LMFDB - Embedded newform 9576.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9576,2,Mod(1,9576)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

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(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
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			$2.64575$ 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)

$f(q)$	$=$	$q+1.64575 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.00000 q^{13} +5.64575 q^{17} +1.00000 q^{19} +6.00000 q^{23} -2.29150 q^{25} -0.354249 q^{29} -9.29150 q^{31} +1.64575 q^{35} -5.29150 q^{37} -12.5830 q^{41} -2.00000 q^{43} -7.64575 q^{47} +1.00000 q^{49} -6.93725 q^{53} -3.29150 q^{55} +6.58301 q^{59} +13.2915 q^{61} -6.58301 q^{65} -6.58301 q^{67} -5.64575 q^{71} +1.29150 q^{73} -2.00000 q^{77} +0.708497 q^{79} -1.06275 q^{83} +9.29150 q^{85} -6.00000 q^{89} -4.00000 q^{91} +1.64575 q^{95} -4.58301 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{19} + 12 q^{23} + 6 q^{25} - 6 q^{29} - 8 q^{31} - 2 q^{35} - 4 q^{41} - 4 q^{43} - 10 q^{47} + 2 q^{49} + 2 q^{53} + 4 q^{55} - 8 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^19 + 12 * q^23 + 6 * q^25 - 6 * q^29 - 8 * q^31 - 2 * q^35 - 4 * q^41 - 4 * q^43 - 10 * q^47 + 2 * q^49 + 2 * q^53 + 4 * q^55 - 8 * q^59 + 16 * q^61 + 8 * q^65 + 8 * q^67 - 6 * q^71 - 8 * q^73 - 4 * q^77 + 12 * q^79 - 18 * q^83 + 8 * q^85 - 12 * q^89 - 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
$5$	1.64575	0.736002	0.368001	−	0.929825i	$-0.380042\pi$
$5$	1.64575	0.736002	0.368001	+	0.929825i	$0.380042\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.64575	1.36930	0.684648	−	0.728874i	$-0.259956\pi$
$17$	5.64575	1.36930	0.684648	+	0.728874i	$0.259956\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	−2.29150	−0.458301
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.354249	−0.0657823	−0.0328912	−	0.999459i	$-0.510471\pi$
$29$	−0.354249	−0.0657823	−0.0328912	+	0.999459i	$0.510471\pi$
$30$	0	0
$31$	−9.29150	−1.66880	−0.834402	−	0.551157i	$-0.814187\pi$
$31$	−9.29150	−1.66880	−0.834402	+	0.551157i	$0.814187\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.64575	0.278183
$36$	0	0
$37$	−5.29150	−0.869918	−0.434959	−	0.900450i	$-0.643237\pi$
$37$	−5.29150	−0.869918	−0.434959	+	0.900450i	$0.643237\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.5830	−1.96514	−0.982568	−	0.185905i	$-0.940478\pi$
$41$	−12.5830	−1.96514	−0.982568	+	0.185905i	$0.940478\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−7.64575	−1.11525	−0.557624	−	0.830094i	$-0.688287\pi$
$47$	−7.64575	−1.11525	−0.557624	+	0.830094i	$0.688287\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.93725	−0.952905	−0.476453	−	0.879200i	$-0.658078\pi$
$53$	−6.93725	−0.952905	−0.476453	+	0.879200i	$0.658078\pi$
$54$	0	0
$55$	−3.29150	−0.443826
$56$	0	0
$57$	0	0
$58$	0	0
$59$	6.58301	0.857034	0.428517	−	0.903534i	$-0.359036\pi$
$59$	6.58301	0.857034	0.428517	+	0.903534i	$0.359036\pi$
$60$	0	0
$61$	13.2915	1.70180	0.850901	−	0.525326i	$-0.176056\pi$
$61$	13.2915	1.70180	0.850901	+	0.525326i	$0.176056\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.58301	−0.816521
$66$	0	0
$67$	−6.58301	−0.804242	−0.402121	−	0.915587i	$-0.631727\pi$
$67$	−6.58301	−0.804242	−0.402121	+	0.915587i	$0.631727\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.64575	−0.670027	−0.335014	−	0.942213i	$-0.608741\pi$
$71$	−5.64575	−0.670027	−0.335014	+	0.942213i	$0.608741\pi$
$72$	0	0
$73$	1.29150	0.151159	0.0755795	−	0.997140i	$-0.475919\pi$
$73$	1.29150	0.151159	0.0755795	+	0.997140i	$0.475919\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.00000	−0.227921
$78$	0	0
$79$	0.708497	0.0797122	0.0398561	−	0.999205i	$-0.487310\pi$
$79$	0.708497	0.0797122	0.0398561	+	0.999205i	$0.487310\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.06275	−0.116652	−0.0583258	−	0.998298i	$-0.518576\pi$
$83$	−1.06275	−0.116652	−0.0583258	+	0.998298i	$0.518576\pi$
$84$	0	0
$85$	9.29150	1.00780
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.64575	0.168851
$96$	0	0
$97$	−4.58301	−0.465334	−0.232667	−	0.972556i	$-0.574745\pi$
