LMFDB - Embedded newform 135.4.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,4,Mod(1,135)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$7.96525785077$
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)

$f(q)$

$=$

$q-2.00000 q^{2} -4.00000 q^{4} +5.00000 q^{5} +24.0000 q^{8} +O(q^{10})$

$q-2.00000 q^{2} -4.00000 q^{4} +5.00000 q^{5} +24.0000 q^{8} -10.0000 q^{10} +10.0000 q^{11} -80.0000 q^{13} -16.0000 q^{16} +7.00000 q^{17} -113.000 q^{19} -20.0000 q^{20} -20.0000 q^{22} -81.0000 q^{23} +25.0000 q^{25} +160.000 q^{26} -220.000 q^{29} -189.000 q^{31} -160.000 q^{32} -14.0000 q^{34} +170.000 q^{37} +226.000 q^{38} +120.000 q^{40} -130.000 q^{41} +10.0000 q^{43} -40.0000 q^{44} +162.000 q^{46} +160.000 q^{47} -343.000 q^{49} -50.0000 q^{50} +320.000 q^{52} +631.000 q^{53} +50.0000 q^{55} +440.000 q^{58} -560.000 q^{59} +229.000 q^{61} +378.000 q^{62} +448.000 q^{64} -400.000 q^{65} +750.000 q^{67} -28.0000 q^{68} +890.000 q^{71} -890.000 q^{73} -340.000 q^{74} +452.000 q^{76} -27.0000 q^{79} -80.0000 q^{80} +260.000 q^{82} +429.000 q^{83} +35.0000 q^{85} -20.0000 q^{86} +240.000 q^{88} -750.000 q^{89} +324.000 q^{92} -320.000 q^{94} -565.000 q^{95} -1480.00 q^{97} +686.000 q^{98} +O(q^{100})$

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−2.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	24.0000	1.06066
$9$	0	0
$10$	−10.0000	−0.316228
$11$	10.0000	0.274101	0.137051	−	0.990564i	$-0.456238\pi$
$11$	10.0000	0.274101	0.137051	+	0.990564i	$0.456238\pi$
$12$	0	0
$13$	−80.0000	−1.70677	−0.853385	−	0.521281i	$-0.825454\pi$
$13$	−80.0000	−1.70677	−0.853385	+	0.521281i	$0.825454\pi$
$14$	0	0
$15$	0	0
$16$	−16.0000	−0.250000
$17$	7.00000	0.0998676	0.0499338	−	0.998753i	$-0.484099\pi$
$17$	7.00000	0.0998676	0.0499338	+	0.998753i	$0.484099\pi$
$18$	0	0
$19$	−113.000	−1.36442	−0.682210	−	0.731156i	$-0.738981\pi$
$19$	−113.000	−1.36442	−0.682210	+	0.731156i	$0.738981\pi$
$20$	−20.0000	−0.223607
$21$	0	0
$22$	−20.0000	−0.193819
$23$	−81.0000	−0.734333	−0.367167	−	0.930155i	$-0.619672\pi$
$23$	−81.0000	−0.734333	−0.367167	+	0.930155i	$0.619672\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	160.000	1.20687
$27$	0	0
$28$	0	0
$29$	−220.000	−1.40872	−0.704362	−	0.709841i	$-0.748767\pi$
$29$	−220.000	−1.40872	−0.704362	+	0.709841i	$0.748767\pi$
$30$	0	0
$31$	−189.000	−1.09501	−0.547506	−	0.836801i	$-0.684423\pi$
$31$	−189.000	−1.09501	−0.547506	+	0.836801i	$0.684423\pi$
$32$	−160.000	−0.883883
$33$	0	0
$34$	−14.0000	−0.0706171
$35$	0	0
$36$	0	0
$37$	170.000	0.755347	0.377673	−	0.925939i	$-0.376724\pi$
$37$	170.000	0.755347	0.377673	+	0.925939i	$0.376724\pi$
$38$	226.000	0.964791
$39$	0	0
$40$	120.000	0.474342
$41$	−130.000	−0.495185	−0.247593	−	0.968864i	$-0.579639\pi$
$41$	−130.000	−0.495185	−0.247593	+	0.968864i	$0.579639\pi$
$42$	0	0
$43$	10.0000	0.0354648	0.0177324	−	0.999843i	$-0.494355\pi$
$43$	10.0000	0.0354648	0.0177324	+	0.999843i	$0.494355\pi$
$44$	−40.0000	−0.137051
$45$	0	0
$46$	162.000	0.519252
$47$	160.000	0.496562	0.248281	−	0.968688i	$-0.420134\pi$
$47$	160.000	0.496562	0.248281	+	0.968688i	$0.420134\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	−50.0000	−0.141421
$51$	0	0
$52$	320.000	0.853385
$53$	631.000	1.63537	0.817684	−	0.575667i	$-0.195258\pi$
$53$	631.000	1.63537	0.817684	+	0.575667i	$0.195258\pi$
$54$	0	0
$55$	50.0000	0.122582
$56$	0	0
