LMFDB - Embedded newform 4320.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4320,2,Mod(1,4320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4320, 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("4320.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4320 = 2^{5} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4320.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:	$34.4953736732$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	4320.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -3.64575 q^{7} +3.64575 q^{11} -1.64575 q^{13} +0.645751 q^{17} -2.64575 q^{19} +6.29150 q^{23} +1.00000 q^{25} -0.354249 q^{29} +2.64575 q^{31} +3.64575 q^{35} +5.29150 q^{37} -5.64575 q^{41} +8.93725 q^{43} -5.29150 q^{47} +6.29150 q^{49} -11.9373 q^{53} -3.64575 q^{55} -1.64575 q^{59} -13.5830 q^{61} +1.64575 q^{65} +6.00000 q^{67} -7.64575 q^{71} +14.9373 q^{73} -13.2915 q^{77} -12.6458 q^{79} +5.00000 q^{83} -0.645751 q^{85} -13.6458 q^{89} +6.00000 q^{91} +2.64575 q^{95} -11.2915 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 2 q^{23} + 2 q^{25} - 6 q^{29} + 2 q^{35} - 6 q^{41} + 2 q^{43} + 2 q^{49} - 8 q^{53} - 2 q^{55} + 2 q^{59} - 6 q^{61} - 2 q^{65} + 12 q^{67} - 10 q^{71}+ \cdots - 12 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 2 * q^23 + 2 * q^25 - 6 * q^29 + 2 * q^35 - 6 * q^41 + 2 * q^43 + 2 * q^49 - 8 * q^53 - 2 * q^55 + 2 * q^59 - 6 * q^61 - 2 * q^65 + 12 * q^67 - 10 * q^71 + 14 * q^73 - 16 * q^77 - 20 * q^79 + 10 * q^83 + 4 * q^85 - 22 * q^89 + 12 * q^91 - 12 * q^97

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.64575	−1.37796	−0.688982	−	0.724778i	$-0.741942\pi$
$7$	−3.64575	−1.37796	−0.688982	+	0.724778i	$0.741942\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.64575	1.09924	0.549618	−	0.835416i	$-0.314773\pi$
$11$	3.64575	1.09924	0.549618	+	0.835416i	$0.314773\pi$
$12$	0	0
$13$	−1.64575	−0.456449	−0.228225	−	0.973609i	$-0.573292\pi$
$13$	−1.64575	−0.456449	−0.228225	+	0.973609i	$0.573292\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.645751	0.156618	0.0783088	−	0.996929i	$-0.475048\pi$
$17$	0.645751	0.156618	0.0783088	+	0.996929i	$0.475048\pi$
$18$	0	0
$19$	−2.64575	−0.606977	−0.303488	−	0.952835i	$-0.598151\pi$
$19$	−2.64575	−0.606977	−0.303488	+	0.952835i	$0.598151\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.29150	1.31187	0.655934	−	0.754818i	$-0.272275\pi$
$23$	6.29150	1.31187	0.655934	+	0.754818i	$0.272275\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.354249	−0.0657823	−0.0328912	−	0.999459i	$-0.510471\pi$
$29$	−0.354249	−0.0657823	−0.0328912	+	0.999459i	$0.510471\pi$
$30$	0	0
$31$	2.64575	0.475191	0.237595	−	0.971364i	$-0.423641\pi$
$31$	2.64575	0.475191	0.237595	+	0.971364i	$0.423641\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.64575	0.616244
$36$	0	0
$37$	5.29150	0.869918	0.434959	−	0.900450i	$-0.356763\pi$
$37$	5.29150	0.869918	0.434959	+	0.900450i	$0.356763\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−5.64575	−0.881718	−0.440859	−	0.897576i	$-0.645326\pi$
$41$	−5.64575	−0.881718	−0.440859	+	0.897576i	$0.645326\pi$
$42$	0	0
$43$	8.93725	1.36292	0.681459	−	0.731856i	$-0.261346\pi$
$43$	8.93725	1.36292	0.681459	+	0.731856i	$0.261346\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.29150	−0.771845	−0.385922	−	0.922531i	$-0.626117\pi$
$47$	−5.29150	−0.771845	−0.385922	+	0.922531i	$0.626117\pi$
$48$	0	0
$49$	6.29150	0.898786
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−11.9373	−1.63971	−0.819854	−	0.572573i	$-0.805945\pi$
$53$	−11.9373	−1.63971	−0.819854	+	0.572573i	$0.805945\pi$
$54$	0	0
$55$	−3.64575	−0.491593
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.64575	−0.214259	−0.107129	−	0.994245i	$-0.534166\pi$
$59$	−1.64575	−0.214259	−0.107129	+	0.994245i	$0.534166\pi$
$60$	0	0
$61$	−13.5830	−1.73913	−0.869563	−	0.493822i	$-0.835599\pi$
$61$	−13.5830	−1.73913	−0.869563	+	0.493822i	$0.835599\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.64575	0.204130
$66$	0	0
$67$	6.00000	0.733017	0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000	0.733017	0.366508	+	0.930415i	$0.380553\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.64575	−0.907384	−0.453692	−	0.891159i	$-0.649893\pi$
$71$	−7.64575	−0.907384	−0.453692	+	0.891159i	$0.649893\pi$
$72$	0	0
$73$	14.9373	1.74827	0.874137	−	0.485680i	$-0.161428\pi$