$97$	−4.58301	−0.465334	−0.232667	+	0.972556i	$0.574745\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−9.64575	−0.959788	−0.479894	−	0.877326i	$-0.659325\pi$
$101$	−9.64575	−0.959788	−0.479894	+	0.877326i	$0.659325\pi$
$102$	0	0
$103$	18.5830	1.83104	0.915519	−	0.402275i	$-0.131780\pi$
$103$	18.5830	1.83104	0.915519	+	0.402275i	$0.131780\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.64575	0.159101	0.0795504	−	0.996831i	$-0.474652\pi$
$107$	1.64575	0.159101	0.0795504	+	0.996831i	$0.474652\pi$
$108$	0	0
$109$	10.0000	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$109$	10.0000	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.06275	0.0999747	0.0499874	−	0.998750i	$-0.484082\pi$
$113$	1.06275	0.0999747	0.0499874	+	0.998750i	$0.484082\pi$
$114$	0	0
$115$	9.87451	0.920803
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.64575	0.517545
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.06275	0.0928526	0.0464263	−	0.998922i	$-0.485217\pi$
$131$	1.06275	0.0928526	0.0464263	+	0.998922i	$0.485217\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.41699	−0.121062	−0.0605310	−	0.998166i	$-0.519279\pi$
$137$	−1.41699	−0.121062	−0.0605310	+	0.998166i	$0.519279\pi$
$138$	0	0
$139$	−11.2915	−0.957733	−0.478866	−	0.877888i	$-0.658952\pi$
$139$	−11.2915	−0.957733	−0.478866	+	0.877888i	$0.658952\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.00000	0.668994
$144$	0	0
$145$	−0.583005	−0.0484160
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−4.70850	−0.385735	−0.192868	−	0.981225i	$-0.561779\pi$
$149$	−4.70850	−0.385735	−0.192868	+	0.981225i	$0.561779\pi$
$150$	0	0
$151$	8.70850	0.708687	0.354344	−	0.935115i	$-0.384704\pi$
$151$	8.70850	0.708687	0.354344	+	0.935115i	$0.384704\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.2915	−1.22824
$156$	0	0
$157$	−19.8745	−1.58616	−0.793079	−	0.609119i	$-0.791523\pi$
$157$	−19.8745	−1.58616	−0.793079	+	0.609119i	$0.791523\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.00000	0.472866
$162$	0	0
$163$	8.58301	0.672273	0.336136	−	0.941813i	$-0.390880\pi$
$163$	8.58301	0.672273	0.336136	+	0.941813i	$0.390880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4.70850	−0.364354	−0.182177	−	0.983266i	$-0.558314\pi$
$167$	−4.70850	−0.364354	−0.182177	+	0.983266i	$0.558314\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−2.29150	−0.173221
$176$	0	0
$177$	0	0
$178$	0	0
$179$	11.5203	0.861065	0.430532	−	0.902575i	$-0.358326\pi$
$179$	11.5203	0.861065	0.430532	+	0.902575i	$0.358326\pi$
$180$	0	0
$181$	−4.58301	−0.340652	−0.170326	−	0.985388i	$-0.554482\pi$
$181$	−4.58301	−0.340652	−0.170326	+	0.985388i	$0.554482\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.70850	−0.640261
$186$	0	0
$187$	−11.2915	−0.825716
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	−14.7085	−1.05874	−0.529370	−	0.848391i	$-0.677572\pi$
$193$	−14.7085	−1.05874	−0.529370	+	0.848391i	$0.677572\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	26.5830	1.89396	0.946980	−	0.321291i	$-0.104117\pi$
$197$	26.5830	1.89396	0.946980	+	0.321291i	$0.104117\pi$
$198$	0	0
$199$	22.5830	1.60087	0.800433	−	0.599422i	$-0.204603\pi$
$199$	22.5830	1.60087	0.800433	+	0.599422i	$0.204603\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.354249	−0.0248634
$204$	0	0
$205$	−20.7085	−1.44634
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2.00000	−0.138343
$210$	0	0
$211$	−22.5830	−1.55468	−0.777339	−	0.629082i	$-0.783431\pi$
$211$	−22.5830	−1.55468	−0.777339	+	0.629082i	$0.783431\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−3.29150	−0.224479
$216$	0	0
$217$	−9.29150	−0.630748
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−22.5830	−1.51910
$222$	0	0
$223$	5.29150	0.354345	0.177173	−	0.984180i	$-0.443305\pi$
$223$	5.29150	0.354345	0.177173	+	0.984180i	$0.443305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.2915	−1.54591	−0.772956	−	0.634460i	$-0.781223\pi$
$227$	−23.2915	−1.54591	−0.772956	+	0.634460i	$0.781223\pi$
$228$	0	0
$229$	12.5830	0.831508	0.415754	−	0.909477i	$-0.363518\pi$
$229$	12.5830	0.831508	0.415754	+	0.909477i	$0.363518\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.70850	0.308464	0.154232	−	0.988035i	$-0.450710\pi$
