LMFDB - Embedded newform 320.4.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [320,4,Mod(1,320)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("320.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(320, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 320 = 2^{6} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	320.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,8,0,-5,0,-4,0,37] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.8806112018$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	320.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} -5.00000 q^{5} -4.00000 q^{7} +37.0000 q^{9} -12.0000 q^{11} +58.0000 q^{13} -40.0000 q^{15} +66.0000 q^{17} +100.000 q^{19} -32.0000 q^{21} +132.000 q^{23} +25.0000 q^{25} +80.0000 q^{27} +90.0000 q^{29} +152.000 q^{31} -96.0000 q^{33} +20.0000 q^{35} +34.0000 q^{37} +464.000 q^{39} -438.000 q^{41} -32.0000 q^{43} -185.000 q^{45} -204.000 q^{47} -327.000 q^{49} +528.000 q^{51} -222.000 q^{53} +60.0000 q^{55} +800.000 q^{57} -420.000 q^{59} -902.000 q^{61} -148.000 q^{63} -290.000 q^{65} +1024.00 q^{67} +1056.00 q^{69} +432.000 q^{71} +362.000 q^{73} +200.000 q^{75} +48.0000 q^{77} -160.000 q^{79} -359.000 q^{81} -72.0000 q^{83} -330.000 q^{85} +720.000 q^{87} +810.000 q^{89} -232.000 q^{91} +1216.00 q^{93} -500.000 q^{95} +1106.00 q^{97} -444.000 q^{99} +O(q^{100})$

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
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	58.0000	1.23741	0.618704	−	0.785624i	$-0.287658\pi$
$13$	58.0000	1.23741	0.618704	+	0.785624i	$0.287658\pi$
$14$	0	0
$15$	−40.0000	−0.688530
$16$	0	0
$17$	66.0000	0.941609	0.470804	−	0.882238i	$-0.343964\pi$
$17$	66.0000	0.941609	0.470804	+	0.882238i	$0.343964\pi$
$18$	0	0
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	0	0
$21$	−32.0000	−0.332522
$22$	0	0
$23$	132.000	1.19669	0.598346	−	0.801238i	$-0.295825\pi$
$23$	132.000	1.19669	0.598346	+	0.801238i	$0.295825\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	90.0000	0.576296	0.288148	−	0.957586i	$-0.406961\pi$
$29$	90.0000	0.576296	0.288148	+	0.957586i	$0.406961\pi$
$30$	0	0
$31$	152.000	0.880645	0.440323	−	0.897840i	$-0.354864\pi$
$31$	152.000	0.880645	0.440323	+	0.897840i	$0.354864\pi$
$32$	0	0
$33$	−96.0000	−0.506408
$34$	0	0
$35$	20.0000	0.0965891
$36$	0	0
$37$	34.0000	0.151069	0.0755347	−	0.997143i	$-0.475934\pi$
$37$	34.0000	0.151069	0.0755347	+	0.997143i	$0.475934\pi$
$38$	0	0
$39$	464.000	1.90511
$40$	0	0
$41$	−438.000	−1.66839	−0.834196	−	0.551467i	$-0.814068\pi$
$41$	−438.000	−1.66839	−0.834196	+	0.551467i	$0.814068\pi$
$42$	0	0
$43$	−32.0000	−0.113487	−0.0567437	−	0.998389i	$-0.518072\pi$
$43$	−32.0000	−0.113487	−0.0567437	+	0.998389i	$0.518072\pi$
$44$	0	0
$45$	−185.000	−0.612848
$46$	0	0
$47$	−204.000	−0.633116	−0.316558	−	0.948573i	$-0.602527\pi$
$47$	−204.000	−0.633116	−0.316558	+	0.948573i	$0.602527\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	528.000	1.44970
$52$	0	0
$53$	−222.000	−0.575359	−0.287680	−	0.957727i	$-0.592884\pi$
$53$	−222.000	−0.575359	−0.287680	+	0.957727i	$0.592884\pi$
$54$	0	0
$55$	60.0000	0.147098
$56$	0	0
$57$	800.000	1.85899
$58$	0	0
$59$	−420.000	−0.926769	−0.463384	−	0.886157i	$-0.653365\pi$
$59$	−420.000	−0.926769	−0.463384	+	0.886157i	$0.653365\pi$
$60$	0	0
$61$	−902.000	−1.89327	−0.946633	−	0.322312i	$-0.895540\pi$
$61$	−902.000	−1.89327	−0.946633	+	0.322312i	$0.895540\pi$
$62$	0	0
$63$	−148.000	−0.295972
$64$	0	0
$65$	−290.000	−0.553386
$66$	0	0
$67$	1024.00	1.86719	0.933593	−	0.358334i	$-0.116655\pi$
$67$	1024.00	1.86719	0.933593	+	0.358334i	$0.116655\pi$
$68$	0	0
$69$	1056.00	1.84243
$70$	0	0
$71$	432.000	0.722098	0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000	0.722098	0.361049	+	0.932547i	$0.382419\pi$
$72$	0	0
$73$	362.000	0.580396	0.290198	−	0.956967i	$-0.406279\pi$
$73$	362.000	0.580396	0.290198	+	0.956967i	$0.406279\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	48.0000	0.0710404
$78$	0	0
$79$	−160.000	−0.227866	−0.113933	−	0.993488i	$-0.536345\pi$
$79$	−160.000	−0.227866	−0.113933	+	0.993488i	$0.536345\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	−72.0000	−0.0952172	−0.0476086	−	0.998866i	$-0.515160\pi$
$83$	−72.0000	−0.0952172	−0.0476086	+	0.998866i	$0.515160\pi$
$84$	0	0
$85$	−330.000	−0.421100
$86$	0	0
$87$	720.000	0.887266
$88$	0	0
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	−232.000	−0.267255
$92$	0	0
$93$	1216.00	1.35584
$94$	0	0
$95$	−500.000	−0.539989
$96$	0	0
$97$	1106.00	1.15770	0.578852	−	0.815433i	$-0.303501\pi$
