LMFDB - Embedded newform 9792.2.a.dc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9792,2,Mod(1,9792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9792.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9792 = 2^{6} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9792.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:	$78.1895136592$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 544)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	9792.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.34017 q^{5} +0.630898 q^{7} -5.70928 q^{11} +3.07838 q^{13} -1.00000 q^{17} -3.41855 q^{19} +5.89269 q^{23} +13.8371 q^{25} +0.340173 q^{29} -6.78765 q^{31} -2.73820 q^{35} -0.340173 q^{37} -4.15676 q^{41} -5.26180 q^{43} -8.68035 q^{47} -6.60197 q^{49} +0.156755 q^{53} +24.7792 q^{55} -12.0989 q^{59} -7.17727 q^{61} -13.3607 q^{65} +6.83710 q^{67} +3.95055 q^{71} -4.83710 q^{73} -3.60197 q^{77} +8.94441 q^{79} -5.26180 q^{83} +4.34017 q^{85} -9.60197 q^{89} +1.94214 q^{91} +14.8371 q^{95} -12.8371 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{5} - 2 q^{7} - 10 q^{11} + 6 q^{13} - 3 q^{17} + 4 q^{19} + 6 q^{23} + 13 q^{25} - 10 q^{29} - 10 q^{31} - 16 q^{35} + 10 q^{37} - 6 q^{41} - 8 q^{43} - 4 q^{47} - q^{49} - 6 q^{53} + 16 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) $ 3 * q - 2 * q^5 - 2 * q^7 - 10 * q^11 + 6 * q^13 - 3 * q^17 + 4 * q^19 + 6 * q^23 + 13 * q^25 - 10 * q^29 - 10 * q^31 - 16 * q^35 + 10 * q^37 - 6 * q^41 - 8 * q^43 - 4 * q^47 - q^49 - 6 * q^53 + 16 * q^55 + 18 * q^61 + 4 * q^65 - 8 * q^67 + 30 * q^71 + 14 * q^73 + 8 * q^77 + 10 * q^79 - 8 * q^83 + 2 * q^85 - 10 * q^89 - 24 * q^91 + 16 * q^95 - 10 * 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
$5$	−4.34017	−1.94098	−0.970492	−	0.241133i	$-0.922481\pi$
$5$	−4.34017	−1.94098	−0.970492	+	0.241133i	$0.922481\pi$
$6$	0	0
$7$	0.630898	0.238457	0.119228	−	0.992867i	$-0.461958\pi$
$7$	0.630898	0.238457	0.119228	+	0.992867i	$0.461958\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.70928	−1.72141	−0.860706	−	0.509103i	$-0.829977\pi$
$11$	−5.70928	−1.72141	−0.860706	+	0.509103i	$0.829977\pi$
$12$	0	0
$13$	3.07838	0.853788	0.426894	−	0.904302i	$-0.359608\pi$
$13$	3.07838	0.853788	0.426894	+	0.904302i	$0.359608\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−3.41855	−0.784269	−0.392135	−	0.919908i	$-0.628263\pi$
$19$	−3.41855	−0.784269	−0.392135	+	0.919908i	$0.628263\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.89269	1.22871	0.614356	−	0.789029i	$-0.289416\pi$
$23$	5.89269	1.22871	0.614356	+	0.789029i	$0.289416\pi$
$24$	0	0
$25$	13.8371	2.76742
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.340173	0.0631685	0.0315843	−	0.999501i	$-0.489945\pi$
$29$	0.340173	0.0631685	0.0315843	+	0.999501i	$0.489945\pi$
$30$	0	0
$31$	−6.78765	−1.21910	−0.609549	−	0.792748i	$-0.708650\pi$
$31$	−6.78765	−1.21910	−0.609549	+	0.792748i	$0.708650\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.73820	−0.462841
$36$	0	0
$37$	−0.340173	−0.0559241	−0.0279620	−	0.999609i	$-0.508902\pi$
$37$	−0.340173	−0.0559241	−0.0279620	+	0.999609i	$0.508902\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.15676	−0.649176	−0.324588	−	0.945855i	$-0.605226\pi$
$41$	−4.15676	−0.649176	−0.324588	+	0.945855i	$0.605226\pi$
$42$	0	0
$43$	−5.26180	−0.802416	−0.401208	−	0.915987i	$-0.631410\pi$
$43$	−5.26180	−0.802416	−0.401208	+	0.915987i	$0.631410\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.68035	−1.26616	−0.633079	−	0.774087i	$-0.718209\pi$
$47$	−8.68035	−1.26616	−0.633079	+	0.774087i	$0.718209\pi$
$48$	0	0
$49$	−6.60197	−0.943138
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.156755	0.0215320	0.0107660	−	0.999942i	$-0.496573\pi$
$53$	0.156755	0.0215320	0.0107660	+	0.999942i	$0.496573\pi$
$54$	0	0
$55$	24.7792	3.34123
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.0989	−1.57514	−0.787571	−	0.616224i	$-0.788662\pi$
$59$	−12.0989	−1.57514	−0.787571	+	0.616224i	$0.788662\pi$
$60$	0	0
$61$	−7.17727	−0.918956	−0.459478	−	0.888189i	$-0.651963\pi$
$61$	−7.17727	−0.918956	−0.459478	+	0.888189i	$0.651963\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−13.3607	−1.65719
$66$	0	0
$67$	6.83710	0.835285	0.417642	−	0.908611i	$-0.362856\pi$
$67$	6.83710	0.835285	0.417642	+	0.908611i	$0.362856\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.95055	0.468844	0.234422	−	0.972135i	$-0.424680\pi$
$71$	3.95055	0.468844	0.234422	+	0.972135i	$0.424680\pi$
