LMFDB - Embedded newform 5760.2.a.ci.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5760,2,Mod(1,5760)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5760, 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("5760.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5760 = 2^{7} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5760.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:	$45.9938315643$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 640)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	5760.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -1.23607 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -1.23607 q^{7} -2.00000 q^{11} +4.47214 q^{13} +4.47214 q^{17} -4.47214 q^{19} -9.23607 q^{23} +1.00000 q^{25} +2.00000 q^{29} +2.47214 q^{31} -1.23607 q^{35} -10.9443 q^{37} -3.52786 q^{41} -5.70820 q^{43} -2.76393 q^{47} -5.47214 q^{49} +8.47214 q^{53} -2.00000 q^{55} +0.472136 q^{59} -6.00000 q^{61} +4.47214 q^{65} +5.70820 q^{67} +6.47214 q^{71} -4.47214 q^{73} +2.47214 q^{77} +4.94427 q^{79} +9.70820 q^{83} +4.47214 q^{85} -2.94427 q^{89} -5.52786 q^{91} -4.47214 q^{95} -7.52786 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 + 2 * q^7	$ 2 q + 2 q^{5} + 2 q^{7} - 4 q^{11} - 14 q^{23} + 2 q^{25} + 4 q^{29} - 4 q^{31} + 2 q^{35} - 4 q^{37} - 16 q^{41} + 2 q^{43} - 10 q^{47} - 2 q^{49} + 8 q^{53} - 4 q^{55} - 8 q^{59} - 12 q^{61} - 2 q^{67} + 4 q^{71} - 4 q^{77} - 8 q^{79} + 6 q^{83} + 12 q^{89} - 20 q^{91} - 24 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 - 4 * q^11 - 14 * q^23 + 2 * q^25 + 4 * q^29 - 4 * q^31 + 2 * q^35 - 4 * q^37 - 16 * q^41 + 2 * q^43 - 10 * q^47 - 2 * q^49 + 8 * q^53 - 4 * q^55 - 8 * q^59 - 12 * q^61 - 2 * q^67 + 4 * q^71 - 4 * q^77 - 8 * q^79 + 6 * q^83 + 12 * q^89 - 20 * q^91 - 24 * 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
$7$	−1.23607	−0.467190	−0.233595	−	0.972334i	$-0.575049\pi$
$7$	−1.23607	−0.467190	−0.233595	+	0.972334i	$0.575049\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	4.47214	1.24035	0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	0	0
$19$	−4.47214	−1.02598	−0.512989	−	0.858395i	$-0.671462\pi$
$19$	−4.47214	−1.02598	−0.512989	+	0.858395i	$0.671462\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.23607	−1.92585	−0.962927	−	0.269763i	$-0.913055\pi$
$23$	−9.23607	−1.92585	−0.962927	+	0.269763i	$0.913055\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.23607	−0.208934
$36$	0	0
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.52786	−0.550960	−0.275480	−	0.961307i	$-0.588837\pi$
$41$	−3.52786	−0.550960	−0.275480	+	0.961307i	$0.588837\pi$
$42$	0	0
$43$	−5.70820	−0.870493	−0.435246	−	0.900311i	$-0.643339\pi$
$43$	−5.70820	−0.870493	−0.435246	+	0.900311i	$0.643339\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.76393	−0.403161	−0.201580	−	0.979472i	$-0.564608\pi$
$47$	−2.76393	−0.403161	−0.201580	+	0.979472i	$0.564608\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.47214	1.16374	0.581869	−	0.813283i	$-0.302322\pi$
$53$	8.47214	1.16374	0.581869	+	0.813283i	$0.302322\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.472136	0.0614669	0.0307334	−	0.999528i	$-0.490216\pi$
$59$	0.472136	0.0614669	0.0307334	+	0.999528i	$0.490216\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.47214	0.554700
$66$	0	0
$67$	5.70820	0.697368	0.348684	−	0.937240i	$-0.386629\pi$
$67$	5.70820	0.697368	0.348684	+	0.937240i	$0.386629\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.47214	0.768101	0.384051	−	0.923312i	$-0.374529\pi$
$71$	6.47214	0.768101	0.384051	+	0.923312i	$0.374529\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.47214	0.281726
$78$	0	0
$79$	4.94427	0.556274	0.278137	−	0.960541i	$-0.410283\pi$
$79$	4.94427	0.556274	0.278137	+	0.960541i	$0.410283\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	9.70820	1.06561	0.532807	−	0.846237i	$-0.321137\pi$
$83$	9.70820	1.06561	0.532807	+	0.846237i	$0.321137\pi$
$84$	0	0
$85$	4.47214	0.485071
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−2.94427	−0.312092	−0.156046	−	0.987750i	$-0.549875\pi$
$89$	−2.94427	−0.312092	−0.156046	+	0.987750i	$0.549875\pi$
$90$	0	0
$91$	−5.52786	−0.579478
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.47214	−0.458831
$96$	0	0
$97$	−7.52786	−0.764339	−0.382169	−	0.924092i	$-0.624823\pi$
$97$	−7.52786	−0.764339	−0.382169	+	0.924092i	$0.624823\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	1.23607	0.121793	0.0608967	−	0.998144i	$-0.480604\pi$
