LMFDB - Embedded newform 9576.2.a.bf.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9576,2,Mod(1,9576)] 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("9576.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9576, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9576.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,-2,0,0,0,0,0,8,0,0,0,-10] 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:	$76.4647449756$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	9576.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.732051 q^{5} -1.00000 q^{7} +3.46410 q^{11} +4.00000 q^{13} -3.26795 q^{17} +1.00000 q^{19} +0.535898 q^{23} -4.46410 q^{25} +2.73205 q^{29} -3.46410 q^{31} -0.732051 q^{35} -7.46410 q^{37} -11.4641 q^{41} -8.92820 q^{43} -5.66025 q^{47} +1.00000 q^{49} -8.19615 q^{53} +2.53590 q^{55} -8.00000 q^{59} +4.53590 q^{61} +2.92820 q^{65} -4.00000 q^{67} +6.19615 q^{71} +0.535898 q^{73} -3.46410 q^{77} -5.46410 q^{79} +0.196152 q^{83} -2.39230 q^{85} +10.3923 q^{89} -4.00000 q^{91} +0.732051 q^{95} +6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7} + 8 q^{13} - 10 q^{17} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{35} - 8 q^{37} - 16 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} - 6 q^{53} + 12 q^{55} - 16 q^{59} + 16 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^13 - 10 * q^17 + 2 * q^19 + 8 * q^23 - 2 * q^25 + 2 * q^29 + 2 * q^35 - 8 * q^37 - 16 * q^41 - 4 * q^43 + 6 * q^47 + 2 * q^49 - 6 * q^53 + 12 * q^55 - 16 * q^59 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 2 * q^71 + 8 * q^73 - 4 * q^79 - 10 * q^83 + 16 * q^85 - 8 * q^91 - 2 * q^95 + 12 * 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
$5$	0.732051	0.327383	0.163692	−	0.986512i	$-0.447660\pi$
$5$	0.732051	0.327383	0.163692	+	0.986512i	$0.447660\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.26795	−0.792594	−0.396297	−	0.918122i	$-0.629705\pi$
$17$	−3.26795	−0.792594	−0.396297	+	0.918122i	$0.629705\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0.535898	0.111743	0.0558713	−	0.998438i	$-0.482206\pi$
$23$	0.535898	0.111743	0.0558713	+	0.998438i	$0.482206\pi$
$24$	0	0
$25$	−4.46410	−0.892820
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.73205	0.507329	0.253665	−	0.967292i	$-0.418364\pi$
$29$	2.73205	0.507329	0.253665	+	0.967292i	$0.418364\pi$
$30$	0	0
$31$	−3.46410	−0.622171	−0.311086	−	0.950382i	$-0.600693\pi$
$31$	−3.46410	−0.622171	−0.311086	+	0.950382i	$0.600693\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	−7.46410	−1.22709	−0.613545	−	0.789659i	$-0.710257\pi$
$37$	−7.46410	−1.22709	−0.613545	+	0.789659i	$0.710257\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−11.4641	−1.79039	−0.895196	−	0.445673i	$-0.852964\pi$
$41$	−11.4641	−1.79039	−0.895196	+	0.445673i	$0.852964\pi$
$42$	0	0
$43$	−8.92820	−1.36154	−0.680769	−	0.732498i	$-0.738354\pi$
$43$	−8.92820	−1.36154	−0.680769	+	0.732498i	$0.738354\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.66025	−0.825633	−0.412816	−	0.910814i	$-0.635455\pi$
$47$	−5.66025	−0.825633	−0.412816	+	0.910814i	$0.635455\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.19615	−1.12583	−0.562914	−	0.826515i	$-0.690320\pi$
$53$	−8.19615	−1.12583	−0.562914	+	0.826515i	$0.690320\pi$
$54$	0	0
$55$	2.53590	0.341940
$56$	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
$61$	4.53590	0.580762	0.290381	−	0.956911i	$-0.406218\pi$
$61$	4.53590	0.580762	0.290381	+	0.956911i	$0.406218\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.92820	0.363199
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.19615	0.735348	0.367674	−	0.929955i	$-0.380154\pi$
$71$	6.19615	0.735348	0.367674	+	0.929955i	$0.380154\pi$
$72$	0	0
$73$	0.535898	0.0627222	0.0313611	−	0.999508i	$-0.490016\pi$
$73$	0.535898	0.0627222	0.0313611	+	0.999508i	$0.490016\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−5.46410	−0.614759	−0.307380	−	0.951587i	$-0.599452\pi$
$79$	−5.46410	−0.614759	−0.307380	+	0.951587i	$0.599452\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0.196152	0.0215305	0.0107653	−	0.999942i	$-0.496573\pi$
$83$	0.196152	0.0215305	0.0107653	+	0.999942i	$0.496573\pi$
$84$	0	0
$85$	−2.39230	−0.259482
$86$	0	0