$233$	4.70850	0.308464	0.154232	+	0.988035i	$0.450710\pi$
$234$	0	0
$235$	−12.5830	−0.820825
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−13.2915	−0.859756	−0.429878	−	0.902887i	$-0.641443\pi$
$239$	−13.2915	−0.859756	−0.429878	+	0.902887i	$0.641443\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.64575	0.105143
$246$	0	0
$247$	−4.00000	−0.254514
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−7.64575	−0.482596	−0.241298	−	0.970451i	$-0.577573\pi$
$251$	−7.64575	−0.482596	−0.241298	+	0.970451i	$0.577573\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2.70850	−0.168951	−0.0844757	−	0.996426i	$-0.526922\pi$
$257$	−2.70850	−0.168951	−0.0844757	+	0.996426i	$0.526922\pi$
$258$	0	0
$259$	−5.29150	−0.328798
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−13.2915	−0.819589	−0.409795	−	0.912178i	$-0.634400\pi$
$263$	−13.2915	−0.819589	−0.409795	+	0.912178i	$0.634400\pi$
$264$	0	0
$265$	−11.4170	−0.701340
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.4575	−1.85703	−0.928514	−	0.371298i	$-0.878913\pi$
$269$	−30.4575	−1.85703	−0.928514	+	0.371298i	$0.878913\pi$
$270$	0	0
$271$	15.2915	0.928893	0.464446	−	0.885601i	$-0.346253\pi$
$271$	15.2915	0.928893	0.464446	+	0.885601i	$0.346253\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.58301	0.276366
$276$	0	0
$277$	1.87451	0.112628	0.0563141	−	0.998413i	$-0.482065\pi$
$277$	1.87451	0.112628	0.0563141	+	0.998413i	$0.482065\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	22.9373	1.36832	0.684161	−	0.729331i	$-0.260169\pi$
$281$	22.9373	1.36832	0.684161	+	0.729331i	$0.260169\pi$
$282$	0	0
$283$	8.00000	0.475551	0.237775	−	0.971320i	$-0.423582\pi$
$283$	8.00000	0.475551	0.237775	+	0.971320i	$0.423582\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−12.5830	−0.742751
$288$	0	0
$289$	14.8745	0.874971
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−24.5830	−1.43615	−0.718077	−	0.695963i	$-0.754978\pi$
$293$	−24.5830	−1.43615	−0.718077	+	0.695963i	$0.754978\pi$
$294$	0	0
$295$	10.8340	0.630779
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−24.0000	−1.38796
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	0	0
$304$	0	0
$305$	21.8745	1.25253
$306$	0	0
$307$	7.87451	0.449422	0.224711	−	0.974425i	$-0.427856\pi$
$307$	7.87451	0.449422	0.224711	+	0.974425i	$0.427856\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	19.6458	1.11401	0.557004	−	0.830510i	$-0.311951\pi$
$311$	19.6458	1.11401	0.557004	+	0.830510i	$0.311951\pi$
$312$	0	0
$313$	6.70850	0.379187	0.189593	−	0.981863i	$-0.439283\pi$
$313$	6.70850	0.379187	0.189593	+	0.981863i	$0.439283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−10.9373	−0.614297	−0.307149	−	0.951662i	$-0.599375\pi$
$317$	−10.9373	−0.614297	−0.307149	+	0.951662i	$0.599375\pi$
$318$	0	0
$319$	0.708497	0.0396682
$320$	0	0
$321$	0	0
$322$	0	0
$323$	5.64575	0.314138
$324$	0	0
$325$	9.16601	0.508439
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−7.64575	−0.421524
$330$	0	0
$331$	28.4575	1.56417	0.782083	−	0.623174i	$-0.214157\pi$
$331$	28.4575	1.56417	0.782083	+	0.623174i	$0.214157\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.8340	−0.591924
$336$	0	0
$337$	−8.58301	−0.467546	−0.233773	−	0.972291i	$-0.575107\pi$
$337$	−8.58301	−0.467546	−0.233773	+	0.972291i	$0.575107\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	18.5830	1.00633
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.8745	−0.637457	−0.318728	−	0.947846i	$-0.603256\pi$
$347$	−11.8745	−0.637457	−0.318728	+	0.947846i	$0.603256\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.3542	0.764000	0.382000	−	0.924162i	$-0.375236\pi$
$353$	14.3542	0.764000	0.382000	+	0.924162i	$0.375236\pi$
$354$	0	0
$355$	−9.29150	−0.493142
$356$	0	0
$357$	0	0
$358$	0	0
$359$	11.4170	0.602566	0.301283	−	0.953535i	$-0.402585\pi$
$359$	11.4170	0.602566	0.301283	+	0.953535i	$0.402585\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.12549	0.111253
$366$	0	0
$367$	25.1660	1.31366	0.656828	−	0.754041i	$-0.271898\pi$
$367$	25.1660	1.31366	0.656828	+	0.754041i	$0.271898\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.93725	−0.360164