$73$	14.9373	1.74827	0.874137	+	0.485680i	$0.161428\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−13.2915	−1.51471
$78$	0	0
$79$	−12.6458	−1.42276	−0.711379	−	0.702809i	$-0.751929\pi$
$79$	−12.6458	−1.42276	−0.711379	+	0.702809i	$0.751929\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.00000	0.548821	0.274411	−	0.961613i	$-0.411517\pi$
$83$	5.00000	0.548821	0.274411	+	0.961613i	$0.411517\pi$
$84$	0	0
$85$	−0.645751	−0.0700416
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−13.6458	−1.44645	−0.723223	−	0.690614i	$-0.757340\pi$
$89$	−13.6458	−1.44645	−0.723223	+	0.690614i	$0.757340\pi$
$90$	0	0
$91$	6.00000	0.628971
$92$	0	0
$93$	0	0
$94$	0	0
$95$	2.64575	0.271448
$96$	0	0
$97$	−11.2915	−1.14648	−0.573239	−	0.819388i	$-0.694313\pi$
$97$	−11.2915	−1.14648	−0.573239	+	0.819388i	$0.694313\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−0.708497	−0.0704981	−0.0352491	−	0.999379i	$-0.511222\pi$
$101$	−0.708497	−0.0704981	−0.0352491	+	0.999379i	$0.511222\pi$
$102$	0	0
$103$	11.2915	1.11258	0.556292	−	0.830987i	$-0.312223\pi$
$103$	11.2915	1.11258	0.556292	+	0.830987i	$0.312223\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	−8.29150	−0.794182	−0.397091	−	0.917779i	$-0.629980\pi$
$109$	−8.29150	−0.794182	−0.397091	+	0.917779i	$0.629980\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.58301	0.619277	0.309639	−	0.950854i	$-0.399792\pi$
$113$	6.58301	0.619277	0.309639	+	0.950854i	$0.399792\pi$
$114$	0	0
$115$	−6.29150	−0.586686
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.35425	−0.215814
$120$	0	0
$121$	2.29150	0.208318
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	0.583005	0.0517333	0.0258667	−	0.999665i	$-0.491765\pi$
$127$	0.583005	0.0517333	0.0258667	+	0.999665i	$0.491765\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−11.8745	−1.03748	−0.518740	−	0.854932i	$-0.673599\pi$
$131$	−11.8745	−1.03748	−0.518740	+	0.854932i	$0.673599\pi$
$132$	0	0
$133$	9.64575	0.836393
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−17.9373	−1.53248	−0.766242	−	0.642553i	$-0.777875\pi$
$137$	−17.9373	−1.53248	−0.766242	+	0.642553i	$0.777875\pi$
$138$	0	0
$139$	1.41699	0.120188	0.0600940	−	0.998193i	$-0.480860\pi$
$139$	1.41699	0.120188	0.0600940	+	0.998193i	$0.480860\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	0.354249	0.0294188
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.06275	−0.250910	−0.125455	−	0.992099i	$-0.540039\pi$
$149$	−3.06275	−0.250910	−0.125455	+	0.992099i	$0.540039\pi$
$150$	0	0
$151$	−6.70850	−0.545930	−0.272965	−	0.962024i	$-0.588004\pi$
$151$	−6.70850	−0.545930	−0.272965	+	0.962024i	$0.588004\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.64575	−0.212512
$156$	0	0
$157$	2.93725	0.234418	0.117209	−	0.993107i	$-0.462605\pi$
$157$	2.93725	0.234418	0.117209	+	0.993107i	$0.462605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−22.9373	−1.80771
$162$	0	0
$163$	9.29150	0.727767	0.363883	−	0.931445i	$-0.381451\pi$
$163$	9.29150	0.727767	0.363883	+	0.931445i	$0.381451\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.70850	0.132207	0.0661037	−	0.997813i	$-0.478943\pi$
$167$	1.70850	0.132207	0.0661037	+	0.997813i	$0.478943\pi$
$168$	0	0
$169$	−10.2915	−0.791654
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−16.6458	−1.26555	−0.632777	−	0.774334i	$-0.718085\pi$
$173$	−16.6458	−1.26555	−0.632777	+	0.774334i	$0.718085\pi$
$174$	0	0
$175$	−3.64575	−0.275593
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3.29150	−0.246018	−0.123009	−	0.992406i	$-0.539254\pi$
$179$	−3.29150	−0.246018	−0.123009	+	0.992406i	$0.539254\pi$
$180$	0	0
$181$	−5.70850	−0.424309	−0.212155	−	0.977236i	$-0.568048\pi$
$181$	−5.70850	−0.424309	−0.212155	+	0.977236i	$0.568048\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.29150	−0.389039
$186$	0	0
$187$	2.35425	0.172160
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.2915	−0.961739	−0.480870	−	0.876792i	$-0.659679\pi$
$191$	−13.2915	−0.961739	−0.480870	+	0.876792i	$0.659679\pi$
$192$	0	0
$193$	−15.5203	−1.11717	−0.558586	−	0.829446i	$-0.688656\pi$
$193$	−15.5203	−1.11717	−0.558586	+	0.829446i	$0.688656\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	25.8118	1.83901	0.919506	−	0.393076i	$-0.128589\pi$
$197$	25.8118	1.83901	0.919506	+	0.393076i	$0.128589\pi$
$198$	0	0
$199$	−25.8745	−1.83420	−0.917098	−	0.398663i	$-0.869474\pi$
$199$	−25.8745	−1.83420	−0.917098	+	0.398663i	$0.869474\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.29150	0.0906457
