LMFDB - Embedded newform 9576.2.a.co.1.3

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:	$5$
Coefficient field:	5.5.135076.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 5x^3 + 4x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 3192)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.420632$ 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)

$f(q)$	$=$	$q-0.841263 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-0.841263 q^{5} +1.00000 q^{7} +4.29330 q^{11} +5.02211 q^{13} -6.86235 q^{17} +1.00000 q^{19} +3.02211 q^{23} -4.29228 q^{25} +2.18084 q^{29} +7.31540 q^{31} -0.841263 q^{35} +3.27119 q^{37} +0.317473 q^{41} +12.7443 q^{47} +1.00000 q^{49} -6.18084 q^{53} -3.61180 q^{55} -14.1963 q^{59} +6.00000 q^{61} -4.22491 q^{65} +1.86088 q^{67} +7.11143 q^{71} -2.74989 q^{73} +4.29330 q^{77} +1.72778 q^{79} +6.72323 q^{83} +5.77304 q^{85} -3.29432 q^{89} +5.02211 q^{91} -0.841263 q^{95} -1.43344 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{5} + 5 q^{7}+O(q^{10}) $ 5 * q - 2 * q^5 + 5 * q^7	$ 5 q - 2 q^{5} + 5 q^{7} - 2 q^{11} + 8 q^{13} + 2 q^{17} + 5 q^{19} - 2 q^{23} + 19 q^{25} - 4 q^{29} - 4 q^{31} - 2 q^{35} + 10 q^{37} + 6 q^{41} + 2 q^{47} + 5 q^{49} - 16 q^{53} - 16 q^{55} + 12 q^{59} + 30 q^{61} - 4 q^{65} + 18 q^{67} + 10 q^{71} + 14 q^{73} - 2 q^{77} - 2 q^{79} + 6 q^{83} + 12 q^{85} - 10 q^{89} + 8 q^{91} - 2 q^{95} + 8 q^{97}+O(q^{100}) $ 5 * q - 2 * q^5 + 5 * q^7 - 2 * q^11 + 8 * q^13 + 2 * q^17 + 5 * q^19 - 2 * q^23 + 19 * q^25 - 4 * q^29 - 4 * q^31 - 2 * q^35 + 10 * q^37 + 6 * q^41 + 2 * q^47 + 5 * q^49 - 16 * q^53 - 16 * q^55 + 12 * q^59 + 30 * q^61 - 4 * q^65 + 18 * q^67 + 10 * q^71 + 14 * q^73 - 2 * q^77 - 2 * q^79 + 6 * q^83 + 12 * q^85 - 10 * q^89 + 8 * q^91 - 2 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

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
$5$	−0.841263	−0.376224	−0.188112	−	0.982148i	$-0.560237\pi$
$5$	−0.841263	−0.376224	−0.188112	+	0.982148i	$0.560237\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.29330	1.29448	0.647239	−	0.762287i	$-0.275924\pi$
$11$	4.29330	1.29448	0.647239	+	0.762287i	$0.275924\pi$
$12$	0	0
$13$	5.02211	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$13$	5.02211	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.86235	−1.66436	−0.832182	−	0.554503i	$-0.812908\pi$
$17$	−6.86235	−1.66436	−0.832182	+	0.554503i	$0.812908\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.02211	0.630153	0.315076	−	0.949066i	$-0.397970\pi$
$23$	3.02211	0.630153	0.315076	+	0.949066i	$0.397970\pi$
$24$	0	0
$25$	−4.29228	−0.858455
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.18084	0.404972	0.202486	−	0.979285i	$-0.435098\pi$
$29$	2.18084	0.404972	0.202486	+	0.979285i	$0.435098\pi$
$30$	0	0
$31$	7.31540	1.31389	0.656943	−	0.753940i	$-0.271849\pi$
$31$	7.31540	1.31389	0.656943	+	0.753940i	$0.271849\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.841263	−0.142199
$36$	0	0
$37$	3.27119	0.537781	0.268890	−	0.963171i	$-0.413343\pi$
$37$	3.27119	0.537781	0.268890	+	0.963171i	$0.413343\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.317473	0.0495810	0.0247905	−	0.999693i	$-0.492108\pi$
$41$	0.317473	0.0495810	0.0247905	+	0.999693i	$0.492108\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12.7443	1.85895	0.929474	−	0.368887i	$-0.120261\pi$
$47$	12.7443	1.85895	0.929474	+	0.368887i	$0.120261\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−6.18084	−0.849004	−0.424502	−	0.905427i	$-0.639551\pi$
$53$	−6.18084	−0.849004	−0.424502	+	0.905427i	$0.639551\pi$
$54$	0	0
$55$	−3.61180	−0.487014
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−14.1963	−1.84821	−0.924103	−	0.382142i	$-0.875186\pi$
$59$	−14.1963	−1.84821	−0.924103	+	0.382142i	$0.875186\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.22491	−0.524036
$66$	0	0
$67$	1.86088	0.227343	0.113672	−	0.993518i	$-0.463739\pi$
$67$	1.86088	0.227343	0.113672	+	0.993518i	$0.463739\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	7.11143	0.843972	0.421986	−	0.906602i	$-0.361333\pi$
$71$	7.11143	0.843972	0.421986	+	0.906602i	$0.361333\pi$
$72$	0	0
$73$	−2.74989	−0.321850	−0.160925	−	0.986967i	$-0.551448\pi$
$73$	−2.74989	−0.321850	−0.160925	+	0.986967i	$0.551448\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.29330	0.489267
$78$	0	0
$79$	1.72778	0.194391	0.0971954	−	0.995265i	$-0.469013\pi$
