LMFDB - Embedded newform 9216.2.a.bk.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9216,2,Mod(1,9216)] 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("9216.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9216, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9216 = 2^{10} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9216.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,-12,0,0,0,0,0,4,0,0,0,0, 0,0,0,0,0,24,0,0,0,0,0,0,0,-12,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,36,0,0,0, 0,0,24,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(67)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.5901305028$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 256)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	9216.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.44949 q^{5} +3.86370 q^{7} +4.73205 q^{11} -0.378937 q^{13} +3.46410 q^{17} -4.73205 q^{19} -1.79315 q^{23} +1.00000 q^{25} -2.44949 q^{29} -5.65685 q^{31} +9.46410 q^{35} +5.27792 q^{37} -6.92820 q^{41} +2.19615 q^{43} +9.79796 q^{47} +7.92820 q^{49} +10.9348 q^{53} +11.5911 q^{55} +7.26795 q^{59} -5.27792 q^{61} -0.928203 q^{65} -6.19615 q^{67} +1.79315 q^{71} -2.53590 q^{73} +18.2832 q^{77} +4.14110 q^{79} -2.19615 q^{83} +8.48528 q^{85} +2.53590 q^{89} -1.46410 q^{91} -11.5911 q^{95} +3.46410 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{11} - 12 q^{19} + 4 q^{25} + 24 q^{35} - 12 q^{43} + 4 q^{49} + 36 q^{59} + 24 q^{65} - 4 q^{67} - 24 q^{73} + 12 q^{83} + 24 q^{89} + 8 q^{91}+O(q^{100}) $ 4 * q + 12 * q^11 - 12 * q^19 + 4 * q^25 + 24 * q^35 - 12 * q^43 + 4 * q^49 + 36 * q^59 + 24 * q^65 - 4 * q^67 - 24 * q^73 + 12 * q^83 + 24 * q^89 + 8 * q^91

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
$5$	2.44949	1.09545	0.547723	−	0.836660i	$-0.315495\pi$
$5$	2.44949	1.09545	0.547723	+	0.836660i	$0.315495\pi$
$6$	0	0
$7$	3.86370	1.46034	0.730171	−	0.683264i	$-0.239440\pi$
$7$	3.86370	1.46034	0.730171	+	0.683264i	$0.239440\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.73205	1.42677	0.713384	−	0.700774i	$-0.247162\pi$
$11$	4.73205	1.42677	0.713384	+	0.700774i	$0.247162\pi$
$12$	0	0
$13$	−0.378937	−0.105098	−0.0525492	−	0.998618i	$-0.516735\pi$
$13$	−0.378937	−0.105098	−0.0525492	+	0.998618i	$0.516735\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410	0.840168	0.420084	−	0.907485i	$-0.362001\pi$
$17$	3.46410	0.840168	0.420084	+	0.907485i	$0.362001\pi$
$18$	0	0
$19$	−4.73205	−1.08561	−0.542803	−	0.839860i	$-0.682637\pi$
$19$	−4.73205	−1.08561	−0.542803	+	0.839860i	$0.682637\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.79315	−0.373898	−0.186949	−	0.982370i	$-0.559860\pi$
$23$	−1.79315	−0.373898	−0.186949	+	0.982370i	$0.559860\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.44949	−0.454859	−0.227429	−	0.973795i	$-0.573032\pi$
$29$	−2.44949	−0.454859	−0.227429	+	0.973795i	$0.573032\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	9.46410	1.59973
$36$	0	0
$37$	5.27792	0.867684	0.433842	−	0.900989i	$-0.357158\pi$
$37$	5.27792	0.867684	0.433842	+	0.900989i	$0.357158\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.92820	−1.08200	−0.541002	−	0.841021i	$-0.681955\pi$
$41$	−6.92820	−1.08200	−0.541002	+	0.841021i	$0.681955\pi$
$42$	0	0
$43$	2.19615	0.334910	0.167455	−	0.985880i	$-0.446445\pi$
$43$	2.19615	0.334910	0.167455	+	0.985880i	$0.446445\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.79796	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$48$	0	0
$49$	7.92820	1.13260
$50$	0	0
$51$	0	0
$52$	0	0
$53$	10.9348	1.50201	0.751003	−	0.660299i	$-0.229570\pi$
$53$	10.9348	1.50201	0.751003	+	0.660299i	$0.229570\pi$
$54$	0	0
$55$	11.5911	1.56294
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.26795	0.946206	0.473103	−	0.881007i	$-0.343134\pi$
$59$	7.26795	0.946206	0.473103	+	0.881007i	$0.343134\pi$
$60$	0	0
$61$	−5.27792	−0.675768	−0.337884	−	0.941188i	$-0.609711\pi$
$61$	−5.27792	−0.675768	−0.337884	+	0.941188i	$0.609711\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.928203	−0.115129
$66$	0	0
$67$	−6.19615	−0.756980	−0.378490	−	0.925605i	$-0.623557\pi$
$67$	−6.19615	−0.756980	−0.378490	+	0.925605i	$0.623557\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	1.79315	0.212808	0.106404	−	0.994323i	$-0.466066\pi$
$71$	1.79315	0.212808	0.106404	+	0.994323i	$0.466066\pi$
$72$	0	0
$73$	−2.53590	−0.296804	−0.148402	−	0.988927i	$-0.547413\pi$
$73$	−2.53590	−0.296804	−0.148402	+	0.988927i	$0.547413\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	18.2832	2.08357
