LMFDB - Embedded newform 2592.2.a.v.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2592,2,Mod(1,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2592.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2592 = 2^{5} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2592.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,-4,0,-8,0,0,0,0] 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:	yes
Analytic conductor:	$20.6972242039$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.18890$ of defining polynomial
Character	$\chi$	$=$	2592.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.64575 q^{5} -0.913701 q^{7} -5.58258 q^{11} +2.58258 q^{13} -1.73205 q^{17} -7.84190 q^{19} +1.58258 q^{23} +2.00000 q^{25} -0.818350 q^{29} -1.82740 q^{31} -2.41742 q^{35} -6.58258 q^{37} -8.75560 q^{41} +7.84190 q^{43} -8.00000 q^{47} -6.16515 q^{49} +8.75560 q^{53} -14.7701 q^{55} -8.00000 q^{59} -10.5826 q^{61} +6.83285 q^{65} +7.84190 q^{67} -9.58258 q^{71} +12.1652 q^{73} +5.10080 q^{77} +14.7701 q^{79} +3.16515 q^{83} -4.58258 q^{85} -8.85095 q^{89} -2.35970 q^{91} -20.7477 q^{95} +13.1652 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{11} - 8 q^{13} - 12 q^{23} + 8 q^{25} - 28 q^{35} - 8 q^{37} - 32 q^{47} + 12 q^{49} - 32 q^{59} - 24 q^{61} - 20 q^{71} + 12 q^{73} - 24 q^{83} - 28 q^{95} + 16 q^{97}+O(q^{100}) $ 4 * q - 4 * q^11 - 8 * q^13 - 12 * q^23 + 8 * q^25 - 28 * q^35 - 8 * q^37 - 32 * q^47 + 12 * q^49 - 32 * q^59 - 24 * q^61 - 20 * q^71 + 12 * q^73 - 24 * q^83 - 28 * q^95 + 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
$5$	2.64575	1.18322	0.591608	−	0.806226i	$-0.298493\pi$
$5$	2.64575	1.18322	0.591608	+	0.806226i	$0.298493\pi$
$6$	0	0
$7$	−0.913701	−0.345346	−0.172673	−	0.984979i	$-0.555240\pi$
$7$	−0.913701	−0.345346	−0.172673	+	0.984979i	$0.555240\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.58258	−1.68321	−0.841605	−	0.540094i	$-0.818389\pi$
$11$	−5.58258	−1.68321	−0.841605	+	0.540094i	$0.818389\pi$
$12$	0	0
$13$	2.58258	0.716278	0.358139	−	0.933668i	$-0.383411\pi$
$13$	2.58258	0.716278	0.358139	+	0.933668i	$0.383411\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.73205	−0.420084	−0.210042	−	0.977692i	$-0.567360\pi$
$17$	−1.73205	−0.420084	−0.210042	+	0.977692i	$0.567360\pi$
$18$	0	0
$19$	−7.84190	−1.79906	−0.899528	−	0.436863i	$-0.856089\pi$
$19$	−7.84190	−1.79906	−0.899528	+	0.436863i	$0.856089\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.58258	0.329990	0.164995	−	0.986294i	$-0.447239\pi$
$23$	1.58258	0.329990	0.164995	+	0.986294i	$0.447239\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.818350	−0.151964	−0.0759819	−	0.997109i	$-0.524209\pi$
$29$	−0.818350	−0.151964	−0.0759819	+	0.997109i	$0.524209\pi$
$30$	0	0
$31$	−1.82740	−0.328211	−0.164105	−	0.986443i	$-0.552474\pi$
$31$	−1.82740	−0.328211	−0.164105	+	0.986443i	$0.552474\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.41742	−0.408619
$36$	0	0
$37$	−6.58258	−1.08217	−0.541084	−	0.840968i	$-0.681986\pi$
$37$	−6.58258	−1.08217	−0.541084	+	0.840968i	$0.681986\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.75560	−1.36740	−0.683698	−	0.729765i	$-0.739629\pi$
$41$	−8.75560	−1.36740	−0.683698	+	0.729765i	$0.739629\pi$
$42$	0	0
$43$	7.84190	1.19588	0.597940	−	0.801541i	$-0.295986\pi$
$43$	7.84190	1.19588	0.597940	+	0.801541i	$0.295986\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−6.16515	−0.880736
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.75560	1.20267	0.601337	−	0.798995i	$-0.294635\pi$
$53$	8.75560	1.20267	0.601337	+	0.798995i	$0.294635\pi$
$54$	0	0
$55$	−14.7701	−1.99160
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−10.5826	−1.35496	−0.677480	−	0.735541i	$-0.736928\pi$
$61$	−10.5826	−1.35496	−0.677480	+	0.735541i	$0.736928\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6.83285	0.847511
$66$	0	0
$67$	7.84190	0.958041	0.479021	−	0.877804i	$-0.340992\pi$
$67$	7.84190	0.958041	0.479021	+	0.877804i	$0.340992\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−9.58258	−1.13724	−0.568621	−	0.822599i	$-0.692523\pi$
$71$	−9.58258	−1.13724	−0.568621	+	0.822599i	$0.692523\pi$
$72$	0	0
$73$	12.1652	1.42382	0.711912	−	0.702269i	$-0.247830\pi$
$73$	12.1652	1.42382	0.711912	+	0.702269i	$0.247830\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	5.10080	0.581290
$78$	0	0
$79$	14.7701	1.66177	0.830883	−	0.556447i	$-0.187836\pi$