$79$	1.72778	0.194391	0.0971954	+	0.995265i	$0.469013\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.72323	0.737970	0.368985	−	0.929435i	$-0.379705\pi$
$83$	6.72323	0.737970	0.368985	+	0.929435i	$0.379705\pi$
$84$	0	0
$85$	5.77304	0.626174
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.29432	−0.349197	−0.174599	−	0.984640i	$-0.555863\pi$
$89$	−3.29432	−0.349197	−0.174599	+	0.984640i	$0.555863\pi$
$90$	0	0
$91$	5.02211	0.526460
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.841263	−0.0863118
$96$	0	0
$97$	−1.43344	−0.145544	−0.0727718	−	0.997349i	$-0.523184\pi$
$97$	−1.43344	−0.145544	−0.0727718	+	0.997349i	$0.523184\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−5.47621	−0.544903	−0.272452	−	0.962169i	$-0.587834\pi$
$101$	−5.47621	−0.544903	−0.272452	+	0.962169i	$0.587834\pi$
$102$	0	0
$103$	16.2902	1.60512	0.802561	−	0.596570i	$-0.203470\pi$
$103$	16.2902	1.60512	0.802561	+	0.596570i	$0.203470\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.5173	−1.69346	−0.846732	−	0.532020i	$-0.821433\pi$
$107$	−17.5173	−1.69346	−0.846732	+	0.532020i	$0.821433\pi$
$108$	0	0
$109$	−2.99793	−0.287150	−0.143575	−	0.989639i	$-0.545860\pi$
$109$	−2.99793	−0.287150	−0.143575	+	0.989639i	$0.545860\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	17.3098	1.62837	0.814186	−	0.580604i	$-0.197184\pi$
$113$	17.3098	1.62837	0.814186	+	0.580604i	$0.197184\pi$
$114$	0	0
$115$	−2.54239	−0.237079
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.86235	−0.629070
$120$	0	0
$121$	7.43242	0.675674
$122$	0	0
$123$	0	0
$124$	0	0
$125$	7.81725	0.699196
$126$	0	0
$127$	−13.4545	−1.19390	−0.596948	−	0.802280i	$-0.703620\pi$
$127$	−13.4545	−1.19390	−0.596948	+	0.802280i	$0.703620\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.3098	−1.33763	−0.668813	−	0.743431i	$-0.733197\pi$
$131$	−15.3098	−1.33763	−0.668813	+	0.743431i	$0.733197\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.0442	−1.37075	−0.685375	−	0.728191i	$-0.740362\pi$
$137$	−16.0442	−1.37075	−0.685375	+	0.728191i	$0.740362\pi$
$138$	0	0
$139$	13.8347	1.17344	0.586720	−	0.809790i	$-0.300419\pi$
$139$	13.8347	1.17344	0.586720	+	0.809790i	$0.300419\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	21.5614	1.80305
$144$	0	0
$145$	−1.83466	−0.152360
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.41133	−0.525237	−0.262618	−	0.964900i	$-0.584586\pi$
$149$	−6.41133	−0.525237	−0.262618	+	0.964900i	$0.584586\pi$
$150$	0	0
$151$	10.0895	0.821069	0.410535	−	0.911845i	$-0.365342\pi$
$151$	10.0895	0.821069	0.410535	+	0.911845i	$0.365342\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−6.15418	−0.494316
$156$	0	0
$157$	19.2215	1.53405	0.767023	−	0.641619i	$-0.221737\pi$
$157$	19.2215	1.53405	0.767023	+	0.641619i	$0.221737\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.02211	0.238175
$162$	0	0
$163$	14.1542	1.10864	0.554321	−	0.832303i	$-0.312978\pi$
$163$	14.1542	1.10864	0.554321	+	0.832303i	$0.312978\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−7.29228	−0.564293	−0.282147	−	0.959371i	$-0.591046\pi$
$167$	−7.29228	−0.564293	−0.282147	+	0.959371i	$0.591046\pi$
$168$	0	0
$169$	12.2215	0.940119
$170$	0	0
$171$	0	0
$172$	0	0
$173$	17.9251	1.36282	0.681411	−	0.731901i	$-0.261367\pi$
$173$	17.9251	1.36282	0.681411	+	0.731901i	$0.261367\pi$
$174$	0	0
$175$	−4.29228	−0.324466
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.58309	0.267813	0.133906	−	0.990994i	$-0.457248\pi$
$179$	3.58309	0.267813	0.133906	+	0.990994i	$0.457248\pi$
$180$	0	0
$181$	−20.9010	−1.55356	−0.776779	−	0.629774i	$-0.783148\pi$
$181$	−20.9010	−1.55356	−0.776779	+	0.629774i	$0.783148\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.75194	−0.202326
$186$	0	0
$187$	−29.4621	−2.15448
$188$	0	0
$189$	0	0
$190$	0	0
$191$	12.5219	0.906052	0.453026	−	0.891497i	$-0.350345\pi$
$191$	12.5219	0.906052	0.453026	+	0.891497i	$0.350345\pi$
$192$	0	0
$193$	22.0422	1.58663	0.793315	−	0.608812i	$-0.208353\pi$
$193$	22.0422	1.58663	0.793315	+	0.608812i	$0.208353\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.49274	−0.605083	−0.302541	−	0.953136i	$-0.597835\pi$
$197$	−8.49274	−0.605083	−0.302541	+	0.953136i	$0.597835\pi$
$198$	0	0
$199$	−4.74989	−0.336711	−0.168355	−	0.985726i	$-0.553846\pi$
$199$	−4.74989	−0.336711	−0.168355	+	0.985726i	$0.553846\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.18084	0.153065
$204$	0	0
$205$	−0.267079	−0.0186536
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.29330	0.296974