$57$	0	0
$58$	440.000	0.996118
$59$	−560.000	−1.23569	−0.617846	−	0.786299i	$-0.711994\pi$
$59$	−560.000	−1.23569	−0.617846	+	0.786299i	$0.711994\pi$
$60$	0	0
$61$	229.000	0.480663	0.240332	−	0.970691i	$-0.422744\pi$
$61$	229.000	0.480663	0.240332	+	0.970691i	$0.422744\pi$
$62$	378.000	0.774291
$63$	0	0
$64$	448.000	0.875000
$65$	−400.000	−0.763291
$66$	0	0
$67$	750.000	1.36757	0.683784	−	0.729684i	$-0.260333\pi$
$67$	750.000	1.36757	0.683784	+	0.729684i	$0.260333\pi$
$68$	−28.0000	−0.0499338
$69$	0	0
$70$	0	0
$71$	890.000	1.48766	0.743828	−	0.668371i	$-0.233008\pi$
$71$	890.000	1.48766	0.743828	+	0.668371i	$0.233008\pi$
$72$	0	0
$73$	−890.000	−1.42694	−0.713470	−	0.700686i	$-0.752878\pi$
$73$	−890.000	−1.42694	−0.713470	+	0.700686i	$0.752878\pi$
$74$	−340.000	−0.534111
$75$	0	0
$76$	452.000	0.682210
$77$	0	0
$78$	0	0
$79$	−27.0000	−0.0384524	−0.0192262	−	0.999815i	$-0.506120\pi$
$79$	−27.0000	−0.0384524	−0.0192262	+	0.999815i	$0.506120\pi$
$80$	−80.0000	−0.111803
$81$	0	0
$82$	260.000	0.350149
$83$	429.000	0.567336	0.283668	−	0.958923i	$-0.408449\pi$
$83$	429.000	0.567336	0.283668	+	0.958923i	$0.408449\pi$
$84$	0	0
$85$	35.0000	0.0446622
$86$	−20.0000	−0.0250774
$87$	0	0
$88$	240.000	0.290728
$89$	−750.000	−0.893257	−0.446628	−	0.894720i	$-0.647375\pi$
$89$	−750.000	−0.893257	−0.446628	+	0.894720i	$0.647375\pi$
$90$	0	0
$91$	0	0
$92$	324.000	0.367167
$93$	0	0
$94$	−320.000	−0.351122
$95$	−565.000	−0.610187
$96$	0	0
$97$	−1480.00	−1.54919	−0.774594	−	0.632459i	$-0.782046\pi$
$97$	−1480.00	−1.54919	−0.774594	+	0.632459i	$0.782046\pi$
$98$	686.000	0.707107
$99$	0	0
$100$	−100.000	−0.100000
$101$	−1500.00	−1.47778	−0.738889	−	0.673827i	$-0.764649\pi$
$101$	−1500.00	−1.47778	−0.738889	+	0.673827i	$0.764649\pi$
$102$	0	0
$103$	−460.000	−0.440050	−0.220025	−	0.975494i	$-0.570614\pi$
$103$	−460.000	−0.440050	−0.220025	+	0.975494i	$0.570614\pi$
$104$	−1920.00	−1.81030
$105$	0	0
$106$	−1262.00	−1.15638
$107$	−420.000	−0.379467	−0.189733	−	0.981836i	$-0.560762\pi$
$107$	−420.000	−0.379467	−0.189733	+	0.981836i	$0.560762\pi$
$108$	0	0
$109$	−607.000	−0.533395	−0.266698	−	0.963780i	$-0.585932\pi$
$109$	−607.000	−0.533395	−0.266698	+	0.963780i	$0.585932\pi$
$110$	−100.000	−0.0866784
$111$	0	0
$112$	0	0
$113$	2170.00	1.80652	0.903259	−	0.429097i	$-0.141168\pi$
$113$	2170.00	1.80652	0.903259	+	0.429097i	$0.141168\pi$
$114$	0	0
$115$	−405.000	−0.328404
$116$	880.000	0.704362
$117$	0	0
$118$	1120.00	0.873766
$119$	0	0
$120$	0	0
$121$	−1231.00	−0.924869
$122$	−458.000	−0.339880
$123$	0	0
$124$	756.000	0.547506
$125$	125.000	0.0894427
$126$	0	0
$127$	−1610.00	−1.12492	−0.562458	−	0.826826i	$-0.690144\pi$
$127$	−1610.00	−1.12492	−0.562458	+	0.826826i	$0.690144\pi$
$128$	384.000	0.265165
$129$	0	0
$130$	800.000	0.539728
$131$	2370.00	1.58067	0.790335	−	0.612674i	$-0.209906\pi$
$131$	2370.00	1.58067	0.790335	+	0.612674i	$0.209906\pi$
$132$	0	0
$133$	0	0
$134$	−1500.00	−0.967017
$135$	0	0
$136$	168.000	0.105926
$137$	1797.00	1.12064	0.560321	−	0.828275i	$-0.310678\pi$
$137$	1797.00	1.12064	0.560321	+	0.828275i	$0.310678\pi$
$138$	0	0
$139$	−124.000	−0.0756658	−0.0378329	−	0.999284i	$-0.512045\pi$
$139$	−124.000	−0.0756658	−0.0378329	+	0.999284i	$0.512045\pi$
$140$	0	0
$141$	0	0
$142$	−1780.00	−1.05193
$143$	−800.000	−0.467828
$144$	0	0
$145$	−1100.00	−0.630000
$146$	1780.00	1.00900
$147$	0	0