$103$	1.23607	0.121793	0.0608967	+	0.998144i	$0.480604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.7639	−1.23394	−0.616968	−	0.786988i	$-0.711639\pi$
$107$	−12.7639	−1.23394	−0.616968	+	0.786988i	$0.711639\pi$
$108$	0	0
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−14.9443	−1.40584	−0.702919	−	0.711269i	$-0.748121\pi$
$113$	−14.9443	−1.40584	−0.702919	+	0.711269i	$0.748121\pi$
$114$	0	0
$115$	−9.23607	−0.861268
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.52786	−0.506738
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−2.18034	−0.193474	−0.0967369	−	0.995310i	$-0.530841\pi$
$127$	−2.18034	−0.193474	−0.0967369	+	0.995310i	$0.530841\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2.94427	−0.257242	−0.128621	−	0.991694i	$-0.541055\pi$
$131$	−2.94427	−0.257242	−0.128621	+	0.991694i	$0.541055\pi$
$132$	0	0
$133$	5.52786	0.479327
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	−11.5279	−0.977781	−0.488890	−	0.872345i	$-0.662598\pi$
$139$	−11.5279	−0.977781	−0.488890	+	0.872345i	$0.662598\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−8.94427	−0.747958
$144$	0	0
$145$	2.00000	0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.9443	0.896590	0.448295	−	0.893886i	$-0.352031\pi$
$149$	10.9443	0.896590	0.448295	+	0.893886i	$0.352031\pi$
$150$	0	0
$151$	3.41641	0.278023	0.139012	−	0.990291i	$-0.455607\pi$
$151$	3.41641	0.278023	0.139012	+	0.990291i	$0.455607\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.47214	0.198567
$156$	0	0
$157$	22.9443	1.83115	0.915576	−	0.402145i	$-0.131735\pi$
$157$	22.9443	1.83115	0.915576	+	0.402145i	$0.131735\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.4164	0.899739
$162$	0	0
$163$	−1.70820	−0.133797	−0.0668984	−	0.997760i	$-0.521310\pi$
$163$	−1.70820	−0.133797	−0.0668984	+	0.997760i	$0.521310\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.6525	−1.28861	−0.644304	−	0.764770i	$-0.722853\pi$
$167$	−16.6525	−1.28861	−0.644304	+	0.764770i	$0.722853\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.94427	−0.527963	−0.263982	−	0.964528i	$-0.585036\pi$
$173$	−6.94427	−0.527963	−0.263982	+	0.964528i	$0.585036\pi$
$174$	0	0
$175$	−1.23607	−0.0934380
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−11.5279	−0.861633	−0.430817	−	0.902440i	$-0.641774\pi$
$179$	−11.5279	−0.861633	−0.430817	+	0.902440i	$0.641774\pi$
$180$	0	0
$181$	−6.94427	−0.516164	−0.258082	−	0.966123i	$-0.583090\pi$
$181$	−6.94427	−0.516164	−0.258082	+	0.966123i	$0.583090\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.9443	−0.804639
$186$	0	0
$187$	−8.94427	−0.654070
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.4164	−1.11549	−0.557746	−	0.830012i	$-0.688334\pi$
$191$	−15.4164	−1.11549	−0.557746	+	0.830012i	$0.688334\pi$
$192$	0	0
$193$	−20.4721	−1.47362	−0.736808	−	0.676102i	$-0.763668\pi$
$193$	−20.4721	−1.47362	−0.736808	+	0.676102i	$0.763668\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	8.47214	0.603615	0.301807	−	0.953369i	$-0.402410\pi$
$197$	8.47214	0.603615	0.301807	+	0.953369i	$0.402410\pi$
$198$	0	0
$199$	−16.9443	−1.20115	−0.600574	−	0.799569i	$-0.705061\pi$
$199$	−16.9443	−1.20115	−0.600574	+	0.799569i	$0.705061\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.47214	−0.173510
$204$	0	0
$205$	−3.52786	−0.246397
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.94427	0.618688
$210$	0	0
$211$	18.9443	1.30418	0.652089	−	0.758143i	$-0.273893\pi$
$211$	18.9443	1.30418	0.652089	+	0.758143i	$0.273893\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.70820	−0.389296
$216$	0	0
$217$	−3.05573	−0.207436
$218$	0	0
$219$	0	0
$220$	0	0
$221$	20.0000	1.34535
$222$	0	0
$223$	13.2361	0.886353	0.443176	−	0.896434i	$-0.353852\pi$
$223$	13.2361	0.886353	0.443176	+	0.896434i	$0.353852\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	12.1803	0.808438	0.404219	−	0.914662i	$-0.367543\pi$
$227$	12.1803	0.808438	0.404219	+	0.914662i	$0.367543\pi$
$228$	0	0
$229$	−14.9443	−0.987545	−0.493773	−	0.869591i	$-0.664382\pi$
$229$	−14.9443	−0.987545	−0.493773	+	0.869591i	$0.664382\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−24.4721	−1.60322	−0.801611	−	0.597845i	$-0.796024\pi$
$233$	−24.4721	−1.60322	−0.801611	+	0.597845i	$0.796024\pi$
$234$	0	0
$235$	−2.76393	−0.180299
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.94427	0.319818	0.159909	−	0.987132i	$-0.448880\pi$