$87$	0	0
$88$	0	0
$89$	10.3923	1.10158	0.550791	−	0.834643i	$-0.314326\pi$
$89$	10.3923	1.10158	0.550791	+	0.834643i	$0.314326\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0.732051	0.0751068
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.5885	−1.45161	−0.725803	−	0.687903i	$-0.758532\pi$
$101$	−14.5885	−1.45161	−0.725803	+	0.687903i	$0.758532\pi$
$102$	0	0
$103$	6.92820	0.682656	0.341328	−	0.939944i	$-0.389123\pi$
$103$	6.92820	0.682656	0.341328	+	0.939944i	$0.389123\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.19615	−0.212310	−0.106155	−	0.994350i	$-0.533854\pi$
$107$	−2.19615	−0.212310	−0.106155	+	0.994350i	$0.533854\pi$
$108$	0	0
$109$	15.8564	1.51877	0.759384	−	0.650643i	$-0.225500\pi$
$109$	15.8564	1.51877	0.759384	+	0.650643i	$0.225500\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.73205	0.257010	0.128505	−	0.991709i	$-0.458982\pi$
$113$	2.73205	0.257010	0.128505	+	0.991709i	$0.458982\pi$
$114$	0	0
$115$	0.392305	0.0365826
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.26795	0.299572
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−6.92820	−0.619677
$126$	0	0
$127$	2.92820	0.259836	0.129918	−	0.991525i	$-0.458529\pi$
$127$	2.92820	0.259836	0.129918	+	0.991525i	$0.458529\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−8.19615	−0.716101	−0.358051	−	0.933702i	$-0.616558\pi$
$131$	−8.19615	−0.716101	−0.358051	+	0.933702i	$0.616558\pi$
$132$	0	0
$133$	−1.00000	−0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	10.9282	0.933659	0.466830	−	0.884347i	$-0.345396\pi$
$137$	10.9282	0.933659	0.466830	+	0.884347i	$0.345396\pi$
$138$	0	0
$139$	2.53590	0.215092	0.107546	−	0.994200i	$-0.465701\pi$
$139$	2.53590	0.215092	0.107546	+	0.994200i	$0.465701\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	13.8564	1.15873
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.46410	0.775329	0.387665	−	0.921800i	$-0.373282\pi$
$149$	9.46410	0.775329	0.387665	+	0.921800i	$0.373282\pi$
$150$	0	0
$151$	−21.4641	−1.74672	−0.873362	−	0.487072i	$-0.838065\pi$
$151$	−21.4641	−1.74672	−0.873362	+	0.487072i	$0.838065\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.53590	−0.203688
$156$	0	0
$157$	14.3923	1.14863	0.574315	−	0.818634i	$-0.305268\pi$
$157$	14.3923	1.14863	0.574315	+	0.818634i	$0.305268\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.535898	−0.0422347
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.39230	0.339887	0.169943	−	0.985454i	$-0.445642\pi$
$167$	4.39230	0.339887	0.169943	+	0.985454i	$0.445642\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.3923	0.790112	0.395056	−	0.918657i	$-0.370725\pi$
$173$	10.3923	0.790112	0.395056	+	0.918657i	$0.370725\pi$
$174$	0	0
$175$	4.46410	0.337454
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−25.5167	−1.90720	−0.953602	−	0.301069i	$-0.902657\pi$
$179$	−25.5167	−1.90720	−0.953602	+	0.301069i	$0.902657\pi$
$180$	0	0
$181$	−23.8564	−1.77323	−0.886616	−	0.462506i	$-0.846951\pi$
$181$	−23.8564	−1.77323	−0.886616	+	0.462506i	$0.846951\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.46410	−0.401729
$186$	0	0
$187$	−11.3205	−0.827838
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.3923	1.33082	0.665410	−	0.746478i	$-0.268257\pi$
$191$	18.3923	1.33082	0.665410	+	0.746478i	$0.268257\pi$
$192$	0	0
$193$	17.3205	1.24676	0.623379	−	0.781920i	$-0.285760\pi$
$193$	17.3205	1.24676	0.623379	+	0.781920i	$0.285760\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.92820	−0.493614	−0.246807	−	0.969065i	$-0.579381\pi$
$197$	−6.92820	−0.493614	−0.246807	+	0.969065i	$0.579381\pi$
$198$	0	0
$199$	−16.7846	−1.18983	−0.594915	−	0.803789i	$-0.702814\pi$
$199$	−16.7846	−1.18983	−0.594915	+	0.803789i	$0.702814\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.73205	−0.191752
$204$	0	0
$205$	−8.39230	−0.586144
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.46410	0.239617
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.53590	−0.445745
$216$	0	0
$217$	3.46410	0.235159
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−13.0718	−0.879304
$222$	0	0
$223$	13.3205	0.892007	0.446004	−	0.895031i	$-0.352847\pi$
$223$	13.3205	0.892007	0.446004	+	0.895031i	$0.352847\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.4641	−0.893644	−0.446822	−	0.894623i	$-0.647444\pi$