$79$	14.7701	1.66177	0.830883	+	0.556447i	$0.187836\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.16515	0.347421	0.173710	−	0.984797i	$-0.444424\pi$
$83$	3.16515	0.347421	0.173710	+	0.984797i	$0.444424\pi$
$84$	0	0
$85$	−4.58258	−0.497050
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−8.85095	−0.938199	−0.469100	−	0.883145i	$-0.655421\pi$
$89$	−8.85095	−0.938199	−0.469100	+	0.883145i	$0.655421\pi$
$90$	0	0
$91$	−2.35970	−0.247364
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−20.7477	−2.12867
$96$	0	0
$97$	13.1652	1.33672	0.668359	−	0.743839i	$-0.266997\pi$
$97$	13.1652	1.33672	0.668359	+	0.743839i	$0.266997\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−13.8564	−1.37876	−0.689382	−	0.724398i	$-0.742118\pi$
$101$	−13.8564	−1.37876	−0.689382	+	0.724398i	$0.742118\pi$
$102$	0	0
$103$	19.3386	1.90549	0.952745	−	0.303772i	$-0.0982460\pi$
$103$	19.3386	1.90549	0.952745	+	0.303772i	$0.0982460\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−11.1652	−1.07938	−0.539688	−	0.841865i	$-0.681458\pi$
$107$	−11.1652	−1.07938	−0.539688	+	0.841865i	$0.681458\pi$
$108$	0	0
$109$	−5.41742	−0.518895	−0.259448	−	0.965757i	$-0.583540\pi$
$109$	−5.41742	−0.518895	−0.259448	+	0.965757i	$0.583540\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.92275	0.180877	0.0904386	−	0.995902i	$-0.471173\pi$
$113$	1.92275	0.180877	0.0904386	+	0.995902i	$0.471173\pi$
$114$	0	0
$115$	4.18710	0.390449
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.58258	0.145074
$120$	0	0
$121$	20.1652	1.83320
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−7.93725	−0.709930
$126$	0	0
$127$	−12.9427	−1.14848	−0.574240	−	0.818687i	$-0.694702\pi$
$127$	−12.9427	−1.14848	−0.574240	+	0.818687i	$0.694702\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.58258	−0.487752	−0.243876	−	0.969806i	$-0.578419\pi$
$131$	−5.58258	−0.487752	−0.243876	+	0.969806i	$0.578419\pi$
$132$	0	0
$133$	7.16515	0.621297
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.66025	0.739895	0.369948	−	0.929053i	$-0.379376\pi$
$137$	8.66025	0.739895	0.369948	+	0.929053i	$0.379376\pi$
$138$	0	0
$139$	−1.82740	−0.154998	−0.0774991	−	0.996992i	$-0.524693\pi$
$139$	−1.82740	−0.154998	−0.0774991	+	0.996992i	$0.524693\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−14.4174	−1.20565
$144$	0	0
$145$	−2.16515	−0.179806
$146$	0	0
$147$	0	0
$148$	0	0
$149$	14.8655	1.21783	0.608913	−	0.793237i	$-0.291606\pi$
$149$	14.8655	1.21783	0.608913	+	0.793237i	$0.291606\pi$
$150$	0	0
$151$	19.3386	1.57375	0.786877	−	0.617110i	$-0.211697\pi$
$151$	19.3386	1.57375	0.786877	+	0.617110i	$0.211697\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.83485	−0.388344
$156$	0	0
$157$	−11.4174	−0.911210	−0.455605	−	0.890182i	$-0.650577\pi$
$157$	−11.4174	−0.911210	−0.455605	+	0.890182i	$0.650577\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.44600	−0.113961
$162$	0	0
$163$	−17.5112	−1.37158	−0.685792	−	0.727798i	$-0.740544\pi$
$163$	−17.5112	−1.37158	−0.685792	+	0.727798i	$0.740544\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.41742	−0.496595	−0.248298	−	0.968684i	$-0.579871\pi$
$167$	−6.41742	−0.496595	−0.248298	+	0.968684i	$0.579871\pi$
$168$	0	0
$169$	−6.33030	−0.486946
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.3296	−1.39357	−0.696785	−	0.717280i	$-0.745387\pi$
$173$	−18.3296	−1.39357	−0.696785	+	0.717280i	$0.745387\pi$
$174$	0	0
$175$	−1.82740	−0.138139
$176$	0	0
$177$	0	0
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−17.4159	−1.28044
$186$	0	0
$187$	9.66930	0.707090
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.58258	−0.114511	−0.0572556	−	0.998360i	$-0.518235\pi$
$191$	−1.58258	−0.114511	−0.0572556	+	0.998360i	$0.518235\pi$
$192$	0	0
$193$	6.16515	0.443777	0.221889	−	0.975072i	$-0.428778\pi$
$193$	6.16515	0.443777	0.221889	+	0.975072i	$0.428778\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.9663	1.42254	0.711269	−	0.702920i	$-0.248121\pi$
$197$	19.9663	1.42254	0.711269	+	0.702920i	$0.248121\pi$
$198$	0	0
$199$	3.65480	0.259082	0.129541	−	0.991574i	$-0.458650\pi$
$199$	3.65480	0.259082	0.129541	+	0.991574i	$0.458650\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.747727	0.0524802
$204$	0	0
$205$	−23.1652	−1.61792
$206$	0	0
$207$	0	0
$208$	0	0
$209$	43.7780	3.02819
$210$	0	0
$211$	6.01450	0.414055	0.207028	−	0.978335i	$-0.433621\pi$
$211$	6.01450	0.414055	0.207028	+	0.978335i	$0.433621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	20.7477	1.41498