$204$	0	0
$205$	5.64575	0.394316
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−9.64575	−0.667211
$210$	0	0
$211$	20.6458	1.42131	0.710656	−	0.703540i	$-0.248398\pi$
$211$	20.6458	1.42131	0.710656	+	0.703540i	$0.248398\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−8.93725	−0.609516
$216$	0	0
$217$	−9.64575	−0.654796
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−1.06275	−0.0714880
$222$	0	0
$223$	−25.2915	−1.69364	−0.846822	−	0.531876i	$-0.821487\pi$
$223$	−25.2915	−1.69364	−0.846822	+	0.531876i	$0.821487\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.8745	0.854511	0.427256	−	0.904131i	$-0.359480\pi$
$227$	12.8745	0.854511	0.427256	+	0.904131i	$0.359480\pi$
$228$	0	0
$229$	11.5830	0.765427	0.382713	−	0.923867i	$-0.374990\pi$
$229$	11.5830	0.765427	0.382713	+	0.923867i	$0.374990\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.41699	−0.223855	−0.111927	−	0.993716i	$-0.535702\pi$
$233$	−3.41699	−0.223855	−0.111927	+	0.993716i	$0.535702\pi$
$234$	0	0
$235$	5.29150	0.345180
$236$	0	0
$237$	0	0
$238$	0	0
$239$	13.2915	0.859756	0.429878	−	0.902887i	$-0.358557\pi$
$239$	13.2915	0.859756	0.429878	+	0.902887i	$0.358557\pi$
$240$	0	0
$241$	−11.7085	−0.754211	−0.377105	−	0.926170i	$-0.623081\pi$
$241$	−11.7085	−0.754211	−0.377105	+	0.926170i	$0.623081\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.29150	−0.401949
$246$	0	0
$247$	4.35425	0.277054
$248$	0	0
$249$	0	0
$250$	0	0
$251$	7.29150	0.460236	0.230118	−	0.973163i	$-0.426089\pi$
$251$	7.29150	0.460236	0.230118	+	0.973163i	$0.426089\pi$
$252$	0	0
$253$	22.9373	1.44205
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−21.2288	−1.32421	−0.662107	−	0.749410i	$-0.730337\pi$
$257$	−21.2288	−1.32421	−0.662107	+	0.749410i	$0.730337\pi$
$258$	0	0
$259$	−19.2915	−1.19872
$260$	0	0
$261$	0	0
$262$	0	0
$263$	10.7085	0.660314	0.330157	−	0.943926i	$-0.392898\pi$
$263$	10.7085	0.660314	0.330157	+	0.943926i	$0.392898\pi$
$264$	0	0
$265$	11.9373	0.733300
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−15.8745	−0.967886	−0.483943	−	0.875100i	$-0.660796\pi$
$269$	−15.8745	−0.967886	−0.483943	+	0.875100i	$0.660796\pi$
$270$	0	0
$271$	−9.35425	−0.568230	−0.284115	−	0.958790i	$-0.591700\pi$
$271$	−9.35425	−0.568230	−0.284115	+	0.958790i	$0.591700\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.64575	0.219847
$276$	0	0
$277$	3.64575	0.219052	0.109526	−	0.993984i	$-0.465067\pi$
$277$	3.64575	0.219052	0.109526	+	0.993984i	$0.465067\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.6458	−0.694727	−0.347364	−	0.937731i	$-0.612923\pi$
$281$	−11.6458	−0.694727	−0.347364	+	0.937731i	$0.612923\pi$
$282$	0	0
$283$	26.3542	1.56660	0.783299	−	0.621645i	$-0.213536\pi$
$283$	26.3542	1.56660	0.783299	+	0.621645i	$0.213536\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	20.5830	1.21498
$288$	0	0
$289$	−16.5830	−0.975471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−29.2288	−1.70756	−0.853781	−	0.520632i	$-0.825696\pi$
$293$	−29.2288	−1.70756	−0.853781	+	0.520632i	$0.825696\pi$
$294$	0	0
$295$	1.64575	0.0958193
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−10.3542	−0.598802
$300$	0	0
$301$	−32.5830	−1.87805
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13.5830	0.777761
$306$	0	0
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	12.2288	0.693429	0.346715	−	0.937971i	$-0.387297\pi$
$311$	12.2288	0.693429	0.346715	+	0.937971i	$0.387297\pi$
$312$	0	0
$313$	27.2915	1.54261	0.771303	−	0.636468i	$-0.219605\pi$
$313$	27.2915	1.54261	0.771303	+	0.636468i	$0.219605\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.9373	−1.00746	−0.503728	−	0.863862i	$-0.668039\pi$
$317$	−17.9373	−1.00746	−0.503728	+	0.863862i	$0.668039\pi$
$318$	0	0
$319$	−1.29150	−0.0723103
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.70850	−0.0950633
$324$	0	0
$325$	−1.64575	−0.0912899
$326$	0	0
$327$	0	0
$328$	0	0
$329$	19.2915	1.06357
$330$	0	0
$331$	7.29150	0.400777	0.200389	−	0.979716i	$-0.435780\pi$
$331$	7.29150	0.400777	0.200389	+	0.979716i	$0.435780\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.00000	−0.327815
$336$	0	0
$337$	−16.2288	−0.884037	−0.442018	−	0.897006i	$-0.645737\pi$
$337$	−16.2288	−0.884037	−0.442018	+	0.897006i	$0.645737\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.64575	0.522347
$342$	0	0
$343$	2.58301	0.139469
