LMFDB - Embedded newform 9360.2.a.cx.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9360,2,Mod(1,9360)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9360, 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("9360.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9360.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:	$74.7399762919$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4680)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.14510$ of defining polynomial
Character	$\chi$	$=$	9360.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.74657 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.74657 q^{7} +2.54364 q^{11} +1.00000 q^{13} +1.74657 q^{17} +2.29021 q^{19} -1.74657 q^{23} +1.00000 q^{25} -4.29021 q^{29} -2.00000 q^{31} +3.74657 q^{35} +8.03677 q^{37} -2.94950 q^{41} -7.49314 q^{43} -3.49314 q^{47} +7.03677 q^{49} +2.54364 q^{53} -2.54364 q^{55} -8.58041 q^{59} -4.03677 q^{61} -1.00000 q^{65} +1.49314 q^{67} +13.5299 q^{71} +9.78334 q^{73} -9.52991 q^{77} -8.94950 q^{79} -1.74657 q^{85} +6.94950 q^{89} -3.74657 q^{91} -2.29021 q^{95} +5.34071 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 - 3 * q^7	$ 3 q - 3 q^{5} - 3 q^{7} + 3 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{19} + 3 q^{23} + 3 q^{25} - 6 q^{31} + 3 q^{35} + 3 q^{37} + 3 q^{41} - 6 q^{43} + 6 q^{47} + 3 q^{53} - 3 q^{55} + 9 q^{61} - 3 q^{65} - 12 q^{67} + 3 q^{71} + 9 q^{77} - 15 q^{79} + 3 q^{85} + 9 q^{89} - 3 q^{91} + 6 q^{95} + 15 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 - 3 * q^7 + 3 * q^11 + 3 * q^13 - 3 * q^17 - 6 * q^19 + 3 * q^23 + 3 * q^25 - 6 * q^31 + 3 * q^35 + 3 * q^37 + 3 * q^41 - 6 * q^43 + 6 * q^47 + 3 * q^53 - 3 * q^55 + 9 * q^61 - 3 * q^65 - 12 * q^67 + 3 * q^71 + 9 * q^77 - 15 * q^79 + 3 * q^85 + 9 * q^89 - 3 * q^91 + 6 * q^95 + 15 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.74657	−1.41607	−0.708035	−	0.706177i	$-0.750418\pi$
$7$	−3.74657	−1.41607	−0.708035	+	0.706177i	$0.750418\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	2.54364	0.766935	0.383468	−	0.923554i	$-0.374730\pi$
$11$	2.54364	0.766935	0.383468	+	0.923554i	$0.374730\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.74657	0.423605	0.211803	−	0.977312i	$-0.432067\pi$
$17$	1.74657	0.423605	0.211803	+	0.977312i	$0.432067\pi$
$18$	0	0
$19$	2.29021	0.525409	0.262705	−	0.964876i	$-0.415386\pi$
$19$	2.29021	0.525409	0.262705	+	0.964876i	$0.415386\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−1.74657	−0.364185	−0.182092	−	0.983281i	$-0.558287\pi$
$23$	−1.74657	−0.364185	−0.182092	+	0.983281i	$0.558287\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.29021	−0.796671	−0.398336	−	0.917240i	$-0.630412\pi$
$29$	−4.29021	−0.796671	−0.398336	+	0.917240i	$0.630412\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.74657	0.633286
$36$	0	0
$37$	8.03677	1.32124	0.660619	−	0.750722i	$-0.270294\pi$
$37$	8.03677	1.32124	0.660619	+	0.750722i	$0.270294\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.94950	−0.460634	−0.230317	−	0.973116i	$-0.573976\pi$
$41$	−2.94950	−0.460634	−0.230317	+	0.973116i	$0.573976\pi$
$42$	0	0
$43$	−7.49314	−1.14269	−0.571346	−	0.820709i	$-0.693579\pi$
$43$	−7.49314	−1.14269	−0.571346	+	0.820709i	$0.693579\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−3.49314	−0.509526	−0.254763	−	0.967003i	$-0.581998\pi$
$47$	−3.49314	−0.509526	−0.254763	+	0.967003i	$0.581998\pi$
$48$	0	0
$49$	7.03677	1.00525
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.54364	0.349395	0.174698	−	0.984622i	$-0.444105\pi$
$53$	2.54364	0.349395	0.174698	+	0.984622i	$0.444105\pi$
$54$	0	0
$55$	−2.54364	−0.342984
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−8.58041	−1.11707	−0.558537	−	0.829479i	$-0.688637\pi$
$59$	−8.58041	−1.11707	−0.558537	+	0.829479i	$0.688637\pi$
$60$	0	0
$61$	−4.03677	−0.516856	−0.258428	−	0.966031i	$-0.583204\pi$
$61$	−4.03677	−0.516856	−0.258428	+	0.966031i	$0.583204\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	1.49314	0.182416	0.0912078	−	0.995832i	$-0.470927\pi$
$67$	1.49314	0.182416	0.0912078	+	0.995832i	$0.470927\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	13.5299	1.60570	0.802852	−	0.596178i	$-0.203315\pi$
$71$	13.5299	1.60570	0.802852	+	0.596178i	$0.203315\pi$
$72$	0	0
$73$	9.78334	1.14505	0.572527	−	0.819886i	$-0.305963\pi$