$78$	0	0
$79$	4.14110	0.465911	0.232955	−	0.972487i	$-0.425160\pi$
$79$	4.14110	0.465911	0.232955	+	0.972487i	$0.425160\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.19615	−0.241059	−0.120530	−	0.992710i	$-0.538459\pi$
$83$	−2.19615	−0.241059	−0.120530	+	0.992710i	$0.538459\pi$
$84$	0	0
$85$	8.48528	0.920358
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.53590	0.268805	0.134402	−	0.990927i	$-0.457089\pi$
$89$	2.53590	0.268805	0.134402	+	0.990927i	$0.457089\pi$
$90$	0	0
$91$	−1.46410	−0.153480
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−11.5911	−1.18922
$96$	0	0
$97$	3.46410	0.351726	0.175863	−	0.984415i	$-0.443728\pi$
$97$	3.46410	0.351726	0.175863	+	0.984415i	$0.443728\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.13681	−0.113117	−0.0565585	−	0.998399i	$-0.518013\pi$
$101$	−1.13681	−0.113117	−0.0565585	+	0.998399i	$0.518013\pi$
$102$	0	0
$103$	9.52056	0.938088	0.469044	−	0.883175i	$-0.344598\pi$
$103$	9.52056	0.938088	0.469044	+	0.883175i	$0.344598\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.5885	1.79701	0.898507	−	0.438959i	$-0.144653\pi$
$107$	18.5885	1.79701	0.898507	+	0.438959i	$0.144653\pi$
$108$	0	0
$109$	18.6622	1.78751	0.893756	−	0.448553i	$-0.148060\pi$
$109$	18.6622	1.78751	0.893756	+	0.448553i	$0.148060\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	−4.39230	−0.409585
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.3843	1.22693
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−9.79796	−0.876356
$126$	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
$131$	0.339746	0.0296837	0.0148419	−	0.999890i	$-0.495276\pi$
$131$	0.339746	0.0296837	0.0148419	+	0.999890i	$0.495276\pi$
$132$	0	0
$133$	−18.2832	−1.58536
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	10.1962	0.864826	0.432413	−	0.901676i	$-0.357662\pi$
$139$	10.1962	0.864826	0.432413	+	0.901676i	$0.357662\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−1.79315	−0.149951
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−19.4201	−1.59095	−0.795476	−	0.605985i	$-0.792779\pi$
$149$	−19.4201	−1.59095	−0.795476	+	0.605985i	$0.792779\pi$
$150$	0	0
$151$	−9.52056	−0.774772	−0.387386	−	0.921918i	$-0.626622\pi$
$151$	−9.52056	−0.774772	−0.387386	+	0.921918i	$0.626622\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−13.8564	−1.11297
$156$	0	0
$157$	10.1769	0.812205	0.406102	−	0.913828i	$-0.366888\pi$
$157$	10.1769	0.812205	0.406102	+	0.913828i	$0.366888\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−6.92820	−0.546019
$162$	0	0
$163$	16.7321	1.31056	0.655278	−	0.755388i	$-0.272552\pi$
$163$	16.7321	1.31056	0.655278	+	0.755388i	$0.272552\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	18.7637	1.45198	0.725990	−	0.687705i	$-0.241382\pi$
$167$	18.7637	1.45198	0.725990	+	0.687705i	$0.241382\pi$
$168$	0	0
$169$	−12.8564	−0.988954
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−10.9348	−0.831355	−0.415678	−	0.909512i	$-0.636456\pi$
$173$	−10.9348	−0.831355	−0.415678	+	0.909512i	$0.636456\pi$
$174$	0	0
$175$	3.86370	0.292069
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2.19615	−0.164148	−0.0820741	−	0.996626i	$-0.526154\pi$
$179$	−2.19615	−0.164148	−0.0820741	+	0.996626i	$0.526154\pi$
$180$	0	0
$181$	−22.8033	−1.69495	−0.847477	−	0.530832i	$-0.821880\pi$
$181$	−22.8033	−1.69495	−0.847477	+	0.530832i	$0.821880\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12.9282	0.950500
$186$	0	0
$187$	16.3923	1.19872
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	−18.3923	−1.32391	−0.661954	−	0.749545i	$-0.730272\pi$
$193$	−18.3923	−1.32391	−0.661954	+	0.749545i	$0.730272\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.44949	0.174519	0.0872595	−	0.996186i	$-0.472189\pi$
$197$	2.44949	0.174519	0.0872595	+	0.996186i	$0.472189\pi$
$198$	0	0
$199$	−15.7322	−1.11523	−0.557614	−	0.830101i	$-0.688283\pi$
$199$	−15.7322	−1.11523	−0.557614	+	0.830101i	$0.688283\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−9.46410	−0.664250
$204$	0	0
$205$	−16.9706	−1.18528
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−22.3923	−1.54891
$210$	0	0
$211$	10.1962	0.701932	0.350966	−	0.936388i	$-0.385853\pi$
$211$	10.1962	0.701932	0.350966	+	0.936388i	$0.385853\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.37945	0.366876
$216$	0	0
$217$	−21.8564	−1.48371
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.31268	−0.0883003
$222$	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
