LMFDB - Embedded newform 5220.2.a.x.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -4.67282 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.67282 q^{7} +0.672824 q^{11} -1.14399 q^{13} -3.52884 q^{17} +5.52884 q^{19} +3.81681 q^{23} +1.00000 q^{25} +1.00000 q^{29} -1.52884 q^{31} -4.67282 q^{35} +7.16246 q^{37} -2.85601 q^{41} +8.96080 q^{43} +6.67282 q^{47} +14.8353 q^{49} -10.4896 q^{53} +0.672824 q^{55} -10.7776 q^{59} -14.4896 q^{61} -1.14399 q^{65} -7.81681 q^{67} -4.48963 q^{71} -4.96080 q^{73} -3.14399 q^{77} +2.38485 q^{79} -14.0185 q^{83} -3.52884 q^{85} +1.63362 q^{89} +5.34565 q^{91} +5.52884 q^{95} -9.32718 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 4 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 4 * q^7	$ 3 q + 3 q^{5} - 4 q^{7} - 8 q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{19} + 3 q^{25} + 3 q^{29} + 4 q^{31} - 4 q^{35} - 10 q^{37} - 10 q^{41} + 14 q^{43} + 10 q^{47} + 3 q^{49} - 10 q^{53} - 8 q^{55} - 8 q^{59} - 22 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{83} - 2 q^{85} - 18 q^{89} - 4 q^{91} + 8 q^{95} - 38 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 4 * q^7 - 8 * q^11 - 2 * q^13 - 2 * q^17 + 8 * q^19 + 3 * q^25 + 3 * q^29 + 4 * q^31 - 4 * q^35 - 10 * q^37 - 10 * q^41 + 14 * q^43 + 10 * q^47 + 3 * q^49 - 10 * q^53 - 8 * q^55 - 8 * q^59 - 22 * q^61 - 2 * q^65 - 12 * q^67 + 8 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^83 - 2 * q^85 - 18 * q^89 - 4 * q^91 + 8 * q^95 - 38 * 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
$7$	−4.67282	−1.76616	−0.883081	−	0.469221i	$-0.844535\pi$
$7$	−4.67282	−1.76616	−0.883081	+	0.469221i	$0.844535\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.672824	0.202864	0.101432	−	0.994842i	$-0.467658\pi$
$11$	0.672824	0.202864	0.101432	+	0.994842i	$0.467658\pi$
$12$	0	0
$13$	−1.14399	−0.317285	−0.158642	−	0.987336i	$-0.550712\pi$
$13$	−1.14399	−0.317285	−0.158642	+	0.987336i	$0.550712\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.52884	−0.855869	−0.427934	−	0.903810i	$-0.640759\pi$
$17$	−3.52884	−0.855869	−0.427934	+	0.903810i	$0.640759\pi$
$18$	0	0
$19$	5.52884	1.26840	0.634201	−	0.773168i	$-0.281329\pi$
$19$	5.52884	1.26840	0.634201	+	0.773168i	$0.281329\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.81681	0.795860	0.397930	−	0.917416i	$-0.369729\pi$
$23$	3.81681	0.795860	0.397930	+	0.917416i	$0.369729\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−1.52884	−0.274587	−0.137294	−	0.990530i	$-0.543840\pi$
$31$	−1.52884	−0.274587	−0.137294	+	0.990530i	$0.543840\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.67282	−0.789851
$36$	0	0
$37$	7.16246	1.17750	0.588750	−	0.808315i	$-0.299620\pi$
$37$	7.16246	1.17750	0.588750	+	0.808315i	$0.299620\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.85601	−0.446034	−0.223017	−	0.974815i	$-0.571591\pi$
$41$	−2.85601	−0.446034	−0.223017	+	0.974815i	$0.571591\pi$
$42$	0	0
$43$	8.96080	1.36651	0.683254	−	0.730180i	$-0.260564\pi$
$43$	8.96080	1.36651	0.683254	+	0.730180i	$0.260564\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.67282	0.973331	0.486666	−	0.873588i	$-0.338213\pi$
$47$	6.67282	0.973331	0.486666	+	0.873588i	$0.338213\pi$
$48$	0	0
$49$	14.8353	2.11933
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.4896	−1.44086	−0.720431	−	0.693527i	$-0.756056\pi$
$53$	−10.4896	−1.44086	−0.720431	+	0.693527i	$0.756056\pi$
$54$	0	0
$55$	0.672824	0.0907235
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.7776	−1.40312	−0.701562	−	0.712608i	$-0.747514\pi$
$59$	−10.7776	−1.40312	−0.701562	+	0.712608i	$0.747514\pi$
$60$	0	0
$61$	−14.4896	−1.85521	−0.927604	−	0.373566i	$-0.878135\pi$
$61$	−14.4896	−1.85521	−0.927604	+	0.373566i	$0.878135\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.14399	−0.141894
$66$	0	0
$67$	−7.81681	−0.954975	−0.477488	−	0.878638i	$-0.658452\pi$
$67$	−7.81681	−0.954975	−0.477488	+	0.878638i	$0.658452\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.48963	−0.532822	−0.266411	−	0.963860i	$-0.585838\pi$
$71$	−4.48963	−0.532822	−0.266411	+	0.963860i	$0.585838\pi$
$72$	0	0
$73$	−4.96080	−0.580617	−0.290309	−	0.956933i	$-0.593758\pi$
$73$	−4.96080	−0.580617	−0.290309	+	0.956933i	$0.593758\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.14399	−0.358291
$78$	0	0
$79$	2.38485	0.268317	0.134158	−	0.990960i	$-0.457167\pi$
$79$	2.38485	0.268317	0.134158	+	0.990960i	$0.457167\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.0185	−1.53873	−0.769364	−	0.638811i	$-0.779427\pi$