$97$	1106.00	1.15770	0.578852	+	0.815433i	$0.303501\pi$
$98$	0	0
$99$	−444.000	−0.450744
$100$	0	0
$101$	258.000	0.254178	0.127089	−	0.991891i	$-0.459437\pi$
$101$	258.000	0.254178	0.127089	+	0.991891i	$0.459437\pi$
$102$	0	0
$103$	−988.000	−0.945151	−0.472575	−	0.881290i	$-0.656676\pi$
$103$	−988.000	−0.945151	−0.472575	+	0.881290i	$0.656676\pi$
$104$	0	0
$105$	160.000	0.148709
$106$	0	0
$107$	24.0000	0.0216838	0.0108419	−	0.999941i	$-0.496549\pi$
$107$	24.0000	0.0216838	0.0108419	+	0.999941i	$0.496549\pi$
$108$	0	0
$109$	−950.000	−0.834803	−0.417401	−	0.908722i	$-0.637059\pi$
$109$	−950.000	−0.834803	−0.417401	+	0.908722i	$0.637059\pi$
$110$	0	0
$111$	272.000	0.232586
$112$	0	0
$113$	−1038.00	−0.864131	−0.432066	−	0.901842i	$-0.642215\pi$
$113$	−1038.00	−0.864131	−0.432066	+	0.901842i	$0.642215\pi$
$114$	0	0
$115$	−660.000	−0.535177
$116$	0	0
$117$	2146.00	1.69571
$118$	0	0
$119$	−264.000	−0.203368
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	−3504.00	−2.56866
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−124.000	−0.0866395	−0.0433198	−	0.999061i	$-0.513793\pi$
$127$	−124.000	−0.0866395	−0.0433198	+	0.999061i	$0.513793\pi$
$128$	0	0
$129$	−256.000	−0.174725
$130$	0	0
$131$	−132.000	−0.0880374	−0.0440187	−	0.999031i	$-0.514016\pi$
$131$	−132.000	−0.0880374	−0.0440187	+	0.999031i	$0.514016\pi$
$132$	0	0
$133$	−400.000	−0.260785
$134$	0	0
$135$	−400.000	−0.255011
$136$	0	0
$137$	−1254.00	−0.782018	−0.391009	−	0.920387i	$-0.627874\pi$
$137$	−1254.00	−0.782018	−0.391009	+	0.920387i	$0.627874\pi$
$138$	0	0
$139$	2860.00	1.74519	0.872597	−	0.488440i	$-0.162434\pi$
$139$	2860.00	1.74519	0.872597	+	0.488440i	$0.162434\pi$
$140$	0	0
$141$	−1632.00	−0.974746
$142$	0	0
$143$	−696.000	−0.407010
$144$	0	0
$145$	−450.000	−0.257727
$146$	0	0
$147$	−2616.00	−1.46778
$148$	0	0
$149$	−750.000	−0.412365	−0.206183	−	0.978514i	$-0.566104\pi$
$149$	−750.000	−0.412365	−0.206183	+	0.978514i	$0.566104\pi$
$150$	0	0
$151$	−448.000	−0.241442	−0.120721	−	0.992686i	$-0.538521\pi$
$151$	−448.000	−0.241442	−0.120721	+	0.992686i	$0.538521\pi$
$152$	0	0
$153$	2442.00	1.29035
$154$	0	0
$155$	−760.000	−0.393837
$156$	0	0
$157$	−2246.00	−1.14172	−0.570861	−	0.821047i	$-0.693390\pi$
$157$	−2246.00	−1.14172	−0.570861	+	0.821047i	$0.693390\pi$
$158$	0	0
$159$	−1776.00	−0.885824
$160$	0	0
$161$	−528.000	−0.258461
$162$	0	0
$163$	568.000	0.272940	0.136470	−	0.990644i	$-0.456424\pi$
$163$	568.000	0.272940	0.136470	+	0.990644i	$0.456424\pi$
$164$	0	0
$165$	480.000	0.226472
$166$	0	0
$167$	−1524.00	−0.706172	−0.353086	−	0.935591i	$-0.614868\pi$
$167$	−1524.00	−0.706172	−0.353086	+	0.935591i	$0.614868\pi$
$168$	0	0
$169$	1167.00	0.531179
$170$	0	0
$171$	3700.00	1.65466
$172$	0	0
$173$	−3702.00	−1.62692	−0.813462	−	0.581618i	$-0.802420\pi$
$173$	−3702.00	−1.62692	−0.813462	+	0.581618i	$0.802420\pi$
$174$	0	0
$175$	−100.000	−0.0431959
$176$	0	0
$177$	−3360.00	−1.42685
$178$	0	0
$179$	−3180.00	−1.32785	−0.663923	−	0.747801i	$-0.731110\pi$
$179$	−3180.00	−1.32785	−0.663923	+	0.747801i	$0.731110\pi$
$180$	0	0
$181$	2098.00	0.861564	0.430782	−	0.902456i	$-0.358238\pi$
$181$	2098.00	0.861564	0.430782	+	0.902456i	$0.358238\pi$
$182$	0	0
$183$	−7216.00	−2.91487
$184$	0	0
$185$	−170.000	−0.0675603
$186$	0	0
$187$	−792.000	−0.309715
$188$	0	0
$189$	−320.000	−0.123156
$190$	0	0
$191$	4392.00	1.66384	0.831921	−	0.554894i	$-0.187241\pi$
$191$	4392.00	1.66384	0.831921	+	0.554894i	$0.187241\pi$
$192$	0	0
$193$	−2158.00	−0.804851	−0.402425	−	0.915453i	$-0.631833\pi$
$193$	−2158.00	−0.804851	−0.402425	+	0.915453i	$0.631833\pi$
$194$	0	0
$195$	−2320.00	−0.851993
$196$	0	0
$197$	1074.00	0.388423	0.194212	−	0.980960i	$-0.437785\pi$
$197$	1074.00	0.388423	0.194212	+	0.980960i	$0.437785\pi$
$198$	0	0
$199$	2840.00	1.01167	0.505835	−	0.862630i	$-0.331185\pi$
$199$	2840.00	1.01167	0.505835	+	0.862630i	$0.331185\pi$
$200$	0	0
$201$	8192.00	2.87472
$202$	0	0
$203$	−360.000	−0.124468
$204$	0	0
$205$	2190.00	0.746128
$206$	0	0
$207$	4884.00	1.63991
$208$	0	0
$209$	−1200.00	−0.397157
$210$	0	0
$211$	2668.00	0.870487	0.435243	−	0.900313i	$-0.356662\pi$
$211$	2668.00	0.870487	0.435243	+	0.900313i	$0.356662\pi$
$212$	0	0
$213$	3456.00	1.11174
$214$	0	0
$215$	160.000	0.0507531
$216$	0	0
$217$	−608.000	−0.190202
$218$	0	0
$219$	2896.00	0.893578
$220$	0	0
$221$	3828.00	1.16515
$222$	0	0
$223$	1772.00	0.532116	0.266058	−	0.963957i	$-0.414279\pi$
$223$	1772.00	0.532116	0.266058	+	0.963957i	$0.414279\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	2784.00	0.814011	0.407006	−	0.913426i	$-0.366573\pi$
