LMFDB - Embedded newform 9576.2.a.ci.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.25857.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 9x + 3 $ x^4 - x^3 - 8x^2 + 9x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1064)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.52280$ 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)

$f(q)$	$=$	$q-1.52280 q^{5} +1.00000 q^{7} -5.50666 q^{11} +1.15829 q^{13} +2.98386 q^{17} +1.00000 q^{19} -0.841710 q^{23} -2.68109 q^{25} +7.95392 q^{29} +6.66728 q^{31} -1.52280 q^{35} +9.49052 q^{37} -9.95392 q^{41} -1.41010 q^{43} -0.635493 q^{47} +1.00000 q^{49} -0.348371 q^{53} +8.38553 q^{55} -14.0961 q^{59} -7.01565 q^{61} -1.76384 q^{65} -3.30044 q^{67} -14.5756 q^{71} -2.26562 q^{73} -5.50666 q^{77} -6.62120 q^{79} -2.33993 q^{83} -4.54382 q^{85} -1.56839 q^{89} +1.15829 q^{91} -1.52280 q^{95} +13.4977 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{5} + 4 q^{7} - 2 q^{11} + 2 q^{13} - 3 q^{17} + 4 q^{19} - 6 q^{23} - 3 q^{25} + 17 q^{31} - q^{35} + 3 q^{37} - 8 q^{41} + 7 q^{43} - 5 q^{47} + 4 q^{49} + 16 q^{53} + 17 q^{55} - 7 q^{59} - q^{61}+ \cdots + 3 q^{97}+O(q^{100}) $ 4 * q - q^5 + 4 * q^7 - 2 * q^11 + 2 * q^13 - 3 * q^17 + 4 * q^19 - 6 * q^23 - 3 * q^25 + 17 * q^31 - q^35 + 3 * q^37 - 8 * q^41 + 7 * q^43 - 5 * q^47 + 4 * q^49 + 16 * q^53 + 17 * q^55 - 7 * q^59 - q^61 + 10 * q^65 + 7 * q^67 - 27 * q^71 - q^73 - 2 * q^77 + 15 * q^79 - 3 * q^83 + q^85 + 9 * q^89 + 2 * q^91 - q^95 + 3 * 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
$5$	−1.52280	−0.681016	−0.340508	−	0.940242i	$-0.610599\pi$
$5$	−1.52280	−0.681016	−0.340508	+	0.940242i	$0.610599\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.50666	−1.66032	−0.830160	−	0.557525i	$-0.811751\pi$
$11$	−5.50666	−1.66032	−0.830160	+	0.557525i	$0.811751\pi$
$12$	0	0
$13$	1.15829	0.321252	0.160626	−	0.987015i	$-0.448649\pi$
$13$	1.15829	0.321252	0.160626	+	0.987015i	$0.448649\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.98386	0.723693	0.361847	−	0.932238i	$-0.382146\pi$
$17$	2.98386	0.723693	0.361847	+	0.932238i	$0.382146\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.841710	−0.175509	−0.0877543	−	0.996142i	$-0.527969\pi$
$23$	−0.841710	−0.175509	−0.0877543	+	0.996142i	$0.527969\pi$
$24$	0	0
$25$	−2.68109	−0.536218
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.95392	1.47701	0.738503	−	0.674250i	$-0.235533\pi$
$29$	7.95392	1.47701	0.738503	+	0.674250i	$0.235533\pi$
$30$	0	0
$31$	6.66728	1.19748	0.598740	−	0.800944i	$-0.295668\pi$
$31$	6.66728	1.19748	0.598740	+	0.800944i	$0.295668\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.52280	−0.257400
$36$	0	0
$37$	9.49052	1.56023	0.780116	−	0.625635i	$-0.215160\pi$
$37$	9.49052	1.56023	0.780116	+	0.625635i	$0.215160\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.95392	−1.55454	−0.777271	−	0.629166i	$-0.783396\pi$
$41$	−9.95392	−1.55454	−0.777271	+	0.629166i	$0.783396\pi$
$42$	0	0
$43$	−1.41010	−0.215039	−0.107519	−	0.994203i	$-0.534291\pi$
$43$	−1.41010	−0.215039	−0.107519	+	0.994203i	$0.534291\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.635493	−0.0926962	−0.0463481	−	0.998925i	$-0.514758\pi$
$47$	−0.635493	−0.0926962	−0.0463481	+	0.998925i	$0.514758\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−0.348371	−0.0478524	−0.0239262	−	0.999714i	$-0.507617\pi$
$53$	−0.348371	−0.0478524	−0.0239262	+	0.999714i	$0.507617\pi$
$54$	0	0
$55$	8.38553	1.13070
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.0961	−1.83515	−0.917576	−	0.397560i	$-0.869857\pi$
$59$	−14.0961	−1.83515	−0.917576	+	0.397560i	$0.869857\pi$
$60$	0	0
$61$	−7.01565	−0.898262	−0.449131	−	0.893466i	$-0.648266\pi$
$61$	−7.01565	−0.898262	−0.449131	+	0.893466i	$0.648266\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.76384	−0.218778
$66$	0	0
$67$	−3.30044	−0.403213	−0.201607	−	0.979467i	$-0.564616\pi$
$67$	−3.30044	−0.403213	−0.201607	+	0.979467i	$0.564616\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.5756	−1.72981	−0.864903	−	0.501939i	$-0.832620\pi$
$71$	−14.5756	−1.72981	−0.864903	+	0.501939i	$0.832620\pi$
$72$	0	0
$73$	−2.26562	−0.265170	−0.132585	−	0.991172i	$-0.542328\pi$
$73$	−2.26562	−0.265170	−0.132585	+	0.991172i	$0.542328\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.50666	−0.627542
$78$	0	0
$79$	−6.62120	−0.744944	−0.372472	−	0.928043i	$-0.621490\pi$