$227$	21.1244	1.40207	0.701036	−	0.713126i	$-0.252721\pi$
$227$	21.1244	1.40207	0.701036	+	0.713126i	$0.252721\pi$
$228$	0	0
$229$	13.0053	0.859416	0.429708	−	0.902968i	$-0.358617\pi$
$229$	13.0053	0.859416	0.429708	+	0.902968i	$0.358617\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.53590	0.166132	0.0830661	−	0.996544i	$-0.473529\pi$
$233$	2.53590	0.166132	0.0830661	+	0.996544i	$0.473529\pi$
$234$	0	0
$235$	24.0000	1.56559
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.7685	−1.73151	−0.865756	−	0.500467i	$-0.833162\pi$
$239$	−26.7685	−1.73151	−0.865756	+	0.500467i	$0.833162\pi$
$240$	0	0
$241$	2.39230	0.154102	0.0770510	−	0.997027i	$-0.475450\pi$
$241$	2.39230	0.154102	0.0770510	+	0.997027i	$0.475450\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	19.4201	1.24070
$246$	0	0
$247$	1.79315	0.114095
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.1244	1.33336	0.666679	−	0.745345i	$-0.267715\pi$
$251$	21.1244	1.33336	0.666679	+	0.745345i	$0.267715\pi$
$252$	0	0
$253$	−8.48528	−0.533465
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−12.9282	−0.806439	−0.403220	−	0.915103i	$-0.632109\pi$
$257$	−12.9282	−0.806439	−0.403220	+	0.915103i	$0.632109\pi$
$258$	0	0
$259$	20.3923	1.26712
$260$	0	0
$261$	0	0
$262$	0	0
$263$	21.3891	1.31891	0.659453	−	0.751746i	$-0.270788\pi$
$263$	21.3891	1.31891	0.659453	+	0.751746i	$0.270788\pi$
$264$	0	0
$265$	26.7846	1.64537
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.34847	−0.448044	−0.224022	−	0.974584i	$-0.571919\pi$
$269$	−7.34847	−0.448044	−0.224022	+	0.974584i	$0.571919\pi$
$270$	0	0
$271$	1.51575	0.0920752	0.0460376	−	0.998940i	$-0.485341\pi$
$271$	1.51575	0.0920752	0.0460376	+	0.998940i	$0.485341\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.73205	0.285353
$276$	0	0
$277$	4.52004	0.271583	0.135792	−	0.990737i	$-0.456642\pi$
$277$	4.52004	0.271583	0.135792	+	0.990737i	$0.456642\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.3923	−0.977883	−0.488941	−	0.872317i	$-0.662617\pi$
$281$	−16.3923	−0.977883	−0.488941	+	0.872317i	$0.662617\pi$
$282$	0	0
$283$	1.80385	0.107228	0.0536138	−	0.998562i	$-0.482926\pi$
$283$	1.80385	0.107228	0.0536138	+	0.998562i	$0.482926\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.7685	−1.58010
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−17.1464	−1.00171	−0.500853	−	0.865533i	$-0.666980\pi$
$293$	−17.1464	−1.00171	−0.500853	+	0.865533i	$0.666980\pi$
$294$	0	0
$295$	17.8028	1.03652
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.679492	0.0392960
$300$	0	0
$301$	8.48528	0.489083
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−12.9282	−0.740267
$306$	0	0
$307$	30.9808	1.76817	0.884083	−	0.467330i	$-0.154784\pi$
$307$	30.9808	1.76817	0.884083	+	0.467330i	$0.154784\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	15.1774	0.860632	0.430316	−	0.902678i	$-0.358402\pi$
$311$	15.1774	0.860632	0.430316	+	0.902678i	$0.358402\pi$
$312$	0	0
$313$	12.7846	0.722629	0.361314	−	0.932444i	$-0.382328\pi$
$313$	12.7846	0.722629	0.361314	+	0.932444i	$0.382328\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.62209	−0.540431	−0.270215	−	0.962800i	$-0.587095\pi$
$317$	−9.62209	−0.540431	−0.270215	+	0.962800i	$0.587095\pi$
$318$	0	0
$319$	−11.5911	−0.648978
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−16.3923	−0.912092
$324$	0	0
$325$	−0.378937	−0.0210197
$326$	0	0
$327$	0	0
$328$	0	0
$329$	37.8564	2.08709
$330$	0	0
$331$	−26.5885	−1.46143	−0.730717	−	0.682681i	$-0.760814\pi$
$331$	−26.5885	−1.46143	−0.730717	+	0.682681i	$0.760814\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.1774	−0.829231
$336$	0	0
$337$	−21.7128	−1.18277	−0.591386	−	0.806389i	$-0.701419\pi$
$337$	−21.7128	−1.18277	−0.591386	+	0.806389i	$0.701419\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−26.7685	−1.44960
$342$	0	0
$343$	3.58630	0.193642
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.26795	0.390164	0.195082	−	0.980787i	$-0.437503\pi$
$347$	7.26795	0.390164	0.195082	+	0.980787i	$0.437503\pi$
$348$	0	0
$349$	−5.83272	−0.312218	−0.156109	−	0.987740i	$-0.549895\pi$
$349$	−5.83272	−0.312218	−0.156109	+	0.987740i	$0.549895\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	0.928203	0.0494033	0.0247016	−	0.999695i	$-0.492136\pi$
$353$	0.928203	0.0494033	0.0247016	+	0.999695i	$0.492136\pi$
$354$	0	0
$355$	4.39230	0.233119
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.8028	−0.939594	−0.469797	−	0.882774i	$-0.655673\pi$
$359$	−17.8028	−0.939594	−0.469797	+	0.882774i	$0.655673\pi$