$83$	−14.0185	−1.53873	−0.769364	+	0.638811i	$0.779427\pi$
$84$	0	0
$85$	−3.52884	−0.382756
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.63362	0.173163	0.0865817	−	0.996245i	$-0.472406\pi$
$89$	1.63362	0.173163	0.0865817	+	0.996245i	$0.472406\pi$
$90$	0	0
$91$	5.34565	0.560376
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.52884	0.567247
$96$	0	0
$97$	−9.32718	−0.947031	−0.473516	−	0.880785i	$-0.657015\pi$
$97$	−9.32718	−0.947031	−0.473516	+	0.880785i	$0.657015\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.3249	1.62439	0.812195	−	0.583386i	$-0.198273\pi$
$101$	16.3249	1.62439	0.812195	+	0.583386i	$0.198273\pi$
$102$	0	0
$103$	−13.5288	−1.33304	−0.666518	−	0.745489i	$-0.732216\pi$
$103$	−13.5288	−1.33304	−0.666518	+	0.745489i	$0.732216\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.672824	−0.0650443	−0.0325222	−	0.999471i	$-0.510354\pi$
$107$	−0.672824	−0.0650443	−0.0325222	+	0.999471i	$0.510354\pi$
$108$	0	0
$109$	13.5473	1.29760	0.648798	−	0.760960i	$-0.275272\pi$
$109$	13.5473	1.29760	0.648798	+	0.760960i	$0.275272\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−20.0185	−1.88318	−0.941590	−	0.336762i	$-0.890668\pi$
$113$	−20.0185	−1.88318	−0.941590	+	0.336762i	$0.890668\pi$
$114$	0	0
$115$	3.81681	0.355919
$116$	0	0
$117$	0	0
$118$	0	0
$119$	16.4896	1.51160
$120$	0	0
$121$	−10.5473	−0.958846
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.03920	−0.269686	−0.134843	−	0.990867i	$-0.543053\pi$
$127$	−3.03920	−0.269686	−0.134843	+	0.990867i	$0.543053\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	3.81681	0.333476	0.166738	−	0.986001i	$-0.446677\pi$
$131$	3.81681	0.333476	0.166738	+	0.986001i	$0.446677\pi$
$132$	0	0
$133$	−25.8353	−2.24020
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−16.1048	−1.37592	−0.687962	−	0.725746i	$-0.741495\pi$
$137$	−16.1048	−1.37592	−0.687962	+	0.725746i	$0.741495\pi$
$138$	0	0
$139$	3.05767	0.259349	0.129674	−	0.991557i	$-0.458607\pi$
$139$	3.05767	0.259349	0.129674	+	0.991557i	$0.458607\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.769701	−0.0643657
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	6.77761	0.551554	0.275777	−	0.961222i	$-0.411065\pi$
$151$	6.77761	0.551554	0.275777	+	0.961222i	$0.411065\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.52884	−0.122799
$156$	0	0
$157$	8.96080	0.715149	0.357575	−	0.933885i	$-0.383604\pi$
$157$	8.96080	0.715149	0.357575	+	0.933885i	$0.383604\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−17.8353	−1.40562
$162$	0	0
$163$	−6.30644	−0.493959	−0.246979	−	0.969021i	$-0.579438\pi$
$163$	−6.30644	−0.493959	−0.246979	+	0.969021i	$0.579438\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−12.7961	−0.990190	−0.495095	−	0.868839i	$-0.664867\pi$
$167$	−12.7961	−0.990190	−0.495095	+	0.868839i	$0.664867\pi$
$168$	0	0
$169$	−11.6913	−0.899330
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−18.1233	−1.37789	−0.688943	−	0.724816i	$-0.741925\pi$
$173$	−18.1233	−1.37789	−0.688943	+	0.724816i	$0.741925\pi$
$174$	0	0
$175$	−4.67282	−0.353232
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−6.28797	−0.469985	−0.234993	−	0.971997i	$-0.575507\pi$
$179$	−6.28797	−0.469985	−0.234993	+	0.971997i	$0.575507\pi$
$180$	0	0
$181$	4.56804	0.339540	0.169770	−	0.985484i	$-0.445698\pi$
$181$	4.56804	0.339540	0.169770	+	0.985484i	$0.445698\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	7.16246	0.526594
$186$	0	0
$187$	−2.37429	−0.173625
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−21.6521	−1.56669	−0.783345	−	0.621587i	$-0.786488\pi$
$191$	−21.6521	−1.56669	−0.783345	+	0.621587i	$0.786488\pi$
$192$	0	0
$193$	−13.2409	−0.953098	−0.476549	−	0.879148i	$-0.658113\pi$
$193$	−13.2409	−0.953098	−0.476549	+	0.879148i	$0.658113\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−7.34565	−0.523356	−0.261678	−	0.965155i	$-0.584276\pi$
$197$	−7.34565	−0.523356	−0.261678	+	0.965155i	$0.584276\pi$
$198$	0	0
$199$	18.7776	1.33111	0.665555	−	0.746349i	$-0.268195\pi$
$199$	18.7776	1.33111	0.665555	+	0.746349i	$0.268195\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−4.67282	−0.327968
$204$	0	0
$205$	−2.85601	−0.199473
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.71993	0.257313
$210$	0	0
$211$	17.6521	1.21522	0.607610	−	0.794235i	$-0.292128\pi$
$211$	17.6521	1.21522	0.607610	+	0.794235i	$0.292128\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.96080	0.611121