$216$	0	0
$217$	1.66970	0.113346
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.47315	−0.300897
$222$	0	0
$223$	−20.2523	−1.35619	−0.678097	−	0.734972i	$-0.737195\pi$
$223$	−20.2523	−1.35619	−0.678097	+	0.734972i	$0.737195\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.7477	−1.11159	−0.555793	−	0.831321i	$-0.687585\pi$
$227$	−16.7477	−1.11159	−0.555793	+	0.831321i	$0.687585\pi$
$228$	0	0
$229$	−23.7477	−1.56929	−0.784647	−	0.619943i	$-0.787156\pi$
$229$	−23.7477	−1.56929	−0.784647	+	0.619943i	$0.787156\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.0707	1.38038	0.690192	−	0.723626i	$-0.257526\pi$
$233$	21.0707	1.38038	0.690192	+	0.723626i	$0.257526\pi$
$234$	0	0
$235$	−21.1660	−1.38072
$236$	0	0
$237$	0	0
$238$	0	0
$239$	27.1652	1.75717	0.878584	−	0.477588i	$-0.158489\pi$
$239$	27.1652	1.75717	0.878584	+	0.477588i	$0.158489\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−16.3115	−1.04210
$246$	0	0
$247$	−20.2523	−1.28862
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.5826	−0.857325	−0.428662	−	0.903465i	$-0.641015\pi$
$251$	−13.5826	−0.857325	−0.428662	+	0.903465i	$0.641015\pi$
$252$	0	0
$253$	−8.83485	−0.555442
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.02355	−0.438117	−0.219059	−	0.975712i	$-0.570299\pi$
$257$	−7.02355	−0.438117	−0.219059	+	0.975712i	$0.570299\pi$
$258$	0	0
$259$	6.01450	0.373723
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.3303	−1.87025	−0.935123	−	0.354322i	$-0.884712\pi$
$263$	−30.3303	−1.87025	−0.935123	+	0.354322i	$0.884712\pi$
$264$	0	0
$265$	23.1652	1.42302
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−21.7937	−1.32878	−0.664391	−	0.747385i	$-0.731309\pi$
$269$	−21.7937	−1.32878	−0.664391	+	0.747385i	$0.731309\pi$
$270$	0	0
$271$	−2.74110	−0.166510	−0.0832550	−	0.996528i	$-0.526532\pi$
$271$	−2.74110	−0.166510	−0.0832550	+	0.996528i	$0.526532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−11.1652	−0.673284
$276$	0	0
$277$	16.3303	0.981193	0.490596	−	0.871387i	$-0.336779\pi$
$277$	16.3303	0.981193	0.490596	+	0.871387i	$0.336779\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−7.21425	−0.430366	−0.215183	−	0.976574i	$-0.569035\pi$
$281$	−7.21425	−0.430366	−0.215183	+	0.976574i	$0.569035\pi$
$282$	0	0
$283$	3.65480	0.217255	0.108628	−	0.994083i	$-0.465354\pi$
$283$	3.65480	0.217255	0.108628	+	0.994083i	$0.465354\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.00000	0.472225
$288$	0	0
$289$	−14.0000	−0.823529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1.00905	−0.0589494	−0.0294747	−	0.999566i	$-0.509383\pi$
$293$	−1.00905	−0.0589494	−0.0294747	+	0.999566i	$0.509383\pi$
$294$	0	0
$295$	−21.1660	−1.23233
$296$	0	0
$297$	0	0
$298$	0	0
$299$	4.08712	0.236364
$300$	0	0
$301$	−7.16515	−0.412992
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−27.9989	−1.60321
$306$	0	0
$307$	19.3386	1.10371	0.551856	−	0.833939i	$-0.313920\pi$
$307$	19.3386	1.10371	0.551856	+	0.833939i	$0.313920\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.16515	0.179479	0.0897396	−	0.995965i	$-0.471397\pi$
$311$	3.16515	0.179479	0.0897396	+	0.995965i	$0.471397\pi$
$312$	0	0
$313$	19.0000	1.07394	0.536972	−	0.843600i	$-0.319568\pi$
$313$	19.0000	1.07394	0.536972	+	0.843600i	$0.319568\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.3477	−1.14284	−0.571419	−	0.820658i	$-0.693607\pi$
$317$	−20.3477	−1.14284	−0.571419	+	0.820658i	$0.693607\pi$
$318$	0	0
$319$	4.56850	0.255787
$320$	0	0
$321$	0	0
$322$	0	0
$323$	13.5826	0.755755
$324$	0	0
$325$	5.16515	0.286511
$326$	0	0
$327$	0	0
$328$	0	0
$329$	7.30960	0.402992
$330$	0	0
$331$	−25.3531	−1.39353	−0.696767	−	0.717298i	$-0.745379\pi$
$331$	−25.3531	−1.39353	−0.696767	+	0.717298i	$0.745379\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	20.7477	1.13357
$336$	0	0
$337$	−1.16515	−0.0634698	−0.0317349	−	0.999496i	$-0.510103\pi$
$337$	−1.16515	−0.0634698	−0.0317349	+	0.999496i	$0.510103\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	10.2016	0.552448
$342$	0	0
$343$	12.0290	0.649505
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.4174	1.41816	0.709081	−	0.705127i	$-0.249110\pi$
$347$	26.4174	1.41816	0.709081	+	0.705127i	$0.249110\pi$
$348$	0	0
$349$	−17.1652	−0.918829	−0.459415	−	0.888222i	$-0.651941\pi$
$349$	−17.1652	−0.918829	−0.459415	+	0.888222i	$0.651941\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.2016	0.542977	0.271488	−	0.962442i	$-0.412484\pi$