$73$	9.78334	1.14505	0.572527	+	0.819886i	$0.305963\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−9.52991	−1.08603
$78$	0	0
$79$	−8.94950	−1.00690	−0.503449	−	0.864025i	$-0.667936\pi$
$79$	−8.94950	−1.00690	−0.503449	+	0.864025i	$0.667936\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−1.74657	−0.189442
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.94950	0.736645	0.368323	−	0.929698i	$-0.379932\pi$
$89$	6.94950	0.736645	0.368323	+	0.929698i	$0.379932\pi$
$90$	0	0
$91$	−3.74657	−0.392747
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.29021	−0.234970
$96$	0	0
$97$	5.34071	0.542267	0.271133	−	0.962542i	$-0.412602\pi$
$97$	5.34071	0.542267	0.271133	+	0.962542i	$0.412602\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−7.78334	−0.774471	−0.387236	−	0.921981i	$-0.626570\pi$
$101$	−7.78334	−0.774471	−0.387236	+	0.921981i	$0.626570\pi$
$102$	0	0
$103$	8.07355	0.795510	0.397755	−	0.917492i	$-0.369789\pi$
$103$	8.07355	0.795510	0.397755	+	0.917492i	$0.369789\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.52991	−0.534597	−0.267298	−	0.963614i	$-0.586131\pi$
$107$	−5.52991	−0.534597	−0.267298	+	0.963614i	$0.586131\pi$
$108$	0	0
$109$	−18.3638	−1.75893	−0.879464	−	0.475965i	$-0.842099\pi$
$109$	−18.3638	−1.75893	−0.879464	+	0.475965i	$0.842099\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	16.8706	1.58705	0.793527	−	0.608535i	$-0.208242\pi$
$113$	16.8706	1.58705	0.793527	+	0.608535i	$0.208242\pi$
$114$	0	0
$115$	1.74657	0.162868
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.54364	−0.599854
$120$	0	0
$121$	−4.52991	−0.411810
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.59414	−0.141457	−0.0707284	−	0.997496i	$-0.522532\pi$
$127$	−1.59414	−0.141457	−0.0707284	+	0.997496i	$0.522532\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−3.78334	−0.330552	−0.165276	−	0.986247i	$-0.552852\pi$
$131$	−3.78334	−0.330552	−0.165276	+	0.986247i	$0.552852\pi$
$132$	0	0
$133$	−8.58041	−0.744016
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.49314	0.811053	0.405527	−	0.914083i	$-0.367088\pi$
$137$	9.49314	0.811053	0.405527	+	0.914083i	$0.367088\pi$
$138$	0	0
$139$	−1.45636	−0.123527	−0.0617635	−	0.998091i	$-0.519672\pi$
$139$	−1.45636	−0.123527	−0.0617635	+	0.998091i	$0.519672\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.54364	0.212710
$144$	0	0
$145$	4.29021	0.356282
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.5299	1.27226	0.636130	−	0.771582i	$-0.280534\pi$
$149$	15.5299	1.27226	0.636130	+	0.771582i	$0.280534\pi$
$150$	0	0
$151$	−18.0735	−1.47080	−0.735402	−	0.677631i	$-0.763007\pi$
$151$	−18.0735	−1.47080	−0.735402	+	0.677631i	$0.763007\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.00000	0.160644
$156$	0	0
$157$	13.4931	1.07687	0.538435	−	0.842667i	$-0.319016\pi$
$157$	13.4931	1.07687	0.538435	+	0.842667i	$0.319016\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	6.54364	0.515711
$162$	0	0
$163$	−4.54364	−0.355885	−0.177943	−	0.984041i	$-0.556944\pi$
$163$	−4.54364	−0.355885	−0.177943	+	0.984041i	$0.556944\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.08727	0.393665	0.196833	−	0.980437i	$-0.436934\pi$
$167$	5.08727	0.393665	0.196833	+	0.980437i	$0.436934\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.07355	−0.613820	−0.306910	−	0.951738i	$-0.599295\pi$
$173$	−8.07355	−0.613820	−0.306910	+	0.951738i	$0.599295\pi$
$174$	0	0
$175$	−3.74657	−0.283214
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.8569	1.78315	0.891574	−	0.452875i	$-0.149602\pi$
$179$	23.8569	1.78315	0.891574	+	0.452875i	$0.149602\pi$
$180$	0	0
$181$	13.9358	1.03584	0.517919	−	0.855430i	$-0.326707\pi$
$181$	13.9358	1.03584	0.517919	+	0.855430i	$0.326707\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−8.03677	−0.590875
$186$	0	0
$187$	4.44264	0.324878
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1.08727	−0.0786724	−0.0393362	−	0.999226i	$-0.512524\pi$
$191$	−1.08727	−0.0786724	−0.0393362	+	0.999226i	$0.512524\pi$
$192$	0	0
$193$	−11.3133	−0.814346	−0.407173	−	0.913351i	$-0.633485\pi$
$193$	−11.3133	−0.814346	−0.407173	+	0.913351i	$0.633485\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.08727	0.219959	0.109980	−	0.993934i	$-0.464921\pi$
$197$	3.08727	0.219959	0.109980	+	0.993934i	$0.464921\pi$
$198$	0	0
$199$	−20.5804	−1.45891	−0.729453	−	0.684031i	$-0.760225\pi$
