LMFDB - Embedded newform 336.4.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,4,Mod(1,336)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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)

$f(q)$

$=$

$q+3.00000 q^{3} -10.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -10.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +12.0000 q^{11} +30.0000 q^{13} -30.0000 q^{15} +34.0000 q^{17} -148.000 q^{19} -21.0000 q^{21} -152.000 q^{23} -25.0000 q^{25} +27.0000 q^{27} -106.000 q^{29} -304.000 q^{31} +36.0000 q^{33} +70.0000 q^{35} -114.000 q^{37} +90.0000 q^{39} +202.000 q^{41} -116.000 q^{43} -90.0000 q^{45} -224.000 q^{47} +49.0000 q^{49} +102.000 q^{51} -274.000 q^{53} -120.000 q^{55} -444.000 q^{57} +660.000 q^{59} +382.000 q^{61} -63.0000 q^{63} -300.000 q^{65} -12.0000 q^{67} -456.000 q^{69} +552.000 q^{71} -614.000 q^{73} -75.0000 q^{75} -84.0000 q^{77} -880.000 q^{79} +81.0000 q^{81} +108.000 q^{83} -340.000 q^{85} -318.000 q^{87} -86.0000 q^{89} -210.000 q^{91} -912.000 q^{93} +1480.00 q^{95} +1426.00 q^{97} +108.000 q^{99} +O(q^{100})$

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
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−10.0000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−10.0000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	12.0000	0.328921	0.164461	−	0.986384i	$-0.447412\pi$
$11$	12.0000	0.328921	0.164461	+	0.986384i	$0.447412\pi$
$12$	0	0
$13$	30.0000	0.640039	0.320019	−	0.947411i	$-0.396311\pi$
$13$	30.0000	0.640039	0.320019	+	0.947411i	$0.396311\pi$
$14$	0	0
$15$	−30.0000	−0.516398
$16$	0	0
$17$	34.0000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	34.0000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−148.000	−1.78703	−0.893514	−	0.449036i	$-0.851768\pi$
$19$	−148.000	−1.78703	−0.893514	+	0.449036i	$0.851768\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	−152.000	−1.37801	−0.689004	−	0.724757i	$-0.741952\pi$
$23$	−152.000	−1.37801	−0.689004	+	0.724757i	$0.741952\pi$
$24$	0	0
$25$	−25.0000	−0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−106.000	−0.678748	−0.339374	−	0.940651i	$-0.610215\pi$
$29$	−106.000	−0.678748	−0.339374	+	0.940651i	$0.610215\pi$
$30$	0	0
$31$	−304.000	−1.76129	−0.880645	−	0.473776i	$-0.842891\pi$
$31$	−304.000	−1.76129	−0.880645	+	0.473776i	$0.842891\pi$
$32$	0	0
$33$	36.0000	0.189903
$34$	0	0
$35$	70.0000	0.338062
$36$	0	0
$37$	−114.000	−0.506527	−0.253263	−	0.967397i	$-0.581504\pi$
$37$	−114.000	−0.506527	−0.253263	+	0.967397i	$0.581504\pi$
$38$	0	0
$39$	90.0000	0.369527
$40$	0	0
$41$	202.000	0.769441	0.384721	−	0.923033i	$-0.374298\pi$
$41$	202.000	0.769441	0.384721	+	0.923033i	$0.374298\pi$
$42$	0	0
$43$	−116.000	−0.411391	−0.205696	−	0.978616i	$-0.565946\pi$
$43$	−116.000	−0.411391	−0.205696	+	0.978616i	$0.565946\pi$
$44$	0	0
$45$	−90.0000	−0.298142
$46$	0	0
$47$	−224.000	−0.695186	−0.347593	−	0.937645i	$-0.613001\pi$
$47$	−224.000	−0.695186	−0.347593	+	0.937645i	$0.613001\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	102.000	0.280056
$52$	0	0
$53$	−274.000	−0.710128	−0.355064	−	0.934842i	$-0.615541\pi$
$53$	−274.000	−0.710128	−0.355064	+	0.934842i	$0.615541\pi$
$54$	0	0
$55$	−120.000	−0.294196
$56$	0	0
$57$	−444.000	−1.03174
$58$	0	0
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	382.000	0.801805	0.400902	−	0.916121i	$-0.368697\pi$
$61$	382.000	0.801805	0.400902	+	0.916121i	$0.368697\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−300.000	−0.572468
$66$	0	0
$67$	−12.0000	−0.0218811	−0.0109405	−	0.999940i	$-0.503483\pi$
$67$	−12.0000	−0.0218811	−0.0109405	+	0.999940i	$0.503483\pi$
$68$	0	0
$69$	−456.000	−0.795593
$70$	0	0
$71$	552.000	0.922681	0.461340	−	0.887223i	$-0.347369\pi$
$71$	552.000	0.922681	0.461340	+	0.887223i	$0.347369\pi$
$72$	0	0
$73$	−614.000	−0.984428	−0.492214	−	0.870474i	$-0.663812\pi$
$73$	−614.000	−0.984428	−0.492214	+	0.870474i	$0.663812\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−84.0000	−0.124321
$78$	0	0
$79$	−880.000	−1.25326	−0.626631	−	0.779316i	$-0.715567\pi$
$79$	−880.000	−1.25326	−0.626631	+	0.779316i	$0.715567\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	108.000	0.142826	0.0714129	−	0.997447i	$-0.477249\pi$
$83$	108.000	0.142826	0.0714129	+	0.997447i	$0.477249\pi$
$84$	0	0
$85$	−340.000	−0.433861
$86$	0	0
$87$	−318.000	−0.391876
