LMFDB - Embedded newform 336.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(1,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.8246417619$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

f(q)

=

q-3.00000 q^{3} -2.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} -52.0000 q^{11} +86.0000 q^{13} +6.00000 q^{15} -30.0000 q^{17} +4.00000 q^{19} -21.0000 q^{21} -120.000 q^{23} -121.000 q^{25} -27.0000 q^{27} +246.000 q^{29} -80.0000 q^{31} +156.000 q^{33} -14.0000 q^{35} -290.000 q^{37} -258.000 q^{39} -374.000 q^{41} -164.000 q^{43} -18.0000 q^{45} -464.000 q^{47} +49.0000 q^{49} +90.0000 q^{51} -162.000 q^{53} +104.000 q^{55} -12.0000 q^{57} -180.000 q^{59} -666.000 q^{61} +63.0000 q^{63} -172.000 q^{65} +628.000 q^{67} +360.000 q^{69} -296.000 q^{71} -518.000 q^{73} +363.000 q^{75} -364.000 q^{77} +1184.00 q^{79} +81.0000 q^{81} -220.000 q^{83} +60.0000 q^{85} -738.000 q^{87} -774.000 q^{89} +602.000 q^{91} +240.000 q^{93} -8.00000 q^{95} -1086.00 q^{97} -468.000 q^{99} +O(q^{100})

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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−2.00000	−0.178885	−0.0894427	−	0.995992i	$-0.528509\pi$
$5$	−2.00000	−0.178885	−0.0894427	+	0.995992i	$0.528509\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−52.0000	−1.42533	−0.712663	−	0.701506i	$-0.752511\pi$
$11$	−52.0000	−1.42533	−0.712663	+	0.701506i	$0.752511\pi$
$12$	0	0
$13$	86.0000	1.83478	0.917389	−	0.397992i	$-0.130293\pi$
$13$	86.0000	1.83478	0.917389	+	0.397992i	$0.130293\pi$
$14$	0	0
$15$	6.00000	0.103280
$16$	0	0
$17$	−30.0000	−0.428004	−0.214002	−	0.976833i	$-0.568650\pi$
$17$	−30.0000	−0.428004	−0.214002	+	0.976833i	$0.568650\pi$
$18$	0	0
$19$	4.00000	0.0482980	0.0241490	−	0.999708i	$-0.492312\pi$
$19$	4.00000	0.0482980	0.0241490	+	0.999708i	$0.492312\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−120.000	−1.08790	−0.543951	−	0.839117i	$-0.683072\pi$
$23$	−120.000	−1.08790	−0.543951	+	0.839117i	$0.683072\pi$
$24$	0	0
$25$	−121.000	−0.968000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	246.000	1.57521	0.787604	−	0.616181i	$-0.211321\pi$
$29$	246.000	1.57521	0.787604	+	0.616181i	$0.211321\pi$
$30$	0	0
$31$	−80.0000	−0.463498	−0.231749	−	0.972776i	$-0.574445\pi$
$31$	−80.0000	−0.463498	−0.231749	+	0.972776i	$0.574445\pi$
$32$	0	0
$33$	156.000	0.822913
$34$	0	0
$35$	−14.0000	−0.0676123
$36$	0	0
$37$	−290.000	−1.28853	−0.644266	−	0.764801i	$-0.722837\pi$
$37$	−290.000	−1.28853	−0.644266	+	0.764801i	$0.722837\pi$
$38$	0	0
$39$	−258.000	−1.05931
$40$	0	0
$41$	−374.000	−1.42461	−0.712305	−	0.701870i	$-0.752349\pi$
$41$	−374.000	−1.42461	−0.712305	+	0.701870i	$0.752349\pi$
$42$	0	0
$43$	−164.000	−0.581622	−0.290811	−	0.956780i	$-0.593925\pi$
$43$	−164.000	−0.581622	−0.290811	+	0.956780i	$0.593925\pi$
$44$	0	0
$45$	−18.0000	−0.0596285
$46$	0	0
$47$	−464.000	−1.44003	−0.720014	−	0.693959i	$-0.755865\pi$
$47$	−464.000	−1.44003	−0.720014	+	0.693959i	$0.755865\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	90.0000	0.247108
$52$	0	0
$53$	−162.000	−0.419857	−0.209928	−	0.977717i	$-0.567323\pi$
$53$	−162.000	−0.419857	−0.209928	+	0.977717i	$0.567323\pi$
$54$	0	0
$55$	104.000	0.254970
$56$	0	0
$57$	−12.0000	−0.0278849
$58$	0	0
$59$	−180.000	−0.397187	−0.198593	−	0.980082i	$-0.563637\pi$
$59$	−180.000	−0.397187	−0.198593	+	0.980082i	$0.563637\pi$
$60$	0	0
$61$	−666.000	−1.39791	−0.698955	−	0.715165i	$-0.746351\pi$
$61$	−666.000	−1.39791	−0.698955	+	0.715165i	$0.746351\pi$
$62$	0	0
$63$	63.0000	0.125988
$64$	0	0
$65$	−172.000	−0.328215
$66$	0	0
$67$	628.000	1.14511	0.572555	−	0.819866i	$-0.305952\pi$
$67$	628.000	1.14511	0.572555	+	0.819866i	$0.305952\pi$
$68$	0	0
$69$	360.000	0.628100
$70$	0	0
$71$	−296.000	−0.494771	−0.247385	−	0.968917i	$-0.579571\pi$
$71$	−296.000	−0.494771	−0.247385	+	0.968917i	$0.579571\pi$
$72$	0	0
$73$	−518.000	−0.830511	−0.415256	−	0.909705i	$-0.636308\pi$
$73$	−518.000	−0.830511	−0.415256	+	0.909705i	$0.636308\pi$
$74$	0	0
$75$	363.000	0.558875
$76$	0	0
$77$	−364.000	−0.538723
$78$	0	0
$79$	1184.00	1.68621	0.843104	−	0.537751i	$-0.180726\pi$
$79$	1184.00	1.68621	0.843104	+	0.537751i	$0.180726\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−220.000	−0.290941	−0.145471	−	0.989363i	$-0.546470\pi$
$83$	−220.000	−0.290941	−0.145471	+	0.989363i	$0.546470\pi$
$84$	0	0
$85$	60.0000	0.0765637
