LMFDB - Embedded newform 7800.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7800,2,Mod(1,7800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7800.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:	$62.2833135766$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3732.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 13x + 19 $ x^3 - x^2 - 13*x + 19
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-3.77576$ of defining polynomial
Character	$\chi$	$=$	7800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.48059 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.48059 q^{7} +1.00000 q^{9} +5.77576 q^{11} +1.00000 q^{13} -3.00000 q^{17} +2.29517 q^{19} +1.48059 q^{21} +4.29517 q^{23} -1.00000 q^{27} +6.55152 q^{29} +5.77576 q^{31} -5.77576 q^{33} -3.25635 q^{37} -1.00000 q^{39} +5.25635 q^{41} -5.25635 q^{43} +5.03210 q^{47} -4.80786 q^{49} +3.00000 q^{51} +3.29517 q^{53} -2.29517 q^{57} -5.03210 q^{59} -4.55152 q^{61} -1.48059 q^{63} -4.22424 q^{67} -4.29517 q^{69} -12.8079 q^{71} +12.8079 q^{73} -8.55152 q^{77} -1.25635 q^{79} +1.00000 q^{81} +12.0709 q^{83} -6.55152 q^{87} +13.5515 q^{89} -1.48059 q^{91} -5.77576 q^{93} -6.96118 q^{97} +5.77576 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9	$ 3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - q^7 + 3 * q^9 + 5 * q^11 + 3 * q^13 - 9 * q^17 - 2 * q^19 + q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 5 * q^31 - 5 * q^33 + 6 * q^37 - 3 * q^39 - 13 * q^47 + 26 * q^49 + 9 * q^51 + q^53 + 2 * q^57 + 13 * q^59 + 11 * q^61 - q^63 - 25 * q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - q^77 + 12 * q^79 + 3 * q^81 + 15 * q^83 + 5 * q^87 + 16 * q^89 - q^91 - 5 * q^93 - 14 * q^97 + 5 * q^99

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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.48059	−0.559610	−0.279805	−	0.960057i	$-0.590270\pi$
$7$	−1.48059	−0.559610	−0.279805	+	0.960057i	$0.590270\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.77576	1.74146	0.870728	−	0.491765i	$-0.163648\pi$
$11$	5.77576	1.74146	0.870728	+	0.491765i	$0.163648\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	2.29517	0.526548	0.263274	−	0.964721i	$-0.415198\pi$
$19$	2.29517	0.526548	0.263274	+	0.964721i	$0.415198\pi$
$20$	0	0
$21$	1.48059	0.323091
$22$	0	0
$23$	4.29517	0.895605	0.447802	−	0.894133i	$-0.352207\pi$
$23$	4.29517	0.895605	0.447802	+	0.894133i	$0.352207\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.55152	1.21659	0.608293	−	0.793713i	$-0.291855\pi$
$29$	6.55152	1.21659	0.608293	+	0.793713i	$0.291855\pi$
$30$	0	0
$31$	5.77576	1.03736	0.518678	−	0.854969i	$-0.326424\pi$
$31$	5.77576	1.03736	0.518678	+	0.854969i	$0.326424\pi$
$32$	0	0
$33$	−5.77576	−1.00543
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.25635	−0.535340	−0.267670	−	0.963511i	$-0.586254\pi$
$37$	−3.25635	−0.535340	−0.267670	+	0.963511i	$0.586254\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	5.25635	0.820903	0.410452	−	0.911882i	$-0.365371\pi$
$41$	5.25635	0.820903	0.410452	+	0.911882i	$0.365371\pi$
$42$	0	0
$43$	−5.25635	−0.801585	−0.400793	−	0.916169i	$-0.631265\pi$
$43$	−5.25635	−0.801585	−0.400793	+	0.916169i	$0.631265\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.03210	0.734008	0.367004	−	0.930219i	$-0.380384\pi$
$47$	5.03210	0.734008	0.367004	+	0.930219i	$0.380384\pi$
$48$	0	0
$49$	−4.80786	−0.686837
$50$	0	0
$51$	3.00000	0.420084
$52$	0	0
$53$	3.29517	0.452626	0.226313	−	0.974055i	$-0.427333\pi$
$53$	3.29517	0.452626	0.226313	+	0.974055i	$0.427333\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.29517	−0.304003
$58$	0	0
$59$	−5.03210	−0.655124	−0.327562	−	0.944830i	$-0.606227\pi$
$59$	−5.03210	−0.655124	−0.327562	+	0.944830i	$0.606227\pi$
$60$	0	0
$61$	−4.55152	−0.582762	−0.291381	−	0.956607i	$-0.594115\pi$
$61$	−4.55152	−0.582762	−0.291381	+	0.956607i	$0.594115\pi$
$62$	0	0
$63$	−1.48059	−0.186537
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.22424	−0.516073	−0.258037	−	0.966135i	$-0.583076\pi$
$67$	−4.22424	−0.516073	−0.258037	+	0.966135i	$0.583076\pi$
$68$	0	0
$69$	−4.29517	−0.517078
$70$	0	0
$71$	−12.8079	−1.52001	−0.760007	−	0.649915i	$-0.774804\pi$
$71$	−12.8079	−1.52001	−0.760007	+	0.649915i	$0.774804\pi$
$72$	0	0
$73$	12.8079	1.49905	0.749523	−	0.661978i	$-0.230283\pi$
$73$	12.8079	1.49905	0.749523	+	0.661978i	$0.230283\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−8.55152	−0.974536