$88$	0	0
$89$	−86.0000	−0.102427	−0.0512134	−	0.998688i	$-0.516309\pi$
$89$	−86.0000	−0.102427	−0.0512134	+	0.998688i	$0.516309\pi$
$90$	0	0
$91$	−210.000	−0.241912
$92$	0	0
$93$	−912.000	−1.01688
$94$	0	0
$95$	1480.00	1.59837
$96$	0	0
$97$	1426.00	1.49266	0.746332	−	0.665574i	$-0.231813\pi$
$97$	1426.00	1.49266	0.746332	+	0.665574i	$0.231813\pi$
$98$	0	0
$99$	108.000	0.109640
$100$	0	0
$101$	−810.000	−0.798000	−0.399000	−	0.916951i	$-0.630643\pi$
$101$	−810.000	−0.798000	−0.399000	+	0.916951i	$0.630643\pi$
$102$	0	0
$103$	−1032.00	−0.987243	−0.493621	−	0.869677i	$-0.664327\pi$
$103$	−1032.00	−0.987243	−0.493621	+	0.869677i	$0.664327\pi$
$104$	0	0
$105$	210.000	0.195180
$106$	0	0
$107$	−900.000	−0.813143	−0.406571	−	0.913619i	$-0.633276\pi$
$107$	−900.000	−0.813143	−0.406571	+	0.913619i	$0.633276\pi$
$108$	0	0
$109$	454.000	0.398948	0.199474	−	0.979903i	$-0.436077\pi$
$109$	454.000	0.398948	0.199474	+	0.979903i	$0.436077\pi$
$110$	0	0
$111$	−342.000	−0.292443
$112$	0	0
$113$	18.0000	0.0149849	0.00749247	−	0.999972i	$-0.497615\pi$
$113$	18.0000	0.0149849	0.00749247	+	0.999972i	$0.497615\pi$
$114$	0	0
$115$	1520.00	1.23253
$116$	0	0
$117$	270.000	0.213346
$118$	0	0
$119$	−238.000	−0.183340
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	606.000	0.444237
$124$	0	0
$125$	1500.00	1.07331
$126$	0	0
$127$	1072.00	0.749013	0.374506	−	0.927224i	$-0.377812\pi$
$127$	1072.00	0.749013	0.374506	+	0.927224i	$0.377812\pi$
$128$	0	0
$129$	−348.000	−0.237517
$130$	0	0
$131$	1084.00	0.722973	0.361487	−	0.932377i	$-0.382269\pi$
$131$	1084.00	0.722973	0.361487	+	0.932377i	$0.382269\pi$
$132$	0	0
$133$	1036.00	0.675433
$134$	0	0
$135$	−270.000	−0.172133
$136$	0	0
$137$	2426.00	1.51290	0.756450	−	0.654052i	$-0.226932\pi$
$137$	2426.00	1.51290	0.756450	+	0.654052i	$0.226932\pi$
$138$	0	0
$139$	−684.000	−0.417382	−0.208691	−	0.977982i	$-0.566920\pi$
$139$	−684.000	−0.417382	−0.208691	+	0.977982i	$0.566920\pi$
$140$	0	0
$141$	−672.000	−0.401366
$142$	0	0
$143$	360.000	0.210522
$144$	0	0
$145$	1060.00	0.607091
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	3598.00	1.97825	0.989126	−	0.147068i	$-0.0469837\pi$
$149$	3598.00	1.97825	0.989126	+	0.147068i	$0.0469837\pi$
$150$	0	0
$151$	−2680.00	−1.44434	−0.722170	−	0.691716i	$-0.756855\pi$
$151$	−2680.00	−1.44434	−0.722170	+	0.691716i	$0.756855\pi$
$152$	0	0
$153$	306.000	0.161690
$154$	0	0
$155$	3040.00	1.57535
$156$	0	0
$157$	1422.00	0.722853	0.361427	−	0.932401i	$-0.382290\pi$
$157$	1422.00	0.722853	0.361427	+	0.932401i	$0.382290\pi$
$158$	0	0
$159$	−822.000	−0.409993
$160$	0	0
$161$	1064.00	0.520838
$162$	0	0
$163$	−220.000	−0.105716	−0.0528581	−	0.998602i	$-0.516833\pi$
$163$	−220.000	−0.105716	−0.0528581	+	0.998602i	$0.516833\pi$
$164$	0	0
$165$	−360.000	−0.169854
$166$	0	0
$167$	−1992.00	−0.923027	−0.461514	−	0.887133i	$-0.652693\pi$
$167$	−1992.00	−0.923027	−0.461514	+	0.887133i	$0.652693\pi$
$168$	0	0
$169$	−1297.00	−0.590350
$170$	0	0
$171$	−1332.00	−0.595676
$172$	0	0
$173$	4286.00	1.88358	0.941788	−	0.336208i	$-0.109145\pi$
$173$	4286.00	1.88358	0.941788	+	0.336208i	$0.109145\pi$
$174$	0	0
$175$	175.000	0.0755929
$176$	0	0
$177$	1980.00	0.840824
$178$	0	0
$179$	4516.00	1.88571	0.942854	−	0.333206i	$-0.108131\pi$
$179$	4516.00	1.88571	0.942854	+	0.333206i	$0.108131\pi$
$180$	0	0
$181$	−2218.00	−0.910843	−0.455422	−	0.890276i	$-0.650511\pi$
$181$	−2218.00	−0.910843	−0.455422	+	0.890276i	$0.650511\pi$
$182$	0	0
$183$	1146.00	0.462922
$184$	0	0
$185$	1140.00	0.453051
$186$	0	0
$187$	408.000	0.159550
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	−3136.00	−1.18803	−0.594013	−	0.804455i	$-0.702457\pi$
$191$	−3136.00	−1.18803	−0.594013	+	0.804455i	$0.702457\pi$
$192$	0	0
$193$	−3198.00	−1.19273	−0.596365	−	0.802713i	$-0.703389\pi$
$193$	−3198.00	−1.19273	−0.596365	+	0.802713i	$0.703389\pi$
$194$	0	0
$195$	−900.000	−0.330515
$196$	0	0
$197$	−3714.00	−1.34321	−0.671603	−	0.740911i	$-0.734394\pi$
$197$	−3714.00	−1.34321	−0.671603	+	0.740911i	$0.734394\pi$
$198$	0	0
$199$	3256.00	1.15986	0.579929	−	0.814667i	$-0.303080\pi$
$199$	3256.00	1.15986	0.579929	+	0.814667i	$0.303080\pi$
$200$	0	0
$201$	−36.0000	−0.0126331
$202$	0	0
$203$	742.000	0.256543
$204$	0	0
$205$	−2020.00	−0.688209
$206$	0	0
$207$	−1368.00	−0.459336
$208$	0	0
$209$	−1776.00	−0.587792
$210$	0	0