$227$	−13.4641	−0.893644	−0.446822	+	0.894623i	$0.647444\pi$
$228$	0	0
$229$	0.928203	0.0613374	0.0306687	−	0.999530i	$-0.490236\pi$
$229$	0.928203	0.0613374	0.0306687	+	0.999530i	$0.490236\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−14.5359	−0.952278	−0.476139	−	0.879370i	$-0.657964\pi$
$233$	−14.5359	−0.952278	−0.476139	+	0.879370i	$0.657964\pi$
$234$	0	0
$235$	−4.14359	−0.270298
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−19.8564	−1.28440	−0.642202	−	0.766535i	$-0.721979\pi$
$239$	−19.8564	−1.28440	−0.642202	+	0.766535i	$0.721979\pi$
$240$	0	0
$241$	15.0718	0.970860	0.485430	−	0.874276i	$-0.338663\pi$
$241$	15.0718	0.970860	0.485430	+	0.874276i	$0.338663\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0.732051	0.0467690
$246$	0	0
$247$	4.00000	0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.7321	−0.677401	−0.338701	−	0.940894i	$-0.609987\pi$
$251$	−10.7321	−0.677401	−0.338701	+	0.940894i	$0.609987\pi$
$252$	0	0
$253$	1.85641	0.116711
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.143594	0.00895712	0.00447856	−	0.999990i	$-0.498574\pi$
$257$	0.143594	0.00895712	0.00447856	+	0.999990i	$0.498574\pi$
$258$	0	0
$259$	7.46410	0.463797
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−16.9282	−1.04384	−0.521919	−	0.852995i	$-0.674784\pi$
$263$	−16.9282	−1.04384	−0.521919	+	0.852995i	$0.674784\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−12.9282	−0.788246	−0.394123	−	0.919058i	$-0.628952\pi$
$269$	−12.9282	−0.788246	−0.394123	+	0.919058i	$0.628952\pi$
$270$	0	0
$271$	23.3205	1.41662	0.708310	−	0.705902i	$-0.249458\pi$
$271$	23.3205	1.41662	0.708310	+	0.705902i	$0.249458\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.4641	−0.932520
$276$	0	0
$277$	22.2487	1.33680	0.668398	−	0.743804i	$-0.266980\pi$
$277$	22.2487	1.33680	0.668398	+	0.743804i	$0.266980\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−18.7321	−1.11746	−0.558730	−	0.829349i	$-0.688711\pi$
$281$	−18.7321	−1.11746	−0.558730	+	0.829349i	$0.688711\pi$
$282$	0	0
$283$	21.8564	1.29923	0.649614	−	0.760264i	$-0.274930\pi$
$283$	21.8564	1.29923	0.649614	+	0.760264i	$0.274930\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.4641	0.676705
$288$	0	0
$289$	−6.32051	−0.371795
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−20.5359	−1.19972	−0.599860	−	0.800105i	$-0.704777\pi$
$293$	−20.5359	−1.19972	−0.599860	+	0.800105i	$0.704777\pi$
$294$	0	0
$295$	−5.85641	−0.340973
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.14359	0.123967
$300$	0	0
$301$	8.92820	0.514613
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.32051	0.190132
$306$	0	0
$307$	−21.3205	−1.21683	−0.608413	−	0.793621i	$-0.708193\pi$
$307$	−21.3205	−1.21683	−0.608413	+	0.793621i	$0.708193\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−25.2679	−1.43281	−0.716407	−	0.697683i	$-0.754215\pi$
$311$	−25.2679	−1.43281	−0.716407	+	0.697683i	$0.754215\pi$
$312$	0	0
$313$	21.3205	1.20511	0.602553	−	0.798079i	$-0.294150\pi$
$313$	21.3205	1.20511	0.602553	+	0.798079i	$0.294150\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.7321	0.827434	0.413717	−	0.910405i	$-0.364230\pi$
$317$	14.7321	0.827434	0.413717	+	0.910405i	$0.364230\pi$
$318$	0	0
$319$	9.46410	0.529888
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.26795	−0.181834
$324$	0	0
$325$	−17.8564	−0.990495
$326$	0	0
$327$	0	0
$328$	0	0
$329$	5.66025	0.312060
$330$	0	0
$331$	−16.3923	−0.901003	−0.450501	−	0.892776i	$-0.648755\pi$
$331$	−16.3923	−0.901003	−0.450501	+	0.892776i	$0.648755\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.92820	−0.159985
$336$	0	0
$337$	16.9282	0.922138	0.461069	−	0.887364i	$-0.347466\pi$
$337$	16.9282	0.922138	0.461069	+	0.887364i	$0.347466\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−8.14359	−0.437171	−0.218586	−	0.975818i	$-0.570144\pi$
$347$	−8.14359	−0.437171	−0.218586	+	0.975818i	$0.570144\pi$
$348$	0	0
$349$	−3.85641	−0.206429	−0.103214	−	0.994659i	$-0.532913\pi$
$349$	−3.85641	−0.206429	−0.103214	+	0.994659i	$0.532913\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	18.1962	0.968483	0.484242	−	0.874934i	$-0.339096\pi$