$86$	0	0
$87$	−738.000	−0.909447
$88$	0	0
$89$	−774.000	−0.921841	−0.460920	−	0.887441i	$-0.652481\pi$
$89$	−774.000	−0.921841	−0.460920	+	0.887441i	$0.652481\pi$
$90$	0	0
$91$	602.000	0.693481
$92$	0	0
$93$	240.000	0.267600
$94$	0	0
$95$	−8.00000	−0.00863982
$96$	0	0
$97$	−1086.00	−1.13677	−0.568385	−	0.822763i	$-0.692431\pi$
$97$	−1086.00	−1.13677	−0.568385	+	0.822763i	$0.692431\pi$
$98$	0	0
$99$	−468.000	−0.475109
$100$	0	0
$101$	−290.000	−0.285704	−0.142852	−	0.989744i	$-0.545627\pi$
$101$	−290.000	−0.285704	−0.142852	+	0.989744i	$0.545627\pi$
$102$	0	0
$103$	88.0000	0.0841835	0.0420917	−	0.999114i	$-0.486598\pi$
$103$	88.0000	0.0841835	0.0420917	+	0.999114i	$0.486598\pi$
$104$	0	0
$105$	42.0000	0.0390360
$106$	0	0
$107$	−372.000	−0.336099	−0.168050	−	0.985779i	$-0.553747\pi$
$107$	−372.000	−0.336099	−0.168050	+	0.985779i	$0.553747\pi$
$108$	0	0
$109$	1430.00	1.25660	0.628299	−	0.777972i	$-0.283752\pi$
$109$	1430.00	1.25660	0.628299	+	0.777972i	$0.283752\pi$
$110$	0	0
$111$	870.000	0.743935
$112$	0	0
$113$	1810.00	1.50682	0.753409	−	0.657552i	$-0.228408\pi$
$113$	1810.00	1.50682	0.753409	+	0.657552i	$0.228408\pi$
$114$	0	0
$115$	240.000	0.194610
$116$	0	0
$117$	774.000	0.611593
$118$	0	0
$119$	−210.000	−0.161770
$120$	0	0
$121$	1373.00	1.03156
$122$	0	0
$123$	1122.00	0.822499
$124$	0	0
$125$	492.000	0.352047
$126$	0	0
$127$	−1168.00	−0.816089	−0.408044	−	0.912962i	$-0.633789\pi$
$127$	−1168.00	−0.816089	−0.408044	+	0.912962i	$0.633789\pi$
$128$	0	0
$129$	492.000	0.335800
$130$	0	0
$131$	1268.00	0.845692	0.422846	−	0.906202i	$-0.361031\pi$
$131$	1268.00	0.845692	0.422846	+	0.906202i	$0.361031\pi$
$132$	0	0
$133$	28.0000	0.0182549
$134$	0	0
$135$	54.0000	0.0344265
$136$	0	0
$137$	474.000	0.295595	0.147798	−	0.989018i	$-0.452782\pi$
$137$	474.000	0.295595	0.147798	+	0.989018i	$0.452782\pi$
$138$	0	0
$139$	2684.00	1.63780	0.818899	−	0.573938i	$-0.194585\pi$
$139$	2684.00	1.63780	0.818899	+	0.573938i	$0.194585\pi$
$140$	0	0
$141$	1392.00	0.831401
$142$	0	0
$143$	−4472.00	−2.61516
$144$	0	0
$145$	−492.000	−0.281782
$146$	0	0
$147$	−147.000	−0.0824786
$148$	0	0
$149$	−1314.00	−0.722464	−0.361232	−	0.932476i	$-0.617644\pi$
$149$	−1314.00	−0.722464	−0.361232	+	0.932476i	$0.617644\pi$
$150$	0	0
$151$	−3000.00	−1.61680	−0.808399	−	0.588635i	$-0.799666\pi$
$151$	−3000.00	−1.61680	−0.808399	+	0.588635i	$0.799666\pi$
$152$	0	0
$153$	−270.000	−0.142668
$154$	0	0
$155$	160.000	0.0829130
$156$	0	0
$157$	774.000	0.393452	0.196726	−	0.980459i	$-0.436969\pi$
$157$	774.000	0.393452	0.196726	+	0.980459i	$0.436969\pi$
$158$	0	0
$159$	486.000	0.242404
$160$	0	0
$161$	−840.000	−0.411188
$162$	0	0
$163$	2292.00	1.10137	0.550685	−	0.834713i	$-0.314367\pi$
$163$	2292.00	1.10137	0.550685	+	0.834713i	$0.314367\pi$
$164$	0	0
$165$	−312.000	−0.147207
$166$	0	0
$167$	1672.00	0.774750	0.387375	−	0.921922i	$-0.373382\pi$
$167$	1672.00	0.774750	0.387375	+	0.921922i	$0.373382\pi$
$168$	0	0
$169$	5199.00	2.36641
$170$	0	0
$171$	36.0000	0.0160993
$172$	0	0
$173$	−2730.00	−1.19976	−0.599879	−	0.800091i	$-0.704785\pi$
$173$	−2730.00	−1.19976	−0.599879	+	0.800091i	$0.704785\pi$
$174$	0	0
$175$	−847.000	−0.365870
$176$	0	0
$177$	540.000	0.229316
$178$	0	0
$179$	−2572.00	−1.07397	−0.536984	−	0.843592i	$-0.680436\pi$
$179$	−2572.00	−1.07397	−0.536984	+	0.843592i	$0.680436\pi$
$180$	0	0
$181$	3214.00	1.31986	0.659930	−	0.751327i	$-0.270586\pi$
$181$	3214.00	1.31986	0.659930	+	0.751327i	$0.270586\pi$
$182$	0	0
$183$	1998.00	0.807084
$184$	0	0
$185$	580.000	0.230500
$186$	0	0
$187$	1560.00	0.610045
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	1520.00	0.575829	0.287915	−	0.957656i	$-0.407038\pi$
$191$	1520.00	0.575829	0.287915	+	0.957656i	$0.407038\pi$
$192$	0	0
$193$	−62.0000	−0.0231236	−0.0115618	−	0.999933i	$-0.503680\pi$
$193$	−62.0000	−0.0231236	−0.0115618	+	0.999933i	$0.503680\pi$
$194$	0	0
$195$	516.000	0.189495
$196$	0	0
$197$	−1074.00	−0.388423	−0.194212	−	0.980960i	$-0.562215\pi$
$197$	−1074.00	−0.388423	−0.194212	+	0.980960i	$0.562215\pi$
$198$	0	0
$199$	−552.000	−0.196634	−0.0983172	−	0.995155i	$-0.531346\pi$
$199$	−552.000	−0.196634	−0.0983172	+	0.995155i	$0.531346\pi$
$200$	0	0
$201$	−1884.00	−0.661130
$202$	0	0
$203$	1722.00	0.595373
$204$	0	0
$205$	748.000	0.254842
$206$	0	0
$207$	−1080.00	−0.362634
$208$	0	0
$209$	−208.000	−0.0688405
$210$	0	0