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	22.8745	1.22444	0.612222	−	0.790686i	$-0.290276\pi$
$349$	22.8745	1.22444	0.612222	+	0.790686i	$0.290276\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.29150	0.494537	0.247268	−	0.968947i	$-0.420467\pi$
$353$	9.29150	0.494537	0.247268	+	0.968947i	$0.420467\pi$
$354$	0	0
$355$	7.64575	0.405794
$356$	0	0
$357$	0	0
$358$	0	0
$359$	28.5830	1.50855	0.754277	−	0.656557i	$-0.227988\pi$
$359$	28.5830	1.50855	0.754277	+	0.656557i	$0.227988\pi$
$360$	0	0
$361$	−12.0000	−0.631579
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−14.9373	−0.781852
$366$	0	0
$367$	−28.3542	−1.48008	−0.740040	−	0.672563i	$-0.765193\pi$
$367$	−28.3542	−1.48008	−0.740040	+	0.672563i	$0.765193\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	43.5203	2.25946
$372$	0	0
$373$	28.0000	1.44979	0.724893	−	0.688862i	$-0.241889\pi$
$373$	28.0000	1.44979	0.724893	+	0.688862i	$0.241889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0.583005	0.0300263
$378$	0	0
$379$	35.2288	1.80958	0.904790	−	0.425858i	$-0.140028\pi$
$379$	35.2288	1.80958	0.904790	+	0.425858i	$0.140028\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−11.0000	−0.562074	−0.281037	−	0.959697i	$-0.590678\pi$
$383$	−11.0000	−0.562074	−0.281037	+	0.959697i	$0.590678\pi$
$384$	0	0
$385$	13.2915	0.677398
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−12.2288	−0.620023	−0.310011	−	0.950733i	$-0.600333\pi$
$389$	−12.2288	−0.620023	−0.310011	+	0.950733i	$0.600333\pi$
$390$	0	0
$391$	4.06275	0.205462
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.6458	0.636277
$396$	0	0
$397$	−34.1033	−1.71159	−0.855797	−	0.517312i	$-0.826933\pi$
$397$	−34.1033	−1.71159	−0.855797	+	0.517312i	$0.826933\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9.29150	0.463995	0.231998	−	0.972716i	$-0.425474\pi$
$401$	9.29150	0.463995	0.231998	+	0.972716i	$0.425474\pi$
$402$	0	0
$403$	−4.35425	−0.216901
$404$	0	0
$405$	0	0
$406$	0	0
$407$	19.2915	0.956244
$408$	0	0
$409$	−7.58301	−0.374955	−0.187478	−	0.982269i	$-0.560031\pi$
$409$	−7.58301	−0.374955	−0.187478	+	0.982269i	$0.560031\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.00000	0.295241
$414$	0	0
$415$	−5.00000	−0.245440
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.35425	−0.212719	−0.106359	−	0.994328i	$-0.533919\pi$
$419$	−4.35425	−0.212719	−0.106359	+	0.994328i	$0.533919\pi$
$420$	0	0
$421$	23.4575	1.14325	0.571625	−	0.820515i	$-0.306313\pi$
$421$	23.4575	1.14325	0.571625	+	0.820515i	$0.306313\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0.645751	0.0313235
$426$	0	0
$427$	49.5203	2.39645
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−3.06275	−0.147527	−0.0737636	−	0.997276i	$-0.523501\pi$
$431$	−3.06275	−0.147527	−0.0737636	+	0.997276i	$0.523501\pi$
$432$	0	0
$433$	15.0627	0.723869	0.361935	−	0.932203i	$-0.382116\pi$
$433$	15.0627	0.723869	0.361935	+	0.932203i	$0.382116\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−16.6458	−0.796274
$438$	0	0
$439$	23.1033	1.10266	0.551329	−	0.834288i	$-0.314121\pi$
$439$	23.1033	1.10266	0.551329	+	0.834288i	$0.314121\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−38.0405	−1.80736	−0.903680	−	0.428209i	$-0.859145\pi$
$443$	−38.0405	−1.80736	−0.903680	+	0.428209i	$0.859145\pi$
$444$	0	0
$445$	13.6458	0.646871
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−37.5203	−1.77069	−0.885345	−	0.464934i	$-0.846078\pi$
$449$	−37.5203	−1.77069	−0.885345	+	0.464934i	$0.846078\pi$
$450$	0	0
$451$	−20.5830	−0.969216
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−6.00000	−0.281284
$456$	0	0
$457$	6.58301	0.307940	0.153970	−	0.988076i	$-0.450794\pi$
$457$	6.58301	0.307940	0.153970	+	0.988076i	$0.450794\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	12.5830	0.586049	0.293024	−	0.956105i	$-0.405338\pi$
$461$	12.5830	0.586049	0.293024	+	0.956105i	$0.405338\pi$
$462$	0	0
$463$	1.29150	0.0600212	0.0300106	−	0.999550i	$-0.490446\pi$
$463$	1.29150	0.0600212	0.0300106	+	0.999550i	$0.490446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.4575	−1.73333	−0.866663	−	0.498893i	$-0.833740\pi$
$467$	−37.4575	−1.73333	−0.866663	+	0.498893i	$0.833740\pi$
$468$	0	0
$469$	−21.8745	−1.01007
$470$	0	0
$471$	0	0
$472$	0	0
$473$	32.5830	1.49817
$474$	0	0
$475$	−2.64575	−0.121395
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.5203	−1.34882	−0.674408	−	0.738359i	$-0.735601\pi$
$479$	−29.5203	−1.34882	−0.674408	+	0.738359i	$0.735601\pi$
