LMFDB - Embedded newform 675.4.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [675,4,Mod(1,675)] 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("675.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(675, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	675.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,0,-7,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.8262892539$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	675.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

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

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	6.00000	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$7$	6.00000	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$8$	−15.0000	−0.662913
$9$	0	0
$10$	0	0
$11$	47.0000	1.28828	0.644138	−	0.764909i	$-0.277216\pi$
$11$	47.0000	1.28828	0.644138	+	0.764909i	$0.277216\pi$
$12$	0	0
$13$	5.00000	0.106673	0.0533366	−	0.998577i	$-0.483014\pi$
$13$	5.00000	0.106673	0.0533366	+	0.998577i	$0.483014\pi$
$14$	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	$0.609780\pi$
$20$	0	0
$21$	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
$25$	0	0
$26$	5.00000	0.0377146
$27$	0	0
$28$	−42.0000	−0.283473
$29$	157.000	1.00532	0.502658	−	0.864485i	$-0.332355\pi$
$29$	157.000	1.00532	0.502658	+	0.864485i	$0.332355\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	70.0000	0.311025	0.155513	−	0.987834i	$-0.450297\pi$
$37$	70.0000	0.311025	0.155513	+	0.987834i	$0.450297\pi$
$38$	−56.0000	−0.239063
$39$	0	0
$40$	0	0
$41$	−140.000	−0.533276	−0.266638	−	0.963797i	$-0.585913\pi$
$41$	−140.000	−0.533276	−0.266638	+	0.963797i	$0.585913\pi$
$42$	0	0
$43$	−397.000	−1.40795	−0.703976	−	0.710224i	$-0.748594\pi$
$43$	−397.000	−1.40795	−0.703976	+	0.710224i	$0.748594\pi$
$44$	−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
$50$	0	0
$51$	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
$55$	0	0
$56$	−90.0000	−0.214763
$57$	0	0
$58$	157.000	0.355433
$59$	−748.000	−1.65053	−0.825265	−	0.564745i	$-0.808974\pi$
$59$	−748.000	−1.65053	−0.825265	+	0.564745i	$0.808974\pi$
$60$	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
$65$	0	0
$66$	0	0
$67$	−492.000	−0.897125	−0.448562	−	0.893751i	$-0.648064\pi$
$67$	−492.000	−0.897125	−0.448562	+	0.893751i	$0.648064\pi$
$68$	917.000	1.63533
$69$	0	0
$70$	0	0
$71$	−32.0000	−0.0534888	−0.0267444	−	0.999642i	$-0.508514\pi$
$71$	−32.0000	−0.0534888	−0.0267444	+	0.999642i	$0.508514\pi$
$72$	0	0
$73$	−970.000	−1.55520	−0.777602	−	0.628757i	$-0.783564\pi$
$73$	−970.000	−1.55520	−0.777602	+	0.628757i	$0.783564\pi$
$74$	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	$0.852886\pi$
$80$	0	0
$81$	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
$85$	0	0
$86$	−397.000	−0.497786
$87$	0	0
$88$	−705.000	−0.854014
$89$	1488.00	1.77222	0.886111	−	0.463474i	$-0.153397\pi$