$372$	0	0
$373$	−26.0000	−1.34623	−0.673114	−	0.739538i	$-0.735044\pi$
$373$	−26.0000	−1.34623	−0.673114	+	0.739538i	$0.735044\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.41699	0.0729789
$378$	0	0
$379$	−3.29150	−0.169073	−0.0845366	−	0.996420i	$-0.526941\pi$
$379$	−3.29150	−0.169073	−0.0845366	+	0.996420i	$0.526941\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	33.8745	1.73091	0.865453	−	0.500990i	$-0.167031\pi$
$383$	33.8745	1.73091	0.865453	+	0.500990i	$0.167031\pi$
$384$	0	0
$385$	−3.29150	−0.167751
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.87451	−0.500657	−0.250329	−	0.968161i	$-0.580539\pi$
$389$	−9.87451	−0.500657	−0.250329	+	0.968161i	$0.580539\pi$
$390$	0	0
$391$	33.8745	1.71311
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.16601	0.0586684
$396$	0	0
$397$	−18.0000	−0.903394	−0.451697	−	0.892171i	$-0.649181\pi$
$397$	−18.0000	−0.903394	−0.451697	+	0.892171i	$0.649181\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−8.35425	−0.417191	−0.208596	−	0.978002i	$-0.566889\pi$
$401$	−8.35425	−0.417191	−0.208596	+	0.978002i	$0.566889\pi$
$402$	0	0
$403$	37.1660	1.85137
$404$	0	0
$405$	0	0
$406$	0	0
$407$	10.5830	0.524580
$408$	0	0
$409$	29.1660	1.44217	0.721083	−	0.692848i	$-0.243645\pi$
$409$	29.1660	1.44217	0.721083	+	0.692848i	$0.243645\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.58301	0.323929
$414$	0	0
$415$	−1.74902	−0.0858558
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−22.9373	−1.12056	−0.560279	−	0.828304i	$-0.689306\pi$
$419$	−22.9373	−1.12056	−0.560279	+	0.828304i	$0.689306\pi$
$420$	0	0
$421$	−11.8745	−0.578728	−0.289364	−	0.957219i	$-0.593444\pi$
$421$	−11.8745	−0.578728	−0.289364	+	0.957219i	$0.593444\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−12.9373	−0.627549
$426$	0	0
$427$	13.2915	0.643221
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.9373	−1.39386	−0.696929	−	0.717140i	$-0.745451\pi$
$431$	−28.9373	−1.39386	−0.696929	+	0.717140i	$0.745451\pi$
$432$	0	0
$433$	−14.0000	−0.672797	−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000	−0.672797	−0.336399	+	0.941720i	$0.609209\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.00000	0.287019
$438$	0	0
$439$	−4.12549	−0.196899	−0.0984495	−	0.995142i	$-0.531388\pi$
$439$	−4.12549	−0.196899	−0.0984495	+	0.995142i	$0.531388\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10.7085	−0.508776	−0.254388	−	0.967102i	$-0.581874\pi$
$443$	−10.7085	−0.508776	−0.254388	+	0.967102i	$0.581874\pi$
$444$	0	0
$445$	−9.87451	−0.468097
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.35425	−0.205490	−0.102745	−	0.994708i	$-0.532763\pi$
$449$	−4.35425	−0.205490	−0.102745	+	0.994708i	$0.532763\pi$
$450$	0	0
$451$	25.1660	1.18502
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.58301	−0.308616
$456$	0	0
$457$	−2.12549	−0.0994263	−0.0497132	−	0.998764i	$-0.515831\pi$
$457$	−2.12549	−0.0994263	−0.0497132	+	0.998764i	$0.515831\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	4.93725	0.229951	0.114975	−	0.993368i	$-0.463321\pi$
$461$	4.93725	0.229951	0.114975	+	0.993368i	$0.463321\pi$
$462$	0	0
$463$	13.1660	0.611876	0.305938	−	0.952051i	$-0.401030\pi$
$463$	13.1660	0.611876	0.305938	+	0.952051i	$0.401030\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.2288	0.843526	0.421763	−	0.906706i	$-0.361411\pi$
$467$	18.2288	0.843526	0.421763	+	0.906706i	$0.361411\pi$
$468$	0	0
$469$	−6.58301	−0.303975
$470$	0	0
$471$	0	0
$472$	0	0
$473$	4.00000	0.183920
$474$	0	0
$475$	−2.29150	−0.105141
$476$	0	0
$477$	0	0
$478$	0	0
$479$	6.22876	0.284599	0.142300	−	0.989824i	$-0.454550\pi$
$479$	6.22876	0.284599	0.142300	+	0.989824i	$0.454550\pi$
$480$	0	0
$481$	21.1660	0.965087
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.54249	−0.342487
$486$	0	0
$487$	−27.7490	−1.25743	−0.628714	−	0.777637i	$-0.716418\pi$
$487$	−27.7490	−1.25743	−0.628714	+	0.777637i	$0.716418\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1.29150	0.0582847	0.0291423	−	0.999575i	$-0.490722\pi$
$491$	1.29150	0.0582847	0.0291423	+	0.999575i	$0.490722\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.64575	−0.253247
$498$	0	0