$210$	0	0
$211$	−14.1280	−0.972609	−0.486304	−	0.873789i	$-0.661655\pi$
$211$	−14.1280	−0.972609	−0.486304	+	0.873789i	$0.661655\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.31540	0.496602
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−34.4634	−2.31826
$222$	0	0
$223$	−27.9442	−1.87128	−0.935640	−	0.352956i	$-0.885177\pi$
$223$	−27.9442	−1.87128	−0.935640	+	0.352956i	$0.885177\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.3807	1.55183	0.775916	−	0.630837i	$-0.217288\pi$
$227$	23.3807	1.55183	0.775916	+	0.630837i	$0.217288\pi$
$228$	0	0
$229$	20.2691	1.33942	0.669711	−	0.742622i	$-0.266418\pi$
$229$	20.2691	1.33942	0.669711	+	0.742622i	$0.266418\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.3133	−1.46180	−0.730898	−	0.682487i	$-0.760898\pi$
$233$	−22.3133	−1.46180	−0.730898	+	0.682487i	$0.760898\pi$
$234$	0	0
$235$	−10.7213	−0.699382
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−14.4665	−0.935761	−0.467881	−	0.883792i	$-0.654982\pi$
$239$	−14.4665	−0.935761	−0.467881	+	0.883792i	$0.654982\pi$
$240$	0	0
$241$	7.92957	0.510788	0.255394	−	0.966837i	$-0.417795\pi$
$241$	7.92957	0.510788	0.255394	+	0.966837i	$0.417795\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.841263	−0.0537463
$246$	0	0
$247$	5.02211	0.319549
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.8096	−1.56597	−0.782984	−	0.622042i	$-0.786303\pi$
$251$	−24.8096	−1.56597	−0.782984	+	0.622042i	$0.786303\pi$
$252$	0	0
$253$	12.9748	0.815719
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.154182	−0.00961761	−0.00480880	−	0.999988i	$-0.501531\pi$
$257$	−0.154182	−0.00961761	−0.00480880	+	0.999988i	$0.501531\pi$
$258$	0	0
$259$	3.27119	0.203262
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−25.2450	−1.55667	−0.778336	−	0.627848i	$-0.783936\pi$
$263$	−25.2450	−1.55667	−0.778336	+	0.627848i	$0.783936\pi$
$264$	0	0
$265$	5.19972	0.319416
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.06736	−0.308963	−0.154481	−	0.987996i	$-0.549371\pi$
$269$	−5.06736	−0.308963	−0.154481	+	0.987996i	$0.549371\pi$
$270$	0	0
$271$	26.6730	1.62027	0.810134	−	0.586245i	$-0.199394\pi$
$271$	26.6730	1.62027	0.810134	+	0.586245i	$0.199394\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−18.4280	−1.11125
$276$	0	0
$277$	5.34062	0.320887	0.160444	−	0.987045i	$-0.448708\pi$
$277$	5.34062	0.320887	0.160444	+	0.987045i	$0.448708\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.50168	0.208893	0.104446	−	0.994531i	$-0.466693\pi$
$281$	3.50168	0.208893	0.104446	+	0.994531i	$0.466693\pi$
$282$	0	0
$283$	7.48000	0.444640	0.222320	−	0.974974i	$-0.428637\pi$
$283$	7.48000	0.444640	0.222320	+	0.974974i	$0.428637\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.317473	0.0187399
$288$	0	0
$289$	30.0918	1.77011
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−8.79410	−0.513757	−0.256878	−	0.966444i	$-0.582694\pi$
$293$	−8.79410	−0.513757	−0.256878	+	0.966444i	$0.582694\pi$
$294$	0	0
$295$	11.9429	0.695341
$296$	0	0
$297$	0	0
$298$	0	0
$299$	15.1773	0.877728
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−5.04758	−0.289024
$306$	0	0
$307$	−10.1963	−0.581936	−0.290968	−	0.956733i	$-0.593977\pi$
$307$	−10.1963	−0.581936	−0.290968	+	0.956733i	$0.593977\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	24.6960	1.40038	0.700190	−	0.713957i	$-0.253099\pi$
$311$	24.6960	1.40038	0.700190	+	0.713957i	$0.253099\pi$
$312$	0	0
$313$	16.8313	0.951361	0.475680	−	0.879618i	$-0.342202\pi$
$313$	16.8313	0.951361	0.475680	+	0.879618i	$0.342202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−16.9749	−0.953408	−0.476704	−	0.879064i	$-0.658169\pi$
$317$	−16.9749	−0.953408	−0.476704	+	0.879064i	$0.658169\pi$
$318$	0	0
$319$	9.36301	0.524228
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−6.86235	−0.381831
$324$	0	0
$325$	−21.5563	−1.19573
$326$	0	0
$327$	0	0
$328$	0	0
$329$	12.7443	0.702617
$330$	0	0
$331$	23.3123	1.28136	0.640680	−	0.767808i	$-0.278652\pi$
$331$	23.3123	1.28136	0.640680	+	0.767808i	$0.278652\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.56549	−0.0855320
$336$	0	0
$337$	−12.3596	−0.673272	−0.336636	−	0.941635i	$-0.609289\pi$
$337$	−12.3596	−0.673272	−0.336636	+	0.941635i	$0.609289\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	31.4072	1.70080
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.7414	1.22082	0.610410	−	0.792086i	$-0.291005\pi$