$216$	0	0
$217$	7.14399	0.484965
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.03694	0.271554
$222$	0	0
$223$	7.45043	0.498918	0.249459	−	0.968385i	$-0.419747\pi$
$223$	7.45043	0.498918	0.249459	+	0.968385i	$0.419747\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.2201	1.60755	0.803773	−	0.594936i	$-0.202822\pi$
$227$	24.2201	1.60755	0.803773	+	0.594936i	$0.202822\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0.287973	0.0188657	0.00943287	−	0.999956i	$-0.496997\pi$
$233$	0.287973	0.0188657	0.00943287	+	0.999956i	$0.496997\pi$
$234$	0	0
$235$	6.67282	0.435287
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−7.51037	−0.485805	−0.242903	−	0.970051i	$-0.578100\pi$
$239$	−7.51037	−0.485805	−0.242903	+	0.970051i	$0.578100\pi$
$240$	0	0
$241$	−21.6706	−1.39592	−0.697962	−	0.716135i	$-0.745909\pi$
$241$	−21.6706	−1.39592	−0.697962	+	0.716135i	$0.745909\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.8353	0.947791
$246$	0	0
$247$	−6.32492	−0.402445
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.6728	0.799902	0.399951	−	0.916537i	$-0.369027\pi$
$251$	12.6728	0.799902	0.399951	+	0.916537i	$0.369027\pi$
$252$	0	0
$253$	2.56804	0.161451
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.9714	−1.05864	−0.529322	−	0.848421i	$-0.677554\pi$
$257$	−16.9714	−1.05864	−0.529322	+	0.848421i	$0.677554\pi$
$258$	0	0
$259$	−33.4689	−2.07966
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.90312	−0.364002	−0.182001	−	0.983298i	$-0.558257\pi$
$263$	−5.90312	−0.364002	−0.182001	+	0.983298i	$0.558257\pi$
$264$	0	0
$265$	−10.4896	−0.644373
$266$	0	0
$267$	0	0
$268$	0	0
$269$	14.8560	0.905787	0.452894	−	0.891565i	$-0.350392\pi$
$269$	14.8560	0.905787	0.452894	+	0.891565i	$0.350392\pi$
$270$	0	0
$271$	22.5944	1.37251	0.686257	−	0.727360i	$-0.259253\pi$
$271$	22.5944	1.37251	0.686257	+	0.727360i	$0.259253\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.672824	0.0405728
$276$	0	0
$277$	27.4689	1.65045	0.825223	−	0.564807i	$-0.191049\pi$
$277$	27.4689	1.65045	0.825223	+	0.564807i	$0.191049\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−17.5473	−1.04678	−0.523392	−	0.852092i	$-0.675334\pi$
$281$	−17.5473	−1.04678	−0.523392	+	0.852092i	$0.675334\pi$
$282$	0	0
$283$	26.5081	1.57574	0.787872	−	0.615839i	$-0.211183\pi$
$283$	26.5081	1.57574	0.787872	+	0.615839i	$0.211183\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.3456	0.787769
$288$	0	0
$289$	−4.54731	−0.267489
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.7098	1.32672	0.663359	−	0.748301i	$-0.269130\pi$
$293$	22.7098	1.32672	0.663359	+	0.748301i	$0.269130\pi$
$294$	0	0
$295$	−10.7776	−0.627497
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−4.36638	−0.252514
$300$	0	0
$301$	−41.8722	−2.41347
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−14.4896	−0.829674
$306$	0	0
$307$	20.9608	1.19630	0.598148	−	0.801386i	$-0.295904\pi$
$307$	20.9608	1.19630	0.598148	+	0.801386i	$0.295904\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.4874	1.55867	0.779333	−	0.626610i	$-0.215558\pi$
$311$	27.4874	1.55867	0.779333	+	0.626610i	$0.215558\pi$
$312$	0	0
$313$	−18.8560	−1.06580	−0.532902	−	0.846177i	$-0.678899\pi$
$313$	−18.8560	−1.06580	−0.532902	+	0.846177i	$0.678899\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.4504	−1.65410	−0.827050	−	0.562128i	$-0.809983\pi$
$317$	−29.4504	−1.65410	−0.827050	+	0.562128i	$0.809983\pi$
$318$	0	0
$319$	0.672824	0.0376709
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−19.5104	−1.08559
$324$	0	0
$325$	−1.14399	−0.0634570
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−31.1809	−1.71906
$330$	0	0
$331$	3.60724	0.198272	0.0991360	−	0.995074i	$-0.468392\pi$
$331$	3.60724	0.198272	0.0991360	+	0.995074i	$0.468392\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.81681	−0.427078
$336$	0	0
$337$	−18.3064	−0.997216	−0.498608	−	0.866828i	$-0.666155\pi$
$337$	−18.3064	−0.997216	−0.498608	+	0.866828i	$0.666155\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.02864	−0.0557039
$342$	0	0
$343$	−36.6129	−1.97691
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.38485	−0.342757	−0.171378	−	0.985205i	$-0.554822\pi$
$347$	−6.38485	−0.342757	−0.171378	+	0.985205i	$0.554822\pi$
$348$	0	0
$349$	17.2672	0.924294	0.462147	−	0.886803i	$-0.347079\pi$
$349$	17.2672	0.924294	0.462147	+	0.886803i	$0.347079\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.91369	0.527652	0.263826	−	0.964570i	$-0.415015\pi$