$78$	0	0
$79$	−1.25635	−0.141350	−0.0706749	−	0.997499i	$-0.522515\pi$
$79$	−1.25635	−0.141350	−0.0706749	+	0.997499i	$0.522515\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.0709	1.32496	0.662478	−	0.749081i	$-0.269505\pi$
$83$	12.0709	1.32496	0.662478	+	0.749081i	$0.269505\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−6.55152	−0.702396
$88$	0	0
$89$	13.5515	1.43646	0.718229	−	0.695807i	$-0.244953\pi$
$89$	13.5515	1.43646	0.718229	+	0.695807i	$0.244953\pi$
$90$	0	0
$91$	−1.48059	−0.155208
$92$	0	0
$93$	−5.77576	−0.598918
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.96118	−0.706800	−0.353400	−	0.935472i	$-0.614975\pi$
$97$	−6.96118	−0.706800	−0.353400	+	0.935472i	$0.614975\pi$
$98$	0	0
$99$	5.77576	0.580485
$100$	0	0
$101$	−2.33399	−0.232241	−0.116121	−	0.993235i	$-0.537046\pi$
$101$	−2.33399	−0.232241	−0.116121	+	0.993235i	$0.537046\pi$
$102$	0	0
$103$	−3.25635	−0.320857	−0.160429	−	0.987047i	$-0.551288\pi$
$103$	−3.25635	−0.320857	−0.160429	+	0.987047i	$0.551288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−20.1419	−1.94719	−0.973593	−	0.228289i	$-0.926687\pi$
$107$	−20.1419	−1.94719	−0.973593	+	0.228289i	$0.926687\pi$
$108$	0	0
$109$	18.8079	1.80147	0.900733	−	0.434373i	$-0.143030\pi$
$109$	18.8079	1.80147	0.900733	+	0.434373i	$0.143030\pi$
$110$	0	0
$111$	3.25635	0.309079
$112$	0	0
$113$	−7.55152	−0.710387	−0.355193	−	0.934793i	$-0.615585\pi$
$113$	−7.55152	−0.710387	−0.355193	+	0.934793i	$0.615585\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	4.44176	0.407176
$120$	0	0
$121$	22.3594	2.03267
$122$	0	0
$123$	−5.25635	−0.473949
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.80786	0.249157	0.124579	−	0.992210i	$-0.460242\pi$
$127$	2.80786	0.249157	0.124579	+	0.992210i	$0.460242\pi$
$128$	0	0
$129$	5.25635	0.462795
$130$	0	0
$131$	2.29517	0.200530	0.100265	−	0.994961i	$-0.468031\pi$
$131$	2.29517	0.200530	0.100265	+	0.994961i	$0.468031\pi$
$132$	0	0
$133$	−3.39820	−0.294661
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.1419	1.20822	0.604110	−	0.796901i	$-0.293529\pi$
$137$	14.1419	1.20822	0.604110	+	0.796901i	$0.293529\pi$
$138$	0	0
$139$	−7.92235	−0.671965	−0.335982	−	0.941868i	$-0.609068\pi$
$139$	−7.92235	−0.671965	−0.335982	+	0.941868i	$0.609068\pi$
$140$	0	0
$141$	−5.03210	−0.423779
$142$	0	0
$143$	5.77576	0.482993
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.80786	0.396546
$148$	0	0
$149$	−18.0642	−1.47988	−0.739939	−	0.672674i	$-0.765146\pi$
$149$	−18.0642	−1.47988	−0.739939	+	0.672674i	$0.765146\pi$
$150$	0	0
$151$	−23.0321	−1.87433	−0.937163	−	0.348892i	$-0.886558\pi$
$151$	−23.0321	−1.87433	−0.937163	+	0.348892i	$0.886558\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	0	0
$156$	0	0
$157$	3.29517	0.262983	0.131492	−	0.991317i	$-0.458023\pi$
$157$	3.29517	0.262983	0.131492	+	0.991317i	$0.458023\pi$
$158$	0	0
$159$	−3.29517	−0.261324
$160$	0	0
$161$	−6.35938	−0.501189
$162$	0	0
$163$	8.21752	0.643646	0.321823	−	0.946800i	$-0.395704\pi$
$163$	8.21752	0.643646	0.321823	+	0.946800i	$0.395704\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.743655	−0.0575457	−0.0287729	−	0.999586i	$-0.509160\pi$
$167$	−0.743655	−0.0575457	−0.0287729	+	0.999586i	$0.509160\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.29517	0.175516
$172$	0	0
$173$	−4.25635	−0.323604	−0.161802	−	0.986823i	$-0.551731\pi$
$173$	−4.25635	−0.323604	−0.161802	+	0.986823i	$0.551731\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.03210	0.378236
$178$	0	0
$179$	9.76904	0.730172	0.365086	−	0.930974i	$-0.381040\pi$
$179$	9.76904	0.730172	0.365086	+	0.930974i	$0.381040\pi$
$180$	0	0
$181$	18.8467	1.40086	0.700432	−	0.713720i	$-0.252991\pi$
$181$	18.8467	1.40086	0.700432	+	0.713720i	$0.252991\pi$
$182$	0	0
$183$	4.55152	0.336458
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−17.3273	−1.26710
$188$	0	0
$189$	1.48059	0.107697
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	18.2175	1.31133	0.655663	−	0.755054i	$-0.272389\pi$
$193$	18.2175	1.31133	0.655663	+	0.755054i	$0.272389\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	9.84669	0.701547	0.350774	−	0.936460i	$-0.385919\pi$
$197$	9.84669	0.701547	0.350774	+	0.936460i	$0.385919\pi$
$198$	0	0
$199$	−6.59034	−0.467177	−0.233588	−	0.972336i	$-0.575047\pi$
$199$	−6.59034	−0.467177	−0.233588	+	0.972336i	$0.575047\pi$
$200$	0	0
$201$	4.22424	0.297955
$202$	0	0
$203$	−9.70009	−0.680813
$204$	0	0
$205$	0	0