$199$	−20.5804	−1.45891	−0.729453	+	0.684031i	$0.760225\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.0735	1.12814
$204$	0	0
$205$	2.94950	0.206002
$206$	0	0
$207$	0	0
$208$	0	0
$209$	5.82545	0.402955
$210$	0	0
$211$	13.3921	0.921953	0.460976	−	0.887412i	$-0.347499\pi$
$211$	13.3921	0.921953	0.460976	+	0.887412i	$0.347499\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	7.49314	0.511028
$216$	0	0
$217$	7.49314	0.508667
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.74657	0.117487
$222$	0	0
$223$	−12.6961	−0.850192	−0.425096	−	0.905148i	$-0.639760\pi$
$223$	−12.6961	−0.850192	−0.425096	+	0.905148i	$0.639760\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	−6.29021	−0.415668	−0.207834	−	0.978164i	$-0.566641\pi$
$229$	−6.29021	−0.415668	−0.207834	+	0.978164i	$0.566641\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	23.6456	1.54907	0.774536	−	0.632529i	$-0.217983\pi$
$233$	23.6456	1.54907	0.774536	+	0.632529i	$0.217983\pi$
$234$	0	0
$235$	3.49314	0.227867
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−22.6172	−1.46298	−0.731492	−	0.681850i	$-0.761176\pi$
$239$	−22.6172	−1.46298	−0.731492	+	0.681850i	$0.761176\pi$
$240$	0	0
$241$	16.1745	1.04189	0.520947	−	0.853589i	$-0.325579\pi$
$241$	16.1745	1.04189	0.520947	+	0.853589i	$0.325579\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−7.03677	−0.449563
$246$	0	0
$247$	2.29021	0.145722
$248$	0	0
$249$	0	0
$250$	0	0
$251$	23.7833	1.50119	0.750596	−	0.660762i	$-0.229767\pi$
$251$	23.7833	1.50119	0.750596	+	0.660762i	$0.229767\pi$
$252$	0	0
$253$	−4.44264	−0.279306
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.6823	0.853481	0.426740	−	0.904374i	$-0.359662\pi$
$257$	13.6823	0.853481	0.426740	+	0.904374i	$0.359662\pi$
$258$	0	0
$259$	−30.1103	−1.87096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	5.37748	0.331590	0.165795	−	0.986160i	$-0.446981\pi$
$263$	5.37748	0.331590	0.165795	+	0.986160i	$0.446981\pi$
$264$	0	0
$265$	−2.54364	−0.156254
$266$	0	0
$267$	0	0
$268$	0	0
$269$	5.30393	0.323386	0.161693	−	0.986841i	$-0.448304\pi$
$269$	5.30393	0.323386	0.161693	+	0.986841i	$0.448304\pi$
$270$	0	0
$271$	3.08727	0.187539	0.0937693	−	0.995594i	$-0.470108\pi$
$271$	3.08727	0.187539	0.0937693	+	0.995594i	$0.470108\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.54364	0.153387
$276$	0	0
$277$	23.3921	1.40550	0.702749	−	0.711438i	$-0.251956\pi$
$277$	23.3921	1.40550	0.702749	+	0.711438i	$0.251956\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	18.5804	1.10841	0.554207	−	0.832379i	$-0.313022\pi$
$281$	18.5804	1.10841	0.554207	+	0.832379i	$0.313022\pi$
$282$	0	0
$283$	−15.5667	−0.925343	−0.462672	−	0.886530i	$-0.653109\pi$
$283$	−15.5667	−0.925343	−0.462672	+	0.886530i	$0.653109\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11.0505	0.652290
$288$	0	0
$289$	−13.9495	−0.820559
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.01373	0.176064	0.0880319	−	0.996118i	$-0.471942\pi$
$293$	3.01373	0.176064	0.0880319	+	0.996118i	$0.471942\pi$
$294$	0	0
$295$	8.58041	0.499571
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1.74657	−0.101007
$300$	0	0
$301$	28.0735	1.61813
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.03677	0.231145
$306$	0	0
$307$	1.05050	0.0599552	0.0299776	−	0.999551i	$-0.490456\pi$
$307$	1.05050	0.0599552	0.0299776	+	0.999551i	$0.490456\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.0598	1.30760	0.653801	−	0.756666i	$-0.273173\pi$
$311$	23.0598	1.30760	0.653801	+	0.756666i	$0.273173\pi$
$312$	0	0
$313$	28.4794	1.60975	0.804876	−	0.593443i	$-0.202232\pi$
$313$	28.4794	1.60975	0.804876	+	0.593443i	$0.202232\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.1608	0.626854	0.313427	−	0.949612i	$-0.398523\pi$
$317$	11.1608	0.626854	0.313427	+	0.949612i	$0.398523\pi$
$318$	0	0
$319$	−10.9127	−0.610995
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	0	0
$328$	0	0
$329$	13.0873	0.721525
$330$	0	0
$331$	8.46475	0.465265	0.232632	−	0.972565i	$-0.425266\pi$
$331$	8.46475	0.465265	0.232632	+	0.972565i	$0.425266\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1.49314	−0.0815788
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−5.08727	−0.275491
$342$	0	0
$343$	−0.137775	−0.00743915
$344$	0	0
$345$	0	0
$346$	0	0
$347$	26.8485	1.44130	0.720651	−	0.693298i	$-0.243843\pi$
$347$	26.8485	1.44130	0.720651	+	0.693298i	$0.243843\pi$
$348$	0	0
$349$	7.88434	0.422039	0.211020	−	0.977482i	$-0.432322\pi$