$148$	−680.000	−0.377673
$149$	−70.0000	−0.0384874	−0.0192437	−	0.999815i	$-0.506126\pi$
$149$	−70.0000	−0.0384874	−0.0192437	+	0.999815i	$0.506126\pi$
$150$	0	0
$151$	2248.00	1.21152	0.605760	−	0.795647i	$-0.292869\pi$
$151$	2248.00	1.21152	0.605760	+	0.795647i	$0.292869\pi$
$152$	−2712.00	−1.44719
$153$	0	0
$154$	0	0
$155$	−945.000	−0.489705
$156$	0	0
$157$	1010.00	0.513419	0.256709	−	0.966489i	$-0.417362\pi$
$157$	1010.00	0.513419	0.256709	+	0.966489i	$0.417362\pi$
$158$	54.0000	0.0271899
$159$	0	0
$160$	−800.000	−0.395285
$161$	0	0
$162$	0	0
$163$	590.000	0.283511	0.141756	−	0.989902i	$-0.454725\pi$
$163$	590.000	0.283511	0.141756	+	0.989902i	$0.454725\pi$
$164$	520.000	0.247593
$165$	0	0
$166$	−858.000	−0.401167
$167$	2403.00	1.11347	0.556736	−	0.830690i	$-0.312054\pi$
$167$	2403.00	1.11347	0.556736	+	0.830690i	$0.312054\pi$
$168$	0	0
$169$	4203.00	1.91306
$170$	−70.0000	−0.0315809
$171$	0	0
$172$	−40.0000	−0.0177324
$173$	801.000	0.352017	0.176008	−	0.984389i	$-0.443681\pi$
$173$	801.000	0.352017	0.176008	+	0.984389i	$0.443681\pi$
$174$	0	0
$175$	0	0
$176$	−160.000	−0.0685253
$177$	0	0
$178$	1500.00	0.631628
$179$	−2360.00	−0.985445	−0.492723	−	0.870186i	$-0.663998\pi$
$179$	−2360.00	−0.985445	−0.492723	+	0.870186i	$0.663998\pi$
$180$	0	0
$181$	1241.00	0.509629	0.254814	−	0.966990i	$-0.417986\pi$
$181$	1241.00	0.509629	0.254814	+	0.966990i	$0.417986\pi$
$182$	0	0
$183$	0	0
$184$	−1944.00	−0.778878
$185$	850.000	0.337801
$186$	0	0
$187$	70.0000	0.0273738
$188$	−640.000	−0.248281
$189$	0	0
$190$	1130.00	0.431467
$191$	−4990.00	−1.89039	−0.945193	−	0.326512i	$-0.894127\pi$
$191$	−4990.00	−1.89039	−0.945193	+	0.326512i	$0.894127\pi$
$192$	0	0
$193$	−2260.00	−0.842893	−0.421447	−	0.906853i	$-0.638477\pi$
$193$	−2260.00	−0.842893	−0.421447	+	0.906853i	$0.638477\pi$
$194$	2960.00	1.09544
$195$	0	0
$196$	1372.00	0.500000
$197$	−2247.00	−0.812650	−0.406325	−	0.913729i	$-0.633190\pi$
$197$	−2247.00	−0.812650	−0.406325	+	0.913729i	$0.633190\pi$
$198$	0	0
$199$	4564.00	1.62580	0.812898	−	0.582406i	$-0.197889\pi$
$199$	4564.00	1.62580	0.812898	+	0.582406i	$0.197889\pi$
$200$	600.000	0.212132
$201$	0	0
$202$	3000.00	1.04495
$203$	0	0
$204$	0	0
$205$	−650.000	−0.221454
$206$	920.000	0.311162
$207$	0	0
$208$	1280.00	0.426692
$209$	−1130.00	−0.373989
$210$	0	0
$211$	4949.00	1.61471	0.807354	−	0.590068i	$-0.200899\pi$
$211$	4949.00	1.61471	0.807354	+	0.590068i	$0.200899\pi$
$212$	−2524.00	−0.817684
$213$	0	0
$214$	840.000	0.268323
$215$	50.0000	0.0158603
$216$	0	0
$217$	0	0
$218$	1214.00	0.377167
$219$	0	0
$220$	−200.000	−0.0612909
$221$	−560.000	−0.170451
$222$	0	0
$223$	3890.00	1.16813	0.584067	−	0.811706i	$-0.301461\pi$
$223$	3890.00	1.16813	0.584067	+	0.811706i	$0.301461\pi$
$224$	0	0
$225$	0	0
$226$	−4340.00	−1.27740
$227$	2453.00	0.717231	0.358615	−	0.933485i	$-0.383249\pi$
$227$	2453.00	0.717231	0.358615	+	0.933485i	$0.383249\pi$
$228$	0	0
$229$	−6213.00	−1.79287	−0.896434	−	0.443178i	$-0.853851\pi$
$229$	−6213.00	−1.79287	−0.896434	+	0.443178i	$0.853851\pi$
$230$	810.000	0.232217
$231$	0	0
$232$	−5280.00	−1.49418
$233$	−3450.00	−0.970030	−0.485015	−	0.874506i	$-0.661186\pi$
$233$	−3450.00	−0.970030	−0.485015	+	0.874506i	$0.661186\pi$
$234$	0	0
$235$	800.000	0.222069
$236$	2240.00	0.617846
$237$	0	0
$238$	0	0
$239$	6490.00	1.75650	0.878249	−	0.478203i	$-0.158712\pi$
$239$	6490.00	1.75650	0.878249	+	0.478203i	$0.158712\pi$