$89$	1488.00	1.77222	0.886111	+	0.463474i	$0.153397\pi$
$90$	0	0
$91$	30.0000	0.0345588
$92$	−21.0000	−0.0237978
$93$	0	0
$94$	−347.000	−0.380748
$95$	0	0
$96$	0	0
$97$	−974.000	−1.01953	−0.509767	−	0.860313i	$-0.670268\pi$
$97$	−974.000	−1.01953	−0.509767	+	0.860313i	$0.670268\pi$
$98$	−307.000	−0.316446
$99$	0	0
$100$	0	0
$101$	1335.00	1.31522	0.657611	−	0.753357i	$-0.271567\pi$
$101$	1335.00	1.31522	0.657611	+	0.753357i	$0.271567\pi$
$102$	0	0
$103$	−686.000	−0.656248	−0.328124	−	0.944635i	$-0.606416\pi$
$103$	−686.000	−0.656248	−0.328124	+	0.944635i	$0.606416\pi$
$104$	−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	$0.599512\pi$
$110$	0	0
$111$	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
$115$	0	0
$116$	−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
$125$	0	0
$126$	0	0
$127$	1646.00	1.15007	0.575035	−	0.818129i	$-0.304988\pi$
$127$	1646.00	1.15007	0.575035	+	0.818129i	$0.304988\pi$
$128$	−1455.00	−1.00473
$129$	0	0
$130$	0	0
$131$	1833.00	1.22252	0.611259	−	0.791430i	$-0.290663\pi$
$131$	1833.00	1.22252	0.611259	+	0.791430i	$0.290663\pi$
$132$	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	$0.602984\pi$
$140$	0	0
$141$	0	0
$142$	−32.0000	−0.0189111
$143$	235.000	0.137424
$144$	0	0
$145$	0	0
$146$	−970.000	−0.549848
$147$	0	0
$148$	−490.000	−0.272147
$149$	2941.00	1.61702	0.808510	−	0.588482i	$-0.200274\pi$
$149$	2941.00	1.61702	0.808510	+	0.588482i	$0.200274\pi$
$150$	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
$155$	0	0
$156$	0	0
$157$	571.000	0.290260	0.145130	−	0.989413i	$-0.453640\pi$
$157$	571.000	0.290260	0.145130	+	0.989413i	$0.453640\pi$
$158$	−1257.00	−0.632921
$159$	0	0
$160$	0	0
$161$	18.0000	0.00881117
$162$	0	0
$163$	−713.000	−0.342616	−0.171308	−	0.985217i	$-0.554799\pi$
$163$	−713.000	−0.342616	−0.171308	+	0.985217i	$0.554799\pi$
$164$	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
$170$	0	0
$171$	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
$175$	0	0
$176$	1927.00	0.825302
$177$	0	0
$178$	1488.00	0.626575
$179$	−1828.00	−0.763302	−0.381651	−	0.924306i	$-0.624644\pi$
$179$	−1828.00	−0.763302	−0.381651	+	0.924306i	$0.624644\pi$
$180$	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
$185$	0	0
$186$	0	0
$187$	−6157.00	−2.40772
$188$	2429.00	0.942303
$189$	0	0
$190$	0	0
$191$	−4826.00	−1.82826	−0.914129	−	0.405424i	$-0.867124\pi$
$191$	−4826.00	−1.82826	−0.914129	+	0.405424i	$0.867124\pi$
$192$	0	0
$193$	−1670.00	−0.622846	−0.311423	−	0.950271i	$-0.600806\pi$
$193$	−1670.00	−0.622846	−0.311423	+	0.950271i	$0.600806\pi$
$194$	−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	$0.217231\pi$
$200$	0	0
$201$	0	0
$202$	1335.00	0.465001
$203$	942.000	0.325692
$204$	0	0
$205$	0	0
$206$	−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
$215$	0	0
$216$	0	0
$217$	1350.00	0.422322
$218$	−700.000	−0.217477
$219$	0	0
$220$	0	0
$221$	−655.000	−0.199367
$222$	0	0