$72$	0	0
$73$	−4.83710	−0.566140	−0.283070	−	0.959099i	$-0.591353\pi$
$73$	−4.83710	−0.566140	−0.283070	+	0.959099i	$0.591353\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.60197	−0.410482
$78$	0	0
$79$	8.94441	1.00632	0.503162	−	0.864192i	$-0.332170\pi$
$79$	8.94441	1.00632	0.503162	+	0.864192i	$0.332170\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.26180	−0.577557	−0.288779	−	0.957396i	$-0.593249\pi$
$83$	−5.26180	−0.577557	−0.288779	+	0.957396i	$0.593249\pi$
$84$	0	0
$85$	4.34017	0.470758
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−9.60197	−1.01781	−0.508903	−	0.860824i	$-0.669949\pi$
$89$	−9.60197	−1.01781	−0.508903	+	0.860824i	$0.669949\pi$
$90$	0	0
$91$	1.94214	0.203592
$92$	0	0
$93$	0	0
$94$	0	0
$95$	14.8371	1.52225
$96$	0	0
$97$	−12.8371	−1.30341	−0.651705	−	0.758472i	$-0.725946\pi$
$97$	−12.8371	−1.30341	−0.651705	+	0.758472i	$0.725946\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.92162	0.489720	0.244860	−	0.969558i	$-0.421258\pi$
$101$	4.92162	0.489720	0.244860	+	0.969558i	$0.421258\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.8082	−1.72158	−0.860790	−	0.508959i	$-0.830030\pi$
$107$	−17.8082	−1.72158	−0.860790	+	0.508959i	$0.830030\pi$
$108$	0	0
$109$	1.81658	0.173997	0.0869985	−	0.996208i	$-0.472272\pi$
$109$	1.81658	0.173997	0.0869985	+	0.996208i	$0.472272\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−10.9939	−1.03422	−0.517108	−	0.855920i	$-0.672991\pi$
$113$	−10.9939	−1.03422	−0.517108	+	0.855920i	$0.672991\pi$
$114$	0	0
$115$	−25.5753	−2.38491
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.630898	−0.0578343
$120$	0	0
$121$	21.5958	1.96326
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−38.3545	−3.43054
$126$	0	0
$127$	6.73820	0.597919	0.298959	−	0.954266i	$-0.403360\pi$
$127$	6.73820	0.597919	0.298959	+	0.954266i	$0.403360\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	6.38962	0.558264	0.279132	−	0.960253i	$-0.409953\pi$
$131$	6.38962	0.558264	0.279132	+	0.960253i	$0.409953\pi$
$132$	0	0
$133$	−2.15676	−0.187014
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.5958	1.24701	0.623503	−	0.781821i	$-0.285709\pi$
$137$	14.5958	1.24701	0.623503	+	0.781821i	$0.285709\pi$
$138$	0	0
$139$	6.60424	0.560164	0.280082	−	0.959976i	$-0.409638\pi$
$139$	6.60424	0.560164	0.280082	+	0.959976i	$0.409638\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−17.5753	−1.46972
$144$	0	0
$145$	−1.47641	−0.122609
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.3607	1.91378	0.956891	−	0.290447i	$-0.0938038\pi$
$149$	23.3607	1.91378	0.956891	+	0.290447i	$0.0938038\pi$
$150$	0	0
$151$	−16.0989	−1.31011	−0.655055	−	0.755581i	$-0.727354\pi$
$151$	−16.0989	−1.31011	−0.655055	+	0.755581i	$0.727354\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	29.4596	2.36625
$156$	0	0
$157$	12.5236	0.999491	0.499746	−	0.866172i	$-0.333427\pi$
$157$	12.5236	0.999491	0.499746	+	0.866172i	$0.333427\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.71769	0.292995
$162$	0	0
$163$	3.55252	0.278255	0.139127	−	0.990274i	$-0.455570\pi$
$163$	3.55252	0.278255	0.139127	+	0.990274i	$0.455570\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.79380	0.138808	0.0694041	−	0.997589i	$-0.477890\pi$
$167$	1.79380	0.138808	0.0694041	+	0.997589i	$0.477890\pi$
$168$	0	0
$169$	−3.52359	−0.271045
$170$	0	0
$171$	0	0
$172$	0	0
$173$	15.1773	1.15391	0.576953	−	0.816777i	$-0.304241\pi$
$173$	15.1773	1.15391	0.576953	+	0.816777i	$0.304241\pi$
$174$	0	0
$175$	8.72979	0.659910
$176$	0	0
$177$	0	0
$178$	0	0
$179$	10.2557	0.766543	0.383272	−	0.923636i	$-0.374797\pi$
$179$	10.2557	0.766543	0.383272	+	0.923636i	$0.374797\pi$
$180$	0	0
$181$	−15.1773	−1.12812	−0.564059	−	0.825735i	$-0.690761\pi$
$181$	−15.1773	−1.12812	−0.564059	+	0.825735i	$0.690761\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.47641	0.108548
$186$	0	0
$187$	5.70928	0.417504
$188$	0	0
$189$	0	0
$190$	0	0
$191$	16.6803	1.20695	0.603474	−	0.797383i	$-0.293783\pi$
$191$	16.6803	1.20695	0.603474	+	0.797383i	$0.293783\pi$
$192$	0	0
$193$	7.84324	0.564569	0.282285	−	0.959331i	$-0.408908\pi$
$193$	7.84324	0.564569	0.282285	+	0.959331i	$0.408908\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.0205	0.927674	0.463837	−	0.885921i	$-0.346472\pi$
$197$	13.0205	0.927674	0.463837	+	0.885921i	$0.346472\pi$
$198$	0	0
$199$	−18.3051	−1.29761	−0.648807	−	0.760953i	$-0.724732\pi$
$199$	−18.3051	−1.29761	−0.648807	+	0.760953i	$0.724732\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.214614	0.0150630