$240$	0	0
$241$	−3401.00	−0.909036	−0.454518	−	0.890738i	$-0.650188\pi$
$241$	−3401.00	−0.909036	−0.454518	+	0.890738i	$0.650188\pi$
$242$	2462.00	0.653981
$243$	0	0
$244$	−916.000	−0.240332
$245$	−1715.00	−0.447214
$246$	0	0
$247$	9040.00	2.32875
$248$	−4536.00	−1.16144
$249$	0	0
$250$	−250.000	−0.0632456
$251$	−4980.00	−1.25233	−0.626165	−	0.779691i	$-0.715376\pi$
$251$	−4980.00	−1.25233	−0.626165	+	0.779691i	$0.715376\pi$
$252$	0	0
$253$	−810.000	−0.201282
$254$	3220.00	0.795436
$255$	0	0
$256$	−4352.00	−1.06250
$257$	3357.00	0.814801	0.407401	−	0.913250i	$-0.366435\pi$
$257$	3357.00	0.814801	0.407401	+	0.913250i	$0.366435\pi$
$258$	0	0
$259$	0	0
$260$	1600.00	0.381645
$261$	0	0
$262$	−4740.00	−1.11770
$263$	−4540.00	−1.06444	−0.532221	−	0.846605i	$-0.678643\pi$
$263$	−4540.00	−1.06444	−0.532221	+	0.846605i	$0.678643\pi$
$264$	0	0
$265$	3155.00	0.731359
$266$	0	0
$267$	0	0
$268$	−3000.00	−0.683784
$269$	8410.00	1.90620	0.953098	−	0.302662i	$-0.0978752\pi$
$269$	8410.00	1.90620	0.953098	+	0.302662i	$0.0978752\pi$
$270$	0	0
$271$	259.000	0.0580558	0.0290279	−	0.999579i	$-0.490759\pi$
$271$	259.000	0.0580558	0.0290279	+	0.999579i	$0.490759\pi$
$272$	−112.000	−0.0249669
$273$	0	0
$274$	−3594.00	−0.792414
$275$	250.000	0.0548202
$276$	0	0
$277$	−4170.00	−0.904516	−0.452258	−	0.891887i	$-0.649381\pi$
$277$	−4170.00	−0.904516	−0.452258	+	0.891887i	$0.649381\pi$
$278$	248.000	0.0535038
$279$	0	0
$280$	0	0
$281$	−1740.00	−0.369394	−0.184697	−	0.982796i	$-0.559130\pi$
$281$	−1740.00	−0.369394	−0.184697	+	0.982796i	$0.559130\pi$
$282$	0	0
$283$	−5070.00	−1.06495	−0.532474	−	0.846446i	$-0.678738\pi$
$283$	−5070.00	−1.06495	−0.532474	+	0.846446i	$0.678738\pi$
$284$	−3560.00	−0.743828
$285$	0	0
$286$	1600.00	0.330804
$287$	0	0
$288$	0	0
$289$	−4864.00	−0.990026
$290$	2200.00	0.445477
$291$	0	0
$292$	3560.00	0.713470
$293$	−159.000	−0.0317027	−0.0158513	−	0.999874i	$-0.505046\pi$
$293$	−159.000	−0.0317027	−0.0158513	+	0.999874i	$0.505046\pi$
$294$	0	0
$295$	−2800.00	−0.552618
$296$	4080.00	0.801166
$297$	0	0
$298$	140.000	0.0272147
$299$	6480.00	1.25334
$300$	0	0
$301$	0	0
$302$	−4496.00	−0.856675
$303$	0	0
$304$	1808.00	0.341105
$305$	1145.00	0.214959
$306$	0	0
$307$	6490.00	1.20653	0.603264	−	0.797542i	$-0.293867\pi$
$307$	6490.00	1.20653	0.603264	+	0.797542i	$0.293867\pi$
$308$	0	0
$309$	0	0
$310$	1890.00	0.346274
$311$	−8220.00	−1.49876	−0.749379	−	0.662142i	$-0.769648\pi$
$311$	−8220.00	−1.49876	−0.749379	+	0.662142i	$0.769648\pi$
$312$	0	0
$313$	−4660.00	−0.841530	−0.420765	−	0.907170i	$-0.638238\pi$
$313$	−4660.00	−0.841530	−0.420765	+	0.907170i	$0.638238\pi$
$314$	−2020.00	−0.363042
$315$	0	0
$316$	108.000	0.0192262
$317$	−6817.00	−1.20783	−0.603913	−	0.797050i	$-0.706393\pi$
$317$	−6817.00	−1.20783	−0.603913	+	0.797050i	$0.706393\pi$
$318$	0	0
$319$	−2200.00	−0.386133
$320$	2240.00	0.391312
$321$	0	0
$322$	0	0
$323$	−791.000	−0.136261
$324$	0	0
$325$	−2000.00	−0.341354
$326$	−1180.00	−0.200473
$327$	0	0
$328$	−3120.00	−0.525223
$329$	0	0
$330$	0	0
$331$	192.000	0.0318830	0.0159415	−	0.999873i	$-0.494925\pi$
$331$	192.000	0.0318830	0.0159415	+	0.999873i	$0.494925\pi$
$332$	−1716.00	−0.283668
$333$	0	0
$334$	−4806.00	−0.787343
$335$	3750.00	0.611595
$336$	0	0
$337$	4840.00	0.782349	0.391174	−	0.920317i	$-0.372069\pi$
$337$	4840.00	0.782349	0.391174	+	0.920317i	$0.372069\pi$