$349$	7.88434	0.422039	0.211020	+	0.977482i	$0.432322\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.5804	0.776037	0.388018	−	0.921652i	$-0.373160\pi$
$353$	14.5804	0.776037	0.388018	+	0.921652i	$0.373160\pi$
$354$	0	0
$355$	−13.5299	−0.718093
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.1608	−0.694602	−0.347301	−	0.937754i	$-0.612902\pi$
$359$	−13.1608	−0.694602	−0.347301	+	0.937754i	$0.612902\pi$
$360$	0	0
$361$	−13.7550	−0.723945
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−9.78334	−0.512084
$366$	0	0
$367$	33.7412	1.76128	0.880639	−	0.473788i	$-0.157114\pi$
$367$	33.7412	1.76128	0.880639	+	0.473788i	$0.157114\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−9.52991	−0.494768
$372$	0	0
$373$	16.7550	0.867539	0.433769	−	0.901024i	$-0.357183\pi$
$373$	16.7550	0.867539	0.433769	+	0.901024i	$0.357183\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−4.29021	−0.220957
$378$	0	0
$379$	−9.20293	−0.472723	−0.236361	−	0.971665i	$-0.575955\pi$
$379$	−9.20293	−0.472723	−0.236361	+	0.971665i	$0.575955\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	9.52991	0.485689
$386$	0	0
$387$	0	0
$388$	0	0
$389$	31.2765	1.58578	0.792890	−	0.609365i	$-0.208575\pi$
$389$	31.2765	1.58578	0.792890	+	0.609365i	$0.208575\pi$
$390$	0	0
$391$	−3.05050	−0.154270
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.94950	0.450298
$396$	0	0
$397$	5.86223	0.294217	0.147108	−	0.989120i	$-0.453003\pi$
$397$	5.86223	0.294217	0.147108	+	0.989120i	$0.453003\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.59414	0.179483	0.0897413	−	0.995965i	$-0.471396\pi$
$401$	3.59414	0.179483	0.0897413	+	0.995965i	$0.471396\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.4426	1.01330
$408$	0	0
$409$	4.68141	0.231481	0.115740	−	0.993279i	$-0.463076\pi$
$409$	4.68141	0.231481	0.115740	+	0.993279i	$0.463076\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	32.1471	1.58186
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.69607	−0.327124	−0.163562	−	0.986533i	$-0.552298\pi$
$419$	−6.69607	−0.327124	−0.163562	+	0.986533i	$0.552298\pi$
$420$	0	0
$421$	2.29021	0.111618	0.0558089	−	0.998441i	$-0.482226\pi$
$421$	2.29021	0.111618	0.0558089	+	0.998441i	$0.482226\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.74657	0.0847210
$426$	0	0
$427$	15.1240	0.731904
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−17.1608	−0.826608	−0.413304	−	0.910593i	$-0.635625\pi$
$431$	−17.1608	−0.826608	−0.413304	+	0.910593i	$0.635625\pi$
$432$	0	0
$433$	8.91273	0.428318	0.214159	−	0.976799i	$-0.431299\pi$
$433$	8.91273	0.428318	0.214159	+	0.976799i	$0.431299\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−4.64464	−0.221676	−0.110838	−	0.993838i	$-0.535354\pi$
$439$	−4.64464	−0.221676	−0.110838	+	0.993838i	$0.535354\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	15.8622	0.753637	0.376819	−	0.926287i	$-0.377018\pi$
$443$	15.8622	0.753637	0.376819	+	0.926287i	$0.377018\pi$
$444$	0	0
$445$	−6.94950	−0.329438
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−4.11032	−0.193978	−0.0969890	−	0.995285i	$-0.530921\pi$
$449$	−4.11032	−0.193978	−0.0969890	+	0.995285i	$0.530921\pi$
$450$	0	0
$451$	−7.50246	−0.353277
$452$	0	0
$453$	0	0
$454$	0	0
$455$	3.74657	0.175642
$456$	0	0
$457$	3.16616	0.148107	0.0740533	−	0.997254i	$-0.476407\pi$
$457$	3.16616	0.148107	0.0740533	+	0.997254i	$0.476407\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.86223	0.273031	0.136516	−	0.990638i	$-0.456410\pi$
$461$	5.86223	0.273031	0.136516	+	0.990638i	$0.456410\pi$
$462$	0	0
$463$	5.84757	0.271760	0.135880	−	0.990725i	$-0.456614\pi$
$463$	5.84757	0.271760	0.135880	+	0.990725i	$0.456614\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.6309	1.46370	0.731852	−	0.681464i	$-0.238656\pi$
$467$	31.6309	1.46370	0.731852	+	0.681464i	$0.238656\pi$
$468$	0	0
$469$	−5.59414	−0.258313
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−19.0598	−0.876371
$474$	0	0
$475$	2.29021	0.105082
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−9.38282	−0.428712	−0.214356	−	0.976756i	$-0.568765\pi$
$479$	−9.38282	−0.428712	−0.214356	+	0.976756i	$0.568765\pi$
$480$	0	0
$481$	8.03677	0.366445
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−5.34071	−0.242509
$486$	0	0
$487$	6.15243	0.278793	0.139397	−	0.990237i	$-0.455484\pi$