$206$	0	0
$207$	4.29517	0.298535
$208$	0	0
$209$	13.2563	0.916961
$210$	0	0
$211$	−2.80786	−0.193301	−0.0966505	−	0.995318i	$-0.530813\pi$
$211$	−2.80786	−0.193301	−0.0966505	+	0.995318i	$0.530813\pi$
$212$	0	0
$213$	12.8079	0.877580
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−8.55152	−0.580515
$218$	0	0
$219$	−12.8079	−0.865475
$220$	0	0
$221$	−3.00000	−0.201802
$222$	0	0
$223$	−10.8855	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$223$	−10.8855	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−18.7369	−1.24361	−0.621807	−	0.783171i	$-0.713601\pi$
$227$	−18.7369	−1.24361	−0.621807	+	0.783171i	$0.713601\pi$
$228$	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
$231$	8.55152	0.562648
$232$	0	0
$233$	−0.448485	−0.0293812	−0.0146906	−	0.999892i	$-0.504676\pi$
$233$	−0.448485	−0.0293812	−0.0146906	+	0.999892i	$0.504676\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.25635	0.0816084
$238$	0	0
$239$	24.9545	1.61417	0.807085	−	0.590436i	$-0.201044\pi$
$239$	24.9545	1.61417	0.807085	+	0.590436i	$0.201044\pi$
$240$	0	0
$241$	5.78248	0.372482	0.186241	−	0.982504i	$-0.440369\pi$
$241$	5.78248	0.372482	0.186241	+	0.982504i	$0.440369\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.29517	0.146038
$248$	0	0
$249$	−12.0709	−0.764964
$250$	0	0
$251$	9.33399	0.589157	0.294578	−	0.955627i	$-0.404821\pi$
$251$	9.33399	0.589157	0.294578	+	0.955627i	$0.404821\pi$
$252$	0	0
$253$	24.8079	1.55966
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.192140	0.0119853	0.00599267	−	0.999982i	$-0.498092\pi$
$257$	0.192140	0.0119853	0.00599267	+	0.999982i	$0.498092\pi$
$258$	0	0
$259$	4.82130	0.299581
$260$	0	0
$261$	6.55152	0.405529
$262$	0	0
$263$	7.25635	0.447445	0.223723	−	0.974653i	$-0.428179\pi$
$263$	7.25635	0.447445	0.223723	+	0.974653i	$0.428179\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−13.5515	−0.829339
$268$	0	0
$269$	−12.9224	−0.787890	−0.393945	−	0.919134i	$-0.628890\pi$
$269$	−12.9224	−0.787890	−0.393945	+	0.919134i	$0.628890\pi$
$270$	0	0
$271$	−23.6224	−1.43496	−0.717481	−	0.696578i	$-0.754705\pi$
$271$	−23.6224	−1.43496	−0.717481	+	0.696578i	$0.754705\pi$
$272$	0	0
$273$	1.48059	0.0896092
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6.96118	0.418256	0.209128	−	0.977888i	$-0.432937\pi$
$277$	6.96118	0.418256	0.209128	+	0.977888i	$0.432937\pi$
$278$	0	0
$279$	5.77576	0.345786
$280$	0	0
$281$	13.4097	0.799953	0.399977	−	0.916525i	$-0.369018\pi$
$281$	13.4097	0.799953	0.399977	+	0.916525i	$0.369018\pi$
$282$	0	0
$283$	9.40966	0.559346	0.279673	−	0.960095i	$-0.409774\pi$
$283$	9.40966	0.559346	0.279673	+	0.960095i	$0.409774\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.78248	−0.459385
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	6.96118	0.408071
$292$	0	0
$293$	17.1030	0.999170	0.499585	−	0.866265i	$-0.333486\pi$
$293$	17.1030	0.999170	0.499585	+	0.866265i	$0.333486\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.77576	−0.335143
$298$	0	0
$299$	4.29517	0.248396
$300$	0	0
$301$	7.78248	0.448575
$302$	0	0
$303$	2.33399	0.134084
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−18.3594	−1.04782	−0.523912	−	0.851772i	$-0.675528\pi$
$307$	−18.3594	−1.04782	−0.523912	+	0.851772i	$0.675528\pi$
$308$	0	0
$309$	3.25635	0.185247
$310$	0	0
$311$	27.6157	1.56594	0.782972	−	0.622057i	$-0.213703\pi$
$311$	27.6157	1.56594	0.782972	+	0.622057i	$0.213703\pi$
$312$	0	0
$313$	21.2952	1.20367	0.601837	−	0.798619i	$-0.294436\pi$
$313$	21.2952	1.20367	0.601837	+	0.798619i	$0.294436\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.372819	−0.0209396	−0.0104698	−	0.999945i	$-0.503333\pi$
$317$	−0.372819	−0.0209396	−0.0104698	+	0.999945i	$0.503333\pi$
$318$	0	0
$319$	37.8400	2.11863
$320$	0	0
$321$	20.1419	1.12421
$322$	0	0
$323$	−6.88551	−0.383120
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−18.8079	−1.04008
$328$	0	0
$329$	−7.45047	−0.410758
$330$	0	0
$331$	9.39820	0.516572	0.258286	−	0.966069i	$-0.416842\pi$
$331$	9.39820	0.516572	0.258286	+	0.966069i	$0.416842\pi$
$332$	0	0
$333$	−3.25635	−0.178447
$334$	0	0
$335$	0	0
$336$	0	0
$337$	5.58836	0.304417	0.152209	−	0.988348i	$-0.451361\pi$
$337$	5.58836	0.304417	0.152209	+	0.988348i	$0.451361\pi$
$338$	0	0
$339$	7.55152	0.410142
$340$	0	0
$341$	33.3594	1.80651
$342$	0	0
$343$	17.4826	0.943970
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−6.89697	−0.370249	−0.185124	−	0.982715i	$-0.559269\pi$