$338$	−8406.00	−1.35274
$339$	0	0
$340$	−140.000	−0.0223311
$341$	−1890.00	−0.300144
$342$	0	0
$343$	0	0
$344$	240.000	0.0376161
$345$	0	0
$346$	−1602.00	−0.248913
$347$	−860.000	−0.133047	−0.0665234	−	0.997785i	$-0.521191\pi$
$347$	−860.000	−0.133047	−0.0665234	+	0.997785i	$0.521191\pi$
$348$	0	0
$349$	5377.00	0.824711	0.412356	−	0.911023i	$-0.364706\pi$
$349$	5377.00	0.824711	0.412356	+	0.911023i	$0.364706\pi$
$350$	0	0
$351$	0	0
$352$	−1600.00	−0.242274
$353$	−8010.00	−1.20773	−0.603866	−	0.797086i	$-0.706374\pi$
$353$	−8010.00	−1.20773	−0.603866	+	0.797086i	$0.706374\pi$
$354$	0	0
$355$	4450.00	0.665300
$356$	3000.00	0.446628
$357$	0	0
$358$	4720.00	0.696815
$359$	12930.0	1.90089	0.950445	−	0.310894i	$-0.100628\pi$
$359$	12930.0	1.90089	0.950445	+	0.310894i	$0.100628\pi$
$360$	0	0
$361$	5910.00	0.861642
$362$	−2482.00	−0.360362
$363$	0	0
$364$	0	0
$365$	−4450.00	−0.638147
$366$	0	0
$367$	−6000.00	−0.853399	−0.426700	−	0.904393i	$-0.640324\pi$
$367$	−6000.00	−0.853399	−0.426700	+	0.904393i	$0.640324\pi$
$368$	1296.00	0.183583
$369$	0	0
$370$	−1700.00	−0.238862
$371$	0	0
$372$	0	0
$373$	−140.000	−0.0194341	−0.00971706	−	0.999953i	$-0.503093\pi$
$373$	−140.000	−0.0194341	−0.00971706	+	0.999953i	$0.503093\pi$
$374$	−140.000	−0.0193562
$375$	0	0
$376$	3840.00	0.526683
$377$	17600.0	2.40437
$378$	0	0
$379$	6217.00	0.842601	0.421301	−	0.906921i	$-0.361574\pi$
$379$	6217.00	0.842601	0.421301	+	0.906921i	$0.361574\pi$
$380$	2260.00	0.305094
$381$	0	0
$382$	9980.00	1.33670
$383$	4551.00	0.607168	0.303584	−	0.952805i	$-0.401817\pi$
$383$	4551.00	0.607168	0.303584	+	0.952805i	$0.401817\pi$
$384$	0	0
$385$	0	0
$386$	4520.00	0.596015
$387$	0	0
$388$	5920.00	0.774594
$389$	−2310.00	−0.301084	−0.150542	−	0.988604i	$-0.548102\pi$
$389$	−2310.00	−0.301084	−0.150542	+	0.988604i	$0.548102\pi$
$390$	0	0
$391$	−567.000	−0.0733361
$392$	−8232.00	−1.06066
$393$	0	0
$394$	4494.00	0.574631
$395$	−135.000	−0.0171964
$396$	0	0
$397$	−2900.00	−0.366617	−0.183308	−	0.983055i	$-0.558681\pi$
$397$	−2900.00	−0.366617	−0.183308	+	0.983055i	$0.558681\pi$
$398$	−9128.00	−1.14961
$399$	0	0
$400$	−400.000	−0.0500000
$401$	2250.00	0.280199	0.140099	−	0.990137i	$-0.455258\pi$
$401$	2250.00	0.280199	0.140099	+	0.990137i	$0.455258\pi$
$402$	0	0
$403$	15120.0	1.86894
$404$	6000.00	0.738889
$405$	0	0
$406$	0	0
$407$	1700.00	0.207041
$408$	0	0
$409$	−11263.0	−1.36166	−0.680831	−	0.732441i	$-0.738381\pi$
$409$	−11263.0	−1.36166	−0.680831	+	0.732441i	$0.738381\pi$
$410$	1300.00	0.156591
$411$	0	0
$412$	1840.00	0.220025
$413$	0	0
$414$	0	0
$415$	2145.00	0.253720
$416$	12800.0	1.50859
$417$	0	0
$418$	2260.00	0.264450
$419$	−6910.00	−0.805670	−0.402835	−	0.915273i	$-0.631975\pi$
$419$	−6910.00	−0.805670	−0.402835	+	0.915273i	$0.631975\pi$
$420$	0	0
$421$	−5249.00	−0.607650	−0.303825	−	0.952728i	$-0.598264\pi$
$421$	−5249.00	−0.607650	−0.303825	+	0.952728i	$0.598264\pi$
$422$	−9898.00	−1.14177
$423$	0	0
$424$	15144.0	1.73457
$425$	175.000	0.0199735
$426$	0	0
$427$	0	0
$428$	1680.00	0.189733
$429$	0	0
$430$	−100.000	−0.0112149
$431$	−11880.0	−1.32770	−0.663851	−	0.747865i	$-0.731079\pi$
$431$	−11880.0	−1.32770	−0.663851	+	0.747865i	$0.731079\pi$
$432$	0	0
$433$	−4280.00	−0.475020	−0.237510	−	0.971385i	$-0.576331\pi$
$433$	−4280.00	−0.475020	−0.237510	+	0.971385i	$0.576331\pi$
$434$	0	0
$435$	0	0
$436$	2428.00	0.266698