$360$	0	0
$361$	3.39230	0.178542
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.21166	−0.325133
$366$	0	0
$367$	−32.4254	−1.69259	−0.846295	−	0.532714i	$-0.821172\pi$
$367$	−32.4254	−1.69259	−0.846295	+	0.532714i	$0.821172\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	42.2487	2.19344
$372$	0	0
$373$	19.9749	1.03426	0.517129	−	0.855907i	$-0.327001\pi$
$373$	19.9749	1.03426	0.517129	+	0.855907i	$0.327001\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.928203	0.0478049
$378$	0	0
$379$	2.87564	0.147712	0.0738560	−	0.997269i	$-0.476469\pi$
$379$	2.87564	0.147712	0.0738560	+	0.997269i	$0.476469\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.5959	1.00130	0.500652	−	0.865648i	$-0.333094\pi$
$383$	19.5959	1.00130	0.500652	+	0.865648i	$0.333094\pi$
$384$	0	0
$385$	44.7846	2.28244
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6.03579	0.306027	0.153013	−	0.988224i	$-0.451102\pi$
$389$	6.03579	0.306027	0.153013	+	0.988224i	$0.451102\pi$
$390$	0	0
$391$	−6.21166	−0.314137
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.1436	0.510380
$396$	0	0
$397$	−27.7023	−1.39034	−0.695168	−	0.718847i	$-0.744670\pi$
$397$	−27.7023	−1.39034	−0.695168	+	0.718847i	$0.744670\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.6077	0.679536	0.339768	−	0.940509i	$-0.389651\pi$
$401$	13.6077	0.679536	0.339768	+	0.940509i	$0.389651\pi$
$402$	0	0
$403$	2.14359	0.106780
$404$	0	0
$405$	0	0
$406$	0	0
$407$	24.9754	1.23798
$408$	0	0
$409$	17.0718	0.844146	0.422073	−	0.906562i	$-0.361303\pi$
$409$	17.0718	0.844146	0.422073	+	0.906562i	$0.361303\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	28.0812	1.38179
$414$	0	0
$415$	−5.37945	−0.264067
$416$	0	0
$417$	0	0
$418$	0	0
$419$	14.1962	0.693527	0.346764	−	0.937953i	$-0.387281\pi$
$419$	14.1962	0.693527	0.346764	+	0.937953i	$0.387281\pi$
$420$	0	0
$421$	−26.1865	−1.27625	−0.638126	−	0.769932i	$-0.720290\pi$
$421$	−26.1865	−1.27625	−0.638126	+	0.769932i	$0.720290\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	−20.3923	−0.986853
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.7685	−1.28939	−0.644697	−	0.764438i	$-0.723017\pi$
$431$	−26.7685	−1.28939	−0.644697	+	0.764438i	$0.723017\pi$
$432$	0	0
$433$	34.3923	1.65279	0.826394	−	0.563092i	$-0.190388\pi$
$433$	34.3923	1.65279	0.826394	+	0.563092i	$0.190388\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.48528	0.405906
$438$	0	0
$439$	−12.1459	−0.579693	−0.289846	−	0.957073i	$-0.593604\pi$
$439$	−12.1459	−0.579693	−0.289846	+	0.957073i	$0.593604\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	14.1962	0.674480	0.337240	−	0.941419i	$-0.390507\pi$
$443$	14.1962	0.674480	0.337240	+	0.941419i	$0.390507\pi$
$444$	0	0
$445$	6.21166	0.294461
$446$	0	0
$447$	0	0
$448$	0	0
$449$	24.2487	1.14437	0.572184	−	0.820125i	$-0.306096\pi$
$449$	24.2487	1.14437	0.572184	+	0.820125i	$0.306096\pi$
$450$	0	0
$451$	−32.7846	−1.54377
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−3.58630	−0.168128
$456$	0	0
$457$	−20.7846	−0.972263	−0.486132	−	0.873886i	$-0.661592\pi$
$457$	−20.7846	−0.972263	−0.486132	+	0.873886i	$0.661592\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−34.1170	−1.58899	−0.794493	−	0.607273i	$-0.792263\pi$
$461$	−34.1170	−1.58899	−0.794493	+	0.607273i	$0.792263\pi$
$462$	0	0
$463$	−38.0822	−1.76983	−0.884916	−	0.465751i	$-0.845784\pi$
$463$	−38.0822	−1.76983	−0.884916	+	0.465751i	$0.845784\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.7321	1.32956	0.664780	−	0.747039i	$-0.268525\pi$
$467$	28.7321	1.32956	0.664780	+	0.747039i	$0.268525\pi$
$468$	0	0
$469$	−23.9401	−1.10545
$470$	0	0
$471$	0	0
$472$	0	0
$473$	10.3923	0.477839
$474$	0	0
$475$	−4.73205	−0.217121
$476$	0	0
$477$	0	0
$478$	0	0
$479$	36.5665	1.67077	0.835383	−	0.549669i	$-0.185246\pi$
$479$	36.5665	1.67077	0.835383	+	0.549669i	$0.185246\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.48528	0.385297
$486$	0	0
$487$	−27.0459	−1.22557	−0.612784	−	0.790251i	$-0.709950\pi$
$487$	−27.0459	−1.22557	−0.612784	+	0.790251i	$0.709950\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	21.8038	0.983994	0.491997	−	0.870597i	$-0.336267\pi$
$491$	21.8038	0.983994	0.491997	+	0.870597i	$0.336267\pi$
$492$	0	0
$493$	−8.48528	−0.382158
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.92820	0.310772
$498$	0	0
$499$	−5.80385	−0.259816	−0.129908	−	0.991526i	$-0.541468\pi$
$499$	−5.80385	−0.259816	−0.129908	+	0.991526i	$0.541468\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−40.9850	−1.82743	−0.913715	−	0.406355i	$-0.866800\pi$