$353$	9.91369	0.527652	0.263826	+	0.964570i	$0.415015\pi$
$354$	0	0
$355$	−4.48963	−0.238285
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.81681	0.201444	0.100722	−	0.994915i	$-0.467885\pi$
$359$	3.81681	0.201444	0.100722	+	0.994915i	$0.467885\pi$
$360$	0	0
$361$	11.5680	0.608844
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.96080	−0.259660
$366$	0	0
$367$	−7.98153	−0.416632	−0.208316	−	0.978062i	$-0.566798\pi$
$367$	−7.98153	−0.416632	−0.208316	+	0.978062i	$0.566798\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	49.0162	2.54479
$372$	0	0
$373$	−26.0369	−1.34814	−0.674071	−	0.738667i	$-0.735456\pi$
$373$	−26.0369	−1.34814	−0.674071	+	0.738667i	$0.735456\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.14399	−0.0589183
$378$	0	0
$379$	28.8824	1.48359	0.741794	−	0.670627i	$-0.233975\pi$
$379$	28.8824	1.48359	0.741794	+	0.670627i	$0.233975\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−0.0968776	−0.00495021	−0.00247511	−	0.999997i	$-0.500788\pi$
$383$	−0.0968776	−0.00495021	−0.00247511	+	0.999997i	$0.500788\pi$
$384$	0	0
$385$	−3.14399	−0.160232
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.740665	0.0375532	0.0187766	−	0.999824i	$-0.494023\pi$
$389$	0.740665	0.0375532	0.0187766	+	0.999824i	$0.494023\pi$
$390$	0	0
$391$	−13.4689	−0.681152
$392$	0	0
$393$	0	0
$394$	0	0
$395$	2.38485	0.119995
$396$	0	0
$397$	−23.9216	−1.20059	−0.600295	−	0.799779i	$-0.704950\pi$
$397$	−23.9216	−1.20059	−0.600295	+	0.799779i	$0.704950\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.54731	0.277019	0.138510	−	0.990361i	$-0.455769\pi$
$401$	5.54731	0.277019	0.138510	+	0.990361i	$0.455769\pi$
$402$	0	0
$403$	1.74897	0.0871224
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.81907	0.238872
$408$	0	0
$409$	−34.8930	−1.72535	−0.862673	−	0.505762i	$-0.831211\pi$
$409$	−34.8930	−1.72535	−0.862673	+	0.505762i	$0.831211\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	50.3619	2.47814
$414$	0	0
$415$	−14.0185	−0.688140
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.1233	−0.592260	−0.296130	−	0.955148i	$-0.595696\pi$
$419$	−12.1233	−0.592260	−0.296130	+	0.955148i	$0.595696\pi$
$420$	0	0
$421$	19.0656	0.929200	0.464600	−	0.885521i	$-0.346198\pi$
$421$	19.0656	0.929200	0.464600	+	0.885521i	$0.346198\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.52884	−0.171174
$426$	0	0
$427$	67.7075	3.27660
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−2.69129	−0.129635	−0.0648176	−	0.997897i	$-0.520647\pi$
$431$	−2.69129	−0.129635	−0.0648176	+	0.997897i	$0.520647\pi$
$432$	0	0
$433$	17.4874	0.840390	0.420195	−	0.907434i	$-0.361962\pi$
$433$	17.4874	0.840390	0.420195	+	0.907434i	$0.361962\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.1025	1.00947
$438$	0	0
$439$	−4.45269	−0.212515	−0.106258	−	0.994339i	$-0.533887\pi$
$439$	−4.45269	−0.212515	−0.106258	+	0.994339i	$0.533887\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−11.6151	−0.551852	−0.275926	−	0.961179i	$-0.588985\pi$
$443$	−11.6151	−0.551852	−0.275926	+	0.961179i	$0.588985\pi$
$444$	0	0
$445$	1.63362	0.0774410
$446$	0	0
$447$	0	0
$448$	0	0
$449$	9.91369	0.467856	0.233928	−	0.972254i	$-0.424842\pi$
$449$	9.91369	0.467856	0.233928	+	0.972254i	$0.424842\pi$
$450$	0	0
$451$	−1.92159	−0.0904843
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.34565	0.250608
$456$	0	0
$457$	32.6050	1.52520	0.762598	−	0.646872i	$-0.223923\pi$
$457$	32.6050	1.52520	0.762598	+	0.646872i	$0.223923\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−5.23030	−0.243599	−0.121800	−	0.992555i	$-0.538867\pi$
$461$	−5.23030	−0.243599	−0.121800	+	0.992555i	$0.538867\pi$
$462$	0	0
$463$	−35.0841	−1.63049	−0.815247	−	0.579113i	$-0.803399\pi$
$463$	−35.0841	−1.63049	−0.815247	+	0.579113i	$0.803399\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−21.3641	−0.988614	−0.494307	−	0.869288i	$-0.664578\pi$
$467$	−21.3641	−0.988614	−0.494307	+	0.869288i	$0.664578\pi$
$468$	0	0
$469$	36.5266	1.68664
$470$	0	0
$471$	0	0
$472$	0	0
$473$	6.02904	0.277215
$474$	0	0
$475$	5.52884	0.253680
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5.03920	−0.230247	−0.115124	−	0.993351i	$-0.536726\pi$
$479$	−5.03920	−0.230247	−0.115124	+	0.993351i	$0.536726\pi$
$480$	0	0
$481$	−8.19376	−0.373603
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.32718	−0.423525
$486$	0	0
$487$	28.8824	1.30879	0.654393	−	0.756155i	$-0.272924\pi$