$227$	2784.00	0.814011	0.407006	+	0.913426i	$0.366573\pi$
$228$	0	0
$229$	−350.000	−0.100998	−0.0504992	−	0.998724i	$-0.516081\pi$
$229$	−350.000	−0.100998	−0.0504992	+	0.998724i	$0.516081\pi$
$230$	0	0
$231$	384.000	0.109374
$232$	0	0
$233$	1962.00	0.551652	0.275826	−	0.961208i	$-0.411049\pi$
$233$	1962.00	0.551652	0.275826	+	0.961208i	$0.411049\pi$
$234$	0	0
$235$	1020.00	0.283138
$236$	0	0
$237$	−1280.00	−0.350823
$238$	0	0
$239$	−4320.00	−1.16919	−0.584597	−	0.811324i	$-0.698748\pi$
$239$	−4320.00	−1.16919	−0.584597	+	0.811324i	$0.698748\pi$
$240$	0	0
$241$	−478.000	−0.127762	−0.0638811	−	0.997958i	$-0.520348\pi$
$241$	−478.000	−0.127762	−0.0638811	+	0.997958i	$0.520348\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	1635.00	0.426352
$246$	0	0
$247$	5800.00	1.49411
$248$	0	0
$249$	−576.000	−0.146596
$250$	0	0
$251$	−2652.00	−0.666903	−0.333452	−	0.942767i	$-0.608213\pi$
$251$	−2652.00	−0.666903	−0.333452	+	0.942767i	$0.608213\pi$
$252$	0	0
$253$	−1584.00	−0.393617
$254$	0	0
$255$	−2640.00	−0.648326
$256$	0	0
$257$	−2334.00	−0.566502	−0.283251	−	0.959046i	$-0.591413\pi$
$257$	−2334.00	−0.566502	−0.283251	+	0.959046i	$0.591413\pi$
$258$	0	0
$259$	−136.000	−0.0326279
$260$	0	0
$261$	3330.00	0.789739
$262$	0	0
$263$	−3948.00	−0.925643	−0.462822	−	0.886451i	$-0.653163\pi$
$263$	−3948.00	−0.925643	−0.462822	+	0.886451i	$0.653163\pi$
$264$	0	0
$265$	1110.00	0.257309
$266$	0	0
$267$	6480.00	1.48528
$268$	0	0
$269$	−1590.00	−0.360387	−0.180193	−	0.983631i	$-0.557672\pi$
$269$	−1590.00	−0.360387	−0.180193	+	0.983631i	$0.557672\pi$
$270$	0	0
$271$	4952.00	1.11001	0.555005	−	0.831847i	$-0.312716\pi$
$271$	4952.00	1.11001	0.555005	+	0.831847i	$0.312716\pi$
$272$	0	0
$273$	−1856.00	−0.411466
$274$	0	0
$275$	−300.000	−0.0657843
$276$	0	0
$277$	−1646.00	−0.357034	−0.178517	−	0.983937i	$-0.557130\pi$
$277$	−1646.00	−0.357034	−0.178517	+	0.983937i	$0.557130\pi$
$278$	0	0
$279$	5624.00	1.20681
$280$	0	0
$281$	−1158.00	−0.245838	−0.122919	−	0.992417i	$-0.539226\pi$
$281$	−1158.00	−0.245838	−0.122919	+	0.992417i	$0.539226\pi$
$282$	0	0
$283$	−6992.00	−1.46866	−0.734331	−	0.678792i	$-0.762504\pi$
$283$	−6992.00	−1.46866	−0.734331	+	0.678792i	$0.762504\pi$
$284$	0	0
$285$	−4000.00	−0.831367
$286$	0	0
$287$	1752.00	0.360339
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	8848.00	1.78240
$292$	0	0
$293$	258.000	0.0514421	0.0257210	−	0.999669i	$-0.491812\pi$
$293$	258.000	0.0514421	0.0257210	+	0.999669i	$0.491812\pi$
$294$	0	0
$295$	2100.00	0.414463
$296$	0	0
$297$	−960.000	−0.187558
$298$	0	0
$299$	7656.00	1.48080
$300$	0	0
$301$	128.000	0.0245110
$302$	0	0
$303$	2064.00	0.391332
$304$	0	0
$305$	4510.00	0.846695
$306$	0	0
$307$	8944.00	1.66274	0.831370	−	0.555720i	$-0.187557\pi$
$307$	8944.00	1.66274	0.831370	+	0.555720i	$0.187557\pi$
$308$	0	0
$309$	−7904.00	−1.45515
$310$	0	0
$311$	1392.00	0.253804	0.126902	−	0.991915i	$-0.459497\pi$
$311$	1392.00	0.253804	0.126902	+	0.991915i	$0.459497\pi$
$312$	0	0
$313$	−5878.00	−1.06148	−0.530742	−	0.847534i	$-0.678087\pi$
$313$	−5878.00	−1.06148	−0.530742	+	0.847534i	$0.678087\pi$
$314$	0	0
$315$	740.000	0.132363
$316$	0	0
$317$	−10326.0	−1.82955	−0.914773	−	0.403969i	$-0.867630\pi$
$317$	−10326.0	−1.82955	−0.914773	+	0.403969i	$0.867630\pi$
$318$	0	0
$319$	−1080.00	−0.189556
$320$	0	0
$321$	192.000	0.0333844
$322$	0	0
$323$	6600.00	1.13695
$324$	0	0
$325$	1450.00	0.247482
$326$	0	0
$327$	−7600.00	−1.28526
$328$	0	0
$329$	816.000	0.136740
$330$	0	0
$331$	4228.00	0.702090	0.351045	−	0.936359i	$-0.385826\pi$
$331$	4228.00	0.702090	0.351045	+	0.936359i	$0.385826\pi$
$332$	0	0
$333$	1258.00	0.207021
$334$	0	0
$335$	−5120.00	−0.835031
$336$	0	0
$337$	1106.00	0.178776	0.0893882	−	0.995997i	$-0.471509\pi$
$337$	1106.00	0.178776	0.0893882	+	0.995997i	$0.471509\pi$
$338$	0	0
$339$	−8304.00	−1.33042
$340$	0	0
$341$	−1824.00	−0.289663
$342$	0	0
$343$	2680.00	0.421885
$344$	0	0
$345$	−5280.00	−0.823958
$346$	0	0
$347$	−9336.00	−1.44433	−0.722165	−	0.691720i	$-0.756853\pi$
$347$	−9336.00	−1.44433	−0.722165	+	0.691720i	$0.756853\pi$
$348$	0	0
$349$	11770.0	1.80525	0.902627	−	0.430424i	$-0.141636\pi$
$349$	11770.0	1.80525	0.902627	+	0.430424i	$0.141636\pi$
$350$	0	0
$351$	4640.00	0.705598
$352$	0	0
$353$	8322.00	1.25477	0.627387	−	0.778707i	$-0.284124\pi$
$353$	8322.00	1.25477	0.627387	+	0.778707i	$0.284124\pi$
$354$	0	0
$355$	−2160.00	−0.322932
$356$	0	0
$357$	−2112.00	−0.313106
$358$	0	0
$359$	10680.0	1.57011	0.785054	−	0.619427i	$-0.212635\pi$