$211$	−1692.00	−0.552048	−0.276024	−	0.961151i	$-0.589017\pi$
$211$	−1692.00	−0.552048	−0.276024	+	0.961151i	$0.589017\pi$
$212$	0	0
$213$	888.000	0.285656
$214$	0	0
$215$	328.000	0.104044
$216$	0	0
$217$	−560.000	−0.175186
$218$	0	0
$219$	1554.00	0.479496
$220$	0	0
$221$	−2580.00	−0.785292
$222$	0	0
$223$	−528.000	−0.158554	−0.0792769	−	0.996853i	$-0.525261\pi$
$223$	−528.000	−0.158554	−0.0792769	+	0.996853i	$0.525261\pi$
$224$	0	0
$225$	−1089.00	−0.322667
$226$	0	0
$227$	1876.00	0.548522	0.274261	−	0.961655i	$-0.411567\pi$
$227$	1876.00	0.548522	0.274261	+	0.961655i	$0.411567\pi$
$228$	0	0
$229$	−5474.00	−1.57962	−0.789808	−	0.613354i	$-0.789820\pi$
$229$	−5474.00	−1.57962	−0.789808	+	0.613354i	$0.789820\pi$
$230$	0	0
$231$	1092.00	0.311032
$232$	0	0
$233$	3418.00	0.961033	0.480516	−	0.876986i	$-0.340449\pi$
$233$	3418.00	0.961033	0.480516	+	0.876986i	$0.340449\pi$
$234$	0	0
$235$	928.000	0.257600
$236$	0	0
$237$	−3552.00	−0.973532
$238$	0	0
$239$	−7360.00	−1.99196	−0.995981	−	0.0895670i	$-0.971452\pi$
$239$	−7360.00	−1.99196	−0.995981	+	0.0895670i	$0.971452\pi$
$240$	0	0
$241$	−2126.00	−0.568248	−0.284124	−	0.958788i	$-0.591703\pi$
$241$	−2126.00	−0.568248	−0.284124	+	0.958788i	$0.591703\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−98.0000	−0.0255551
$246$	0	0
$247$	344.000	0.0886162
$248$	0	0
$249$	660.000	0.167975
$250$	0	0
$251$	7788.00	1.95846	0.979231	−	0.202745i	$-0.0649864\pi$
$251$	7788.00	1.95846	0.979231	+	0.202745i	$0.0649864\pi$
$252$	0	0
$253$	6240.00	1.55061
$254$	0	0
$255$	−180.000	−0.0442041
$256$	0	0
$257$	−3470.00	−0.842228	−0.421114	−	0.907008i	$-0.638361\pi$
$257$	−3470.00	−0.842228	−0.421114	+	0.907008i	$0.638361\pi$
$258$	0	0
$259$	−2030.00	−0.487020
$260$	0	0
$261$	2214.00	0.525070
$262$	0	0
$263$	3096.00	0.725884	0.362942	−	0.931812i	$-0.381772\pi$
$263$	3096.00	0.725884	0.362942	+	0.931812i	$0.381772\pi$
$264$	0	0
$265$	324.000	0.0751063
$266$	0	0
$267$	2322.00	0.532225
$268$	0	0
$269$	−3274.00	−0.742079	−0.371040	−	0.928617i	$-0.620999\pi$
$269$	−3274.00	−0.742079	−0.371040	+	0.928617i	$0.620999\pi$
$270$	0	0
$271$	−960.000	−0.215188	−0.107594	−	0.994195i	$-0.534315\pi$
$271$	−960.000	−0.215188	−0.107594	+	0.994195i	$0.534315\pi$
$272$	0	0
$273$	−1806.00	−0.400381
$274$	0	0
$275$	6292.00	1.37972
$276$	0	0
$277$	910.000	0.197388	0.0986942	−	0.995118i	$-0.468533\pi$
$277$	910.000	0.197388	0.0986942	+	0.995118i	$0.468533\pi$
$278$	0	0
$279$	−720.000	−0.154499
$280$	0	0
$281$	−6486.00	−1.37695	−0.688474	−	0.725261i	$-0.741719\pi$
$281$	−6486.00	−1.37695	−0.688474	+	0.725261i	$0.741719\pi$
$282$	0	0
$283$	−3796.00	−0.797346	−0.398673	−	0.917093i	$-0.630529\pi$
$283$	−3796.00	−0.797346	−0.398673	+	0.917093i	$0.630529\pi$
$284$	0	0
$285$	24.0000	0.00498820
$286$	0	0
$287$	−2618.00	−0.538452
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	3258.00	0.656314
$292$	0	0
$293$	−6882.00	−1.37219	−0.686093	−	0.727513i	$-0.740676\pi$
$293$	−6882.00	−1.37219	−0.686093	+	0.727513i	$0.740676\pi$
$294$	0	0
$295$	360.000	0.0710509
$296$	0	0
$297$	1404.00	0.274304
$298$	0	0
$299$	−10320.0	−1.99606
$300$	0	0
$301$	−1148.00	−0.219833
$302$	0	0
$303$	870.000	0.164951
$304$	0	0
$305$	1332.00	0.250066
$306$	0	0
$307$	−7228.00	−1.34373	−0.671863	−	0.740676i	$-0.734506\pi$
$307$	−7228.00	−1.34373	−0.671863	+	0.740676i	$0.734506\pi$
$308$	0	0
$309$	−264.000	−0.0486034
$310$	0	0
$311$	−7912.00	−1.44260	−0.721300	−	0.692623i	$-0.756455\pi$
$311$	−7912.00	−1.44260	−0.721300	+	0.692623i	$0.756455\pi$
$312$	0	0
$313$	2218.00	0.400539	0.200270	−	0.979741i	$-0.435818\pi$
$313$	2218.00	0.400539	0.200270	+	0.979741i	$0.435818\pi$
$314$	0	0
$315$	−126.000	−0.0225374
$316$	0	0
$317$	8118.00	1.43834	0.719168	−	0.694837i	$-0.244523\pi$
$317$	8118.00	1.43834	0.719168	+	0.694837i	$0.244523\pi$
$318$	0	0
$319$	−12792.0	−2.24519
$320$	0	0
$321$	1116.00	0.194047
$322$	0	0
$323$	−120.000	−0.0206718
$324$	0	0
$325$	−10406.0	−1.77606
$326$	0	0
$327$	−4290.00	−0.725497
$328$	0	0
$329$	−3248.00	−0.544280
$330$	0	0
$331$	10780.0	1.79010	0.895049	−	0.445968i	$-0.147140\pi$
$331$	10780.0	1.79010	0.895049	+	0.445968i	$0.147140\pi$
$332$	0	0
$333$	−2610.00	−0.429511
$334$	0	0
$335$	−1256.00	−0.204844
$336$	0	0
$337$	3122.00	0.504647	0.252324	−	0.967643i	$-0.418805\pi$
$337$	3122.00	0.504647	0.252324	+	0.967643i	$0.418805\pi$
$338$	0	0
$339$	−5430.00	−0.869962
$340$	0	0