$487$	28.8824	1.30879	0.654393	+	0.756155i	$0.272924\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.16246	−0.232978	−0.116489	−	0.993192i	$-0.537164\pi$
$491$	−5.16246	−0.232978	−0.116489	+	0.993192i	$0.537164\pi$
$492$	0	0
$493$	−3.52884	−0.158931
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.9793	0.941049
$498$	0	0
$499$	−27.0946	−1.21292	−0.606461	−	0.795113i	$-0.707411\pi$
$499$	−27.0946	−1.21292	−0.606461	+	0.795113i	$0.707411\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.7305	−1.14727	−0.573633	−	0.819112i	$-0.694466\pi$
$503$	−25.7305	−1.14727	−0.573633	+	0.819112i	$0.694466\pi$
$504$	0	0
$505$	16.3249	0.726449
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11.8353	−0.524590	−0.262295	−	0.964988i	$-0.584479\pi$
$509$	−11.8353	−0.524590	−0.262295	+	0.964988i	$0.584479\pi$
$510$	0	0
$511$	23.1809	1.02546
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−13.5288	−0.596152
$516$	0	0
$517$	4.48963	0.197454
$518$	0	0
$519$	0	0
$520$	0	0
$521$	0.164719	0.00721646	0.00360823	−	0.999993i	$-0.498851\pi$
$521$	0.164719	0.00721646	0.00360823	+	0.999993i	$0.498851\pi$
$522$	0	0
$523$	−11.6072	−0.507549	−0.253775	−	0.967263i	$-0.581672\pi$
$523$	−11.6072	−0.507549	−0.253775	+	0.967263i	$0.581672\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.39502	0.235011
$528$	0	0
$529$	−8.43196	−0.366607
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.26724	0.141520
$534$	0	0
$535$	−0.672824	−0.0290887
$536$	0	0
$537$	0	0
$538$	0	0
$539$	9.98153	0.429935
$540$	0	0
$541$	−35.5922	−1.53023	−0.765113	−	0.643896i	$-0.777317\pi$
$541$	−35.5922	−1.53023	−0.765113	+	0.643896i	$0.777317\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.5473	0.580303
$546$	0	0
$547$	−44.3803	−1.89757	−0.948783	−	0.315929i	$-0.897684\pi$
$547$	−44.3803	−1.89757	−0.948783	+	0.315929i	$0.897684\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.52884	0.235536
$552$	0	0
$553$	−11.1440	−0.473891
$554$	0	0
$555$	0	0
$556$	0	0
$557$	26.0739	1.10479	0.552393	−	0.833584i	$-0.313715\pi$
$557$	26.0739	1.10479	0.552393	+	0.833584i	$0.313715\pi$
$558$	0	0
$559$	−10.2510	−0.433572
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.4218	0.692096	0.346048	−	0.938217i	$-0.387523\pi$
$563$	16.4218	0.692096	0.346048	+	0.938217i	$0.387523\pi$
$564$	0	0
$565$	−20.0185	−0.842183
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.79834	0.326923	0.163462	−	0.986550i	$-0.447734\pi$
$569$	7.79834	0.326923	0.163462	+	0.986550i	$0.447734\pi$
$570$	0	0
$571$	−33.3826	−1.39702	−0.698509	−	0.715601i	$-0.746153\pi$
$571$	−33.3826	−1.39702	−0.698509	+	0.715601i	$0.746153\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.81681	0.159172
$576$	0	0
$577$	−8.10478	−0.337407	−0.168703	−	0.985667i	$-0.553958\pi$
$577$	−8.10478	−0.337407	−0.168703	+	0.985667i	$0.553958\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	65.5058	2.71764
$582$	0	0
$583$	−7.05767	−0.292299
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.5865	−1.17989	−0.589946	−	0.807443i	$-0.700851\pi$
$587$	−28.5865	−1.17989	−0.589946	+	0.807443i	$0.700851\pi$
$588$	0	0
$589$	−8.45269	−0.348287
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−22.4896	−0.923539	−0.461769	−	0.887000i	$-0.652785\pi$
$593$	−22.4896	−0.923539	−0.461769	+	0.887000i	$0.652785\pi$
$594$	0	0
$595$	16.4896	0.676009
$596$	0	0
$597$	0	0
$598$	0	0
$599$	31.7305	1.29647	0.648237	−	0.761439i	$-0.275507\pi$
$599$	31.7305	1.29647	0.648237	+	0.761439i	$0.275507\pi$
$600$	0	0
$601$	4.56804	0.186334	0.0931671	−	0.995650i	$-0.470301\pi$
$601$	4.56804	0.186334	0.0931671	+	0.995650i	$0.470301\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.5473	−0.428809
$606$	0	0
$607$	−12.7882	−0.519056	−0.259528	−	0.965736i	$-0.583567\pi$
$607$	−12.7882	−0.519056	−0.259528	+	0.965736i	$0.583567\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−7.63362	−0.308823
$612$	0	0
$613$	−6.97927	−0.281890	−0.140945	−	0.990017i	$-0.545014\pi$
$613$	−6.97927	−0.281890	−0.140945	+	0.990017i	$0.545014\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.9193	−1.24477	−0.622383	−	0.782713i	$-0.713835\pi$
$617$	−30.9193	−1.24477	−0.622383	+	0.782713i	$0.713835\pi$
$618$	0	0
$619$	−20.7098	−0.832396	−0.416198	−	0.909274i	$-0.636638\pi$
$619$	−20.7098	−0.832396	−0.416198	+	0.909274i	$0.636638\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−7.63362	−0.305835
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−25.2751	−1.00779
