LMFDB - Embedded newform 135.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [135,4,Mod(1,135)] 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("135.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	135.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$7.96525785077$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	135.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} -7.00000 q^{4} +5.00000 q^{5} -6.00000 q^{7} -15.0000 q^{8} +5.00000 q^{10} -47.0000 q^{11} -5.00000 q^{13} -6.00000 q^{14} +41.0000 q^{16} -131.000 q^{17} -56.0000 q^{19} -35.0000 q^{20} -47.0000 q^{22} +3.00000 q^{23} +25.0000 q^{25} -5.00000 q^{26} +42.0000 q^{28} -157.000 q^{29} +225.000 q^{31} +161.000 q^{32} -131.000 q^{34} -30.0000 q^{35} -70.0000 q^{37} -56.0000 q^{38} -75.0000 q^{40} +140.000 q^{41} +397.000 q^{43} +329.000 q^{44} +3.00000 q^{46} -347.000 q^{47} -307.000 q^{49} +25.0000 q^{50} +35.0000 q^{52} +4.00000 q^{53} -235.000 q^{55} +90.0000 q^{56} -157.000 q^{58} +748.000 q^{59} -338.000 q^{61} +225.000 q^{62} -167.000 q^{64} -25.0000 q^{65} +492.000 q^{67} +917.000 q^{68} -30.0000 q^{70} +32.0000 q^{71} +970.000 q^{73} -70.0000 q^{74} +392.000 q^{76} +282.000 q^{77} -1257.00 q^{79} +205.000 q^{80} +140.000 q^{82} -102.000 q^{83} -655.000 q^{85} +397.000 q^{86} +705.000 q^{88} -1488.00 q^{89} +30.0000 q^{91} -21.0000 q^{92} -347.000 q^{94} -280.000 q^{95} +974.000 q^{97} -307.000 q^{98} +O(q^{100})\)

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 $

$2$	1.00000	0.353553	0.176777	−	0.984251i	$-0.443433\pi$
$2$	1.00000	0.353553	0.176777	+	0.984251i	$0.443433\pi$
$3$	0	0
$3$	0	0
$4$	−7.00000	−0.875000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−6.00000	−0.323970	−0.161985	−	0.986793i	$-0.551790\pi$
$7$	−6.00000	−0.323970	−0.161985	+	0.986793i	$0.551790\pi$
$8$	−15.0000	−0.662913
$9$	0	0
$10$	5.00000	0.158114
$11$	−47.0000	−1.28828	−0.644138	−	0.764909i	$-0.722784\pi$
$11$	−47.0000	−1.28828	−0.644138	+	0.764909i	$0.722784\pi$
$12$	0	0
$13$	−5.00000	−0.106673	−0.0533366	−	0.998577i	$-0.516986\pi$
$13$	−5.00000	−0.106673	−0.0533366	+	0.998577i	$0.516986\pi$
$14$	−6.00000	−0.114541
$15$	0	0
$16$	41.0000	0.640625
$17$	−131.000	−1.86895	−0.934475	−	0.356027i	$-0.884131\pi$
$17$	−131.000	−1.86895	−0.934475	+	0.356027i	$0.884131\pi$
$18$	0	0
$19$	−56.0000	−0.676173	−0.338086	−	0.941115i	$-0.609780\pi$
$19$	−56.0000	−0.676173	−0.338086	+	0.941115i	$0.609780\pi$
$20$	−35.0000	−0.391312
$21$	0	0
$22$	−47.0000	−0.455474
$23$	3.00000	0.0271975	0.0135988	−	0.999908i	$-0.495671\pi$
$23$	3.00000	0.0271975	0.0135988	+	0.999908i	$0.495671\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−5.00000	−0.0377146
$27$	0	0
$28$	42.0000	0.283473
$29$	−157.000	−1.00532	−0.502658	−	0.864485i	$-0.667645\pi$
$29$	−157.000	−1.00532	−0.502658	+	0.864485i	$0.667645\pi$
$30$	0	0
$31$	225.000	1.30359	0.651793	−	0.758397i	$-0.274017\pi$
$31$	225.000	1.30359	0.651793	+	0.758397i	$0.274017\pi$
$32$	161.000	0.889408
$33$	0	0
$34$	−131.000	−0.660774
$35$	−30.0000	−0.144884
$36$	0	0
$37$	−70.0000	−0.311025	−0.155513	−	0.987834i	$-0.549703\pi$
$37$	−70.0000	−0.311025	−0.155513	+	0.987834i	$0.549703\pi$
$38$	−56.0000	−0.239063
$39$	0	0
$40$	−75.0000	−0.296464
$41$	140.000	0.533276	0.266638	−	0.963797i	$-0.414087\pi$
$41$	140.000	0.533276	0.266638	+	0.963797i	$0.414087\pi$
$42$	0	0
$43$	397.000	1.40795	0.703976	−	0.710224i	$-0.251406\pi$
$43$	397.000	1.40795	0.703976	+	0.710224i	$0.251406\pi$
$44$	329.000	1.12724
$45$	0	0
$46$	3.00000	0.00961578
$47$	−347.000	−1.07692	−0.538459	−	0.842652i	$-0.680993\pi$
$47$	−347.000	−1.07692	−0.538459	+	0.842652i	$0.680993\pi$
$48$	0	0
$49$	−307.000	−0.895044
$50$	25.0000	0.0707107
$51$	0	0
$52$	35.0000	0.0933390
$53$	4.00000	0.0103668	0.00518342	−	0.999987i	$-0.498350\pi$
$53$	4.00000	0.0103668	0.00518342	+	0.999987i	$0.498350\pi$
$54$	0	0
$55$	−235.000	−0.576134
$56$	90.0000	0.214763
$57$	0	0
$58$	−157.000	−0.355433
$59$	748.000	1.65053	0.825265	−	0.564745i	$-0.191026\pi$
$59$	748.000	1.65053	0.825265	+	0.564745i	$0.191026\pi$
$60$	0	0
$61$	−338.000	−0.709450	−0.354725	−	0.934971i	$-0.615426\pi$
$61$	−338.000	−0.709450	−0.354725	+	0.934971i	$0.615426\pi$
$62$	225.000	0.460888
$63$	0	0
$64$	−167.000	−0.326172
$65$	−25.0000	−0.0477057
$66$	0	0
$67$	492.000	0.897125	0.448562	−	0.893751i	$-0.351936\pi$
$67$	492.000	0.897125	0.448562	+	0.893751i	$0.351936\pi$
$68$	917.000	1.63533
$69$	0	0
$70$	−30.0000	−0.0512241
$71$	32.0000	0.0534888	0.0267444	−	0.999642i	$-0.491486\pi$
$71$	32.0000	0.0534888	0.0267444	+	0.999642i	$0.491486\pi$
$72$	0	0
$73$	970.000	1.55520	0.777602	−	0.628757i	$-0.216436\pi$
$73$	970.000	1.55520	0.777602	+	0.628757i	$0.216436\pi$
$74$	−70.0000	−0.109964
$75$	0	0
$76$	392.000	0.591651