$341$	4160.00	0.660635
$342$	0	0
$343$	343.000	0.0539949
$344$	0	0
$345$	−720.000	−0.112358
$346$	0	0
$347$	828.000	0.128096	0.0640481	−	0.997947i	$-0.479599\pi$
$347$	828.000	0.128096	0.0640481	+	0.997947i	$0.479599\pi$
$348$	0	0
$349$	4614.00	0.707684	0.353842	−	0.935305i	$-0.384875\pi$
$349$	4614.00	0.707684	0.353842	+	0.935305i	$0.384875\pi$
$350$	0	0
$351$	−2322.00	−0.353103
$352$	0	0
$353$	9458.00	1.42606	0.713029	−	0.701134i	$-0.247323\pi$
$353$	9458.00	1.42606	0.713029	+	0.701134i	$0.247323\pi$
$354$	0	0
$355$	592.000	0.0885073
$356$	0	0
$357$	630.000	0.0933981
$358$	0	0
$359$	−2952.00	−0.433985	−0.216992	−	0.976173i	$-0.569625\pi$
$359$	−2952.00	−0.433985	−0.216992	+	0.976173i	$0.569625\pi$
$360$	0	0
$361$	−6843.00	−0.997667
$362$	0	0
$363$	−4119.00	−0.595569
$364$	0	0
$365$	1036.00	0.148566
$366$	0	0
$367$	10592.0	1.50653	0.753267	−	0.657715i	$-0.228477\pi$
$367$	10592.0	1.50653	0.753267	+	0.657715i	$0.228477\pi$
$368$	0	0
$369$	−3366.00	−0.474870
$370$	0	0
$371$	−1134.00	−0.158691
$372$	0	0
$373$	6478.00	0.899244	0.449622	−	0.893219i	$-0.351559\pi$
$373$	6478.00	0.899244	0.449622	+	0.893219i	$0.351559\pi$
$374$	0	0
$375$	−1476.00	−0.203254
$376$	0	0
$377$	21156.0	2.89016
$378$	0	0
$379$	−5780.00	−0.783374	−0.391687	−	0.920099i	$-0.628108\pi$
$379$	−5780.00	−0.783374	−0.391687	+	0.920099i	$0.628108\pi$
$380$	0	0
$381$	3504.00	0.471169
$382$	0	0
$383$	6912.00	0.922158	0.461079	−	0.887359i	$-0.347462\pi$
$383$	6912.00	0.922158	0.461079	+	0.887359i	$0.347462\pi$
$384$	0	0
$385$	728.000	0.0963697
$386$	0	0
$387$	−1476.00	−0.193874
$388$	0	0
$389$	−9010.00	−1.17436	−0.587179	−	0.809457i	$-0.699761\pi$
$389$	−9010.00	−1.17436	−0.587179	+	0.809457i	$0.699761\pi$
$390$	0	0
$391$	3600.00	0.465626
$392$	0	0
$393$	−3804.00	−0.488261
$394$	0	0
$395$	−2368.00	−0.301638
$396$	0	0
$397$	10774.0	1.36204	0.681022	−	0.732263i	$-0.261536\pi$
$397$	10774.0	1.36204	0.681022	+	0.732263i	$0.261536\pi$
$398$	0	0
$399$	−84.0000	−0.0105395
$400$	0	0
$401$	−78.0000	−0.00971355	−0.00485678	−	0.999988i	$-0.501546\pi$
$401$	−78.0000	−0.00971355	−0.00485678	+	0.999988i	$0.501546\pi$
$402$	0	0
$403$	−6880.00	−0.850415
$404$	0	0
$405$	−162.000	−0.0198762
$406$	0	0
$407$	15080.0	1.83658
$408$	0	0
$409$	−15254.0	−1.84416	−0.922080	−	0.386998i	$-0.873512\pi$
$409$	−15254.0	−1.84416	−0.922080	+	0.386998i	$0.873512\pi$
$410$	0	0
$411$	−1422.00	−0.170662
$412$	0	0
$413$	−1260.00	−0.150122
$414$	0	0
$415$	440.000	0.0520452
$416$	0	0
$417$	−8052.00	−0.945583
$418$	0	0
$419$	7316.00	0.853007	0.426504	−	0.904486i	$-0.359745\pi$
$419$	7316.00	0.853007	0.426504	+	0.904486i	$0.359745\pi$
$420$	0	0
$421$	−11330.0	−1.31162	−0.655808	−	0.754928i	$-0.727672\pi$
$421$	−11330.0	−1.31162	−0.655808	+	0.754928i	$0.727672\pi$
$422$	0	0
$423$	−4176.00	−0.480010
$424$	0	0
$425$	3630.00	0.414308
$426$	0	0
$427$	−4662.00	−0.528361
$428$	0	0
$429$	13416.0	1.50986
$430$	0	0
$431$	6016.00	0.672345	0.336172	−	0.941801i	$-0.390868\pi$
$431$	6016.00	0.672345	0.336172	+	0.941801i	$0.390868\pi$
$432$	0	0
$433$	−13550.0	−1.50386	−0.751930	−	0.659243i	$-0.770877\pi$
$433$	−13550.0	−1.50386	−0.751930	+	0.659243i	$0.770877\pi$
$434$	0	0
$435$	1476.00	0.162687
$436$	0	0
$437$	−480.000	−0.0525435
$438$	0	0
$439$	2760.00	0.300063	0.150031	−	0.988681i	$-0.452063\pi$
$439$	2760.00	0.300063	0.150031	+	0.988681i	$0.452063\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	3036.00	0.325609	0.162804	−	0.986658i	$-0.447946\pi$
$443$	3036.00	0.325609	0.162804	+	0.986658i	$0.447946\pi$
$444$	0	0
$445$	1548.00	0.164904
$446$	0	0
$447$	3942.00	0.417115
$448$	0	0
$449$	12962.0	1.36239	0.681197	−	0.732100i	$-0.261460\pi$
$449$	12962.0	1.36239	0.681197	+	0.732100i	$0.261460\pi$
$450$	0	0
$451$	19448.0	2.03053
$452$	0	0
$453$	9000.00	0.933459
$454$	0	0
$455$	−1204.00	−0.124054
$456$	0	0
$457$	11866.0	1.21459	0.607295	−	0.794476i	$-0.292254\pi$
$457$	11866.0	1.21459	0.607295	+	0.794476i	$0.292254\pi$
$458$	0	0
$459$	810.000	0.0823694
$460$	0	0
$461$	10998.0	1.11112	0.555562	−	0.831475i	$-0.312503\pi$
$461$	10998.0	1.11112	0.555562	+	0.831475i	$0.312503\pi$
$462$	0	0
$463$	9088.00	0.912214	0.456107	−	0.889925i	$-0.349243\pi$
$463$	9088.00	0.912214	0.456107	+	0.889925i	$0.349243\pi$
$464$	0	0
$465$	−480.000	−0.0478698
$466$	0	0
$467$	−18236.0	−1.80698	−0.903492	−	0.428605i	$-0.859005\pi$
$467$	−18236.0	−1.80698	−0.903492	+	0.428605i	$0.859005\pi$