$223$	5956.00	1.78853	0.894267	−	0.447533i	$-0.147697\pi$
$223$	5956.00	1.78853	0.894267	+	0.447533i	$0.147697\pi$
$224$	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	$0.284378\pi$
$230$	0	0
$231$	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
$235$	0	0
$236$	5236.00	1.44421
$237$	0	0
$238$	−786.000	−0.214071
$239$	−1546.00	−0.418420	−0.209210	−	0.977871i	$-0.567089\pi$
$239$	−1546.00	−0.418420	−0.209210	+	0.977871i	$0.567089\pi$
$240$	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
$245$	0	0
$246$	0	0
$247$	−280.000	−0.0721294
$248$	−3375.00	−0.864164
$249$	0	0
$250$	0	0
$251$	−1221.00	−0.307047	−0.153524	−	0.988145i	$-0.549062\pi$
$251$	−1221.00	−0.307047	−0.153524	+	0.988145i	$0.549062\pi$
$252$	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
$260$	0	0
$261$	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
$265$	0	0
$266$	−336.000	−0.0774492
$267$	0	0
$268$	3444.00	0.784984
$269$	2231.00	0.505675	0.252837	−	0.967509i	$-0.418636\pi$
$269$	2231.00	0.505675	0.252837	+	0.967509i	$0.418636\pi$
$270$	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
$275$	0	0
$276$	0	0
$277$	−6450.00	−1.39907	−0.699536	−	0.714597i	$-0.746610\pi$
$277$	−6450.00	−1.39907	−0.699536	+	0.714597i	$0.746610\pi$
$278$	−1042.00	−0.224802
$279$	0	0
$280$	0	0
$281$	1050.00	0.222910	0.111455	−	0.993769i	$-0.464449\pi$
$281$	1050.00	0.222910	0.111455	+	0.993769i	$0.464449\pi$
$282$	0	0
$283$	1584.00	0.332717	0.166359	−	0.986065i	$-0.446799\pi$
$283$	1584.00	0.332717	0.166359	+	0.986065i	$0.446799\pi$
$284$	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
$290$	0	0
$291$	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
$295$	0	0
$296$	−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
$305$	0	0
$306$	0	0
$307$	4343.00	0.807388	0.403694	−	0.914894i	$-0.367726\pi$
$307$	4343.00	0.807388	0.403694	+	0.914894i	$0.367726\pi$
$308$	−1974.00	−0.365192
$309$	0	0
$310$	0	0
$311$	2124.00	0.387270	0.193635	−	0.981074i	$-0.437972\pi$
$311$	2124.00	0.387270	0.193635	+	0.981074i	$0.437972\pi$
$312$	0	0
$313$	7516.00	1.35728	0.678641	−	0.734470i	$-0.262569\pi$
$313$	7516.00	1.35728	0.678641	+	0.734470i	$0.262569\pi$
$314$	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
$320$	0	0
$321$	0	0
$322$	18.0000	0.00311522
$323$	7336.00	1.26373
$324$	0	0
$325$	0	0
$326$	−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
$335$	0	0
$336$	0	0
$337$	−904.000	−0.146125	−0.0730623	−	0.997327i	$-0.523277\pi$
$337$	−904.000	−0.146125	−0.0730623	+	0.997327i	$0.523277\pi$
$338$	−2172.00	−0.349530
$339$	0	0
$340$	0	0
$341$	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	$0.610959\pi$
$350$	0	0
$351$	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
$355$	0	0
$356$	−10416.0	−1.55069
$357$	0	0
$358$	−1828.00	−0.269868
$359$	−2928.00	−0.430457	−0.215228	−	0.976564i	$-0.569050\pi$
$359$	−2928.00	−0.430457	−0.215228	+	0.976564i	$0.569050\pi$