$79$	−6.62120	−0.744944	−0.372472	+	0.928043i	$0.621490\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.33993	−0.256841	−0.128421	−	0.991720i	$-0.540991\pi$
$83$	−2.33993	−0.256841	−0.128421	+	0.991720i	$0.540991\pi$
$84$	0	0
$85$	−4.54382	−0.492846
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.56839	−0.166249	−0.0831246	−	0.996539i	$-0.526490\pi$
$89$	−1.56839	−0.166249	−0.0831246	+	0.996539i	$0.526490\pi$
$90$	0	0
$91$	1.15829	0.121422
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.52280	−0.156236
$96$	0	0
$97$	13.4977	1.37049	0.685244	−	0.728314i	$-0.259696\pi$
$97$	13.4977	1.37049	0.685244	+	0.728314i	$0.259696\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0.552739	0.0549996	0.0274998	−	0.999622i	$-0.491245\pi$
$101$	0.552739	0.0549996	0.0274998	+	0.999622i	$0.491245\pi$
$102$	0	0
$103$	−3.56606	−0.351374	−0.175687	−	0.984446i	$-0.556215\pi$
$103$	−3.56606	−0.351374	−0.175687	+	0.984446i	$0.556215\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0205	1.54876	0.774382	−	0.632718i	$-0.218061\pi$
$107$	16.0205	1.54876	0.774382	+	0.632718i	$0.218061\pi$
$108$	0	0
$109$	14.2268	1.36268	0.681338	−	0.731969i	$-0.261398\pi$
$109$	14.2268	1.36268	0.681338	+	0.731969i	$0.261398\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.1735	−1.33333	−0.666664	−	0.745359i	$-0.732278\pi$
$113$	−14.1735	−1.33333	−0.666664	+	0.745359i	$0.732278\pi$
$114$	0	0
$115$	1.28175	0.119524
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.98386	0.273530
$120$	0	0
$121$	19.3233	1.75666
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.6967	1.04619
$126$	0	0
$127$	13.8850	1.23209	0.616046	−	0.787710i	$-0.288734\pi$
$127$	13.8850	1.23209	0.616046	+	0.787710i	$0.288734\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.30511	0.463509	0.231755	−	0.972774i	$-0.425553\pi$
$131$	5.30511	0.463509	0.231755	+	0.972774i	$0.425553\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.06221	0.0907511	0.0453756	−	0.998970i	$-0.485552\pi$
$137$	1.06221	0.0907511	0.0453756	+	0.998970i	$0.485552\pi$
$138$	0	0
$139$	18.8407	1.59805	0.799025	−	0.601297i	$-0.205349\pi$
$139$	18.8407	1.59805	0.799025	+	0.601297i	$0.205349\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.37831	−0.533381
$144$	0	0
$145$	−12.1122	−1.00586
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.0661	1.88965	0.944826	−	0.327574i	$-0.106231\pi$
$149$	23.0661	1.88965	0.944826	+	0.327574i	$0.106231\pi$
$150$	0	0
$151$	−1.01614	−0.0826920	−0.0413460	−	0.999145i	$-0.513165\pi$
$151$	−1.01614	−0.0826920	−0.0413460	+	0.999145i	$0.513165\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−10.1529	−0.815502
$156$	0	0
$157$	5.06662	0.404360	0.202180	−	0.979348i	$-0.435197\pi$
$157$	5.06662	0.404360	0.202180	+	0.979348i	$0.435197\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−0.841710	−0.0663360
$162$	0	0
$163$	−12.6794	−0.993126	−0.496563	−	0.868001i	$-0.665405\pi$
$163$	−12.6794	−0.993126	−0.496563	+	0.868001i	$0.665405\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	15.3827	1.19035	0.595175	−	0.803596i	$-0.297083\pi$
$167$	15.3827	1.19035	0.595175	+	0.803596i	$0.297083\pi$
$168$	0	0
$169$	−11.6584	−0.896797
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.3299	1.16551	0.582755	−	0.812648i	$-0.301975\pi$
$173$	15.3299	1.16551	0.582755	+	0.812648i	$0.301975\pi$
$174$	0	0
$175$	−2.68109	−0.202671
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15.3640	1.14836	0.574181	−	0.818729i	$-0.305321\pi$
$179$	15.3640	1.14836	0.574181	+	0.818729i	$0.305321\pi$
$180$	0	0
$181$	−3.99719	−0.297108	−0.148554	−	0.988904i	$-0.547462\pi$
$181$	−3.99719	−0.297108	−0.148554	+	0.988904i	$0.547462\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−14.4521	−1.06254
$186$	0	0
$187$	−16.4311	−1.20156
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.79330	0.708618	0.354309	−	0.935128i	$-0.384716\pi$
$191$	9.79330	0.708618	0.354309	+	0.935128i	$0.384716\pi$
$192$	0	0
$193$	−20.1206	−1.44832	−0.724158	−	0.689634i	$-0.757771\pi$
$193$	−20.1206	−1.44832	−0.724158	+	0.689634i	$0.757771\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.5223	1.17717	0.588583	−	0.808437i	$-0.299686\pi$
$197$	16.5223	1.17717	0.588583	+	0.808437i	$0.299686\pi$
$198$	0	0
$199$	12.8574	0.911434	0.455717	−	0.890125i	$-0.349383\pi$
$199$	12.8574	0.911434	0.455717	+	0.890125i	$0.349383\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.95392	0.558256
$204$	0	0
$205$	15.1578	1.05867