$359$	10680.0	1.57011	0.785054	+	0.619427i	$0.212635\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	−9496.00	−1.37303
$364$	0	0
$365$	−1810.00	−0.259561
$366$	0	0
$367$	−5884.00	−0.836900	−0.418450	−	0.908240i	$-0.637426\pi$
$367$	−5884.00	−0.836900	−0.418450	+	0.908240i	$0.637426\pi$
$368$	0	0
$369$	−16206.0	−2.28632
$370$	0	0
$371$	888.000	0.124266
$372$	0	0
$373$	2098.00	0.291234	0.145617	−	0.989341i	$-0.453483\pi$
$373$	2098.00	0.291234	0.145617	+	0.989341i	$0.453483\pi$
$374$	0	0
$375$	−1000.00	−0.137706
$376$	0	0
$377$	5220.00	0.713113
$378$	0	0
$379$	−3860.00	−0.523153	−0.261576	−	0.965183i	$-0.584242\pi$
$379$	−3860.00	−0.523153	−0.261576	+	0.965183i	$0.584242\pi$
$380$	0	0
$381$	−992.000	−0.133390
$382$	0	0
$383$	−9588.00	−1.27917	−0.639587	−	0.768718i	$-0.720895\pi$
$383$	−9588.00	−1.27917	−0.639587	+	0.768718i	$0.720895\pi$
$384$	0	0
$385$	−240.000	−0.0317702
$386$	0	0
$387$	−1184.00	−0.155520
$388$	0	0
$389$	13410.0	1.74785	0.873925	−	0.486060i	$-0.161566\pi$
$389$	13410.0	1.74785	0.873925	+	0.486060i	$0.161566\pi$
$390$	0	0
$391$	8712.00	1.12682
$392$	0	0
$393$	−1056.00	−0.135542
$394$	0	0
$395$	800.000	0.101905
$396$	0	0
$397$	13114.0	1.65787	0.828933	−	0.559348i	$-0.188948\pi$
$397$	13114.0	1.65787	0.828933	+	0.559348i	$0.188948\pi$
$398$	0	0
$399$	−3200.00	−0.401505
$400$	0	0
$401$	−5838.00	−0.727022	−0.363511	−	0.931590i	$-0.618422\pi$
$401$	−5838.00	−0.727022	−0.363511	+	0.931590i	$0.618422\pi$
$402$	0	0
$403$	8816.00	1.08972
$404$	0	0
$405$	1795.00	0.220233
$406$	0	0
$407$	−408.000	−0.0496899
$408$	0	0
$409$	9530.00	1.15215	0.576074	−	0.817398i	$-0.304584\pi$
$409$	9530.00	1.15215	0.576074	+	0.817398i	$0.304584\pi$
$410$	0	0
$411$	−10032.0	−1.20400
$412$	0	0
$413$	1680.00	0.200163
$414$	0	0
$415$	360.000	0.0425824
$416$	0	0
$417$	22880.0	2.68690
$418$	0	0
$419$	−7260.00	−0.846478	−0.423239	−	0.906018i	$-0.639107\pi$
$419$	−7260.00	−0.846478	−0.423239	+	0.906018i	$0.639107\pi$
$420$	0	0
$421$	−12062.0	−1.39636	−0.698178	−	0.715924i	$-0.746006\pi$
$421$	−12062.0	−1.39636	−0.698178	+	0.715924i	$0.746006\pi$
$422$	0	0
$423$	−7548.00	−0.867604
$424$	0	0
$425$	1650.00	0.188322
$426$	0	0
$427$	3608.00	0.408907
$428$	0	0
$429$	−5568.00	−0.626633
$430$	0	0
$431$	−13608.0	−1.52082	−0.760411	−	0.649442i	$-0.775002\pi$
$431$	−13608.0	−1.52082	−0.760411	+	0.649442i	$0.775002\pi$
$432$	0	0
$433$	−3838.00	−0.425964	−0.212982	−	0.977056i	$-0.568318\pi$
$433$	−3838.00	−0.425964	−0.212982	+	0.977056i	$0.568318\pi$
$434$	0	0
$435$	−3600.00	−0.396797
$436$	0	0
$437$	13200.0	1.44495
$438$	0	0
$439$	7400.00	0.804516	0.402258	−	0.915526i	$-0.368225\pi$
$439$	7400.00	0.804516	0.402258	+	0.915526i	$0.368225\pi$
$440$	0	0
$441$	−12099.0	−1.30645
$442$	0	0
$443$	−8352.00	−0.895746	−0.447873	−	0.894097i	$-0.647818\pi$
$443$	−8352.00	−0.895746	−0.447873	+	0.894097i	$0.647818\pi$
$444$	0	0
$445$	−4050.00	−0.431435
$446$	0	0
$447$	−6000.00	−0.634878
$448$	0	0
$449$	10770.0	1.13200	0.566000	−	0.824405i	$-0.308490\pi$
$449$	10770.0	1.13200	0.566000	+	0.824405i	$0.308490\pi$
$450$	0	0
$451$	5256.00	0.548770
$452$	0	0
$453$	−3584.00	−0.371724
$454$	0	0
$455$	1160.00	0.119520
$456$	0	0
$457$	−6694.00	−0.685191	−0.342595	−	0.939483i	$-0.611306\pi$
$457$	−6694.00	−0.685191	−0.342595	+	0.939483i	$0.611306\pi$
$458$	0	0
$459$	5280.00	0.536927
$460$	0	0
$461$	3018.00	0.304907	0.152454	−	0.988311i	$-0.451283\pi$
$461$	3018.00	0.304907	0.152454	+	0.988311i	$0.451283\pi$
$462$	0	0
$463$	14492.0	1.45464	0.727322	−	0.686296i	$-0.240765\pi$
$463$	14492.0	1.45464	0.727322	+	0.686296i	$0.240765\pi$
$464$	0	0
$465$	−6080.00	−0.606351
$466$	0	0
$467$	−7776.00	−0.770515	−0.385257	−	0.922809i	$-0.625887\pi$
$467$	−7776.00	−0.770515	−0.385257	+	0.922809i	$0.625887\pi$
$468$	0	0
$469$	−4096.00	−0.403274
$470$	0	0
$471$	−17968.0	−1.75780
$472$	0	0
$473$	384.000	0.0373284
$474$	0	0
$475$	2500.00	0.241490
$476$	0	0
$477$	−8214.00	−0.788455
$478$	0	0
$479$	−13680.0	−1.30492	−0.652458	−	0.757825i	$-0.726262\pi$
$479$	−13680.0	−1.30492	−0.652458	+	0.757825i	$0.726262\pi$
$480$	0	0
$481$	1972.00	0.186934
$482$	0	0
$483$	−4224.00	−0.397927
$484$	0	0
$485$	−5530.00	−0.517741
$486$	0	0
$487$	7916.00	0.736567	0.368284	−	0.929714i	$-0.379946\pi$
$487$	7916.00	0.736567	0.368284	+	0.929714i	$0.379946\pi$
$488$	0	0
$489$	4544.00	0.420218
$490$	0	0
$491$	−13932.0	−1.28053	−0.640267	−	0.768152i	$-0.721176\pi$
$491$	−13932.0	−1.28053	−0.640267	+	0.768152i	$0.721176\pi$
$492$	0	0
$493$	5940.00	0.542645
$494$	0	0
$495$	2220.00	0.201579