$630$	0	0
$631$	9.67508	0.385159	0.192580	−	0.981281i	$-0.438315\pi$
$631$	9.67508	0.385159	0.192580	+	0.981281i	$0.438315\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.03920	−0.120607
$636$	0	0
$637$	−16.9714	−0.672430
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−23.5058	−0.928425	−0.464213	−	0.885724i	$-0.653663\pi$
$641$	−23.5058	−0.928425	−0.464213	+	0.885724i	$0.653663\pi$
$642$	0	0
$643$	40.6235	1.60203	0.801016	−	0.598643i	$-0.204293\pi$
$643$	40.6235	1.60203	0.801016	+	0.598643i	$0.204293\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.2280	0.402106	0.201053	−	0.979580i	$-0.435564\pi$
$647$	10.2280	0.402106	0.201053	+	0.979580i	$0.435564\pi$
$648$	0	0
$649$	−7.25143	−0.284644
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.2280	1.10465	0.552324	−	0.833629i	$-0.313741\pi$
$653$	28.2280	1.10465	0.552324	+	0.833629i	$0.313741\pi$
$654$	0	0
$655$	3.81681	0.149135
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0471	0.586152	0.293076	−	0.956089i	$-0.405321\pi$
$659$	15.0471	0.586152	0.293076	+	0.956089i	$0.405321\pi$
$660$	0	0
$661$	13.8767	0.539743	0.269871	−	0.962896i	$-0.413019\pi$
$661$	13.8767	0.539743	0.269871	+	0.962896i	$0.413019\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−25.8353	−1.00185
$666$	0	0
$667$	3.81681	0.147787
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.74897	−0.376355
$672$	0	0
$673$	15.3456	0.591531	0.295766	−	0.955261i	$-0.404425\pi$
$673$	15.3456	0.591531	0.295766	+	0.955261i	$0.404425\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	50.5530	1.94291	0.971454	−	0.237228i	$-0.0762390\pi$
$677$	50.5530	1.94291	0.971454	+	0.237228i	$0.0762390\pi$
$678$	0	0
$679$	43.5843	1.67261
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−41.6027	−1.59188	−0.795942	−	0.605373i	$-0.793024\pi$
$683$	−41.6027	−1.59188	−0.795942	+	0.605373i	$0.793024\pi$
$684$	0	0
$685$	−16.1048	−0.615332
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−41.6627	−1.58492	−0.792461	−	0.609922i	$-0.791201\pi$
$691$	−41.6627	−1.58492	−0.792461	+	0.609922i	$0.791201\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.05767	0.115984
$696$	0	0
$697$	10.0784	0.381747
$698$	0	0
$699$	0	0
$700$	0	0
$701$	44.8145	1.69262	0.846311	−	0.532689i	$-0.178818\pi$
$701$	44.8145	1.69262	0.846311	+	0.532689i	$0.178818\pi$
$702$	0	0
$703$	39.6001	1.49354
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−76.2835	−2.86893
$708$	0	0
$709$	23.4320	0.880006	0.440003	−	0.897996i	$-0.354977\pi$
$709$	23.4320	0.880006	0.440003	+	0.897996i	$0.354977\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.83528	−0.218533
$714$	0	0
$715$	−0.769701	−0.0287852
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−32.3328	−1.20581	−0.602905	−	0.797813i	$-0.705990\pi$
$719$	−32.3328	−1.20581	−0.602905	+	0.797813i	$0.705990\pi$
$720$	0	0
$721$	63.2179	2.35436
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	4.01847	0.149037	0.0745184	−	0.997220i	$-0.476258\pi$
$727$	4.01847	0.149037	0.0745184	+	0.997220i	$0.476258\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−31.6212	−1.16955
$732$	0	0
$733$	−3.49189	−0.128976	−0.0644880	−	0.997918i	$-0.520541\pi$
$733$	−3.49189	−0.128976	−0.0644880	+	0.997918i	$0.520541\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.25934	−0.193730
$738$	0	0
$739$	−45.4090	−1.67040	−0.835198	−	0.549949i	$-0.814647\pi$
$739$	−45.4090	−1.67040	−0.835198	+	0.549949i	$0.814647\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−40.6683	−1.49198	−0.745988	−	0.665960i	$-0.768022\pi$
$743$	−40.6683	−1.49198	−0.745988	+	0.665960i	$0.768022\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.14399	0.114879
$750$	0	0
$751$	49.0268	1.78901	0.894506	−	0.447055i	$-0.147527\pi$
$751$	49.0268	1.78901	0.894506	+	0.447055i	$0.147527\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.77761	0.246662
$756$	0	0
$757$	−51.7260	−1.88001	−0.940006	−	0.341157i	$-0.889181\pi$
$757$	−51.7260	−1.88001	−0.940006	+	0.341157i	$0.889181\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−34.2307	−1.24086	−0.620431	−	0.784261i	$-0.713042\pi$
$761$	−34.2307	−1.24086	−0.620431	+	0.784261i	$0.713042\pi$
$762$	0	0
$763$	−63.3042	−2.29177
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.3294	0.445190
$768$	0	0
$769$	22.5971	0.814871	0.407436	−	0.913234i	$-0.366423\pi$
$769$	22.5971	0.814871	0.407436	+	0.913234i	$0.366423\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−13.8538	−0.498285	−0.249142	−	0.968467i	$-0.580149\pi$