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−5.50666	−0.380904
$210$	0	0
$211$	−15.1555	−1.04335	−0.521673	−	0.853146i	$-0.674692\pi$
$211$	−15.1555	−1.04335	−0.521673	+	0.853146i	$0.674692\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.14730	0.146445
$216$	0	0
$217$	6.66728	0.452605
$218$	0	0
$219$	0	0
$220$	0	0
$221$	3.45618	0.232488
$222$	0	0
$223$	12.8094	0.857783	0.428892	−	0.903356i	$-0.358904\pi$
$223$	12.8094	0.857783	0.428892	+	0.903356i	$0.358904\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.2539	1.60979	0.804893	−	0.593419i	$-0.202222\pi$
$227$	24.2539	1.60979	0.804893	+	0.593419i	$0.202222\pi$
$228$	0	0
$229$	−17.1945	−1.13624	−0.568122	−	0.822945i	$-0.692330\pi$
$229$	−17.1945	−1.13624	−0.568122	+	0.822945i	$0.692330\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	28.0056	1.83471	0.917355	−	0.398071i	$-0.130320\pi$
$233$	28.0056	1.83471	0.917355	+	0.398071i	$0.130320\pi$
$234$	0	0
$235$	0.967727	0.0631275
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.460583	0.0297926	0.0148963	−	0.999889i	$-0.495258\pi$
$239$	0.460583	0.0297926	0.0148963	+	0.999889i	$0.495258\pi$
$240$	0	0
$241$	−2.16673	−0.139571	−0.0697856	−	0.997562i	$-0.522232\pi$
$241$	−2.16673	−0.139571	−0.0697856	+	0.997562i	$0.522232\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.52280	−0.0972880
$246$	0	0
$247$	1.15829	0.0737002
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.0705	−1.77180	−0.885898	−	0.463880i	$-0.846457\pi$
$251$	−28.0705	−1.77180	−0.885898	+	0.463880i	$0.846457\pi$
$252$	0	0
$253$	4.63501	0.291401
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.63124	0.101754	0.0508769	−	0.998705i	$-0.483798\pi$
$257$	1.63124	0.101754	0.0508769	+	0.998705i	$0.483798\pi$
$258$	0	0
$259$	9.49052	0.589712
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−3.56791	−0.220007	−0.110003	−	0.993931i	$-0.535086\pi$
$263$	−3.56791	−0.220007	−0.110003	+	0.993931i	$0.535086\pi$
$264$	0	0
$265$	0.530498	0.0325882
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.6422	−0.953723	−0.476862	−	0.878978i	$-0.658226\pi$
$269$	−15.6422	−0.953723	−0.476862	+	0.878978i	$0.658226\pi$
$270$	0	0
$271$	32.1461	1.95273	0.976367	−	0.216117i	$-0.0693393\pi$
$271$	32.1461	1.95273	0.976367	+	0.216117i	$0.0693393\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	14.7638	0.890293
$276$	0	0
$277$	5.01332	0.301221	0.150611	−	0.988593i	$-0.451876\pi$
$277$	5.01332	0.301221	0.150611	+	0.988593i	$0.451876\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.0481	0.957352	0.478676	−	0.877992i	$-0.341117\pi$
$281$	16.0481	0.957352	0.478676	+	0.877992i	$0.341117\pi$
$282$	0	0
$283$	1.41314	0.0840024	0.0420012	−	0.999118i	$-0.486627\pi$
$283$	1.41314	0.0840024	0.0420012	+	0.999118i	$0.486627\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.95392	−0.587561
$288$	0	0
$289$	−8.09656	−0.476268
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−10.9006	−0.636821	−0.318411	−	0.947953i	$-0.603149\pi$
$293$	−10.9006	−0.636821	−0.318411	+	0.947953i	$0.603149\pi$
$294$	0	0
$295$	21.4655	1.24977
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−0.974944	−0.0563825
$300$	0	0
$301$	−1.41010	−0.0812769
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.6834	0.611731
$306$	0	0
$307$	−4.15170	−0.236950	−0.118475	−	0.992957i	$-0.537801\pi$
$307$	−4.15170	−0.236950	−0.118475	+	0.992957i	$0.537801\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.00770	−0.283961	−0.141980	−	0.989869i	$-0.545347\pi$
$311$	−5.00770	−0.283961	−0.141980	+	0.989869i	$0.545347\pi$
$312$	0	0
$313$	19.2556	1.08839	0.544195	−	0.838959i	$-0.316835\pi$
$313$	19.2556	1.08839	0.544195	+	0.838959i	$0.316835\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.95818	0.503141	0.251571	−	0.967839i	$-0.419053\pi$
$317$	8.95818	0.503141	0.251571	+	0.967839i	$0.419053\pi$
$318$	0	0
$319$	−43.7996	−2.45230
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.98386	0.166027
$324$	0	0
$325$	−3.10548	−0.172261
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.635493	−0.0350359
$330$	0	0
$331$	17.9588	0.987105	0.493553	−	0.869716i	$-0.335698\pi$
$331$	17.9588	0.987105	0.493553	+	0.869716i	$0.335698\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.02591	0.274595
$336$	0	0
$337$	−6.75381	−0.367903	−0.183952	−	0.982935i	$-0.558889\pi$
$337$	−6.75381	−0.367903	−0.183952	+	0.982935i	$0.558889\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−36.7145	−1.98820