$239$	4.94427	0.319818	0.159909	+	0.987132i	$0.448880\pi$
$240$	0	0
$241$	−18.3607	−1.18272	−0.591358	−	0.806409i	$-0.701408\pi$
$241$	−18.3607	−1.18272	−0.591358	+	0.806409i	$0.701408\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−5.47214	−0.349602
$246$	0	0
$247$	−20.0000	−1.27257
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2.00000	−0.126239	−0.0631194	−	0.998006i	$-0.520105\pi$
$251$	−2.00000	−0.126239	−0.0631194	+	0.998006i	$0.520105\pi$
$252$	0	0
$253$	18.4721	1.16133
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.9443	−0.932198	−0.466099	−	0.884733i	$-0.654341\pi$
$257$	−14.9443	−0.932198	−0.466099	+	0.884733i	$0.654341\pi$
$258$	0	0
$259$	13.5279	0.840581
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4.29180	−0.264643	−0.132322	−	0.991207i	$-0.542243\pi$
$263$	−4.29180	−0.264643	−0.132322	+	0.991207i	$0.542243\pi$
$264$	0	0
$265$	8.47214	0.520439
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−11.8885	−0.724857	−0.362429	−	0.932012i	$-0.618052\pi$
$269$	−11.8885	−0.724857	−0.362429	+	0.932012i	$0.618052\pi$
$270$	0	0
$271$	−20.3607	−1.23682	−0.618412	−	0.785854i	$-0.712224\pi$
$271$	−20.3607	−1.23682	−0.618412	+	0.785854i	$0.712224\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	6.94427	0.417241	0.208620	−	0.977997i	$-0.433103\pi$
$277$	6.94427	0.417241	0.208620	+	0.977997i	$0.433103\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−12.4721	−0.744025	−0.372013	−	0.928228i	$-0.621332\pi$
$281$	−12.4721	−0.744025	−0.372013	+	0.928228i	$0.621332\pi$
$282$	0	0
$283$	−21.7082	−1.29042	−0.645209	−	0.764006i	$-0.723230\pi$
$283$	−21.7082	−1.29042	−0.645209	+	0.764006i	$0.723230\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.36068	0.257403
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−11.8885	−0.694536	−0.347268	−	0.937766i	$-0.612891\pi$
$293$	−11.8885	−0.694536	−0.347268	+	0.937766i	$0.612891\pi$
$294$	0	0
$295$	0.472136	0.0274888
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−41.3050	−2.38873
$300$	0	0
$301$	7.05573	0.406685
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.00000	−0.343559
$306$	0	0
$307$	13.7082	0.782369	0.391184	−	0.920312i	$-0.372066\pi$
$307$	13.7082	0.782369	0.391184	+	0.920312i	$0.372066\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	11.4164	0.647365	0.323683	−	0.946166i	$-0.395079\pi$
$311$	11.4164	0.647365	0.323683	+	0.946166i	$0.395079\pi$
$312$	0	0
$313$	24.8328	1.40363	0.701817	−	0.712357i	$-0.252372\pi$
$313$	24.8328	1.40363	0.701817	+	0.712357i	$0.252372\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.5279	0.647469	0.323735	−	0.946148i	$-0.395061\pi$
$317$	11.5279	0.647469	0.323735	+	0.946148i	$0.395061\pi$
$318$	0	0
$319$	−4.00000	−0.223957
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.0000	−1.11283
$324$	0	0
$325$	4.47214	0.248069
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.41641	0.188353
$330$	0	0
$331$	19.8885	1.09317	0.546587	−	0.837403i	$-0.315927\pi$
$331$	19.8885	1.09317	0.546587	+	0.837403i	$0.315927\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	5.70820	0.311872
$336$	0	0
$337$	22.9443	1.24985	0.624927	−	0.780683i	$-0.285129\pi$
$337$	22.9443	1.24985	0.624927	+	0.780683i	$0.285129\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−4.94427	−0.267747
$342$	0	0
$343$	15.4164	0.832408
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.7639	−0.685204	−0.342602	−	0.939481i	$-0.611308\pi$
$347$	−12.7639	−0.685204	−0.342602	+	0.939481i	$0.611308\pi$
$348$	0	0
$349$	2.94427	0.157603	0.0788016	−	0.996890i	$-0.474891\pi$
$349$	2.94427	0.157603	0.0788016	+	0.996890i	$0.474891\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.9443	−1.22120	−0.610600	−	0.791939i	$-0.709072\pi$
$353$	−22.9443	−1.22120	−0.610600	+	0.791939i	$0.709072\pi$
$354$	0	0
$355$	6.47214	0.343505
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−32.9443	−1.73873	−0.869366	−	0.494169i	$-0.835473\pi$
$359$	−32.9443	−1.73873	−0.869366	+	0.494169i	$0.835473\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.47214	−0.234082
$366$	0	0
$367$	31.7082	1.65515	0.827577	−	0.561352i	$-0.189718\pi$
$367$	31.7082	1.65515	0.827577	+	0.561352i	$0.189718\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−10.4721	−0.543686
$372$	0	0
$373$	22.9443	1.18801	0.594005	−	0.804462i	$-0.297546\pi$
$373$	22.9443	1.18801	0.594005	+	0.804462i	$0.297546\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.94427	0.460653