$77$	282.000	0.417362
$78$	0	0
$79$	−1257.00	−1.79017	−0.895086	−	0.445894i	$-0.852886\pi$
$79$	−1257.00	−1.79017	−0.895086	+	0.445894i	$0.852886\pi$
$80$	205.000	0.286496
$81$	0	0
$82$	140.000	0.188542
$83$	−102.000	−0.134891	−0.0674455	−	0.997723i	$-0.521485\pi$
$83$	−102.000	−0.134891	−0.0674455	+	0.997723i	$0.521485\pi$
$84$	0	0
$85$	−655.000	−0.835820
$86$	397.000	0.497786
$87$	0	0
$88$	705.000	0.854014
$89$	−1488.00	−1.77222	−0.886111	−	0.463474i	$-0.846603\pi$
$89$	−1488.00	−1.77222	−0.886111	+	0.463474i	$0.846603\pi$
$90$	0	0
$91$	30.0000	0.0345588
$92$	−21.0000	−0.0237978
$93$	0	0
$94$	−347.000	−0.380748
$95$	−280.000	−0.302394
$96$	0	0
$97$	974.000	1.01953	0.509767	−	0.860313i	$-0.329732\pi$
$97$	974.000	1.01953	0.509767	+	0.860313i	$0.329732\pi$
$98$	−307.000	−0.316446
$99$	0	0
$100$	−175.000	−0.175000
$101$	−1335.00	−1.31522	−0.657611	−	0.753357i	$-0.728433\pi$
$101$	−1335.00	−1.31522	−0.657611	+	0.753357i	$0.728433\pi$
$102$	0	0
$103$	686.000	0.656248	0.328124	−	0.944635i	$-0.393584\pi$
$103$	686.000	0.656248	0.328124	+	0.944635i	$0.393584\pi$
$104$	75.0000	0.0707150
$105$	0	0
$106$	4.00000	0.00366523
$107$	−1098.00	−0.992034	−0.496017	−	0.868313i	$-0.665205\pi$
$107$	−1098.00	−0.992034	−0.496017	+	0.868313i	$0.665205\pi$
$108$	0	0
$109$	−700.000	−0.615118	−0.307559	−	0.951529i	$-0.599512\pi$
$109$	−700.000	−0.615118	−0.307559	+	0.951529i	$0.599512\pi$
$110$	−235.000	−0.203694
$111$	0	0
$112$	−246.000	−0.207543
$113$	−1055.00	−0.878284	−0.439142	−	0.898418i	$-0.644717\pi$
$113$	−1055.00	−0.878284	−0.439142	+	0.898418i	$0.644717\pi$
$114$	0	0
$115$	15.0000	0.0121631
$116$	1099.00	0.879652
$117$	0	0
$118$	748.000	0.583551
$119$	786.000	0.605483
$120$	0	0
$121$	878.000	0.659654
$122$	−338.000	−0.250829
$123$	0	0
$124$	−1575.00	−1.14064
$125$	125.000	0.0894427
$126$	0	0
$127$	−1646.00	−1.15007	−0.575035	−	0.818129i	$-0.695012\pi$
$127$	−1646.00	−1.15007	−0.575035	+	0.818129i	$0.695012\pi$
$128$	−1455.00	−1.00473
$129$	0	0
$130$	−25.0000	−0.0168665
$131$	−1833.00	−1.22252	−0.611259	−	0.791430i	$-0.709337\pi$
$131$	−1833.00	−1.22252	−0.611259	+	0.791430i	$0.709337\pi$
$132$	0	0
$133$	336.000	0.219059
$134$	492.000	0.317182
$135$	0	0
$136$	1965.00	1.23895
$137$	1098.00	0.684733	0.342367	−	0.939566i	$-0.388771\pi$
$137$	1098.00	0.684733	0.342367	+	0.939566i	$0.388771\pi$
$138$	0	0
$139$	−1042.00	−0.635837	−0.317918	−	0.948118i	$-0.602984\pi$
$139$	−1042.00	−0.635837	−0.317918	+	0.948118i	$0.602984\pi$
$140$	210.000	0.126773
$141$	0	0
$142$	32.0000	0.0189111
$143$	235.000	0.137424
$144$	0	0
$145$	−785.000	−0.449591
$146$	970.000	0.549848
$147$	0	0
$148$	490.000	0.272147
$149$	−2941.00	−1.61702	−0.808510	−	0.588482i	$-0.799726\pi$
$149$	−2941.00	−1.61702	−0.808510	+	0.588482i	$0.799726\pi$
$150$	0	0
$151$	511.000	0.275395	0.137697	−	0.990474i	$-0.456030\pi$
$151$	511.000	0.275395	0.137697	+	0.990474i	$0.456030\pi$
$152$	840.000	0.448243
$153$	0	0
$154$	282.000	0.147560
$155$	1125.00	0.582982
$156$	0	0
$157$	−571.000	−0.290260	−0.145130	−	0.989413i	$-0.546360\pi$
$157$	−571.000	−0.290260	−0.145130	+	0.989413i	$0.546360\pi$
$158$	−1257.00	−0.632921
$159$	0	0
$160$	805.000	0.397755
$161$	−18.0000	−0.00881117
$162$	0	0
$163$	713.000	0.342616	0.171308	−	0.985217i	$-0.445201\pi$
$163$	713.000	0.342616	0.171308	+	0.985217i	$0.445201\pi$
$164$	−980.000	−0.466617
$165$	0	0
$166$	−102.000	−0.0476912
$167$	1596.00	0.739534	0.369767	−	0.929125i	$-0.379437\pi$
$167$	1596.00	0.739534	0.369767	+	0.929125i	$0.379437\pi$
$168$	0	0
$169$	−2172.00	−0.988621
$170$	−655.000	−0.295507
$171$	0	0
$172$	−2779.00	−1.23196
$173$	4134.00	1.81678	0.908388	−	0.418129i	$-0.137314\pi$
$173$	4134.00	1.81678	0.908388	+	0.418129i	$0.137314\pi$
$174$	0	0
$175$	−150.000	−0.0647939
$176$	−1927.00	−0.825302
$177$	0	0
$178$	−1488.00	−0.626575
$179$	1828.00	0.763302	0.381651	−	0.924306i	$-0.375356\pi$
$179$	1828.00	0.763302	0.381651	+	0.924306i	$0.375356\pi$
$180$	0	0
$181$	−520.000	−0.213543	−0.106772	−	0.994284i	$-0.534051\pi$
$181$	−520.000	−0.213543	−0.106772	+	0.994284i	$0.534051\pi$
$182$	30.0000	0.0122184
$183$	0	0
$184$	−45.0000	−0.0180296
$185$	−350.000	−0.139095
$186$	0	0
$187$	6157.00	2.40772
$188$	2429.00	0.942303
$189$	0	0
$190$	−280.000	−0.106912
$191$	4826.00	1.82826	0.914129	−	0.405424i	$-0.132876\pi$
$191$	4826.00	1.82826	0.914129	+	0.405424i	$0.132876\pi$
$192$	0	0
$193$	1670.00	0.622846	0.311423	−	0.950271i	$-0.399194\pi$
$193$	1670.00	0.622846	0.311423	+	0.950271i	$0.399194\pi$
$194$	974.000	0.360459
$195$	0	0
$196$	2149.00	0.783163
$197$	−1380.00	−0.499091	−0.249546	−	0.968363i	$-0.580281\pi$
$197$	−1380.00	−0.499091	−0.249546	+	0.968363i	$0.580281\pi$
$198$	0	0
$199$	4357.00	1.55206	0.776029	−	0.630697i	$-0.217231\pi$
$199$	4357.00	1.55206	0.776029	+	0.630697i	$0.217231\pi$