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.0033	1.39593	0.697965	−	0.716132i	$-0.254089\pi$
$347$	26.0033	1.39593	0.697965	+	0.716132i	$0.254089\pi$
$348$	0	0
$349$	20.4634	1.09538	0.547690	−	0.836681i	$-0.315507\pi$
$349$	20.4634	1.09538	0.547690	+	0.836681i	$0.315507\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.9696	0.849975	0.424987	−	0.905199i	$-0.360279\pi$
$353$	15.9696	0.849975	0.424987	+	0.905199i	$0.360279\pi$
$354$	0	0
$355$	22.1957	1.17803
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.6038	−0.717984	−0.358992	−	0.933341i	$-0.616879\pi$
$359$	−13.6038	−0.717984	−0.358992	+	0.933341i	$0.616879\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.45008	0.180585
$366$	0	0
$367$	−24.9655	−1.30319	−0.651595	−	0.758567i	$-0.725900\pi$
$367$	−24.9655	−1.30319	−0.651595	+	0.758567i	$0.725900\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−0.348371	−0.0180865
$372$	0	0
$373$	15.3965	0.797202	0.398601	−	0.917125i	$-0.369496\pi$
$373$	15.3965	0.797202	0.398601	+	0.917125i	$0.369496\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	9.21295	0.474491
$378$	0	0
$379$	−0.932900	−0.0479198	−0.0239599	−	0.999713i	$-0.507627\pi$
$379$	−0.932900	−0.0479198	−0.0239599	+	0.999713i	$0.507627\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.841710	−0.0430094	−0.0215047	−	0.999769i	$-0.506846\pi$
$383$	−0.841710	−0.0430094	−0.0215047	+	0.999769i	$0.506846\pi$
$384$	0	0
$385$	8.38553	0.427366
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.75492	0.190382	0.0951911	−	0.995459i	$-0.469654\pi$
$389$	3.75492	0.190382	0.0951911	+	0.995459i	$0.469654\pi$
$390$	0	0
$391$	−2.51155	−0.127014
$392$	0	0
$393$	0	0
$394$	0	0
$395$	10.0828	0.507318
$396$	0	0
$397$	28.7052	1.44067	0.720336	−	0.693626i	$-0.243988\pi$
$397$	28.7052	1.44067	0.720336	+	0.693626i	$0.243988\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.9660	−0.947118	−0.473559	−	0.880762i	$-0.657031\pi$
$401$	−18.9660	−0.947118	−0.473559	+	0.880762i	$0.657031\pi$
$402$	0	0
$403$	7.72265	0.384693
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−52.2611	−2.59049
$408$	0	0
$409$	−3.08716	−0.152650	−0.0763250	−	0.997083i	$-0.524319\pi$
$409$	−3.08716	−0.152650	−0.0763250	+	0.997083i	$0.524319\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.0961	−0.693623
$414$	0	0
$415$	3.56325	0.174913
$416$	0	0
$417$	0	0
$418$	0	0
$419$	11.2219	0.548224	0.274112	−	0.961698i	$-0.411616\pi$
$419$	11.2219	0.548224	0.274112	+	0.961698i	$0.411616\pi$
$420$	0	0
$421$	5.58146	0.272024	0.136012	−	0.990707i	$-0.456571\pi$
$421$	5.58146	0.272024	0.136012	+	0.990707i	$0.456571\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−8.00000	−0.388057
$426$	0	0
$427$	−7.01565	−0.339511
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−0.555071	−0.0267368	−0.0133684	−	0.999911i	$-0.504255\pi$
$431$	−0.555071	−0.0267368	−0.0133684	+	0.999911i	$0.504255\pi$
$432$	0	0
$433$	−11.8215	−0.568107	−0.284053	−	0.958808i	$-0.591679\pi$
$433$	−11.8215	−0.568107	−0.284053	+	0.958808i	$0.591679\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−0.841710	−0.0402644
$438$	0	0
$439$	−26.3432	−1.25729	−0.628647	−	0.777691i	$-0.716391\pi$
$439$	−26.3432	−1.25729	−0.628647	+	0.777691i	$0.716391\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.0246	0.761350	0.380675	−	0.924709i	$-0.375692\pi$
$443$	16.0246	0.761350	0.380675	+	0.924709i	$0.375692\pi$
$444$	0	0
$445$	2.38834	0.113218
$446$	0	0
$447$	0	0
$448$	0	0
$449$	33.4589	1.57902	0.789511	−	0.613736i	$-0.210334\pi$
$449$	33.4589	1.57902	0.789511	+	0.613736i	$0.210334\pi$
$450$	0	0
$451$	54.8129	2.58104
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.76384	−0.0826902
$456$	0	0
$457$	40.1051	1.87604	0.938020	−	0.346582i	$-0.112658\pi$
$457$	40.1051	1.87604	0.938020	+	0.346582i	$0.112658\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15.9719	−0.743886	−0.371943	−	0.928256i	$-0.621308\pi$
$461$	−15.9719	−0.743886	−0.371943	+	0.928256i	$0.621308\pi$
$462$	0	0
$463$	16.8094	0.781201	0.390600	−	0.920560i	$-0.372267\pi$
$463$	16.8094	0.781201	0.390600	+	0.920560i	$0.372267\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	35.5178	1.64357	0.821784	−	0.569799i	$-0.192979\pi$
$467$	35.5178	1.64357	0.821784	+	0.569799i	$0.192979\pi$
$468$	0	0
$469$	−3.30044	−0.152400
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.76495	0.357033
$474$	0	0
$475$	−2.68109	−0.123017