$480$	0	0
$481$	−8.70850	−0.397073
$482$	0	0
$483$	0	0
$484$	0	0
$485$	11.2915	0.512721
$486$	0	0
$487$	−15.5203	−0.703290	−0.351645	−	0.936133i	$-0.614378\pi$
$487$	−15.5203	−0.703290	−0.351645	+	0.936133i	$0.614378\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	16.5830	0.748381	0.374190	−	0.927352i	$-0.377921\pi$
$491$	16.5830	0.748381	0.374190	+	0.927352i	$0.377921\pi$
$492$	0	0
$493$	−0.228757	−0.0103027
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.8745	1.25034
$498$	0	0
$499$	43.8118	1.96128	0.980642	−	0.195810i	$-0.0627335\pi$
$499$	43.8118	1.96128	0.980642	+	0.195810i	$0.0627335\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−33.4575	−1.49180	−0.745898	−	0.666060i	$-0.767980\pi$
$503$	−33.4575	−1.49180	−0.745898	+	0.666060i	$0.767980\pi$
$504$	0	0
$505$	0.708497	0.0315277
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.7490	1.40725	0.703625	−	0.710571i	$-0.251563\pi$
$509$	31.7490	1.40725	0.703625	+	0.710571i	$0.251563\pi$
$510$	0	0
$511$	−54.4575	−2.40906
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−11.2915	−0.497563
$516$	0	0
$517$	−19.2915	−0.848439
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−20.2288	−0.886238	−0.443119	−	0.896463i	$-0.646128\pi$
$521$	−20.2288	−0.886238	−0.443119	+	0.896463i	$0.646128\pi$
$522$	0	0
$523$	−38.6863	−1.69163	−0.845816	−	0.533474i	$-0.820886\pi$
$523$	−38.6863	−1.69163	−0.845816	+	0.533474i	$0.820886\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.70850	0.0744233
$528$	0	0
$529$	16.5830	0.721000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	9.29150	0.402460
$534$	0	0
$535$	6.00000	0.259403
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22.9373	0.987977
$540$	0	0
$541$	−9.41699	−0.404868	−0.202434	−	0.979296i	$-0.564885\pi$
$541$	−9.41699	−0.404868	−0.202434	+	0.979296i	$0.564885\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.29150	0.355169
$546$	0	0
$547$	−24.9373	−1.06624	−0.533120	−	0.846040i	$-0.678981\pi$
$547$	−24.9373	−1.06624	−0.533120	+	0.846040i	$0.678981\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0.937254	0.0399284
$552$	0	0
$553$	46.1033	1.96051
$554$	0	0
$555$	0	0
$556$	0	0
$557$	22.7085	0.962190	0.481095	−	0.876668i	$-0.340239\pi$
$557$	22.7085	0.962190	0.481095	+	0.876668i	$0.340239\pi$
$558$	0	0
$559$	−14.7085	−0.622103
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	0	0
$565$	−6.58301	−0.276949
$566$	0	0
$567$	0	0
$568$	0	0
$569$	36.1033	1.51353	0.756764	−	0.653688i	$-0.226779\pi$
$569$	36.1033	1.51353	0.756764	+	0.653688i	$0.226779\pi$
$570$	0	0
$571$	20.0627	0.839600	0.419800	−	0.907617i	$-0.362100\pi$
$571$	20.0627	0.839600	0.419800	+	0.907617i	$0.362100\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.29150	0.262374
$576$	0	0
$577$	31.3948	1.30698	0.653491	−	0.756935i	$-0.273304\pi$
$577$	31.3948	1.30698	0.653491	+	0.756935i	$0.273304\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−18.2288	−0.756256
$582$	0	0
$583$	−43.5203	−1.80242
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23.5830	−0.973375	−0.486687	−	0.873576i	$-0.661795\pi$
$587$	−23.5830	−0.973375	−0.486687	+	0.873576i	$0.661795\pi$
$588$	0	0
$589$	−7.00000	−0.288430
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−42.3948	−1.74094	−0.870472	−	0.492218i	$-0.836186\pi$
$593$	−42.3948	−1.74094	−0.870472	+	0.492218i	$0.836186\pi$
$594$	0	0
$595$	2.35425	0.0965148
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−26.8118	−1.09550	−0.547749	−	0.836642i	$-0.684515\pi$
$599$	−26.8118	−1.09550	−0.547749	+	0.836642i	$0.684515\pi$
$600$	0	0
$601$	−32.2915	−1.31720	−0.658599	−	0.752494i	$-0.728851\pi$
$601$	−32.2915	−1.31720	−0.658599	+	0.752494i	$0.728851\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.29150	−0.0931628
$606$	0	0
$607$	−16.2288	−0.658705	−0.329352	−	0.944207i	$-0.606830\pi$
$607$	−16.2288	−0.658705	−0.329352	+	0.944207i	$0.606830\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.70850	0.352308
$612$	0	0
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.81176	−0.233973	−0.116986	−	0.993134i	$-0.537323\pi$
$617$	−5.81176	−0.233973	−0.116986	+	0.993134i	$0.537323\pi$
$618$	0	0
$619$	−13.8745	−0.557664	−0.278832	−	0.960340i	$-0.589947\pi$
$619$	−13.8745	−0.557664	−0.278832	+	0.960340i	$0.589947\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	49.7490	1.99315
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.41699	0.136245