$503$	−40.9850	−1.82743	−0.913715	+	0.406355i	$0.866800\pi$
$504$	0	0
$505$	−2.78461	−0.123914
$506$	0	0
$507$	0	0
$508$	0	0
$509$	13.5601	0.601042	0.300521	−	0.953775i	$-0.402839\pi$
$509$	13.5601	0.601042	0.300521	+	0.953775i	$0.402839\pi$
$510$	0	0
$511$	−9.79796	−0.433436
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23.3205	1.02762
$516$	0	0
$517$	46.3644	2.03911
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.0000	−0.525730	−0.262865	−	0.964833i	$-0.584667\pi$
$521$	−12.0000	−0.525730	−0.262865	+	0.964833i	$0.584667\pi$
$522$	0	0
$523$	19.2679	0.842529	0.421264	−	0.906938i	$-0.361586\pi$
$523$	19.2679	0.842529	0.421264	+	0.906938i	$0.361586\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−19.5959	−0.853612
$528$	0	0
$529$	−19.7846	−0.860200
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.62536	0.113717
$534$	0	0
$535$	45.5322	1.96853
$536$	0	0
$537$	0	0
$538$	0	0
$539$	37.5167	1.61596
$540$	0	0
$541$	14.3180	0.615579	0.307789	−	0.951454i	$-0.400411\pi$
$541$	14.3180	0.615579	0.307789	+	0.951454i	$0.400411\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	45.7128	1.95812
$546$	0	0
$547$	−0.339746	−0.0145265	−0.00726324	−	0.999974i	$-0.502312\pi$
$547$	−0.339746	−0.0145265	−0.00726324	+	0.999974i	$0.502312\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	11.5911	0.493798
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15.8338	0.670898	0.335449	−	0.942058i	$-0.391112\pi$
$557$	15.8338	0.670898	0.335449	+	0.942058i	$0.391112\pi$
$558$	0	0
$559$	−0.832204	−0.0351985
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.8756	0.626934	0.313467	−	0.949599i	$-0.398510\pi$
$563$	14.8756	0.626934	0.313467	+	0.949599i	$0.398510\pi$
$564$	0	0
$565$	31.6675	1.33226
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−30.1962	−1.26367	−0.631835	−	0.775103i	$-0.717698\pi$
$571$	−30.1962	−1.26367	−0.631835	+	0.775103i	$0.717698\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.79315	−0.0747796
$576$	0	0
$577$	−43.8564	−1.82577	−0.912883	−	0.408221i	$-0.866149\pi$
$577$	−43.8564	−1.82577	−0.912883	+	0.408221i	$0.866149\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.48528	−0.352029
$582$	0	0
$583$	51.7439	2.14301
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.12436	−0.376602	−0.188301	−	0.982111i	$-0.560298\pi$
$587$	−9.12436	−0.376602	−0.188301	+	0.982111i	$0.560298\pi$
$588$	0	0
$589$	26.7685	1.10298
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−19.8564	−0.815405	−0.407702	−	0.913115i	$-0.633670\pi$
$593$	−19.8564	−0.815405	−0.407702	+	0.913115i	$0.633670\pi$
$594$	0	0
$595$	32.7846	1.34404
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−4.41851	−0.180535	−0.0902676	−	0.995918i	$-0.528772\pi$
$599$	−4.41851	−0.180535	−0.0902676	+	0.995918i	$0.528772\pi$
$600$	0	0
$601$	−40.3923	−1.64764	−0.823818	−	0.566854i	$-0.808160\pi$
$601$	−40.3923	−1.64764	−0.823818	+	0.566854i	$0.808160\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	27.9053	1.13451
$606$	0	0
$607$	8.28221	0.336165	0.168082	−	0.985773i	$-0.446243\pi$
$607$	8.28221	0.336165	0.168082	+	0.985773i	$0.446243\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.71281	−0.150204
$612$	0	0
$613$	21.2875	0.859795	0.429898	−	0.902878i	$-0.358550\pi$
$613$	21.2875	0.859795	0.429898	+	0.902878i	$0.358550\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.3205	0.455746	0.227873	−	0.973691i	$-0.426823\pi$
$617$	11.3205	0.455746	0.227873	+	0.973691i	$0.426823\pi$
$618$	0	0
$619$	−26.5885	−1.06868	−0.534340	−	0.845270i	$-0.679440\pi$
$619$	−26.5885	−1.06868	−0.534340	+	0.845270i	$0.679440\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	9.79796	0.392547
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	18.2832	0.729001
$630$	0	0
$631$	−1.23835	−0.0492979	−0.0246489	−	0.999696i	$-0.507847\pi$
$631$	−1.23835	−0.0492979	−0.0246489	+	0.999696i	$0.507847\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.8564	0.549875
$636$	0	0
$637$	−3.00429	−0.119034
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.6077	−0.537472	−0.268736	−	0.963214i	$-0.586606\pi$
$641$	−13.6077	−0.537472	−0.268736	+	0.963214i	$0.586606\pi$
$642$	0	0
$643$	9.80385	0.386626	0.193313	−	0.981137i	$-0.438077\pi$
$643$	9.80385	0.386626	0.193313	+	0.981137i	$0.438077\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.79315	0.0704960	0.0352480	−	0.999379i	$-0.488778\pi$
$647$	1.79315	0.0704960	0.0352480	+	0.999379i	$0.488778\pi$
$648$	0	0
$649$	34.3923	1.35002