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−30.2336	−1.38141	−0.690703	−	0.723138i	$-0.742699\pi$
$479$	−30.2336	−1.38141	−0.690703	+	0.723138i	$0.742699\pi$
$480$	0	0
$481$	10.9928	0.501228
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−20.5543	−0.933324
$486$	0	0
$487$	−4.06847	−0.184360	−0.0921799	−	0.995742i	$-0.529383\pi$
$487$	−4.06847	−0.184360	−0.0921799	+	0.995742i	$0.529383\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	26.6046	1.20065	0.600324	−	0.799757i	$-0.295038\pi$
$491$	26.6046	1.20065	0.600324	+	0.799757i	$0.295038\pi$
$492$	0	0
$493$	23.7334	1.06890
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−14.5756	−0.653805
$498$	0	0
$499$	−14.9098	−0.667453	−0.333726	−	0.942670i	$-0.608306\pi$
$499$	−14.9098	−0.667453	−0.333726	+	0.942670i	$0.608306\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−17.4054	−0.776070	−0.388035	−	0.921645i	$-0.626846\pi$
$503$	−17.4054	−0.776070	−0.388035	+	0.921645i	$0.626846\pi$
$504$	0	0
$505$	−0.841710	−0.0374556
$506$	0	0
$507$	0	0
$508$	0	0
$509$	35.7414	1.58421	0.792104	−	0.610386i	$-0.208986\pi$
$509$	35.7414	1.58421	0.792104	+	0.610386i	$0.208986\pi$
$510$	0	0
$511$	−2.26562	−0.100225
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.43039	0.239292
$516$	0	0
$517$	3.49944	0.153905
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.905290	0.0396615	0.0198307	−	0.999803i	$-0.493687\pi$
$521$	0.905290	0.0396615	0.0198307	+	0.999803i	$0.493687\pi$
$522$	0	0
$523$	−29.3182	−1.28199	−0.640997	−	0.767543i	$-0.721479\pi$
$523$	−29.3182	−1.28199	−0.640997	+	0.767543i	$0.721479\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	19.8943	0.866608
$528$	0	0
$529$	−22.2915	−0.969197
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.5295	−0.499399
$534$	0	0
$535$	−24.3960	−1.05473
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5.50666	−0.237189
$540$	0	0
$541$	13.6521	0.586950	0.293475	−	0.955967i	$-0.405188\pi$
$541$	13.6521	0.586950	0.293475	+	0.955967i	$0.405188\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−21.6645	−0.928004
$546$	0	0
$547$	21.0489	0.899985	0.449993	−	0.893032i	$-0.351427\pi$
$547$	21.0489	0.899985	0.449993	+	0.893032i	$0.351427\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	7.95392	0.338848
$552$	0	0
$553$	−6.62120	−0.281562
$554$	0	0
$555$	0	0
$556$	0	0
$557$	43.6750	1.85057	0.925284	−	0.379275i	$-0.123827\pi$
$557$	43.6750	1.85057	0.925284	+	0.379275i	$0.123827\pi$
$558$	0	0
$559$	−1.63331	−0.0690816
$560$	0	0
$561$	0	0
$562$	0	0
$563$	14.6201	0.616163	0.308082	−	0.951360i	$-0.400313\pi$
$563$	14.6201	0.616163	0.308082	+	0.951360i	$0.400313\pi$
$564$	0	0
$565$	21.5833	0.908017
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.54311	0.0646907	0.0323453	−	0.999477i	$-0.489702\pi$
$569$	1.54311	0.0646907	0.0323453	+	0.999477i	$0.489702\pi$
$570$	0	0
$571$	3.23031	0.135184	0.0675921	−	0.997713i	$-0.478468\pi$
$571$	3.23031	0.135184	0.0675921	+	0.997713i	$0.478468\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.25670	0.0941108
$576$	0	0
$577$	12.0678	0.502391	0.251195	−	0.967936i	$-0.419176\pi$
$577$	12.0678	0.502391	0.251195	+	0.967936i	$0.419176\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.33993	−0.0970768
$582$	0	0
$583$	1.91836	0.0794503
$584$	0	0
$585$	0	0
$586$	0	0
$587$	18.9165	0.780768	0.390384	−	0.920652i	$-0.372342\pi$
$587$	18.9165	0.780768	0.390384	+	0.920652i	$0.372342\pi$
$588$	0	0
$589$	6.66728	0.274721
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.9703	1.76458	0.882289	−	0.470708i	$-0.156002\pi$
$593$	42.9703	1.76458	0.882289	+	0.470708i	$0.156002\pi$
$594$	0	0
$595$	−4.54382	−0.186278
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−45.3930	−1.85471	−0.927354	−	0.374186i	$-0.877922\pi$
$599$	−45.3930	−1.85471	−0.927354	+	0.374186i	$0.877922\pi$
$600$	0	0
$601$	−24.9712	−1.01859	−0.509297	−	0.860591i	$-0.670095\pi$
$601$	−24.9712	−1.01859	−0.509297	+	0.860591i	$0.670095\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−29.4255	−1.19632
$606$	0	0
$607$	−16.8131	−0.682424	−0.341212	−	0.939986i	$-0.610837\pi$
$607$	−16.8131	−0.682424	−0.341212	+	0.939986i	$0.610837\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.736085	−0.0297788
$612$	0	0
$613$	0.894115	0.0361130	0.0180565	−	0.999837i	$-0.494252\pi$
$613$	0.894115	0.0361130	0.0180565	+	0.999837i	$0.494252\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−13.4396	−0.541056	−0.270528	−	0.962712i	$-0.587198\pi$