$353$	18.1962	0.968483	0.484242	+	0.874934i	$0.339096\pi$
$354$	0	0
$355$	4.53590	0.240740
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.4641	−0.605052	−0.302526	−	0.953141i	$-0.597830\pi$
$359$	−11.4641	−0.605052	−0.302526	+	0.953141i	$0.597830\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0.392305	0.0205342
$366$	0	0
$367$	1.85641	0.0969036	0.0484518	−	0.998826i	$-0.484571\pi$
$367$	1.85641	0.0969036	0.0484518	+	0.998826i	$0.484571\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	8.19615	0.425523
$372$	0	0
$373$	−29.7128	−1.53847	−0.769236	−	0.638965i	$-0.779363\pi$
$373$	−29.7128	−1.53847	−0.769236	+	0.638965i	$0.779363\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.9282	0.562831
$378$	0	0
$379$	18.2487	0.937373	0.468687	−	0.883364i	$-0.344727\pi$
$379$	18.2487	0.937373	0.468687	+	0.883364i	$0.344727\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−14.5359	−0.742750	−0.371375	−	0.928483i	$-0.621114\pi$
$383$	−14.5359	−0.742750	−0.371375	+	0.928483i	$0.621114\pi$
$384$	0	0
$385$	−2.53590	−0.129241
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.46410	−0.479849	−0.239924	−	0.970792i	$-0.577123\pi$
$389$	−9.46410	−0.479849	−0.239924	+	0.970792i	$0.577123\pi$
$390$	0	0
$391$	−1.75129	−0.0885665
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.0526	−1.10125	−0.550626	−	0.834752i	$-0.685611\pi$
$401$	−22.0526	−1.10125	−0.550626	+	0.834752i	$0.685611\pi$
$402$	0	0
$403$	−13.8564	−0.690237
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−25.8564	−1.28165
$408$	0	0
$409$	−19.7128	−0.974736	−0.487368	−	0.873197i	$-0.662043\pi$
$409$	−19.7128	−0.974736	−0.487368	+	0.873197i	$0.662043\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	0.143594	0.00704873
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−0.196152	−0.00958267	−0.00479134	−	0.999989i	$-0.501525\pi$
$419$	−0.196152	−0.00958267	−0.00479134	+	0.999989i	$0.501525\pi$
$420$	0	0
$421$	9.32051	0.454254	0.227127	−	0.973865i	$-0.427067\pi$
$421$	9.32051	0.454254	0.227127	+	0.973865i	$0.427067\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	14.5885	0.707644
$426$	0	0
$427$	−4.53590	−0.219508
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−16.0526	−0.773225	−0.386612	−	0.922242i	$-0.626355\pi$
$431$	−16.0526	−0.773225	−0.386612	+	0.922242i	$0.626355\pi$
$432$	0	0
$433$	8.92820	0.429062	0.214531	−	0.976717i	$-0.431178\pi$
$433$	8.92820	0.429062	0.214531	+	0.976717i	$0.431178\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.535898	0.0256355
$438$	0	0
$439$	4.53590	0.216487	0.108243	−	0.994124i	$-0.465477\pi$
$439$	4.53590	0.216487	0.108243	+	0.994124i	$0.465477\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−7.85641	−0.373269	−0.186635	−	0.982429i	$-0.559758\pi$
$443$	−7.85641	−0.373269	−0.186635	+	0.982429i	$0.559758\pi$
$444$	0	0
$445$	7.60770	0.360639
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.41154	−0.161001	−0.0805003	−	0.996755i	$-0.525652\pi$
$449$	−3.41154	−0.161001	−0.0805003	+	0.996755i	$0.525652\pi$
$450$	0	0
$451$	−39.7128	−1.87000
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.92820	−0.137276
$456$	0	0
$457$	12.3923	0.579688	0.289844	−	0.957074i	$-0.406397\pi$
$457$	12.3923	0.579688	0.289844	+	0.957074i	$0.406397\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−41.5167	−1.93362	−0.966812	−	0.255490i	$-0.917763\pi$
$461$	−41.5167	−1.93362	−0.966812	+	0.255490i	$0.917763\pi$
$462$	0	0
$463$	−13.8564	−0.643962	−0.321981	−	0.946746i	$-0.604349\pi$
$463$	−13.8564	−0.643962	−0.321981	+	0.946746i	$0.604349\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.73205	0.311522	0.155761	−	0.987795i	$-0.450217\pi$
$467$	6.73205	0.311522	0.155761	+	0.987795i	$0.450217\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−30.9282	−1.42208
$474$	0	0
$475$	−4.46410	−0.204827
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.2679	−1.33729	−0.668643	−	0.743583i	$-0.733125\pi$
$479$	−29.2679	−1.33729	−0.668643	+	0.743583i	$0.733125\pi$
$480$	0	0
$481$	−29.8564	−1.36133
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4.39230	0.199444
$486$	0	0
$487$	−6.14359	−0.278393	−0.139196	−	0.990265i	$-0.544452\pi$