$378$	0	0
$379$	−24.4721	−1.25705	−0.628525	−	0.777790i	$-0.716341\pi$
$379$	−24.4721	−1.25705	−0.628525	+	0.777790i	$0.716341\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.7082	1.21143	0.605716	−	0.795681i	$-0.292887\pi$
$383$	23.7082	1.21143	0.605716	+	0.795681i	$0.292887\pi$
$384$	0	0
$385$	2.47214	0.125992
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.8885	1.21120	0.605599	−	0.795770i	$-0.292934\pi$
$389$	23.8885	1.21120	0.605599	+	0.795770i	$0.292934\pi$
$390$	0	0
$391$	−41.3050	−2.08888
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.94427	0.248773
$396$	0	0
$397$	−3.52786	−0.177058	−0.0885292	−	0.996074i	$-0.528217\pi$
$397$	−3.52786	−0.177058	−0.0885292	+	0.996074i	$0.528217\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	0	0
$403$	11.0557	0.550725
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.8885	1.08497
$408$	0	0
$409$	−9.41641	−0.465611	−0.232806	−	0.972523i	$-0.574791\pi$
$409$	−9.41641	−0.465611	−0.232806	+	0.972523i	$0.574791\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−0.583592	−0.0287167
$414$	0	0
$415$	9.70820	0.476557
$416$	0	0
$417$	0	0
$418$	0	0
$419$	4.47214	0.218478	0.109239	−	0.994016i	$-0.465159\pi$
$419$	4.47214	0.218478	0.109239	+	0.994016i	$0.465159\pi$
$420$	0	0
$421$	24.8328	1.21028	0.605139	−	0.796120i	$-0.293118\pi$
$421$	24.8328	1.21028	0.605139	+	0.796120i	$0.293118\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.47214	0.216930
$426$	0	0
$427$	7.41641	0.358905
$428$	0	0
$429$	0	0
$430$	0	0
$431$	20.3607	0.980739	0.490370	−	0.871515i	$-0.336862\pi$
$431$	20.3607	0.980739	0.490370	+	0.871515i	$0.336862\pi$
$432$	0	0
$433$	8.47214	0.407145	0.203572	−	0.979060i	$-0.434745\pi$
$433$	8.47214	0.407145	0.203572	+	0.979060i	$0.434745\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	41.3050	1.97588
$438$	0	0
$439$	−32.9443	−1.57234	−0.786172	−	0.618008i	$-0.787940\pi$
$439$	−32.9443	−1.57234	−0.786172	+	0.618008i	$0.787940\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	26.6525	1.26630	0.633149	−	0.774030i	$-0.281762\pi$
$443$	26.6525	1.26630	0.633149	+	0.774030i	$0.281762\pi$
$444$	0	0
$445$	−2.94427	−0.139572
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−25.4164	−1.19947	−0.599737	−	0.800197i	$-0.704728\pi$
$449$	−25.4164	−1.19947	−0.599737	+	0.800197i	$0.704728\pi$
$450$	0	0
$451$	7.05573	0.332241
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.52786	−0.259150
$456$	0	0
$457$	−31.8885	−1.49168	−0.745842	−	0.666123i	$-0.767952\pi$
$457$	−31.8885	−1.49168	−0.745842	+	0.666123i	$0.767952\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	29.7771	1.38686	0.693429	−	0.720525i	$-0.256099\pi$
$461$	29.7771	1.38686	0.693429	+	0.720525i	$0.256099\pi$
$462$	0	0
$463$	39.1246	1.81827	0.909137	−	0.416496i	$-0.136742\pi$
$463$	39.1246	1.81827	0.909137	+	0.416496i	$0.136742\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7.59675	−0.351536	−0.175768	−	0.984432i	$-0.556241\pi$
$467$	−7.59675	−0.351536	−0.175768	+	0.984432i	$0.556241\pi$
$468$	0	0
$469$	−7.05573	−0.325803
$470$	0	0
$471$	0	0
$472$	0	0
$473$	11.4164	0.524927
$474$	0	0
$475$	−4.47214	−0.205196
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−4.94427	−0.225910	−0.112955	−	0.993600i	$-0.536032\pi$
$479$	−4.94427	−0.225910	−0.112955	+	0.993600i	$0.536032\pi$
$480$	0	0
$481$	−48.9443	−2.23167
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.52786	−0.341823
$486$	0	0
$487$	−17.2361	−0.781041	−0.390520	−	0.920594i	$-0.627705\pi$
$487$	−17.2361	−0.781041	−0.390520	+	0.920594i	$0.627705\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−38.9443	−1.75753	−0.878765	−	0.477254i	$-0.841632\pi$
$491$	−38.9443	−1.75753	−0.878765	+	0.477254i	$0.841632\pi$
$492$	0	0
$493$	8.94427	0.402830
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.00000	−0.358849
$498$	0	0
$499$	−41.4164	−1.85405	−0.927027	−	0.374996i	$-0.877644\pi$
$499$	−41.4164	−1.85405	−0.927027	+	0.374996i	$0.877644\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.1246	−1.56613	−0.783065	−	0.621940i	$-0.786345\pi$
$503$	−35.1246	−1.56613	−0.783065	+	0.621940i	$0.786345\pi$
$504$	0	0
$505$	10.0000	0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−36.8328	−1.63259	−0.816293	−	0.577638i	$-0.803974\pi$
$509$	−36.8328	−1.63259	−0.816293	+	0.577638i	$0.803974\pi$
$510$	0	0
$511$	5.52786	0.244538
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.23607	0.0544677