$360$	0	0
$361$	−3723.00	−0.542790
$362$	−520.000	−0.0754989
$363$	0	0
$364$	−210.000	−0.0302390
$365$	0	0
$366$	0	0
$367$	−9102.00	−1.29461	−0.647303	−	0.762233i	$-0.724103\pi$
$367$	−9102.00	−1.29461	−0.647303	+	0.762233i	$0.724103\pi$
$368$	123.000	0.0174234
$369$	0	0
$370$	0	0
$371$	24.0000	0.00335854
$372$	0	0
$373$	8183.00	1.13592	0.567962	−	0.823055i	$-0.307732\pi$
$373$	8183.00	1.13592	0.567962	+	0.823055i	$0.307732\pi$
$374$	−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	$0.363498\pi$
$380$	0	0
$381$	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
$385$	0	0
$386$	−1670.00	−0.220209
$387$	0	0
$388$	6818.00	0.892092
$389$	−8991.00	−1.17188	−0.585941	−	0.810354i	$-0.699275\pi$
$389$	−8991.00	−1.17188	−0.585941	+	0.810354i	$0.699275\pi$
$390$	0	0
$391$	−393.000	−0.0508309
$392$	4605.00	0.593336
$393$	0	0
$394$	−1380.00	−0.176455
$395$	0	0
$396$	0	0
$397$	12449.0	1.57380	0.786898	−	0.617082i	$-0.211686\pi$
$397$	12449.0	1.57380	0.786898	+	0.617082i	$0.211686\pi$
$398$	4357.00	0.548735
$399$	0	0
$400$	0	0
$401$	−8076.00	−1.00573	−0.502863	−	0.864366i	$-0.667720\pi$
$401$	−8076.00	−1.00573	−0.502863	+	0.864366i	$0.667720\pi$
$402$	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	$0.554781\pi$
$410$	0	0
$411$	0	0
$412$	4802.00	0.574217
$413$	−4488.00	−0.534722
$414$	0	0
$415$	0	0
$416$	805.000	0.0948759
$417$	0	0
$418$	−2632.00	−0.307979
$419$	4777.00	0.556973	0.278487	−	0.960440i	$-0.410167\pi$
$419$	4777.00	0.556973	0.278487	+	0.960440i	$0.410167\pi$
$420$	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
$425$	0	0
$426$	0	0
$427$	−2028.00	−0.229840
$428$	7686.00	0.868030
$429$	0	0
$430$	0	0
$431$	−10680.0	−1.19359	−0.596795	−	0.802394i	$-0.703560\pi$
$431$	−10680.0	−1.19359	−0.596795	+	0.802394i	$0.703560\pi$
$432$	0	0
$433$	−11566.0	−1.28366	−0.641832	−	0.766845i	$-0.721825\pi$
$433$	−11566.0	−1.28366	−0.641832	+	0.766845i	$0.721825\pi$
$434$	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	$0.525081\pi$
$440$	0	0
$441$	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
$445$	0	0
$446$	5956.00	0.632343
$447$	0	0
$448$	−1002.00	−0.105670
$449$	14894.0	1.56546	0.782730	−	0.622362i	$-0.213827\pi$
$449$	14894.0	1.56546	0.782730	+	0.622362i	$0.213827\pi$
$450$	0	0
$451$	−6580.00	−0.687007
$452$	7385.00	0.768498
$453$	0	0
$454$	4940.00	0.510673
$455$	0	0
$456$	0	0
$457$	−16204.0	−1.65862	−0.829312	−	0.558786i	$-0.811267\pi$
$457$	−16204.0	−1.65862	−0.829312	+	0.558786i	$0.811267\pi$
$458$	4344.00	0.443192
$459$	0	0
$460$	0	0
$461$	5082.00	0.513432	0.256716	−	0.966487i	$-0.417359\pi$
$461$	5082.00	0.513432	0.256716	+	0.966487i	$0.417359\pi$
$462$	0	0
$463$	10326.0	1.03648	0.518240	−	0.855235i	$-0.326588\pi$
$463$	10326.0	1.03648	0.518240	+	0.855235i	$0.326588\pi$
$464$	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
$470$	0	0
$471$	0	0
$472$	11220.0	1.09416
$473$	−18659.0	−1.81383
