137	0.377333
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−36.0000	−1.19668
$906$	0	0
$907$	4.24264	0.140875	0.0704373	−	0.997516i	$-0.477561\pi$
$907$	4.24264	0.140875	0.0704373	+	0.997516i	$0.477561\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	33.9411	1.12452	0.562260	−	0.826961i	$-0.309932\pi$
$911$	33.9411	1.12452	0.562260	+	0.826961i	$0.309932\pi$
$912$	0	0
$913$	18.0000	0.595713
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.00000	−0.132092
$918$	0	0
$919$	−2.82843	−0.0933012	−0.0466506	−	0.998911i	$-0.514855\pi$
$919$	−2.82843	−0.0933012	−0.0466506	+	0.998911i	$0.514855\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−50.9117	−1.67578
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.0000	0.787414	0.393707	−	0.919236i	$-0.371192\pi$
$929$	24.0000	0.787414	0.393707	+	0.919236i	$0.371192\pi$
$930$	0	0
$931$	4.24264	0.139047
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.0000	0.653372	0.326686	−	0.945133i	$-0.394068\pi$
$937$	20.0000	0.653372	0.326686	+	0.945133i	$0.394068\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	50.9117	1.65791
$944$	0	0
$945$	0	0
$946$	0	0
$947$	41.0122	1.33272	0.666359	−	0.745631i	$-0.267852\pi$
$947$	41.0122	1.33272	0.666359	+	0.745631i	$0.267852\pi$
$948$	0	0
$949$	72.0000	2.33722
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	33.9411	1.09831
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.9706	−0.548008
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	48.0000	1.54517
$966$	0	0
$967$	31.1127	1.00052	0.500258	−	0.865876i	$-0.333238\pi$
$967$	31.1127	1.00052	0.500258	+	0.865876i	$0.333238\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	29.6985	0.953070	0.476535	−	0.879156i	$-0.341893\pi$
$971$	29.6985	0.953070	0.476535	+	0.879156i	$0.341893\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.0000	1.53566	0.767828	−	0.640656i	$-0.221338\pi$
$977$	48.0000	1.53566	0.767828	+	0.640656i	$0.221338\pi$
$978$	0	0
$979$	50.9117	1.62714
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.4264	1.35319	0.676596	−	0.736354i	$-0.263454\pi$
$983$	42.4264	1.35319	0.676596	+	0.736354i	$0.263454\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−36.0000	−1.14473
$990$	0	0
$991$	45.2548	1.43757	0.718784	−	0.695234i	$-0.244699\pi$
$991$	45.2548	1.43757	0.718784	+	0.695234i	$0.244699\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	39.5980	1.25534
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4608.2.a.p.1.1		2
3.2	odd	2		512.2.a.b.1.2	yes	2
4.3	odd	2	inner	4608.2.a.p.1.2		2
8.3	odd	2		4608.2.a.c.1.2		2
8.5	even	2		4608.2.a.c.1.1		2
12.11	even	2		512.2.a.b.1.1	✓	2
16.3	odd	4		4608.2.d.j.2305.1		4
16.5	even	4		4608.2.d.j.2305.4		4
16.11	odd	4		4608.2.d.j.2305.3		4
16.13	even	4		4608.2.d.j.2305.2		4
24.5	odd	2		512.2.a.e.1.1	yes	2
24.11	even	2		512.2.a.e.1.2	yes	2
48.5	odd	4		512.2.b.e.257.1		4
48.11	even	4		512.2.b.e.257.3		4
48.29	odd	4		512.2.b.e.257.4		4
48.35	even	4		512.2.b.e.257.2		4
96.5	odd	8		1024.2.e.h.769.1		4
96.11	even	8		1024.2.e.h.769.2		4
96.29	odd	8		1024.2.e.n.257.2		4
96.35	even	8		1024.2.e.h.257.2		4
96.53	odd	8		1024.2.e.n.769.2		4
96.59	even	8		1024.2.e.n.769.1		4
96.77	odd	8		1024.2.e.h.257.1		4
96.83	even	8		1024.2.e.n.257.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
512.2.a.b.1.1	✓	2	12.11	even	2
512.2.a.b.1.2	yes	2	3.2	odd	2
512.2.a.e.1.1	yes	2	24.5	odd	2
512.2.a.e.1.2	yes	2	24.11	even	2
512.2.b.e.257.1		4	48.5	odd	4
512.2.b.e.257.2		4	48.35	even	4
512.2.b.e.257.3		4	48.11	even	4
512.2.b.e.257.4		4	48.29	odd	4
1024.2.e.h.257.1		4	96.77	odd	8
1024.2.e.h.257.2		4	96.35	even	8
1024.2.e.h.769.1		4	96.5	odd	8
1024.2.e.h.769.2		4	96.11	even	8
1024.2.e.n.257.1		4	96.83	even	8
1024.2.e.n.257.2		4	96.29	odd	8
1024.2.e.n.769.1		4	96.59	even	8
1024.2.e.n.769.2		4	96.53	odd	8
4608.2.a.c.1.1		2	8.5	even	2
4608.2.a.c.1.2		2	8.3	odd	2
4608.2.a.p.1.1		2	1.1	even	1	trivial
4608.2.a.p.1.2		2	4.3	odd	2	inner
4608.2.d.j.2305.1		4	16.3	odd	4
4608.2.d.j.2305.2		4	16.13	even	4
4608.2.d.j.2305.3		4	16.11	odd	4
4608.2.d.j.2305.4		4	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{5} -2.82843 q^{7} +4.24264 q^{11} -6.00000 q^{13} +4.24264 q^{19} -8.48528 q^{23} -1.00000 q^{25} +2.00000 q^{29} +5.65685 q^{31} -5.65685 q^{35} -6.00000 q^{37} -6.00000 q^{41} +4.24264 q^{43} +1.00000 q^{49} -2.00000 q^{53} +8.48528 q^{55} +1.41421 q^{59} -6.00000 q^{61} -12.0000 q^{65} -12.7279 q^{67} +8.48528 q^{71} -12.0000 q^{73} -12.0000 q^{77} -5.65685 q^{79} +4.24264 q^{83} +12.0000 q^{89} +16.9706 q^{91} +8.48528 q^{95} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{5} - 12 q^{13} - 2 q^{25} + 4 q^{29} - 12 q^{37} - 12 q^{41} + 2 q^{49} - 4 q^{53} - 12 q^{61} - 24 q^{65} - 24 q^{73} - 24 q^{77} + 24 q^{89} - 16 q^{97}+O(q^{100}) \) 2 * q + 4 * q^5 - 12 * q^13 - 2 * q^25 + 4 * q^29 - 12 * q^37 - 12 * q^41 + 2 * q^49 - 4 * q^53 - 12 * q^61 - 24 * q^65 - 24 * q^73 - 24 * q^77 + 24 * q^89 - 16 * q^97

Introduction

L-functions

Modular forms

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Database

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