$516$	0	0
$517$	5.52786	0.243115
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19.8885	−0.871333	−0.435666	−	0.900108i	$-0.643487\pi$
$521$	−19.8885	−0.871333	−0.435666	+	0.900108i	$0.643487\pi$
$522$	0	0
$523$	38.0689	1.66464	0.832318	−	0.554298i	$-0.187013\pi$
$523$	38.0689	1.66464	0.832318	+	0.554298i	$0.187013\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	11.0557	0.481595
$528$	0	0
$529$	62.3050	2.70891
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−15.7771	−0.683382
$534$	0	0
$535$	−12.7639	−0.551833
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.9443	0.471403
$540$	0	0
$541$	−11.8885	−0.511128	−0.255564	−	0.966792i	$-0.582261\pi$
$541$	−11.8885	−0.511128	−0.255564	+	0.966792i	$0.582261\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.94427	−0.126119
$546$	0	0
$547$	10.6525	0.455467	0.227733	−	0.973724i	$-0.426869\pi$
$547$	10.6525	0.455467	0.227733	+	0.973724i	$0.426869\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8.94427	−0.381039
$552$	0	0
$553$	−6.11146	−0.259886
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−30.9443	−1.31115	−0.655575	−	0.755130i	$-0.727574\pi$
$557$	−30.9443	−1.31115	−0.655575	+	0.755130i	$0.727574\pi$
$558$	0	0
$559$	−25.5279	−1.07971
$560$	0	0
$561$	0	0
$562$	0	0
$563$	17.7082	0.746312	0.373156	−	0.927769i	$-0.378276\pi$
$563$	17.7082	0.746312	0.373156	+	0.927769i	$0.378276\pi$
$564$	0	0
$565$	−14.9443	−0.628710
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−10.3607	−0.434342	−0.217171	−	0.976134i	$-0.569683\pi$
$569$	−10.3607	−0.434342	−0.217171	+	0.976134i	$0.569683\pi$
$570$	0	0
$571$	−36.8328	−1.54141	−0.770703	−	0.637195i	$-0.780095\pi$
$571$	−36.8328	−1.54141	−0.770703	+	0.637195i	$0.780095\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−9.23607	−0.385171
$576$	0	0
$577$	−15.8885	−0.661449	−0.330724	−	0.943727i	$-0.607293\pi$
$577$	−15.8885	−0.661449	−0.330724	+	0.943727i	$0.607293\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12.0000	−0.497844
$582$	0	0
$583$	−16.9443	−0.701760
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−22.6525	−0.934968	−0.467484	−	0.884002i	$-0.654839\pi$
$587$	−22.6525	−0.934968	−0.467484	+	0.884002i	$0.654839\pi$
$588$	0	0
$589$	−11.0557	−0.455543
$590$	0	0
$591$	0	0
$592$	0	0
$593$	14.0000	0.574911	0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000	0.574911	0.287456	+	0.957794i	$0.407191\pi$
$594$	0	0
$595$	−5.52786	−0.226620
$596$	0	0
$597$	0	0
$598$	0	0
$599$	13.8885	0.567471	0.283735	−	0.958903i	$-0.408426\pi$
$599$	13.8885	0.567471	0.283735	+	0.958903i	$0.408426\pi$
$600$	0	0
$601$	22.3607	0.912111	0.456056	−	0.889951i	$-0.349262\pi$
$601$	22.3607	0.912111	0.456056	+	0.889951i	$0.349262\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−7.00000	−0.284590
$606$	0	0
$607$	−47.1246	−1.91273	−0.956364	−	0.292176i	$-0.905621\pi$
$607$	−47.1246	−1.91273	−0.956364	+	0.292176i	$0.905621\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−12.3607	−0.500060
$612$	0	0
$613$	28.4721	1.14998	0.574989	−	0.818161i	$-0.305006\pi$
$613$	28.4721	1.14998	0.574989	+	0.818161i	$0.305006\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.41641	0.0570224	0.0285112	−	0.999593i	$-0.490923\pi$
$617$	1.41641	0.0570224	0.0285112	+	0.999593i	$0.490923\pi$
$618$	0	0
$619$	−11.5279	−0.463344	−0.231672	−	0.972794i	$-0.574420\pi$
$619$	−11.5279	−0.463344	−0.231672	+	0.972794i	$0.574420\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.63932	0.145806
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−48.9443	−1.95154
$630$	0	0
$631$	25.5279	1.01625	0.508124	−	0.861284i	$-0.330339\pi$
$631$	25.5279	1.01625	0.508124	+	0.861284i	$0.330339\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−2.18034	−0.0865241
$636$	0	0
$637$	−24.4721	−0.969621
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.47214	0.334629	0.167315	−	0.985904i	$-0.446490\pi$
$641$	8.47214	0.334629	0.167315	+	0.985904i	$0.446490\pi$
$642$	0	0
$643$	0.180340	0.00711191	0.00355596	−	0.999994i	$-0.498868\pi$
$643$	0.180340	0.00711191	0.00355596	+	0.999994i	$0.498868\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.2361	0.677620	0.338810	−	0.940855i	$-0.389976\pi$
$647$	17.2361	0.677620	0.338810	+	0.940855i	$0.389976\pi$
$648$	0	0
$649$	−0.944272	−0.0370659
$650$	0	0
$651$	0	0
$652$	0	0
$653$	22.5836	0.883764	0.441882	−	0.897073i	$-0.354311\pi$
$653$	22.5836	0.883764	0.441882	+	0.897073i	$0.354311\pi$