$617$	−13.4396	−0.541056	−0.270528	+	0.962712i	$0.587198\pi$
$618$	0	0
$619$	2.96332	0.119106	0.0595530	−	0.998225i	$-0.481032\pi$
$619$	2.96332	0.119106	0.0595530	+	0.998225i	$0.481032\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.56839	−0.0628363
$624$	0	0
$625$	−4.40633	−0.176253
$626$	0	0
$627$	0	0
$628$	0	0
$629$	28.3184	1.12913
$630$	0	0
$631$	−17.6849	−0.704023	−0.352012	−	0.935996i	$-0.614502\pi$
$631$	−17.6849	−0.704023	−0.352012	+	0.935996i	$0.614502\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−21.1440	−0.839074
$636$	0	0
$637$	1.15829	0.0458931
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−13.6002	−0.537174	−0.268587	−	0.963255i	$-0.586557\pi$
$641$	−13.6002	−0.537174	−0.268587	+	0.963255i	$0.586557\pi$
$642$	0	0
$643$	−13.5492	−0.534328	−0.267164	−	0.963651i	$-0.586087\pi$
$643$	−13.5492	−0.534328	−0.267164	+	0.963651i	$0.586087\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.0396	−0.394697	−0.197349	−	0.980333i	$-0.563233\pi$
$647$	−10.0396	−0.394697	−0.197349	+	0.980333i	$0.563233\pi$
$648$	0	0
$649$	77.6223	3.04694
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.3408	−1.26559	−0.632797	−	0.774318i	$-0.718093\pi$
$653$	−32.3408	−1.26559	−0.632797	+	0.774318i	$0.718093\pi$
$654$	0	0
$655$	−8.07860	−0.315657
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−30.5883	−1.19155	−0.595776	−	0.803151i	$-0.703155\pi$
$659$	−30.5883	−1.19155	−0.595776	+	0.803151i	$0.703155\pi$
$660$	0	0
$661$	4.03597	0.156981	0.0784905	−	0.996915i	$-0.474990\pi$
$661$	4.03597	0.156981	0.0784905	+	0.996915i	$0.474990\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.52280	−0.0590516
$666$	0	0
$667$	−6.69489	−0.259227
$668$	0	0
$669$	0	0
$670$	0	0
$671$	38.6328	1.49140
$672$	0	0
$673$	−9.29434	−0.358270	−0.179135	−	0.983824i	$-0.557330\pi$
$673$	−9.29434	−0.358270	−0.179135	+	0.983824i	$0.557330\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32.4111	1.24566	0.622829	−	0.782358i	$-0.285983\pi$
$677$	32.4111	1.24566	0.622829	+	0.782358i	$0.285983\pi$
$678$	0	0
$679$	13.4977	0.517996
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−29.1840	−1.11669	−0.558347	−	0.829608i	$-0.688564\pi$
$683$	−29.1840	−1.11669	−0.558347	+	0.829608i	$0.688564\pi$
$684$	0	0
$685$	−1.61754	−0.0618029
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−0.403514	−0.0153727
$690$	0	0
$691$	−43.9489	−1.67190	−0.835948	−	0.548809i	$-0.815081\pi$
$691$	−43.9489	−1.67190	−0.835948	+	0.548809i	$0.815081\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.6906	−1.08830
$696$	0	0
$697$	−29.7011	−1.12501
$698$	0	0
$699$	0	0
$700$	0	0
$701$	13.5446	0.511571	0.255785	−	0.966734i	$-0.417666\pi$
$701$	13.5446	0.511571	0.255785	+	0.966734i	$0.417666\pi$
$702$	0	0
$703$	9.49052	0.357942
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.552739	0.0207879
$708$	0	0
$709$	26.2496	0.985825	0.492913	−	0.870079i	$-0.335932\pi$
$709$	26.2496	0.985825	0.492913	+	0.870079i	$0.335932\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.61192	−0.210168
$714$	0	0
$715$	9.71288	0.363241
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−21.9069	−0.816989	−0.408494	−	0.912761i	$-0.633946\pi$
$719$	−21.9069	−0.816989	−0.408494	+	0.912761i	$0.633946\pi$
$720$	0	0
$721$	−3.56606	−0.132807
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−21.3252	−0.791997
$726$	0	0
$727$	29.5100	1.09446	0.547232	−	0.836981i	$-0.315682\pi$
$727$	29.5100	1.09446	0.547232	+	0.836981i	$0.315682\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−4.20755	−0.155622
$732$	0	0
$733$	35.1131	1.29693	0.648465	−	0.761244i	$-0.275411\pi$
$733$	35.1131	1.29693	0.648465	+	0.761244i	$0.275411\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	18.1744	0.669464
$738$	0	0
$739$	40.2785	1.48167	0.740833	−	0.671689i	$-0.234431\pi$
$739$	40.2785	1.48167	0.740833	+	0.671689i	$0.234431\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.19082	0.263805	0.131903	−	0.991263i	$-0.457891\pi$
$743$	7.19082	0.263805	0.131903	+	0.991263i	$0.457891\pi$
$744$	0	0
$745$	−35.1251	−1.28688
$746$	0	0
$747$	0	0
$748$	0	0
$749$	16.0205	0.585378
$750$	0	0
$751$	0.902222	0.0329226	0.0164613	−	0.999865i	$-0.494760\pi$
$751$	0.902222	0.0329226	0.0164613	+	0.999865i	$0.494760\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	1.54737	0.0563146
$756$	0	0
$757$	−49.3719	−1.79445	−0.897227	−	0.441570i	$-0.854422\pi$