$474$	0	0
$475$	0	0
$476$	5502.00	0.529798
$477$	0	0
$478$	−1546.00	−0.147934
$479$	15576.0	1.48577	0.742887	−	0.669417i	$-0.233456\pi$
$479$	15576.0	1.48577	0.742887	+	0.669417i	$0.233456\pi$
$480$	0	0
$481$	350.000	0.0331780
$482$	−3659.00	−0.345774
$483$	0	0
$484$	−6146.00	−0.577198
$485$	0	0
$486$	0	0
$487$	−10220.0	−0.950949	−0.475475	−	0.879729i	$-0.657724\pi$
$487$	−10220.0	−0.950949	−0.475475	+	0.879729i	$0.657724\pi$
$488$	5070.00	0.470304
$489$	0	0
$490$	0	0
$491$	2692.00	0.247430	0.123715	−	0.992318i	$-0.460519\pi$
$491$	2692.00	0.247430	0.123715	+	0.992318i	$0.460519\pi$
$492$	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	$0.416756\pi$
$500$	0	0
$501$	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
$505$	0	0
$506$	141.000	0.0123878
$507$	0	0
$508$	−11522.0	−1.00631
$509$	5849.00	0.509337	0.254668	−	0.967028i	$-0.418034\pi$
$509$	5849.00	0.509337	0.254668	+	0.967028i	$0.418034\pi$
$510$	0	0
$511$	−5820.00	−0.503839
$512$	11521.0	0.994455
$513$	0	0
$514$	−6255.00	−0.536763
$515$	0	0
$516$	0	0
$517$	−16309.0	−1.38737
$518$	420.000	0.0356250
$519$	0	0
$520$	0	0
$521$	17032.0	1.43222	0.716109	−	0.697989i	$-0.245921\pi$
$521$	17032.0	1.43222	0.716109	+	0.697989i	$0.245921\pi$
$522$	0	0
$523$	−4147.00	−0.346722	−0.173361	−	0.984858i	$-0.555463\pi$
$523$	−4147.00	−0.346722	−0.173361	+	0.984858i	$0.555463\pi$
$524$	−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
$530$	0	0
$531$	0	0
$532$	2352.00	0.191677
$533$	−700.000	−0.0568862
$534$	0	0
$535$	0	0
$536$	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
$545$	0	0
$546$	0	0
$547$	13751.0	1.07486	0.537432	−	0.843307i	$-0.319395\pi$
$547$	13751.0	1.07486	0.537432	+	0.843307i	$0.319395\pi$
$548$	−7686.00	−0.599142
$549$	0	0
$550$	0	0
$551$	−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
$560$	0	0
$561$	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
$565$	0	0
$566$	1584.00	0.117633
$567$	0	0
$568$	480.000	0.0354584
$569$	2760.00	0.203348	0.101674	−	0.994818i	$-0.467580\pi$
$569$	2760.00	0.203348	0.101674	+	0.994818i	$0.467580\pi$
$570$	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
$575$	0	0
$576$	0	0
$577$	−6944.00	−0.501010	−0.250505	−	0.968115i	$-0.580597\pi$
$577$	−6944.00	−0.501010	−0.250505	+	0.968115i	$0.580597\pi$
$578$	12248.0	0.881401
$579$	0	0
$580$	0	0
$581$	−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
$590$	0	0
$591$	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
$595$	0	0
$596$	−20587.0	−1.41489
$597$	0	0
$598$	15.0000	0.00102575
$599$	9490.00	0.647330	0.323665	−	0.946172i	$-0.395085\pi$
$599$	9490.00	0.647330	0.323665	+	0.946172i	$0.395085\pi$
$600$	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
$605$	0	0
$606$	0	0
$607$	518.000	0.0346375	0.0173188	−	0.999850i	$-0.494487\pi$
$607$	518.000	0.0346375	0.0173188	+	0.999850i	$0.494487\pi$
$608$	−9016.00	−0.601393
$609$	0	0
$610$	0	0