$496$	0	0
$497$	−1728.00	−0.155959
$498$	0	0
$499$	8260.00	0.741019	0.370509	−	0.928829i	$-0.379183\pi$
$499$	8260.00	0.741019	0.370509	+	0.928829i	$0.379183\pi$
$500$	0	0
$501$	−12192.0	−1.08722
$502$	0	0
$503$	−11148.0	−0.988200	−0.494100	−	0.869405i	$-0.664502\pi$
$503$	−11148.0	−0.988200	−0.494100	+	0.869405i	$0.664502\pi$
$504$	0	0
$505$	−1290.00	−0.113672
$506$	0	0
$507$	9336.00	0.817803
$508$	0	0
$509$	9690.00	0.843815	0.421907	−	0.906639i	$-0.361361\pi$
$509$	9690.00	0.843815	0.421907	+	0.906639i	$0.361361\pi$
$510$	0	0
$511$	−1448.00	−0.125354
$512$	0	0
$513$	8000.00	0.688516
$514$	0	0
$515$	4940.00	0.422684
$516$	0	0
$517$	2448.00	0.208245
$518$	0	0
$519$	−29616.0	−2.50481
$520$	0	0
$521$	−16038.0	−1.34863	−0.674316	−	0.738443i	$-0.735562\pi$
$521$	−16038.0	−1.34863	−0.674316	+	0.738443i	$0.735562\pi$
$522$	0	0
$523$	−992.000	−0.0829391	−0.0414695	−	0.999140i	$-0.513204\pi$
$523$	−992.000	−0.0829391	−0.0414695	+	0.999140i	$0.513204\pi$
$524$	0	0
$525$	−800.000	−0.0665045
$526$	0	0
$527$	10032.0	0.829223
$528$	0	0
$529$	5257.00	0.432070
$530$	0	0
$531$	−15540.0	−1.27002
$532$	0	0
$533$	−25404.0	−2.06448
$534$	0	0
$535$	−120.000	−0.00969729
$536$	0	0
$537$	−25440.0	−2.04435
$538$	0	0
$539$	3924.00	0.313578
$540$	0	0
$541$	−7142.00	−0.567576	−0.283788	−	0.958887i	$-0.591591\pi$
$541$	−7142.00	−0.567576	−0.283788	+	0.958887i	$0.591591\pi$
$542$	0	0
$543$	16784.0	1.32646
$544$	0	0
$545$	4750.00	0.373335
$546$	0	0
$547$	−7616.00	−0.595314	−0.297657	−	0.954673i	$-0.596205\pi$
$547$	−7616.00	−0.595314	−0.297657	+	0.954673i	$0.596205\pi$
$548$	0	0
$549$	−33374.0	−2.59448
$550$	0	0
$551$	9000.00	0.695849
$552$	0	0
$553$	640.000	0.0492144
$554$	0	0
$555$	−1360.00	−0.104016
$556$	0	0
$557$	10314.0	0.784593	0.392296	−	0.919839i	$-0.371681\pi$
$557$	10314.0	0.784593	0.392296	+	0.919839i	$0.371681\pi$
$558$	0	0
$559$	−1856.00	−0.140430
$560$	0	0
$561$	−6336.00	−0.476838
$562$	0	0
$563$	7128.00	0.533587	0.266793	−	0.963754i	$-0.414036\pi$
$563$	7128.00	0.533587	0.266793	+	0.963754i	$0.414036\pi$
$564$	0	0
$565$	5190.00	0.386451
$566$	0	0
$567$	1436.00	0.106360
$568$	0	0
$569$	2010.00	0.148091	0.0740453	−	0.997255i	$-0.476409\pi$
$569$	2010.00	0.148091	0.0740453	+	0.997255i	$0.476409\pi$
$570$	0	0
$571$	23188.0	1.69945	0.849726	−	0.527224i	$-0.176767\pi$
$571$	23188.0	1.69945	0.849726	+	0.527224i	$0.176767\pi$
$572$	0	0
$573$	35136.0	2.56165
$574$	0	0
$575$	3300.00	0.239338
$576$	0	0
$577$	22466.0	1.62092	0.810461	−	0.585793i	$-0.199217\pi$
$577$	22466.0	1.62092	0.810461	+	0.585793i	$0.199217\pi$
$578$	0	0
$579$	−17264.0	−1.23915
$580$	0	0
$581$	288.000	0.0205650
$582$	0	0
$583$	2664.00	0.189248
$584$	0	0
$585$	−10730.0	−0.758343
$586$	0	0
$587$	−22776.0	−1.60148	−0.800738	−	0.599015i	$-0.795559\pi$
$587$	−22776.0	−1.60148	−0.800738	+	0.599015i	$0.795559\pi$
$588$	0	0
$589$	15200.0	1.06334
$590$	0	0
$591$	8592.00	0.598016
$592$	0	0
$593$	−21198.0	−1.46796	−0.733978	−	0.679174i	$-0.762338\pi$
$593$	−21198.0	−1.46796	−0.733978	+	0.679174i	$0.762338\pi$
$594$	0	0
$595$	1320.00	0.0909491
$596$	0	0
$597$	22720.0	1.55757
$598$	0	0
$599$	15960.0	1.08866	0.544330	−	0.838871i	$-0.316784\pi$
$599$	15960.0	1.08866	0.544330	+	0.838871i	$0.316784\pi$
$600$	0	0
$601$	5882.00	0.399221	0.199610	−	0.979875i	$-0.436032\pi$
$601$	5882.00	0.399221	0.199610	+	0.979875i	$0.436032\pi$
$602$	0	0
$603$	37888.0	2.55874
$604$	0	0
$605$	5935.00	0.398830
$606$	0	0
$607$	8516.00	0.569446	0.284723	−	0.958610i	$-0.408098\pi$
$607$	8516.00	0.569446	0.284723	+	0.958610i	$0.408098\pi$
$608$	0	0
$609$	−2880.00	−0.191631
$610$	0	0
$611$	−11832.0	−0.783423
$612$	0	0
$613$	−8462.00	−0.557548	−0.278774	−	0.960357i	$-0.589928\pi$
$613$	−8462.00	−0.557548	−0.278774	+	0.960357i	$0.589928\pi$
$614$	0	0
$615$	17520.0	1.14874
$616$	0	0
$617$	−11094.0	−0.723870	−0.361935	−	0.932203i	$-0.617884\pi$
$617$	−11094.0	−0.723870	−0.361935	+	0.932203i	$0.617884\pi$
$618$	0	0
$619$	−2180.00	−0.141553	−0.0707767	−	0.997492i	$-0.522548\pi$
$619$	−2180.00	−0.141553	−0.0707767	+	0.997492i	$0.522548\pi$
$620$	0	0
$621$	10560.0	0.682380
$622$	0	0
$623$	−3240.00	−0.208359
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−9600.00	−0.611463
$628$	0	0
$629$	2244.00	0.142248
$630$	0	0
$631$	−26848.0	−1.69382	−0.846911	−	0.531734i	$-0.821541\pi$
$631$	−26848.0	−1.69382	−0.846911	+	0.531734i	$0.821541\pi$
$632$	0	0
$633$	21344.0	1.34020
$634$	0	0
$635$	620.000	0.0387464
$636$	0	0
$637$	−18966.0	−1.17969
$638$	0	0
$639$	15984.0	0.989542
$640$	0	0