$487$	6.15243	0.278793	0.139397	+	0.990237i	$0.455484\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.3039	0.600398	0.300199	−	0.953877i	$-0.402947\pi$
$491$	13.3039	0.600398	0.300199	+	0.953877i	$0.402947\pi$
$492$	0	0
$493$	−7.49314	−0.337474
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−50.6907	−2.27379
$498$	0	0
$499$	−0.391207	−0.0175128	−0.00875641	−	0.999962i	$-0.502787\pi$
$499$	−0.391207	−0.0175128	−0.00875641	+	0.999962i	$0.502787\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−3.78334	−0.168691	−0.0843454	−	0.996437i	$-0.526880\pi$
$503$	−3.78334	−0.168691	−0.0843454	+	0.996437i	$0.526880\pi$
$504$	0	0
$505$	7.78334	0.346354
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.0230	−0.488588	−0.244294	−	0.969701i	$-0.578556\pi$
$509$	−11.0230	−0.488588	−0.244294	+	0.969701i	$0.578556\pi$
$510$	0	0
$511$	−36.6540	−1.62148
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.07355	−0.355763
$516$	0	0
$517$	−8.88527	−0.390774
$518$	0	0
$519$	0	0
$520$	0	0
$521$	38.4794	1.68581	0.842907	−	0.538060i	$-0.180842\pi$
$521$	38.4794	1.68581	0.842907	+	0.538060i	$0.180842\pi$
$522$	0	0
$523$	34.9863	1.52984	0.764921	−	0.644124i	$-0.222778\pi$
$523$	34.9863	1.52984	0.764921	+	0.644124i	$0.222778\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3.49314	−0.152163
$528$	0	0
$529$	−19.9495	−0.867370
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.94950	−0.127757
$534$	0	0
$535$	5.52991	0.239079
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17.8990	0.770964
$540$	0	0
$541$	−27.6823	−1.19016	−0.595078	−	0.803668i	$-0.702879\pi$
$541$	−27.6823	−1.19016	−0.595078	+	0.803668i	$0.702879\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	18.3638	0.786617
$546$	0	0
$547$	−23.0598	−0.985967	−0.492983	−	0.870039i	$-0.664094\pi$
$547$	−23.0598	−0.985967	−0.492983	+	0.870039i	$0.664094\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−9.82545	−0.418578
$552$	0	0
$553$	33.5299	1.42584
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.82545	−0.331575	−0.165788	−	0.986161i	$-0.553017\pi$
$557$	−7.82545	−0.331575	−0.165788	+	0.986161i	$0.553017\pi$
$558$	0	0
$559$	−7.49314	−0.316926
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−36.0829	−1.52071	−0.760356	−	0.649507i	$-0.774975\pi$
$563$	−36.0829	−1.52071	−0.760356	+	0.649507i	$0.774975\pi$
$564$	0	0
$565$	−16.8706	−0.709752
$566$	0	0
$567$	0	0
$568$	0	0
$569$	23.4931	0.984884	0.492442	−	0.870345i	$-0.336104\pi$
$569$	23.4931	0.984884	0.492442	+	0.870345i	$0.336104\pi$
$570$	0	0
$571$	28.7917	1.20490	0.602448	−	0.798158i	$-0.294192\pi$
$571$	28.7917	1.20490	0.602448	+	0.798158i	$0.294192\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.74657	−0.0728369
$576$	0	0
$577$	−33.4143	−1.39105	−0.695527	−	0.718500i	$-0.744829\pi$
$577$	−33.4143	−1.39105	−0.695527	+	0.718500i	$0.744829\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6.47009	0.267964
$584$	0	0
$585$	0	0
$586$	0	0
$587$	20.0000	0.825488	0.412744	−	0.910847i	$-0.364570\pi$
$587$	20.0000	0.825488	0.412744	+	0.910847i	$0.364570\pi$
$588$	0	0
$589$	−4.58041	−0.188733
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−37.7138	−1.54872	−0.774360	−	0.632746i	$-0.781928\pi$
$593$	−37.7138	−1.54872	−0.774360	+	0.632746i	$0.781928\pi$
$594$	0	0
$595$	6.54364	0.268263
$596$	0	0
$597$	0	0
$598$	0	0
$599$	19.0598	0.778763	0.389382	−	0.921077i	$-0.372689\pi$
$599$	19.0598	0.778763	0.389382	+	0.921077i	$0.372689\pi$
$600$	0	0
$601$	−15.7612	−0.642914	−0.321457	−	0.946924i	$-0.604173\pi$
$601$	−15.7612	−0.642914	−0.321457	+	0.946924i	$0.604173\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.52991	0.184167
$606$	0	0
$607$	15.4196	0.625862	0.312931	−	0.949776i	$-0.398689\pi$
$607$	15.4196	0.625862	0.312931	+	0.949776i	$0.398689\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−3.49314	−0.141317
$612$	0	0
$613$	−1.86223	−0.0752146	−0.0376073	−	0.999293i	$-0.511974\pi$
$613$	−1.86223	−0.0752146	−0.0376073	+	0.999293i	$0.511974\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−35.1608	−1.41552	−0.707761	−	0.706452i	$-0.750294\pi$
$617$	−35.1608	−1.41552	−0.707761	+	0.706452i	$0.750294\pi$
$618$	0	0
$619$	3.88434	0.156125	0.0780625	−	0.996948i	$-0.475127\pi$
$619$	3.88434	0.156125	0.0780625	+	0.996948i	$0.475127\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−26.0368	−1.04314
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	14.0368	0.559683
$630$	0	0