$468$	0	0
$469$	4396.00	0.432811
$470$	0	0
$471$	−2322.00	−0.227159
$472$	0	0
$473$	8528.00	0.829002
$474$	0	0
$475$	−484.000	−0.0467525
$476$	0	0
$477$	−1458.00	−0.139952
$478$	0	0
$479$	11424.0	1.08972	0.544860	−	0.838527i	$-0.316583\pi$
$479$	11424.0	1.08972	0.544860	+	0.838527i	$0.316583\pi$
$480$	0	0
$481$	−24940.0	−2.36417
$482$	0	0
$483$	2520.00	0.237400
$484$	0	0
$485$	2172.00	0.203351
$486$	0	0
$487$	8536.00	0.794257	0.397128	−	0.917763i	$-0.370007\pi$
$487$	8536.00	0.794257	0.397128	+	0.917763i	$0.370007\pi$
$488$	0	0
$489$	−6876.00	−0.635876
$490$	0	0
$491$	18732.0	1.72172	0.860859	−	0.508844i	$-0.169927\pi$
$491$	18732.0	1.72172	0.860859	+	0.508844i	$0.169927\pi$
$492$	0	0
$493$	−7380.00	−0.674196
$494$	0	0
$495$	936.000	0.0849900
$496$	0	0
$497$	−2072.00	−0.187006
$498$	0	0
$499$	21700.0	1.94674	0.973372	−	0.229231i	$-0.0736210\pi$
$499$	21700.0	1.94674	0.973372	+	0.229231i	$0.0736210\pi$
$500$	0	0
$501$	−5016.00	−0.447302
$502$	0	0
$503$	1048.00	0.0928986	0.0464493	−	0.998921i	$-0.485209\pi$
$503$	1048.00	0.0928986	0.0464493	+	0.998921i	$0.485209\pi$
$504$	0	0
$505$	580.000	0.0511082
$506$	0	0
$507$	−15597.0	−1.36625
$508$	0	0
$509$	−8890.00	−0.774150	−0.387075	−	0.922048i	$-0.626515\pi$
$509$	−8890.00	−0.774150	−0.387075	+	0.922048i	$0.626515\pi$
$510$	0	0
$511$	−3626.00	−0.313904
$512$	0	0
$513$	−108.000	−0.00929496
$514$	0	0
$515$	−176.000	−0.0150592
$516$	0	0
$517$	24128.0	2.05251
$518$	0	0
$519$	8190.00	0.692680
$520$	0	0
$521$	13962.0	1.17406	0.587031	−	0.809564i	$-0.300297\pi$
$521$	13962.0	1.17406	0.587031	+	0.809564i	$0.300297\pi$
$522$	0	0
$523$	−16420.0	−1.37284	−0.686421	−	0.727204i	$-0.740819\pi$
$523$	−16420.0	−1.37284	−0.686421	+	0.727204i	$0.740819\pi$
$524$	0	0
$525$	2541.00	0.211235
$526$	0	0
$527$	2400.00	0.198379
$528$	0	0
$529$	2233.00	0.183529
$530$	0	0
$531$	−1620.00	−0.132396
$532$	0	0
$533$	−32164.0	−2.61384
$534$	0	0
$535$	744.000	0.0601232
$536$	0	0
$537$	7716.00	0.620056
$538$	0	0
$539$	−2548.00	−0.203618
$540$	0	0
$541$	−14554.0	−1.15661	−0.578304	−	0.815821i	$-0.696285\pi$
$541$	−14554.0	−1.15661	−0.578304	+	0.815821i	$0.696285\pi$
$542$	0	0
$543$	−9642.00	−0.762022
$544$	0	0
$545$	−2860.00	−0.224787
$546$	0	0
$547$	−588.000	−0.0459617	−0.0229809	−	0.999736i	$-0.507316\pi$
$547$	−588.000	−0.0459617	−0.0229809	+	0.999736i	$0.507316\pi$
$548$	0	0
$549$	−5994.00	−0.465970
$550$	0	0
$551$	984.000	0.0760795
$552$	0	0
$553$	8288.00	0.637327
$554$	0	0
$555$	−1740.00	−0.133079
$556$	0	0
$557$	10726.0	0.815934	0.407967	−	0.912997i	$-0.366238\pi$
$557$	10726.0	0.815934	0.407967	+	0.912997i	$0.366238\pi$
$558$	0	0
$559$	−14104.0	−1.06715
$560$	0	0
$561$	−4680.00	−0.352210
$562$	0	0
$563$	740.000	0.0553948	0.0276974	−	0.999616i	$-0.491183\pi$
$563$	740.000	0.0553948	0.0276974	+	0.999616i	$0.491183\pi$
$564$	0	0
$565$	−3620.00	−0.269548
$566$	0	0
$567$	567.000	0.0419961
$568$	0	0
$569$	17386.0	1.28095	0.640474	−	0.767980i	$-0.278738\pi$
$569$	17386.0	1.28095	0.640474	+	0.767980i	$0.278738\pi$
$570$	0	0
$571$	−1108.00	−0.0812055	−0.0406028	−	0.999175i	$-0.512928\pi$
$571$	−1108.00	−0.0812055	−0.0406028	+	0.999175i	$0.512928\pi$
$572$	0	0
$573$	−4560.00	−0.332455
$574$	0	0
$575$	14520.0	1.05309
$576$	0	0
$577$	−13694.0	−0.988022	−0.494011	−	0.869456i	$-0.664470\pi$
$577$	−13694.0	−0.988022	−0.494011	+	0.869456i	$0.664470\pi$
$578$	0	0
$579$	186.000	0.0133504
$580$	0	0
$581$	−1540.00	−0.109966
$582$	0	0
$583$	8424.00	0.598433
$584$	0	0
$585$	−1548.00	−0.109405
$586$	0	0
$587$	2844.00	0.199973	0.0999867	−	0.994989i	$-0.468120\pi$
$587$	2844.00	0.199973	0.0999867	+	0.994989i	$0.468120\pi$
$588$	0	0
$589$	−320.000	−0.0223860
$590$	0	0
$591$	3222.00	0.224256
$592$	0	0
$593$	9410.00	0.651640	0.325820	−	0.945432i	$-0.394360\pi$
$593$	9410.00	0.651640	0.325820	+	0.945432i	$0.394360\pi$
$594$	0	0
$595$	420.000	0.0289384
$596$	0	0
$597$	1656.00	0.113527
$598$	0	0
$599$	14952.0	1.01990	0.509952	−	0.860203i	$-0.329663\pi$
$599$	14952.0	1.01990	0.509952	+	0.860203i	$0.329663\pi$
$600$	0	0
$601$	2570.00	0.174430	0.0872150	−	0.996190i	$-0.472203\pi$
$601$	2570.00	0.174430	0.0872150	+	0.996190i	$0.472203\pi$
$602$	0	0
$603$	5652.00	0.381704
$604$	0	0
$605$	−2746.00	−0.184530
$606$	0	0
$607$	8176.00	0.546711	0.273356	−	0.961913i	$-0.411866\pi$
$607$	8176.00	0.546711	0.273356	+	0.961913i	$0.411866\pi$
$608$	0	0