$211$	4612.00	1.50475	0.752377	−	0.658733i	$-0.228907\pi$
$211$	4612.00	1.50475	0.752377	+	0.658733i	$0.228907\pi$
$212$	0	0
$213$	1656.00	0.532710
$214$	0	0
$215$	1160.00	0.367960
$216$	0	0
$217$	2128.00	0.665705
$218$	0	0
$219$	−1842.00	−0.568360
$220$	0	0
$221$	1020.00	0.310464
$222$	0	0
$223$	−2976.00	−0.893667	−0.446833	−	0.894617i	$-0.647448\pi$
$223$	−2976.00	−0.893667	−0.446833	+	0.894617i	$0.647448\pi$
$224$	0	0
$225$	−225.000	−0.0666667
$226$	0	0
$227$	1084.00	0.316950	0.158475	−	0.987363i	$-0.449342\pi$
$227$	1084.00	0.316950	0.158475	+	0.987363i	$0.449342\pi$
$228$	0	0
$229$	−2746.00	−0.792405	−0.396203	−	0.918163i	$-0.629672\pi$
$229$	−2746.00	−0.792405	−0.396203	+	0.918163i	$0.629672\pi$
$230$	0	0
$231$	−252.000	−0.0717765
$232$	0	0
$233$	−5062.00	−1.42327	−0.711637	−	0.702548i	$-0.752046\pi$
$233$	−5062.00	−1.42327	−0.711637	+	0.702548i	$0.752046\pi$
$234$	0	0
$235$	2240.00	0.621794
$236$	0	0
$237$	−2640.00	−0.723571
$238$	0	0
$239$	−320.000	−0.0866070	−0.0433035	−	0.999062i	$-0.513788\pi$
$239$	−320.000	−0.0866070	−0.0433035	+	0.999062i	$0.513788\pi$
$240$	0	0
$241$	−6974.00	−1.86404	−0.932022	−	0.362400i	$-0.881957\pi$
$241$	−6974.00	−1.86404	−0.932022	+	0.362400i	$0.881957\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−490.000	−0.127775
$246$	0	0
$247$	−4440.00	−1.14377
$248$	0	0
$249$	324.000	0.0824605
$250$	0	0
$251$	356.000	0.0895240	0.0447620	−	0.998998i	$-0.485747\pi$
$251$	356.000	0.0895240	0.0447620	+	0.998998i	$0.485747\pi$
$252$	0	0
$253$	−1824.00	−0.453257
$254$	0	0
$255$	−1020.00	−0.250490
$256$	0	0
$257$	4770.00	1.15776	0.578880	−	0.815413i	$-0.303490\pi$
$257$	4770.00	1.15776	0.578880	+	0.815413i	$0.303490\pi$
$258$	0	0
$259$	798.000	0.191449
$260$	0	0
$261$	−954.000	−0.226249
$262$	0	0
$263$	−2712.00	−0.635852	−0.317926	−	0.948116i	$-0.602986\pi$
$263$	−2712.00	−0.635852	−0.317926	+	0.948116i	$0.602986\pi$
$264$	0	0
$265$	2740.00	0.635158
$266$	0	0
$267$	−258.000	−0.0591361
$268$	0	0
$269$	6942.00	1.57346	0.786731	−	0.617296i	$-0.211772\pi$
$269$	6942.00	1.57346	0.786731	+	0.617296i	$0.211772\pi$
$270$	0	0
$271$	4720.00	1.05801	0.529003	−	0.848620i	$-0.322566\pi$
$271$	4720.00	1.05801	0.529003	+	0.848620i	$0.322566\pi$
$272$	0	0
$273$	−630.000	−0.139668
$274$	0	0
$275$	−300.000	−0.0657843
$276$	0	0
$277$	−1490.00	−0.323196	−0.161598	−	0.986857i	$-0.551665\pi$
$277$	−1490.00	−0.323196	−0.161598	+	0.986857i	$0.551665\pi$
$278$	0	0
$279$	−2736.00	−0.587097
$280$	0	0
$281$	−7926.00	−1.68265	−0.841327	−	0.540527i	$-0.818225\pi$
$281$	−7926.00	−1.68265	−0.841327	+	0.540527i	$0.818225\pi$
$282$	0	0
$283$	8596.00	1.80558	0.902790	−	0.430082i	$-0.141515\pi$
$283$	8596.00	1.80558	0.902790	+	0.430082i	$0.141515\pi$
$284$	0	0
$285$	4440.00	0.922817
$286$	0	0
$287$	−1414.00	−0.290822
$288$	0	0
$289$	−3757.00	−0.764706
$290$	0	0
$291$	4278.00	0.861790
$292$	0	0
$293$	438.000	0.0873319	0.0436659	−	0.999046i	$-0.486096\pi$
$293$	438.000	0.0873319	0.0436659	+	0.999046i	$0.486096\pi$
$294$	0	0
$295$	−6600.00	−1.30260
$296$	0	0
$297$	324.000	0.0633010
$298$	0	0
$299$	−4560.00	−0.881979
$300$	0	0
$301$	812.000	0.155491
$302$	0	0
$303$	−2430.00	−0.460726
$304$	0	0
$305$	−3820.00	−0.717156
$306$	0	0
$307$	−5140.00	−0.955555	−0.477777	−	0.878481i	$-0.658557\pi$
$307$	−5140.00	−0.955555	−0.477777	+	0.878481i	$0.658557\pi$
$308$	0	0
$309$	−3096.00	−0.569985
$310$	0	0
$311$	−4296.00	−0.783292	−0.391646	−	0.920116i	$-0.628094\pi$
$311$	−4296.00	−0.783292	−0.391646	+	0.920116i	$0.628094\pi$
$312$	0	0
$313$	2714.00	0.490110	0.245055	−	0.969509i	$-0.421194\pi$
$313$	2714.00	0.490110	0.245055	+	0.969509i	$0.421194\pi$
$314$	0	0
$315$	630.000	0.112687
$316$	0	0
$317$	−4314.00	−0.764348	−0.382174	−	0.924090i	$-0.624825\pi$
$317$	−4314.00	−0.764348	−0.382174	+	0.924090i	$0.624825\pi$
$318$	0	0
$319$	−1272.00	−0.223255
$320$	0	0
$321$	−2700.00	−0.469468
$322$	0	0
$323$	−5032.00	−0.866836
$324$	0	0
$325$	−750.000	−0.128008
$326$	0	0
$327$	1362.00	0.230333
$328$	0	0
$329$	1568.00	0.262756
$330$	0	0
$331$	2284.00	0.379275	0.189637	−	0.981854i	$-0.439269\pi$
$331$	2284.00	0.379275	0.189637	+	0.981854i	$0.439269\pi$
$332$	0	0
$333$	−1026.00	−0.168842
$334$	0	0
$335$	120.000	0.0195710
$336$	0	0
$337$	9650.00	1.55985	0.779924	−	0.625874i	$-0.215258\pi$
$337$	9650.00	1.55985	0.779924	+	0.625874i	$0.215258\pi$
$338$	0	0
$339$	54.0000	0.00865156