$641$	26322.0	1.62193	0.810965	−	0.585095i	$-0.198943\pi$
$641$	26322.0	1.62193	0.810965	+	0.585095i	$0.198943\pi$
$642$	0	0
$643$	10168.0	0.623619	0.311809	−	0.950145i	$-0.399065\pi$
$643$	10168.0	0.623619	0.311809	+	0.950145i	$0.399065\pi$
$644$	0	0
$645$	1280.00	0.0781395
$646$	0	0
$647$	−23604.0	−1.43426	−0.717132	−	0.696937i	$-0.754546\pi$
$647$	−23604.0	−1.43426	−0.717132	+	0.696937i	$0.754546\pi$
$648$	0	0
$649$	5040.00	0.304834
$650$	0	0
$651$	−4864.00	−0.292834
$652$	0	0
$653$	−16422.0	−0.984139	−0.492069	−	0.870556i	$-0.663759\pi$
$653$	−16422.0	−0.984139	−0.492069	+	0.870556i	$0.663759\pi$
$654$	0	0
$655$	660.000	0.0393715
$656$	0	0
$657$	13394.0	0.795357
$658$	0	0
$659$	26100.0	1.54281	0.771405	−	0.636345i	$-0.219554\pi$
$659$	26100.0	1.54281	0.771405	+	0.636345i	$0.219554\pi$
$660$	0	0
$661$	3058.00	0.179943	0.0899716	−	0.995944i	$-0.471322\pi$
$661$	3058.00	0.179943	0.0899716	+	0.995944i	$0.471322\pi$
$662$	0	0
$663$	30624.0	1.79387
$664$	0	0
$665$	2000.00	0.116627
$666$	0	0
$667$	11880.0	0.689648
$668$	0	0
$669$	14176.0	0.819246
$670$	0	0
$671$	10824.0	0.622736
$672$	0	0
$673$	10802.0	0.618702	0.309351	−	0.950948i	$-0.399888\pi$
$673$	10802.0	0.618702	0.309351	+	0.950948i	$0.399888\pi$
$674$	0	0
$675$	2000.00	0.114044
$676$	0	0
$677$	10674.0	0.605960	0.302980	−	0.952997i	$-0.402018\pi$
$677$	10674.0	0.605960	0.302980	+	0.952997i	$0.402018\pi$
$678$	0	0
$679$	−4424.00	−0.250041
$680$	0	0
$681$	22272.0	1.25325
$682$	0	0
$683$	28608.0	1.60272	0.801358	−	0.598185i	$-0.204111\pi$
$683$	28608.0	1.60272	0.801358	+	0.598185i	$0.204111\pi$
$684$	0	0
$685$	6270.00	0.349729
$686$	0	0
$687$	−2800.00	−0.155497
$688$	0	0
$689$	−12876.0	−0.711954
$690$	0	0
$691$	2428.00	0.133669	0.0668346	−	0.997764i	$-0.478710\pi$
$691$	2428.00	0.133669	0.0668346	+	0.997764i	$0.478710\pi$
$692$	0	0
$693$	1776.00	0.0973516
$694$	0	0
$695$	−14300.0	−0.780475
$696$	0	0
$697$	−28908.0	−1.57097
$698$	0	0
$699$	15696.0	0.849324
$700$	0	0
$701$	6618.00	0.356574	0.178287	−	0.983979i	$-0.442944\pi$
$701$	6618.00	0.356574	0.178287	+	0.983979i	$0.442944\pi$
$702$	0	0
$703$	3400.00	0.182409
$704$	0	0
$705$	8160.00	0.435920
$706$	0	0
$707$	−1032.00	−0.0548972
$708$	0	0
$709$	−20510.0	−1.08642	−0.543208	−	0.839598i	$-0.682791\pi$
$709$	−20510.0	−1.08642	−0.543208	+	0.839598i	$0.682791\pi$
$710$	0	0
$711$	−5920.00	−0.312261
$712$	0	0
$713$	20064.0	1.05386
$714$	0	0
$715$	3480.00	0.182020
$716$	0	0
$717$	−34560.0	−1.80009
$718$	0	0
$719$	31680.0	1.64321	0.821603	−	0.570061i	$-0.193080\pi$
$719$	31680.0	1.64321	0.821603	+	0.570061i	$0.193080\pi$
$720$	0	0
$721$	3952.00	0.204133
$722$	0	0
$723$	−3824.00	−0.196703
$724$	0	0
$725$	2250.00	0.115259
$726$	0	0
$727$	13196.0	0.673195	0.336597	−	0.941649i	$-0.390724\pi$
$727$	13196.0	0.673195	0.336597	+	0.941649i	$0.390724\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	−2112.00	−0.106861
$732$	0	0
$733$	−8102.00	−0.408259	−0.204130	−	0.978944i	$-0.565436\pi$
$733$	−8102.00	−0.408259	−0.204130	+	0.978944i	$0.565436\pi$
$734$	0	0
$735$	13080.0	0.656412
$736$	0	0
$737$	−12288.0	−0.614158
$738$	0	0
$739$	12580.0	0.626201	0.313101	−	0.949720i	$-0.398632\pi$
$739$	12580.0	0.626201	0.313101	+	0.949720i	$0.398632\pi$
$740$	0	0
$741$	46400.0	2.30033
$742$	0	0
$743$	29892.0	1.47595	0.737975	−	0.674828i	$-0.235782\pi$
$743$	29892.0	1.47595	0.737975	+	0.674828i	$0.235782\pi$
$744$	0	0
$745$	3750.00	0.184415
$746$	0	0
$747$	−2664.00	−0.130483
$748$	0	0
$749$	−96.0000	−0.00468326
$750$	0	0
$751$	−40408.0	−1.96339	−0.981697	−	0.190450i	$-0.939005\pi$
$751$	−40408.0	−1.96339	−0.981697	+	0.190450i	$0.939005\pi$
$752$	0	0
$753$	−21216.0	−1.02676
$754$	0	0
$755$	2240.00	0.107976
$756$	0	0
$757$	−32366.0	−1.55398	−0.776990	−	0.629513i	$-0.783254\pi$
$757$	−32366.0	−1.55398	−0.776990	+	0.629513i	$0.783254\pi$
$758$	0	0
$759$	−12672.0	−0.606014
$760$	0	0
$761$	−17238.0	−0.821126	−0.410563	−	0.911832i	$-0.634668\pi$
$761$	−17238.0	−0.821126	−0.410563	+	0.911832i	$0.634668\pi$
$762$	0	0
$763$	3800.00	0.180300
$764$	0	0
$765$	−12210.0	−0.577063
$766$	0	0
$767$	−24360.0	−1.14679
$768$	0	0
$769$	10850.0	0.508792	0.254396	−	0.967100i	$-0.418123\pi$
$769$	10850.0	0.508792	0.254396	+	0.967100i	$0.418123\pi$
$770$	0	0
$771$	−18672.0	−0.872186
$772$	0	0
$773$	−9102.00	−0.423514	−0.211757	−	0.977322i	$-0.567919\pi$
$773$	−9102.00	−0.423514	−0.211757	+	0.977322i	$0.567919\pi$
$774$	0	0
$775$	3800.00	0.176129
$776$	0	0
$777$	−1088.00	−0.0502340
$778$	0	0
$779$	−43800.0	−2.01450
$780$	0	0
$781$	−5184.00	−0.237514
$782$	0	0
$783$	7200.00	0.328617