$347$	22.7414	1.22082	0.610410	+	0.792086i	$0.291005\pi$
$348$	0	0
$349$	33.2596	1.78034	0.890172	−	0.455625i	$-0.150584\pi$
$349$	33.2596	1.78034	0.890172	+	0.455625i	$0.150584\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.90993	0.208104	0.104052	−	0.994572i	$-0.466819\pi$
$353$	3.90993	0.208104	0.104052	+	0.994572i	$0.466819\pi$
$354$	0	0
$355$	−5.98259	−0.317523
$356$	0	0
$357$	0	0
$358$	0	0
$359$	16.2436	0.857307	0.428654	−	0.903469i	$-0.358988\pi$
$359$	16.2436	0.857307	0.428654	+	0.903469i	$0.358988\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.31338	0.121088
$366$	0	0
$367$	−13.4465	−0.701899	−0.350950	−	0.936394i	$-0.614141\pi$
$367$	−13.4465	−0.701899	−0.350950	+	0.936394i	$0.614141\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.18084	−0.320893
$372$	0	0
$373$	10.6846	0.553227	0.276614	−	0.960981i	$-0.410788\pi$
$373$	10.6846	0.553227	0.276614	+	0.960981i	$0.410788\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	10.9524	0.564078
$378$	0	0
$379$	0.727761	0.0373826	0.0186913	−	0.999825i	$-0.494050\pi$
$379$	0.727761	0.0373826	0.0186913	+	0.999825i	$0.494050\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	30.6197	1.56459	0.782296	−	0.622907i	$-0.214049\pi$
$383$	30.6197	1.56459	0.782296	+	0.622907i	$0.214049\pi$
$384$	0	0
$385$	−3.61180	−0.184074
$386$	0	0
$387$	0	0
$388$	0	0
$389$	16.0827	0.815426	0.407713	−	0.913110i	$-0.366326\pi$
$389$	16.0827	0.815426	0.407713	+	0.913110i	$0.366326\pi$
$390$	0	0
$391$	−20.7387	−1.04880
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−1.45352	−0.0731346
$396$	0	0
$397$	−13.1290	−0.658925	−0.329462	−	0.944169i	$-0.606868\pi$
$397$	−13.1290	−0.658925	−0.329462	+	0.944169i	$0.606868\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	16.8844	0.843168	0.421584	−	0.906789i	$-0.361474\pi$
$401$	16.8844	0.843168	0.421584	+	0.906789i	$0.361474\pi$
$402$	0	0
$403$	36.7387	1.83009
$404$	0	0
$405$	0	0
$406$	0	0
$407$	14.0442	0.696146
$408$	0	0
$409$	−7.05434	−0.348815	−0.174407	−	0.984674i	$-0.555801\pi$
$409$	−7.05434	−0.348815	−0.174407	+	0.984674i	$0.555801\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−14.1963	−0.698557
$414$	0	0
$415$	−5.65601	−0.277642
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.45543	0.0711023	0.0355511	−	0.999368i	$-0.488681\pi$
$419$	1.45543	0.0711023	0.0355511	+	0.999368i	$0.488681\pi$
$420$	0	0
$421$	−2.36508	−0.115267	−0.0576334	−	0.998338i	$-0.518355\pi$
$421$	−2.36508	−0.115267	−0.0576334	+	0.998338i	$0.518355\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	29.4551	1.42878
$426$	0	0
$427$	6.00000	0.290360
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.8382	−0.907403	−0.453701	−	0.891154i	$-0.649897\pi$
$431$	−18.8382	−0.907403	−0.453701	+	0.891154i	$0.649897\pi$
$432$	0	0
$433$	15.2899	0.734787	0.367394	−	0.930066i	$-0.380250\pi$
$433$	15.2899	0.734787	0.367394	+	0.930066i	$0.380250\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.02211	0.144567
$438$	0	0
$439$	29.1235	1.38999	0.694996	−	0.719014i	$-0.255406\pi$
$439$	29.1235	1.38999	0.694996	+	0.719014i	$0.255406\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.796446	0.0378403	0.0189202	−	0.999821i	$-0.493977\pi$
$443$	0.796446	0.0378403	0.0189202	+	0.999821i	$0.493977\pi$
$444$	0	0
$445$	2.77139	0.131377
$446$	0	0
$447$	0	0
$448$	0	0
$449$	38.1060	1.79833	0.899166	−	0.437608i	$-0.144174\pi$
$449$	38.1060	1.79833	0.899166	+	0.437608i	$0.144174\pi$
$450$	0	0
$451$	1.36301	0.0641815
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−4.22491	−0.198067
$456$	0	0
$457$	−4.08140	−0.190920	−0.0954600	−	0.995433i	$-0.530432\pi$
$457$	−4.08140	−0.190920	−0.0954600	+	0.995433i	$0.530432\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	17.4812	0.814180	0.407090	−	0.913388i	$-0.366544\pi$
$461$	17.4812	0.814180	0.407090	+	0.913388i	$0.366544\pi$
$462$	0	0
$463$	−33.9496	−1.57777	−0.788886	−	0.614540i	$-0.789342\pi$
$463$	−33.9496	−1.57777	−0.788886	+	0.614540i	$0.789342\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	22.5914	1.04541	0.522703	−	0.852515i	$-0.324924\pi$
$467$	22.5914	1.04541	0.522703	+	0.852515i	$0.324924\pi$
$468$	0	0
$469$	1.86088	0.0859276
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4.29228	−0.196943
$476$	0	0
$477$	0	0
$478$	0	0
$479$	13.3421	0.609614	0.304807	−	0.952414i	$-0.401408\pi$
$479$	13.3421	0.609614	0.304807	+	0.952414i	$0.401408\pi$
$480$	0	0
$481$	16.4283	0.749065