$631$	24.5530	0.977438	0.488719	−	0.872441i	$-0.337464\pi$
$631$	24.5530	0.977438	0.488719	+	0.872441i	$0.337464\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.59414	0.0632614
$636$	0	0
$637$	7.03677	0.278807
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.17455	0.0858895	0.0429448	−	0.999077i	$-0.486326\pi$
$641$	2.17455	0.0858895	0.0429448	+	0.999077i	$0.486326\pi$
$642$	0	0
$643$	4.03677	0.159195	0.0795974	−	0.996827i	$-0.474637\pi$
$643$	4.03677	0.159195	0.0795974	+	0.996827i	$0.474637\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	42.7054	1.67892	0.839461	−	0.543420i	$-0.182871\pi$
$647$	42.7054	1.67892	0.839461	+	0.543420i	$0.182871\pi$
$648$	0	0
$649$	−21.8255	−0.856724
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7.49314	−0.293229	−0.146615	−	0.989194i	$-0.546838\pi$
$653$	−7.49314	−0.293229	−0.146615	+	0.989194i	$0.546838\pi$
$654$	0	0
$655$	3.78334	0.147827
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−15.5078	−0.604098	−0.302049	−	0.953292i	$-0.597671\pi$
$659$	−15.5078	−0.604098	−0.302049	+	0.953292i	$0.597671\pi$
$660$	0	0
$661$	21.5520	0.838277	0.419138	−	0.907922i	$-0.362332\pi$
$661$	21.5520	0.838277	0.419138	+	0.907922i	$0.362332\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	8.58041	0.332734
$666$	0	0
$667$	7.49314	0.290135
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.2681	−0.396395
$672$	0	0
$673$	−5.76869	−0.222367	−0.111183	−	0.993800i	$-0.535464\pi$
$673$	−5.76869	−0.222367	−0.111183	+	0.993800i	$0.535464\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.8485	−0.724407	−0.362203	−	0.932099i	$-0.617975\pi$
$677$	−18.8485	−0.724407	−0.362203	+	0.932099i	$0.617975\pi$
$678$	0	0
$679$	−20.0093	−0.767887
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.9863	0.573434	0.286717	−	0.958015i	$-0.407436\pi$
$683$	14.9863	0.573434	0.286717	+	0.958015i	$0.407436\pi$
$684$	0	0
$685$	−9.49314	−0.362714
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.54364	0.0969049
$690$	0	0
$691$	0.927381	0.0352792	0.0176396	−	0.999844i	$-0.494385\pi$
$691$	0.927381	0.0352792	0.0176396	+	0.999844i	$0.494385\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1.45636	0.0552430
$696$	0	0
$697$	−5.15150	−0.195127
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.02838	0.340997	0.170499	−	0.985358i	$-0.445462\pi$
$701$	9.02838	0.340997	0.170499	+	0.985358i	$0.445462\pi$
$702$	0	0
$703$	18.4059	0.694190
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.1608	1.09671
$708$	0	0
$709$	−21.2765	−0.799055	−0.399527	−	0.916721i	$-0.630826\pi$
$709$	−21.2765	−0.799055	−0.399527	+	0.916721i	$0.630826\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.49314	0.130819
$714$	0	0
$715$	−2.54364	−0.0951266
$716$	0	0
$717$	0	0
$718$	0	0
$719$	7.76869	0.289723	0.144862	−	0.989452i	$-0.453726\pi$
$719$	7.76869	0.289723	0.144862	+	0.989452i	$0.453726\pi$
$720$	0	0
$721$	−30.2481	−1.12650
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4.29021	−0.159334
$726$	0	0
$727$	−29.8148	−1.10577	−0.552884	−	0.833258i	$-0.686473\pi$
$727$	−29.8148	−1.10577	−0.552884	+	0.833258i	$0.686473\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.0873	−0.484050
$732$	0	0
$733$	−2.94950	−0.108942	−0.0544711	−	0.998515i	$-0.517347\pi$
$733$	−2.94950	−0.108942	−0.0544711	+	0.998515i	$0.517347\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.79800	0.139901
$738$	0	0
$739$	20.7696	0.764023	0.382011	−	0.924158i	$-0.375232\pi$
$739$	20.7696	0.764023	0.382011	+	0.924158i	$0.375232\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.3784	−1.62809	−0.814043	−	0.580805i	$-0.802738\pi$
$743$	−44.3784	−1.62809	−0.814043	+	0.580805i	$0.802738\pi$
$744$	0	0
$745$	−15.5299	−0.568972
$746$	0	0
$747$	0	0
$748$	0	0
$749$	20.7182	0.757026
$750$	0	0
$751$	40.8653	1.49120	0.745598	−	0.666396i	$-0.232164\pi$
$751$	40.8653	1.49120	0.745598	+	0.666396i	$0.232164\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	18.0735	0.657764
$756$	0	0
$757$	−40.1745	−1.46017	−0.730084	−	0.683357i	$-0.760519\pi$
$757$	−40.1745	−1.46017	−0.730084	+	0.683357i	$0.760519\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.40586	0.159712	0.0798562	−	0.996806i	$-0.474554\pi$
$761$	4.40586	0.159712	0.0798562	+	0.996806i	$0.474554\pi$
$762$	0	0
$763$	68.8011	2.49077
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.58041	−0.309821
$768$	0	0
$769$	−18.3491	−0.661685	−0.330843	−	0.943686i	$-0.607333\pi$