$353$	10.2016	0.542977	0.271488	+	0.962442i	$0.412484\pi$
$354$	0	0
$355$	−25.3531	−1.34560
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−23.9129	−1.26207	−0.631037	−	0.775753i	$-0.717370\pi$
$359$	−23.9129	−1.26207	−0.631037	+	0.775753i	$0.717370\pi$
$360$	0	0
$361$	42.4955	2.23660
$362$	0	0
$363$	0	0
$364$	0	0
$365$	32.1860	1.68469
$366$	0	0
$367$	5.48220	0.286169	0.143084	−	0.989710i	$-0.454298\pi$
$367$	5.48220	0.286169	0.143084	+	0.989710i	$0.454298\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.11345	−0.108848
$378$	0	0
$379$	15.6838	0.805623	0.402812	−	0.915283i	$-0.368033\pi$
$379$	15.6838	0.805623	0.402812	+	0.915283i	$0.368033\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−6.41742	−0.327915	−0.163958	−	0.986467i	$-0.552426\pi$
$383$	−6.41742	−0.327915	−0.163958	+	0.986467i	$0.552426\pi$
$384$	0	0
$385$	13.4955	0.687792
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−1.44600	−0.0733151	−0.0366576	−	0.999328i	$-0.511671\pi$
$389$	−1.44600	−0.0733151	−0.0366576	+	0.999328i	$0.511671\pi$
$390$	0	0
$391$	−2.74110	−0.138623
$392$	0	0
$393$	0	0
$394$	0	0
$395$	39.0780	1.96623
$396$	0	0
$397$	−9.74773	−0.489224	−0.244612	−	0.969621i	$-0.578661\pi$
$397$	−9.74773	−0.489224	−0.244612	+	0.969621i	$0.578661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.4877	0.523729	0.261864	−	0.965105i	$-0.415663\pi$
$401$	10.4877	0.523729	0.261864	+	0.965105i	$0.415663\pi$
$402$	0	0
$403$	−4.71940	−0.235090
$404$	0	0
$405$	0	0
$406$	0	0
$407$	36.7477	1.82152
$408$	0	0
$409$	−2.16515	−0.107060	−0.0535299	−	0.998566i	$-0.517047\pi$
$409$	−2.16515	−0.107060	−0.0535299	+	0.998566i	$0.517047\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	7.30960	0.359682
$414$	0	0
$415$	8.37420	0.411073
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.1652	0.545453	0.272727	−	0.962092i	$-0.412075\pi$
$419$	11.1652	0.545453	0.272727	+	0.962092i	$0.412075\pi$
$420$	0	0
$421$	−8.58258	−0.418289	−0.209145	−	0.977885i	$-0.567068\pi$
$421$	−8.58258	−0.418289	−0.209145	+	0.977885i	$0.567068\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.46410	−0.168034
$426$	0	0
$427$	9.66930	0.467930
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.8348	1.00358	0.501790	−	0.864990i	$-0.332675\pi$
$431$	20.8348	1.00358	0.501790	+	0.864990i	$0.332675\pi$
$432$	0	0
$433$	13.3303	0.640613	0.320307	−	0.947314i	$-0.396214\pi$
$433$	13.3303	0.640613	0.320307	+	0.947314i	$0.396214\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−12.4104	−0.593670
$438$	0	0
$439$	21.1660	1.01020	0.505099	−	0.863061i	$-0.331456\pi$
$439$	21.1660	1.01020	0.505099	+	0.863061i	$0.331456\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	36.6606	1.74180	0.870899	−	0.491462i	$-0.163537\pi$
$443$	36.6606	1.74180	0.870899	+	0.491462i	$0.163537\pi$
$444$	0	0
$445$	−23.4174	−1.11009
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.9572	−0.894646	−0.447323	−	0.894372i	$-0.647623\pi$
$449$	−18.9572	−0.894646	−0.447323	+	0.894372i	$0.647623\pi$
$450$	0	0
$451$	48.8788	2.30161
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.24318	−0.292685
$456$	0	0
$457$	−10.1652	−0.475506	−0.237753	−	0.971326i	$-0.576411\pi$
$457$	−10.1652	−0.475506	−0.237753	+	0.971326i	$0.576411\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	26.2668	1.22337	0.611684	−	0.791102i	$-0.290493\pi$
$461$	26.2668	1.22337	0.611684	+	0.791102i	$0.290493\pi$
$462$	0	0
$463$	17.5112	0.813815	0.406907	−	0.913469i	$-0.366607\pi$
$463$	17.5112	0.813815	0.406907	+	0.913469i	$0.366607\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−19.1652	−0.886857	−0.443429	−	0.896310i	$-0.646238\pi$
$467$	−19.1652	−0.886857	−0.443429	+	0.896310i	$0.646238\pi$
$468$	0	0
$469$	−7.16515	−0.330856
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−43.7780	−2.01292
$474$	0	0
$475$	−15.6838	−0.719622
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.74773	0.216929	0.108465	−	0.994100i	$-0.465407\pi$
$479$	4.74773	0.216929	0.108465	+	0.994100i	$0.465407\pi$
$480$	0	0
$481$	−17.0000	−0.775133
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.8317	1.58163
$486$	0	0
$487$	29.5402	1.33859	0.669297	−	0.742995i	$-0.266595\pi$
$487$	29.5402	1.33859	0.669297	+	0.742995i	$0.266595\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.58258	−0.251938	−0.125969	−	0.992034i	$-0.540204\pi$
$491$	−5.58258	−0.251938	−0.125969	+	0.992034i	$0.540204\pi$
$492$	0	0
$493$	1.41742	0.0638376