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.20590	0.0547571
$486$	0	0
$487$	−0.905117	−0.0410148	−0.0205074	−	0.999790i	$-0.506528\pi$
$487$	−0.905117	−0.0410148	−0.0205074	+	0.999790i	$0.506528\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.330492	0.0149149	0.00745745	−	0.999972i	$-0.497626\pi$
$491$	0.330492	0.0149149	0.00745745	+	0.999972i	$0.497626\pi$
$492$	0	0
$493$	−14.9657	−0.674021
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7.11143	0.318991
$498$	0	0
$499$	−2.01741	−0.0903117	−0.0451559	−	0.998980i	$-0.514378\pi$
$499$	−2.01741	−0.0903117	−0.0451559	+	0.998980i	$0.514378\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−2.19695	−0.0979573	−0.0489786	−	0.998800i	$-0.515597\pi$
$503$	−2.19695	−0.0979573	−0.0489786	+	0.998800i	$0.515597\pi$
$504$	0	0
$505$	4.60693	0.205006
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.70905	0.164401	0.0822003	−	0.996616i	$-0.473805\pi$
$509$	3.70905	0.164401	0.0822003	+	0.996616i	$0.473805\pi$
$510$	0	0
$511$	−2.74989	−0.121648
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.7044	−0.603886
$516$	0	0
$517$	54.7151	2.40637
$518$	0	0
$519$	0	0
$520$	0	0
$521$	36.9707	1.61971	0.809857	−	0.586627i	$-0.199545\pi$
$521$	36.9707	1.61971	0.809857	+	0.586627i	$0.199545\pi$
$522$	0	0
$523$	5.88798	0.257464	0.128732	−	0.991679i	$-0.458909\pi$
$523$	5.88798	0.257464	0.128732	+	0.991679i	$0.458909\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−50.2008	−2.18678
$528$	0	0
$529$	−13.8669	−0.602908
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.59438	0.0690604
$534$	0	0
$535$	14.7367	0.637123
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.29330	0.184925
$540$	0	0
$541$	−26.9639	−1.15927	−0.579635	−	0.814876i	$-0.696805\pi$
$541$	−26.9639	−1.15927	−0.579635	+	0.814876i	$0.696805\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.52205	0.108033
$546$	0	0
$547$	−30.9506	−1.32335	−0.661677	−	0.749789i	$-0.730155\pi$
$547$	−30.9506	−1.32335	−0.661677	+	0.749789i	$0.730155\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.18084	0.0929070
$552$	0	0
$553$	1.72778	0.0734728
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19.5383	0.827863	0.413932	−	0.910308i	$-0.364155\pi$
$557$	19.5383	0.827863	0.413932	+	0.910308i	$0.364155\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	24.7499	1.04308	0.521542	−	0.853226i	$-0.325357\pi$
$563$	24.7499	1.04308	0.521542	+	0.853226i	$0.325357\pi$
$564$	0	0
$565$	−14.5621	−0.612633
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−7.03452	−0.294902	−0.147451	−	0.989069i	$-0.547107\pi$
$569$	−7.03452	−0.294902	−0.147451	+	0.989069i	$0.547107\pi$
$570$	0	0
$571$	5.06941	0.212148	0.106074	−	0.994358i	$-0.466172\pi$
$571$	5.06941	0.212148	0.106074	+	0.994358i	$0.466172\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−12.9717	−0.540958
$576$	0	0
$577$	26.6267	1.10848	0.554241	−	0.832356i	$-0.313008\pi$
$577$	26.6267	1.10848	0.554241	+	0.832356i	$0.313008\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	6.72323	0.278927
$582$	0	0
$583$	−26.5362	−1.09902
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.04275	0.125588	0.0627938	−	0.998027i	$-0.479999\pi$
$587$	3.04275	0.125588	0.0627938	+	0.998027i	$0.479999\pi$
$588$	0	0
$589$	7.31540	0.301426
$590$	0	0
$591$	0	0
$592$	0	0
$593$	42.6664	1.75210	0.876049	−	0.482223i	$-0.160170\pi$
$593$	42.6664	1.75210	0.876049	+	0.482223i	$0.160170\pi$
$594$	0	0
$595$	5.77304	0.236672
$596$	0	0
$597$	0	0
$598$	0	0
$599$	36.7878	1.50311	0.751554	−	0.659672i	$-0.229305\pi$
$599$	36.7878	1.50311	0.751554	+	0.659672i	$0.229305\pi$
$600$	0	0
$601$	−37.3789	−1.52472	−0.762359	−	0.647155i	$-0.775959\pi$
$601$	−37.3789	−1.52472	−0.762359	+	0.647155i	$0.775959\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.25262	−0.254205
$606$	0	0
$607$	5.51175	0.223715	0.111858	−	0.993724i	$-0.464320\pi$
$607$	5.51175	0.223715	0.111858	+	0.993724i	$0.464320\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	64.0033	2.58930
$612$	0	0
$613$	−5.92513	−0.239314	−0.119657	−	0.992815i	$-0.538179\pi$
$613$	−5.92513	−0.239314	−0.119657	+	0.992815i	$0.538179\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	25.9517	1.04477	0.522387	−	0.852708i	$-0.325042\pi$
$617$	25.9517	1.04477	0.522387	+	0.852708i	$0.325042\pi$
$618$	0	0
$619$	13.2419	0.532236	0.266118	−	0.963940i	$-0.414259\pi$
$619$	13.2419	0.532236	0.266118	+	0.963940i	$0.414259\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.29432	−0.131984