$773$	−13.8538	−0.498285	−0.249142	+	0.968467i	$0.580149\pi$
$774$	0	0
$775$	−1.52884	−0.0549175
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−15.7904	−0.565751
$780$	0	0
$781$	−3.02073	−0.108090
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.96080	0.319825
$786$	0	0
$787$	20.0969	0.716376	0.358188	−	0.933649i	$-0.383395\pi$
$787$	20.0969	0.716376	0.358188	+	0.933649i	$0.383395\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	93.5428	3.32600
$792$	0	0
$793$	16.5759	0.588629
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−43.6890	−1.54754	−0.773772	−	0.633464i	$-0.781633\pi$
$797$	−43.6890	−1.54754	−0.773772	+	0.633464i	$0.781633\pi$
$798$	0	0
$799$	−23.5473	−0.833044
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.33774	−0.117786
$804$	0	0
$805$	−17.8353	−0.628611
$806$	0	0
$807$	0	0
$808$	0	0
$809$	18.4403	0.648325	0.324163	−	0.946001i	$-0.394918\pi$
$809$	18.4403	0.648325	0.324163	+	0.946001i	$0.394918\pi$
$810$	0	0
$811$	15.0577	0.528746	0.264373	−	0.964420i	$-0.414835\pi$
$811$	15.0577	0.528746	0.264373	+	0.964420i	$0.414835\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.30644	−0.220905
$816$	0	0
$817$	49.5428	1.73328
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−50.9793	−1.77919	−0.889594	−	0.456751i	$-0.849013\pi$
$821$	−50.9793	−1.77919	−0.889594	+	0.456751i	$0.849013\pi$
$822$	0	0
$823$	−2.84545	−0.0991861	−0.0495930	−	0.998770i	$-0.515792\pi$
$823$	−2.84545	−0.0991861	−0.0495930	+	0.998770i	$0.515792\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	15.0761	0.524249	0.262124	−	0.965034i	$-0.415577\pi$
$827$	15.0761	0.524249	0.262124	+	0.965034i	$0.415577\pi$
$828$	0	0
$829$	−34.6992	−1.20515	−0.602577	−	0.798061i	$-0.705859\pi$
$829$	−34.6992	−1.20515	−0.602577	+	0.798061i	$0.705859\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−52.3513	−1.81386
$834$	0	0
$835$	−12.7961	−0.442827
$836$	0	0
$837$	0	0
$838$	0	0
$839$	14.7882	0.510544	0.255272	−	0.966869i	$-0.417835\pi$
$839$	14.7882	0.510544	0.255272	+	0.966869i	$0.417835\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.6913	−0.402193
$846$	0	0
$847$	49.2857	1.69348
$848$	0	0
$849$	0	0
$850$	0	0
$851$	27.3377	0.937126
$852$	0	0
$853$	35.7753	1.22492	0.612462	−	0.790500i	$-0.290179\pi$
$853$	35.7753	1.22492	0.612462	+	0.790500i	$0.290179\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.4112	0.423959	0.211980	−	0.977274i	$-0.432009\pi$
$857$	12.4112	0.423959	0.211980	+	0.977274i	$0.432009\pi$
$858$	0	0
$859$	−0.0599353	−0.00204497	−0.00102248	−	0.999999i	$-0.500325\pi$
$859$	−0.0599353	−0.00204497	−0.00102248	+	0.999999i	$0.500325\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44.5530	1.51660	0.758300	−	0.651906i	$-0.226030\pi$
$863$	44.5530	1.51660	0.758300	+	0.651906i	$0.226030\pi$
$864$	0	0
$865$	−18.1233	−0.616209
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.60458	0.0544318
$870$	0	0
$871$	8.94233	0.302999
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.67282	−0.157970
$876$	0	0
$877$	29.9137	1.01011	0.505057	−	0.863086i	$-0.331472\pi$
$877$	29.9137	1.01011	0.505057	+	0.863086i	$0.331472\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	32.2386	1.08615	0.543073	−	0.839685i	$-0.317261\pi$
$881$	32.2386	1.08615	0.543073	+	0.839685i	$0.317261\pi$
$882$	0	0
$883$	36.7098	1.23538	0.617691	−	0.786421i	$-0.288068\pi$
$883$	36.7098	1.23538	0.617691	+	0.786421i	$0.288068\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.63588	0.0885042	0.0442521	−	0.999020i	$-0.485910\pi$
$887$	2.63588	0.0885042	0.0442521	+	0.999020i	$0.485910\pi$
$888$	0	0
$889$	14.2017	0.476308
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.8930	1.23458
$894$	0	0
$895$	−6.28797	−0.210184
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−1.52884	−0.0509896
$900$	0	0
$901$	37.0162	1.23319
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.56804	0.151847
$906$	0	0
$907$	16.0185	0.531885	0.265942	−	0.963989i	$-0.414317\pi$
$907$	16.0185	0.531885	0.265942	+	0.963989i	$0.414317\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−37.9480	−1.25727	−0.628636	−	0.777700i	$-0.716387\pi$
$911$	−37.9480	−1.25727	−0.628636	+	0.777700i	$0.716387\pi$
$912$	0	0
$913$	−9.43196	−0.312152
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−17.8353	−0.588973
$918$	0	0
$919$	−7.58425	−0.250181	−0.125091	−	0.992145i	$-0.539922\pi$