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.75560	0.392743
$498$	0	0
$499$	−18.0435	−0.807738	−0.403869	−	0.914817i	$-0.632335\pi$
$499$	−18.0435	−0.807738	−0.403869	+	0.914817i	$0.632335\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.16515	−0.141127	−0.0705636	−	0.997507i	$-0.522480\pi$
$503$	−3.16515	−0.141127	−0.0705636	+	0.997507i	$0.522480\pi$
$504$	0	0
$505$	−36.6606	−1.63138
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−1.44600	−0.0640928	−0.0320464	−	0.999486i	$-0.510202\pi$
$509$	−1.44600	−0.0640928	−0.0320464	+	0.999486i	$0.510202\pi$
$510$	0	0
$511$	−11.1153	−0.491712
$512$	0	0
$513$	0	0
$514$	0	0
$515$	51.1652	2.25461
$516$	0	0
$517$	44.6606	1.96417
$518$	0	0
$519$	0	0
$520$	0	0
$521$	26.2668	1.15077	0.575385	−	0.817883i	$-0.304852\pi$
$521$	26.2668	1.15077	0.575385	+	0.817883i	$0.304852\pi$
$522$	0	0
$523$	37.3821	1.63461	0.817303	−	0.576208i	$-0.195468\pi$
$523$	37.3821	1.63461	0.817303	+	0.576208i	$0.195468\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.16515	0.137876
$528$	0	0
$529$	−20.4955	−0.891107
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−22.6120	−0.979435
$534$	0	0
$535$	−29.5402	−1.27713
$536$	0	0
$537$	0	0
$538$	0	0
$539$	34.4174	1.48246
$540$	0	0
$541$	−42.9129	−1.84497	−0.922484	−	0.386034i	$-0.873845\pi$
$541$	−42.9129	−1.84497	−0.922484	+	0.386034i	$0.873845\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−14.3332	−0.613965
$546$	0	0
$547$	−13.8564	−0.592457	−0.296229	−	0.955117i	$-0.595729\pi$
$547$	−13.8564	−0.592457	−0.296229	+	0.955117i	$0.595729\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.41742	0.273391
$552$	0	0
$553$	−13.4955	−0.573885
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.5312	−1.20890	−0.604452	−	0.796641i	$-0.706608\pi$
$557$	−28.5312	−1.20890	−0.604452	+	0.796641i	$0.706608\pi$
$558$	0	0
$559$	20.2523	0.856581
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.0000	1.01148	0.505740	−	0.862686i	$-0.331220\pi$
$563$	24.0000	1.01148	0.505740	+	0.862686i	$0.331220\pi$
$564$	0	0
$565$	5.08712	0.214017
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.2723	−1.31100	−0.655501	−	0.755195i	$-0.727542\pi$
$569$	−31.2723	−1.31100	−0.655501	+	0.755195i	$0.727542\pi$
$570$	0	0
$571$	21.1660	0.885770	0.442885	−	0.896578i	$-0.353955\pi$
$571$	21.1660	0.885770	0.442885	+	0.896578i	$0.353955\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.16515	0.131996
$576$	0	0
$577$	−0.165151	−0.00687534	−0.00343767	−	0.999994i	$-0.501094\pi$
$577$	−0.165151	−0.00687534	−0.00343767	+	0.999994i	$0.501094\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.89200	−0.119980
$582$	0	0
$583$	−48.8788	−2.02435
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.5826	−0.560613	−0.280306	−	0.959911i	$-0.590436\pi$
$587$	−13.5826	−0.560613	−0.280306	+	0.959911i	$0.590436\pi$
$588$	0	0
$589$	14.3303	0.590470
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−12.1244	−0.497888	−0.248944	−	0.968518i	$-0.580083\pi$
$593$	−12.1244	−0.497888	−0.248944	+	0.968518i	$0.580083\pi$
$594$	0	0
$595$	4.18710	0.171654
$596$	0	0
$597$	0	0
$598$	0	0
$599$	12.8348	0.524418	0.262209	−	0.965011i	$-0.415549\pi$
$599$	12.8348	0.524418	0.262209	+	0.965011i	$0.415549\pi$
$600$	0	0
$601$	40.4955	1.65184	0.825922	−	0.563784i	$-0.190655\pi$
$601$	40.4955	1.65184	0.825922	+	0.563784i	$0.190655\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	53.3520	2.16907
$606$	0	0
$607$	6.39590	0.259602	0.129801	−	0.991540i	$-0.458566\pi$
$607$	6.39590	0.259602	0.129801	+	0.991540i	$0.458566\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.6606	−0.835839
$612$	0	0
$613$	30.0000	1.21169	0.605844	−	0.795583i	$-0.292835\pi$
$613$	30.0000	1.21169	0.605844	+	0.795583i	$0.292835\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	21.0707	0.848273	0.424136	−	0.905598i	$-0.360578\pi$
$617$	21.0707	0.848273	0.424136	+	0.905598i	$0.360578\pi$
$618$	0	0
$619$	8.37420	0.336588	0.168294	−	0.985737i	$-0.446174\pi$
$619$	8.37420	0.336588	0.168294	+	0.985737i	$0.446174\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	8.08712	0.324004
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.4014	0.454602
$630$	0	0
$631$	12.0290	0.478867	0.239434	−	0.970913i	$-0.423038\pi$
$631$	12.0290	0.478867	0.239434	+	0.970913i	$0.423038\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−34.2432	−1.35890
$636$	0	0
$637$	−15.9220	−0.630851
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.92275	−0.0759441	−0.0379721	−	0.999279i	$-0.512090\pi$