$347$	−6.89697	−0.370249	−0.185124	+	0.982715i	$0.559269\pi$
$348$	0	0
$349$	13.3340	0.713752	0.356876	−	0.934152i	$-0.383842\pi$
$349$	13.3340	0.713752	0.356876	+	0.934152i	$0.383842\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−36.3594	−1.93521	−0.967607	−	0.252461i	$-0.918760\pi$
$353$	−36.3594	−1.93521	−0.967607	+	0.252461i	$0.918760\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−4.44176	−0.235083
$358$	0	0
$359$	−3.03210	−0.160028	−0.0800141	−	0.996794i	$-0.525497\pi$
$359$	−3.03210	−0.160028	−0.0800141	+	0.996794i	$0.525497\pi$
$360$	0	0
$361$	−13.7322	−0.722747
$362$	0	0
$363$	−22.3594	−1.17356
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−21.1807	−1.10562	−0.552811	−	0.833307i	$-0.686445\pi$
$367$	−21.1807	−1.10562	−0.552811	+	0.833307i	$0.686445\pi$
$368$	0	0
$369$	5.25635	0.273634
$370$	0	0
$371$	−4.87879	−0.253294
$372$	0	0
$373$	22.1787	1.14837	0.574185	−	0.818726i	$-0.305319\pi$
$373$	22.1787	1.14837	0.574185	+	0.818726i	$0.305319\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.55152	0.337420
$378$	0	0
$379$	−17.8400	−0.916377	−0.458189	−	0.888855i	$-0.651502\pi$
$379$	−17.8400	−0.916377	−0.458189	+	0.888855i	$0.651502\pi$
$380$	0	0
$381$	−2.80786	−0.143851
$382$	0	0
$383$	20.2175	1.03307	0.516534	−	0.856267i	$-0.327222\pi$
$383$	20.2175	1.03307	0.516534	+	0.856267i	$0.327222\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−5.25635	−0.267195
$388$	0	0
$389$	−18.6545	−0.945823	−0.472911	−	0.881110i	$-0.656797\pi$
$389$	−18.6545	−0.945823	−0.472911	+	0.881110i	$0.656797\pi$
$390$	0	0
$391$	−12.8855	−0.651648
$392$	0	0
$393$	−2.29517	−0.115776
$394$	0	0
$395$	0	0
$396$	0	0
$397$	21.6292	1.08554	0.542768	−	0.839882i	$-0.317376\pi$
$397$	21.6292	1.08554	0.542768	+	0.839882i	$0.317376\pi$
$398$	0	0
$399$	3.39820	0.170123
$400$	0	0
$401$	33.0274	1.64931	0.824654	−	0.565638i	$-0.191370\pi$
$401$	33.0274	1.64931	0.824654	+	0.565638i	$0.191370\pi$
$402$	0	0
$403$	5.77576	0.287711
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−18.8079	−0.932271
$408$	0	0
$409$	31.0274	1.53420	0.767102	−	0.641525i	$-0.221698\pi$
$409$	31.0274	1.53420	0.767102	+	0.641525i	$0.221698\pi$
$410$	0	0
$411$	−14.1419	−0.697566
$412$	0	0
$413$	7.45047	0.366614
$414$	0	0
$415$	0	0
$416$	0	0
$417$	7.92235	0.387959
$418$	0	0
$419$	16.7437	0.817981	0.408991	−	0.912539i	$-0.365881\pi$
$419$	16.7437	0.817981	0.408991	+	0.912539i	$0.365881\pi$
$420$	0	0
$421$	23.6934	1.15474	0.577372	−	0.816481i	$-0.304078\pi$
$421$	23.6934	1.15474	0.577372	+	0.816481i	$0.304078\pi$
$422$	0	0
$423$	5.03210	0.244669
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.73892	0.326119
$428$	0	0
$429$	−5.77576	−0.278856
$430$	0	0
$431$	30.8855	1.48770	0.743851	−	0.668345i	$-0.232997\pi$
$431$	30.8855	1.48770	0.743851	+	0.668345i	$0.232997\pi$
$432$	0	0
$433$	20.1419	0.967956	0.483978	−	0.875080i	$-0.339192\pi$
$433$	20.1419	0.967956	0.483978	+	0.875080i	$0.339192\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.85814	0.471579
$438$	0	0
$439$	35.2429	1.68205	0.841026	−	0.540995i	$-0.181952\pi$
$439$	35.2429	1.68205	0.841026	+	0.540995i	$0.181952\pi$
$440$	0	0
$441$	−4.80786	−0.228946
$442$	0	0
$443$	13.8467	0.657876	0.328938	−	0.944352i	$-0.393309\pi$
$443$	13.8467	0.657876	0.328938	+	0.944352i	$0.393309\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	18.0642	0.854408
$448$	0	0
$449$	−24.4350	−1.15316	−0.576580	−	0.817040i	$-0.695613\pi$
$449$	−24.4350	−1.15316	−0.576580	+	0.817040i	$0.695613\pi$
$450$	0	0
$451$	30.3594	1.42957
$452$	0	0
$453$	23.0321	1.08214
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−14.4370	−0.675336	−0.337668	−	0.941265i	$-0.609638\pi$
$457$	−14.4370	−0.675336	−0.337668	+	0.941265i	$0.609638\pi$
$458$	0	0
$459$	3.00000	0.140028
$460$	0	0
$461$	39.6157	1.84509	0.922544	−	0.385892i	$-0.126106\pi$
$461$	39.6157	1.84509	0.922544	+	0.385892i	$0.126106\pi$
$462$	0	0
$463$	33.9176	1.57629	0.788143	−	0.615493i	$-0.211043\pi$
$463$	33.9176	1.57629	0.788143	+	0.615493i	$0.211043\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.4739	−1.27134	−0.635669	−	0.771961i	$-0.719276\pi$
$467$	−27.4739	−1.27134	−0.635669	+	0.771961i	$0.719276\pi$
$468$	0	0
$469$	6.25436	0.288800
$470$	0	0
$471$	−3.29517	−0.151833
$472$	0	0
$473$	−30.3594	−1.39593
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.29517	0.150875
$478$	0	0
$479$	1.26307	0.0577110	0.0288555	−	0.999584i	$-0.490814\pi$