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−13.5203	−0.602839	−0.301419	−	0.953492i	$-0.597460\pi$
$503$	−13.5203	−0.602839	−0.301419	+	0.953492i	$0.597460\pi$
$504$	0	0
$505$	−15.8745	−0.706406
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−10.7085	−0.474646	−0.237323	−	0.971431i	$-0.576270\pi$
$509$	−10.7085	−0.474646	−0.237323	+	0.971431i	$0.576270\pi$
$510$	0	0
$511$	1.29150	0.0571327
$512$	0	0
$513$	0	0
$514$	0	0
$515$	30.5830	1.34765
$516$	0	0
$517$	15.2915	0.672520
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−15.8745	−0.695475	−0.347737	−	0.937592i	$-0.613050\pi$
$521$	−15.8745	−0.695475	−0.347737	+	0.937592i	$0.613050\pi$
$522$	0	0
$523$	−7.87451	−0.344328	−0.172164	−	0.985068i	$-0.555076\pi$
$523$	−7.87451	−0.344328	−0.172164	+	0.985068i	$0.555076\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−52.4575	−2.28509
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	50.3320	2.18012
$534$	0	0
$535$	2.70850	0.117099
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−2.00000	−0.0861461
$540$	0	0
$541$	−27.1660	−1.16796	−0.583979	−	0.811769i	$-0.698505\pi$
$541$	−27.1660	−1.16796	−0.583979	+	0.811769i	$0.698505\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	16.4575	0.704962
$546$	0	0
$547$	4.70850	0.201321	0.100660	−	0.994921i	$-0.467904\pi$
$547$	4.70850	0.201321	0.100660	+	0.994921i	$0.467904\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.354249	−0.0150915
$552$	0	0
$553$	0.708497	0.0301284
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−21.8745	−0.926853	−0.463426	−	0.886135i	$-0.653380\pi$
$557$	−21.8745	−0.926853	−0.463426	+	0.886135i	$0.653380\pi$
$558$	0	0
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−9.87451	−0.416161	−0.208080	−	0.978112i	$-0.566722\pi$
$563$	−9.87451	−0.416161	−0.208080	+	0.978112i	$0.566722\pi$
$564$	0	0
$565$	1.74902	0.0735816
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.6458	−0.991281	−0.495641	−	0.868528i	$-0.665067\pi$
$569$	−23.6458	−0.991281	−0.495641	+	0.868528i	$0.665067\pi$
$570$	0	0
$571$	−23.7490	−0.993865	−0.496933	−	0.867789i	$-0.665540\pi$
$571$	−23.7490	−0.993865	−0.496933	+	0.867789i	$0.665540\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−13.7490	−0.573374
$576$	0	0
$577$	33.0405	1.37549	0.687747	−	0.725950i	$-0.258600\pi$
$577$	33.0405	1.37549	0.687747	+	0.725950i	$0.258600\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−1.06275	−0.0440901
$582$	0	0
$583$	13.8745	0.574623
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.06275	−0.0438642	−0.0219321	−	0.999759i	$-0.506982\pi$
$587$	−1.06275	−0.0438642	−0.0219321	+	0.999759i	$0.506982\pi$
$588$	0	0
$589$	−9.29150	−0.382850
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26.3542	−1.08224	−0.541120	−	0.840946i	$-0.681999\pi$
$593$	−26.3542	−1.08224	−0.541120	+	0.840946i	$0.681999\pi$
$594$	0	0
$595$	9.29150	0.380914
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−0.479741	−0.0196017	−0.00980084	−	0.999952i	$-0.503120\pi$
$599$	−0.479741	−0.0196017	−0.00980084	+	0.999952i	$0.503120\pi$
$600$	0	0
$601$	40.3320	1.64518	0.822589	−	0.568637i	$-0.192529\pi$
$601$	40.3320	1.64518	0.822589	+	0.568637i	$0.192529\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−11.5203	−0.468365
$606$	0	0
$607$	22.5830	0.916616	0.458308	−	0.888793i	$-0.348456\pi$
$607$	22.5830	0.916616	0.458308	+	0.888793i	$0.348456\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	30.5830	1.23726
$612$	0	0
$613$	−20.7085	−0.836408	−0.418204	−	0.908353i	$-0.637340\pi$
$613$	−20.7085	−0.836408	−0.418204	+	0.908353i	$0.637340\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−37.1660	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$617$	−37.1660	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$618$	0	0
$619$	−6.12549	−0.246204	−0.123102	−	0.992394i	$-0.539284\pi$
$619$	−6.12549	−0.246204	−0.123102	+	0.992394i	$0.539284\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−8.29150	−0.331660
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−29.8745	−1.19117
$630$	0	0
$631$	34.0000	1.35352	0.676759	−	0.736204i	$-0.263384\pi$
$631$	34.0000	1.35352	0.676759	+	0.736204i	$0.263384\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.58301	−0.261239