$437$	9153.00	1.00194
$438$	0	0
$439$	6463.00	0.702647	0.351324	−	0.936254i	$-0.385732\pi$
$439$	6463.00	0.702647	0.351324	+	0.936254i	$0.385732\pi$
$440$	1200.00	0.130018
$441$	0	0
$442$	1120.00	0.120527
$443$	11721.0	1.25707	0.628534	−	0.777782i	$-0.283655\pi$
$443$	11721.0	1.25707	0.628534	+	0.777782i	$0.283655\pi$
$444$	0	0
$445$	−3750.00	−0.399477
$446$	−7780.00	−0.825995
$447$	0	0
$448$	0	0
$449$	−2180.00	−0.229133	−0.114566	−	0.993416i	$-0.536548\pi$
$449$	−2180.00	−0.229133	−0.114566	+	0.993416i	$0.536548\pi$
$450$	0	0
$451$	−1300.00	−0.135731
$452$	−8680.00	−0.903259
$453$	0	0
$454$	−4906.00	−0.507159
$455$	0	0
$456$	0	0
$457$	−17840.0	−1.82608	−0.913042	−	0.407866i	$-0.866273\pi$
$457$	−17840.0	−1.82608	−0.913042	+	0.407866i	$0.866273\pi$
$458$	12426.0	1.26775
$459$	0	0
$460$	1620.00	0.164202
$461$	−2250.00	−0.227317	−0.113658	−	0.993520i	$-0.536257\pi$
$461$	−2250.00	−0.227317	−0.113658	+	0.993520i	$0.536257\pi$
$462$	0	0
$463$	1230.00	0.123462	0.0617310	−	0.998093i	$-0.480338\pi$
$463$	1230.00	0.123462	0.0617310	+	0.998093i	$0.480338\pi$
$464$	3520.00	0.352181
$465$	0	0
$466$	6900.00	0.685915
$467$	5813.00	0.576003	0.288002	−	0.957630i	$-0.407009\pi$
$467$	5813.00	0.576003	0.288002	+	0.957630i	$0.407009\pi$
$468$	0	0
$469$	0	0
$470$	−1600.00	−0.157027
$471$	0	0
$472$	−13440.0	−1.31065
$473$	100.000	0.00972094
$474$	0	0
$475$	−2825.00	−0.272884
$476$	0	0
$477$	0	0
$478$	−12980.0	−1.24203
$479$	−6750.00	−0.643873	−0.321937	−	0.946761i	$-0.604334\pi$
$479$	−6750.00	−0.643873	−0.321937	+	0.946761i	$0.604334\pi$
$480$	0	0
$481$	−13600.0	−1.28920
$482$	6802.00	0.642785
$483$	0	0
$484$	4924.00	0.462434
$485$	−7400.00	−0.692818
$486$	0	0
$487$	−6610.00	−0.615047	−0.307523	−	0.951541i	$-0.599500\pi$
$487$	−6610.00	−0.615047	−0.307523	+	0.951541i	$0.599500\pi$
$488$	5496.00	0.509820
$489$	0	0
$490$	3430.00	0.316228
$491$	−4990.00	−0.458647	−0.229323	−	0.973350i	$-0.573651\pi$
$491$	−4990.00	−0.458647	−0.229323	+	0.973350i	$0.573651\pi$
$492$	0	0
$493$	−1540.00	−0.140686
$494$	−18080.0	−1.64668
$495$	0	0
$496$	3024.00	0.273753
$497$	0	0
$498$	0	0
$499$	1483.00	0.133042	0.0665212	−	0.997785i	$-0.478810\pi$
$499$	1483.00	0.133042	0.0665212	+	0.997785i	$0.478810\pi$
$500$	−500.000	−0.0447214
$501$	0	0
$502$	9960.00	0.885531
$503$	−11641.0	−1.03190	−0.515951	−	0.856618i	$-0.672561\pi$
$503$	−11641.0	−1.03190	−0.515951	+	0.856618i	$0.672561\pi$
$504$	0	0
$505$	−7500.00	−0.660882
$506$	1620.00	0.142328
$507$	0	0
$508$	6440.00	0.562458
$509$	2620.00	0.228152	0.114076	−	0.993472i	$-0.463609\pi$
$509$	2620.00	0.228152	0.114076	+	0.993472i	$0.463609\pi$
$510$	0	0
$511$	0	0
$512$	5632.00	0.486136
$513$	0	0
$514$	−6714.00	−0.576151
$515$	−2300.00	−0.196796
$516$	0	0
$517$	1600.00	0.136108
$518$	0	0
$519$	0	0
$520$	−9600.00	−0.809592
$521$	−13690.0	−1.15119	−0.575595	−	0.817735i	$-0.695229\pi$
$521$	−13690.0	−1.15119	−0.575595	+	0.817735i	$0.695229\pi$
$522$	0	0
$523$	−10220.0	−0.854473	−0.427237	−	0.904140i	$-0.640513\pi$
$523$	−10220.0	−0.854473	−0.427237	+	0.904140i	$0.640513\pi$
$524$	−9480.00	−0.790335
$525$	0	0
$526$	9080.00	0.752675
$527$	−1323.00	−0.109356
$528$	0	0
$529$	−5606.00	−0.460754
$530$	−6310.00	−0.517149
$531$	0	0
$532$	0	0
$533$	10400.0	0.845167
$534$	0	0
$535$	−2100.00	−0.169703
$536$	18000.0	1.45053
$537$	0	0
$538$	−16820.0	−1.34788
$539$	−3430.00	−0.274101
$540$	0	0