$479$	1.26307	0.0577110	0.0288555	+	0.999584i	$0.490814\pi$
$480$	0	0
$481$	−3.25635	−0.148477
$482$	0	0
$483$	6.35938	0.289362
$484$	0	0
$485$	0	0
$486$	0	0
$487$	22.4303	1.01641	0.508207	−	0.861235i	$-0.330308\pi$
$487$	22.4303	1.01641	0.508207	+	0.861235i	$0.330308\pi$
$488$	0	0
$489$	−8.21752	−0.371609
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	−19.6545	−0.885196
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.9632	0.850614
$498$	0	0
$499$	−1.70009	−0.0761066	−0.0380533	−	0.999276i	$-0.512116\pi$
$499$	−1.70009	−0.0761066	−0.0380533	+	0.999276i	$0.512116\pi$
$500$	0	0
$501$	0.743655	0.0332240
$502$	0	0
$503$	−7.19214	−0.320682	−0.160341	−	0.987062i	$-0.551259\pi$
$503$	−7.19214	−0.320682	−0.160341	+	0.987062i	$0.551259\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−20.1399	−0.892684	−0.446342	−	0.894862i	$-0.647274\pi$
$509$	−20.1399	−0.892684	−0.446342	+	0.894862i	$0.647274\pi$
$510$	0	0
$511$	−18.9632	−0.838881
$512$	0	0
$513$	−2.29517	−0.101334
$514$	0	0
$515$	0	0
$516$	0	0
$517$	29.0642	1.27824
$518$	0	0
$519$	4.25635	0.186833
$520$	0	0
$521$	−29.1030	−1.27503	−0.637513	−	0.770439i	$-0.720037\pi$
$521$	−29.1030	−1.27503	−0.637513	+	0.770439i	$0.720037\pi$
$522$	0	0
$523$	36.1419	1.58037	0.790186	−	0.612866i	$-0.209984\pi$
$523$	36.1419	1.58037	0.790186	+	0.612866i	$0.209984\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−17.3273	−0.754788
$528$	0	0
$529$	−4.55152	−0.197892
$530$	0	0
$531$	−5.03210	−0.218375
$532$	0	0
$533$	5.25635	0.227678
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9.76904	−0.421565
$538$	0	0
$539$	−27.7690	−1.19610
$540$	0	0
$541$	8.80786	0.378679	0.189340	−	0.981912i	$-0.439365\pi$
$541$	8.80786	0.378679	0.189340	+	0.981912i	$0.439365\pi$
$542$	0	0
$543$	−18.8467	−0.808789
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2.07567	−0.0887490	−0.0443745	−	0.999015i	$-0.514129\pi$
$547$	−2.07567	−0.0887490	−0.0443745	+	0.999015i	$0.514129\pi$
$548$	0	0
$549$	−4.55152	−0.194254
$550$	0	0
$551$	15.0368	0.640591
$552$	0	0
$553$	1.86013	0.0791007
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.2175	0.602416	0.301208	−	0.953559i	$-0.402610\pi$
$557$	14.2175	0.602416	0.301208	+	0.953559i	$0.402610\pi$
$558$	0	0
$559$	−5.25635	−0.222320
$560$	0	0
$561$	17.3273	0.731558
$562$	0	0
$563$	−37.9109	−1.59775	−0.798877	−	0.601495i	$-0.794572\pi$
$563$	−37.9109	−1.59775	−0.798877	+	0.601495i	$0.794572\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1.48059	−0.0621788
$568$	0	0
$569$	−34.2563	−1.43610	−0.718050	−	0.695991i	$-0.754965\pi$
$569$	−34.2563	−1.43610	−0.718050	+	0.695991i	$0.754965\pi$
$570$	0	0
$571$	−6.37282	−0.266694	−0.133347	−	0.991069i	$-0.542573\pi$
$571$	−6.37282	−0.266694	−0.133347	+	0.991069i	$0.542573\pi$
$572$	0	0
$573$	−6.00000	−0.250654
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.1672	1.71381	0.856907	−	0.515471i	$-0.172383\pi$
$577$	41.1672	1.71381	0.856907	+	0.515471i	$0.172383\pi$
$578$	0	0
$579$	−18.2175	−0.757094
$580$	0	0
$581$	−17.8721	−0.741458
$582$	0	0
$583$	19.0321	0.788229
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.6593	−1.01780	−0.508899	−	0.860826i	$-0.669947\pi$
$587$	−24.6593	−1.01780	−0.508899	+	0.860826i	$0.669947\pi$
$588$	0	0
$589$	13.2563	0.546218
$590$	0	0
$591$	−9.84669	−0.405039
$592$	0	0
$593$	−22.4350	−0.921297	−0.460648	−	0.887583i	$-0.652383\pi$
$593$	−22.4350	−0.921297	−0.460648	+	0.887583i	$0.652383\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.59034	0.269725
$598$	0	0
$599$	21.9885	0.898427	0.449214	−	0.893424i	$-0.351704\pi$
$599$	21.9885	0.898427	0.449214	+	0.893424i	$0.351704\pi$
$600$	0	0
$601$	−10.3982	−0.424151	−0.212076	−	0.977253i	$-0.568022\pi$
$601$	−10.3982	−0.424151	−0.212076	+	0.977253i	$0.568022\pi$
$602$	0	0
$603$	−4.22424	−0.172024
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−9.47387	−0.384532	−0.192266	−	0.981343i	$-0.561584\pi$
$607$	−9.47387	−0.384532	−0.192266	+	0.981343i	$0.561584\pi$
$608$	0	0
$609$	9.70009	0.393068
$610$	0	0
$611$	5.03210	0.203577
$612$	0	0
$613$	−5.10303	−0.206109	−0.103055	−	0.994676i	$-0.532862\pi$
$613$	−5.10303	−0.206109	−0.103055	+	0.994676i	$0.532862\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.4739	−1.66967	−0.834837	−	0.550496i	$-0.814438\pi$
$617$	−41.4739	−1.66967	−0.834837	+	0.550496i	$0.814438\pi$
$618$	0	0
$619$	28.8721	1.16047	0.580233	−	0.814450i	$-0.302961\pi$
$619$	28.8721	1.16047	0.580233	+	0.814450i	$0.302961\pi$