$784$	0	0
$785$	11230.0	0.510593
$786$	0	0
$787$	25504.0	1.15517	0.577585	−	0.816330i	$-0.303995\pi$
$787$	25504.0	1.15517	0.577585	+	0.816330i	$0.303995\pi$
$788$	0	0
$789$	−31584.0	−1.42512
$790$	0	0
$791$	4152.00	0.186635
$792$	0	0
$793$	−52316.0	−2.34274
$794$	0	0
$795$	8880.00	0.396152
$796$	0	0
$797$	−14166.0	−0.629593	−0.314796	−	0.949159i	$-0.601936\pi$
$797$	−14166.0	−0.629593	−0.314796	+	0.949159i	$0.601936\pi$
$798$	0	0
$799$	−13464.0	−0.596148
$800$	0	0
$801$	29970.0	1.32202
$802$	0	0
$803$	−4344.00	−0.190905
$804$	0	0
$805$	2640.00	0.115587
$806$	0	0
$807$	−12720.0	−0.554852
$808$	0	0
$809$	33210.0	1.44327	0.721633	−	0.692276i	$-0.243392\pi$
$809$	33210.0	1.44327	0.721633	+	0.692276i	$0.243392\pi$
$810$	0	0
$811$	−39212.0	−1.69780	−0.848902	−	0.528550i	$-0.822736\pi$
$811$	−39212.0	−1.69780	−0.848902	+	0.528550i	$0.822736\pi$
$812$	0	0
$813$	39616.0	1.70897
$814$	0	0
$815$	−2840.00	−0.122062
$816$	0	0
$817$	−3200.00	−0.137030
$818$	0	0
$819$	−8584.00	−0.366238
$820$	0	0
$821$	−6222.00	−0.264494	−0.132247	−	0.991217i	$-0.542219\pi$
$821$	−6222.00	−0.264494	−0.132247	+	0.991217i	$0.542219\pi$
$822$	0	0
$823$	31172.0	1.32028	0.660138	−	0.751144i	$-0.270498\pi$
$823$	31172.0	1.32028	0.660138	+	0.751144i	$0.270498\pi$
$824$	0	0
$825$	−2400.00	−0.101282
$826$	0	0
$827$	264.000	0.0111006	0.00555029	−	0.999985i	$-0.498233\pi$
$827$	264.000	0.0111006	0.00555029	+	0.999985i	$0.498233\pi$
$828$	0	0
$829$	29050.0	1.21707	0.608533	−	0.793528i	$-0.291758\pi$
$829$	29050.0	1.21707	0.608533	+	0.793528i	$0.291758\pi$
$830$	0	0
$831$	−13168.0	−0.549691
$832$	0	0
$833$	−21582.0	−0.897685
$834$	0	0
$835$	7620.00	0.315810
$836$	0	0
$837$	12160.0	0.502164
$838$	0	0
$839$	−21720.0	−0.893752	−0.446876	−	0.894596i	$-0.647463\pi$
$839$	−21720.0	−0.893752	−0.446876	+	0.894596i	$0.647463\pi$
$840$	0	0
$841$	−16289.0	−0.667883
$842$	0	0
$843$	−9264.00	−0.378492
$844$	0	0
$845$	−5835.00	−0.237550
$846$	0	0
$847$	4748.00	0.192613
$848$	0	0
$849$	−55936.0	−2.26115
$850$	0	0
$851$	4488.00	0.180783
$852$	0	0
$853$	6658.00	0.267252	0.133626	−	0.991032i	$-0.457338\pi$
$853$	6658.00	0.267252	0.133626	+	0.991032i	$0.457338\pi$
$854$	0	0
$855$	−18500.0	−0.739984
$856$	0	0
$857$	−13974.0	−0.556993	−0.278496	−	0.960437i	$-0.589836\pi$
$857$	−13974.0	−0.556993	−0.278496	+	0.960437i	$0.589836\pi$
$858$	0	0
$859$	−23780.0	−0.944544	−0.472272	−	0.881453i	$-0.656566\pi$
$859$	−23780.0	−0.944544	−0.472272	+	0.881453i	$0.656566\pi$
$860$	0	0
$861$	14016.0	0.554778
$862$	0	0
$863$	−12228.0	−0.482324	−0.241162	−	0.970485i	$-0.577529\pi$
$863$	−12228.0	−0.482324	−0.241162	+	0.970485i	$0.577529\pi$
$864$	0	0
$865$	18510.0	0.727583
$866$	0	0
$867$	−4456.00	−0.174549
$868$	0	0
$869$	1920.00	0.0749500
$870$	0	0
$871$	59392.0	2.31047
$872$	0	0
$873$	40922.0	1.58648
$874$	0	0
$875$	500.000	0.0193178
$876$	0	0
$877$	−11606.0	−0.446872	−0.223436	−	0.974719i	$-0.571727\pi$
$877$	−11606.0	−0.446872	−0.223436	+	0.974719i	$0.571727\pi$
$878$	0	0
$879$	2064.00	0.0792002
$880$	0	0
$881$	−32958.0	−1.26037	−0.630183	−	0.776446i	$-0.717020\pi$
$881$	−32958.0	−1.26037	−0.630183	+	0.776446i	$0.717020\pi$
$882$	0	0
$883$	−8072.00	−0.307638	−0.153819	−	0.988099i	$-0.549157\pi$
$883$	−8072.00	−0.307638	−0.153819	+	0.988099i	$0.549157\pi$
$884$	0	0
$885$	16800.0	0.638108
$886$	0	0
$887$	15756.0	0.596431	0.298216	−	0.954498i	$-0.403609\pi$
$887$	15756.0	0.596431	0.298216	+	0.954498i	$0.403609\pi$
$888$	0	0
$889$	496.000	0.0187124
$890$	0	0
$891$	4308.00	0.161979
$892$	0	0
$893$	−20400.0	−0.764457
$894$	0	0
$895$	15900.0	0.593831
$896$	0	0
$897$	61248.0	2.27983
$898$	0	0
$899$	13680.0	0.507512
$900$	0	0
$901$	−14652.0	−0.541763
$902$	0	0
$903$	1024.00	0.0377371
$904$	0	0
$905$	−10490.0	−0.385303
$906$	0	0
$907$	−18776.0	−0.687372	−0.343686	−	0.939085i	$-0.611676\pi$
$907$	−18776.0	−0.687372	−0.343686	+	0.939085i	$0.611676\pi$
$908$	0	0
$909$	9546.00	0.348318
$910$	0	0
$911$	−20568.0	−0.748022	−0.374011	−	0.927424i	$-0.622018\pi$
$911$	−20568.0	−0.748022	−0.374011	+	0.927424i	$0.622018\pi$
$912$	0	0
$913$	864.000	0.0313190
$914$	0	0
$915$	36080.0	1.30357
$916$	0	0
$917$	528.000	0.0190143
$918$	0	0
$919$	−6280.00	−0.225417	−0.112708	−	0.993628i	$-0.535953\pi$
$919$	−6280.00	−0.225417	−0.112708	+	0.993628i	$0.535953\pi$
$920$	0	0
$921$	71552.0	2.55996
$922$	0	0
$923$	25056.0	0.893530
$924$	0	0
$925$	850.000	0.0302139
$926$	0	0
$927$	−36556.0	−1.29521
$928$	0	0
$929$	−20430.0	−0.721514	−0.360757	−	0.932660i	$-0.617482\pi$
$929$	−20430.0	−0.721514	−0.360757	+	0.932660i	$0.617482\pi$