$630$	0	0
$631$	20.5203	0.816899	0.408449	−	0.912781i	$-0.366070\pi$
$631$	20.5203	0.816899	0.408449	+	0.912781i	$0.366070\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.583005	−0.0231359
$636$	0	0
$637$	−10.3542	−0.410250
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.16601	−0.0460546	−0.0230273	−	0.999735i	$-0.507330\pi$
$641$	−1.16601	−0.0460546	−0.0230273	+	0.999735i	$0.507330\pi$
$642$	0	0
$643$	−2.22876	−0.0878936	−0.0439468	−	0.999034i	$-0.513993\pi$
$643$	−2.22876	−0.0878936	−0.0439468	+	0.999034i	$0.513993\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.00000	0.353827	0.176913	−	0.984226i	$-0.443389\pi$
$647$	9.00000	0.353827	0.176913	+	0.984226i	$0.443389\pi$
$648$	0	0
$649$	−6.00000	−0.235521
$650$	0	0
$651$	0	0
$652$	0	0
$653$	19.9373	0.780205	0.390103	−	0.920771i	$-0.372440\pi$
$653$	19.9373	0.780205	0.390103	+	0.920771i	$0.372440\pi$
$654$	0	0
$655$	11.8745	0.463975
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−38.3542	−1.49407	−0.747035	−	0.664785i	$-0.768523\pi$
$659$	−38.3542	−1.49407	−0.747035	+	0.664785i	$0.768523\pi$
$660$	0	0
$661$	−45.1660	−1.75675	−0.878377	−	0.477968i	$-0.841373\pi$
$661$	−45.1660	−1.75675	−0.878377	+	0.477968i	$0.841373\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−9.64575	−0.374046
$666$	0	0
$667$	−2.22876	−0.0862978
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−49.5203	−1.91171
$672$	0	0
$673$	19.7712	0.762125	0.381063	−	0.924549i	$-0.375558\pi$
$673$	19.7712	0.762125	0.381063	+	0.924549i	$0.375558\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	26.5830	1.02167	0.510834	−	0.859679i	$-0.329337\pi$
$677$	26.5830	1.02167	0.510834	+	0.859679i	$0.329337\pi$
$678$	0	0
$679$	41.1660	1.57981
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−6.41699	−0.245539	−0.122770	−	0.992435i	$-0.539178\pi$
$683$	−6.41699	−0.245539	−0.122770	+	0.992435i	$0.539178\pi$
$684$	0	0
$685$	17.9373	0.685347
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.6458	0.748444
$690$	0	0
$691$	−12.7712	−0.485841	−0.242920	−	0.970046i	$-0.578105\pi$
$691$	−12.7712	−0.485841	−0.242920	+	0.970046i	$0.578105\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−1.41699	−0.0537497
$696$	0	0
$697$	−3.64575	−0.138093
$698$	0	0
$699$	0	0
$700$	0	0
$701$	39.8745	1.50604	0.753020	−	0.657998i	$-0.228596\pi$
$701$	39.8745	1.50604	0.753020	+	0.657998i	$0.228596\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.58301	0.0971439
$708$	0	0
$709$	−47.7490	−1.79325	−0.896626	−	0.442789i	$-0.853989\pi$
$709$	−47.7490	−1.79325	−0.896626	+	0.442789i	$0.853989\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.6458	0.623388
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.4170	1.17166	0.585828	−	0.810435i	$-0.300769\pi$
$719$	31.4170	1.17166	0.585828	+	0.810435i	$0.300769\pi$
$720$	0	0
$721$	−41.1660	−1.53310
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−0.354249	−0.0131565
$726$	0	0
$727$	13.8745	0.514577	0.257289	−	0.966335i	$-0.417171\pi$
$727$	13.8745	0.514577	0.257289	+	0.966335i	$0.417171\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	5.77124	0.213457
$732$	0	0
$733$	39.2915	1.45126	0.725632	−	0.688083i	$-0.241547\pi$
$733$	39.2915	1.45126	0.725632	+	0.688083i	$0.241547\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.8745	0.805758
$738$	0	0
$739$	31.6863	1.16560	0.582799	−	0.812616i	$-0.301957\pi$
$739$	31.6863	1.16560	0.582799	+	0.812616i	$0.301957\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−13.1660	−0.483014	−0.241507	−	0.970399i	$-0.577642\pi$
$743$	−13.1660	−0.483014	−0.241507	+	0.970399i	$0.577642\pi$
$744$	0	0
$745$	3.06275	0.112210
$746$	0	0
$747$	0	0
$748$	0	0
$749$	21.8745	0.799277
$750$	0	0
$751$	17.9373	0.654540	0.327270	−	0.944931i	$-0.393871\pi$
$751$	17.9373	0.654540	0.327270	+	0.944931i	$0.393871\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.70850	0.244147
$756$	0	0
$757$	−7.41699	−0.269575	−0.134788	−	0.990875i	$-0.543035\pi$
$757$	−7.41699	−0.269575	−0.134788	+	0.990875i	$0.543035\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.3542	1.17284	0.586420	−	0.810007i	$-0.300537\pi$
$761$	32.3542	1.17284	0.586420	+	0.810007i	$0.300537\pi$
$762$	0	0
$763$	30.2288	1.09435
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.70850	0.0977982
$768$	0	0
$769$	16.1660	0.582961	0.291481	−	0.956577i	$-0.405852\pi$