$641$	−1.92275	−0.0759441	−0.0379721	+	0.999279i	$0.512090\pi$
$642$	0	0
$643$	−5.48220	−0.216197	−0.108098	−	0.994140i	$-0.534476\pi$
$643$	−5.48220	−0.216197	−0.108098	+	0.994140i	$0.534476\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.9129	1.25462	0.627312	−	0.778768i	$-0.284155\pi$
$647$	31.9129	1.25462	0.627312	+	0.778768i	$0.284155\pi$
$648$	0	0
$649$	44.6606	1.75308
$650$	0	0
$651$	0	0
$652$	0	0
$653$	36.4684	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$653$	36.4684	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$654$	0	0
$655$	−14.7701	−0.577116
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.25227	0.282508	0.141254	−	0.989973i	$-0.454887\pi$
$659$	7.25227	0.282508	0.141254	+	0.989973i	$0.454887\pi$
$660$	0	0
$661$	15.7477	0.612516	0.306258	−	0.951949i	$-0.400923\pi$
$661$	15.7477	0.612516	0.306258	+	0.951949i	$0.400923\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	18.9572	0.735129
$666$	0	0
$667$	−1.29510	−0.0501465
$668$	0	0
$669$	0	0
$670$	0	0
$671$	59.0780	2.28068
$672$	0	0
$673$	−40.1652	−1.54825	−0.774126	−	0.633031i	$-0.781811\pi$
$673$	−40.1652	−1.54825	−0.774126	+	0.633031i	$0.781811\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−16.0652	−0.617436	−0.308718	−	0.951154i	$-0.599900\pi$
$677$	−16.0652	−0.617436	−0.308718	+	0.951154i	$0.599900\pi$
$678$	0	0
$679$	−12.0290	−0.461631
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	22.9129	0.875456
$686$	0	0
$687$	0	0
$688$	0	0
$689$	22.6120	0.861449
$690$	0	0
$691$	−21.6983	−0.825443	−0.412721	−	0.910857i	$-0.635422\pi$
$691$	−21.6983	−0.825443	−0.412721	+	0.910857i	$0.635422\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.83485	−0.183396
$696$	0	0
$697$	15.1652	0.574421
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20.3477	−0.768521	−0.384260	−	0.923225i	$-0.625543\pi$
$701$	−20.3477	−0.768521	−0.384260	+	0.923225i	$0.625543\pi$
$702$	0	0
$703$	51.6199	1.94688
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.6606	0.476151
$708$	0	0
$709$	8.25227	0.309921	0.154960	−	0.987921i	$-0.450475\pi$
$709$	8.25227	0.309921	0.154960	+	0.987921i	$0.450475\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.89200	−0.108306
$714$	0	0
$715$	−38.1449	−1.42654
$716$	0	0
$717$	0	0
$718$	0	0
$719$	47.9129	1.78685	0.893424	−	0.449214i	$-0.148296\pi$
$719$	47.9129	1.78685	0.893424	+	0.449214i	$0.148296\pi$
$720$	0	0
$721$	−17.6697	−0.658054
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−1.63670	−0.0607855
$726$	0	0
$727$	−16.5975	−0.615567	−0.307784	−	0.951456i	$-0.599587\pi$
$727$	−16.5975	−0.615567	−0.307784	+	0.951456i	$0.599587\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.5826	−0.502370
$732$	0	0
$733$	−4.33030	−0.159943	−0.0799717	−	0.996797i	$-0.525483\pi$
$733$	−4.33030	−0.159943	−0.0799717	+	0.996797i	$0.525483\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−43.7780	−1.61258
$738$	0	0
$739$	−42.3320	−1.55721	−0.778604	−	0.627515i	$-0.784072\pi$
$739$	−42.3320	−1.55721	−0.778604	+	0.627515i	$0.784072\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.33030	0.232236	0.116118	−	0.993235i	$-0.462955\pi$
$743$	6.33030	0.232236	0.116118	+	0.993235i	$0.462955\pi$
$744$	0	0
$745$	39.3303	1.44095
$746$	0	0
$747$	0	0
$748$	0	0
$749$	10.2016	0.372759
$750$	0	0
$751$	−37.7635	−1.37801	−0.689005	−	0.724756i	$-0.741952\pi$
$751$	−37.7635	−1.37801	−0.689005	+	0.724756i	$0.741952\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	51.1652	1.86209
$756$	0	0
$757$	18.0000	0.654221	0.327111	−	0.944986i	$-0.393925\pi$
$757$	18.0000	0.654221	0.327111	+	0.944986i	$0.393925\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	22.5167	0.816228	0.408114	−	0.912931i	$-0.366187\pi$
$761$	22.5167	0.816228	0.408114	+	0.912931i	$0.366187\pi$
$762$	0	0
$763$	4.94990	0.179199
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−20.6606	−0.746011
$768$	0	0
$769$	22.1652	0.799296	0.399648	−	0.916669i	$-0.369132\pi$
$769$	22.1652	0.799296	0.399648	+	0.916669i	$0.369132\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−9.95536	−0.358069	−0.179035	−	0.983843i	$-0.557297\pi$
$773$	−9.95536	−0.358069	−0.179035	+	0.983843i	$0.557297\pi$
$774$	0	0
$775$	−3.65480	−0.131284
$776$	0	0
$777$	0	0
$778$	0	0
$779$	68.6606	2.46002
$780$	0	0
$781$	53.4955	1.91422
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−30.2077	−1.07816
$786$	0	0