$340$	0	0
$341$	−3648.00	−0.579326
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	4560.00	0.711600
$346$	0	0
$347$	−4068.00	−0.629342	−0.314671	−	0.949201i	$-0.601894\pi$
$347$	−4068.00	−0.629342	−0.314671	+	0.949201i	$0.601894\pi$
$348$	0	0
$349$	−3922.00	−0.601547	−0.300773	−	0.953696i	$-0.597245\pi$
$349$	−3922.00	−0.601547	−0.300773	+	0.953696i	$0.597245\pi$
$350$	0	0
$351$	810.000	0.123176
$352$	0	0
$353$	−414.000	−0.0624221	−0.0312110	−	0.999513i	$-0.509936\pi$
$353$	−414.000	−0.0624221	−0.0312110	+	0.999513i	$0.509936\pi$
$354$	0	0
$355$	−5520.00	−0.825271
$356$	0	0
$357$	−714.000	−0.105851
$358$	0	0
$359$	−5160.00	−0.758592	−0.379296	−	0.925275i	$-0.623834\pi$
$359$	−5160.00	−0.758592	−0.379296	+	0.925275i	$0.623834\pi$
$360$	0	0
$361$	15045.0	2.19347
$362$	0	0
$363$	−3561.00	−0.514887
$364$	0	0
$365$	6140.00	0.880499
$366$	0	0
$367$	11056.0	1.57253	0.786265	−	0.617889i	$-0.212012\pi$
$367$	11056.0	1.57253	0.786265	+	0.617889i	$0.212012\pi$
$368$	0	0
$369$	1818.00	0.256480
$370$	0	0
$371$	1918.00	0.268403
$372$	0	0
$373$	−7762.00	−1.07748	−0.538741	−	0.842471i	$-0.681100\pi$
$373$	−7762.00	−1.07748	−0.538741	+	0.842471i	$0.681100\pi$
$374$	0	0
$375$	4500.00	0.619677
$376$	0	0
$377$	−3180.00	−0.434425
$378$	0	0
$379$	−12228.0	−1.65728	−0.828641	−	0.559780i	$-0.810886\pi$
$379$	−12228.0	−1.65728	−0.828641	+	0.559780i	$0.810886\pi$
$380$	0	0
$381$	3216.00	0.432443
$382$	0	0
$383$	−336.000	−0.0448271	−0.0224136	−	0.999749i	$-0.507135\pi$
$383$	−336.000	−0.0448271	−0.0224136	+	0.999749i	$0.507135\pi$
$384$	0	0
$385$	840.000	0.111196
$386$	0	0
$387$	−1044.00	−0.137130
$388$	0	0
$389$	1966.00	0.256247	0.128124	−	0.991758i	$-0.459105\pi$
$389$	1966.00	0.256247	0.128124	+	0.991758i	$0.459105\pi$
$390$	0	0
$391$	−5168.00	−0.668432
$392$	0	0
$393$	3252.00	0.417409
$394$	0	0
$395$	8800.00	1.12095
$396$	0	0
$397$	4606.00	0.582288	0.291144	−	0.956679i	$-0.405964\pi$
$397$	4606.00	0.582288	0.291144	+	0.956679i	$0.405964\pi$
$398$	0	0
$399$	3108.00	0.389961
$400$	0	0
$401$	−13038.0	−1.62366	−0.811829	−	0.583896i	$-0.801528\pi$
$401$	−13038.0	−1.62366	−0.811829	+	0.583896i	$0.801528\pi$
$402$	0	0
$403$	−9120.00	−1.12729
$404$	0	0
$405$	−810.000	−0.0993808
$406$	0	0
$407$	−1368.00	−0.166607
$408$	0	0
$409$	7290.00	0.881338	0.440669	−	0.897670i	$-0.354741\pi$
$409$	7290.00	0.881338	0.440669	+	0.897670i	$0.354741\pi$
$410$	0	0
$411$	7278.00	0.873473
$412$	0	0
$413$	−4620.00	−0.550449
$414$	0	0
$415$	−1080.00	−0.127747
$416$	0	0
$417$	−2052.00	−0.240976
$418$	0	0
$419$	5836.00	0.680447	0.340223	−	0.940345i	$-0.389497\pi$
$419$	5836.00	0.680447	0.340223	+	0.940345i	$0.389497\pi$
$420$	0	0
$421$	2398.00	0.277604	0.138802	−	0.990320i	$-0.455675\pi$
$421$	2398.00	0.277604	0.138802	+	0.990320i	$0.455675\pi$
$422$	0	0
$423$	−2016.00	−0.231729
$424$	0	0
$425$	−850.000	−0.0970143
$426$	0	0
$427$	−2674.00	−0.303054
$428$	0	0
$429$	1080.00	0.121545
$430$	0	0
$431$	17568.0	1.96339	0.981695	−	0.190462i	$-0.0609985\pi$
$431$	17568.0	1.96339	0.981695	+	0.190462i	$0.0609985\pi$
$432$	0	0
$433$	4354.00	0.483233	0.241616	−	0.970372i	$-0.422322\pi$
$433$	4354.00	0.483233	0.241616	+	0.970372i	$0.422322\pi$
$434$	0	0
$435$	3180.00	0.350504
$436$	0	0
$437$	22496.0	2.46254
$438$	0	0
$439$	14616.0	1.58903	0.794514	−	0.607245i	$-0.207725\pi$
$439$	14616.0	1.58903	0.794514	+	0.607245i	$0.207725\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	−14516.0	−1.55683	−0.778415	−	0.627750i	$-0.783976\pi$
$443$	−14516.0	−1.55683	−0.778415	+	0.627750i	$0.783976\pi$
$444$	0	0
$445$	860.000	0.0916133
$446$	0	0
$447$	10794.0	1.14214
$448$	0	0
$449$	−14462.0	−1.52005	−0.760027	−	0.649892i	$-0.774814\pi$
$449$	−14462.0	−1.52005	−0.760027	+	0.649892i	$0.774814\pi$
$450$	0	0
$451$	2424.00	0.253086
$452$	0	0
$453$	−8040.00	−0.833890
$454$	0	0
$455$	2100.00	0.216373
$456$	0	0
$457$	3610.00	0.369516	0.184758	−	0.982784i	$-0.440850\pi$
$457$	3610.00	0.369516	0.184758	+	0.982784i	$0.440850\pi$
$458$	0	0
$459$	918.000	0.0933520
$460$	0	0
$461$	−3138.00	−0.317031	−0.158515	−	0.987356i	$-0.550671\pi$
$461$	−3138.00	−0.317031	−0.158515	+	0.987356i	$0.550671\pi$
$462$	0	0
$463$	−6896.00	−0.692191	−0.346095	−	0.938199i	$-0.612493\pi$
$463$	−6896.00	−0.692191	−0.346095	+	0.938199i	$0.612493\pi$
$464$	0	0
$465$	9120.00	0.909527
$466$	0	0