$654$	0	0
$655$	−2.94427	−0.115042
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.41641	−0.210993	−0.105497	−	0.994420i	$-0.533643\pi$
$659$	−5.41641	−0.210993	−0.105497	+	0.994420i	$0.533643\pi$
$660$	0	0
$661$	0.111456	0.00433514	0.00216757	−	0.999998i	$-0.499310\pi$
$661$	0.111456	0.00433514	0.00216757	+	0.999998i	$0.499310\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	5.52786	0.214361
$666$	0	0
$667$	−18.4721	−0.715244
$668$	0	0
$669$	0	0
$670$	0	0
$671$	12.0000	0.463255
$672$	0	0
$673$	39.3050	1.51509	0.757547	−	0.652780i	$-0.226398\pi$
$673$	39.3050	1.51509	0.757547	+	0.652780i	$0.226398\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−14.3607	−0.551926	−0.275963	−	0.961168i	$-0.588997\pi$
$677$	−14.3607	−0.551926	−0.275963	+	0.961168i	$0.588997\pi$
$678$	0	0
$679$	9.30495	0.357091
$680$	0	0
$681$	0	0
$682$	0	0
$683$	37.7082	1.44286	0.721432	−	0.692485i	$-0.243484\pi$
$683$	37.7082	1.44286	0.721432	+	0.692485i	$0.243484\pi$
$684$	0	0
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	37.8885	1.44344
$690$	0	0
$691$	30.0000	1.14125	0.570627	−	0.821209i	$-0.306700\pi$
$691$	30.0000	1.14125	0.570627	+	0.821209i	$0.306700\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.5279	−0.437277
$696$	0	0
$697$	−15.7771	−0.597600
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.9443	1.01767	0.508836	−	0.860864i	$-0.330076\pi$
$701$	26.9443	1.01767	0.508836	+	0.860864i	$0.330076\pi$
$702$	0	0
$703$	48.9443	1.84597
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.3607	−0.464871
$708$	0	0
$709$	−18.0000	−0.676004	−0.338002	−	0.941145i	$-0.609751\pi$
$709$	−18.0000	−0.676004	−0.338002	+	0.941145i	$0.609751\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−22.8328	−0.855096
$714$	0	0
$715$	−8.94427	−0.334497
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−30.8328	−1.14987	−0.574935	−	0.818199i	$-0.694973\pi$
$719$	−30.8328	−1.14987	−0.574935	+	0.818199i	$0.694973\pi$
$720$	0	0
$721$	−1.52786	−0.0569006
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	11.7082	0.434233	0.217117	−	0.976146i	$-0.430335\pi$
$727$	11.7082	0.434233	0.217117	+	0.976146i	$0.430335\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−25.5279	−0.944182
$732$	0	0
$733$	16.1115	0.595090	0.297545	−	0.954708i	$-0.403832\pi$
$733$	16.1115	0.595090	0.297545	+	0.954708i	$0.403832\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−11.4164	−0.420529
$738$	0	0
$739$	49.1935	1.80961	0.904806	−	0.425824i	$-0.140016\pi$
$739$	49.1935	1.80961	0.904806	+	0.425824i	$0.140016\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.652476	0.0239370	0.0119685	−	0.999928i	$-0.496190\pi$
$743$	0.652476	0.0239370	0.0119685	+	0.999928i	$0.496190\pi$
$744$	0	0
$745$	10.9443	0.400967
$746$	0	0
$747$	0	0
$748$	0	0
$749$	15.7771	0.576482
$750$	0	0
$751$	28.3607	1.03490	0.517448	−	0.855715i	$-0.326882\pi$
$751$	28.3607	1.03490	0.517448	+	0.855715i	$0.326882\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.41641	0.124336
$756$	0	0
$757$	−15.8885	−0.577479	−0.288739	−	0.957408i	$-0.593236\pi$
$757$	−15.8885	−0.577479	−0.288739	+	0.957408i	$0.593236\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	31.8885	1.15596	0.577979	−	0.816051i	$-0.303841\pi$
$761$	31.8885	1.15596	0.577979	+	0.816051i	$0.303841\pi$
$762$	0	0
$763$	3.63932	0.131752
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.11146	0.0762403
$768$	0	0
$769$	2.94427	0.106173	0.0530866	−	0.998590i	$-0.483094\pi$
$769$	2.94427	0.106173	0.0530866	+	0.998590i	$0.483094\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	26.3607	0.948128	0.474064	−	0.880490i	$-0.342787\pi$
$773$	26.3607	0.948128	0.474064	+	0.880490i	$0.342787\pi$
$774$	0	0
$775$	2.47214	0.0888017
$776$	0	0
$777$	0	0
$778$	0	0
$779$	15.7771	0.565273
$780$	0	0
$781$	−12.9443	−0.463182
$782$	0	0
$783$	0	0
$784$	0	0
$785$	22.9443	0.818916
$786$	0	0
$787$	−14.0689	−0.501502	−0.250751	−	0.968052i	$-0.580677\pi$
$787$	−14.0689	−0.501502	−0.250751	+	0.968052i	$0.580677\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	18.4721	0.656794
$792$	0	0
$793$	−26.8328	−0.952861
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−1.41641	−0.0501717	−0.0250859	−	0.999685i	$-0.507986\pi$
$797$	−1.41641	−0.0501717	−0.0250859	+	0.999685i	$0.507986\pi$
$798$	0	0
$799$	−12.3607	−0.437289
$800$	0	0
$801$	0	0
$802$	0	0
$803$	8.94427	0.315637
$804$	0	0
$805$	11.4164	0.402376
$806$	0	0
$807$	0	0
$808$	0	0