$204$	0	0
$205$	18.0410	1.26004
$206$	0	0
$207$	0	0
$208$	0	0
$209$	19.5174	1.35005
$210$	0	0
$211$	6.38962	0.439880	0.219940	−	0.975513i	$-0.429414\pi$
$211$	6.38962	0.439880	0.219940	+	0.975513i	$0.429414\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	22.8371	1.55748
$216$	0	0
$217$	−4.28231	−0.290702
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.07838	−0.207074
$222$	0	0
$223$	−15.6163	−1.04575	−0.522874	−	0.852410i	$-0.675140\pi$
$223$	−15.6163	−1.04575	−0.522874	+	0.852410i	$0.675140\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.65756	0.176388	0.0881942	−	0.996103i	$-0.471890\pi$
$227$	2.65756	0.176388	0.0881942	+	0.996103i	$0.471890\pi$
$228$	0	0
$229$	5.60197	0.370188	0.185094	−	0.982721i	$-0.440741\pi$
$229$	5.60197	0.370188	0.185094	+	0.982721i	$0.440741\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.2039	1.38912	0.694558	−	0.719437i	$-0.255600\pi$
$233$	21.2039	1.38912	0.694558	+	0.719437i	$0.255600\pi$
$234$	0	0
$235$	37.6742	2.45759
$236$	0	0
$237$	0	0
$238$	0	0
$239$	22.1568	1.43320	0.716601	−	0.697484i	$-0.245697\pi$
$239$	22.1568	1.43320	0.716601	+	0.697484i	$0.245697\pi$
$240$	0	0
$241$	6.68035	0.430319	0.215159	−	0.976579i	$-0.430973\pi$
$241$	6.68035	0.430319	0.215159	+	0.976579i	$0.430973\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	28.6537	1.83062
$246$	0	0
$247$	−10.5236	−0.669600
$248$	0	0
$249$	0	0
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	−33.6430	−2.11512
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.92162	−0.307002	−0.153501	−	0.988148i	$-0.549055\pi$
$257$	−4.92162	−0.307002	−0.153501	+	0.988148i	$0.549055\pi$
$258$	0	0
$259$	−0.214614	−0.0133355
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.09890	−0.499399	−0.249700	−	0.968323i	$-0.580332\pi$
$263$	−8.09890	−0.499399	−0.249700	+	0.968323i	$0.580332\pi$
$264$	0	0
$265$	−0.680346	−0.0417933
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.5380	−1.25222	−0.626111	−	0.779734i	$-0.715354\pi$
$269$	−20.5380	−1.25222	−0.626111	+	0.779734i	$0.715354\pi$
$270$	0	0
$271$	7.51745	0.456652	0.228326	−	0.973585i	$-0.426675\pi$
$271$	7.51745	0.456652	0.228326	+	0.973585i	$0.426675\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−78.9998	−4.76387
$276$	0	0
$277$	25.6475	1.54101	0.770506	−	0.637433i	$-0.220004\pi$
$277$	25.6475	1.54101	0.770506	+	0.637433i	$0.220004\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	16.5236	0.985715	0.492857	−	0.870110i	$-0.335952\pi$
$281$	16.5236	0.985715	0.492857	+	0.870110i	$0.335952\pi$
$282$	0	0
$283$	−19.6514	−1.16816	−0.584078	−	0.811698i	$-0.698544\pi$
$283$	−19.6514	−1.16816	−0.584078	+	0.811698i	$0.698544\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.62249	−0.154801
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	9.68649	0.565891	0.282945	−	0.959136i	$-0.408688\pi$
$293$	9.68649	0.565891	0.282945	+	0.959136i	$0.408688\pi$
$294$	0	0
$295$	52.5113	3.05733
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.1399	1.04906
$300$	0	0
$301$	−3.31965	−0.191342
$302$	0	0
$303$	0	0
$304$	0	0
$305$	31.1506	1.78368
$306$	0	0
$307$	22.1568	1.26455	0.632276	−	0.774743i	$-0.282121\pi$
$307$	22.1568	1.26455	0.632276	+	0.774743i	$0.282121\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	16.6309	0.943052	0.471526	−	0.881852i	$-0.343703\pi$
$311$	16.6309	0.943052	0.471526	+	0.881852i	$0.343703\pi$
$312$	0	0
$313$	19.7275	1.11507	0.557533	−	0.830155i	$-0.311748\pi$
$313$	19.7275	1.11507	0.557533	+	0.830155i	$0.311748\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−20.3402	−1.14242	−0.571209	−	0.820805i	$-0.693525\pi$
$317$	−20.3402	−1.14242	−0.571209	+	0.820805i	$0.693525\pi$
$318$	0	0
$319$	−1.94214	−0.108739
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.41855	0.190213
$324$	0	0
$325$	42.5958	2.36279
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−5.47641	−0.301924
$330$	0	0
$331$	−8.94828	−0.491842	−0.245921	−	0.969290i	$-0.579090\pi$
$331$	−8.94828	−0.491842	−0.245921	+	0.969290i	$0.579090\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−29.6742	−1.62127
$336$	0	0
$337$	12.5236	0.682203	0.341102	−	0.940026i	$-0.389200\pi$
$337$	12.5236	0.682203	0.341102	+	0.940026i	$0.389200\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	38.7526	2.09857
$342$	0	0
$343$	−8.58145	−0.463355
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.7503	−1.48971	−0.744857	−	0.667224i	$-0.767483\pi$
$347$	−27.7503	−1.48971	−0.744857	+	0.667224i	$0.767483\pi$