$467$	1356.00	0.134364	0.0671822	−	0.997741i	$-0.478599\pi$
$467$	1356.00	0.134364	0.0671822	+	0.997741i	$0.478599\pi$
$468$	0	0
$469$	84.0000	0.00827028
$470$	0	0
$471$	4266.00	0.417339
$472$	0	0
$473$	−1392.00	−0.135315
$474$	0	0
$475$	3700.00	0.357406
$476$	0	0
$477$	−2466.00	−0.236709
$478$	0	0
$479$	−9136.00	−0.871471	−0.435735	−	0.900075i	$-0.643512\pi$
$479$	−9136.00	−0.871471	−0.435735	+	0.900075i	$0.643512\pi$
$480$	0	0
$481$	−3420.00	−0.324197
$482$	0	0
$483$	3192.00	0.300706
$484$	0	0
$485$	−14260.0	−1.33508
$486$	0	0
$487$	−4072.00	−0.378891	−0.189446	−	0.981891i	$-0.560669\pi$
$487$	−4072.00	−0.378891	−0.189446	+	0.981891i	$0.560669\pi$
$488$	0	0
$489$	−660.000	−0.0610352
$490$	0	0
$491$	−12420.0	−1.14156	−0.570781	−	0.821102i	$-0.693359\pi$
$491$	−12420.0	−1.14156	−0.570781	+	0.821102i	$0.693359\pi$
$492$	0	0
$493$	−3604.00	−0.329241
$494$	0	0
$495$	−1080.00	−0.0980654
$496$	0	0
$497$	−3864.00	−0.348741
$498$	0	0
$499$	13892.0	1.24628	0.623138	−	0.782112i	$-0.285858\pi$
$499$	13892.0	1.24628	0.623138	+	0.782112i	$0.285858\pi$
$500$	0	0
$501$	−5976.00	−0.532910
$502$	0	0
$503$	20408.0	1.80904	0.904521	−	0.426430i	$-0.140229\pi$
$503$	20408.0	1.80904	0.904521	+	0.426430i	$0.140229\pi$
$504$	0	0
$505$	8100.00	0.713753
$506$	0	0
$507$	−3891.00	−0.340839
$508$	0	0
$509$	−3666.00	−0.319239	−0.159619	−	0.987179i	$-0.551027\pi$
$509$	−3666.00	−0.319239	−0.159619	+	0.987179i	$0.551027\pi$
$510$	0	0
$511$	4298.00	0.372079
$512$	0	0
$513$	−3996.00	−0.343914
$514$	0	0
$515$	10320.0	0.883017
$516$	0	0
$517$	−2688.00	−0.228662
$518$	0	0
$519$	12858.0	1.08748
$520$	0	0
$521$	−5046.00	−0.424317	−0.212159	−	0.977235i	$-0.568049\pi$
$521$	−5046.00	−0.424317	−0.212159	+	0.977235i	$0.568049\pi$
$522$	0	0
$523$	18628.0	1.55745	0.778724	−	0.627366i	$-0.215867\pi$
$523$	18628.0	1.55745	0.778724	+	0.627366i	$0.215867\pi$
$524$	0	0
$525$	525.000	0.0436436
$526$	0	0
$527$	−10336.0	−0.854351
$528$	0	0
$529$	10937.0	0.898907
$530$	0	0
$531$	5940.00	0.485450
$532$	0	0
$533$	6060.00	0.492472
$534$	0	0
$535$	9000.00	0.727297
$536$	0	0
$537$	13548.0	1.08871
$538$	0	0
$539$	588.000	0.0469888
$540$	0	0
$541$	−5594.00	−0.444556	−0.222278	−	0.974983i	$-0.571349\pi$
$541$	−5594.00	−0.444556	−0.222278	+	0.974983i	$0.571349\pi$
$542$	0	0
$543$	−6654.00	−0.525876
$544$	0	0
$545$	−4540.00	−0.356830
$546$	0	0
$547$	−13372.0	−1.04524	−0.522619	−	0.852566i	$-0.675045\pi$
$547$	−13372.0	−1.04524	−0.522619	+	0.852566i	$0.675045\pi$
$548$	0	0
$549$	3438.00	0.267268
$550$	0	0
$551$	15688.0	1.21294
$552$	0	0
$553$	6160.00	0.473689
$554$	0	0
$555$	3420.00	0.261569
$556$	0	0
$557$	15478.0	1.17742	0.588711	−	0.808344i	$-0.299636\pi$
$557$	15478.0	1.17742	0.588711	+	0.808344i	$0.299636\pi$
$558$	0	0
$559$	−3480.00	−0.263306
$560$	0	0
$561$	1224.00	0.0921164
$562$	0	0
$563$	−12964.0	−0.970457	−0.485229	−	0.874387i	$-0.661264\pi$
$563$	−12964.0	−0.970457	−0.485229	+	0.874387i	$0.661264\pi$
$564$	0	0
$565$	−180.000	−0.0134029
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	10506.0	0.774050	0.387025	−	0.922069i	$-0.373503\pi$
$569$	10506.0	0.774050	0.387025	+	0.922069i	$0.373503\pi$
$570$	0	0
$571$	6268.00	0.459383	0.229691	−	0.973263i	$-0.426228\pi$
$571$	6268.00	0.459383	0.229691	+	0.973263i	$0.426228\pi$
$572$	0	0
$573$	−9408.00	−0.685907
$574$	0	0
$575$	3800.00	0.275602
$576$	0	0
$577$	−5790.00	−0.417748	−0.208874	−	0.977943i	$-0.566980\pi$
$577$	−5790.00	−0.417748	−0.208874	+	0.977943i	$0.566980\pi$
$578$	0	0
$579$	−9594.00	−0.688624
$580$	0	0
$581$	−756.000	−0.0539831
$582$	0	0
$583$	−3288.00	−0.233576
$584$	0	0
$585$	−2700.00	−0.190823
$586$	0	0
$587$	−8332.00	−0.585858	−0.292929	−	0.956134i	$-0.594630\pi$
$587$	−8332.00	−0.585858	−0.292929	+	0.956134i	$0.594630\pi$
$588$	0	0
$589$	44992.0	3.14748
$590$	0	0
$591$	−11142.0	−0.775500
$592$	0	0
$593$	−23310.0	−1.61421	−0.807105	−	0.590407i	$-0.798967\pi$
$593$	−23310.0	−1.61421	−0.807105	+	0.590407i	$0.798967\pi$
$594$	0	0
$595$	2380.00	0.163984
$596$	0	0
$597$	9768.00	0.669644
$598$	0	0
$599$	−18184.0	−1.24036	−0.620182	−	0.784458i	$-0.712941\pi$
$599$	−18184.0	−1.24036	−0.620182	+	0.784458i	$0.712941\pi$
$600$	0	0
$601$	−9398.00	−0.637858	−0.318929	−	0.947779i	$-0.603323\pi$
$601$	−9398.00	−0.637858	−0.318929	+	0.947779i	$0.603323\pi$
$602$	0	0
$603$	−108.000	−0.00729370
$604$	0	0
$605$	11870.0	0.797660
$606$	0	0