$757$	−49.3719	−1.79445	−0.897227	+	0.441570i	$0.854422\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	25.7582	0.933734	0.466867	−	0.884327i	$-0.345383\pi$
$761$	25.7582	0.933734	0.466867	+	0.884327i	$0.345383\pi$
$762$	0	0
$763$	14.2268	0.515043
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−16.3273	−0.589546
$768$	0	0
$769$	47.7844	1.72315	0.861574	−	0.507631i	$-0.169479\pi$
$769$	47.7844	1.72315	0.861574	+	0.507631i	$0.169479\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.5911	0.956416	0.478208	−	0.878247i	$-0.341287\pi$
$773$	26.5911	0.956416	0.478208	+	0.878247i	$0.341287\pi$
$774$	0	0
$775$	−17.8756	−0.642109
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.95392	−0.356636
$780$	0	0
$781$	80.2629	2.87203
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−7.71543	−0.275376
$786$	0	0
$787$	51.9142	1.85054	0.925270	−	0.379309i	$-0.123838\pi$
$787$	51.9142	1.85054	0.925270	+	0.379309i	$0.123838\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.1735	−0.503950
$792$	0	0
$793$	−8.12616	−0.288569
$794$	0	0
$795$	0	0
$796$	0	0
$797$	20.2226	0.716320	0.358160	−	0.933660i	$-0.383404\pi$
$797$	20.2226	0.716320	0.358160	+	0.933660i	$0.383404\pi$
$798$	0	0
$799$	−1.89622	−0.0670836
$800$	0	0
$801$	0	0
$802$	0	0
$803$	12.4760	0.440268
$804$	0	0
$805$	1.28175	0.0451759
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.2186	−0.570215	−0.285107	−	0.958496i	$-0.592029\pi$
$809$	−16.2186	−0.570215	−0.285107	+	0.958496i	$0.592029\pi$
$810$	0	0
$811$	−15.3107	−0.537632	−0.268816	−	0.963191i	$-0.586632\pi$
$811$	−15.3107	−0.537632	−0.268816	+	0.963191i	$0.586632\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	19.3081	0.676335
$816$	0	0
$817$	−1.41010	−0.0493332
$818$	0	0
$819$	0	0
$820$	0	0
$821$	28.8945	1.00843	0.504213	−	0.863579i	$-0.331783\pi$
$821$	28.8945	1.00843	0.504213	+	0.863579i	$0.331783\pi$
$822$	0	0
$823$	−55.3879	−1.93070	−0.965350	−	0.260960i	$-0.915961\pi$
$823$	−55.3879	−1.93070	−0.965350	+	0.260960i	$0.915961\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.8094	−0.514975	−0.257487	−	0.966282i	$-0.582895\pi$
$827$	−14.8094	−0.514975	−0.257487	+	0.966282i	$0.582895\pi$
$828$	0	0
$829$	8.75100	0.303935	0.151967	−	0.988386i	$-0.451439\pi$
$829$	8.75100	0.303935	0.151967	+	0.988386i	$0.451439\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	2.98386	0.103385
$834$	0	0
$835$	−23.4248	−0.810647
$836$	0	0
$837$	0	0
$838$	0	0
$839$	37.9859	1.31142	0.655709	−	0.755014i	$-0.272370\pi$
$839$	37.9859	1.31142	0.655709	+	0.755014i	$0.272370\pi$
$840$	0	0
$841$	34.2649	1.18155
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.7533	0.610733
$846$	0	0
$847$	19.3233	0.663957
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−7.98827	−0.273834
$852$	0	0
$853$	−6.37246	−0.218189	−0.109094	−	0.994031i	$-0.534795\pi$
$853$	−6.37246	−0.218189	−0.109094	+	0.994031i	$0.534795\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	58.3717	1.99394	0.996970	−	0.0777879i	$-0.0247857\pi$
$857$	58.3717	1.99394	0.996970	+	0.0777879i	$0.0247857\pi$
$858$	0	0
$859$	−14.4372	−0.492592	−0.246296	−	0.969195i	$-0.579214\pi$
$859$	−14.4372	−0.492592	−0.246296	+	0.969195i	$0.579214\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−35.4975	−1.20835	−0.604174	−	0.796852i	$-0.706497\pi$
$863$	−35.4975	−1.20835	−0.604174	+	0.796852i	$0.706497\pi$
$864$	0	0
$865$	−23.3443	−0.793731
$866$	0	0
$867$	0	0
$868$	0	0
$869$	36.4607	1.23685
$870$	0	0
$871$	−3.82287	−0.129533
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.6967	0.395422
$876$	0	0
$877$	12.3429	0.416789	0.208394	−	0.978045i	$-0.433176\pi$
$877$	12.3429	0.416789	0.208394	+	0.978045i	$0.433176\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	28.5321	0.961270	0.480635	−	0.876921i	$-0.340406\pi$
$881$	28.5321	0.961270	0.480635	+	0.876921i	$0.340406\pi$
$882$	0	0
$883$	−35.2677	−1.18685	−0.593426	−	0.804888i	$-0.702225\pi$
$883$	−35.2677	−1.18685	−0.593426	+	0.804888i	$0.702225\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−14.2146	−0.477280	−0.238640	−	0.971108i	$-0.576702\pi$
$887$	−14.2146	−0.477280	−0.238640	+	0.971108i	$0.576702\pi$
$888$	0	0
$889$	13.8850	0.465687
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−0.635493	−0.0212660
$894$	0	0
$895$	−23.3963	−0.782052
$896$	0	0
$897$	0	0
$898$	0	0
$899$	53.0310	1.76868
$900$	0	0
$901$	−1.03949	−0.0346304
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.08690	0.202336
$906$	0	0
$907$	0.342749	0.0113808	0.00569040	−	0.999984i	$-0.498189\pi$