$769$	−18.3491	−0.661685	−0.330843	+	0.943686i	$0.607333\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	37.0598	1.33295	0.666475	−	0.745528i	$-0.267803\pi$
$773$	37.0598	1.33295	0.666475	+	0.745528i	$0.267803\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−6.75496	−0.242022
$780$	0	0
$781$	34.4152	1.23147
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−13.4931	−0.481591
$786$	0	0
$787$	−9.92645	−0.353840	−0.176920	−	0.984225i	$-0.556613\pi$
$787$	−9.92645	−0.353840	−0.176920	+	0.984225i	$0.556613\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−63.2069	−2.24738
$792$	0	0
$793$	−4.03677	−0.143350
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.7045	−0.981342	−0.490671	−	0.871345i	$-0.663248\pi$
$797$	−27.7045	−0.981342	−0.490671	+	0.871345i	$0.663248\pi$
$798$	0	0
$799$	−6.10100	−0.215838
$800$	0	0
$801$	0	0
$802$	0	0
$803$	24.8853	0.878182
$804$	0	0
$805$	−6.54364	−0.230633
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.36282	0.329179	0.164590	−	0.986362i	$-0.447370\pi$
$809$	9.36282	0.329179	0.164590	+	0.986362i	$0.447370\pi$
$810$	0	0
$811$	4.04211	0.141938	0.0709688	−	0.997479i	$-0.477391\pi$
$811$	4.04211	0.141938	0.0709688	+	0.997479i	$0.477391\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.54364	0.159157
$816$	0	0
$817$	−17.1608	−0.600381
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.6770	−1.24514	−0.622568	−	0.782566i	$-0.713911\pi$
$821$	−35.6770	−1.24514	−0.622568	+	0.782566i	$0.713911\pi$
$822$	0	0
$823$	−7.18828	−0.250568	−0.125284	−	0.992121i	$-0.539984\pi$
$823$	−7.18828	−0.250568	−0.125284	+	0.992121i	$0.539984\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.0873	1.28965	0.644825	−	0.764330i	$-0.276930\pi$
$827$	37.0873	1.28965	0.644825	+	0.764330i	$0.276930\pi$
$828$	0	0
$829$	6.14709	0.213497	0.106749	−	0.994286i	$-0.465956\pi$
$829$	6.14709	0.213497	0.106749	+	0.994286i	$0.465956\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	12.2902	0.425830
$834$	0	0
$835$	−5.08727	−0.176052
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−22.4701	−0.775754	−0.387877	−	0.921711i	$-0.626791\pi$
$839$	−22.4701	−0.775754	−0.387877	+	0.921711i	$0.626791\pi$
$840$	0	0
$841$	−10.5941	−0.365315
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	16.9716	0.583152
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−14.0368	−0.481174
$852$	0	0
$853$	−21.0505	−0.720755	−0.360378	−	0.932806i	$-0.617352\pi$
$853$	−21.0505	−0.720755	−0.360378	+	0.932806i	$0.617352\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.60253	0.225538	0.112769	−	0.993621i	$-0.464028\pi$
$857$	6.60253	0.225538	0.112769	+	0.993621i	$0.464028\pi$
$858$	0	0
$859$	−38.1839	−1.30282	−0.651408	−	0.758727i	$-0.725822\pi$
$859$	−38.1839	−1.30282	−0.651408	+	0.758727i	$0.725822\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0.885272	0.0301350	0.0150675	−	0.999886i	$-0.495204\pi$
$863$	0.885272	0.0301350	0.0150675	+	0.999886i	$0.495204\pi$
$864$	0	0
$865$	8.07355	0.274509
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−22.7643	−0.772225
$870$	0	0
$871$	1.49314	0.0505930
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.74657	0.126657
$876$	0	0
$877$	44.3216	1.49664	0.748318	−	0.663340i	$-0.230862\pi$
$877$	44.3216	1.49664	0.748318	+	0.663340i	$0.230862\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.6540	1.10014	0.550070	−	0.835119i	$-0.314601\pi$
$881$	32.6540	1.10014	0.550070	+	0.835119i	$0.314601\pi$
$882$	0	0
$883$	−16.2206	−0.545868	−0.272934	−	0.962033i	$-0.587994\pi$
$883$	−16.2206	−0.545868	−0.272934	+	0.962033i	$0.587994\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.0084	−0.571086	−0.285543	−	0.958366i	$-0.592174\pi$
$887$	−17.0084	−0.571086	−0.285543	+	0.958366i	$0.592174\pi$
$888$	0	0
$889$	5.97255	0.200313
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.00000	−0.267710
$894$	0	0
$895$	−23.8569	−0.797448
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8.58041	0.286173
$900$	0	0
$901$	4.44264	0.148006
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−13.9358	−0.463241
$906$	0	0
$907$	−6.68141	−0.221853	−0.110926	−	0.993829i	$-0.535382\pi$
$907$	−6.68141	−0.221853	−0.110926	+	0.993829i	$0.535382\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.5941	−0.980498	−0.490249	−	0.871583i	$-0.663094\pi$