$769$	16.1660	0.582961	0.291481	+	0.956577i	$0.405852\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	22.7712	0.819025	0.409512	−	0.912305i	$-0.365699\pi$
$773$	22.7712	0.819025	0.409512	+	0.912305i	$0.365699\pi$
$774$	0	0
$775$	2.64575	0.0950382
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.9373	0.535183
$780$	0	0
$781$	−27.8745	−0.997428
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−2.93725	−0.104835
$786$	0	0
$787$	−22.8118	−0.813151	−0.406576	−	0.913617i	$-0.633277\pi$
$787$	−22.8118	−0.813151	−0.406576	+	0.913617i	$0.633277\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	22.3542	0.793823
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−23.8118	−0.843456	−0.421728	−	0.906722i	$-0.638576\pi$
$797$	−23.8118	−0.843456	−0.421728	+	0.906722i	$0.638576\pi$
$798$	0	0
$799$	−3.41699	−0.120885
$800$	0	0
$801$	0	0
$802$	0	0
$803$	54.4575	1.92176
$804$	0	0
$805$	22.9373	0.808432
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.93725	0.173585	0.0867923	−	0.996226i	$-0.472338\pi$
$809$	4.93725	0.173585	0.0867923	+	0.996226i	$0.472338\pi$
$810$	0	0
$811$	13.2915	0.466728	0.233364	−	0.972389i	$-0.425027\pi$
$811$	13.2915	0.466728	0.233364	+	0.972389i	$0.425027\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−9.29150	−0.325467
$816$	0	0
$817$	−23.6458	−0.827260
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.41699	0.119254	0.0596270	−	0.998221i	$-0.481009\pi$
$821$	3.41699	0.119254	0.0596270	+	0.998221i	$0.481009\pi$
$822$	0	0
$823$	−6.10326	−0.212746	−0.106373	−	0.994326i	$-0.533924\pi$
$823$	−6.10326	−0.212746	−0.106373	+	0.994326i	$0.533924\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	28.0405	0.975064	0.487532	−	0.873105i	$-0.337897\pi$
$827$	28.0405	0.975064	0.487532	+	0.873105i	$0.337897\pi$
$828$	0	0
$829$	36.4575	1.26622	0.633111	−	0.774061i	$-0.281778\pi$
$829$	36.4575	1.26622	0.633111	+	0.774061i	$0.281778\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	4.06275	0.140766
$834$	0	0
$835$	−1.70850	−0.0591250
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.45751	0.153890	0.0769452	−	0.997035i	$-0.475483\pi$
$839$	4.45751	0.153890	0.0769452	+	0.997035i	$0.475483\pi$
$840$	0	0
$841$	−28.8745	−0.995673
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.2915	0.354038
$846$	0	0
$847$	−8.35425	−0.287055
$848$	0	0
$849$	0	0
$850$	0	0
$851$	33.2915	1.14122
$852$	0	0
$853$	7.29150	0.249656	0.124828	−	0.992178i	$-0.460162\pi$
$853$	7.29150	0.249656	0.124828	+	0.992178i	$0.460162\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−11.8118	−0.403482	−0.201741	−	0.979439i	$-0.564660\pi$
$857$	−11.8118	−0.403482	−0.201741	+	0.979439i	$0.564660\pi$
$858$	0	0
$859$	51.9373	1.77208	0.886038	−	0.463612i	$-0.153447\pi$
$859$	51.9373	1.77208	0.886038	+	0.463612i	$0.153447\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−18.4170	−0.626922	−0.313461	−	0.949601i	$-0.601488\pi$
$863$	−18.4170	−0.626922	−0.313461	+	0.949601i	$0.601488\pi$
$864$	0	0
$865$	16.6458	0.565973
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−46.1033	−1.56395
$870$	0	0
$871$	−9.87451	−0.334585
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.64575	0.123249
$876$	0	0
$877$	−27.6458	−0.933531	−0.466765	−	0.884381i	$-0.654581\pi$
$877$	−27.6458	−0.933531	−0.466765	+	0.884381i	$0.654581\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	41.9778	1.41427	0.707134	−	0.707080i	$-0.249988\pi$
$881$	41.9778	1.41427	0.707134	+	0.707080i	$0.249988\pi$
$882$	0	0
$883$	−21.0627	−0.708818	−0.354409	−	0.935091i	$-0.615318\pi$
$883$	−21.0627	−0.708818	−0.354409	+	0.935091i	$0.615318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	38.7490	1.30106	0.650532	−	0.759479i	$-0.274546\pi$
$887$	38.7490	1.30106	0.650532	+	0.759479i	$0.274546\pi$
$888$	0	0
$889$	−2.12549	−0.0712867
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.0000	0.468492
$894$	0	0
$895$	3.29150	0.110023
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.937254	−0.0312592
$900$	0	0
$901$	−7.70850	−0.256807
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.70850	0.189757
$906$	0	0
$907$	−6.12549	−0.203394	−0.101697	−	0.994815i	$-0.532427\pi$
$907$	−6.12549	−0.203394	−0.101697	+	0.994815i	$0.532427\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−34.1033	−1.12989	−0.564946	−	0.825128i	$-0.691103\pi$
$911$	−34.1033	−1.12989	−0.564946	+	0.825128i	$0.691103\pi$
$912$	0	0
$913$	18.2288	0.603284
$914$	0	0
$915$	0	0
$916$	0	0
$917$	43.2915	1.42961
$918$	0	0