$607$	20192.0	1.35019	0.675097	−	0.737729i	$-0.264102\pi$
$607$	20192.0	1.35019	0.675097	+	0.737729i	$0.264102\pi$
$608$	0	0
$609$	2226.00	0.148115
$610$	0	0
$611$	−6720.00	−0.444946
$612$	0	0
$613$	11758.0	0.774716	0.387358	−	0.921929i	$-0.373388\pi$
$613$	11758.0	0.774716	0.387358	+	0.921929i	$0.373388\pi$
$614$	0	0
$615$	−6060.00	−0.397338
$616$	0	0
$617$	−2310.00	−0.150725	−0.0753623	−	0.997156i	$-0.524011\pi$
$617$	−2310.00	−0.150725	−0.0753623	+	0.997156i	$0.524011\pi$
$618$	0	0
$619$	−19356.0	−1.25684	−0.628419	−	0.777875i	$-0.716298\pi$
$619$	−19356.0	−1.25684	−0.628419	+	0.777875i	$0.716298\pi$
$620$	0	0
$621$	−4104.00	−0.265198
$622$	0	0
$623$	602.000	0.0387137
$624$	0	0
$625$	−11875.0	−0.760000
$626$	0	0
$627$	−5328.00	−0.339362
$628$	0	0
$629$	−3876.00	−0.245701
$630$	0	0
$631$	7768.00	0.490078	0.245039	−	0.969513i	$-0.421199\pi$
$631$	7768.00	0.490078	0.245039	+	0.969513i	$0.421199\pi$
$632$	0	0
$633$	13836.0	0.868770
$634$	0	0
$635$	−10720.0	−0.669937
$636$	0	0
$637$	1470.00	0.0914341
$638$	0	0
$639$	4968.00	0.307560
$640$	0	0
$641$	−11102.0	−0.684091	−0.342046	−	0.939683i	$-0.611120\pi$
$641$	−11102.0	−0.684091	−0.342046	+	0.939683i	$0.611120\pi$
$642$	0	0
$643$	−22612.0	−1.38683	−0.693414	−	0.720540i	$-0.743894\pi$
$643$	−22612.0	−1.38683	−0.693414	+	0.720540i	$0.743894\pi$
$644$	0	0
$645$	3480.00	0.212442
$646$	0	0
$647$	1656.00	0.100625	0.0503123	−	0.998734i	$-0.483978\pi$
$647$	1656.00	0.100625	0.0503123	+	0.998734i	$0.483978\pi$
$648$	0	0
$649$	7920.00	0.479025
$650$	0	0
$651$	6384.00	0.384345
$652$	0	0
$653$	16070.0	0.963044	0.481522	−	0.876434i	$-0.340084\pi$
$653$	16070.0	0.963044	0.481522	+	0.876434i	$0.340084\pi$
$654$	0	0
$655$	−10840.0	−0.646647
$656$	0	0
$657$	−5526.00	−0.328143
$658$	0	0
$659$	−21740.0	−1.28508	−0.642542	−	0.766251i	$-0.722120\pi$
$659$	−21740.0	−1.28508	−0.642542	+	0.766251i	$0.722120\pi$
$660$	0	0
$661$	−31642.0	−1.86192	−0.930962	−	0.365117i	$-0.881029\pi$
$661$	−31642.0	−1.86192	−0.930962	+	0.365117i	$0.881029\pi$
$662$	0	0
$663$	3060.00	0.179247
$664$	0	0
$665$	−10360.0	−0.604126
$666$	0	0
$667$	16112.0	0.935321
$668$	0	0
$669$	−8928.00	−0.515959
$670$	0	0
$671$	4584.00	0.263731
$672$	0	0
$673$	−27518.0	−1.57614	−0.788069	−	0.615587i	$-0.788919\pi$
$673$	−27518.0	−1.57614	−0.788069	+	0.615587i	$0.788919\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−7290.00	−0.413851	−0.206926	−	0.978357i	$-0.566346\pi$
$677$	−7290.00	−0.413851	−0.206926	+	0.978357i	$0.566346\pi$
$678$	0	0
$679$	−9982.00	−0.564174
$680$	0	0
$681$	3252.00	0.182991
$682$	0	0
$683$	−3684.00	−0.206390	−0.103195	−	0.994661i	$-0.532907\pi$
$683$	−3684.00	−0.206390	−0.103195	+	0.994661i	$0.532907\pi$
$684$	0	0
$685$	−24260.0	−1.35318
$686$	0	0
$687$	−8238.00	−0.457495
$688$	0	0
$689$	−8220.00	−0.454510
$690$	0	0
$691$	−24788.0	−1.36466	−0.682330	−	0.731044i	$-0.739033\pi$
$691$	−24788.0	−1.36466	−0.682330	+	0.731044i	$0.739033\pi$
$692$	0	0
$693$	−756.000	−0.0414402
$694$	0	0
$695$	6840.00	0.373318
$696$	0	0
$697$	6868.00	0.373234
$698$	0	0
$699$	−15186.0	−0.821727
$700$	0	0
$701$	15574.0	0.839118	0.419559	−	0.907728i	$-0.362185\pi$
$701$	15574.0	0.839118	0.419559	+	0.907728i	$0.362185\pi$
$702$	0	0
$703$	16872.0	0.905177
$704$	0	0
$705$	6720.00	0.358993
$706$	0	0
$707$	5670.00	0.301616
$708$	0	0
$709$	−24338.0	−1.28919	−0.644593	−	0.764526i	$-0.722973\pi$
$709$	−24338.0	−1.28919	−0.644593	+	0.764526i	$0.722973\pi$
$710$	0	0
$711$	−7920.00	−0.417754
$712$	0	0
$713$	46208.0	2.42707
$714$	0	0
$715$	−3600.00	−0.188297
$716$	0	0
$717$	−960.000	−0.0500026
$718$	0	0
$719$	−17760.0	−0.921191	−0.460595	−	0.887610i	$-0.652364\pi$
$719$	−17760.0	−0.921191	−0.460595	+	0.887610i	$0.652364\pi$
$720$	0	0
$721$	7224.00	0.373143
$722$	0	0
$723$	−20922.0	−1.07621
$724$	0	0
$725$	2650.00	0.135750
$726$	0	0
$727$	−3704.00	−0.188960	−0.0944799	−	0.995527i	$-0.530119\pi$
$727$	−3704.00	−0.188960	−0.0944799	+	0.995527i	$0.530119\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−3944.00	−0.199554
$732$	0	0
$733$	−32354.0	−1.63032	−0.815158	−	0.579238i	$-0.803350\pi$
$733$	−32354.0	−1.63032	−0.815158	+	0.579238i	$0.803350\pi$
$734$	0	0
$735$	−1470.00	−0.0737711
$736$	0	0
$737$	−144.000	−0.00719716
$738$	0	0
$739$	−8860.00	−0.441029	−0.220514	−	0.975384i	$-0.570774\pi$
$739$	−8860.00	−0.441029	−0.220514	+	0.975384i	$0.570774\pi$