$200$	−375.000	−0.132583
$201$	0	0
$202$	−1335.00	−0.465001
$203$	942.000	0.325692
$204$	0	0
$205$	700.000	0.238488
$206$	686.000	0.232019
$207$	0	0
$208$	−205.000	−0.0683375
$209$	2632.00	0.871097
$210$	0	0
$211$	−4162.00	−1.35793	−0.678967	−	0.734169i	$-0.737572\pi$
$211$	−4162.00	−1.35793	−0.678967	+	0.734169i	$0.737572\pi$
$212$	−28.0000	−0.00907098
$213$	0	0
$214$	−1098.00	−0.350737
$215$	1985.00	0.629655
$216$	0	0
$217$	−1350.00	−0.422322
$218$	−700.000	−0.217477
$219$	0	0
$220$	1645.00	0.504118
$221$	655.000	0.199367
$222$	0	0
$223$	−5956.00	−1.78853	−0.894267	−	0.447533i	$-0.852303\pi$
$223$	−5956.00	−1.78853	−0.894267	+	0.447533i	$0.852303\pi$
$224$	−966.000	−0.288141
$225$	0	0
$226$	−1055.00	−0.310520
$227$	4940.00	1.44440	0.722201	−	0.691683i	$-0.243130\pi$
$227$	4940.00	1.44440	0.722201	+	0.691683i	$0.243130\pi$
$228$	0	0
$229$	4344.00	1.25354	0.626768	−	0.779206i	$-0.284378\pi$
$229$	4344.00	1.25354	0.626768	+	0.779206i	$0.284378\pi$
$230$	15.0000	0.00430031
$231$	0	0
$232$	2355.00	0.666437
$233$	−5202.00	−1.46264	−0.731318	−	0.682036i	$-0.761095\pi$
$233$	−5202.00	−1.46264	−0.731318	+	0.682036i	$0.761095\pi$
$234$	0	0
$235$	−1735.00	−0.481612
$236$	−5236.00	−1.44421
$237$	0	0
$238$	786.000	0.214071
$239$	1546.00	0.418420	0.209210	−	0.977871i	$-0.432911\pi$
$239$	1546.00	0.418420	0.209210	+	0.977871i	$0.432911\pi$
$240$	0	0
$241$	−3659.00	−0.977995	−0.488998	−	0.872285i	$-0.662637\pi$
$241$	−3659.00	−0.977995	−0.488998	+	0.872285i	$0.662637\pi$
$242$	878.000	0.233223
$243$	0	0
$244$	2366.00	0.620769
$245$	−1535.00	−0.400276
$246$	0	0
$247$	280.000	0.0721294
$248$	−3375.00	−0.864164
$249$	0	0
$250$	125.000	0.0316228
$251$	1221.00	0.307047	0.153524	−	0.988145i	$-0.450938\pi$
$251$	1221.00	0.307047	0.153524	+	0.988145i	$0.450938\pi$
$252$	0	0
$253$	−141.000	−0.0350379
$254$	−1646.00	−0.406611
$255$	0	0
$256$	−119.000	−0.0290527
$257$	−6255.00	−1.51820	−0.759098	−	0.650977i	$-0.774360\pi$
$257$	−6255.00	−1.51820	−0.759098	+	0.650977i	$0.774360\pi$
$258$	0	0
$259$	420.000	0.100763
$260$	175.000	0.0417425
$261$	0	0
$262$	−1833.00	−0.432226
$263$	836.000	0.196007	0.0980037	−	0.995186i	$-0.468754\pi$
$263$	836.000	0.196007	0.0980037	+	0.995186i	$0.468754\pi$
$264$	0	0
$265$	20.0000	0.00463619
$266$	336.000	0.0774492
$267$	0	0
$268$	−3444.00	−0.784984
$269$	−2231.00	−0.505675	−0.252837	−	0.967509i	$-0.581364\pi$
$269$	−2231.00	−0.505675	−0.252837	+	0.967509i	$0.581364\pi$
$270$	0	0
$271$	−4832.00	−1.08311	−0.541556	−	0.840665i	$-0.682164\pi$
$271$	−4832.00	−1.08311	−0.541556	+	0.840665i	$0.682164\pi$
$272$	−5371.00	−1.19730
$273$	0	0
$274$	1098.00	0.242090
$275$	−1175.00	−0.257655
$276$	0	0
$277$	6450.00	1.39907	0.699536	−	0.714597i	$-0.253390\pi$
$277$	6450.00	1.39907	0.699536	+	0.714597i	$0.253390\pi$
$278$	−1042.00	−0.224802
$279$	0	0
$280$	450.000	0.0960452
$281$	−1050.00	−0.222910	−0.111455	−	0.993769i	$-0.535551\pi$
$281$	−1050.00	−0.222910	−0.111455	+	0.993769i	$0.535551\pi$
$282$	0	0
$283$	−1584.00	−0.332717	−0.166359	−	0.986065i	$-0.553201\pi$
$283$	−1584.00	−0.332717	−0.166359	+	0.986065i	$0.553201\pi$
$284$	−224.000	−0.0468027
$285$	0	0
$286$	235.000	0.0485869
$287$	−840.000	−0.172765
$288$	0	0
$289$	12248.0	2.49298
$290$	−785.000	−0.158954
$291$	0	0
$292$	−6790.00	−1.36080
$293$	6594.00	1.31476	0.657382	−	0.753558i	$-0.271664\pi$
$293$	6594.00	1.31476	0.657382	+	0.753558i	$0.271664\pi$
$294$	0	0
$295$	3740.00	0.738140
$296$	1050.00	0.206182
$297$	0	0
$298$	−2941.00	−0.571703
$299$	−15.0000	−0.00290125
$300$	0	0
$301$	−2382.00	−0.456134
$302$	511.000	0.0973667
$303$	0	0
$304$	−2296.00	−0.433173
$305$	−1690.00	−0.317276
$306$	0	0
$307$	−4343.00	−0.807388	−0.403694	−	0.914894i	$-0.632274\pi$
$307$	−4343.00	−0.807388	−0.403694	+	0.914894i	$0.632274\pi$
$308$	−1974.00	−0.365192
$309$	0	0
$310$	1125.00	0.206115
$311$	−2124.00	−0.387270	−0.193635	−	0.981074i	$-0.562028\pi$
$311$	−2124.00	−0.387270	−0.193635	+	0.981074i	$0.562028\pi$
$312$	0	0
$313$	−7516.00	−1.35728	−0.678641	−	0.734470i	$-0.737431\pi$
$313$	−7516.00	−1.35728	−0.678641	+	0.734470i	$0.737431\pi$
$314$	−571.000	−0.102622
$315$	0	0
$316$	8799.00	1.56640
$317$	−6880.00	−1.21899	−0.609494	−	0.792791i	$-0.708627\pi$
$317$	−6880.00	−1.21899	−0.609494	+	0.792791i	$0.708627\pi$
$318$	0	0
$319$	7379.00	1.29512
$320$	−835.000	−0.145868
$321$	0	0
$322$	−18.0000	−0.00311522
$323$	7336.00	1.26373
$324$	0	0
$325$	−125.000	−0.0213346
$326$	713.000	0.121133
$327$	0	0
$328$	−2100.00	−0.353516
$329$	2082.00	0.348889
$330$	0	0
$331$	−4986.00	−0.827962	−0.413981	−	0.910286i	$-0.635862\pi$
$331$	−4986.00	−0.827962	−0.413981	+	0.910286i	$0.635862\pi$
$332$	714.000	0.118030
$333$	0	0
$334$	1596.00	0.261465
$335$	2460.00	0.401206
$336$	0	0