$787$	15.1515	0.540093	0.270046	−	0.962847i	$-0.412961\pi$
$787$	15.1515	0.540093	0.270046	+	0.962847i	$0.412961\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.75682	−0.0624653
$792$	0	0
$793$	−27.3303	−0.970528
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−44.5964	−1.57968	−0.789842	−	0.613310i	$-0.789838\pi$
$797$	−44.5964	−1.57968	−0.789842	+	0.613310i	$0.789838\pi$
$798$	0	0
$799$	13.8564	0.490204
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−67.9129	−2.39659
$804$	0	0
$805$	−3.82576	−0.134840
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.1287	1.58664	0.793320	−	0.608805i	$-0.208351\pi$
$809$	45.1287	1.58664	0.793320	+	0.608805i	$0.208351\pi$
$810$	0	0
$811$	−3.65480	−0.128337	−0.0641687	−	0.997939i	$-0.520440\pi$
$811$	−3.65480	−0.128337	−0.0641687	+	0.997939i	$0.520440\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−46.3303	−1.62288
$816$	0	0
$817$	−61.4955	−2.15145
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−9.76465	−0.340789	−0.170394	−	0.985376i	$-0.554504\pi$
$821$	−9.76465	−0.340789	−0.170394	+	0.985376i	$0.554504\pi$
$822$	0	0
$823$	24.0580	0.838610	0.419305	−	0.907846i	$-0.362274\pi$
$823$	24.0580	0.838610	0.419305	+	0.907846i	$0.362274\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.74773	−0.304188	−0.152094	−	0.988366i	$-0.548602\pi$
$827$	−8.74773	−0.304188	−0.152094	+	0.988366i	$0.548602\pi$
$828$	0	0
$829$	8.33030	0.289323	0.144662	−	0.989481i	$-0.453791\pi$
$829$	8.33030	0.289323	0.144662	+	0.989481i	$0.453791\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.6784	0.369983
$834$	0	0
$835$	−16.9789	−0.587579
$836$	0	0
$837$	0	0
$838$	0	0
$839$	44.7477	1.54486	0.772432	−	0.635098i	$-0.219040\pi$
$839$	44.7477	1.54486	0.772432	+	0.635098i	$0.219040\pi$
$840$	0	0
$841$	−28.3303	−0.976907
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−16.7484	−0.576163
$846$	0	0
$847$	−18.4249	−0.633087
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−10.4174	−0.357105
$852$	0	0
$853$	−0.330303	−0.0113094	−0.00565468	−	0.999984i	$-0.501800\pi$
$853$	−0.330303	−0.0113094	−0.00565468	+	0.999984i	$0.501800\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24.1534	−0.825063	−0.412532	−	0.910943i	$-0.635355\pi$
$857$	−24.1534	−0.825063	−0.412532	+	0.910943i	$0.635355\pi$
$858$	0	0
$859$	−48.8788	−1.66772	−0.833862	−	0.551973i	$-0.813875\pi$
$859$	−48.8788	−1.66772	−0.833862	+	0.551973i	$0.813875\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−49.4083	−1.68188	−0.840940	−	0.541129i	$-0.817997\pi$
$863$	−49.4083	−1.68188	−0.840940	+	0.541129i	$0.817997\pi$
$864$	0	0
$865$	−48.4955	−1.64889
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−82.4552	−2.79710
$870$	0	0
$871$	20.2523	0.686223
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.25227	0.245172
$876$	0	0
$877$	−40.0780	−1.35334	−0.676669	−	0.736287i	$-0.736577\pi$
$877$	−40.0780	−1.35334	−0.676669	+	0.736287i	$0.736577\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−1.44600	−0.0487170	−0.0243585	−	0.999703i	$-0.507754\pi$
$881$	−1.44600	−0.0487170	−0.0243585	+	0.999703i	$0.507754\pi$
$882$	0	0
$883$	9.13701	0.307485	0.153742	−	0.988111i	$-0.450867\pi$
$883$	9.13701	0.307485	0.153742	+	0.988111i	$0.450867\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.5826	0.858979	0.429489	−	0.903072i	$-0.358694\pi$
$887$	25.5826	0.858979	0.429489	+	0.903072i	$0.358694\pi$
$888$	0	0
$889$	11.8258	0.396623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	62.7352	2.09935
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.49545	0.0498762
$900$	0	0
$901$	−15.1652	−0.505224
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.29150	0.175895
$906$	0	0
$907$	−22.9934	−0.763484	−0.381742	−	0.924269i	$-0.624676\pi$
$907$	−22.9934	−0.763484	−0.381742	+	0.924269i	$0.624676\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−22.4174	−0.742722	−0.371361	−	0.928488i	$-0.621109\pi$
$911$	−22.4174	−0.742722	−0.371361	+	0.928488i	$0.621109\pi$
$912$	0	0
$913$	−17.6697	−0.584782
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.10080	0.168443
$918$	0	0
$919$	−30.4539	−1.00458	−0.502291	−	0.864699i	$-0.667509\pi$
$919$	−30.4539	−1.00458	−0.502291	+	0.864699i	$0.667509\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−24.7477	−0.814581
$924$	0	0
$925$	−13.1652	−0.432868
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.5885	−0.511441	−0.255720	−	0.966751i	$-0.582313\pi$
$929$	−15.5885	−0.511441	−0.255720	+	0.966751i	$0.582313\pi$