$541$	−2778.00	−0.220768	−0.110384	−	0.993889i	$-0.535208\pi$
$541$	−2778.00	−0.220768	−0.110384	+	0.993889i	$0.535208\pi$
$542$	−518.000	−0.0410517
$543$	0	0
$544$	−1120.00	−0.0882713
$545$	−3035.00	−0.238541
$546$	0	0
$547$	−12830.0	−1.00287	−0.501436	−	0.865195i	$-0.667195\pi$
$547$	−12830.0	−1.00287	−0.501436	+	0.865195i	$0.667195\pi$
$548$	−7188.00	−0.560321
$549$	0	0
$550$	−500.000	−0.0387638
$551$	24860.0	1.92209
$552$	0	0
$553$	0	0
$554$	8340.00	0.639590
$555$	0	0
$556$	496.000	0.0378329
$557$	−4950.00	−0.376550	−0.188275	−	0.982116i	$-0.560290\pi$
$557$	−4950.00	−0.376550	−0.188275	+	0.982116i	$0.560290\pi$
$558$	0	0
$559$	−800.000	−0.0605302
$560$	0	0
$561$	0	0
$562$	3480.00	0.261201
$563$	6540.00	0.489570	0.244785	−	0.969577i	$-0.421283\pi$
$563$	6540.00	0.489570	0.244785	+	0.969577i	$0.421283\pi$
$564$	0	0
$565$	10850.0	0.807899
$566$	10140.0	0.753032
$567$	0	0
$568$	21360.0	1.57790
$569$	−15240.0	−1.12284	−0.561418	−	0.827532i	$-0.689744\pi$
$569$	−15240.0	−1.12284	−0.561418	+	0.827532i	$0.689744\pi$
$570$	0	0
$571$	−5281.00	−0.387045	−0.193523	−	0.981096i	$-0.561991\pi$
$571$	−5281.00	−0.387045	−0.193523	+	0.981096i	$0.561991\pi$
$572$	3200.00	0.233914
$573$	0	0
$574$	0	0
$575$	−2025.00	−0.146867
$576$	0	0
$577$	−10510.0	−0.758296	−0.379148	−	0.925336i	$-0.623783\pi$
$577$	−10510.0	−0.758296	−0.379148	+	0.925336i	$0.623783\pi$
$578$	9728.00	0.700054
$579$	0	0
$580$	4400.00	0.315000
$581$	0	0
$582$	0	0
$583$	6310.00	0.448256
$584$	−21360.0	−1.51350
$585$	0	0
$586$	318.000	0.0224172
$587$	4107.00	0.288780	0.144390	−	0.989521i	$-0.453878\pi$
$587$	4107.00	0.288780	0.144390	+	0.989521i	$0.453878\pi$
$588$	0	0
$589$	21357.0	1.49406
$590$	5600.00	0.390760
$591$	0	0
$592$	−2720.00	−0.188837
$593$	26129.0	1.80943	0.904713	−	0.426022i	$-0.140085\pi$
$593$	26129.0	1.80943	0.904713	+	0.426022i	$0.140085\pi$
$594$	0	0
$595$	0	0
$596$	280.000	0.0192437
$597$	0	0
$598$	−12960.0	−0.886244
$599$	−4360.00	−0.297404	−0.148702	−	0.988882i	$-0.547509\pi$
$599$	−4360.00	−0.297404	−0.148702	+	0.988882i	$0.547509\pi$
$600$	0	0
$601$	−16639.0	−1.12932	−0.564658	−	0.825325i	$-0.690992\pi$
$601$	−16639.0	−1.12932	−0.564658	+	0.825325i	$0.690992\pi$
$602$	0	0
$603$	0	0
$604$	−8992.00	−0.605760
$605$	−6155.00	−0.413614
$606$	0	0
$607$	490.000	0.0327652	0.0163826	−	0.999866i	$-0.494785\pi$
$607$	490.000	0.0327652	0.0163826	+	0.999866i	$0.494785\pi$
$608$	18080.0	1.20599
$609$	0	0
$610$	−2290.00	−0.151999
$611$	−12800.0	−0.847516
$612$	0	0
$613$	18400.0	1.21235	0.606174	−	0.795332i	$-0.292704\pi$
$613$	18400.0	1.21235	0.606174	+	0.795332i	$0.292704\pi$
$614$	−12980.0	−0.853144
$615$	0	0
$616$	0	0
$617$	7827.00	0.510702	0.255351	−	0.966848i	$-0.417809\pi$
$617$	7827.00	0.510702	0.255351	+	0.966848i	$0.417809\pi$
$618$	0	0
$619$	−19756.0	−1.28281	−0.641406	−	0.767202i	$-0.721649\pi$
$619$	−19756.0	−1.28281	−0.641406	+	0.767202i	$0.721649\pi$
$620$	3780.00	0.244852
$621$	0	0
$622$	16440.0	1.05978
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	9320.00	0.595051
$627$	0	0
$628$	−4040.00	−0.256709
$629$	1190.00	0.0754347
$630$	0	0
$631$	9829.00	0.620105	0.310053	−	0.950719i	$-0.399653\pi$
$631$	9829.00	0.620105	0.310053	+	0.950719i	$0.399653\pi$
$632$	−648.000	−0.0407849
$633$	0	0
$634$	13634.0	0.854062
$635$	−8050.00	−0.503078
$636$	0	0
$637$	27440.0	1.70677
$638$	4400.00	0.273037
$639$	0	0
$640$	1920.00	0.118585