$337$	904.000	0.146125	0.0730623	−	0.997327i	$-0.476723\pi$
$337$	904.000	0.146125	0.0730623	+	0.997327i	$0.476723\pi$
$338$	−2172.00	−0.349530
$339$	0	0
$340$	4585.00	0.731343
$341$	−10575.0	−1.67938
$342$	0	0
$343$	3900.00	0.613936
$344$	−5955.00	−0.933349
$345$	0	0
$346$	4134.00	0.642327
$347$	8860.00	1.37069	0.685345	−	0.728218i	$-0.259651\pi$
$347$	8860.00	1.37069	0.685345	+	0.728218i	$0.259651\pi$
$348$	0	0
$349$	−4454.00	−0.683144	−0.341572	−	0.939856i	$-0.610959\pi$
$349$	−4454.00	−0.683144	−0.341572	+	0.939856i	$0.610959\pi$
$350$	−150.000	−0.0229081
$351$	0	0
$352$	−7567.00	−1.14580
$353$	−8781.00	−1.32398	−0.661991	−	0.749512i	$-0.730288\pi$
$353$	−8781.00	−1.32398	−0.661991	+	0.749512i	$0.730288\pi$
$354$	0	0
$355$	160.000	0.0239209
$356$	10416.0	1.55069
$357$	0	0
$358$	1828.00	0.269868
$359$	2928.00	0.430457	0.215228	−	0.976564i	$-0.430950\pi$
$359$	2928.00	0.430457	0.215228	+	0.976564i	$0.430950\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	−520.000	−0.0754989
$363$	0	0
$364$	−210.000	−0.0302390
$365$	4850.00	0.695508
$366$	0	0
$367$	9102.00	1.29461	0.647303	−	0.762233i	$-0.275897\pi$
$367$	9102.00	1.29461	0.647303	+	0.762233i	$0.275897\pi$
$368$	123.000	0.0174234
$369$	0	0
$370$	−350.000	−0.0491774
$371$	−24.0000	−0.00335854
$372$	0	0
$373$	−8183.00	−1.13592	−0.567962	−	0.823055i	$-0.692268\pi$
$373$	−8183.00	−1.13592	−0.567962	+	0.823055i	$0.692268\pi$
$374$	6157.00	0.851259
$375$	0	0
$376$	5205.00	0.713903
$377$	785.000	0.107240
$378$	0	0
$379$	6136.00	0.831623	0.415812	−	0.909451i	$-0.363498\pi$
$379$	6136.00	0.831623	0.415812	+	0.909451i	$0.363498\pi$
$380$	1960.00	0.264594
$381$	0	0
$382$	4826.00	0.646386
$383$	5643.00	0.752856	0.376428	−	0.926446i	$-0.377152\pi$
$383$	5643.00	0.752856	0.376428	+	0.926446i	$0.377152\pi$
$384$	0	0
$385$	1410.00	0.186650
$386$	1670.00	0.220209
$387$	0	0
$388$	−6818.00	−0.892092
$389$	8991.00	1.17188	0.585941	−	0.810354i	$-0.300725\pi$
$389$	8991.00	1.17188	0.585941	+	0.810354i	$0.300725\pi$
$390$	0	0
$391$	−393.000	−0.0508309
$392$	4605.00	0.593336
$393$	0	0
$394$	−1380.00	−0.176455
$395$	−6285.00	−0.800589
$396$	0	0
$397$	−12449.0	−1.57380	−0.786898	−	0.617082i	$-0.788314\pi$
$397$	−12449.0	−1.57380	−0.786898	+	0.617082i	$0.788314\pi$
$398$	4357.00	0.548735
$399$	0	0
$400$	1025.00	0.128125
$401$	8076.00	1.00573	0.502863	−	0.864366i	$-0.332280\pi$
$401$	8076.00	1.00573	0.502863	+	0.864366i	$0.332280\pi$
$402$	0	0
$403$	−1125.00	−0.139058
$404$	9345.00	1.15082
$405$	0	0
$406$	942.000	0.115149
$407$	3290.00	0.400686
$408$	0	0
$409$	−2833.00	−0.342501	−0.171250	−	0.985228i	$-0.554781\pi$
$409$	−2833.00	−0.342501	−0.171250	+	0.985228i	$0.554781\pi$
$410$	700.000	0.0843184
$411$	0	0
$412$	−4802.00	−0.574217
$413$	−4488.00	−0.534722
$414$	0	0
$415$	−510.000	−0.0603251
$416$	−805.000	−0.0948759
$417$	0	0
$418$	2632.00	0.307979
$419$	−4777.00	−0.556973	−0.278487	−	0.960440i	$-0.589833\pi$
$419$	−4777.00	−0.556973	−0.278487	+	0.960440i	$0.589833\pi$
$420$	0	0
$421$	−6464.00	−0.748304	−0.374152	−	0.927367i	$-0.622066\pi$
$421$	−6464.00	−0.748304	−0.374152	+	0.927367i	$0.622066\pi$
$422$	−4162.00	−0.480102
$423$	0	0
$424$	−60.0000	−0.00687231
$425$	−3275.00	−0.373790
$426$	0	0
$427$	2028.00	0.229840
$428$	7686.00	0.868030
$429$	0	0
$430$	1985.00	0.222617
$431$	10680.0	1.19359	0.596795	−	0.802394i	$-0.296440\pi$
$431$	10680.0	1.19359	0.596795	+	0.802394i	$0.296440\pi$
$432$	0	0
$433$	11566.0	1.28366	0.641832	−	0.766845i	$-0.278175\pi$
$433$	11566.0	1.28366	0.641832	+	0.766845i	$0.278175\pi$
$434$	−1350.00	−0.149314
$435$	0	0
$436$	4900.00	0.538228
$437$	−168.000	−0.0183902
$438$	0	0
$439$	−1448.00	−0.157424	−0.0787122	−	0.996897i	$-0.525081\pi$
$439$	−1448.00	−0.157424	−0.0787122	+	0.996897i	$0.525081\pi$
$440$	3525.00	0.381927
$441$	0	0
$442$	655.000	0.0704868
$443$	−2376.00	−0.254824	−0.127412	−	0.991850i	$-0.540667\pi$
$443$	−2376.00	−0.254824	−0.127412	+	0.991850i	$0.540667\pi$
$444$	0	0
$445$	−7440.00	−0.792561
$446$	−5956.00	−0.632343
$447$	0	0
$448$	1002.00	0.105670
$449$	−14894.0	−1.56546	−0.782730	−	0.622362i	$-0.786173\pi$
$449$	−14894.0	−1.56546	−0.782730	+	0.622362i	$0.786173\pi$
$450$	0	0
$451$	−6580.00	−0.687007
$452$	7385.00	0.768498
$453$	0	0
$454$	4940.00	0.510673
$455$	150.000	0.0154552
$456$	0	0
$457$	16204.0	1.65862	0.829312	−	0.558786i	$-0.188733\pi$
$457$	16204.0	1.65862	0.829312	+	0.558786i	$0.188733\pi$
$458$	4344.00	0.443192
$459$	0	0
$460$	−105.000	−0.0106427
$461$	−5082.00	−0.513432	−0.256716	−	0.966487i	$-0.582641\pi$
$461$	−5082.00	−0.513432	−0.256716	+	0.966487i	$0.582641\pi$
$462$	0	0
$463$	−10326.0	−1.03648	−0.518240	−	0.855235i	$-0.673412\pi$
$463$	−10326.0	−1.03648	−0.518240	+	0.855235i	$0.673412\pi$
$464$	−6437.00	−0.644031
$465$	0	0
$466$	−5202.00	−0.517120