$919$	−7.58425	−0.250181	−0.125091	+	0.992145i	$0.539922\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.13608	0.169056
$924$	0	0
$925$	7.16246	0.235500
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−26.4033	−0.866265	−0.433132	−	0.901330i	$-0.642592\pi$
$929$	−26.4033	−0.866265	−0.433132	+	0.901330i	$0.642592\pi$
$930$	0	0
$931$	82.0219	2.68816
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−2.37429	−0.0776474
$936$	0	0
$937$	44.6050	1.45718	0.728591	−	0.684949i	$-0.240176\pi$
$937$	44.6050	1.45718	0.728591	+	0.684949i	$0.240176\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	9.26724	0.302103	0.151052	−	0.988526i	$-0.451734\pi$
$941$	9.26724	0.302103	0.151052	+	0.988526i	$0.451734\pi$
$942$	0	0
$943$	−10.9009	−0.354981
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.4218	0.923584	0.461792	−	0.886988i	$-0.347207\pi$
$947$	28.4218	0.923584	0.461792	+	0.886988i	$0.347207\pi$
$948$	0	0
$949$	5.67508	0.184221
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−55.2100	−1.78843	−0.894213	−	0.447642i	$-0.852264\pi$
$953$	−55.2100	−1.78843	−0.894213	+	0.447642i	$0.852264\pi$
$954$	0	0
$955$	−21.6521	−0.700645
$956$	0	0
$957$	0	0
$958$	0	0
$959$	75.2548	2.43010
$960$	0	0
$961$	−28.6627	−0.924602
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.2409	−0.426238
$966$	0	0
$967$	−34.1496	−1.09818	−0.549089	−	0.835764i	$-0.685025\pi$
$967$	−34.1496	−1.09818	−0.549089	+	0.835764i	$0.685025\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	41.4090	1.32888	0.664438	−	0.747343i	$-0.268671\pi$
$971$	41.4090	1.32888	0.664438	+	0.747343i	$0.268671\pi$
$972$	0	0
$973$	−14.2880	−0.458051
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.9506	0.446320	0.223160	−	0.974782i	$-0.428363\pi$
$977$	13.9506	0.446320	0.223160	+	0.974782i	$0.428363\pi$
$978$	0	0
$979$	1.09914	0.0351286
$980$	0	0
$981$	0	0
$982$	0	0
$983$	39.6521	1.26471	0.632353	−	0.774681i	$-0.282089\pi$
$983$	39.6521	1.26471	0.632353	+	0.774681i	$0.282089\pi$
$984$	0	0
$985$	−7.34565	−0.234052
$986$	0	0
$987$	0	0
$988$	0	0
$989$	34.2017	1.08755
$990$	0	0
$991$	2.93442	0.0932149	0.0466075	−	0.998913i	$-0.485159\pi$
$991$	2.93442	0.0932149	0.0466075	+	0.998913i	$0.485159\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18.7776	0.595290
$996$	0	0
$997$	12.9977	0.411643	0.205821	−	0.978590i	$-0.434013\pi$
$997$	12.9977	0.411643	0.205821	+	0.978590i	$0.434013\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.x.1.1		3
3.2	odd	2		580.2.a.c.1.1	✓	3
12.11	even	2		2320.2.a.m.1.3		3
15.2	even	4		2900.2.c.f.349.5		6
15.8	even	4		2900.2.c.f.349.2		6
15.14	odd	2		2900.2.a.g.1.3		3
24.5	odd	2		9280.2.a.bk.1.3		3
24.11	even	2		9280.2.a.bw.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.c.1.1	✓	3	3.2	odd	2
2320.2.a.m.1.3		3	12.11	even	2
2900.2.a.g.1.3		3	15.14	odd	2
2900.2.c.f.349.2		6	15.8	even	4
2900.2.c.f.349.5		6	15.2	even	4
5220.2.a.x.1.1		3	1.1	even	1	trivial
9280.2.a.bk.1.3		3	24.5	odd	2
9280.2.a.bw.1.1		3	24.11	even	2

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

Level:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.67282 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.67282 q^{7} +0.672824 q^{11} -1.14399 q^{13} -3.52884 q^{17} +5.52884 q^{19} +3.81681 q^{23} +1.00000 q^{25} +1.00000 q^{29} -1.52884 q^{31} -4.67282 q^{35} +7.16246 q^{37} -2.85601 q^{41} +8.96080 q^{43} +6.67282 q^{47} +14.8353 q^{49} -10.4896 q^{53} +0.672824 q^{55} -10.7776 q^{59} -14.4896 q^{61} -1.14399 q^{65} -7.81681 q^{67} -4.48963 q^{71} -4.96080 q^{73} -3.14399 q^{77} +2.38485 q^{79} -14.0185 q^{83} -3.52884 q^{85} +1.63362 q^{89} +5.34565 q^{91} +5.52884 q^{95} -9.32718 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 4 * q^7	\( 3 q + 3 q^{5} - 4 q^{7} - 8 q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{19} + 3 q^{25} + 3 q^{29} + 4 q^{31} - 4 q^{35} - 10 q^{37} - 10 q^{41} + 14 q^{43} + 10 q^{47} + 3 q^{49} - 10 q^{53} - 8 q^{55} - 8 q^{59} - 22 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{83} - 2 q^{85} - 18 q^{89} - 4 q^{91} + 8 q^{95} - 38 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 4 * q^7 - 8 * q^11 - 2 * q^13 - 2 * q^17 + 8 * q^19 + 3 * q^25 + 3 * q^29 + 4 * q^31 - 4 * q^35 - 10 * q^37 - 10 * q^41 + 14 * q^43 + 10 * q^47 + 3 * q^49 - 10 * q^53 - 8 * q^55 - 8 * q^59 - 22 * q^61 - 2 * q^65 - 12 * q^67 + 8 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^83 - 2 * q^85 - 18 * q^89 - 4 * q^91 + 8 * q^95 - 38 * q^97

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