$609$	−5166.00	−0.343739
$610$	0	0
$611$	−39904.0	−2.64213
$612$	0	0
$613$	8862.00	0.583903	0.291952	−	0.956433i	$-0.405695\pi$
$613$	8862.00	0.583903	0.291952	+	0.956433i	$0.405695\pi$
$614$	0	0
$615$	−2244.00	−0.147133
$616$	0	0
$617$	−1126.00	−0.0734701	−0.0367351	−	0.999325i	$-0.511696\pi$
$617$	−1126.00	−0.0734701	−0.0367351	+	0.999325i	$0.511696\pi$
$618$	0	0
$619$	−9892.00	−0.642315	−0.321158	−	0.947026i	$-0.604072\pi$
$619$	−9892.00	−0.642315	−0.321158	+	0.947026i	$0.604072\pi$
$620$	0	0
$621$	3240.00	0.209367
$622$	0	0
$623$	−5418.00	−0.348423
$624$	0	0
$625$	14141.0	0.905024
$626$	0	0
$627$	624.000	0.0397451
$628$	0	0
$629$	8700.00	0.551497
$630$	0	0
$631$	−11256.0	−0.710134	−0.355067	−	0.934841i	$-0.615542\pi$
$631$	−11256.0	−0.710134	−0.355067	+	0.934841i	$0.615542\pi$
$632$	0	0
$633$	5076.00	0.318725
$634$	0	0
$635$	2336.00	0.145986
$636$	0	0
$637$	4214.00	0.262111
$638$	0	0
$639$	−2664.00	−0.164924
$640$	0	0
$641$	−5694.00	−0.350857	−0.175429	−	0.984492i	$-0.556131\pi$
$641$	−5694.00	−0.350857	−0.175429	+	0.984492i	$0.556131\pi$
$642$	0	0
$643$	−30028.0	−1.84166	−0.920831	−	0.389962i	$-0.872488\pi$
$643$	−30028.0	−1.84166	−0.920831	+	0.389962i	$0.872488\pi$
$644$	0	0
$645$	−984.000	−0.0600697
$646$	0	0
$647$	−18680.0	−1.13506	−0.567532	−	0.823351i	$-0.692102\pi$
$647$	−18680.0	−1.13506	−0.567532	+	0.823351i	$0.692102\pi$
$648$	0	0
$649$	9360.00	0.566120
$650$	0	0
$651$	1680.00	0.101143
$652$	0	0
$653$	−3034.00	−0.181822	−0.0909109	−	0.995859i	$-0.528978\pi$
$653$	−3034.00	−0.181822	−0.0909109	+	0.995859i	$0.528978\pi$
$654$	0	0
$655$	−2536.00	−0.151282
$656$	0	0
$657$	−4662.00	−0.276837
$658$	0	0
$659$	−26508.0	−1.56693	−0.783464	−	0.621438i	$-0.786549\pi$
$659$	−26508.0	−1.56693	−0.783464	+	0.621438i	$0.786549\pi$
$660$	0	0
$661$	−24658.0	−1.45096	−0.725480	−	0.688243i	$-0.758382\pi$
$661$	−24658.0	−1.45096	−0.725480	+	0.688243i	$0.758382\pi$
$662$	0	0
$663$	7740.00	0.453389
$664$	0	0
$665$	−56.0000	−0.00326554
$666$	0	0
$667$	−29520.0	−1.71367
$668$	0	0
$669$	1584.00	0.0915411
$670$	0	0
$671$	34632.0	1.99248
$672$	0	0
$673$	23266.0	1.33260	0.666299	−	0.745685i	$-0.267877\pi$
$673$	23266.0	1.33260	0.666299	+	0.745685i	$0.267877\pi$
$674$	0	0
$675$	3267.00	0.186292
$676$	0	0
$677$	5694.00	0.323247	0.161623	−	0.986852i	$-0.448327\pi$
$677$	5694.00	0.323247	0.161623	+	0.986852i	$0.448327\pi$
$678$	0	0
$679$	−7602.00	−0.429658
$680$	0	0
$681$	−5628.00	−0.316689
$682$	0	0
$683$	14796.0	0.828921	0.414461	−	0.910067i	$-0.363970\pi$
$683$	14796.0	0.828921	0.414461	+	0.910067i	$0.363970\pi$
$684$	0	0
$685$	−948.000	−0.0528777
$686$	0	0
$687$	16422.0	0.911992
$688$	0	0
$689$	−13932.0	−0.770344
$690$	0	0
$691$	−4540.00	−0.249942	−0.124971	−	0.992160i	$-0.539884\pi$
$691$	−4540.00	−0.249942	−0.124971	+	0.992160i	$0.539884\pi$
$692$	0	0
$693$	−3276.00	−0.179574
$694$	0	0
$695$	−5368.00	−0.292978
$696$	0	0
$697$	11220.0	0.609739
$698$	0	0
$699$	−10254.0	−0.554853
$700$	0	0
$701$	−18666.0	−1.00571	−0.502857	−	0.864370i	$-0.667718\pi$
$701$	−18666.0	−1.00571	−0.502857	+	0.864370i	$0.667718\pi$
$702$	0	0
$703$	−1160.00	−0.0622336
$704$	0	0
$705$	−2784.00	−0.148726
$706$	0	0
$707$	−2030.00	−0.107986
$708$	0	0
$709$	10974.0	0.581294	0.290647	−	0.956830i	$-0.406130\pi$
$709$	10974.0	0.581294	0.290647	+	0.956830i	$0.406130\pi$
$710$	0	0
$711$	10656.0	0.562069
$712$	0	0
$713$	9600.00	0.504240
$714$	0	0
$715$	8944.00	0.467813
$716$	0	0
$717$	22080.0	1.15006
$718$	0	0
$719$	−28240.0	−1.46478	−0.732388	−	0.680887i	$-0.761594\pi$
$719$	−28240.0	−1.46478	−0.732388	+	0.680887i	$0.761594\pi$
$720$	0	0
$721$	616.000	0.0318184
$722$	0	0
$723$	6378.00	0.328078
$724$	0	0
$725$	−29766.0	−1.52480
$726$	0	0
$727$	−6232.00	−0.317926	−0.158963	−	0.987285i	$-0.550815\pi$
$727$	−6232.00	−0.317926	−0.158963	+	0.987285i	$0.550815\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	4920.00	0.248937
$732$	0	0
$733$	21638.0	1.09034	0.545169	−	0.838326i	$-0.316465\pi$
$733$	21638.0	1.09034	0.545169	+	0.838326i	$0.316465\pi$
$734$	0	0
$735$	294.000	0.0147542
$736$	0	0
$737$	−32656.0	−1.63216
$738$	0	0
$739$	−24364.0	−1.21278	−0.606390	−	0.795167i	$-0.707383\pi$
$739$	−24364.0	−1.21278	−0.606390	+	0.795167i	$0.707383\pi$
$740$	0	0
$741$	−1032.00	−0.0511626
$742$	0	0
$743$	−3112.00	−0.153658	−0.0768292	−	0.997044i	$-0.524480\pi$
$743$	−3112.00	−0.153658	−0.0768292	+	0.997044i	$0.524480\pi$