$624$	0	0
$625$	14.8850	0.595400
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−22.4481	−0.895063
$630$	0	0
$631$	18.8313	0.749662	0.374831	−	0.927093i	$-0.377701\pi$
$631$	18.8313	0.749662	0.374831	+	0.927093i	$0.377701\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.3188	0.449173
$636$	0	0
$637$	5.02211	0.198983
$638$	0	0
$639$	0	0
$640$	0	0
$641$	6.97908	0.275657	0.137829	−	0.990456i	$-0.455988\pi$
$641$	6.97908	0.275657	0.137829	+	0.990456i	$0.455988\pi$
$642$	0	0
$643$	8.31952	0.328090	0.164045	−	0.986453i	$-0.447546\pi$
$643$	8.31952	0.328090	0.164045	+	0.986453i	$0.447546\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.4160	−1.51029	−0.755144	−	0.655559i	$-0.772433\pi$
$647$	−38.4160	−1.51029	−0.755144	+	0.655559i	$0.772433\pi$
$648$	0	0
$649$	−60.9492	−2.39246
$650$	0	0
$651$	0	0
$652$	0	0
$653$	25.5936	1.00156	0.500778	−	0.865576i	$-0.333047\pi$
$653$	25.5936	1.00156	0.500778	+	0.865576i	$0.333047\pi$
$654$	0	0
$655$	12.8796	0.503247
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.5397	−0.605341	−0.302671	−	0.953095i	$-0.597878\pi$
$659$	−15.5397	−0.605341	−0.302671	+	0.953095i	$0.597878\pi$
$660$	0	0
$661$	−3.01797	−0.117385	−0.0586927	−	0.998276i	$-0.518693\pi$
$661$	−3.01797	−0.117385	−0.0586927	+	0.998276i	$0.518693\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.841263	−0.0326228
$666$	0	0
$667$	6.59074	0.255194
$668$	0	0
$669$	0	0
$670$	0	0
$671$	25.7598	0.994446
$672$	0	0
$673$	−28.3485	−1.09275	−0.546377	−	0.837539i	$-0.683993\pi$
$673$	−28.3485	−1.09275	−0.546377	+	0.837539i	$0.683993\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−1.01901	−0.0391639	−0.0195819	−	0.999808i	$-0.506234\pi$
$677$	−1.01901	−0.0391639	−0.0195819	+	0.999808i	$0.506234\pi$
$678$	0	0
$679$	−1.43344	−0.0550103
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.22345	−0.199870	−0.0999349	−	0.994994i	$-0.531863\pi$
$683$	−5.22345	−0.199870	−0.0999349	+	0.994994i	$0.531863\pi$
$684$	0	0
$685$	13.4974	0.515709
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−31.0408	−1.18256
$690$	0	0
$691$	−28.2299	−1.07392	−0.536958	−	0.843609i	$-0.680427\pi$
$691$	−28.2299	−1.07392	−0.536958	+	0.843609i	$0.680427\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.6386	−0.441477
$696$	0	0
$697$	−2.17861	−0.0825208
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.31750	0.351917	0.175958	−	0.984398i	$-0.443698\pi$
$701$	9.31750	0.351917	0.175958	+	0.984398i	$0.443698\pi$
$702$	0	0
$703$	3.27119	0.123375
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.47621	−0.205954
$708$	0	0
$709$	4.95711	0.186168	0.0930841	−	0.995658i	$-0.470327\pi$
$709$	4.95711	0.186168	0.0930841	+	0.995658i	$0.470327\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	22.1079	0.827948
$714$	0	0
$715$	−18.1388	−0.678353
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17.5186	−0.653335	−0.326667	−	0.945139i	$-0.605926\pi$
$719$	−17.5186	−0.653335	−0.326667	+	0.945139i	$0.605926\pi$
$720$	0	0
$721$	16.2902	0.606679
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−9.36078	−0.347651
$726$	0	0
$727$	−17.0526	−0.632444	−0.316222	−	0.948685i	$-0.602414\pi$
$727$	−17.0526	−0.632444	−0.316222	+	0.948685i	$0.602414\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−10.0794	−0.372290	−0.186145	−	0.982522i	$-0.559599\pi$
$733$	−10.0794	−0.372290	−0.186145	+	0.982522i	$0.559599\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.98933	0.294291
$738$	0	0
$739$	14.5208	0.534158	0.267079	−	0.963675i	$-0.413942\pi$
$739$	14.5208	0.534158	0.267079	+	0.963675i	$0.413942\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.7940	−1.34984	−0.674920	−	0.737891i	$-0.735822\pi$
$743$	−36.7940	−1.34984	−0.674920	+	0.737891i	$0.735822\pi$
$744$	0	0
$745$	5.39362	0.197607
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−17.5173	−0.640069
$750$	0	0
$751$	−25.6422	−0.935699	−0.467849	−	0.883808i	$-0.654971\pi$
$751$	−25.6422	−0.935699	−0.467849	+	0.883808i	$0.654971\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−8.48790	−0.308906
$756$	0	0
$757$	37.1650	1.35078	0.675392	−	0.737459i	$-0.263975\pi$
$757$	37.1650	1.35078	0.675392	+	0.737459i	$0.263975\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3.40887	0.123571	0.0617857	−	0.998089i	$-0.480320\pi$
$761$	3.40887	0.123571	0.0617857	+	0.998089i	$0.480320\pi$
$762$	0	0
$763$	−2.99793	−0.108532