$467$	4184.00	0.414588	0.207294	−	0.978279i	$-0.433534\pi$
$467$	4184.00	0.414588	0.207294	+	0.978279i	$0.433534\pi$
$468$	0	0
$469$	−2952.00	−0.290641
$470$	−1735.00	−0.170276
$471$	0	0
$472$	−11220.0	−1.09416
$473$	−18659.0	−1.81383
$474$	0	0
$475$	−1400.00	−0.135235
$476$	−5502.00	−0.529798
$477$	0	0
$478$	1546.00	0.147934
$479$	−15576.0	−1.48577	−0.742887	−	0.669417i	$-0.766544\pi$
$479$	−15576.0	−1.48577	−0.742887	+	0.669417i	$0.766544\pi$
$480$	0	0
$481$	350.000	0.0331780
$482$	−3659.00	−0.345774
$483$	0	0
$484$	−6146.00	−0.577198
$485$	4870.00	0.455949
$486$	0	0
$487$	10220.0	0.950949	0.475475	−	0.879729i	$-0.342276\pi$
$487$	10220.0	0.950949	0.475475	+	0.879729i	$0.342276\pi$
$488$	5070.00	0.470304
$489$	0	0
$490$	−1535.00	−0.141519
$491$	−2692.00	−0.247430	−0.123715	−	0.992318i	$-0.539481\pi$
$491$	−2692.00	−0.247430	−0.123715	+	0.992318i	$0.539481\pi$
$492$	0	0
$493$	20567.0	1.87889
$494$	280.000	0.0255016
$495$	0	0
$496$	9225.00	0.835110
$497$	−192.000	−0.0173287
$498$	0	0
$499$	5764.00	0.517098	0.258549	−	0.965998i	$-0.416756\pi$
$499$	5764.00	0.517098	0.258549	+	0.965998i	$0.416756\pi$
$500$	−875.000	−0.0782624
$501$	0	0
$502$	1221.00	0.108558
$503$	−2437.00	−0.216025	−0.108012	−	0.994150i	$-0.534449\pi$
$503$	−2437.00	−0.216025	−0.108012	+	0.994150i	$0.534449\pi$
$504$	0	0
$505$	−6675.00	−0.588185
$506$	−141.000	−0.0123878
$507$	0	0
$508$	11522.0	1.00631
$509$	−5849.00	−0.509337	−0.254668	−	0.967028i	$-0.581966\pi$
$509$	−5849.00	−0.509337	−0.254668	+	0.967028i	$0.581966\pi$
$510$	0	0
$511$	−5820.00	−0.503839
$512$	11521.0	0.994455
$513$	0	0
$514$	−6255.00	−0.536763
$515$	3430.00	0.293483
$516$	0	0
$517$	16309.0	1.38737
$518$	420.000	0.0356250
$519$	0	0
$520$	375.000	0.0316247
$521$	−17032.0	−1.43222	−0.716109	−	0.697989i	$-0.754079\pi$
$521$	−17032.0	−1.43222	−0.716109	+	0.697989i	$0.754079\pi$
$522$	0	0
$523$	4147.00	0.346722	0.173361	−	0.984858i	$-0.444537\pi$
$523$	4147.00	0.346722	0.173361	+	0.984858i	$0.444537\pi$
$524$	12831.0	1.06970
$525$	0	0
$526$	836.000	0.0692991
$527$	−29475.0	−2.43634
$528$	0	0
$529$	−12158.0	−0.999260
$530$	20.0000	0.00163914
$531$	0	0
$532$	−2352.00	−0.191677
$533$	−700.000	−0.0568862
$534$	0	0
$535$	−5490.00	−0.443651
$536$	−7380.00	−0.594715
$537$	0	0
$538$	−2231.00	−0.178783
$539$	14429.0	1.15306
$540$	0	0
$541$	−3942.00	−0.313271	−0.156636	−	0.987656i	$-0.550065\pi$
$541$	−3942.00	−0.313271	−0.156636	+	0.987656i	$0.550065\pi$
$542$	−4832.00	−0.382938
$543$	0	0
$544$	−21091.0	−1.66226
$545$	−3500.00	−0.275089
$546$	0	0
$547$	−13751.0	−1.07486	−0.537432	−	0.843307i	$-0.680605\pi$
$547$	−13751.0	−1.07486	−0.537432	+	0.843307i	$0.680605\pi$
$548$	−7686.00	−0.599142
$549$	0	0
$550$	−1175.00	−0.0910949
$551$	8792.00	0.679767
$552$	0	0
$553$	7542.00	0.579961
$554$	6450.00	0.494647
$555$	0	0
$556$	7294.00	0.556357
$557$	7944.00	0.604305	0.302153	−	0.953260i	$-0.402295\pi$
$557$	7944.00	0.604305	0.302153	+	0.953260i	$0.402295\pi$
$558$	0	0
$559$	−1985.00	−0.150191
$560$	−1230.00	−0.0928160
$561$	0	0
$562$	−1050.00	−0.0788106
$563$	−6702.00	−0.501697	−0.250849	−	0.968026i	$-0.580710\pi$
$563$	−6702.00	−0.501697	−0.250849	+	0.968026i	$0.580710\pi$
$564$	0	0
$565$	−5275.00	−0.392780
$566$	−1584.00	−0.117633
$567$	0	0
$568$	−480.000	−0.0354584
$569$	−2760.00	−0.203348	−0.101674	−	0.994818i	$-0.532420\pi$
$569$	−2760.00	−0.203348	−0.101674	+	0.994818i	$0.532420\pi$
$570$	0	0
$571$	8930.00	0.654481	0.327241	−	0.944941i	$-0.393881\pi$
$571$	8930.00	0.654481	0.327241	+	0.944941i	$0.393881\pi$
$572$	−1645.00	−0.120246
$573$	0	0
$574$	−840.000	−0.0610817
$575$	75.0000	0.00543951
$576$	0	0
$577$	6944.00	0.501010	0.250505	−	0.968115i	$-0.419403\pi$
$577$	6944.00	0.501010	0.250505	+	0.968115i	$0.419403\pi$
$578$	12248.0	0.881401
$579$	0	0
$580$	5495.00	0.393392
$581$	612.000	0.0437006
$582$	0	0
$583$	−188.000	−0.0133553
$584$	−14550.0	−1.03096
$585$	0	0
$586$	6594.00	0.464839
$587$	−4206.00	−0.295741	−0.147871	−	0.989007i	$-0.547242\pi$
$587$	−4206.00	−0.295741	−0.147871	+	0.989007i	$0.547242\pi$
$588$	0	0
$589$	−12600.0	−0.881450
$590$	3740.00	0.260972
$591$	0	0
$592$	−2870.00	−0.199250
$593$	−6571.00	−0.455040	−0.227520	−	0.973773i	$-0.573062\pi$
$593$	−6571.00	−0.455040	−0.227520	+	0.973773i	$0.573062\pi$
$594$	0	0
$595$	3930.00	0.270780
$596$	20587.0	1.41489
$597$	0	0
$598$	−15.0000	−0.00102575
$599$	−9490.00	−0.647330	−0.323665	−	0.946172i	$-0.604915\pi$
$599$	−9490.00	−0.647330	−0.323665	+	0.946172i	$0.604915\pi$
$600$	0	0
$601$	11861.0	0.805025	0.402513	−	0.915414i	$-0.368137\pi$
$601$	11861.0	0.805025	0.402513	+	0.915414i	$0.368137\pi$
$602$	−2382.00	−0.161268
$603$	0	0
$604$	−3577.00	−0.240970
$605$	4390.00	0.295006
$606$	0	0
$607$	−518.000	−0.0346375	−0.0173188	−	0.999850i	$-0.505513\pi$