$809$	47.8885	1.68367	0.841836	−	0.539734i	$-0.181475\pi$
$809$	47.8885	1.68367	0.841836	+	0.539734i	$0.181475\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.70820	−0.0598358
$816$	0	0
$817$	25.5279	0.893107
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−38.9443	−1.35916	−0.679582	−	0.733599i	$-0.737839\pi$
$821$	−38.9443	−1.35916	−0.679582	+	0.733599i	$0.737839\pi$
$822$	0	0
$823$	−11.7082	−0.408122	−0.204061	−	0.978958i	$-0.565414\pi$
$823$	−11.7082	−0.408122	−0.204061	+	0.978958i	$0.565414\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.1803	0.562646	0.281323	−	0.959613i	$-0.409227\pi$
$827$	16.1803	0.562646	0.281323	+	0.959613i	$0.409227\pi$
$828$	0	0
$829$	−38.0000	−1.31979	−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000	−1.31979	−0.659897	+	0.751356i	$0.729400\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−24.4721	−0.847909
$834$	0	0
$835$	−16.6525	−0.576283
$836$	0	0
$837$	0	0
$838$	0	0
$839$	4.00000	0.138095	0.0690477	−	0.997613i	$-0.478004\pi$
$839$	4.00000	0.138095	0.0690477	+	0.997613i	$0.478004\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.00000	0.240807
$846$	0	0
$847$	8.65248	0.297303
$848$	0	0
$849$	0	0
$850$	0	0
$851$	101.082	3.46505
$852$	0	0
$853$	−16.4721	−0.563995	−0.281998	−	0.959415i	$-0.590997\pi$
$853$	−16.4721	−0.563995	−0.281998	+	0.959415i	$0.590997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	28.8328	0.984910	0.492455	−	0.870338i	$-0.336100\pi$
$857$	28.8328	0.984910	0.492455	+	0.870338i	$0.336100\pi$
$858$	0	0
$859$	−22.5836	−0.770542	−0.385271	−	0.922803i	$-0.625892\pi$
$859$	−22.5836	−0.770542	−0.385271	+	0.922803i	$0.625892\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31.7082	1.07936	0.539680	−	0.841870i	$-0.318545\pi$
$863$	31.7082	1.07936	0.539680	+	0.841870i	$0.318545\pi$
$864$	0	0
$865$	−6.94427	−0.236112
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−9.88854	−0.335446
$870$	0	0
$871$	25.5279	0.864979
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.23607	−0.0417867
$876$	0	0
$877$	16.1115	0.544045	0.272023	−	0.962291i	$-0.412307\pi$
$877$	16.1115	0.544045	0.272023	+	0.962291i	$0.412307\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	15.5279	0.523147	0.261574	−	0.965184i	$-0.415759\pi$
$881$	15.5279	0.523147	0.261574	+	0.965184i	$0.415759\pi$
$882$	0	0
$883$	14.2918	0.480957	0.240479	−	0.970654i	$-0.422696\pi$
$883$	14.2918	0.480957	0.240479	+	0.970654i	$0.422696\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−37.5967	−1.26238	−0.631188	−	0.775630i	$-0.717432\pi$
$887$	−37.5967	−1.26238	−0.631188	+	0.775630i	$0.717432\pi$
$888$	0	0
$889$	2.69505	0.0903890
$890$	0	0
$891$	0	0
$892$	0	0
$893$	12.3607	0.413634
$894$	0	0
$895$	−11.5279	−0.385334
$896$	0	0
$897$	0	0
$898$	0	0
$899$	4.94427	0.164901
$900$	0	0
$901$	37.8885	1.26225
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.94427	−0.230835
$906$	0	0
$907$	4.76393	0.158184	0.0790919	−	0.996867i	$-0.474798\pi$
$907$	4.76393	0.158184	0.0790919	+	0.996867i	$0.474798\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.30495	0.308287	0.154143	−	0.988048i	$-0.450738\pi$
$911$	9.30495	0.308287	0.154143	+	0.988048i	$0.450738\pi$
$912$	0	0
$913$	−19.4164	−0.642589
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.63932	0.120181
$918$	0	0
$919$	−29.8885	−0.985932	−0.492966	−	0.870049i	$-0.664087\pi$
$919$	−29.8885	−0.985932	−0.492966	+	0.870049i	$0.664087\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	28.9443	0.952712
$924$	0	0
$925$	−10.9443	−0.359845
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.5279	−0.509453	−0.254726	−	0.967013i	$-0.581985\pi$
$929$	−15.5279	−0.509453	−0.254726	+	0.967013i	$0.581985\pi$
$930$	0	0
$931$	24.4721	0.802042
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.94427	−0.292509
$936$	0	0
$937$	−12.4721	−0.407447	−0.203723	−	0.979028i	$-0.565304\pi$
$937$	−12.4721	−0.407447	−0.203723	+	0.979028i	$0.565304\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	8.83282	0.287942	0.143971	−	0.989582i	$-0.454013\pi$
$941$	8.83282	0.287942	0.143971	+	0.989582i	$0.454013\pi$
$942$	0	0
$943$	32.5836	1.06107
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44.1803	1.43567	0.717834	−	0.696214i	$-0.245134\pi$
$947$	44.1803	1.43567	0.717834	+	0.696214i	$0.245134\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	25.7771	0.835002	0.417501	−	0.908677i	$-0.362906\pi$
$953$	25.7771	0.835002	0.417501	+	0.908677i	$0.362906\pi$