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−71.2956	−2.57433
$768$	0	0
$769$	−30.0575	−1.08390	−0.541951	−	0.840410i	$-0.682314\pi$
$769$	−30.0575	−1.08390	−0.541951	+	0.840410i	$0.682314\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−34.5138	−1.24138	−0.620688	−	0.784058i	$-0.713147\pi$
$773$	−34.5138	−1.24138	−0.620688	+	0.784058i	$0.713147\pi$
$774$	0	0
$775$	−31.3997	−1.12791
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0.317473	0.0113747
$780$	0	0
$781$	30.5315	1.09250
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−16.1704	−0.577146
$786$	0	0
$787$	−48.0507	−1.71282	−0.856412	−	0.516293i	$-0.827312\pi$
$787$	−48.0507	−1.71282	−0.856412	+	0.516293i	$0.827312\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17.3098	0.615467
$792$	0	0
$793$	30.1326	1.07004
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.7796	1.76328	0.881642	−	0.471920i	$-0.156439\pi$
$797$	49.7796	1.76328	0.881642	+	0.471920i	$0.156439\pi$
$798$	0	0
$799$	−87.4559	−3.09397
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11.8061	−0.416628
$804$	0	0
$805$	−2.54239	−0.0896074
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.2721	0.642411	0.321206	−	0.947010i	$-0.395912\pi$
$809$	18.2721	0.642411	0.321206	+	0.947010i	$0.395912\pi$
$810$	0	0
$811$	50.0318	1.75686	0.878428	−	0.477875i	$-0.158593\pi$
$811$	50.0318	1.75686	0.878428	+	0.477875i	$0.158593\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−11.9074	−0.417098
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	14.8040	0.516664	0.258332	−	0.966056i	$-0.416827\pi$
$821$	14.8040	0.516664	0.258332	+	0.966056i	$0.416827\pi$
$822$	0	0
$823$	34.1017	1.18871	0.594356	−	0.804202i	$-0.297407\pi$
$823$	34.1017	1.18871	0.594356	+	0.804202i	$0.297407\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.4583	−0.467991	−0.233996	−	0.972238i	$-0.575180\pi$
$827$	−13.4583	−0.467991	−0.233996	+	0.972238i	$0.575180\pi$
$828$	0	0
$829$	32.9472	1.14430	0.572152	−	0.820147i	$-0.306109\pi$
$829$	32.9472	1.14430	0.572152	+	0.820147i	$0.306109\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−6.86235	−0.237766
$834$	0	0
$835$	6.13472	0.212301
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−47.0149	−1.62314	−0.811568	−	0.584258i	$-0.801385\pi$
$839$	−47.0149	−1.62314	−0.811568	+	0.584258i	$0.801385\pi$
$840$	0	0
$841$	−24.2439	−0.835997
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.2815	−0.353696
$846$	0	0
$847$	7.43242	0.255381
$848$	0	0
$849$	0	0
$850$	0	0
$851$	9.88589	0.338884
$852$	0	0
$853$	−24.0380	−0.823046	−0.411523	−	0.911399i	$-0.635003\pi$
$853$	−24.0380	−0.823046	−0.411523	+	0.911399i	$0.635003\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−28.0177	−0.957066	−0.478533	−	0.878070i	$-0.658831\pi$
$857$	−28.0177	−0.957066	−0.478533	+	0.878070i	$0.658831\pi$
$858$	0	0
$859$	−54.2302	−1.85031	−0.925156	−	0.379587i	$-0.876066\pi$
$859$	−54.2302	−1.85031	−0.925156	+	0.379587i	$0.876066\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	18.2448	0.621060	0.310530	−	0.950564i	$-0.399494\pi$
$863$	18.2448	0.621060	0.310530	+	0.950564i	$0.399494\pi$
$864$	0	0
$865$	−15.0798	−0.512727
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7.41789	0.251635
$870$	0	0
$871$	9.34555	0.316662
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.81725	0.264271
$876$	0	0
$877$	22.4907	0.759457	0.379728	−	0.925098i	$-0.376017\pi$
$877$	22.4907	0.759457	0.379728	+	0.925098i	$0.376017\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	38.4061	1.29393	0.646966	−	0.762518i	$-0.276037\pi$
$881$	38.4061	1.29393	0.646966	+	0.762518i	$0.276037\pi$
$882$	0	0
$883$	50.3316	1.69379	0.846897	−	0.531757i	$-0.178468\pi$
$883$	50.3316	1.69379	0.846897	+	0.531757i	$0.178468\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−49.9484	−1.67710	−0.838551	−	0.544822i	$-0.816597\pi$
$887$	−49.9484	−1.67710	−0.838551	+	0.544822i	$0.816597\pi$
$888$	0	0
$889$	−13.4545	−0.451250
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.7443	0.426472
$894$	0	0
$895$	−3.01432	−0.100758
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.9537	0.532087
$900$	0	0
$901$	42.4151	1.41305
$902$	0	0
$903$	0	0
$904$	0	0
$905$	17.5832	0.584486
$906$	0	0
$907$	−19.7223	−0.654870	−0.327435	−	0.944874i	$-0.606184\pi$
$907$	−19.7223	−0.654870	−0.327435	+	0.944874i	$0.606184\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.9506	0.826650	0.413325	−	0.910584i	$-0.364367\pi$