$611$	−1735.00	−0.114878
$612$	0	0
$613$	−15163.0	−0.999067	−0.499533	−	0.866295i	$-0.666495\pi$
$613$	−15163.0	−0.999067	−0.499533	+	0.866295i	$0.666495\pi$
$614$	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	$0.582628\pi$
$620$	0	0
$621$	0	0
$622$	2124.00	0.136921
$623$	8928.00	0.574146
$624$	0	0
$625$	0	0
$626$	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
$635$	0	0
$636$	0	0
$637$	−1535.00	−0.0954771
$638$	7379.00	0.457896
$639$	0	0
$640$	0	0
$641$	−4830.00	−0.297619	−0.148809	−	0.988866i	$-0.547544\pi$
$641$	−4830.00	−0.297619	−0.148809	+	0.988866i	$0.547544\pi$
$642$	0	0
$643$	−12549.0	−0.769649	−0.384824	−	0.922990i	$-0.625738\pi$
$643$	−12549.0	−0.769649	−0.384824	+	0.922990i	$0.625738\pi$
$644$	−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
$650$	0	0
$651$	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
$655$	0	0
$656$	−5740.00	−0.341630
$657$	0	0
$658$	−2082.00	−0.123351
$659$	−21472.0	−1.26924	−0.634621	−	0.772824i	$-0.718844\pi$
$659$	−21472.0	−1.26924	−0.634621	+	0.772824i	$0.718844\pi$
$660$	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
$665$	0	0
$666$	0	0
$667$	471.000	0.0273421
$668$	−11172.0	−0.647092
$669$	0	0
$670$	0	0
$671$	−15886.0	−0.913968
$672$	0	0
$673$	6006.00	0.344003	0.172002	−	0.985097i	$-0.444977\pi$
$673$	6006.00	0.344003	0.172002	+	0.985097i	$0.444977\pi$
$674$	−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
$680$	0	0
$681$	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
$685$	0	0
$686$	−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
$695$	0	0
$696$	0	0
$697$	18340.0	0.996667
$698$	−4454.00	−0.241528
$699$	0	0
$700$	0	0
$701$	−30251.0	−1.62991	−0.814953	−	0.579527i	$-0.803237\pi$
$701$	−30251.0	−1.62991	−0.814953	+	0.579527i	$0.803237\pi$
$702$	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	$0.333903\pi$
$710$	0	0
$711$	0	0
$712$	−22320.0	−1.17483
$713$	675.000	0.0354543
$714$	0	0
$715$	0	0
$716$	12796.0	0.667890
$717$	0	0
$718$	−2928.00	−0.152189
$719$	−31890.0	−1.65410	−0.827049	−	0.562130i	$-0.809982\pi$
$719$	−31890.0	−1.65410	−0.827049	+	0.562130i	$0.809982\pi$
$720$	0	0
$721$	−4116.00	−0.212605
$722$	−3723.00	−0.191905
$723$	0	0
$724$	3640.00	0.186850
$725$	0	0
$726$	0	0
$727$	11452.0	0.584224	0.292112	−	0.956384i	$-0.405642\pi$
$727$	11452.0	0.584224	0.292112	+	0.956384i	$0.405642\pi$
$728$	−450.000	−0.0229095
$729$	0	0
$730$	0	0
$731$	52007.0	2.63139
$732$	0	0
$733$	−7094.00	−0.357466	−0.178733	−	0.983898i	$-0.557200\pi$
$733$	−7094.00	−0.357466	−0.178733	+	0.983898i	$0.557200\pi$
$734$	−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	$0.525378\pi$
$740$	0	0
$741$	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
$745$	0	0
$746$	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
$755$	0	0
$756$	0	0
$757$	−21349.0	−1.02502	−0.512512	−	0.858680i	$-0.671285\pi$
$757$	−21349.0	−1.02502	−0.512512	+	0.858680i	$0.671285\pi$