$487$	−6.14359	−0.278393	−0.139196	+	0.990265i	$0.544452\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.85641	0.354555	0.177277	−	0.984161i	$-0.443271\pi$
$491$	7.85641	0.354555	0.177277	+	0.984161i	$0.443271\pi$
$492$	0	0
$493$	−8.92820	−0.402106
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.19615	−0.277935
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	5.66025	0.252378	0.126189	−	0.992006i	$-0.459725\pi$
$503$	5.66025	0.252378	0.126189	+	0.992006i	$0.459725\pi$
$504$	0	0
$505$	−10.6795	−0.475231
$506$	0	0
$507$	0	0
$508$	0	0
$509$	14.7846	0.655316	0.327658	−	0.944796i	$-0.393741\pi$
$509$	14.7846	0.655316	0.327658	+	0.944796i	$0.393741\pi$
$510$	0	0
$511$	−0.535898	−0.0237067
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.07180	0.223490
$516$	0	0
$517$	−19.6077	−0.862345
$518$	0	0
$519$	0	0
$520$	0	0
$521$	17.7128	0.776012	0.388006	−	0.921657i	$-0.373164\pi$
$521$	17.7128	0.776012	0.388006	+	0.921657i	$0.373164\pi$
$522$	0	0
$523$	−0.535898	−0.0234332	−0.0117166	−	0.999931i	$-0.503730\pi$
$523$	−0.535898	−0.0234332	−0.0117166	+	0.999931i	$0.503730\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.3205	0.493129
$528$	0	0
$529$	−22.7128	−0.987514
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−45.8564	−1.98626
$534$	0	0
$535$	−1.60770	−0.0695067
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.46410	0.149209
$540$	0	0
$541$	31.8564	1.36961	0.684807	−	0.728725i	$-0.259887\pi$
$541$	31.8564	1.36961	0.684807	+	0.728725i	$0.259887\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11.6077	0.497219
$546$	0	0
$547$	40.1051	1.71477	0.857386	−	0.514675i	$-0.172087\pi$
$547$	40.1051	1.71477	0.857386	+	0.514675i	$0.172087\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.73205	0.116389
$552$	0	0
$553$	5.46410	0.232357
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.4641	0.570492	0.285246	−	0.958454i	$-0.407925\pi$
$557$	13.4641	0.570492	0.285246	+	0.958454i	$0.407925\pi$
$558$	0	0
$559$	−35.7128	−1.51049
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−14.5359	−0.612615	−0.306308	−	0.951933i	$-0.599094\pi$
$563$	−14.5359	−0.612615	−0.306308	+	0.951933i	$0.599094\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.0526	0.924491	0.462246	−	0.886752i	$-0.347044\pi$
$569$	22.0526	0.924491	0.462246	+	0.886752i	$0.347044\pi$
$570$	0	0
$571$	−17.0718	−0.714432	−0.357216	−	0.934022i	$-0.616274\pi$
$571$	−17.0718	−0.714432	−0.357216	+	0.934022i	$0.616274\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−2.39230	−0.0997660
$576$	0	0
$577$	45.3205	1.88672	0.943359	−	0.331775i	$-0.107647\pi$
$577$	45.3205	1.88672	0.943359	+	0.331775i	$0.107647\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−0.196152	−0.00813777
$582$	0	0
$583$	−28.3923	−1.17589
$584$	0	0
$585$	0	0
$586$	0	0
$587$	22.0526	0.910207	0.455103	−	0.890439i	$-0.349602\pi$
$587$	22.0526	0.910207	0.455103	+	0.890439i	$0.349602\pi$
$588$	0	0
$589$	−3.46410	−0.142736
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−30.9808	−1.27223	−0.636114	−	0.771595i	$-0.719459\pi$
$593$	−30.9808	−1.27223	−0.636114	+	0.771595i	$0.719459\pi$
$594$	0	0
$595$	2.39230	0.0980749
$596$	0	0
$597$	0	0
$598$	0	0
$599$	20.0526	0.819325	0.409663	−	0.912237i	$-0.365646\pi$
$599$	20.0526	0.819325	0.409663	+	0.912237i	$0.365646\pi$
$600$	0	0
$601$	40.6410	1.65778	0.828891	−	0.559410i	$-0.188972\pi$
$601$	40.6410	1.65778	0.828891	+	0.559410i	$0.188972\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.732051	0.0297621
$606$	0	0
$607$	−24.7846	−1.00598	−0.502988	−	0.864293i	$-0.667766\pi$
$607$	−24.7846	−1.00598	−0.502988	+	0.864293i	$0.667766\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−22.6410	−0.915957
$612$	0	0
$613$	7.60770	0.307272	0.153636	−	0.988128i	$-0.450902\pi$
$613$	7.60770	0.307272	0.153636	+	0.988128i	$0.450902\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.7128	−1.43774	−0.718872	−	0.695143i	$-0.755341\pi$
$617$	−35.7128	−1.43774	−0.718872	+	0.695143i	$0.755341\pi$
$618$	0	0
$619$	−13.4641	−0.541168	−0.270584	−	0.962696i	$-0.587217\pi$
$619$	−13.4641	−0.541168	−0.270584	+	0.962696i	$0.587217\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−10.3923	−0.416359
$624$	0	0
$625$	17.2487	0.689948
$626$	0	0
$627$	0	0
$628$	0	0
$629$	24.3923	0.972585
$630$	0	0