$930$	0	0
$931$	−32700.0	−1.15113
$932$	0	0
$933$	11136.0	0.390757
$934$	0	0
$935$	3960.00	0.138509
$936$	0	0
$937$	8906.00	0.310508	0.155254	−	0.987875i	$-0.450380\pi$
$937$	8906.00	0.310508	0.155254	+	0.987875i	$0.450380\pi$
$938$	0	0
$939$	−47024.0	−1.63426
$940$	0	0
$941$	17418.0	0.603412	0.301706	−	0.953401i	$-0.402444\pi$
$941$	17418.0	0.603412	0.301706	+	0.953401i	$0.402444\pi$
$942$	0	0
$943$	−57816.0	−1.99655
$944$	0	0
$945$	1600.00	0.0550773
$946$	0	0
$947$	2544.00	0.0872956	0.0436478	−	0.999047i	$-0.486102\pi$
$947$	2544.00	0.0872956	0.0436478	+	0.999047i	$0.486102\pi$
$948$	0	0
$949$	20996.0	0.718187
$950$	0	0
$951$	−82608.0	−2.81677
$952$	0	0
$953$	15402.0	0.523525	0.261763	−	0.965132i	$-0.415696\pi$
$953$	15402.0	0.523525	0.261763	+	0.965132i	$0.415696\pi$
$954$	0	0
$955$	−21960.0	−0.744093
$956$	0	0
$957$	−8640.00	−0.291841
$958$	0	0
$959$	5016.00	0.168900
$960$	0	0
$961$	−6687.00	−0.224464
$962$	0	0
$963$	888.000	0.0297148
$964$	0	0
$965$	10790.0	0.359940
$966$	0	0
$967$	−49444.0	−1.64427	−0.822136	−	0.569291i	$-0.807218\pi$
$967$	−49444.0	−1.64427	−0.822136	+	0.569291i	$0.807218\pi$
$968$	0	0
$969$	52800.0	1.75044
$970$	0	0
$971$	25188.0	0.832463	0.416231	−	0.909259i	$-0.363351\pi$
$971$	25188.0	0.832463	0.416231	+	0.909259i	$0.363351\pi$
$972$	0	0
$973$	−11440.0	−0.376927
$974$	0	0
$975$	11600.0	0.381023
$976$	0	0
$977$	2946.00	0.0964697	0.0482348	−	0.998836i	$-0.484640\pi$
$977$	2946.00	0.0964697	0.0482348	+	0.998836i	$0.484640\pi$
$978$	0	0
$979$	−9720.00	−0.317316
$980$	0	0
$981$	−35150.0	−1.14399
$982$	0	0
$983$	15012.0	0.487089	0.243544	−	0.969890i	$-0.421690\pi$
$983$	15012.0	0.487089	0.243544	+	0.969890i	$0.421690\pi$
$984$	0	0
$985$	−5370.00	−0.173708
$986$	0	0
$987$	6528.00	0.210525
$988$	0	0
$989$	−4224.00	−0.135809
$990$	0	0
$991$	−5128.00	−0.164376	−0.0821878	−	0.996617i	$-0.526191\pi$
$991$	−5128.00	−0.164376	−0.0821878	+	0.996617i	$0.526191\pi$
$992$	0	0
$993$	33824.0	1.08094
$994$	0	0
$995$	−14200.0	−0.452432
$996$	0	0
$997$	49714.0	1.57920	0.789598	−	0.613625i	$-0.210289\pi$
$997$	49714.0	1.57920	0.789598	+	0.613625i	$0.210289\pi$
$998$	0	0
$999$	2720.00	0.0861431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	320.4.a.m.1.1		1
4.3	odd	2		320.4.a.b.1.1		1
5.4	even	2		1600.4.a.d.1.1		1
8.3	odd	2		80.4.a.f.1.1		1
8.5	even	2		10.4.a.a.1.1	✓	1
16.3	odd	4		1280.4.d.g.641.1		2
16.5	even	4		1280.4.d.j.641.1		2
16.11	odd	4		1280.4.d.g.641.2		2
16.13	even	4		1280.4.d.j.641.2		2
20.19	odd	2		1600.4.a.bx.1.1		1
24.5	odd	2		90.4.a.a.1.1		1
24.11	even	2		720.4.a.j.1.1		1
40.3	even	4		400.4.c.c.49.2		2
40.13	odd	4		50.4.b.a.49.1		2
40.19	odd	2		400.4.a.b.1.1		1
40.27	even	4		400.4.c.c.49.1		2
40.29	even	2		50.4.a.c.1.1		1
40.37	odd	4		50.4.b.a.49.2		2
56.5	odd	6		490.4.e.a.361.1		2
56.13	odd	2		490.4.a.o.1.1		1
56.37	even	6		490.4.e.i.361.1		2
56.45	odd	6		490.4.e.a.471.1		2
56.53	even	6		490.4.e.i.471.1		2
72.5	odd	6		810.4.e.w.541.1		2
72.13	even	6		810.4.e.c.541.1		2
72.29	odd	6		810.4.e.w.271.1		2
72.61	even	6		810.4.e.c.271.1		2
88.21	odd	2		1210.4.a.b.1.1		1
104.77	even	2		1690.4.a.a.1.1		1
120.29	odd	2		450.4.a.q.1.1		1
120.53	even	4		450.4.c.d.199.2		2
120.77	even	4		450.4.c.d.199.1		2
280.69	odd	2		2450.4.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.4.a.a.1.1	✓	1	8.5	even	2
50.4.a.c.1.1		1	40.29	even	2
50.4.b.a.49.1		2	40.13	odd	4
50.4.b.a.49.2		2	40.37	odd	4
80.4.a.f.1.1		1	8.3	odd	2
90.4.a.a.1.1		1	24.5	odd	2
320.4.a.b.1.1		1	4.3	odd	2
320.4.a.m.1.1		1	1.1	even	1	trivial
400.4.a.b.1.1		1	40.19	odd	2
400.4.c.c.49.1		2	40.27	even	4
400.4.c.c.49.2		2	40.3	even	4
450.4.a.q.1.1		1	120.29	odd	2
450.4.c.d.199.1		2	120.77	even	4
450.4.c.d.199.2		2	120.53	even	4
490.4.a.o.1.1		1	56.13	odd	2
490.4.e.a.361.1		2	56.5	odd	6
490.4.e.a.471.1		2	56.45	odd	6
490.4.e.i.361.1		2	56.37	even	6
490.4.e.i.471.1		2	56.53	even	6
720.4.a.j.1.1		1	24.11	even	2
810.4.e.c.271.1		2	72.61	even	6
810.4.e.c.541.1		2	72.13	even	6
810.4.e.w.271.1		2	72.29	odd	6
810.4.e.w.541.1		2	72.5	odd	6
1210.4.a.b.1.1		1	88.21	odd	2
1280.4.d.g.641.1		2	16.3	odd	4
1280.4.d.g.641.2		2	16.11	odd	4
1280.4.d.j.641.1		2	16.5	even	4
1280.4.d.j.641.2		2	16.13	even	4
1600.4.a.d.1.1		1	5.4	even	2
1600.4.a.bx.1.1		1	20.19	odd	2
1690.4.a.a.1.1		1	104.77	even	2
2450.4.a.b.1.1		1	280.69	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 320 = 2^{6} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	320.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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