$636$	0	0
$637$	−4.00000	−0.158486
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−11.6458	−0.459980	−0.229990	−	0.973193i	$-0.573869\pi$
$641$	−11.6458	−0.459980	−0.229990	+	0.973193i	$0.573869\pi$
$642$	0	0
$643$	−21.1660	−0.834706	−0.417353	−	0.908744i	$-0.637042\pi$
$643$	−21.1660	−0.834706	−0.417353	+	0.908744i	$0.637042\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.2288	−0.873903	−0.436951	−	0.899485i	$-0.643942\pi$
$647$	−22.2288	−0.873903	−0.436951	+	0.899485i	$0.643942\pi$
$648$	0	0
$649$	−13.1660	−0.516811
$650$	0	0
$651$	0	0
$652$	0	0
$653$	5.16601	0.202162	0.101081	−	0.994878i	$-0.467770\pi$
$653$	5.16601	0.202162	0.101081	+	0.994878i	$0.467770\pi$
$654$	0	0
$655$	1.74902	0.0683397
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−20.9373	−0.815600	−0.407800	−	0.913071i	$-0.633704\pi$
$659$	−20.9373	−0.815600	−0.407800	+	0.913071i	$0.633704\pi$
$660$	0	0
$661$	−13.4170	−0.521861	−0.260930	−	0.965358i	$-0.584029\pi$
$661$	−13.4170	−0.521861	−0.260930	+	0.965358i	$0.584029\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.64575	0.0638195
$666$	0	0
$667$	−2.12549	−0.0822994
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−26.5830	−1.02623
$672$	0	0
$673$	30.4575	1.17405	0.587025	−	0.809568i	$-0.300299\pi$
$673$	30.4575	1.17405	0.587025	+	0.809568i	$0.300299\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−22.4575	−0.863112	−0.431556	−	0.902086i	$-0.642035\pi$
$677$	−22.4575	−0.863112	−0.431556	+	0.902086i	$0.642035\pi$
$678$	0	0
$679$	−4.58301	−0.175880
$680$	0	0
$681$	0	0
$682$	0	0
$683$	34.1033	1.30493	0.652463	−	0.757821i	$-0.273736\pi$
$683$	34.1033	1.30493	0.652463	+	0.757821i	$0.273736\pi$
$684$	0	0
$685$	−2.33202	−0.0891019
$686$	0	0
$687$	0	0
$688$	0	0
$689$	27.7490	1.05715
$690$	0	0
$691$	−36.7085	−1.39646	−0.698229	−	0.715875i	$-0.746028\pi$
$691$	−36.7085	−1.39646	−0.698229	+	0.715875i	$0.746028\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−18.5830	−0.704894
$696$	0	0
$697$	−71.0405	−2.69085
$698$	0	0
$699$	0	0
$700$	0	0
$701$	25.1660	0.950507	0.475254	−	0.879849i	$-0.342356\pi$
$701$	25.1660	0.950507	0.475254	+	0.879849i	$0.342356\pi$
$702$	0	0
$703$	−5.29150	−0.199573
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−9.64575	−0.362766
$708$	0	0
$709$	31.0405	1.16575	0.582876	−	0.812561i	$-0.301928\pi$
$709$	31.0405	1.16575	0.582876	+	0.812561i	$0.301928\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−55.7490	−2.08782
$714$	0	0
$715$	13.1660	0.492381
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−25.0627	−0.934683	−0.467341	−	0.884077i	$-0.654788\pi$
$719$	−25.0627	−0.934683	−0.467341	+	0.884077i	$0.654788\pi$
$720$	0	0
$721$	18.5830	0.692067
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.811762	0.0301481
$726$	0	0
$727$	−41.8745	−1.55304	−0.776520	−	0.630093i	$-0.783017\pi$
$727$	−41.8745	−1.55304	−0.776520	+	0.630093i	$0.783017\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−11.2915	−0.417631
$732$	0	0
$733$	11.8745	0.438595	0.219297	−	0.975658i	$-0.429623\pi$
$733$	11.8745	0.438595	0.219297	+	0.975658i	$0.429623\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	13.1660	0.484976
$738$	0	0
$739$	26.0000	0.956425	0.478213	−	0.878244i	$-0.341285\pi$
$739$	26.0000	0.956425	0.478213	+	0.878244i	$0.341285\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	33.6458	1.23434	0.617171	−	0.786829i	$-0.288278\pi$
$743$	33.6458	1.23434	0.617171	+	0.786829i	$0.288278\pi$
$744$	0	0
$745$	−7.74902	−0.283902
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.64575	0.0601344
$750$	0	0
$751$	44.4575	1.62228	0.811139	−	0.584854i	$-0.198848\pi$
$751$	44.4575	1.62228	0.811139	+	0.584854i	$0.198848\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.3320	0.521596
$756$	0	0
$757$	−52.3320	−1.90204	−0.951020	−	0.309130i	$-0.899962\pi$
$757$	−52.3320	−1.90204	−0.951020	+	0.309130i	$0.899962\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.9373	−1.04897	−0.524487	−	0.851418i	$-0.675743\pi$
$761$	−28.9373	−1.04897	−0.524487	+	0.851418i	$0.675743\pi$
$762$	0	0
$763$	10.0000	0.362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−26.3320	−0.950794