$348$	0	0
$349$	−8.15676	−0.436621	−0.218311	−	0.975879i	$-0.570055\pi$
$349$	−8.15676	−0.436621	−0.218311	+	0.975879i	$0.570055\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−16.8371	−0.896148	−0.448074	−	0.893996i	$-0.647890\pi$
$353$	−16.8371	−0.896148	−0.448074	+	0.893996i	$0.647890\pi$
$354$	0	0
$355$	−17.1461	−0.910019
$356$	0	0
$357$	0	0
$358$	0	0
$359$	9.20847	0.486005	0.243002	−	0.970026i	$-0.421868\pi$
$359$	9.20847	0.486005	0.243002	+	0.970026i	$0.421868\pi$
$360$	0	0
$361$	−7.31351	−0.384922
$362$	0	0
$363$	0	0
$364$	0	0
$365$	20.9939	1.09887
$366$	0	0
$367$	−20.0494	−1.04657	−0.523286	−	0.852157i	$-0.675294\pi$
$367$	−20.0494	−1.04657	−0.523286	+	0.852157i	$0.675294\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.0988967	0.00513446
$372$	0	0
$373$	−5.54864	−0.287298	−0.143649	−	0.989629i	$-0.545884\pi$
$373$	−5.54864	−0.287298	−0.143649	+	0.989629i	$0.545884\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.04718	0.0539326
$378$	0	0
$379$	−1.70928	−0.0877996	−0.0438998	−	0.999036i	$-0.513978\pi$
$379$	−1.70928	−0.0877996	−0.0438998	+	0.999036i	$0.513978\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.4247	−0.532677	−0.266338	−	0.963880i	$-0.585814\pi$
$383$	−10.4247	−0.532677	−0.266338	+	0.963880i	$0.585814\pi$
$384$	0	0
$385$	15.6332	0.796740
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−22.9627	−1.16425	−0.582127	−	0.813098i	$-0.697779\pi$
$389$	−22.9627	−1.16425	−0.582127	+	0.813098i	$0.697779\pi$
$390$	0	0
$391$	−5.89269	−0.298006
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−38.8203	−1.95326
$396$	0	0
$397$	−1.70086	−0.0853640	−0.0426820	−	0.999089i	$-0.513590\pi$
$397$	−1.70086	−0.0853640	−0.0426820	+	0.999089i	$0.513590\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	22.9939	1.14826	0.574129	−	0.818765i	$-0.305341\pi$
$401$	22.9939	1.14826	0.574129	+	0.818765i	$0.305341\pi$
$402$	0	0
$403$	−20.8950	−1.04085
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.94214	0.0962684
$408$	0	0
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−7.63317	−0.375603
$414$	0	0
$415$	22.8371	1.12103
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−14.7565	−0.720900	−0.360450	−	0.932779i	$-0.617377\pi$
$419$	−14.7565	−0.720900	−0.360450	+	0.932779i	$0.617377\pi$
$420$	0	0
$421$	35.0784	1.70962	0.854808	−	0.518945i	$-0.173675\pi$
$421$	35.0784	1.70962	0.854808	+	0.518945i	$0.173675\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−13.8371	−0.671198
$426$	0	0
$427$	−4.52813	−0.219131
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−26.3051	−1.26707	−0.633536	−	0.773713i	$-0.718397\pi$
$431$	−26.3051	−1.26707	−0.633536	+	0.773713i	$0.718397\pi$
$432$	0	0
$433$	−14.9627	−0.719060	−0.359530	−	0.933134i	$-0.617063\pi$
$433$	−14.9627	−0.719060	−0.359530	+	0.933134i	$0.617063\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−20.1445	−0.963641
$438$	0	0
$439$	17.0966	0.815978	0.407989	−	0.912987i	$-0.366230\pi$
$439$	17.0966	0.815978	0.407989	+	0.912987i	$0.366230\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.52359	0.119899	0.0599497	−	0.998201i	$-0.480906\pi$
$443$	2.52359	0.119899	0.0599497	+	0.998201i	$0.480906\pi$
$444$	0	0
$445$	41.6742	1.97555
$446$	0	0
$447$	0	0
$448$	0	0
$449$	27.6742	1.30603	0.653013	−	0.757347i	$-0.273505\pi$
$449$	27.6742	1.30603	0.653013	+	0.757347i	$0.273505\pi$
$450$	0	0
$451$	23.7321	1.11750
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−8.42923	−0.395168
$456$	0	0
$457$	16.8059	0.786147	0.393074	−	0.919507i	$-0.371412\pi$
$457$	16.8059	0.786147	0.393074	+	0.919507i	$0.371412\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.2039	1.36016	0.680081	−	0.733137i	$-0.261944\pi$
$461$	29.2039	1.36016	0.680081	+	0.733137i	$0.261944\pi$
$462$	0	0
$463$	−8.19779	−0.380984	−0.190492	−	0.981689i	$-0.561008\pi$
$463$	−8.19779	−0.380984	−0.190492	+	0.981689i	$0.561008\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	30.5692	1.41457	0.707286	−	0.706927i	$-0.249919\pi$
$467$	30.5692	1.41457	0.707286	+	0.706927i	$0.249919\pi$
$468$	0	0
$469$	4.31351	0.199179
$470$	0	0
$471$	0	0
$472$	0	0
$473$	30.0410	1.38129
$474$	0	0
$475$	−47.3028	−2.17040
$476$	0	0
$477$	0	0
$478$	0	0
$479$	37.9916	1.73588	0.867940	−	0.496669i	$-0.165444\pi$
$479$	37.9916	1.73588	0.867940	+	0.496669i	$0.165444\pi$
$480$	0	0
$481$	−1.04718	−0.0477473
$482$	0	0
$483$	0	0
$484$	0	0
$485$	55.7152	2.52990
$486$	0	0
$487$	−15.0023	−0.679818	−0.339909	−	0.940458i	$-0.610396\pi$