$607$	−518.000	−0.0346375	−0.0173188	+	0.999850i	$0.505513\pi$
$608$	−9016.00	−0.601393
$609$	0	0
$610$	−1690.00	−0.112174
$611$	1735.00	0.114878
$612$	0	0
$613$	15163.0	0.999067	0.499533	−	0.866295i	$-0.333505\pi$
$613$	15163.0	0.999067	0.499533	+	0.866295i	$0.333505\pi$
$614$	−4343.00	−0.285455
$615$	0	0
$616$	−4230.00	−0.276675
$617$	19011.0	1.24044	0.620222	−	0.784426i	$-0.287042\pi$
$617$	19011.0	1.24044	0.620222	+	0.784426i	$0.287042\pi$
$618$	0	0
$619$	−7906.00	−0.513359	−0.256679	−	0.966497i	$-0.582628\pi$
$619$	−7906.00	−0.513359	−0.256679	+	0.966497i	$0.582628\pi$
$620$	−7875.00	−0.510109
$621$	0	0
$622$	−2124.00	−0.136921
$623$	8928.00	0.574146
$624$	0	0
$625$	625.000	0.0400000
$626$	−7516.00	−0.479872
$627$	0	0
$628$	3997.00	0.253977
$629$	9170.00	0.581291
$630$	0	0
$631$	−3416.00	−0.215513	−0.107757	−	0.994177i	$-0.534367\pi$
$631$	−3416.00	−0.215513	−0.107757	+	0.994177i	$0.534367\pi$
$632$	18855.0	1.18673
$633$	0	0
$634$	−6880.00	−0.430977
$635$	−8230.00	−0.514327
$636$	0	0
$637$	1535.00	0.0954771
$638$	7379.00	0.457896
$639$	0	0
$640$	−7275.00	−0.449328
$641$	4830.00	0.297619	0.148809	−	0.988866i	$-0.452456\pi$
$641$	4830.00	0.297619	0.148809	+	0.988866i	$0.452456\pi$
$642$	0	0
$643$	12549.0	0.769649	0.384824	−	0.922990i	$-0.374262\pi$
$643$	12549.0	0.769649	0.384824	+	0.922990i	$0.374262\pi$
$644$	126.000	0.00770978
$645$	0	0
$646$	7336.00	0.446797
$647$	8164.00	0.496074	0.248037	−	0.968751i	$-0.420215\pi$
$647$	8164.00	0.496074	0.248037	+	0.968751i	$0.420215\pi$
$648$	0	0
$649$	−35156.0	−2.12634
$650$	−125.000	−0.00754293
$651$	0	0
$652$	−4991.00	−0.299789
$653$	23768.0	1.42437	0.712185	−	0.701992i	$-0.247706\pi$
$653$	23768.0	1.42437	0.712185	+	0.701992i	$0.247706\pi$
$654$	0	0
$655$	−9165.00	−0.546727
$656$	5740.00	0.341630
$657$	0	0
$658$	2082.00	0.123351
$659$	21472.0	1.26924	0.634621	−	0.772824i	$-0.281156\pi$
$659$	21472.0	1.26924	0.634621	+	0.772824i	$0.281156\pi$
$660$	0	0
$661$	12982.0	0.763905	0.381953	−	0.924182i	$-0.375252\pi$
$661$	12982.0	0.763905	0.381953	+	0.924182i	$0.375252\pi$
$662$	−4986.00	−0.292729
$663$	0	0
$664$	1530.00	0.0894210
$665$	1680.00	0.0979663
$666$	0	0
$667$	−471.000	−0.0273421
$668$	−11172.0	−0.647092
$669$	0	0
$670$	2460.00	0.141848
$671$	15886.0	0.913968
$672$	0	0
$673$	−6006.00	−0.344003	−0.172002	−	0.985097i	$-0.555023\pi$
$673$	−6006.00	−0.344003	−0.172002	+	0.985097i	$0.555023\pi$
$674$	904.000	0.0516629
$675$	0	0
$676$	15204.0	0.865043
$677$	1164.00	0.0660800	0.0330400	−	0.999454i	$-0.489481\pi$
$677$	1164.00	0.0660800	0.0330400	+	0.999454i	$0.489481\pi$
$678$	0	0
$679$	−5844.00	−0.330298
$680$	9825.00	0.554076
$681$	0	0
$682$	−10575.0	−0.593750
$683$	26496.0	1.48439	0.742197	−	0.670182i	$-0.233784\pi$
$683$	26496.0	1.48439	0.742197	+	0.670182i	$0.233784\pi$
$684$	0	0
$685$	5490.00	0.306222
$686$	3900.00	0.217059
$687$	0	0
$688$	16277.0	0.901969
$689$	−20.0000	−0.00110586
$690$	0	0
$691$	17110.0	0.941961	0.470981	−	0.882144i	$-0.343900\pi$
$691$	17110.0	0.941961	0.470981	+	0.882144i	$0.343900\pi$
$692$	−28938.0	−1.58968
$693$	0	0
$694$	8860.00	0.484612
$695$	−5210.00	−0.284355
$696$	0	0
$697$	−18340.0	−0.996667
$698$	−4454.00	−0.241528
$699$	0	0
$700$	1050.00	0.0566947
$701$	30251.0	1.62991	0.814953	−	0.579527i	$-0.196763\pi$
$701$	30251.0	1.62991	0.814953	+	0.579527i	$0.196763\pi$
$702$	0	0
$703$	3920.00	0.210307
$704$	7849.00	0.420199
$705$	0	0
$706$	−8781.00	−0.468098
$707$	8010.00	0.426092
$708$	0	0
$709$	18820.0	0.996897	0.498448	−	0.866919i	$-0.333903\pi$
$709$	18820.0	0.996897	0.498448	+	0.866919i	$0.333903\pi$
$710$	160.000	0.00845731
$711$	0	0
$712$	22320.0	1.17483
$713$	675.000	0.0354543
$714$	0	0
$715$	1175.00	0.0614581
$716$	−12796.0	−0.667890
$717$	0	0
$718$	2928.00	0.152189
$719$	31890.0	1.65410	0.827049	−	0.562130i	$-0.190018\pi$
$719$	31890.0	1.65410	0.827049	+	0.562130i	$0.190018\pi$
$720$	0	0
$721$	−4116.00	−0.212605
$722$	−3723.00	−0.191905
$723$	0	0
$724$	3640.00	0.186850
$725$	−3925.00	−0.201063
$726$	0	0
$727$	−11452.0	−0.584224	−0.292112	−	0.956384i	$-0.594358\pi$
$727$	−11452.0	−0.584224	−0.292112	+	0.956384i	$0.594358\pi$
$728$	−450.000	−0.0229095
$729$	0	0
$730$	4850.00	0.245899
$731$	−52007.0	−2.63139
$732$	0	0
$733$	7094.00	0.357466	0.178733	−	0.983898i	$-0.442800\pi$
$733$	7094.00	0.357466	0.178733	+	0.983898i	$0.442800\pi$
$734$	9102.00	0.457712
$735$	0	0
$736$	483.000	0.0241897
$737$	−23124.0	−1.15574
$738$	0	0
$739$	−3200.00	−0.159288	−0.0796440	−	0.996823i	$-0.525378\pi$
$739$	−3200.00	−0.159288	−0.0796440	+	0.996823i	$0.525378\pi$
$740$	2450.00	0.121708
$741$	0	0
$742$	−24.0000	−0.00118742
$743$	−20831.0	−1.02855	−0.514277	−	0.857624i	$-0.671940\pi$
$743$	−20831.0	−1.02855	−0.514277	+	0.857624i	$0.671940\pi$
$744$	0	0