$768$	0	0
$769$	−22.4575	−0.809839	−0.404919	−	0.914352i	$-0.632700\pi$
$769$	−22.4575	−0.809839	−0.404919	+	0.914352i	$0.632700\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.5830	−0.452579	−0.226290	−	0.974060i	$-0.572660\pi$
$773$	−12.5830	−0.452579	−0.226290	+	0.974060i	$0.572660\pi$
$774$	0	0
$775$	21.2915	0.764813
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.5830	−0.450833
$780$	0	0
$781$	11.2915	0.404042
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−32.7085	−1.16742
$786$	0	0
$787$	−18.4575	−0.657939	−0.328970	−	0.944340i	$-0.606701\pi$
$787$	−18.4575	−0.657939	−0.328970	+	0.944340i	$0.606701\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.06275	0.0377869
$792$	0	0
$793$	−53.1660	−1.88798
$794$	0	0
$795$	0	0
$796$	0	0
$797$	47.1660	1.67071	0.835353	−	0.549714i	$-0.185263\pi$
$797$	47.1660	1.67071	0.835353	+	0.549714i	$0.185263\pi$
$798$	0	0
$799$	−43.1660	−1.52710
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−2.58301	−0.0911523
$804$	0	0
$805$	9.87451	0.348031
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.7490	−0.834971	−0.417485	−	0.908684i	$-0.637089\pi$
$809$	−23.7490	−0.834971	−0.417485	+	0.908684i	$0.637089\pi$
$810$	0	0
$811$	−34.3320	−1.20556	−0.602780	−	0.797907i	$-0.705940\pi$
$811$	−34.3320	−1.20556	−0.602780	+	0.797907i	$0.705940\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.1255	0.494794
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	7.29150	0.254475	0.127238	−	0.991872i	$-0.459389\pi$
$821$	7.29150	0.254475	0.127238	+	0.991872i	$0.459389\pi$
$822$	0	0
$823$	16.5830	0.578047	0.289024	−	0.957322i	$-0.406669\pi$
$823$	16.5830	0.578047	0.289024	+	0.957322i	$0.406669\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.7712	0.409326	0.204663	−	0.978832i	$-0.434390\pi$
$827$	11.7712	0.409326	0.204663	+	0.978832i	$0.434390\pi$
$828$	0	0
$829$	35.1660	1.22137	0.610683	−	0.791875i	$-0.290895\pi$
$829$	35.1660	1.22137	0.610683	+	0.791875i	$0.290895\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	5.64575	0.195614
$834$	0	0
$835$	−7.74902	−0.268166
$836$	0	0
$837$	0	0
$838$	0	0
$839$	30.5830	1.05584	0.527921	−	0.849293i	$-0.322972\pi$
$839$	30.5830	1.05584	0.527921	+	0.849293i	$0.322972\pi$
$840$	0	0
$841$	−28.8745	−0.995673
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.93725	0.169847
$846$	0	0
$847$	−7.00000	−0.240523
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−31.7490	−1.08834
$852$	0	0
$853$	−52.5830	−1.80041	−0.900204	−	0.435469i	$-0.856583\pi$
$853$	−52.5830	−1.80041	−0.900204	+	0.435469i	$0.856583\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	13.7490	0.469657	0.234829	−	0.972037i	$-0.424547\pi$
$857$	13.7490	0.469657	0.234829	+	0.972037i	$0.424547\pi$
$858$	0	0
$859$	−38.5830	−1.31644	−0.658218	−	0.752828i	$-0.728689\pi$
$859$	−38.5830	−1.31644	−0.658218	+	0.752828i	$0.728689\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−16.2288	−0.552433	−0.276217	−	0.961095i	$-0.589081\pi$
$863$	−16.2288	−0.552433	−0.276217	+	0.961095i	$0.589081\pi$
$864$	0	0
$865$	−23.0405	−0.783401
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−1.41699	−0.0480683
$870$	0	0
$871$	26.3320	0.892226
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−12.0000	−0.405674
$876$	0	0
$877$	−8.58301	−0.289827	−0.144914	−	0.989444i	$-0.546290\pi$
$877$	−8.58301	−0.289827	−0.144914	+	0.989444i	$0.546290\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	51.5203	1.73576	0.867881	−	0.496772i	$-0.165482\pi$
$881$	51.5203	1.73576	0.867881	+	0.496772i	$0.165482\pi$
$882$	0	0
$883$	−1.16601	−0.0392394	−0.0196197	−	0.999808i	$-0.506246\pi$
$883$	−1.16601	−0.0392394	−0.0196197	+	0.999808i	$0.506246\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.8745	−1.27170	−0.635851	−	0.771812i	$-0.719351\pi$
$887$	−37.8745	−1.27170	−0.635851	+	0.771812i	$0.719351\pi$
$888$	0	0
$889$	−4.00000	−0.134156
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−7.64575	−0.255855
$894$	0	0
$895$	18.9595	0.633746
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3.29150	0.109778
$900$	0	0
$901$	−39.1660	−1.30481
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−7.54249	−0.250721