$487$	−15.0023	−0.679818	−0.339909	+	0.940458i	$0.610396\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.465732	0.0210182	0.0105091	−	0.999945i	$-0.496655\pi$
$491$	0.465732	0.0210182	0.0105091	+	0.999945i	$0.496655\pi$
$492$	0	0
$493$	−0.340173	−0.0153206
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.49239	0.111799
$498$	0	0
$499$	−30.1217	−1.34843	−0.674216	−	0.738534i	$-0.735518\pi$
$499$	−30.1217	−1.34843	−0.674216	+	0.738534i	$0.735518\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−39.8348	−1.77615	−0.888074	−	0.459701i	$-0.847957\pi$
$503$	−39.8348	−1.77615	−0.888074	+	0.459701i	$0.847957\pi$
$504$	0	0
$505$	−21.3607	−0.950538
$506$	0	0
$507$	0	0
$508$	0	0
$509$	19.6742	0.872044	0.436022	−	0.899936i	$-0.356387\pi$
$509$	19.6742	0.872044	0.436022	+	0.899936i	$0.356387\pi$
$510$	0	0
$511$	−3.05172	−0.135000
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	49.5585	2.17958
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.8904	−0.915226	−0.457613	−	0.889151i	$-0.651296\pi$
$521$	−20.8904	−0.915226	−0.457613	+	0.889151i	$0.651296\pi$
$522$	0	0
$523$	−20.3135	−0.888248	−0.444124	−	0.895965i	$-0.646485\pi$
$523$	−20.3135	−0.888248	−0.444124	+	0.895965i	$0.646485\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.78765	0.295675
$528$	0	0
$529$	11.7238	0.509732
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−12.7961	−0.554259
$534$	0	0
$535$	77.2905	3.34156
$536$	0	0
$537$	0	0
$538$	0	0
$539$	37.6925	1.62353
$540$	0	0
$541$	−5.18956	−0.223117	−0.111558	−	0.993758i	$-0.535584\pi$
$541$	−5.18956	−0.223117	−0.111558	+	0.993758i	$0.535584\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.88428	−0.337726
$546$	0	0
$547$	−29.2267	−1.24964	−0.624822	−	0.780767i	$-0.714829\pi$
$547$	−29.2267	−1.24964	−0.624822	+	0.780767i	$0.714829\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1.16290	−0.0495411
$552$	0	0
$553$	5.64301	0.239965
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.5486	0.574074	0.287037	−	0.957919i	$-0.407330\pi$
$557$	13.5486	0.574074	0.287037	+	0.957919i	$0.407330\pi$
$558$	0	0
$559$	−16.1978	−0.685094
$560$	0	0
$561$	0	0
$562$	0	0
$563$	46.0866	1.94232	0.971160	−	0.238431i	$-0.0766330\pi$
$563$	46.0866	1.94232	0.971160	+	0.238431i	$0.0766330\pi$
$564$	0	0
$565$	47.7152	2.00740
$566$	0	0
$567$	0	0
$568$	0	0
$569$	43.8720	1.83921	0.919605	−	0.392845i	$-0.128509\pi$
$569$	43.8720	1.83921	0.919605	+	0.392845i	$0.128509\pi$
$570$	0	0
$571$	16.3486	0.684167	0.342083	−	0.939670i	$-0.388867\pi$
$571$	16.3486	0.684167	0.342083	+	0.939670i	$0.388867\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	81.5378	3.40036
$576$	0	0
$577$	19.3919	0.807295	0.403647	−	0.914915i	$-0.367742\pi$
$577$	19.3919	0.807295	0.403647	+	0.914915i	$0.367742\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.31965	−0.137722
$582$	0	0
$583$	−0.894960	−0.0370655
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.5814	0.849487	0.424744	−	0.905314i	$-0.360364\pi$
$587$	20.5814	0.849487	0.424744	+	0.905314i	$0.360364\pi$
$588$	0	0
$589$	23.2039	0.956102
$590$	0	0
$591$	0	0
$592$	0	0
$593$	4.21008	0.172887	0.0864436	−	0.996257i	$-0.472450\pi$
$593$	4.21008	0.172887	0.0864436	+	0.996257i	$0.472450\pi$
$594$	0	0
$595$	2.73820	0.112255
$596$	0	0
$597$	0	0
$598$	0	0
$599$	6.21008	0.253737	0.126868	−	0.991920i	$-0.459507\pi$
$599$	6.21008	0.253737	0.126868	+	0.991920i	$0.459507\pi$
$600$	0	0
$601$	4.95282	0.202030	0.101015	−	0.994885i	$-0.467791\pi$
$601$	4.95282	0.202030	0.101015	+	0.994885i	$0.467791\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−93.7296	−3.81065
$606$	0	0
$607$	−4.26406	−0.173073	−0.0865365	−	0.996249i	$-0.527580\pi$
$607$	−4.26406	−0.173073	−0.0865365	+	0.996249i	$0.527580\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−26.7214	−1.08103
$612$	0	0
$613$	−16.7214	−0.675370	−0.337685	−	0.941259i	$-0.609644\pi$
$613$	−16.7214	−0.675370	−0.337685	+	0.941259i	$0.609644\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.15676	−0.167345	−0.0836723	−	0.996493i	$-0.526665\pi$
$617$	−4.15676	−0.167345	−0.0836723	+	0.996493i	$0.526665\pi$
$618$	0	0
$619$	−19.2846	−0.775113	−0.387556	−	0.921846i	$-0.626681\pi$
$619$	−19.2846	−0.775113	−0.387556	+	0.921846i	$0.626681\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−6.05786	−0.242703
$624$	0	0
$625$	97.2799	3.89119
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.340173	0.0135636
$630$	0	0
$631$	−13.6286	−0.542547	−0.271274	−	0.962502i	$-0.587445\pi$