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−21.6937	−0.848939	−0.424470	−	0.905442i	$-0.639539\pi$
$653$	−21.6937	−0.848939	−0.424470	+	0.905442i	$0.639539\pi$
$654$	0	0
$655$	0.832204	0.0325169
$656$	0	0
$657$	0	0
$658$	0	0
$659$	25.5167	0.993988	0.496994	−	0.867754i	$-0.334437\pi$
$659$	25.5167	0.993988	0.496994	+	0.867754i	$0.334437\pi$
$660$	0	0
$661$	−25.8348	−1.00486	−0.502428	−	0.864619i	$-0.667560\pi$
$661$	−25.8348	−1.00486	−0.502428	+	0.864619i	$0.667560\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−44.7846	−1.73667
$666$	0	0
$667$	4.39230	0.170071
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.9754	−0.964163
$672$	0	0
$673$	41.3205	1.59279	0.796394	−	0.604778i	$-0.206738\pi$
$673$	41.3205	1.59279	0.796394	+	0.604778i	$0.206738\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	39.9769	1.53644	0.768219	−	0.640187i	$-0.221143\pi$
$677$	39.9769	1.53644	0.768219	+	0.640187i	$0.221143\pi$
$678$	0	0
$679$	13.3843	0.513641
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.3397	−0.472167	−0.236084	−	0.971733i	$-0.575864\pi$
$683$	−12.3397	−0.472167	−0.236084	+	0.971733i	$0.575864\pi$
$684$	0	0
$685$	−29.3939	−1.12308
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−4.14359	−0.157858
$690$	0	0
$691$	−39.3731	−1.49782	−0.748911	−	0.662671i	$-0.769423\pi$
$691$	−39.3731	−1.49782	−0.748911	+	0.662671i	$0.769423\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	24.9754	0.947370
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13.2084	−0.498874	−0.249437	−	0.968391i	$-0.580246\pi$
$701$	−13.2084	−0.498874	−0.249437	+	0.968391i	$0.580246\pi$
$702$	0	0
$703$	−24.9754	−0.941964
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.39230	−0.165190
$708$	0	0
$709$	−41.0865	−1.54304	−0.771518	−	0.636207i	$-0.780502\pi$
$709$	−41.0865	−1.54304	−0.771518	+	0.636207i	$0.780502\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.1436	0.379881
$714$	0	0
$715$	−4.39230	−0.164263
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−12.4233	−0.463311	−0.231656	−	0.972798i	$-0.574414\pi$
$719$	−12.4233	−0.463311	−0.231656	+	0.972798i	$0.574414\pi$
$720$	0	0
$721$	36.7846	1.36993
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.44949	−0.0909718
$726$	0	0
$727$	−23.4596	−0.870069	−0.435035	−	0.900414i	$-0.643264\pi$
$727$	−23.4596	−0.870069	−0.435035	+	0.900414i	$0.643264\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7.60770	0.281381
$732$	0	0
$733$	20.9358	0.773281	0.386641	−	0.922230i	$-0.373635\pi$
$733$	20.9358	0.773281	0.386641	+	0.922230i	$0.373635\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−29.3205	−1.08003
$738$	0	0
$739$	−2.98076	−0.109649	−0.0548246	−	0.998496i	$-0.517460\pi$
$739$	−2.98076	−0.109649	−0.0548246	+	0.998496i	$0.517460\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−15.1774	−0.556805	−0.278403	−	0.960464i	$-0.589805\pi$
$743$	−15.1774	−0.556805	−0.278403	+	0.960464i	$0.589805\pi$
$744$	0	0
$745$	−47.5692	−1.74280
$746$	0	0
$747$	0	0
$748$	0	0
$749$	71.8203	2.62426
$750$	0	0
$751$	−4.14110	−0.151111	−0.0755555	−	0.997142i	$-0.524073\pi$
$751$	−4.14110	−0.151111	−0.0755555	+	0.997142i	$0.524073\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−23.3205	−0.848720
$756$	0	0
$757$	30.7338	1.11704	0.558519	−	0.829492i	$-0.311370\pi$
$757$	30.7338	1.11704	0.558519	+	0.829492i	$0.311370\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.6410	−1.25574	−0.627868	−	0.778320i	$-0.716072\pi$
$761$	−34.6410	−1.25574	−0.627868	+	0.778320i	$0.716072\pi$
$762$	0	0
$763$	72.1051	2.61038
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.75410	−0.0994447
$768$	0	0
$769$	−2.39230	−0.0862687	−0.0431344	−	0.999069i	$-0.513734\pi$
$769$	−2.39230	−0.0862687	−0.0431344	+	0.999069i	$0.513734\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20.7327	−0.745704	−0.372852	−	0.927891i	$-0.621620\pi$
$773$	−20.7327	−0.745704	−0.372852	+	0.927891i	$0.621620\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	32.7846	1.17463
$780$	0	0
$781$	8.48528	0.303627
$782$	0	0
$783$	0	0
$784$	0	0
$785$	24.9282	0.889726
$786$	0	0
$787$	−35.6603	−1.27115	−0.635575	−	0.772039i	$-0.719237\pi$
$787$	−35.6603	−1.27115	−0.635575	+	0.772039i	$0.719237\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	49.9507	1.77604
$792$	0	0
$793$	2.00000	0.0710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.5013	1.68258	0.841290	−	0.540584i	$-0.181797\pi$
$797$	47.5013	1.68258	0.841290	+	0.540584i	$0.181797\pi$
$798$	0	0
$799$	33.9411	1.20075