$631$	−20.9282	−0.833139	−0.416569	−	0.909104i	$-0.636768\pi$
$631$	−20.9282	−0.833139	−0.416569	+	0.909104i	$0.636768\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	2.14359	0.0850659
$636$	0	0
$637$	4.00000	0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	7.12436	0.281395	0.140698	−	0.990053i	$-0.455065\pi$
$641$	7.12436	0.281395	0.140698	+	0.990053i	$0.455065\pi$
$642$	0	0
$643$	41.5692	1.63933	0.819665	−	0.572843i	$-0.194160\pi$
$643$	41.5692	1.63933	0.819665	+	0.572843i	$0.194160\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.1244	0.751856	0.375928	−	0.926649i	$-0.377324\pi$
$647$	19.1244	0.751856	0.375928	+	0.926649i	$0.377324\pi$
$648$	0	0
$649$	−27.7128	−1.08782
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.0718	0.511539	0.255769	−	0.966738i	$-0.417671\pi$
$653$	13.0718	0.511539	0.255769	+	0.966738i	$0.417671\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	0	0
$657$	0	0
$658$	0	0
$659$	16.7321	0.651788	0.325894	−	0.945406i	$-0.394335\pi$
$659$	16.7321	0.651788	0.325894	+	0.945406i	$0.394335\pi$
$660$	0	0
$661$	26.6410	1.03622	0.518108	−	0.855315i	$-0.326637\pi$
$661$	26.6410	1.03622	0.518108	+	0.855315i	$0.326637\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.732051	−0.0283877
$666$	0	0
$667$	1.46410	0.0566902
$668$	0	0
$669$	0	0
$670$	0	0
$671$	15.7128	0.606586
$672$	0	0
$673$	−32.2487	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$673$	−32.2487	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−4.92820	−0.189406	−0.0947031	−	0.995506i	$-0.530190\pi$
$677$	−4.92820	−0.189406	−0.0947031	+	0.995506i	$0.530190\pi$
$678$	0	0
$679$	−6.00000	−0.230259
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.2679	0.584212	0.292106	−	0.956386i	$-0.405644\pi$
$683$	15.2679	0.584212	0.292106	+	0.956386i	$0.405644\pi$
$684$	0	0
$685$	8.00000	0.305664
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−32.7846	−1.24899
$690$	0	0
$691$	−4.67949	−0.178016	−0.0890081	−	0.996031i	$-0.528370\pi$
$691$	−4.67949	−0.178016	−0.0890081	+	0.996031i	$0.528370\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.85641	0.0704175
$696$	0	0
$697$	37.4641	1.41905
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−12.0000	−0.453234	−0.226617	−	0.973984i	$-0.572767\pi$
$701$	−12.0000	−0.453234	−0.226617	+	0.973984i	$0.572767\pi$
$702$	0	0
$703$	−7.46410	−0.281514
$704$	0	0
$705$	0	0
$706$	0	0
$707$	14.5885	0.548655
$708$	0	0
$709$	14.2487	0.535122	0.267561	−	0.963541i	$-0.413782\pi$
$709$	14.2487	0.535122	0.267561	+	0.963541i	$0.413782\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.85641	−0.0695230
$714$	0	0
$715$	10.1436	0.379349
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.41154	0.276404	0.138202	−	0.990404i	$-0.455868\pi$
$719$	7.41154	0.276404	0.138202	+	0.990404i	$0.455868\pi$
$720$	0	0
$721$	−6.92820	−0.258020
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−12.1962	−0.452954
$726$	0	0
$727$	−8.39230	−0.311253	−0.155627	−	0.987816i	$-0.549740\pi$
$727$	−8.39230	−0.311253	−0.155627	+	0.987816i	$0.549740\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	29.1769	1.07915
$732$	0	0
$733$	7.46410	0.275693	0.137846	−	0.990454i	$-0.455982\pi$
$733$	7.46410	0.275693	0.137846	+	0.990454i	$0.455982\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.8564	−0.510407
$738$	0	0
$739$	27.0718	0.995852	0.497926	−	0.867219i	$-0.334095\pi$
$739$	27.0718	0.995852	0.497926	+	0.867219i	$0.334095\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	34.3013	1.25839	0.629196	−	0.777247i	$-0.283384\pi$
$743$	34.3013	1.25839	0.629196	+	0.777247i	$0.283384\pi$
$744$	0	0
$745$	6.92820	0.253830
$746$	0	0
$747$	0	0
$748$	0	0
$749$	2.19615	0.0802457
$750$	0	0
$751$	−16.3923	−0.598164	−0.299082	−	0.954227i	$-0.596680\pi$
$751$	−16.3923	−0.598164	−0.299082	+	0.954227i	$0.596680\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−15.7128	−0.571848
$756$	0	0
$757$	−5.71281	−0.207636	−0.103818	−	0.994596i	$-0.533106\pi$
$757$	−5.71281	−0.207636	−0.103818	+	0.994596i	$0.533106\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−7.26795	−0.263463	−0.131731	−	0.991285i	$-0.542054\pi$
$761$	−7.26795	−0.263463	−0.131731	+	0.991285i	$0.542054\pi$
$762$	0	0
$763$	−15.8564	−0.574040