$744$	0	0
$745$	2628.00	0.129238
$746$	0	0
$747$	−1980.00	−0.0969805
$748$	0	0
$749$	−2604.00	−0.127033
$750$	0	0
$751$	20576.0	0.999772	0.499886	−	0.866091i	$-0.333375\pi$
$751$	20576.0	0.999772	0.499886	+	0.866091i	$0.333375\pi$
$752$	0	0
$753$	−23364.0	−1.13072
$754$	0	0
$755$	6000.00	0.289222
$756$	0	0
$757$	−4754.00	−0.228252	−0.114126	−	0.993466i	$-0.536407\pi$
$757$	−4754.00	−0.228252	−0.114126	+	0.993466i	$0.536407\pi$
$758$	0	0
$759$	−18720.0	−0.895248
$760$	0	0
$761$	−10950.0	−0.521599	−0.260800	−	0.965393i	$-0.583986\pi$
$761$	−10950.0	−0.521599	−0.260800	+	0.965393i	$0.583986\pi$
$762$	0	0
$763$	10010.0	0.474949
$764$	0	0
$765$	540.000	0.0255212
$766$	0	0
$767$	−15480.0	−0.728749
$768$	0	0
$769$	−28798.0	−1.35043	−0.675216	−	0.737620i	$-0.735950\pi$
$769$	−28798.0	−1.35043	−0.675216	+	0.737620i	$0.735950\pi$
$770$	0	0
$771$	10410.0	0.486261
$772$	0	0
$773$	−12866.0	−0.598652	−0.299326	−	0.954151i	$-0.596762\pi$
$773$	−12866.0	−0.598652	−0.299326	+	0.954151i	$0.596762\pi$
$774$	0	0
$775$	9680.00	0.448666
$776$	0	0
$777$	6090.00	0.281181
$778$	0	0
$779$	−1496.00	−0.0688059
$780$	0	0
$781$	15392.0	0.705210
$782$	0	0
$783$	−6642.00	−0.303149
$784$	0	0
$785$	−1548.00	−0.0703828
$786$	0	0
$787$	13156.0	0.595884	0.297942	−	0.954584i	$-0.403700\pi$
$787$	13156.0	0.595884	0.297942	+	0.954584i	$0.403700\pi$
$788$	0	0
$789$	−9288.00	−0.419089
$790$	0	0
$791$	12670.0	0.569524
$792$	0	0
$793$	−57276.0	−2.56486
$794$	0	0
$795$	−972.000	−0.0433626
$796$	0	0
$797$	20454.0	0.909056	0.454528	−	0.890732i	$-0.349808\pi$
$797$	20454.0	0.909056	0.454528	+	0.890732i	$0.349808\pi$
$798$	0	0
$799$	13920.0	0.616338
$800$	0	0
$801$	−6966.00	−0.307280
$802$	0	0
$803$	26936.0	1.18375
$804$	0	0
$805$	1680.00	0.0735556
$806$	0	0
$807$	9822.00	0.428440
$808$	0	0
$809$	15706.0	0.682563	0.341282	−	0.939961i	$-0.389139\pi$
$809$	15706.0	0.682563	0.341282	+	0.939961i	$0.389139\pi$
$810$	0	0
$811$	−6532.00	−0.282823	−0.141412	−	0.989951i	$-0.545164\pi$
$811$	−6532.00	−0.282823	−0.141412	+	0.989951i	$0.545164\pi$
$812$	0	0
$813$	2880.00	0.124239
$814$	0	0
$815$	−4584.00	−0.197019
$816$	0	0
$817$	−656.000	−0.0280912
$818$	0	0
$819$	5418.00	0.231160
$820$	0	0
$821$	−46754.0	−1.98749	−0.993743	−	0.111692i	$-0.964373\pi$
$821$	−46754.0	−1.98749	−0.993743	+	0.111692i	$0.964373\pi$
$822$	0	0
$823$	−22008.0	−0.932139	−0.466070	−	0.884748i	$-0.654330\pi$
$823$	−22008.0	−0.932139	−0.466070	+	0.884748i	$0.654330\pi$
$824$	0	0
$825$	−18876.0	−0.796579
$826$	0	0
$827$	−45412.0	−1.90947	−0.954734	−	0.297461i	$-0.903860\pi$
$827$	−45412.0	−1.90947	−0.954734	+	0.297461i	$0.903860\pi$
$828$	0	0
$829$	13670.0	0.572713	0.286356	−	0.958123i	$-0.407556\pi$
$829$	13670.0	0.572713	0.286356	+	0.958123i	$0.407556\pi$
$830$	0	0
$831$	−2730.00	−0.113962
$832$	0	0
$833$	−1470.00	−0.0611434
$834$	0	0
$835$	−3344.00	−0.138591
$836$	0	0
$837$	2160.00	0.0892001
$838$	0	0
$839$	26568.0	1.09324	0.546621	−	0.837380i	$-0.315914\pi$
$839$	26568.0	1.09324	0.546621	+	0.837380i	$0.315914\pi$
$840$	0	0
$841$	36127.0	1.48128
$842$	0	0
$843$	19458.0	0.794981
$844$	0	0
$845$	−10398.0	−0.423316
$846$	0	0
$847$	9611.00	0.389891
$848$	0	0
$849$	11388.0	0.460348
$850$	0	0
$851$	34800.0	1.40180
$852$	0	0
$853$	1070.00	0.0429497	0.0214749	−	0.999769i	$-0.493164\pi$
$853$	1070.00	0.0429497	0.0214749	+	0.999769i	$0.493164\pi$
$854$	0	0
$855$	−72.0000	−0.00287994
$856$	0	0
$857$	42906.0	1.71020	0.855100	−	0.518463i	$-0.173496\pi$
$857$	42906.0	1.71020	0.855100	+	0.518463i	$0.173496\pi$
$858$	0	0
$859$	−11252.0	−0.446930	−0.223465	−	0.974712i	$-0.571737\pi$
$859$	−11252.0	−0.446930	−0.223465	+	0.974712i	$0.571737\pi$
$860$	0	0
$861$	7854.00	0.310875
$862$	0	0
$863$	−29264.0	−1.15430	−0.577148	−	0.816639i	$-0.695835\pi$
$863$	−29264.0	−1.15430	−0.577148	+	0.816639i	$0.695835\pi$
$864$	0	0
$865$	5460.00	0.214619
$866$	0	0
$867$	12039.0	0.471587
$868$	0	0
$869$	−61568.0	−2.40340
$870$	0	0
$871$	54008.0	2.10102
$872$	0	0
$873$	−9774.00	−0.378923
$874$	0	0
$875$	3444.00	0.133061
$876$	0	0
$877$	25782.0	0.992698	0.496349	−	0.868123i	$-0.334674\pi$
$877$	25782.0	0.992698	0.496349	+	0.868123i	$0.334674\pi$
$878$	0	0
$879$	20646.0	0.792232
$880$	0	0
$881$	−9054.00	−0.346240	−0.173120	−	0.984901i	$-0.555385\pi$
$881$	−9054.00	−0.346240	−0.173120	+	0.984901i	$0.555385\pi$
$882$	0	0
$883$	−8092.00	−0.308400	−0.154200	−	0.988040i	$-0.549280\pi$