$758$	6136.00	0.294023
$759$	0	0
$760$	0	0
$761$	−3702.00	−0.176343	−0.0881717	−	0.996105i	$-0.528102\pi$
$761$	−3702.00	−0.176343	−0.0881717	+	0.996105i	$0.528102\pi$
$762$	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	$0.510398\pi$
$770$	0	0
$771$	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
$775$	0	0
$776$	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
$785$	0	0
$786$	0	0
$787$	−30493.0	−1.38114	−0.690571	−	0.723265i	$-0.742641\pi$
$787$	−30493.0	−1.38114	−0.690571	+	0.723265i	$0.742641\pi$
$788$	9660.00	0.436705
$789$	0	0
$790$	0	0
$791$	−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
$800$	0	0
$801$	0	0
$802$	−8076.00	−0.355578
$803$	−45590.0	−2.00353
$804$	0	0
$805$	0	0
$806$	1125.00	0.0491643
$807$	0	0
$808$	−20025.0	−0.871878
$809$	15304.0	0.665093	0.332546	−	0.943087i	$-0.392092\pi$
$809$	15304.0	0.665093	0.332546	+	0.943087i	$0.392092\pi$
$810$	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
$815$	0	0
$816$	0	0
$817$	22232.0	0.952019
$818$	−2833.00	−0.121092
$819$	0	0
$820$	0	0
$821$	−25098.0	−1.06690	−0.533451	−	0.845831i	$-0.679105\pi$
$821$	−25098.0	−1.06690	−0.533451	+	0.845831i	$0.679105\pi$
$822$	0	0
$823$	43492.0	1.84208	0.921042	−	0.389462i	$-0.127339\pi$
$823$	43492.0	1.84208	0.921042	+	0.389462i	$0.127339\pi$
$824$	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	$0.667400\pi$
$830$	0	0
$831$	0	0
$832$	−835.000	−0.0347938
$833$	40217.0	1.67279
$834$	0	0
$835$	0	0
$836$	18424.0	0.762210
$837$	0	0
$838$	4777.00	0.196920
$839$	34606.0	1.42399	0.711997	−	0.702182i	$-0.247791\pi$
$839$	34606.0	1.42399	0.711997	+	0.702182i	$0.247791\pi$
$840$	0	0
$841$	260.000	0.0106605
$842$	−6464.00	−0.264566
$843$	0	0
$844$	29134.0	1.18819
$845$	0	0
$846$	0	0
$847$	5268.00	0.213708
$848$	164.000	0.00664125
$849$	0	0
$850$	0	0
$851$	210.000	0.00845912
$852$	0	0
$853$	18477.0	0.741665	0.370833	−	0.928700i	$-0.379072\pi$
$853$	18477.0	0.741665	0.370833	+	0.928700i	$0.379072\pi$
$854$	−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	$0.356785\pi$
$860$	0	0
$861$	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
$865$	0	0
$866$	−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
$875$	0	0
$876$	0	0
$877$	−7593.00	−0.292357	−0.146179	−	0.989258i	$-0.546697\pi$
$877$	−7593.00	−0.292357	−0.146179	+	0.989258i	$0.546697\pi$
$878$	−1448.00	−0.0556579
$879$	0	0
$880$	0	0
$881$	3038.00	0.116178	0.0580890	−	0.998311i	$-0.481499\pi$
$881$	3038.00	0.116178	0.0580890	+	0.998311i	$0.481499\pi$
$882$	0	0
$883$	16732.0	0.637686	0.318843	−	0.947808i	$-0.396706\pi$
$883$	16732.0	0.637686	0.318843	+	0.947808i	$0.396706\pi$
$884$	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
$890$	0	0
$891$	0	0
$892$	−41692.0	−1.56497
$893$	19432.0	0.728183
$894$	0	0
$895$	0	0
$896$	−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
$905$	0	0
$906$	0	0
$907$	38487.0	1.40897	0.704487	−	0.709717i	$-0.251177\pi$