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−12.0000	−0.423471
$804$	0	0
$805$	−16.9706	−0.598134
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−17.0718	−0.600212	−0.300106	−	0.953906i	$-0.597022\pi$
$809$	−17.0718	−0.600212	−0.300106	+	0.953906i	$0.597022\pi$
$810$	0	0
$811$	46.9808	1.64972	0.824859	−	0.565339i	$-0.191255\pi$
$811$	46.9808	1.64972	0.824859	+	0.565339i	$0.191255\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	40.9850	1.43564
$816$	0	0
$817$	−10.3923	−0.363581
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.2571	−0.986178	−0.493089	−	0.869979i	$-0.664132\pi$
$821$	−28.2571	−0.986178	−0.493089	+	0.869979i	$0.664132\pi$
$822$	0	0
$823$	15.7322	0.548391	0.274195	−	0.961674i	$-0.411589\pi$
$823$	15.7322	0.548391	0.274195	+	0.961674i	$0.411589\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.6603	−0.822748	−0.411374	−	0.911467i	$-0.634951\pi$
$827$	−23.6603	−0.822748	−0.411374	+	0.911467i	$0.634951\pi$
$828$	0	0
$829$	3.00429	0.104343	0.0521717	−	0.998638i	$-0.483386\pi$
$829$	3.00429	0.104343	0.0521717	+	0.998638i	$0.483386\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	27.4641	0.951575
$834$	0	0
$835$	45.9615	1.59056
$836$	0	0
$837$	0	0
$838$	0	0
$839$	38.3596	1.32432	0.662161	−	0.749362i	$-0.269640\pi$
$839$	38.3596	1.32432	0.662161	+	0.749362i	$0.269640\pi$
$840$	0	0
$841$	−23.0000	−0.793103
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−31.4916	−1.08335
$846$	0	0
$847$	44.0165	1.51242
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.46410	−0.324425
$852$	0	0
$853$	−13.9663	−0.478196	−0.239098	−	0.970995i	$-0.576852\pi$
$853$	−13.9663	−0.478196	−0.239098	+	0.970995i	$0.576852\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	25.8564	0.883238	0.441619	−	0.897203i	$-0.354404\pi$
$857$	25.8564	0.883238	0.441619	+	0.897203i	$0.354404\pi$
$858$	0	0
$859$	20.4449	0.697570	0.348785	−	0.937203i	$-0.386594\pi$
$859$	20.4449	0.697570	0.348785	+	0.937203i	$0.386594\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−2.62536	−0.0893681	−0.0446841	−	0.999001i	$-0.514228\pi$
$863$	−2.62536	−0.0893681	−0.0446841	+	0.999001i	$0.514228\pi$
$864$	0	0
$865$	−26.7846	−0.910704
$866$	0	0
$867$	0	0
$868$	0	0
$869$	19.5959	0.664746
$870$	0	0
$871$	2.34795	0.0795574
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−37.8564	−1.27978
$876$	0	0
$877$	22.8033	0.770012	0.385006	−	0.922914i	$-0.374199\pi$
$877$	22.8033	0.770012	0.385006	+	0.922914i	$0.374199\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	31.8564	1.07327	0.536635	−	0.843815i	$-0.319695\pi$
$881$	31.8564	1.07327	0.536635	+	0.843815i	$0.319695\pi$
$882$	0	0
$883$	57.1244	1.92239	0.961194	−	0.275874i	$-0.0889672\pi$
$883$	57.1244	1.92239	0.961194	+	0.275874i	$0.0889672\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.1774	0.509608	0.254804	−	0.966993i	$-0.417989\pi$
$887$	15.1774	0.509608	0.254804	+	0.966993i	$0.417989\pi$
$888$	0	0
$889$	21.8564	0.733040
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−46.3644	−1.55153
$894$	0	0
$895$	−5.37945	−0.179815
$896$	0	0
$897$	0	0
$898$	0	0
$899$	13.8564	0.462137
$900$	0	0
$901$	37.8792	1.26194
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−55.8564	−1.85673
$906$	0	0
$907$	29.9090	0.993111	0.496555	−	0.868005i	$-0.334598\pi$
$907$	29.9090	0.993111	0.496555	+	0.868005i	$0.334598\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.1432	−0.799899	−0.399949	−	0.916537i	$-0.630972\pi$
$911$	−24.1432	−0.799899	−0.399949	+	0.916537i	$0.630972\pi$
$912$	0	0
$913$	−10.3923	−0.343935
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.31268	0.0433484
$918$	0	0
$919$	18.3576	0.605560	0.302780	−	0.953060i	$-0.402085\pi$
$919$	18.3576	0.605560	0.302780	+	0.953060i	$0.402085\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.679492	−0.0223657
$924$	0	0
$925$	5.27792	0.173537
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−3.46410	−0.113653	−0.0568267	−	0.998384i	$-0.518098\pi$
$929$	−3.46410	−0.113653	−0.0568267	+	0.998384i	$0.518098\pi$
$930$	0	0
$931$	−37.5167	−1.22956
$932$	0	0
$933$	0	0
$934$	0	0
$935$	40.1528	1.31314
$936$	0	0
$937$	9.17691	0.299797	0.149898	−	0.988701i	$-0.452105\pi$
$937$	9.17691	0.299797	0.149898	+	0.988701i	$0.452105\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	26.5927	0.866896	0.433448	−	0.901179i	$-0.357297\pi$
$941$	26.5927	0.866896	0.433448	+	0.901179i	$0.357297\pi$
$942$	0	0
$943$	12.4233	0.404559
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−19.2679	−0.626124	−0.313062	−	0.949733i	$-0.601355\pi$