$883$	−8092.00	−0.308400	−0.154200	+	0.988040i	$0.549280\pi$
$884$	0	0
$885$	−1080.00	−0.0410212
$886$	0	0
$887$	−11944.0	−0.452131	−0.226066	−	0.974112i	$-0.572586\pi$
$887$	−11944.0	−0.452131	−0.226066	+	0.974112i	$0.572586\pi$
$888$	0	0
$889$	−8176.00	−0.308452
$890$	0	0
$891$	−4212.00	−0.158370
$892$	0	0
$893$	−1856.00	−0.0695506
$894$	0	0
$895$	5144.00	0.192117
$896$	0	0
$897$	30960.0	1.15242
$898$	0	0
$899$	−19680.0	−0.730105
$900$	0	0
$901$	4860.00	0.179700
$902$	0	0
$903$	3444.00	0.126920
$904$	0	0
$905$	−6428.00	−0.236104
$906$	0	0
$907$	31804.0	1.16432	0.582158	−	0.813076i	$-0.302209\pi$
$907$	31804.0	1.16432	0.582158	+	0.813076i	$0.302209\pi$
$908$	0	0
$909$	−2610.00	−0.0952346
$910$	0	0
$911$	−27840.0	−1.01249	−0.506246	−	0.862389i	$-0.668967\pi$
$911$	−27840.0	−1.01249	−0.506246	+	0.862389i	$0.668967\pi$
$912$	0	0
$913$	11440.0	0.414686
$914$	0	0
$915$	−3996.00	−0.144376
$916$	0	0
$917$	8876.00	0.319642
$918$	0	0
$919$	−12536.0	−0.449972	−0.224986	−	0.974362i	$-0.572234\pi$
$919$	−12536.0	−0.449972	−0.224986	+	0.974362i	$0.572234\pi$
$920$	0	0
$921$	21684.0	0.775800
$922$	0	0
$923$	−25456.0	−0.907795
$924$	0	0
$925$	35090.0	1.24730
$926$	0	0
$927$	792.000	0.0280612
$928$	0	0
$929$	−25870.0	−0.913635	−0.456818	−	0.889560i	$-0.651011\pi$
$929$	−25870.0	−0.913635	−0.456818	+	0.889560i	$0.651011\pi$
$930$	0	0
$931$	196.000	0.00689972
$932$	0	0
$933$	23736.0	0.832885
$934$	0	0
$935$	−3120.00	−0.109128
$936$	0	0
$937$	−6086.00	−0.212189	−0.106094	−	0.994356i	$-0.533835\pi$
$937$	−6086.00	−0.212189	−0.106094	+	0.994356i	$0.533835\pi$
$938$	0	0
$939$	−6654.00	−0.231251
$940$	0	0
$941$	3894.00	0.134900	0.0674499	−	0.997723i	$-0.478514\pi$
$941$	3894.00	0.134900	0.0674499	+	0.997723i	$0.478514\pi$
$942$	0	0
$943$	44880.0	1.54983
$944$	0	0
$945$	378.000	0.0130120
$946$	0	0
$947$	20692.0	0.710031	0.355016	−	0.934860i	$-0.384476\pi$
$947$	20692.0	0.710031	0.355016	+	0.934860i	$0.384476\pi$
$948$	0	0
$949$	−44548.0	−1.52380
$950$	0	0
$951$	−24354.0	−0.830423
$952$	0	0
$953$	46986.0	1.59709	0.798545	−	0.601936i	$-0.205604\pi$
$953$	46986.0	1.59709	0.798545	+	0.601936i	$0.205604\pi$
$954$	0	0
$955$	−3040.00	−0.103007
$956$	0	0
$957$	38376.0	1.29626
$958$	0	0
$959$	3318.00	0.111725
$960$	0	0
$961$	−23391.0	−0.785170
$962$	0	0
$963$	−3348.00	−0.112033
$964$	0	0
$965$	124.000	0.00413648
$966$	0	0
$967$	−53960.0	−1.79445	−0.897227	−	0.441570i	$-0.854422\pi$
$967$	−53960.0	−1.79445	−0.897227	+	0.441570i	$0.854422\pi$
$968$	0	0
$969$	360.000	0.0119348
$970$	0	0
$971$	7068.00	0.233597	0.116799	−	0.993156i	$-0.462737\pi$
$971$	7068.00	0.233597	0.116799	+	0.993156i	$0.462737\pi$
$972$	0	0
$973$	18788.0	0.619029
$974$	0	0
$975$	31218.0	1.02541
$976$	0	0
$977$	54130.0	1.77254	0.886270	−	0.463168i	$-0.153288\pi$
$977$	54130.0	1.77254	0.886270	+	0.463168i	$0.153288\pi$
$978$	0	0
$979$	40248.0	1.31392
$980$	0	0
$981$	12870.0	0.418866
$982$	0	0
$983$	23064.0	0.748349	0.374175	−	0.927358i	$-0.377926\pi$
$983$	23064.0	0.748349	0.374175	+	0.927358i	$0.377926\pi$
$984$	0	0
$985$	2148.00	0.0694832
$986$	0	0
$987$	9744.00	0.314240
$988$	0	0
$989$	19680.0	0.632748
$990$	0	0
$991$	10768.0	0.345163	0.172582	−	0.984995i	$-0.444789\pi$
$991$	10768.0	0.345163	0.172582	+	0.984995i	$0.444789\pi$
$992$	0	0
$993$	−32340.0	−1.03351
$994$	0	0
$995$	1104.00	0.0351750
$996$	0	0
$997$	12766.0	0.405520	0.202760	−	0.979228i	$-0.435009\pi$
$997$	12766.0	0.405520	0.202760	+	0.979228i	$0.435009\pi$
$998$	0	0
$999$	7830.00	0.247978

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.a.c.1.1		1
3.2	odd	2		1008.4.a.l.1.1		1
4.3	odd	2		168.4.a.f.1.1	✓	1
7.6	odd	2		2352.4.a.bb.1.1		1
8.3	odd	2		1344.4.a.g.1.1		1
8.5	even	2		1344.4.a.v.1.1		1
12.11	even	2		504.4.a.c.1.1		1
28.27	even	2		1176.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.f.1.1	✓	1	4.3	odd	2
336.4.a.c.1.1		1	1.1	even	1	trivial
504.4.a.c.1.1		1	12.11	even	2
1008.4.a.l.1.1		1	3.2	odd	2
1176.4.a.e.1.1		1	28.27	even	2
1344.4.a.g.1.1		1	8.3	odd	2
1344.4.a.v.1.1		1	8.5	even	2
2352.4.a.bb.1.1		1	7.6	odd	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 $\F_{q}$
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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	336.4.a.c.1.1

Level	$336$
Weight	$4$
Character	336.1
Self dual	yes
Analytic conductor	$19.825$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$