$631$	−13.6286	−0.542547	−0.271274	+	0.962502i	$0.587445\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−29.2450	−1.16055
$636$	0	0
$637$	−20.3234	−0.805241
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−31.2450	−1.23410	−0.617051	−	0.786923i	$-0.711673\pi$
$641$	−31.2450	−1.23410	−0.617051	+	0.786923i	$0.711673\pi$
$642$	0	0
$643$	−37.0577	−1.46141	−0.730706	−	0.682692i	$-0.760809\pi$
$643$	−37.0577	−1.46141	−0.730706	+	0.682692i	$0.760809\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−49.5052	−1.94625	−0.973124	−	0.230280i	$-0.926036\pi$
$647$	−49.5052	−1.94625	−0.973124	+	0.230280i	$0.926036\pi$
$648$	0	0
$649$	69.0759	2.71147
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.50307	0.0588197	0.0294099	−	0.999567i	$-0.490637\pi$
$653$	1.50307	0.0588197	0.0294099	+	0.999567i	$0.490637\pi$
$654$	0	0
$655$	−27.7321	−1.08358
$656$	0	0
$657$	0	0
$658$	0	0
$659$	29.7275	1.15802	0.579010	−	0.815320i	$-0.303439\pi$
$659$	29.7275	1.15802	0.579010	+	0.815320i	$0.303439\pi$
$660$	0	0
$661$	−1.94668	−0.0757169	−0.0378585	−	0.999283i	$-0.512054\pi$
$661$	−1.94668	−0.0757169	−0.0378585	+	0.999283i	$0.512054\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	9.36069	0.362992
$666$	0	0
$667$	2.00453	0.0776159
$668$	0	0
$669$	0	0
$670$	0	0
$671$	40.9770	1.58190
$672$	0	0
$673$	8.63931	0.333021	0.166510	−	0.986040i	$-0.446750\pi$
$673$	8.63931	0.333021	0.166510	+	0.986040i	$0.446750\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.2183	1.43042	0.715208	−	0.698912i	$-0.246332\pi$
$677$	37.2183	1.43042	0.715208	+	0.698912i	$0.246332\pi$
$678$	0	0
$679$	−8.09890	−0.310807
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−24.8143	−0.949493	−0.474747	−	0.880122i	$-0.657460\pi$
$683$	−24.8143	−0.949493	−0.474747	+	0.880122i	$0.657460\pi$
$684$	0	0
$685$	−63.3484	−2.42042
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0.482553	0.0183838
$690$	0	0
$691$	24.2329	0.921862	0.460931	−	0.887436i	$-0.347516\pi$
$691$	24.2329	0.921862	0.460931	+	0.887436i	$0.347516\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−28.6635	−1.08727
$696$	0	0
$697$	4.15676	0.157448
$698$	0	0
$699$	0	0
$700$	0	0
$701$	22.2823	0.841591	0.420796	−	0.907155i	$-0.361751\pi$
$701$	22.2823	0.841591	0.420796	+	0.907155i	$0.361751\pi$
$702$	0	0
$703$	1.16290	0.0438595
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.10504	0.116777
$708$	0	0
$709$	−31.8043	−1.19444	−0.597218	−	0.802079i	$-0.703727\pi$
$709$	−31.8043	−1.19444	−0.597218	+	0.802079i	$0.703727\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−39.9976	−1.49792
$714$	0	0
$715$	76.2799	2.85271
$716$	0	0
$717$	0	0
$718$	0	0
$719$	47.2990	1.76395	0.881977	−	0.471293i	$-0.156213\pi$
$719$	47.2990	1.76395	0.881977	+	0.471293i	$0.156213\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.70701	0.174814
$726$	0	0
$727$	−0.680346	−0.0252326	−0.0126163	−	0.999920i	$-0.504016\pi$
$727$	−0.680346	−0.0252326	−0.0126163	+	0.999920i	$0.504016\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.26180	0.194615
$732$	0	0
$733$	24.0410	0.887976	0.443988	−	0.896033i	$-0.353563\pi$
$733$	24.0410	0.887976	0.443988	+	0.896033i	$0.353563\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−39.0349	−1.43787
$738$	0	0
$739$	31.1050	1.14422	0.572109	−	0.820178i	$-0.306126\pi$
$739$	31.1050	1.14422	0.572109	+	0.820178i	$0.306126\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.5730	−0.681379	−0.340689	−	0.940176i	$-0.610660\pi$
$743$	−18.5730	−0.681379	−0.340689	+	0.940176i	$0.610660\pi$
$744$	0	0
$745$	−101.389	−3.71462
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−11.2351	−0.410523
$750$	0	0
$751$	15.9337	0.581430	0.290715	−	0.956810i	$-0.406107\pi$
$751$	15.9337	0.581430	0.290715	+	0.956810i	$0.406107\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	69.8720	2.54290
$756$	0	0
$757$	−9.43293	−0.342846	−0.171423	−	0.985198i	$-0.554836\pi$
$757$	−9.43293	−0.342846	−0.171423	+	0.985198i	$0.554836\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	42.2823	1.53273	0.766366	−	0.642404i	$-0.222063\pi$
$761$	42.2823	1.53273	0.766366	+	0.642404i	$0.222063\pi$
$762$	0	0
$763$	1.14608	0.0414908
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−37.2450	−1.34484
$768$	0	0
$769$	−1.54864	−0.0558455	−0.0279228	−	0.999610i	$-0.508889\pi$
$769$	−1.54864	−0.0558455	−0.0279228	+	0.999610i	$0.508889\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−5.60197	−0.201489	−0.100744	−	0.994912i	$-0.532122\pi$