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−32.0000	−1.15545
$768$	0	0
$769$	39.4641	1.42311	0.711556	−	0.702629i	$-0.247991\pi$
$769$	39.4641	1.42311	0.711556	+	0.702629i	$0.247991\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	9.60770	0.345565	0.172782	−	0.984960i	$-0.444724\pi$
$773$	9.60770	0.345565	0.172782	+	0.984960i	$0.444724\pi$
$774$	0	0
$775$	15.4641	0.555487
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−11.4641	−0.410744
$780$	0	0
$781$	21.4641	0.768046
$782$	0	0
$783$	0	0
$784$	0	0
$785$	10.5359	0.376042
$786$	0	0
$787$	16.5359	0.589441	0.294721	−	0.955583i	$-0.404773\pi$
$787$	16.5359	0.589441	0.294721	+	0.955583i	$0.404773\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−2.73205	−0.0971405
$792$	0	0
$793$	18.1436	0.644298
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.2487	−1.28400	−0.641998	−	0.766707i	$-0.721894\pi$
$797$	−36.2487	−1.28400	−0.641998	+	0.766707i	$0.721894\pi$
$798$	0	0
$799$	18.4974	0.654392
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.85641	0.0655112
$804$	0	0
$805$	−0.392305	−0.0138269
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38.9282	1.36864	0.684321	−	0.729181i	$-0.260099\pi$
$809$	38.9282	1.36864	0.684321	+	0.729181i	$0.260099\pi$
$810$	0	0
$811$	−27.7128	−0.973128	−0.486564	−	0.873645i	$-0.661750\pi$
$811$	−27.7128	−0.973128	−0.486564	+	0.873645i	$0.661750\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−7.32051	−0.256426
$816$	0	0
$817$	−8.92820	−0.312358
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.3205	0.674290	0.337145	−	0.941453i	$-0.390539\pi$
$821$	19.3205	0.674290	0.337145	+	0.941453i	$0.390539\pi$
$822$	0	0
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−45.5167	−1.58277	−0.791385	−	0.611318i	$-0.790639\pi$
$827$	−45.5167	−1.58277	−0.791385	+	0.611318i	$0.790639\pi$
$828$	0	0
$829$	39.0718	1.35702	0.678510	−	0.734591i	$-0.262626\pi$
$829$	39.0718	1.35702	0.678510	+	0.734591i	$0.262626\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.26795	−0.113228
$834$	0	0
$835$	3.21539	0.111273
$836$	0	0
$837$	0	0
$838$	0	0
$839$	32.7846	1.13185	0.565925	−	0.824457i	$-0.308519\pi$
$839$	32.7846	1.13185	0.565925	+	0.824457i	$0.308519\pi$
$840$	0	0
$841$	−21.5359	−0.742617
$842$	0	0
$843$	0	0
$844$	0	0
$845$	2.19615	0.0755499
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−4.00000	−0.137118
$852$	0	0
$853$	−19.0718	−0.653006	−0.326503	−	0.945196i	$-0.605870\pi$
$853$	−19.0718	−0.653006	−0.326503	+	0.945196i	$0.605870\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.3923	−0.491632	−0.245816	−	0.969317i	$-0.579056\pi$
$857$	−14.3923	−0.491632	−0.245816	+	0.969317i	$0.579056\pi$
$858$	0	0
$859$	−40.7846	−1.39155	−0.695776	−	0.718258i	$-0.744940\pi$
$859$	−40.7846	−1.39155	−0.695776	+	0.718258i	$0.744940\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	16.0526	0.546435	0.273218	−	0.961952i	$-0.411912\pi$
$863$	16.0526	0.546435	0.273218	+	0.961952i	$0.411912\pi$
$864$	0	0
$865$	7.60770	0.258669
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−18.9282	−0.642095
$870$	0	0
$871$	−16.0000	−0.542139
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.92820	0.234216
$876$	0	0
$877$	13.2154	0.446252	0.223126	−	0.974790i	$-0.428374\pi$
$877$	13.2154	0.446252	0.223126	+	0.974790i	$0.428374\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17.5167	−0.590151	−0.295076	−	0.955474i	$-0.595345\pi$
$881$	−17.5167	−0.590151	−0.295076	+	0.955474i	$0.595345\pi$
$882$	0	0
$883$	−39.7128	−1.33644	−0.668221	−	0.743963i	$-0.732944\pi$
$883$	−39.7128	−1.33644	−0.668221	+	0.743963i	$0.732944\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	36.1051	1.21229	0.606146	−	0.795354i	$-0.292715\pi$
$887$	36.1051	1.21229	0.606146	+	0.795354i	$0.292715\pi$
$888$	0	0
$889$	−2.92820	−0.0982088
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5.66025	−0.189413
$894$	0	0
$895$	−18.6795	−0.624387
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−9.46410	−0.315645
$900$	0	0
$901$	26.7846	0.892325
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17.4641	−0.580526
$906$	0	0
$907$	38.6410	1.28305	0.641527	−	0.767101i	$-0.278301\pi$