$919$	1.41699	0.0467423	0.0233712	−	0.999727i	$-0.492560\pi$
$919$	1.41699	0.0467423	0.0233712	+	0.999727i	$0.492560\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.5830	0.414175
$924$	0	0
$925$	5.29150	0.173984
$926$	0	0
$927$	0	0
$928$	0	0
$929$	0.479741	0.0157398	0.00786989	−	0.999969i	$-0.497495\pi$
$929$	0.479741	0.0157398	0.00786989	+	0.999969i	$0.497495\pi$
$930$	0	0
$931$	−16.6458	−0.545542
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.35425	−0.0769922
$936$	0	0
$937$	−38.5830	−1.26045	−0.630226	−	0.776412i	$-0.717038\pi$
$937$	−38.5830	−1.26045	−0.630226	+	0.776412i	$0.717038\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	20.5830	0.670987	0.335493	−	0.942043i	$-0.391097\pi$
$941$	20.5830	0.670987	0.335493	+	0.942043i	$0.391097\pi$
$942$	0	0
$943$	−35.5203	−1.15670
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30.7490	−0.999209	−0.499604	−	0.866254i	$-0.666521\pi$
$947$	−30.7490	−0.999209	−0.499604	+	0.866254i	$0.666521\pi$
$948$	0	0
$949$	−24.5830	−0.797998
$950$	0	0
$951$	0	0
$952$	0	0
$953$	4.58301	0.148458	0.0742291	−	0.997241i	$-0.476350\pi$
$953$	4.58301	0.148458	0.0742291	+	0.997241i	$0.476350\pi$
$954$	0	0
$955$	13.2915	0.430103
$956$	0	0
$957$	0	0
$958$	0	0
$959$	65.3948	2.11171
$960$	0	0
$961$	−24.0000	−0.774194
$962$	0	0
$963$	0	0
$964$	0	0
$965$	15.5203	0.499615
$966$	0	0
$967$	55.8745	1.79680	0.898402	−	0.439175i	$-0.144729\pi$
$967$	55.8745	1.79680	0.898402	+	0.439175i	$0.144729\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−22.4575	−0.720696	−0.360348	−	0.932818i	$-0.617342\pi$
$971$	−22.4575	−0.720696	−0.360348	+	0.932818i	$0.617342\pi$
$972$	0	0
$973$	−5.16601	−0.165615
$974$	0	0
$975$	0	0
$976$	0	0
$977$	58.4575	1.87022	0.935111	−	0.354356i	$-0.115300\pi$
$977$	58.4575	1.87022	0.935111	+	0.354356i	$0.115300\pi$
$978$	0	0
$979$	−49.7490	−1.58999
$980$	0	0
$981$	0	0
$982$	0	0
$983$	60.1660	1.91900	0.959499	−	0.281713i	$-0.0909024\pi$
$983$	60.1660	1.91900	0.959499	+	0.281713i	$0.0909024\pi$
$984$	0	0
$985$	−25.8118	−0.822431
$986$	0	0
$987$	0	0
$988$	0	0
$989$	56.2288	1.78797
$990$	0	0
$991$	−37.8118	−1.20113	−0.600565	−	0.799576i	$-0.705058\pi$
$991$	−37.8118	−1.20113	−0.600565	+	0.799576i	$0.705058\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	25.8745	0.820277
$996$	0	0
$997$	−10.9373	−0.346386	−0.173193	−	0.984888i	$-0.555408\pi$
$997$	−10.9373	−0.346386	−0.173193	+	0.984888i	$0.555408\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4320.2.a.p.1.1	✓	2
3.2	odd	2		4320.2.a.z.1.1	yes	2
4.3	odd	2		4320.2.a.t.1.2	yes	2
8.3	odd	2		8640.2.a.dg.1.2		2
8.5	even	2		8640.2.a.cz.1.1		2
12.11	even	2		4320.2.a.bd.1.2	yes	2
24.5	odd	2		8640.2.a.cl.1.1		2
24.11	even	2		8640.2.a.cs.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4320.2.a.p.1.1	✓	2	1.1	even	1	trivial
4320.2.a.t.1.2	yes	2	4.3	odd	2
4320.2.a.z.1.1	yes	2	3.2	odd	2
4320.2.a.bd.1.2	yes	2	12.11	even	2
8640.2.a.cl.1.1		2	24.5	odd	2
8640.2.a.cs.1.2		2	24.11	even	2
8640.2.a.cz.1.1		2	8.5	even	2
8640.2.a.dg.1.2		2	8.3	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:	\( N \)	\(=\)	\( 4320 = 2^{5} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4320.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.64575 q^{7} +3.64575 q^{11} -1.64575 q^{13} +0.645751 q^{17} -2.64575 q^{19} +6.29150 q^{23} +1.00000 q^{25} -0.354249 q^{29} +2.64575 q^{31} +3.64575 q^{35} +5.29150 q^{37} -5.64575 q^{41} +8.93725 q^{43} -5.29150 q^{47} +6.29150 q^{49} -11.9373 q^{53} -3.64575 q^{55} -1.64575 q^{59} -13.5830 q^{61} +1.64575 q^{65} +6.00000 q^{67} -7.64575 q^{71} +14.9373 q^{73} -13.2915 q^{77} -12.6458 q^{79} +5.00000 q^{83} -0.645751 q^{85} -13.6458 q^{89} +6.00000 q^{91} +2.64575 q^{95} -11.2915 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 2 q^{7} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 2 q^{23} + 2 q^{25} - 6 q^{29} + 2 q^{35} - 6 q^{41} + 2 q^{43} + 2 q^{49} - 8 q^{53} - 2 q^{55} + 2 q^{59} - 6 q^{61} - 2 q^{65} + 12 q^{67} - 10 q^{71}+ \cdots - 12 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 2 * q^7 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 2 * q^23 + 2 * q^25 - 6 * q^29 + 2 * q^35 - 6 * q^41 + 2 * q^43 + 2 * q^49 - 8 * q^53 - 2 * q^55 + 2 * q^59 - 6 * q^61 - 2 * q^65 + 12 * q^67 - 10 * q^71 + 14 * q^73 - 16 * q^77 - 20 * q^79 + 10 * q^83 + 4 * q^85 - 22 * q^89 + 12 * q^91 - 12 * q^97

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