$773$	−5.60197	−0.201489	−0.100744	+	0.994912i	$0.532122\pi$
$774$	0	0
$775$	−93.9214	−3.37376
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.2101	0.509129
$780$	0	0
$781$	−22.5548	−0.807074
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−54.3545	−1.94000
$786$	0	0
$787$	49.3256	1.75827	0.879134	−	0.476574i	$-0.158122\pi$
$787$	49.3256	1.75827	0.879134	+	0.476574i	$0.158122\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−6.93600	−0.246616
$792$	0	0
$793$	−22.0944	−0.784594
$794$	0	0
$795$	0	0
$796$	0	0
$797$	35.3074	1.25065	0.625326	−	0.780364i	$-0.284966\pi$
$797$	35.3074	1.25065	0.625326	+	0.780364i	$0.284966\pi$
$798$	0	0
$799$	8.68035	0.307089
$800$	0	0
$801$	0	0
$802$	0	0
$803$	27.6163	0.974560
$804$	0	0
$805$	−16.1354	−0.568698
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−45.1506	−1.58741	−0.793705	−	0.608302i	$-0.791851\pi$
$809$	−45.1506	−1.58741	−0.793705	+	0.608302i	$0.791851\pi$
$810$	0	0
$811$	−19.0166	−0.667765	−0.333882	−	0.942615i	$-0.608359\pi$
$811$	−19.0166	−0.667765	−0.333882	+	0.942615i	$0.608359\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−15.4186	−0.540088
$816$	0	0
$817$	17.9877	0.629310
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.2122	−1.75242	−0.876208	−	0.481932i	$-0.839935\pi$
$821$	−50.2122	−1.75242	−0.876208	+	0.481932i	$0.839935\pi$
$822$	0	0
$823$	2.30510	0.0803508	0.0401754	−	0.999193i	$-0.487208\pi$
$823$	2.30510	0.0803508	0.0401754	+	0.999193i	$0.487208\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−9.34244	−0.324868	−0.162434	−	0.986719i	$-0.551935\pi$
$827$	−9.34244	−0.324868	−0.162434	+	0.986719i	$0.551935\pi$
$828$	0	0
$829$	12.5236	0.434962	0.217481	−	0.976065i	$-0.430216\pi$
$829$	12.5236	0.434962	0.217481	+	0.976065i	$0.430216\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	6.60197	0.228745
$834$	0	0
$835$	−7.78539	−0.269424
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.5669	0.813620	0.406810	−	0.913513i	$-0.366641\pi$
$839$	23.5669	0.813620	0.406810	+	0.913513i	$0.366641\pi$
$840$	0	0
$841$	−28.8843	−0.996010
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.2930	0.526095
$846$	0	0
$847$	13.6248	0.468152
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.00453	−0.0687146
$852$	0	0
$853$	−4.28685	−0.146779	−0.0733895	−	0.997303i	$-0.523382\pi$
$853$	−4.28685	−0.146779	−0.0733895	+	0.997303i	$0.523382\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−14.6803	−0.501471	−0.250736	−	0.968056i	$-0.580672\pi$
$857$	−14.6803	−0.501471	−0.250736	+	0.968056i	$0.580672\pi$
$858$	0	0
$859$	49.1461	1.67684	0.838421	−	0.545023i	$-0.183479\pi$
$859$	49.1461	1.67684	0.838421	+	0.545023i	$0.183479\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.8843	−0.949192	−0.474596	−	0.880204i	$-0.657406\pi$
$863$	−27.8843	−0.949192	−0.474596	+	0.880204i	$0.657406\pi$
$864$	0	0
$865$	−65.8720	−2.23972
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−51.0661	−1.73230
$870$	0	0
$871$	21.0472	0.713157
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−24.1978	−0.818035
$876$	0	0
$877$	−9.86991	−0.333283	−0.166642	−	0.986018i	$-0.553292\pi$
$877$	−9.86991	−0.333283	−0.166642	+	0.986018i	$0.553292\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	46.8248	1.57757	0.788784	−	0.614670i	$-0.210711\pi$
$881$	46.8248	1.57757	0.788784	+	0.614670i	$0.210711\pi$
$882$	0	0
$883$	4.84939	0.163195	0.0815974	−	0.996665i	$-0.473998\pi$
$883$	4.84939	0.163195	0.0815974	+	0.996665i	$0.473998\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.5029	1.42711	0.713554	−	0.700600i	$-0.247084\pi$
$887$	42.5029	1.42711	0.713554	+	0.700600i	$0.247084\pi$
$888$	0	0
$889$	4.25112	0.142578
$890$	0	0
$891$	0	0
$892$	0	0
$893$	29.6742	0.993009
$894$	0	0
$895$	−44.5113	−1.48785
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.30898	−0.0770087
$900$	0	0
$901$	−0.156755	−0.00522228
$902$	0	0
$903$	0	0
$904$	0	0
$905$	65.8720	2.18966
$906$	0	0
$907$	36.0105	1.19571	0.597855	−	0.801605i	$-0.296020\pi$
$907$	36.0105	1.19571	0.597855	+	0.801605i	$0.296020\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−43.7815	−1.45055	−0.725273	−	0.688461i	$-0.758286\pi$
$911$	−43.7815	−1.45055	−0.725273	+	0.688461i	$0.758286\pi$
$912$	0	0
$913$	30.0410	0.994213
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.03120	0.133122
$918$	0	0
$919$	13.1917	0.435152	0.217576	−	0.976043i	$-0.430185\pi$
$919$	13.1917	0.435152	0.217576	+	0.976043i	$0.430185\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.1613	0.400294