$620$	0	0
$621$	−4.29517	−0.172359
$622$	0	0
$623$	−20.0642	−0.803855
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13.2563	−0.529407
$628$	0	0
$629$	9.76904	0.389517
$630$	0	0
$631$	−18.6660	−0.743082	−0.371541	−	0.928417i	$-0.621170\pi$
$631$	−18.6660	−0.743082	−0.371541	+	0.928417i	$0.621170\pi$
$632$	0	0
$633$	2.80786	0.111602
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.80786	−0.190494
$638$	0	0
$639$	−12.8079	−0.506671
$640$	0	0
$641$	26.0254	1.02794	0.513970	−	0.857808i	$-0.328174\pi$
$641$	26.0254	1.02794	0.513970	+	0.857808i	$0.328174\pi$
$642$	0	0
$643$	−11.3206	−0.446439	−0.223219	−	0.974768i	$-0.571657\pi$
$643$	−11.3206	−0.446439	−0.223219	+	0.974768i	$0.571657\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.7322	−0.421926	−0.210963	−	0.977494i	$-0.567660\pi$
$647$	−10.7322	−0.421926	−0.210963	+	0.977494i	$0.567660\pi$
$648$	0	0
$649$	−29.0642	−1.14087
$650$	0	0
$651$	8.55152	0.335160
$652$	0	0
$653$	9.00000	0.352197	0.176099	−	0.984373i	$-0.443652\pi$
$653$	9.00000	0.352197	0.176099	+	0.984373i	$0.443652\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.8079	0.499682
$658$	0	0
$659$	−25.6914	−1.00079	−0.500397	−	0.865796i	$-0.666813\pi$
$659$	−25.6914	−1.00079	−0.500397	+	0.865796i	$0.666813\pi$
$660$	0	0
$661$	12.6660	0.492651	0.246325	−	0.969187i	$-0.420777\pi$
$661$	12.6660	0.492651	0.246325	+	0.969187i	$0.420777\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	28.1399	1.08958
$668$	0	0
$669$	10.8855	0.420858
$670$	0	0
$671$	−26.2884	−1.01485
$672$	0	0
$673$	−27.2817	−1.05163	−0.525817	−	0.850598i	$-0.676240\pi$
$673$	−27.2817	−1.05163	−0.525817	+	0.850598i	$0.676240\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25.4097	0.976573	0.488286	−	0.872684i	$-0.337622\pi$
$677$	25.4097	0.976573	0.488286	+	0.872684i	$0.337622\pi$
$678$	0	0
$679$	10.3066	0.395532
$680$	0	0
$681$	18.7369	0.718001
$682$	0	0
$683$	9.55625	0.365660	0.182830	−	0.983145i	$-0.441474\pi$
$683$	9.55625	0.365660	0.182830	+	0.983145i	$0.441474\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.00000	−0.0763048
$688$	0	0
$689$	3.29517	0.125536
$690$	0	0
$691$	26.0575	0.991273	0.495637	−	0.868530i	$-0.334935\pi$
$691$	26.0575	0.991273	0.495637	+	0.868530i	$0.334935\pi$
$692$	0	0
$693$	−8.55152	−0.324845
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−15.7690	−0.597295
$698$	0	0
$699$	0.448485	0.0169633
$700$	0	0
$701$	−49.4992	−1.86956	−0.934780	−	0.355226i	$-0.884404\pi$
$701$	−49.4992	−1.86956	−0.934780	+	0.355226i	$0.884404\pi$
$702$	0	0
$703$	−7.47387	−0.281882
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.45568	0.129964
$708$	0	0
$709$	7.33399	0.275434	0.137717	−	0.990472i	$-0.456024\pi$
$709$	7.33399	0.275434	0.137717	+	0.990472i	$0.456024\pi$
$710$	0	0
$711$	−1.25635	−0.0471166
$712$	0	0
$713$	24.8079	0.929062
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24.9545	−0.931941
$718$	0	0
$719$	−30.9497	−1.15423	−0.577115	−	0.816663i	$-0.695821\pi$
$719$	−30.9497	−1.15423	−0.577115	+	0.816663i	$0.695821\pi$
$720$	0	0
$721$	4.82130	0.179555
$722$	0	0
$723$	−5.78248	−0.215053
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.6660	0.988987	0.494494	−	0.869181i	$-0.335354\pi$
$727$	26.6660	0.988987	0.494494	+	0.869181i	$0.335354\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	15.7690	0.583239
$732$	0	0
$733$	27.6934	1.02288	0.511439	−	0.859320i	$-0.329113\pi$
$733$	27.6934	1.02288	0.511439	+	0.859320i	$0.329113\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−24.3982	−0.898719
$738$	0	0
$739$	−1.91761	−0.0705405	−0.0352703	−	0.999378i	$-0.511229\pi$
$739$	−1.91761	−0.0705405	−0.0352703	+	0.999378i	$0.511229\pi$
$740$	0	0
$741$	−2.29517	−0.0843152
$742$	0	0
$743$	−50.2884	−1.84490	−0.922452	−	0.386112i	$-0.873818\pi$
$743$	−50.2884	−1.84490	−0.922452	+	0.386112i	$0.873818\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.0709	0.441652
$748$	0	0
$749$	29.8218	1.08966
$750$	0	0
$751$	13.4758	0.491741	0.245870	−	0.969303i	$-0.420926\pi$
$751$	13.4758	0.491741	0.245870	+	0.969303i	$0.420926\pi$
$752$	0	0
$753$	−9.33399	−0.340150
$754$	0	0
$755$	0	0
$756$	0	0
$757$	24.1030	0.876040	0.438020	−	0.898965i	$-0.355680\pi$
$757$	24.1030	0.876040	0.438020	+	0.898965i	$0.355680\pi$
$758$	0	0
$759$	−24.8079	−0.900468
$760$	0	0
$761$	28.2175	1.02288	0.511442	−	0.859318i	$-0.329111\pi$
$761$	28.2175	1.02288	0.511442	+	0.859318i	$0.329111\pi$
$762$	0	0