$745$	−14705.0	−0.723154
$746$	−8183.00	−0.401610
$747$	0	0
$748$	−43099.0	−2.10676
$749$	6588.00	0.321389
$750$	0	0
$751$	−15605.0	−0.758235	−0.379118	−	0.925349i	$-0.623772\pi$
$751$	−15605.0	−0.758235	−0.379118	+	0.925349i	$0.623772\pi$
$752$	−14227.0	−0.689901
$753$	0	0
$754$	785.000	0.0379151
$755$	2555.00	0.123160
$756$	0	0
$757$	21349.0	1.02502	0.512512	−	0.858680i	$-0.328715\pi$
$757$	21349.0	1.02502	0.512512	+	0.858680i	$0.328715\pi$
$758$	6136.00	0.294023
$759$	0	0
$760$	4200.00	0.200461
$761$	3702.00	0.176343	0.0881717	−	0.996105i	$-0.471898\pi$
$761$	3702.00	0.176343	0.0881717	+	0.996105i	$0.471898\pi$
$762$	0	0
$763$	4200.00	0.199279
$764$	−33782.0	−1.59972
$765$	0	0
$766$	5643.00	0.266175
$767$	−3740.00	−0.176067
$768$	0	0
$769$	−1393.00	−0.0653223	−0.0326612	−	0.999466i	$-0.510398\pi$
$769$	−1393.00	−0.0653223	−0.0326612	+	0.999466i	$0.510398\pi$
$770$	1410.00	0.0659907
$771$	0	0
$772$	−11690.0	−0.544990
$773$	−6906.00	−0.321334	−0.160667	−	0.987009i	$-0.551365\pi$
$773$	−6906.00	−0.321334	−0.160667	+	0.987009i	$0.551365\pi$
$774$	0	0
$775$	5625.00	0.260717
$776$	−14610.0	−0.675861
$777$	0	0
$778$	8991.00	0.414323
$779$	−7840.00	−0.360587
$780$	0	0
$781$	−1504.00	−0.0689083
$782$	−393.000	−0.0179714
$783$	0	0
$784$	−12587.0	−0.573387
$785$	−2855.00	−0.129808
$786$	0	0
$787$	30493.0	1.38114	0.690571	−	0.723265i	$-0.257359\pi$
$787$	30493.0	1.38114	0.690571	+	0.723265i	$0.257359\pi$
$788$	9660.00	0.436705
$789$	0	0
$790$	−6285.00	−0.283051
$791$	6330.00	0.284537
$792$	0	0
$793$	1690.00	0.0756793
$794$	−12449.0	−0.556421
$795$	0	0
$796$	−30499.0	−1.35805
$797$	33488.0	1.48834	0.744169	−	0.667991i	$-0.232846\pi$
$797$	33488.0	1.48834	0.744169	+	0.667991i	$0.232846\pi$
$798$	0	0
$799$	45457.0	2.01271
$800$	4025.00	0.177882
$801$	0	0
$802$	8076.00	0.355578
$803$	−45590.0	−2.00353
$804$	0	0
$805$	−90.0000	−0.00394048
$806$	−1125.00	−0.0491643
$807$	0	0
$808$	20025.0	0.871878
$809$	−15304.0	−0.665093	−0.332546	−	0.943087i	$-0.607908\pi$
$809$	−15304.0	−0.665093	−0.332546	+	0.943087i	$0.607908\pi$
$810$	0	0
$811$	−40122.0	−1.73721	−0.868603	−	0.495509i	$-0.834982\pi$
$811$	−40122.0	−1.73721	−0.868603	+	0.495509i	$0.834982\pi$
$812$	−6594.00	−0.284980
$813$	0	0
$814$	3290.00	0.141664
$815$	3565.00	0.153223
$816$	0	0
$817$	−22232.0	−0.952019
$818$	−2833.00	−0.121092
$819$	0	0
$820$	−4900.00	−0.208677
$821$	25098.0	1.06690	0.533451	−	0.845831i	$-0.320895\pi$
$821$	25098.0	1.06690	0.533451	+	0.845831i	$0.320895\pi$
$822$	0	0
$823$	−43492.0	−1.84208	−0.921042	−	0.389462i	$-0.872661\pi$
$823$	−43492.0	−1.84208	−0.921042	+	0.389462i	$0.872661\pi$
$824$	−10290.0	−0.435035
$825$	0	0
$826$	−4488.00	−0.189053
$827$	11206.0	0.471186	0.235593	−	0.971852i	$-0.424297\pi$
$827$	11206.0	0.471186	0.235593	+	0.971852i	$0.424297\pi$
$828$	0	0
$829$	−23964.0	−1.00399	−0.501993	−	0.864872i	$-0.667400\pi$
$829$	−23964.0	−1.00399	−0.501993	+	0.864872i	$0.667400\pi$
$830$	−510.000	−0.0213281
$831$	0	0
$832$	835.000	0.0347938
$833$	40217.0	1.67279
$834$	0	0
$835$	7980.00	0.330730
$836$	−18424.0	−0.762210
$837$	0	0
$838$	−4777.00	−0.196920
$839$	−34606.0	−1.42399	−0.711997	−	0.702182i	$-0.752209\pi$
$839$	−34606.0	−1.42399	−0.711997	+	0.702182i	$0.752209\pi$
$840$	0	0
$841$	260.000	0.0106605
$842$	−6464.00	−0.264566
$843$	0	0
$844$	29134.0	1.18819
$845$	−10860.0	−0.442125
$846$	0	0
$847$	−5268.00	−0.213708
$848$	164.000	0.00664125
$849$	0	0
$850$	−3275.00	−0.132155
$851$	−210.000	−0.00845912
$852$	0	0
$853$	−18477.0	−0.741665	−0.370833	−	0.928700i	$-0.620928\pi$
$853$	−18477.0	−0.741665	−0.370833	+	0.928700i	$0.620928\pi$
$854$	2028.00	0.0812608
$855$	0	0
$856$	16470.0	0.657632
$857$	−41342.0	−1.64786	−0.823930	−	0.566692i	$-0.808223\pi$
$857$	−41342.0	−1.64786	−0.823930	+	0.566692i	$0.808223\pi$
$858$	0	0
$859$	21898.0	0.869791	0.434895	−	0.900481i	$-0.356785\pi$
$859$	21898.0	0.869791	0.434895	+	0.900481i	$0.356785\pi$
$860$	−13895.0	−0.550948
$861$	0	0
$862$	10680.0	0.421998
$863$	−18487.0	−0.729206	−0.364603	−	0.931163i	$-0.618795\pi$
$863$	−18487.0	−0.729206	−0.364603	+	0.931163i	$0.618795\pi$
$864$	0	0
$865$	20670.0	0.812487
$866$	11566.0	0.453844
$867$	0	0
$868$	9450.00	0.369532
$869$	59079.0	2.30623
$870$	0	0
$871$	−2460.00	−0.0956991
$872$	10500.0	0.407769
$873$	0	0
$874$	−168.000	−0.00650193
$875$	−750.000	−0.0289767
$876$	0	0
$877$	7593.00	0.292357	0.146179	−	0.989258i	$-0.453303\pi$
$877$	7593.00	0.292357	0.146179	+	0.989258i	$0.453303\pi$
$878$	−1448.00	−0.0556579
$879$	0	0
$880$	−9635.00	−0.369086
$881$	−3038.00	−0.116178	−0.0580890	−	0.998311i	$-0.518501\pi$
$881$	−3038.00	−0.116178	−0.0580890	+	0.998311i	$0.518501\pi$
$882$	0	0
$883$	−16732.0	−0.637686	−0.318843	−	0.947808i	$-0.603294\pi$
$883$	−16732.0	−0.637686	−0.318843	+	0.947808i	$0.603294\pi$