$907$	38.6410	1.28305	0.641527	+	0.767101i	$0.278301\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−48.0526	−1.59205	−0.796026	−	0.605262i	$-0.793068\pi$
$911$	−48.0526	−1.59205	−0.796026	+	0.605262i	$0.793068\pi$
$912$	0	0
$913$	0.679492	0.0224879
$914$	0	0
$915$	0	0
$916$	0	0
$917$	8.19615	0.270661
$918$	0	0
$919$	13.8564	0.457081	0.228540	−	0.973534i	$-0.426605\pi$
$919$	13.8564	0.457081	0.228540	+	0.973534i	$0.426605\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.7846	0.815795
$924$	0	0
$925$	33.3205	1.09557
$926$	0	0
$927$	0	0
$928$	0	0
$929$	21.8038	0.715361	0.357681	−	0.933844i	$-0.383568\pi$
$929$	21.8038	0.715361	0.357681	+	0.933844i	$0.383568\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.28719	−0.271020
$936$	0	0
$937$	39.8564	1.30205	0.651026	−	0.759055i	$-0.274339\pi$
$937$	39.8564	1.30205	0.651026	+	0.759055i	$0.274339\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−40.6410	−1.32486	−0.662430	−	0.749124i	$-0.730475\pi$
$941$	−40.6410	−1.32486	−0.662430	+	0.749124i	$0.730475\pi$
$942$	0	0
$943$	−6.14359	−0.200063
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−57.0333	−1.85333	−0.926667	−	0.375883i	$-0.877339\pi$
$947$	−57.0333	−1.85333	−0.926667	+	0.375883i	$0.877339\pi$
$948$	0	0
$949$	2.14359	0.0695840
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−3.41154	−0.110511	−0.0552554	−	0.998472i	$-0.517597\pi$
$953$	−3.41154	−0.110511	−0.0552554	+	0.998472i	$0.517597\pi$
$954$	0	0
$955$	13.4641	0.435688
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−10.9282	−0.352890
$960$	0	0
$961$	−19.0000	−0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.6795	0.408167
$966$	0	0
$967$	−38.7846	−1.24723	−0.623614	−	0.781732i	$-0.714336\pi$
$967$	−38.7846	−1.24723	−0.623614	+	0.781732i	$0.714336\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.0718	1.31806	0.659028	−	0.752118i	$-0.270968\pi$
$971$	41.0718	1.31806	0.659028	+	0.752118i	$0.270968\pi$
$972$	0	0
$973$	−2.53590	−0.0812972
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.12436	−0.0999570	−0.0499785	−	0.998750i	$-0.515915\pi$
$977$	−3.12436	−0.0999570	−0.0499785	+	0.998750i	$0.515915\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	0	0
$981$	0	0
$982$	0	0
$983$	12.7846	0.407766	0.203883	−	0.978995i	$-0.434644\pi$
$983$	12.7846	0.407766	0.203883	+	0.978995i	$0.434644\pi$
$984$	0	0
$985$	−5.07180	−0.161601
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.78461	−0.152142
$990$	0	0
$991$	−52.7846	−1.67676	−0.838379	−	0.545087i	$-0.816496\pi$
$991$	−52.7846	−1.67676	−0.838379	+	0.545087i	$0.816496\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12.2872	−0.389530
$996$	0	0
$997$	−46.1051	−1.46016	−0.730082	−	0.683360i	$-0.760518\pi$
$997$	−46.1051	−1.46016	−0.730082	+	0.683360i	$0.760518\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.bf.1.2	✓	2
3.2	odd	2		9576.2.a.br.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bf.1.2	✓	2	1.1	even	1	trivial
9576.2.a.br.1.1	yes	2	3.2	odd	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:	\( N \)	\(=\)	\( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9576.a (trivial)

\(f(q)\)	\(=\)	\(q+0.732051 q^{5} -1.00000 q^{7} +3.46410 q^{11} +4.00000 q^{13} -3.26795 q^{17} +1.00000 q^{19} +0.535898 q^{23} -4.46410 q^{25} +2.73205 q^{29} -3.46410 q^{31} -0.732051 q^{35} -7.46410 q^{37} -11.4641 q^{41} -8.92820 q^{43} -5.66025 q^{47} +1.00000 q^{49} -8.19615 q^{53} +2.53590 q^{55} -8.00000 q^{59} +4.53590 q^{61} +2.92820 q^{65} -4.00000 q^{67} +6.19615 q^{71} +0.535898 q^{73} -3.46410 q^{77} -5.46410 q^{79} +0.196152 q^{83} -2.39230 q^{85} +10.3923 q^{89} -4.00000 q^{91} +0.732051 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7} + 8 q^{13} - 10 q^{17} + 2 q^{19} + 8 q^{23} - 2 q^{25} + 2 q^{29} + 2 q^{35} - 8 q^{37} - 16 q^{41} - 4 q^{43} + 6 q^{47} + 2 q^{49} - 6 q^{53} + 12 q^{55} - 16 q^{59} + 16 q^{61}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 8 * q^13 - 10 * q^17 + 2 * q^19 + 8 * q^23 - 2 * q^25 + 2 * q^29 + 2 * q^35 - 8 * q^37 - 16 * q^41 - 4 * q^43 + 6 * q^47 + 2 * q^49 - 6 * q^53 + 12 * q^55 - 16 * q^59 + 16 * q^61 - 8 * q^65 - 8 * q^67 + 2 * q^71 + 8 * q^73 - 4 * q^79 - 10 * q^83 + 16 * q^85 - 8 * q^91 - 2 * q^95 + 12 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more