$911$	−29.5941	−0.980498	−0.490249	+	0.871583i	$0.663094\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	14.1745	0.468085
$918$	0	0
$919$	42.6172	1.40581	0.702906	−	0.711283i	$-0.251886\pi$
$919$	42.6172	1.40581	0.702906	+	0.711283i	$0.251886\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	13.5299	0.445342
$924$	0	0
$925$	8.03677	0.264247
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12.1103	0.397327	0.198663	−	0.980068i	$-0.436340\pi$
$929$	12.1103	0.397327	0.198663	+	0.980068i	$0.436340\pi$
$930$	0	0
$931$	16.1157	0.528169
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.44264	−0.145290
$936$	0	0
$937$	−35.4657	−1.15861	−0.579307	−	0.815110i	$-0.696677\pi$
$937$	−35.4657	−1.15861	−0.579307	+	0.815110i	$0.696677\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	45.0691	1.46921	0.734606	−	0.678494i	$-0.237367\pi$
$941$	45.0691	1.46921	0.734606	+	0.678494i	$0.237367\pi$
$942$	0	0
$943$	5.15150	0.167756
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.07355	−0.132372	−0.0661862	−	0.997807i	$-0.521083\pi$
$947$	−4.07355	−0.132372	−0.0661862	+	0.997807i	$0.521083\pi$
$948$	0	0
$949$	9.78334	0.317581
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35.7191	1.15706	0.578528	−	0.815663i	$-0.303627\pi$
$953$	35.7191	1.15706	0.578528	+	0.815663i	$0.303627\pi$
$954$	0	0
$955$	1.08727	0.0351834
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−35.5667	−1.14851
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	0	0
$964$	0	0
$965$	11.3133	0.364186
$966$	0	0
$967$	−45.4236	−1.46072	−0.730362	−	0.683060i	$-0.760649\pi$
$967$	−45.4236	−1.46072	−0.730362	+	0.683060i	$0.760649\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.7833	1.01998	0.509988	−	0.860182i	$-0.329650\pi$
$971$	31.7833	1.01998	0.509988	+	0.860182i	$0.329650\pi$
$972$	0	0
$973$	5.45636	0.174923
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−61.5667	−1.96969	−0.984846	−	0.173429i	$-0.944515\pi$
$977$	−61.5667	−1.96969	−0.984846	+	0.173429i	$0.944515\pi$
$978$	0	0
$979$	17.6770	0.564960
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.39214	0.171982	0.0859912	−	0.996296i	$-0.472594\pi$
$983$	5.39214	0.171982	0.0859912	+	0.996296i	$0.472594\pi$
$984$	0	0
$985$	−3.08727	−0.0983687
$986$	0	0
$987$	0	0
$988$	0	0
$989$	13.0873	0.416151
$990$	0	0
$991$	−3.19760	−0.101575	−0.0507875	−	0.998709i	$-0.516173\pi$
$991$	−3.19760	−0.101575	−0.0507875	+	0.998709i	$0.516173\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	20.5804	0.652443
$996$	0	0
$997$	−14.2313	−0.450710	−0.225355	−	0.974277i	$-0.572354\pi$
$997$	−14.2313	−0.450710	−0.225355	+	0.974277i	$0.572354\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9360.2.a.cx.1.1		3
3.2	odd	2		9360.2.a.dc.1.1		3
4.3	odd	2		4680.2.a.bi.1.3	✓	3
12.11	even	2		4680.2.a.bk.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4680.2.a.bi.1.3	✓	3	4.3	odd	2
4680.2.a.bk.1.3	yes	3	12.11	even	2
9360.2.a.cx.1.1		3	1.1	even	1	trivial
9360.2.a.dc.1.1		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9360 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9360.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.74657 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.74657 q^{7} +2.54364 q^{11} +1.00000 q^{13} +1.74657 q^{17} +2.29021 q^{19} -1.74657 q^{23} +1.00000 q^{25} -4.29021 q^{29} -2.00000 q^{31} +3.74657 q^{35} +8.03677 q^{37} -2.94950 q^{41} -7.49314 q^{43} -3.49314 q^{47} +7.03677 q^{49} +2.54364 q^{53} -2.54364 q^{55} -8.58041 q^{59} -4.03677 q^{61} -1.00000 q^{65} +1.49314 q^{67} +13.5299 q^{71} +9.78334 q^{73} -9.52991 q^{77} -8.94950 q^{79} -1.74657 q^{85} +6.94950 q^{89} -3.74657 q^{91} -2.29021 q^{95} +5.34071 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 - 3 * q^7	\( 3 q - 3 q^{5} - 3 q^{7} + 3 q^{11} + 3 q^{13} - 3 q^{17} - 6 q^{19} + 3 q^{23} + 3 q^{25} - 6 q^{31} + 3 q^{35} + 3 q^{37} + 3 q^{41} - 6 q^{43} + 6 q^{47} + 3 q^{53} - 3 q^{55} + 9 q^{61} - 3 q^{65} - 12 q^{67} + 3 q^{71} + 9 q^{77} - 15 q^{79} + 3 q^{85} + 9 q^{89} - 3 q^{91} + 6 q^{95} + 15 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 - 3 * q^7 + 3 * q^11 + 3 * q^13 - 3 * q^17 - 6 * q^19 + 3 * q^23 + 3 * q^25 - 6 * q^31 + 3 * q^35 + 3 * q^37 + 3 * q^41 - 6 * q^43 + 6 * q^47 + 3 * q^53 - 3 * q^55 + 9 * q^61 - 3 * q^65 - 12 * q^67 + 3 * q^71 + 9 * q^77 - 15 * q^79 + 3 * q^85 + 9 * q^89 - 3 * q^91 + 6 * q^95 + 15 * q^97

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