$763$	−27.8467	−1.00812
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.03210	−0.181699
$768$	0	0
$769$	−32.9612	−1.18861	−0.594305	−	0.804240i	$-0.702573\pi$
$769$	−32.9612	−1.18861	−0.594305	+	0.804240i	$0.702573\pi$
$770$	0	0
$771$	−0.192140	−0.00691974
$772$	0	0
$773$	8.88353	0.319518	0.159759	−	0.987156i	$-0.448928\pi$
$773$	8.88353	0.319518	0.159759	+	0.987156i	$0.448928\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.82130	−0.172963
$778$	0	0
$779$	12.0642	0.432245
$780$	0	0
$781$	−73.9751	−2.64704
$782$	0	0
$783$	−6.55152	−0.234132
$784$	0	0
$785$	0	0
$786$	0	0
$787$	12.8788	0.459079	0.229540	−	0.973299i	$-0.426278\pi$
$787$	12.8788	0.459079	0.229540	+	0.973299i	$0.426278\pi$
$788$	0	0
$789$	−7.25635	−0.258333
$790$	0	0
$791$	11.1807	0.397539
$792$	0	0
$793$	−4.55152	−0.161629
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.78050	0.0984902	0.0492451	−	0.998787i	$-0.484318\pi$
$797$	2.78050	0.0984902	0.0492451	+	0.998787i	$0.484318\pi$
$798$	0	0
$799$	−15.0963	−0.534069
$800$	0	0
$801$	13.5515	0.478819
$802$	0	0
$803$	73.9751	2.61052
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.9224	0.454888
$808$	0	0
$809$	8.14186	0.286252	0.143126	−	0.989704i	$-0.454285\pi$
$809$	8.14186	0.286252	0.143126	+	0.989704i	$0.454285\pi$
$810$	0	0
$811$	−18.7255	−0.657540	−0.328770	−	0.944410i	$-0.606634\pi$
$811$	−18.7255	−0.657540	−0.328770	+	0.944410i	$0.606634\pi$
$812$	0	0
$813$	23.6224	0.828475
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−12.0642	−0.422073
$818$	0	0
$819$	−1.48059	−0.0517359
$820$	0	0
$821$	−23.5401	−0.821554	−0.410777	−	0.911736i	$-0.634742\pi$
$821$	−23.5401	−0.821554	−0.410777	+	0.911736i	$0.634742\pi$
$822$	0	0
$823$	−51.7576	−1.80416	−0.902078	−	0.431573i	$-0.857959\pi$
$823$	−51.7576	−1.80416	−0.902078	+	0.431573i	$0.857959\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.5333	1.30516	0.652581	−	0.757719i	$-0.273686\pi$
$827$	37.5333	1.30516	0.652581	+	0.757719i	$0.273686\pi$
$828$	0	0
$829$	37.8835	1.31575	0.657875	−	0.753127i	$-0.271456\pi$
$829$	37.8835	1.31575	0.657875	+	0.753127i	$0.271456\pi$
$830$	0	0
$831$	−6.96118	−0.241480
$832$	0	0
$833$	14.4236	0.499747
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−5.77576	−0.199639
$838$	0	0
$839$	−0.0891094	−0.00307640	−0.00153820	−	0.999999i	$-0.500490\pi$
$839$	−0.0891094	−0.00307640	−0.00153820	+	0.999999i	$0.500490\pi$
$840$	0	0
$841$	13.9224	0.480081
$842$	0	0
$843$	−13.4097	−0.461853
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−33.1050	−1.13750
$848$	0	0
$849$	−9.40966	−0.322939
$850$	0	0
$851$	−13.9866	−0.479453
$852$	0	0
$853$	19.2583	0.659393	0.329696	−	0.944087i	$-0.393054\pi$
$853$	19.2583	0.659393	0.329696	+	0.944087i	$0.393054\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.84470	−0.0630138	−0.0315069	−	0.999504i	$-0.510031\pi$
$857$	−1.84470	−0.0630138	−0.0315069	+	0.999504i	$0.510031\pi$
$858$	0	0
$859$	49.3848	1.68499	0.842493	−	0.538707i	$-0.181087\pi$
$859$	49.3848	1.68499	0.842493	+	0.538707i	$0.181087\pi$
$860$	0	0
$861$	7.78248	0.265226
$862$	0	0
$863$	−29.1720	−0.993026	−0.496513	−	0.868029i	$-0.665386\pi$
$863$	−29.1720	−0.993026	−0.496513	+	0.868029i	$0.665386\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8.00000	0.271694
$868$	0	0
$869$	−7.25635	−0.246155
$870$	0	0
$871$	−4.22424	−0.143133
$872$	0	0
$873$	−6.96118	−0.235600
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.1419	−0.747677	−0.373839	−	0.927494i	$-0.621959\pi$
$877$	−22.1419	−0.747677	−0.373839	+	0.927494i	$0.621959\pi$
$878$	0	0
$879$	−17.1030	−0.576871
$880$	0	0
$881$	−16.1145	−0.542911	−0.271455	−	0.962451i	$-0.587505\pi$
$881$	−16.1145	−0.542911	−0.271455	+	0.962451i	$0.587505\pi$
$882$	0	0
$883$	59.1672	1.99114	0.995568	−	0.0940444i	$-0.0299796\pi$
$883$	59.1672	1.99114	0.995568	+	0.0940444i	$0.0299796\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	34.5654	1.16059	0.580297	−	0.814405i	$-0.302937\pi$
$887$	34.5654	1.16059	0.580297	+	0.814405i	$0.302937\pi$
$888$	0	0
$889$	−4.15728	−0.139431
$890$	0	0
$891$	5.77576	0.193495
$892$	0	0
$893$	11.5495	0.386490
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−4.29517	−0.143412
$898$	0	0
$899$	37.8400	1.26203
$900$	0	0
$901$	−9.88551	−0.329334
$902$	0	0
$903$	−7.78248	−0.258985
$904$	0	0
$905$	0	0
$906$	0	0
$907$	4.74167	0.157445	0.0787223	−	0.996897i	$-0.474916\pi$
$907$	4.74167	0.157445	0.0787223	+	0.996897i	$0.474916\pi$
$908$	0	0