$911$	24.9506	0.826650	0.413325	+	0.910584i	$0.364367\pi$
$912$	0	0
$913$	28.8648	0.955287
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15.3098	−0.505575
$918$	0	0
$919$	−53.2616	−1.75694	−0.878469	−	0.477799i	$-0.841435\pi$
$919$	−53.2616	−1.75694	−0.878469	+	0.477799i	$0.841435\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	35.7144	1.17555
$924$	0	0
$925$	−14.0409	−0.461661
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−5.94038	−0.194898	−0.0974488	−	0.995241i	$-0.531068\pi$
$929$	−5.94038	−0.194898	−0.0974488	+	0.995241i	$0.531068\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	24.7854	0.810569
$936$	0	0
$937$	−31.0632	−1.01479	−0.507396	−	0.861713i	$-0.669392\pi$
$937$	−31.0632	−1.01479	−0.507396	+	0.861713i	$0.669392\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.84377	−0.190502	−0.0952508	−	0.995453i	$-0.530365\pi$
$941$	−5.84377	−0.190502	−0.0952508	+	0.995453i	$0.530365\pi$
$942$	0	0
$943$	0.959438	0.0312436
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−15.1623	−0.492708	−0.246354	−	0.969180i	$-0.579233\pi$
$947$	−15.1623	−0.492708	−0.246354	+	0.969180i	$0.579233\pi$
$948$	0	0
$949$	−13.8102	−0.448299
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−19.3479	−0.626738	−0.313369	−	0.949631i	$-0.601458\pi$
$953$	−19.3479	−0.626738	−0.313369	+	0.949631i	$0.601458\pi$
$954$	0	0
$955$	−10.5342	−0.340879
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16.0442	−0.518095
$960$	0	0
$961$	22.5151	0.726295
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−18.5433	−0.596929
$966$	0	0
$967$	−41.3415	−1.32945	−0.664726	−	0.747087i	$-0.731452\pi$
$967$	−41.3415	−1.32945	−0.664726	+	0.747087i	$0.731452\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	6.78913	0.217873	0.108937	−	0.994049i	$-0.465255\pi$
$971$	6.78913	0.217873	0.108937	+	0.994049i	$0.465255\pi$
$972$	0	0
$973$	13.8347	0.443519
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.6078	0.563323	0.281662	−	0.959514i	$-0.409114\pi$
$977$	17.6078	0.563323	0.281662	+	0.959514i	$0.409114\pi$
$978$	0	0
$979$	−14.1435	−0.452029
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.3352	−0.903751	−0.451876	−	0.892081i	$-0.649245\pi$
$983$	−28.3352	−0.903751	−0.451876	+	0.892081i	$0.649245\pi$
$984$	0	0
$985$	7.14463	0.227647
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	28.4470	0.903649	0.451825	−	0.892107i	$-0.350773\pi$
$991$	28.4470	0.903649	0.451825	+	0.892107i	$0.350773\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	3.99591	0.126679
$996$	0	0
$997$	41.8061	1.32401	0.662006	−	0.749498i	$-0.269705\pi$
$997$	41.8061	1.32401	0.662006	+	0.749498i	$0.269705\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9576.2.a.co.1.3		5
3.2	odd	2		3192.2.a.ba.1.3	✓	5
12.11	even	2		6384.2.a.ce.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3192.2.a.ba.1.3	✓	5	3.2	odd	2
6384.2.a.ce.1.3		5	12.11	even	2
9576.2.a.co.1.3		5	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)

\(f(q)\)	\(=\)	\(q-0.841263 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-0.841263 q^{5} +1.00000 q^{7} +4.29330 q^{11} +5.02211 q^{13} -6.86235 q^{17} +1.00000 q^{19} +3.02211 q^{23} -4.29228 q^{25} +2.18084 q^{29} +7.31540 q^{31} -0.841263 q^{35} +3.27119 q^{37} +0.317473 q^{41} +12.7443 q^{47} +1.00000 q^{49} -6.18084 q^{53} -3.61180 q^{55} -14.1963 q^{59} +6.00000 q^{61} -4.22491 q^{65} +1.86088 q^{67} +7.11143 q^{71} -2.74989 q^{73} +4.29330 q^{77} +1.72778 q^{79} +6.72323 q^{83} +5.77304 q^{85} -3.29432 q^{89} +5.02211 q^{91} -0.841263 q^{95} -1.43344 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{5} + 5 q^{7}+O(q^{10}) \) 5 * q - 2 * q^5 + 5 * q^7	\( 5 q - 2 q^{5} + 5 q^{7} - 2 q^{11} + 8 q^{13} + 2 q^{17} + 5 q^{19} - 2 q^{23} + 19 q^{25} - 4 q^{29} - 4 q^{31} - 2 q^{35} + 10 q^{37} + 6 q^{41} + 2 q^{47} + 5 q^{49} - 16 q^{53} - 16 q^{55} + 12 q^{59} + 30 q^{61} - 4 q^{65} + 18 q^{67} + 10 q^{71} + 14 q^{73} - 2 q^{77} - 2 q^{79} + 6 q^{83} + 12 q^{85} - 10 q^{89} + 8 q^{91} - 2 q^{95} + 8 q^{97}+O(q^{100}) \) 5 * q - 2 * q^5 + 5 * q^7 - 2 * q^11 + 8 * q^13 + 2 * q^17 + 5 * q^19 - 2 * q^23 + 19 * q^25 - 4 * q^29 - 4 * q^31 - 2 * q^35 + 10 * q^37 + 6 * q^41 + 2 * q^47 + 5 * q^49 - 16 * q^53 - 16 * q^55 + 12 * q^59 + 30 * q^61 - 4 * q^65 + 18 * q^67 + 10 * q^71 + 14 * q^73 - 2 * q^77 - 2 * q^79 + 6 * q^83 + 12 * q^85 - 10 * q^89 + 8 * q^91 - 2 * q^95 + 8 * q^97

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