$907$	0.342749	0.0113808	0.00569040	+	0.999984i	$0.498189\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−38.8036	−1.28562	−0.642810	−	0.766026i	$-0.722232\pi$
$911$	−38.8036	−1.28562	−0.642810	+	0.766026i	$0.722232\pi$
$912$	0	0
$913$	12.8852	0.426439
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.30511	0.175190
$918$	0	0
$919$	5.52158	0.182140	0.0910700	−	0.995844i	$-0.470971\pi$
$919$	5.52158	0.182140	0.0910700	+	0.995844i	$0.470971\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−16.8828	−0.555704
$924$	0	0
$925$	−25.4449	−0.836624
$926$	0	0
$927$	0	0
$928$	0	0
$929$	15.7718	0.517455	0.258727	−	0.965950i	$-0.416697\pi$
$929$	15.7718	0.517455	0.258727	+	0.965950i	$0.416697\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	25.0213	0.818283
$936$	0	0
$937$	29.4650	0.962579	0.481290	−	0.876562i	$-0.340169\pi$
$937$	29.4650	0.962579	0.481290	+	0.876562i	$0.340169\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−1.12235	−0.0365876	−0.0182938	−	0.999833i	$-0.505823\pi$
$941$	−1.12235	−0.0365876	−0.0182938	+	0.999833i	$0.505823\pi$
$942$	0	0
$943$	8.37831	0.272835
$944$	0	0
$945$	0	0
$946$	0	0
$947$	21.8952	0.711497	0.355748	−	0.934582i	$-0.384226\pi$
$947$	21.8952	0.711497	0.355748	+	0.934582i	$0.384226\pi$
$948$	0	0
$949$	−2.62424	−0.0851865
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.88912	0.0611947	0.0305974	−	0.999532i	$-0.490259\pi$
$953$	1.88912	0.0611947	0.0305974	+	0.999532i	$0.490259\pi$
$954$	0	0
$955$	−14.9132	−0.482580
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.06221	0.0343007
$960$	0	0
$961$	13.4527	0.433957
$962$	0	0
$963$	0	0
$964$	0	0
$965$	30.6397	0.986326
$966$	0	0
$967$	−19.9874	−0.642752	−0.321376	−	0.946952i	$-0.604145\pi$
$967$	−19.9874	−0.642752	−0.321376	+	0.946952i	$0.604145\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	48.0256	1.54121	0.770607	−	0.637311i	$-0.219953\pi$
$971$	48.0256	1.54121	0.770607	+	0.637311i	$0.219953\pi$
$972$	0	0
$973$	18.8407	0.604007
$974$	0	0
$975$	0	0
$976$	0	0
$977$	14.9221	0.477401	0.238701	−	0.971093i	$-0.423279\pi$
$977$	14.9221	0.477401	0.238701	+	0.971093i	$0.423279\pi$
$978$	0	0
$979$	8.63661	0.276027
$980$	0	0
$981$	0	0
$982$	0	0
$983$	15.0194	0.479045	0.239523	−	0.970891i	$-0.423009\pi$
$983$	15.0194	0.479045	0.239523	+	0.970891i	$0.423009\pi$
$984$	0	0
$985$	−25.1601	−0.801669
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.18690	0.0377411
$990$	0	0
$991$	40.2089	1.27728	0.638640	−	0.769506i	$-0.279497\pi$
$991$	40.2089	1.27728	0.638640	+	0.769506i	$0.279497\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−19.5792	−0.620701
$996$	0	0
$997$	−25.0968	−0.794824	−0.397412	−	0.917640i	$-0.630092\pi$
$997$	−25.0968	−0.794824	−0.397412	+	0.917640i	$0.630092\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.ci.1.2		4
3.2	odd	2		1064.2.a.h.1.2	✓	4
12.11	even	2		2128.2.a.t.1.4		4
21.20	even	2		7448.2.a.bj.1.3		4
24.5	odd	2		8512.2.a.bq.1.3		4
24.11	even	2		8512.2.a.bu.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1064.2.a.h.1.2	✓	4	3.2	odd	2
2128.2.a.t.1.4		4	12.11	even	2
7448.2.a.bj.1.3		4	21.20	even	2
8512.2.a.bq.1.3		4	24.5	odd	2
8512.2.a.bu.1.1		4	24.11	even	2
9576.2.a.ci.1.2		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.52280 q^{5} +1.00000 q^{7} -5.50666 q^{11} +1.15829 q^{13} +2.98386 q^{17} +1.00000 q^{19} -0.841710 q^{23} -2.68109 q^{25} +7.95392 q^{29} +6.66728 q^{31} -1.52280 q^{35} +9.49052 q^{37} -9.95392 q^{41} -1.41010 q^{43} -0.635493 q^{47} +1.00000 q^{49} -0.348371 q^{53} +8.38553 q^{55} -14.0961 q^{59} -7.01565 q^{61} -1.76384 q^{65} -3.30044 q^{67} -14.5756 q^{71} -2.26562 q^{73} -5.50666 q^{77} -6.62120 q^{79} -2.33993 q^{83} -4.54382 q^{85} -1.56839 q^{89} +1.15829 q^{91} -1.52280 q^{95} +13.4977 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{5} + 4 q^{7} - 2 q^{11} + 2 q^{13} - 3 q^{17} + 4 q^{19} - 6 q^{23} - 3 q^{25} + 17 q^{31} - q^{35} + 3 q^{37} - 8 q^{41} + 7 q^{43} - 5 q^{47} + 4 q^{49} + 16 q^{53} + 17 q^{55} - 7 q^{59} - q^{61}+ \cdots + 3 q^{97}+O(q^{100}) \) 4 * q - q^5 + 4 * q^7 - 2 * q^11 + 2 * q^13 - 3 * q^17 + 4 * q^19 - 6 * q^23 - 3 * q^25 + 17 * q^31 - q^35 + 3 * q^37 - 8 * q^41 + 7 * q^43 - 5 * q^47 + 4 * q^49 + 16 * q^53 + 17 * q^55 - 7 * q^59 - q^61 + 10 * q^65 + 7 * q^67 - 27 * q^71 - q^73 - 2 * q^77 + 15 * q^79 - 3 * q^83 + q^85 + 9 * q^89 + 2 * q^91 - q^95 + 3 * q^97

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