$740$	0	0
$741$	−13320.0	−0.660354
$742$	0	0
$743$	−8168.00	−0.403304	−0.201652	−	0.979457i	$-0.564631\pi$
$743$	−8168.00	−0.403304	−0.201652	+	0.979457i	$0.564631\pi$
$744$	0	0
$745$	−35980.0	−1.76940
$746$	0	0
$747$	972.000	0.0476086
$748$	0	0
$749$	6300.00	0.307339
$750$	0	0
$751$	25600.0	1.24388	0.621942	−	0.783063i	$-0.286344\pi$
$751$	25600.0	1.24388	0.621942	+	0.783063i	$0.286344\pi$
$752$	0	0
$753$	1068.00	0.0516867
$754$	0	0
$755$	26800.0	1.29186
$756$	0	0
$757$	14878.0	0.714333	0.357167	−	0.934041i	$-0.383743\pi$
$757$	14878.0	0.714333	0.357167	+	0.934041i	$0.383743\pi$
$758$	0	0
$759$	−5472.00	−0.261688
$760$	0	0
$761$	9690.00	0.461580	0.230790	−	0.973004i	$-0.425869\pi$
$761$	9690.00	0.461580	0.230790	+	0.973004i	$0.425869\pi$
$762$	0	0
$763$	−3178.00	−0.150788
$764$	0	0
$765$	−3060.00	−0.144620
$766$	0	0
$767$	19800.0	0.932121
$768$	0	0
$769$	−10670.0	−0.500351	−0.250176	−	0.968200i	$-0.580488\pi$
$769$	−10670.0	−0.500351	−0.250176	+	0.968200i	$0.580488\pi$
$770$	0	0
$771$	14310.0	0.668433
$772$	0	0
$773$	2326.00	0.108228	0.0541141	−	0.998535i	$-0.482767\pi$
$773$	2326.00	0.108228	0.0541141	+	0.998535i	$0.482767\pi$
$774$	0	0
$775$	7600.00	0.352258
$776$	0	0
$777$	2394.00	0.110533
$778$	0	0
$779$	−29896.0	−1.37501
$780$	0	0
$781$	6624.00	0.303490
$782$	0	0
$783$	−2862.00	−0.130625
$784$	0	0
$785$	−14220.0	−0.646540
$786$	0	0
$787$	−10612.0	−0.480657	−0.240328	−	0.970692i	$-0.577255\pi$
$787$	−10612.0	−0.480657	−0.240328	+	0.970692i	$0.577255\pi$
$788$	0	0
$789$	−8136.00	−0.367109
$790$	0	0
$791$	−126.000	−0.00566377
$792$	0	0
$793$	11460.0	0.513186
$794$	0	0
$795$	8220.00	0.366709
$796$	0	0
$797$	41166.0	1.82958	0.914790	−	0.403931i	$-0.132356\pi$
$797$	41166.0	1.82958	0.914790	+	0.403931i	$0.132356\pi$
$798$	0	0
$799$	−7616.00	−0.337215
$800$	0	0
$801$	−774.000	−0.0341423
$802$	0	0
$803$	−7368.00	−0.323800
$804$	0	0
$805$	−10640.0	−0.465852
$806$	0	0
$807$	20826.0	0.908439
$808$	0	0
$809$	−34182.0	−1.48551	−0.742753	−	0.669565i	$-0.766481\pi$
$809$	−34182.0	−1.48551	−0.742753	+	0.669565i	$0.766481\pi$
$810$	0	0
$811$	10308.0	0.446317	0.223158	−	0.974782i	$-0.428363\pi$
$811$	10308.0	0.446317	0.223158	+	0.974782i	$0.428363\pi$
$812$	0	0
$813$	14160.0	0.610840
$814$	0	0
$815$	2200.00	0.0945554
$816$	0	0
$817$	17168.0	0.735168
$818$	0	0
$819$	−1890.00	−0.0806373
$820$	0	0
$821$	−37362.0	−1.58824	−0.794119	−	0.607763i	$-0.792067\pi$
$821$	−37362.0	−1.58824	−0.794119	+	0.607763i	$0.792067\pi$
$822$	0	0
$823$	30808.0	1.30486	0.652430	−	0.757849i	$-0.273750\pi$
$823$	30808.0	1.30486	0.652430	+	0.757849i	$0.273750\pi$
$824$	0	0
$825$	−900.000	−0.0379806
$826$	0	0
$827$	43596.0	1.83311	0.916555	−	0.399909i	$-0.130958\pi$
$827$	43596.0	1.83311	0.916555	+	0.399909i	$0.130958\pi$
$828$	0	0
$829$	−8578.00	−0.359380	−0.179690	−	0.983723i	$-0.557510\pi$
$829$	−8578.00	−0.359380	−0.179690	+	0.983723i	$0.557510\pi$
$830$	0	0
$831$	−4470.00	−0.186598
$832$	0	0
$833$	1666.00	0.0692959
$834$	0	0
$835$	19920.0	0.825581
$836$	0	0
$837$	−8208.00	−0.338961
$838$	0	0
$839$	41304.0	1.69961	0.849805	−	0.527098i	$-0.176720\pi$
$839$	41304.0	1.69961	0.849805	+	0.527098i	$0.176720\pi$
$840$	0	0
$841$	−13153.0	−0.539301
$842$	0	0
$843$	−23778.0	−0.971480
$844$	0	0
$845$	12970.0	0.528026
$846$	0	0
$847$	8309.00	0.337073
$848$	0	0
$849$	25788.0	1.04245
$850$	0	0
$851$	17328.0	0.697998
$852$	0	0
$853$	−11498.0	−0.461529	−0.230764	−	0.973010i	$-0.574123\pi$
$853$	−11498.0	−0.461529	−0.230764	+	0.973010i	$0.574123\pi$
$854$	0	0
$855$	13320.0	0.532789
$856$	0	0
$857$	21210.0	0.845414	0.422707	−	0.906266i	$-0.361080\pi$
$857$	21210.0	0.845414	0.422707	+	0.906266i	$0.361080\pi$
$858$	0	0
$859$	−3580.00	−0.142198	−0.0710990	−	0.997469i	$-0.522651\pi$
$859$	−3580.00	−0.142198	−0.0710990	+	0.997469i	$0.522651\pi$
$860$	0	0
$861$	−4242.00	−0.167906
$862$	0	0
$863$	10144.0	0.400123	0.200061	−	0.979783i	$-0.435886\pi$
$863$	10144.0	0.400123	0.200061	+	0.979783i	$0.435886\pi$
$864$	0	0
$865$	−42860.0	−1.68472
$866$	0	0
$867$	−11271.0	−0.441503
$868$	0	0
$869$	−10560.0	−0.412225
$870$	0	0
$871$	−360.000	−0.0140047
$872$	0	0
$873$	12834.0	0.497555
$874$	0	0
$875$	−10500.0	−0.405674
$876$	0	0
$877$	45926.0	1.76831	0.884157	−	0.467190i	$-0.154734\pi$
$877$	45926.0	1.76831	0.884157	+	0.467190i	$0.154734\pi$
$878$	0	0
$879$	1314.00	0.0504211
$880$	0	0