$909$	−2.33399	−0.0774137
$910$	0	0
$911$	−0.450469	−0.0149247	−0.00746235	−	0.999972i	$-0.502375\pi$
$911$	−0.450469	−0.0149247	−0.00746235	+	0.999972i	$0.502375\pi$
$912$	0	0
$913$	69.7188	2.30735
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−3.39820	−0.112218
$918$	0	0
$919$	−19.5515	−0.644945	−0.322472	−	0.946579i	$-0.604514\pi$
$919$	−19.5515	−0.644945	−0.322472	+	0.946579i	$0.604514\pi$
$920$	0	0
$921$	18.3594	0.604962
$922$	0	0
$923$	−12.8079	−0.421576
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−3.25635	−0.106952
$928$	0	0
$929$	−25.6934	−0.842972	−0.421486	−	0.906835i	$-0.638491\pi$
$929$	−25.6934	−0.842972	−0.421486	+	0.906835i	$0.638491\pi$
$930$	0	0
$931$	−11.0349	−0.361653
$932$	0	0
$933$	−27.6157	−0.904098
$934$	0	0
$935$	0	0
$936$	0	0
$937$	35.6545	1.16478	0.582392	−	0.812908i	$-0.302117\pi$
$937$	35.6545	1.16478	0.582392	+	0.812908i	$0.302117\pi$
$938$	0	0
$939$	−21.2952	−0.694942
$940$	0	0
$941$	−5.26979	−0.171790	−0.0858951	−	0.996304i	$-0.527375\pi$
$941$	−5.26979	−0.171790	−0.0858951	+	0.996304i	$0.527375\pi$
$942$	0	0
$943$	22.5769	0.735205
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−39.9042	−1.29671	−0.648356	−	0.761338i	$-0.724543\pi$
$947$	−39.9042	−1.29671	−0.648356	+	0.761338i	$0.724543\pi$
$948$	0	0
$949$	12.8079	0.415761
$950$	0	0
$951$	0.372819	0.0120895
$952$	0	0
$953$	−30.2563	−0.980099	−0.490050	−	0.871695i	$-0.663021\pi$
$953$	−30.2563	−0.980099	−0.490050	+	0.871695i	$0.663021\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−37.8400	−1.22319
$958$	0	0
$959$	−20.9383	−0.676132
$960$	0	0
$961$	2.35938	0.0761089
$962$	0	0
$963$	−20.1419	−0.649062
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−29.3915	−0.945166	−0.472583	−	0.881286i	$-0.656678\pi$
$967$	−29.3915	−0.945166	−0.472583	+	0.881286i	$0.656678\pi$
$968$	0	0
$969$	6.88551	0.221194
$970$	0	0
$971$	−13.4739	−0.432397	−0.216198	−	0.976349i	$-0.569366\pi$
$971$	−13.4739	−0.432397	−0.216198	+	0.976349i	$0.569366\pi$
$972$	0	0
$973$	11.7297	0.376038
$974$	0	0
$975$	0	0
$976$	0	0
$977$	31.6934	1.01396	0.506980	−	0.861958i	$-0.330762\pi$
$977$	31.6934	1.01396	0.506980	+	0.861958i	$0.330762\pi$
$978$	0	0
$979$	78.2703	2.50153
$980$	0	0
$981$	18.8079	0.600489
$982$	0	0
$983$	−13.9311	−0.444332	−0.222166	−	0.975009i	$-0.571313\pi$
$983$	−13.9311	−0.444332	−0.222166	+	0.975009i	$0.571313\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	7.45047	0.237151
$988$	0	0
$989$	−22.5769	−0.717904
$990$	0	0
$991$	50.6296	1.60830	0.804152	−	0.594424i	$-0.202620\pi$
$991$	50.6296	1.60830	0.804152	+	0.594424i	$0.202620\pi$
$992$	0	0
$993$	−9.39820	−0.298243
$994$	0	0
$995$	0	0
$996$	0	0
$997$	38.4002	1.21615	0.608073	−	0.793881i	$-0.291943\pi$
$997$	38.4002	1.21615	0.608073	+	0.793881i	$0.291943\pi$
$998$	0	0
$999$	3.25635	0.103026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7800.2.a.bh.1.2	✓	3
5.4	even	2		7800.2.a.bs.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7800.2.a.bh.1.2	✓	3	1.1	even	1	trivial
7800.2.a.bs.1.2	yes	3	5.4	even	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:	\( N \)	\(=\)	\( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7800.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.48059 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.48059 q^{7} +1.00000 q^{9} +5.77576 q^{11} +1.00000 q^{13} -3.00000 q^{17} +2.29517 q^{19} +1.48059 q^{21} +4.29517 q^{23} -1.00000 q^{27} +6.55152 q^{29} +5.77576 q^{31} -5.77576 q^{33} -3.25635 q^{37} -1.00000 q^{39} +5.25635 q^{41} -5.25635 q^{43} +5.03210 q^{47} -4.80786 q^{49} +3.00000 q^{51} +3.29517 q^{53} -2.29517 q^{57} -5.03210 q^{59} -4.55152 q^{61} -1.48059 q^{63} -4.22424 q^{67} -4.29517 q^{69} -12.8079 q^{71} +12.8079 q^{73} -8.55152 q^{77} -1.25635 q^{79} +1.00000 q^{81} +12.0709 q^{83} -6.55152 q^{87} +13.5515 q^{89} -1.48059 q^{91} -5.77576 q^{93} -6.96118 q^{97} +5.77576 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9	\( 3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - q^7 + 3 * q^9 + 5 * q^11 + 3 * q^13 - 9 * q^17 - 2 * q^19 + q^21 + 4 * q^23 - 3 * q^27 - 5 * q^29 + 5 * q^31 - 5 * q^33 + 6 * q^37 - 3 * q^39 - 13 * q^47 + 26 * q^49 + 9 * q^51 + q^53 + 2 * q^57 + 13 * q^59 + 11 * q^61 - q^63 - 25 * q^67 - 4 * q^69 + 2 * q^71 - 2 * q^73 - q^77 + 12 * q^79 + 3 * q^81 + 15 * q^83 + 5 * q^87 + 16 * q^89 - q^91 - 5 * q^93 - 14 * q^97 + 5 * q^99

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