$954$	0	0
$955$	−15.4164	−0.498863
$956$	0	0
$957$	0	0
$958$	0	0
$959$	2.47214	0.0798294
$960$	0	0
$961$	−24.8885	−0.802856
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.4721	−0.659021
$966$	0	0
$967$	32.6525	1.05003	0.525016	−	0.851092i	$-0.324059\pi$
$967$	32.6525	1.05003	0.525016	+	0.851092i	$0.324059\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.8885	−0.381522	−0.190761	−	0.981637i	$-0.561095\pi$
$971$	−11.8885	−0.381522	−0.190761	+	0.981637i	$0.561095\pi$
$972$	0	0
$973$	14.2492	0.456809
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−23.3050	−0.745591	−0.372796	−	0.927913i	$-0.621601\pi$
$977$	−23.3050	−0.745591	−0.372796	+	0.927913i	$0.621601\pi$
$978$	0	0
$979$	5.88854	0.188199
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−37.0132	−1.18054	−0.590268	−	0.807207i	$-0.700978\pi$
$983$	−37.0132	−1.18054	−0.590268	+	0.807207i	$0.700978\pi$
$984$	0	0
$985$	8.47214	0.269945
$986$	0	0
$987$	0	0
$988$	0	0
$989$	52.7214	1.67644
$990$	0	0
$991$	26.4721	0.840915	0.420458	−	0.907312i	$-0.361870\pi$
$991$	26.4721	0.840915	0.420458	+	0.907312i	$0.361870\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−16.9443	−0.537170
$996$	0	0
$997$	−61.4164	−1.94508	−0.972539	−	0.232742i	$-0.925230\pi$
$997$	−61.4164	−1.94508	−0.972539	+	0.232742i	$0.925230\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5760.2.a.ci.1.1		2
3.2	odd	2		640.2.a.k.1.2	yes	2
4.3	odd	2		5760.2.a.ch.1.2		2
8.3	odd	2		5760.2.a.bw.1.2		2
8.5	even	2		5760.2.a.cd.1.1		2
12.11	even	2		640.2.a.i.1.1	✓	2
15.2	even	4		3200.2.c.w.2049.1		4
15.8	even	4		3200.2.c.w.2049.4		4
15.14	odd	2		3200.2.a.be.1.1		2
24.5	odd	2		640.2.a.j.1.1	yes	2
24.11	even	2		640.2.a.l.1.2	yes	2
48.5	odd	4		1280.2.d.k.641.1		4
48.11	even	4		1280.2.d.m.641.4		4
48.29	odd	4		1280.2.d.k.641.4		4
48.35	even	4		1280.2.d.m.641.1		4
60.23	odd	4		3200.2.c.u.2049.1		4
60.47	odd	4		3200.2.c.u.2049.4		4
60.59	even	2		3200.2.a.bl.1.2		2
120.29	odd	2		3200.2.a.bk.1.2		2
120.53	even	4		3200.2.c.v.2049.1		4
120.59	even	2		3200.2.a.bf.1.1		2
120.77	even	4		3200.2.c.v.2049.4		4
120.83	odd	4		3200.2.c.x.2049.4		4
120.107	odd	4		3200.2.c.x.2049.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.a.i.1.1	✓	2	12.11	even	2
640.2.a.j.1.1	yes	2	24.5	odd	2
640.2.a.k.1.2	yes	2	3.2	odd	2
640.2.a.l.1.2	yes	2	24.11	even	2
1280.2.d.k.641.1		4	48.5	odd	4
1280.2.d.k.641.4		4	48.29	odd	4
1280.2.d.m.641.1		4	48.35	even	4
1280.2.d.m.641.4		4	48.11	even	4
3200.2.a.be.1.1		2	15.14	odd	2
3200.2.a.bf.1.1		2	120.59	even	2
3200.2.a.bk.1.2		2	120.29	odd	2
3200.2.a.bl.1.2		2	60.59	even	2
3200.2.c.u.2049.1		4	60.23	odd	4
3200.2.c.u.2049.4		4	60.47	odd	4
3200.2.c.v.2049.1		4	120.53	even	4
3200.2.c.v.2049.4		4	120.77	even	4
3200.2.c.w.2049.1		4	15.2	even	4
3200.2.c.w.2049.4		4	15.8	even	4
3200.2.c.x.2049.1		4	120.107	odd	4
3200.2.c.x.2049.4		4	120.83	odd	4
5760.2.a.bw.1.2		2	8.3	odd	2
5760.2.a.cd.1.1		2	8.5	even	2
5760.2.a.ch.1.2		2	4.3	odd	2
5760.2.a.ci.1.1		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5760 = 2^{7} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5760.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -1.23607 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -1.23607 q^{7} -2.00000 q^{11} +4.47214 q^{13} +4.47214 q^{17} -4.47214 q^{19} -9.23607 q^{23} +1.00000 q^{25} +2.00000 q^{29} +2.47214 q^{31} -1.23607 q^{35} -10.9443 q^{37} -3.52786 q^{41} -5.70820 q^{43} -2.76393 q^{47} -5.47214 q^{49} +8.47214 q^{53} -2.00000 q^{55} +0.472136 q^{59} -6.00000 q^{61} +4.47214 q^{65} +5.70820 q^{67} +6.47214 q^{71} -4.47214 q^{73} +2.47214 q^{77} +4.94427 q^{79} +9.70820 q^{83} +4.47214 q^{85} -2.94427 q^{89} -5.52786 q^{91} -4.47214 q^{95} -7.52786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 + 2 * q^7	\( 2 q + 2 q^{5} + 2 q^{7} - 4 q^{11} - 14 q^{23} + 2 q^{25} + 4 q^{29} - 4 q^{31} + 2 q^{35} - 4 q^{37} - 16 q^{41} + 2 q^{43} - 10 q^{47} - 2 q^{49} + 8 q^{53} - 4 q^{55} - 8 q^{59} - 12 q^{61} - 2 q^{67} + 4 q^{71} - 4 q^{77} - 8 q^{79} + 6 q^{83} + 12 q^{89} - 20 q^{91} - 24 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 - 4 * q^11 - 14 * q^23 + 2 * q^25 + 4 * q^29 - 4 * q^31 + 2 * q^35 - 4 * q^37 - 16 * q^41 + 2 * q^43 - 10 * q^47 - 2 * q^49 + 8 * q^53 - 4 * q^55 - 8 * q^59 - 12 * q^61 - 2 * q^67 + 4 * q^71 - 4 * q^77 - 8 * q^79 + 6 * q^83 + 12 * q^89 - 20 * q^91 - 24 * q^97

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