$881$	−9038.00	−0.345628	−0.172814	−	0.984955i	$-0.555286\pi$
$881$	−9038.00	−0.345628	−0.172814	+	0.984955i	$0.555286\pi$
$882$	0	0
$883$	30692.0	1.16973	0.584863	−	0.811132i	$-0.301148\pi$
$883$	30692.0	1.16973	0.584863	+	0.811132i	$0.301148\pi$
$884$	0	0
$885$	−19800.0	−0.752056
$886$	0	0
$887$	−22104.0	−0.836730	−0.418365	−	0.908279i	$-0.637397\pi$
$887$	−22104.0	−0.836730	−0.418365	+	0.908279i	$0.637397\pi$
$888$	0	0
$889$	−7504.00	−0.283100
$890$	0	0
$891$	972.000	0.0365468
$892$	0	0
$893$	33152.0	1.24232
$894$	0	0
$895$	−45160.0	−1.68663
$896$	0	0
$897$	−13680.0	−0.509211
$898$	0	0
$899$	32224.0	1.19547
$900$	0	0
$901$	−9316.00	−0.344463
$902$	0	0
$903$	2436.00	0.0897730
$904$	0	0
$905$	22180.0	0.814683
$906$	0	0
$907$	−42212.0	−1.54534	−0.772672	−	0.634806i	$-0.781080\pi$
$907$	−42212.0	−1.54534	−0.772672	+	0.634806i	$0.781080\pi$
$908$	0	0
$909$	−7290.00	−0.266000
$910$	0	0
$911$	24512.0	0.891459	0.445729	−	0.895168i	$-0.352944\pi$
$911$	24512.0	0.891459	0.445729	+	0.895168i	$0.352944\pi$
$912$	0	0
$913$	1296.00	0.0469785
$914$	0	0
$915$	−11460.0	−0.414050
$916$	0	0
$917$	−7588.00	−0.273258
$918$	0	0
$919$	−19912.0	−0.714729	−0.357365	−	0.933965i	$-0.616325\pi$
$919$	−19912.0	−0.714729	−0.357365	+	0.933965i	$0.616325\pi$
$920$	0	0
$921$	−15420.0	−0.551690
$922$	0	0
$923$	16560.0	0.590552
$924$	0	0
$925$	2850.00	0.101305
$926$	0	0
$927$	−9288.00	−0.329081
$928$	0	0
$929$	6802.00	0.240222	0.120111	−	0.992760i	$-0.461675\pi$
$929$	6802.00	0.240222	0.120111	+	0.992760i	$0.461675\pi$
$930$	0	0
$931$	−7252.00	−0.255290
$932$	0	0
$933$	−12888.0	−0.452234
$934$	0	0
$935$	−4080.00	−0.142706
$936$	0	0
$937$	55482.0	1.93438	0.967192	−	0.254046i	$-0.0817616\pi$
$937$	55482.0	1.93438	0.967192	+	0.254046i	$0.0817616\pi$
$938$	0	0
$939$	8142.00	0.282965
$940$	0	0
$941$	3630.00	0.125754	0.0628771	−	0.998021i	$-0.479972\pi$
$941$	3630.00	0.125754	0.0628771	+	0.998021i	$0.479972\pi$
$942$	0	0
$943$	−30704.0	−1.06030
$944$	0	0
$945$	1890.00	0.0650600
$946$	0	0
$947$	32372.0	1.11082	0.555411	−	0.831576i	$-0.312561\pi$
$947$	32372.0	1.11082	0.555411	+	0.831576i	$0.312561\pi$
$948$	0	0
$949$	−18420.0	−0.630072
$950$	0	0
$951$	−12942.0	−0.441297
$952$	0	0
$953$	−43286.0	−1.47132	−0.735662	−	0.677349i	$-0.763129\pi$
$953$	−43286.0	−1.47132	−0.735662	+	0.677349i	$0.763129\pi$
$954$	0	0
$955$	31360.0	1.06260
$956$	0	0
$957$	−3816.00	−0.128896
$958$	0	0
$959$	−16982.0	−0.571822
$960$	0	0
$961$	62625.0	2.10214
$962$	0	0
$963$	−8100.00	−0.271048
$964$	0	0
$965$	31980.0	1.06681
$966$	0	0
$967$	15864.0	0.527561	0.263781	−	0.964583i	$-0.415030\pi$
$967$	15864.0	0.527561	0.263781	+	0.964583i	$0.415030\pi$
$968$	0	0
$969$	−15096.0	−0.500468
$970$	0	0
$971$	9956.00	0.329046	0.164523	−	0.986373i	$-0.447392\pi$
$971$	9956.00	0.329046	0.164523	+	0.986373i	$0.447392\pi$
$972$	0	0
$973$	4788.00	0.157756
$974$	0	0
$975$	−2250.00	−0.0739053
$976$	0	0
$977$	28818.0	0.943674	0.471837	−	0.881686i	$-0.343591\pi$
$977$	28818.0	0.943674	0.471837	+	0.881686i	$0.343591\pi$
$978$	0	0
$979$	−1032.00	−0.0336904
$980$	0	0
$981$	4086.00	0.132983
$982$	0	0
$983$	−40920.0	−1.32772	−0.663858	−	0.747858i	$-0.731082\pi$
$983$	−40920.0	−1.32772	−0.663858	+	0.747858i	$0.731082\pi$
$984$	0	0
$985$	37140.0	1.20140
$986$	0	0
$987$	4704.00	0.151702
$988$	0	0
$989$	17632.0	0.566901
$990$	0	0
$991$	−3072.00	−0.0984715	−0.0492358	−	0.998787i	$-0.515679\pi$
$991$	−3072.00	−0.0984715	−0.0492358	+	0.998787i	$0.515679\pi$
$992$	0	0
$993$	6852.00	0.218974
$994$	0	0
$995$	−32560.0	−1.03741
$996$	0	0
$997$	−11306.0	−0.359142	−0.179571	−	0.983745i	$-0.557471\pi$
$997$	−11306.0	−0.359142	−0.179571	+	0.983745i	$0.557471\pi$
$998$	0	0
$999$	−3078.00	−0.0974811

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.a.g.1.1		1
3.2	odd	2		1008.4.a.p.1.1		1
4.3	odd	2		168.4.a.a.1.1	✓	1
7.6	odd	2		2352.4.a.n.1.1		1
8.3	odd	2		1344.4.a.y.1.1		1
8.5	even	2		1344.4.a.j.1.1		1
12.11	even	2		504.4.a.f.1.1		1
28.27	even	2		1176.4.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.4.a.a.1.1	✓	1	4.3	odd	2
336.4.a.g.1.1		1	1.1	even	1	trivial
504.4.a.f.1.1		1	12.11	even	2
1008.4.a.p.1.1		1	3.2	odd	2
1176.4.a.m.1.1		1	28.27	even	2
1344.4.a.j.1.1		1	8.5	even	2
1344.4.a.y.1.1		1	8.3	odd	2
2352.4.a.n.1.1		1	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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