$947$	−19.2679	−0.626124	−0.313062	+	0.949733i	$0.601355\pi$
$948$	0	0
$949$	0.960947	0.0311936
$950$	0	0
$951$	0	0
$952$	0	0
$953$	53.5692	1.73528	0.867639	−	0.497195i	$-0.165637\pi$
$953$	53.5692	1.73528	0.867639	+	0.497195i	$0.165637\pi$
$954$	0	0
$955$	−41.5692	−1.34515
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−46.3644	−1.49719
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−45.0518	−1.45027
$966$	0	0
$967$	−27.0459	−0.869738	−0.434869	−	0.900494i	$-0.643205\pi$
$967$	−27.0459	−0.869738	−0.434869	+	0.900494i	$0.643205\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.3731	1.26354	0.631771	−	0.775155i	$-0.282328\pi$
$971$	39.3731	1.26354	0.631771	+	0.775155i	$0.282328\pi$
$972$	0	0
$973$	39.3949	1.26294
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−44.5359	−1.42483	−0.712415	−	0.701759i	$-0.752399\pi$
$977$	−44.5359	−1.42483	−0.712415	+	0.701759i	$0.752399\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−52.7048	−1.68102	−0.840512	−	0.541793i	$-0.817745\pi$
$983$	−52.7048	−1.68102	−0.840512	+	0.541793i	$0.817745\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−3.93803	−0.125222
$990$	0	0
$991$	61.8193	1.96375	0.981877	−	0.189521i	$-0.0606936\pi$
$991$	61.8193	1.96375	0.981877	+	0.189521i	$0.0606936\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−38.5359	−1.22167
$996$	0	0
$997$	−21.4906	−0.680614	−0.340307	−	0.940314i	$-0.610531\pi$
$997$	−21.4906	−0.680614	−0.340307	+	0.940314i	$0.610531\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9216.2.a.bk.1.4		4
3.2	odd	2		1024.2.a.g.1.3		4
4.3	odd	2		9216.2.a.bb.1.3		4
8.3	odd	2	inner	9216.2.a.bk.1.1		4
8.5	even	2		9216.2.a.bb.1.2		4
12.11	even	2		1024.2.a.j.1.1		4
24.5	odd	2		1024.2.a.j.1.2		4
24.11	even	2		1024.2.a.g.1.4		4
32.3	odd	8		2304.2.k.k.577.2		8
32.5	even	8		2304.2.k.f.1729.4		8
32.11	odd	8		2304.2.k.k.1729.1		8
32.13	even	8		2304.2.k.f.577.3		8
32.19	odd	8		2304.2.k.f.577.4		8
32.21	even	8		2304.2.k.k.1729.2		8
32.27	odd	8		2304.2.k.f.1729.3		8
32.29	even	8		2304.2.k.k.577.1		8
48.5	odd	4		1024.2.b.h.513.3		8
48.11	even	4		1024.2.b.h.513.5		8
48.29	odd	4		1024.2.b.h.513.6		8
48.35	even	4		1024.2.b.h.513.4		8
96.5	odd	8		256.2.e.a.193.2	yes	8
96.11	even	8		256.2.e.b.193.2	yes	8
96.29	odd	8		256.2.e.b.65.3	yes	8
96.35	even	8		256.2.e.b.65.2	yes	8
96.53	odd	8		256.2.e.b.193.3	yes	8
96.59	even	8		256.2.e.a.193.3	yes	8
96.77	odd	8		256.2.e.a.65.2	✓	8
96.83	even	8		256.2.e.a.65.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
256.2.e.a.65.2	✓	8	96.77	odd	8
256.2.e.a.65.3	yes	8	96.83	even	8
256.2.e.a.193.2	yes	8	96.5	odd	8
256.2.e.a.193.3	yes	8	96.59	even	8
256.2.e.b.65.2	yes	8	96.35	even	8
256.2.e.b.65.3	yes	8	96.29	odd	8
256.2.e.b.193.2	yes	8	96.11	even	8
256.2.e.b.193.3	yes	8	96.53	odd	8
1024.2.a.g.1.3		4	3.2	odd	2
1024.2.a.g.1.4		4	24.11	even	2
1024.2.a.j.1.1		4	12.11	even	2
1024.2.a.j.1.2		4	24.5	odd	2
1024.2.b.h.513.3		8	48.5	odd	4
1024.2.b.h.513.4		8	48.35	even	4
1024.2.b.h.513.5		8	48.11	even	4
1024.2.b.h.513.6		8	48.29	odd	4
2304.2.k.f.577.3		8	32.13	even	8
2304.2.k.f.577.4		8	32.19	odd	8
2304.2.k.f.1729.3		8	32.27	odd	8
2304.2.k.f.1729.4		8	32.5	even	8
2304.2.k.k.577.1		8	32.29	even	8
2304.2.k.k.577.2		8	32.3	odd	8
2304.2.k.k.1729.1		8	32.11	odd	8
2304.2.k.k.1729.2		8	32.21	even	8
9216.2.a.bb.1.2		4	8.5	even	2
9216.2.a.bb.1.3		4	4.3	odd	2
9216.2.a.bk.1.1		4	8.3	odd	2	inner
9216.2.a.bk.1.4		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9216 = 2^{10} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9216.a (trivial)

\(f(q)\)	\(=\)	\(q+2.44949 q^{5} +3.86370 q^{7} +4.73205 q^{11} -0.378937 q^{13} +3.46410 q^{17} -4.73205 q^{19} -1.79315 q^{23} +1.00000 q^{25} -2.44949 q^{29} -5.65685 q^{31} +9.46410 q^{35} +5.27792 q^{37} -6.92820 q^{41} +2.19615 q^{43} +9.79796 q^{47} +7.92820 q^{49} +10.9348 q^{53} +11.5911 q^{55} +7.26795 q^{59} -5.27792 q^{61} -0.928203 q^{65} -6.19615 q^{67} +1.79315 q^{71} -2.53590 q^{73} +18.2832 q^{77} +4.14110 q^{79} -2.19615 q^{83} +8.48528 q^{85} +2.53590 q^{89} -1.46410 q^{91} -11.5911 q^{95} +3.46410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{11} - 12 q^{19} + 4 q^{25} + 24 q^{35} - 12 q^{43} + 4 q^{49} + 36 q^{59} + 24 q^{65} - 4 q^{67} - 24 q^{73} + 12 q^{83} + 24 q^{89} + 8 q^{91}+O(q^{100}) \) 4 * q + 12 * q^11 - 12 * q^19 + 4 * q^25 + 24 * q^35 - 12 * q^43 + 4 * q^49 + 36 * q^59 + 24 * q^65 - 4 * q^67 - 24 * q^73 + 12 * q^83 + 24 * q^89 + 8 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more