$907$	38487.0	1.40897	0.704487	+	0.709717i	$0.251177\pi$
$908$	−34580.0	−1.26385
$909$	0	0
$910$	0	0
$911$	5120.00	0.186205	0.0931027	−	0.995657i	$-0.470321\pi$
$911$	5120.00	0.186205	0.0931027	+	0.995657i	$0.470321\pi$
$912$	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	$0.331910\pi$
$920$	0	0
$921$	0	0
$922$	5082.00	0.181526
$923$	−160.000	−0.00570581
$924$	0	0
$925$	0	0
$926$	10326.0	0.366451
$927$	0	0
$928$	25277.0	0.894136
$929$	12856.0	0.454028	0.227014	−	0.973892i	$-0.427104\pi$
$929$	12856.0	0.454028	0.227014	+	0.973892i	$0.427104\pi$
$930$	0	0
$931$	17192.0	0.605204
$932$	36414.0	1.27981
$933$	0	0
$934$	4184.00	0.146579
$935$	0	0
$936$	0	0
$937$	1374.00	0.0479046	0.0239523	−	0.999713i	$-0.492375\pi$
$937$	1374.00	0.0479046	0.0239523	+	0.999713i	$0.492375\pi$
$938$	−2952.00	−0.102757
$939$	0	0
$940$	0	0
$941$	−8543.00	−0.295955	−0.147978	−	0.988991i	$-0.547276\pi$
$941$	−8543.00	−0.295955	−0.147978	+	0.988991i	$0.547276\pi$
$942$	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
$950$	0	0
$951$	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
$955$	0	0
$956$	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
$965$	0	0
$966$	0	0
$967$	−3854.00	−0.128166	−0.0640829	−	0.997945i	$-0.520412\pi$
$967$	−3854.00	−0.128166	−0.0640829	+	0.997945i	$0.520412\pi$
$968$	−13170.0	−0.437293
$969$	0	0
$970$	0	0
$971$	−12933.0	−0.427435	−0.213718	−	0.976895i	$-0.568557\pi$
$971$	−12933.0	−0.427435	−0.213718	+	0.976895i	$0.568557\pi$
$972$	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
$980$	0	0
$981$	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
$985$	0	0
$986$	−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
$995$	0	0
$996$	0	0
$997$	58179.0	1.84809	0.924046	−	0.382282i	$-0.124862\pi$
$997$	58179.0	1.84809	0.924046	+	0.382282i	$0.124862\pi$
$998$	5764.00	0.182822
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	675.4.a.h.1.1		1
3.2	odd	2		675.4.a.c.1.1		1
5.2	odd	4		675.4.b.f.649.2		2
5.3	odd	4		675.4.b.f.649.1		2
5.4	even	2		135.4.a.b.1.1	✓	1
15.2	even	4		675.4.b.e.649.1		2
15.8	even	4		675.4.b.e.649.2		2
15.14	odd	2		135.4.a.c.1.1	yes	1
20.19	odd	2		2160.4.a.f.1.1		1
45.4	even	6		405.4.e.h.136.1		2
45.14	odd	6		405.4.e.f.136.1		2
45.29	odd	6		405.4.e.f.271.1		2
45.34	even	6		405.4.e.h.271.1		2
60.59	even	2		2160.4.a.p.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.b.1.1	✓	1	5.4	even	2
135.4.a.c.1.1	yes	1	15.14	odd	2
405.4.e.f.136.1		2	45.14	odd	6
405.4.e.f.271.1		2	45.29	odd	6
405.4.e.h.136.1		2	45.4	even	6
405.4.e.h.271.1		2	45.34	even	6
675.4.a.c.1.1		1	3.2	odd	2
675.4.a.h.1.1		1	1.1	even	1	trivial
675.4.b.e.649.1		2	15.2	even	4
675.4.b.e.649.2		2	15.8	even	4
675.4.b.f.649.1		2	5.3	odd	4
675.4.b.f.649.2		2	5.2	odd	4
2160.4.a.f.1.1		1	20.19	odd	2
2160.4.a.p.1.1		1	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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