$924$	0	0
$925$	−4.70701	−0.154765
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−6.68035	−0.219175	−0.109588	−	0.993977i	$-0.534953\pi$
$929$	−6.68035	−0.219175	−0.109588	+	0.993977i	$0.534953\pi$
$930$	0	0
$931$	22.5692	0.739674
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−24.7792	−0.810368
$936$	0	0
$937$	−26.3135	−0.859625	−0.429812	−	0.902918i	$-0.641420\pi$
$937$	−26.3135	−0.859625	−0.429812	+	0.902918i	$0.641420\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.44975	0.177657	0.0888283	−	0.996047i	$-0.471688\pi$
$941$	5.44975	0.177657	0.0888283	+	0.996047i	$0.471688\pi$
$942$	0	0
$943$	−24.4945	−0.797650
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.2784	−0.918926	−0.459463	−	0.888197i	$-0.651958\pi$
$947$	−28.2784	−0.918926	−0.459463	+	0.888197i	$0.651958\pi$
$948$	0	0
$949$	−14.8904	−0.483364
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.71154	0.0878355	0.0439177	−	0.999035i	$-0.486016\pi$
$953$	2.71154	0.0878355	0.0439177	+	0.999035i	$0.486016\pi$
$954$	0	0
$955$	−72.3956	−2.34267
$956$	0	0
$957$	0	0
$958$	0	0
$959$	9.20847	0.297357
$960$	0	0
$961$	15.0722	0.486201
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−34.0410	−1.09582
$966$	0	0
$967$	−48.9770	−1.57500	−0.787498	−	0.616318i	$-0.788624\pi$
$967$	−48.9770	−1.57500	−0.787498	+	0.616318i	$0.788624\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	3.47187	0.111418	0.0557089	−	0.998447i	$-0.482258\pi$
$971$	3.47187	0.111418	0.0557089	+	0.998447i	$0.482258\pi$
$972$	0	0
$973$	4.16660	0.133575
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.8248	−0.602259	−0.301130	−	0.953583i	$-0.597364\pi$
$977$	−18.8248	−0.602259	−0.301130	+	0.953583i	$0.597364\pi$
$978$	0	0
$979$	54.8203	1.75206
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−11.3158	−0.360917	−0.180459	−	0.983583i	$-0.557758\pi$
$983$	−11.3158	−0.360917	−0.180459	+	0.983583i	$0.557758\pi$
$984$	0	0
$985$	−56.5113	−1.80060
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−31.0061	−0.985938
$990$	0	0
$991$	−40.9977	−1.30234	−0.651168	−	0.758934i	$-0.725721\pi$
$991$	−40.9977	−1.30234	−0.651168	+	0.758934i	$0.725721\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	79.4473	2.51865
$996$	0	0
$997$	48.6537	1.54088	0.770439	−	0.637514i	$-0.220037\pi$
$997$	48.6537	1.54088	0.770439	+	0.637514i	$0.220037\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9792.2.a.dc.1.1		3
3.2	odd	2		1088.2.a.v.1.1		3
4.3	odd	2		9792.2.a.dd.1.1		3
8.3	odd	2		4896.2.a.bf.1.3		3
8.5	even	2		4896.2.a.be.1.3		3
12.11	even	2		1088.2.a.u.1.3		3
24.5	odd	2		544.2.a.i.1.3	✓	3
24.11	even	2		544.2.a.j.1.1	yes	3
408.101	odd	2		9248.2.a.u.1.1		3
408.203	even	2		9248.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
544.2.a.i.1.3	✓	3	24.5	odd	2
544.2.a.j.1.1	yes	3	24.11	even	2
1088.2.a.u.1.3		3	12.11	even	2
1088.2.a.v.1.1		3	3.2	odd	2
4896.2.a.be.1.3		3	8.5	even	2
4896.2.a.bf.1.3		3	8.3	odd	2
9248.2.a.t.1.3		3	408.203	even	2
9248.2.a.u.1.1		3	408.101	odd	2
9792.2.a.dc.1.1		3	1.1	even	1	trivial
9792.2.a.dd.1.1		3	4.3	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:	\( N \)	\(=\)	\( 9792 = 2^{6} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9792.a (trivial)

\(f(q)\)	\(=\)	\(q-4.34017 q^{5} +0.630898 q^{7} -5.70928 q^{11} +3.07838 q^{13} -1.00000 q^{17} -3.41855 q^{19} +5.89269 q^{23} +13.8371 q^{25} +0.340173 q^{29} -6.78765 q^{31} -2.73820 q^{35} -0.340173 q^{37} -4.15676 q^{41} -5.26180 q^{43} -8.68035 q^{47} -6.60197 q^{49} +0.156755 q^{53} +24.7792 q^{55} -12.0989 q^{59} -7.17727 q^{61} -13.3607 q^{65} +6.83710 q^{67} +3.95055 q^{71} -4.83710 q^{73} -3.60197 q^{77} +8.94441 q^{79} -5.26180 q^{83} +4.34017 q^{85} -9.60197 q^{89} +1.94214 q^{91} +14.8371 q^{95} -12.8371 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{5} - 2 q^{7} - 10 q^{11} + 6 q^{13} - 3 q^{17} + 4 q^{19} + 6 q^{23} + 13 q^{25} - 10 q^{29} - 10 q^{31} - 16 q^{35} + 10 q^{37} - 6 q^{41} - 8 q^{43} - 4 q^{47} - q^{49} - 6 q^{53} + 16 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) \) 3 * q - 2 * q^5 - 2 * q^7 - 10 * q^11 + 6 * q^13 - 3 * q^17 + 4 * q^19 + 6 * q^23 + 13 * q^25 - 10 * q^29 - 10 * q^31 - 16 * q^35 + 10 * q^37 - 6 * q^41 - 8 * q^43 - 4 * q^47 - q^49 - 6 * q^53 + 16 * q^55 + 18 * q^61 + 4 * q^65 - 8 * q^67 + 30 * q^71 + 14 * q^73 + 8 * q^77 + 10 * q^79 - 8 * q^83 + 2 * q^85 - 10 * q^89 - 24 * q^91 + 16 * q^95 - 10 * q^97

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