$930$	0	0
$931$	48.3465	1.58449
$932$	0	0
$933$	0	0
$934$	0	0
$935$	25.5826	0.836640
$936$	0	0
$937$	2.16515	0.0707324	0.0353662	−	0.999374i	$-0.488740\pi$
$937$	2.16515	0.0707324	0.0353662	+	0.999374i	$0.488740\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−8.12795	−0.264964	−0.132482	−	0.991185i	$-0.542295\pi$
$941$	−8.12795	−0.264964	−0.132482	+	0.991185i	$0.542295\pi$
$942$	0	0
$943$	−13.8564	−0.451227
$944$	0	0
$945$	0	0
$946$	0	0
$947$	47.0780	1.52983	0.764915	−	0.644131i	$-0.222781\pi$
$947$	47.0780	1.52983	0.764915	+	0.644131i	$0.222781\pi$
$948$	0	0
$949$	31.4174	1.01985
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.7835	1.58025	0.790126	−	0.612945i	$-0.210015\pi$
$953$	48.7835	1.58025	0.790126	+	0.612945i	$0.210015\pi$
$954$	0	0
$955$	−4.18710	−0.135491
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−7.91288	−0.255520
$960$	0	0
$961$	−27.6606	−0.892278
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.3115	0.525084
$966$	0	0
$967$	30.4539	0.979332	0.489666	−	0.871910i	$-0.337119\pi$
$967$	30.4539	0.979332	0.489666	+	0.871910i	$0.337119\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0.747727	0.0239957	0.0119979	−	0.999928i	$-0.496181\pi$
$971$	0.747727	0.0239957	0.0119979	+	0.999928i	$0.496181\pi$
$972$	0	0
$973$	1.66970	0.0535280
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.9572	0.606495	0.303247	−	0.952912i	$-0.401929\pi$
$977$	18.9572	0.606495	0.303247	+	0.952912i	$0.401929\pi$
$978$	0	0
$979$	49.4111	1.57919
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57.4955	−1.83382	−0.916910	−	0.399094i	$-0.869325\pi$
$983$	−57.4955	−1.83382	−0.916910	+	0.399094i	$0.869325\pi$
$984$	0	0
$985$	52.8258	1.68317
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.4104	0.394628
$990$	0	0
$991$	−12.9427	−0.411139	−0.205569	−	0.978643i	$-0.565905\pi$
$991$	−12.9427	−0.411139	−0.205569	+	0.978643i	$0.565905\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	9.66970	0.306550
$996$	0	0
$997$	10.2523	0.324693	0.162346	−	0.986734i	$-0.448094\pi$
$997$	10.2523	0.324693	0.162346	+	0.986734i	$0.448094\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2592.2.a.v.1.4	yes	4
3.2	odd	2		2592.2.a.w.1.2	yes	4
4.3	odd	2		2592.2.a.w.1.3	yes	4
8.3	odd	2		5184.2.a.cd.1.1		4
8.5	even	2		5184.2.a.ce.1.2		4
9.2	odd	6		2592.2.i.bg.1729.3		8
9.4	even	3		2592.2.i.bh.865.1		8
9.5	odd	6		2592.2.i.bg.865.3		8
9.7	even	3		2592.2.i.bh.1729.1		8
12.11	even	2	inner	2592.2.a.v.1.1	✓	4
24.5	odd	2		5184.2.a.cd.1.4		4
24.11	even	2		5184.2.a.ce.1.3		4
36.7	odd	6		2592.2.i.bg.1729.2		8
36.11	even	6		2592.2.i.bh.1729.4		8
36.23	even	6		2592.2.i.bh.865.4		8
36.31	odd	6		2592.2.i.bg.865.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2592.2.a.v.1.1	✓	4	12.11	even	2	inner
2592.2.a.v.1.4	yes	4	1.1	even	1	trivial
2592.2.a.w.1.2	yes	4	3.2	odd	2
2592.2.a.w.1.3	yes	4	4.3	odd	2
2592.2.i.bg.865.2		8	36.31	odd	6
2592.2.i.bg.865.3		8	9.5	odd	6
2592.2.i.bg.1729.2		8	36.7	odd	6
2592.2.i.bg.1729.3		8	9.2	odd	6
2592.2.i.bh.865.1		8	9.4	even	3
2592.2.i.bh.865.4		8	36.23	even	6
2592.2.i.bh.1729.1		8	9.7	even	3
2592.2.i.bh.1729.4		8	36.11	even	6
5184.2.a.cd.1.1		4	8.3	odd	2
5184.2.a.cd.1.4		4	24.5	odd	2
5184.2.a.ce.1.2		4	8.5	even	2
5184.2.a.ce.1.3		4	24.11	even	2

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:	\( N \)	\(=\)	\( 2592 = 2^{5} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2592.a (trivial)

\(f(q)\)	\(=\)	\(q+2.64575 q^{5} -0.913701 q^{7} -5.58258 q^{11} +2.58258 q^{13} -1.73205 q^{17} -7.84190 q^{19} +1.58258 q^{23} +2.00000 q^{25} -0.818350 q^{29} -1.82740 q^{31} -2.41742 q^{35} -6.58258 q^{37} -8.75560 q^{41} +7.84190 q^{43} -8.00000 q^{47} -6.16515 q^{49} +8.75560 q^{53} -14.7701 q^{55} -8.00000 q^{59} -10.5826 q^{61} +6.83285 q^{65} +7.84190 q^{67} -9.58258 q^{71} +12.1652 q^{73} +5.10080 q^{77} +14.7701 q^{79} +3.16515 q^{83} -4.58258 q^{85} -8.85095 q^{89} -2.35970 q^{91} -20.7477 q^{95} +13.1652 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{11} - 8 q^{13} - 12 q^{23} + 8 q^{25} - 28 q^{35} - 8 q^{37} - 32 q^{47} + 12 q^{49} - 32 q^{59} - 24 q^{61} - 20 q^{71} + 12 q^{73} - 24 q^{83} - 28 q^{95} + 16 q^{97}+O(q^{100}) \) 4 * q - 4 * q^11 - 8 * q^13 - 12 * q^23 + 8 * q^25 - 28 * q^35 - 8 * q^37 - 32 * q^47 + 12 * q^49 - 32 * q^59 - 24 * q^61 - 20 * q^71 + 12 * q^73 - 24 * q^83 - 28 * q^95 + 16 * q^97

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