$884$	−4585.00	−0.174446
$885$	0	0
$886$	−2376.00	−0.0900940
$887$	−8031.00	−0.304007	−0.152004	−	0.988380i	$-0.548573\pi$
$887$	−8031.00	−0.304007	−0.152004	+	0.988380i	$0.548573\pi$
$888$	0	0
$889$	9876.00	0.372588
$890$	−7440.00	−0.280213
$891$	0	0
$892$	41692.0	1.56497
$893$	19432.0	0.728183
$894$	0	0
$895$	9140.00	0.341359
$896$	8730.00	0.325501
$897$	0	0
$898$	−14894.0	−0.553474
$899$	−35325.0	−1.31052
$900$	0	0
$901$	−524.000	−0.0193751
$902$	−6580.00	−0.242894
$903$	0	0
$904$	15825.0	0.582225
$905$	−2600.00	−0.0954994
$906$	0	0
$907$	−38487.0	−1.40897	−0.704487	−	0.709717i	$-0.748823\pi$
$907$	−38487.0	−1.40897	−0.704487	+	0.709717i	$0.748823\pi$
$908$	−34580.0	−1.26385
$909$	0	0
$910$	150.000	0.00546423
$911$	−5120.00	−0.186205	−0.0931027	−	0.995657i	$-0.529679\pi$
$911$	−5120.00	−0.186205	−0.0931027	+	0.995657i	$0.529679\pi$
$912$	0	0
$913$	4794.00	0.173777
$914$	16204.0	0.586412
$915$	0	0
$916$	−30408.0	−1.09684
$917$	10998.0	0.396059
$918$	0	0
$919$	28075.0	1.00774	0.503868	−	0.863781i	$-0.331910\pi$
$919$	28075.0	1.00774	0.503868	+	0.863781i	$0.331910\pi$
$920$	−225.000	−0.00806308
$921$	0	0
$922$	−5082.00	−0.181526
$923$	−160.000	−0.00570581
$924$	0	0
$925$	−1750.00	−0.0622050
$926$	−10326.0	−0.366451
$927$	0	0
$928$	−25277.0	−0.894136
$929$	−12856.0	−0.454028	−0.227014	−	0.973892i	$-0.572896\pi$
$929$	−12856.0	−0.454028	−0.227014	+	0.973892i	$0.572896\pi$
$930$	0	0
$931$	17192.0	0.605204
$932$	36414.0	1.27981
$933$	0	0
$934$	4184.00	0.146579
$935$	30785.0	1.07677
$936$	0	0
$937$	−1374.00	−0.0479046	−0.0239523	−	0.999713i	$-0.507625\pi$
$937$	−1374.00	−0.0479046	−0.0239523	+	0.999713i	$0.507625\pi$
$938$	−2952.00	−0.102757
$939$	0	0
$940$	12145.0	0.421411
$941$	8543.00	0.295955	0.147978	−	0.988991i	$-0.452724\pi$
$941$	8543.00	0.295955	0.147978	+	0.988991i	$0.452724\pi$
$942$	0	0
$943$	420.000	0.0145038
$944$	30668.0	1.05737
$945$	0	0
$946$	−18659.0	−0.641286
$947$	13506.0	0.463449	0.231724	−	0.972781i	$-0.425563\pi$
$947$	13506.0	0.463449	0.231724	+	0.972781i	$0.425563\pi$
$948$	0	0
$949$	−4850.00	−0.165898
$950$	−1400.00	−0.0478126
$951$	0	0
$952$	−11790.0	−0.401382
$953$	21775.0	0.740148	0.370074	−	0.929002i	$-0.379332\pi$
$953$	21775.0	0.740148	0.370074	+	0.929002i	$0.379332\pi$
$954$	0	0
$955$	24130.0	0.817621
$956$	−10822.0	−0.366118
$957$	0	0
$958$	−15576.0	−0.525300
$959$	−6588.00	−0.221833
$960$	0	0
$961$	20834.0	0.699339
$962$	350.000	0.0117302
$963$	0	0
$964$	25613.0	0.855746
$965$	8350.00	0.278545
$966$	0	0
$967$	3854.00	0.128166	0.0640829	−	0.997945i	$-0.479588\pi$
$967$	3854.00	0.128166	0.0640829	+	0.997945i	$0.479588\pi$
$968$	−13170.0	−0.437293
$969$	0	0
$970$	4870.00	0.161202
$971$	12933.0	0.427435	0.213718	−	0.976895i	$-0.431443\pi$
$971$	12933.0	0.427435	0.213718	+	0.976895i	$0.431443\pi$
$972$	0	0
$973$	6252.00	0.205992
$974$	10220.0	0.336211
$975$	0	0
$976$	−13858.0	−0.454492
$977$	17521.0	0.573743	0.286871	−	0.957969i	$-0.407385\pi$
$977$	17521.0	0.573743	0.286871	+	0.957969i	$0.407385\pi$
$978$	0	0
$979$	69936.0	2.28311
$980$	10745.0	0.350241
$981$	0	0
$982$	−2692.00	−0.0874798
$983$	−12573.0	−0.407952	−0.203976	−	0.978976i	$-0.565386\pi$
$983$	−12573.0	−0.407952	−0.203976	+	0.978976i	$0.565386\pi$
$984$	0	0
$985$	−6900.00	−0.223200
$986$	20567.0	0.664287
$987$	0	0
$988$	−1960.00	−0.0631133
$989$	1191.00	0.0382928
$990$	0	0
$991$	8945.00	0.286728	0.143364	−	0.989670i	$-0.454208\pi$
$991$	8945.00	0.286728	0.143364	+	0.989670i	$0.454208\pi$
$992$	36225.0	1.15942
$993$	0	0
$994$	−192.000	−0.00612663
$995$	21785.0	0.694101
$996$	0	0
$997$	−58179.0	−1.84809	−0.924046	−	0.382282i	$-0.875138\pi$
$997$	−58179.0	−1.84809	−0.924046	+	0.382282i	$0.875138\pi$
$998$	5764.00	0.182822
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.4.a.c.1.1	yes	1
3.2	odd	2		135.4.a.b.1.1	✓	1
4.3	odd	2		2160.4.a.p.1.1		1
5.2	odd	4		675.4.b.e.649.2		2
5.3	odd	4		675.4.b.e.649.1		2
5.4	even	2		675.4.a.c.1.1		1
9.2	odd	6		405.4.e.h.271.1		2
9.4	even	3		405.4.e.f.136.1		2
9.5	odd	6		405.4.e.h.136.1		2
9.7	even	3		405.4.e.f.271.1		2
12.11	even	2		2160.4.a.f.1.1		1
15.2	even	4		675.4.b.f.649.1		2
15.8	even	4		675.4.b.f.649.2		2
15.14	odd	2		675.4.a.h.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.b.1.1	✓	1	3.2	odd	2
135.4.a.c.1.1	yes	1	1.1	even	1	trivial
405.4.e.f.136.1		2	9.4	even	3
405.4.e.f.271.1		2	9.7	even	3
405.4.e.h.136.1		2	9.5	odd	6
405.4.e.h.271.1		2	9.2	odd	6
675.4.a.c.1.1		1	5.4	even	2
675.4.a.h.1.1		1	15.14	odd	2
675.4.b.e.649.1		2	5.3	odd	4
675.4.b.e.649.2		2	5.2	odd	4
675.4.b.f.649.1		2	15.2	even	4
675.4.b.f.649.2		2	15.8	even	4
2160.4.a.f.1.1		1	12.11	even	2
2160.4.a.p.1.1		1	4.3	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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