$906$	0	0
$907$	17.1660	0.569988	0.284994	−	0.958529i	$-0.408008\pi$
$907$	17.1660	0.569988	0.284994	+	0.958529i	$0.408008\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.06275	0.101473	0.0507367	−	0.998712i	$-0.483843\pi$
$911$	3.06275	0.101473	0.0507367	+	0.998712i	$0.483843\pi$
$912$	0	0
$913$	2.12549	0.0703435
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.06275	0.0350950
$918$	0	0
$919$	24.0000	0.791687	0.395843	−	0.918318i	$-0.370452\pi$
$919$	24.0000	0.791687	0.395843	+	0.918318i	$0.370452\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	22.5830	0.743329
$924$	0	0
$925$	12.1255	0.398684
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−22.3542	−0.733419	−0.366710	−	0.930335i	$-0.619516\pi$
$929$	−22.3542	−0.733419	−0.366710	+	0.930335i	$0.619516\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.5830	−0.607729
$936$	0	0
$937$	−38.0000	−1.24141	−0.620703	−	0.784046i	$-0.713153\pi$
$937$	−38.0000	−1.24141	−0.620703	+	0.784046i	$0.713153\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.29150	0.172498	0.0862490	−	0.996274i	$-0.472512\pi$
$941$	5.29150	0.172498	0.0862490	+	0.996274i	$0.472512\pi$
$942$	0	0
$943$	−75.4980	−2.45855
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.00000	0.194974	0.0974869	−	0.995237i	$-0.468920\pi$
$947$	6.00000	0.194974	0.0974869	+	0.995237i	$0.468920\pi$
$948$	0	0
$949$	−5.16601	−0.167696
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−17.5203	−0.567537	−0.283768	−	0.958893i	$-0.591585\pi$
$953$	−17.5203	−0.567537	−0.283768	+	0.958893i	$0.591585\pi$
$954$	0	0
$955$	−9.87451	−0.319532
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.41699	−0.0457571
$960$	0	0
$961$	55.3320	1.78490
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−24.2065	−0.779236
$966$	0	0
$967$	43.1660	1.38813	0.694063	−	0.719915i	$-0.255819\pi$
$967$	43.1660	1.38813	0.694063	+	0.719915i	$0.255819\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	26.5830	0.853089	0.426545	−	0.904467i	$-0.359731\pi$
$971$	26.5830	0.853089	0.426545	+	0.904467i	$0.359731\pi$
$972$	0	0
$973$	−11.2915	−0.361989
$974$	0	0
$975$	0	0
$976$	0	0
$977$	59.3948	1.90021	0.950103	−	0.311935i	$-0.100977\pi$
$977$	59.3948	1.90021	0.950103	+	0.311935i	$0.100977\pi$
$978$	0	0
$979$	12.0000	0.383522
$980$	0	0
$981$	0	0
$982$	0	0
$983$	21.4170	0.683096	0.341548	−	0.939864i	$-0.389049\pi$
$983$	21.4170	0.683096	0.341548	+	0.939864i	$0.389049\pi$
$984$	0	0
$985$	43.7490	1.39396
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	0	0
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	37.1660	1.17824
$996$	0	0
$997$	7.87451	0.249388	0.124694	−	0.992195i	$-0.460205\pi$
$997$	7.87451	0.249388	0.124694	+	0.992195i	$0.460205\pi$
$998$	0	0
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9576.2.a.bh.1.2	✓	2	1.1	even	1	trivial
9576.2.a.bu.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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.64575 q^{5} +1.00000 q^{7} -2.00000 q^{11} -4.00000 q^{13} +5.64575 q^{17} +1.00000 q^{19} +6.00000 q^{23} -2.29150 q^{25} -0.354249 q^{29} -9.29150 q^{31} +1.64575 q^{35} -5.29150 q^{37} -12.5830 q^{41} -2.00000 q^{43} -7.64575 q^{47} +1.00000 q^{49} -6.93725 q^{53} -3.29150 q^{55} +6.58301 q^{59} +13.2915 q^{61} -6.58301 q^{65} -6.58301 q^{67} -5.64575 q^{71} +1.29150 q^{73} -2.00000 q^{77} +0.708497 q^{79} -1.06275 q^{83} +9.29150 q^{85} -6.00000 q^{89} -4.00000 q^{91} +1.64575 q^{95} -4.58301 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7} - 4 q^{11} - 8 q^{13} + 6 q^{17} + 2 q^{19} + 12 q^{23} + 6 q^{25} - 6 q^{29} - 8 q^{31} - 2 q^{35} - 4 q^{41} - 4 q^{43} - 10 q^{47} + 2 q^{49} + 2 q^{53} + 4 q^{55} - 8 q^{59}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 - 4 * q^11 - 8 * q^13 + 6 * q^17 + 2 * q^19 + 12 * q^23 + 6 * q^25 - 6 * q^29 - 8 * q^31 - 2 * q^35 - 4 * q^41 - 4 * q^43 - 10 * q^47 + 2 * q^49 + 2 * q^53 + 4 * q^55 - 8 * q^59 + 16 * q^61 + 8 * q^65 + 8 * q^67 - 6 * q^71 - 8 * q^73 - 4 * q^77 + 12 * q^79 - 18 * q^83 + 8 * q^85 - 12 * q^89 - 8 * q^91 - 2 * q^95 + 12 * q^97

Introduction

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