$641$	−6000.00	−0.369713	−0.184856	−	0.982766i	$-0.559182\pi$
$641$	−6000.00	−0.369713	−0.184856	+	0.982766i	$0.559182\pi$
$642$	0	0
$643$	8280.00	0.507825	0.253912	−	0.967227i	$-0.418283\pi$
$643$	8280.00	0.507825	0.253912	+	0.967227i	$0.418283\pi$
$644$	0	0
$645$	0	0
$646$	1582.00	0.0963513
$647$	−16637.0	−1.01092	−0.505462	−	0.862849i	$-0.668678\pi$
$647$	−16637.0	−1.01092	−0.505462	+	0.862849i	$0.668678\pi$
$648$	0	0
$649$	−5600.00	−0.338705
$650$	4000.00	0.241374
$651$	0	0
$652$	−2360.00	−0.141756
$653$	19751.0	1.18364	0.591820	−	0.806070i	$-0.298410\pi$
$653$	19751.0	1.18364	0.591820	+	0.806070i	$0.298410\pi$
$654$	0	0
$655$	11850.0	0.706897
$656$	2080.00	0.123796
$657$	0	0
$658$	0	0
$659$	14260.0	0.842930	0.421465	−	0.906845i	$-0.361516\pi$
$659$	14260.0	0.842930	0.421465	+	0.906845i	$0.361516\pi$
$660$	0	0
$661$	22318.0	1.31327	0.656634	−	0.754210i	$-0.271980\pi$
$661$	22318.0	1.31327	0.656634	+	0.754210i	$0.271980\pi$
$662$	−384.000	−0.0225447
$663$	0	0
$664$	10296.0	0.601750
$665$	0	0
$666$	0	0
$667$	17820.0	1.03447
$668$	−9612.00	−0.556736
$669$	0	0
$670$	−7500.00	−0.432463
$671$	2290.00	0.131750
$672$	0	0
$673$	20040.0	1.14782	0.573912	−	0.818917i	$-0.305425\pi$
$673$	20040.0	1.14782	0.573912	+	0.818917i	$0.305425\pi$
$674$	−9680.00	−0.553204
$675$	0	0
$676$	−16812.0	−0.956532
$677$	−2310.00	−0.131138	−0.0655691	−	0.997848i	$-0.520886\pi$
$677$	−2310.00	−0.131138	−0.0655691	+	0.997848i	$0.520886\pi$
$678$	0	0
$679$	0	0
$680$	840.000	0.0473714
$681$	0	0
$682$	3780.00	0.212234
$683$	−26739.0	−1.49801	−0.749004	−	0.662566i	$-0.769468\pi$
$683$	−26739.0	−1.49801	−0.749004	+	0.662566i	$0.769468\pi$
$684$	0	0
$685$	8985.00	0.501167
$686$	0	0
$687$	0	0
$688$	−160.000	−0.00886620
$689$	−50480.0	−2.79120
$690$	0	0
$691$	5101.00	0.280827	0.140413	−	0.990093i	$-0.455157\pi$
$691$	5101.00	0.280827	0.140413	+	0.990093i	$0.455157\pi$
$692$	−3204.00	−0.176008
$693$	0	0
$694$	1720.00	0.0940783
$695$	−620.000	−0.0338388
$696$	0	0
$697$	−910.000	−0.0494530
$698$	−10754.0	−0.583159
$699$	0	0
$700$	0	0
$701$	26030.0	1.40248	0.701241	−	0.712925i	$-0.252630\pi$
$701$	26030.0	1.40248	0.701241	+	0.712925i	$0.252630\pi$
$702$	0	0
$703$	−19210.0	−1.03061
$704$	4480.00	0.239839
$705$	0	0
$706$	16020.0	0.853995
$707$	0	0
$708$	0	0
$709$	−3854.00	−0.204147	−0.102073	−	0.994777i	$-0.532548\pi$
$709$	−3854.00	−0.204147	−0.102073	+	0.994777i	$0.532548\pi$
$710$	−8900.00	−0.470438
$711$	0	0
$712$	−18000.0	−0.947442
$713$	15309.0	0.804105
$714$	0	0
$715$	−4000.00	−0.209219
$716$	9440.00	0.492723
$717$	0	0
$718$	−25860.0	−1.34413
$719$	−870.000	−0.0451259	−0.0225630	−	0.999745i	$-0.507183\pi$
$719$	−870.000	−0.0451259	−0.0225630	+	0.999745i	$0.507183\pi$
$720$	0	0
$721$	0	0
$722$	−11820.0	−0.609273
$723$	0	0
$724$	−4964.00	−0.254814
$725$	−5500.00	−0.281745
$726$	0	0
$727$	35780.0	1.82532	0.912659	−	0.408721i	$-0.134025\pi$
$727$	35780.0	1.82532	0.912659	+	0.408721i	$0.134025\pi$
$728$	0	0
$729$	0	0
$730$	8900.00	0.451238
$731$	70.0000	0.00354178
$732$	0	0
$733$	−3400.00	−0.171326	−0.0856629	−	0.996324i	$-0.527301\pi$
$733$	−3400.00	−0.171326	−0.0856629	+	0.996324i	$0.527301\pi$
$734$	12000.0	0.603444
$735$	0	0
$736$	12960.0	0.649065
$737$	7500.00	0.374852
$738$	0	0
$739$	−683.000	−0.0339981	−0.0169990	−	0.999856i	$-0.505411\pi$
$739$	−683.000	−0.0339981	−0.0169990	+	0.999856i	$0.505411\pi$
$740$	−3400.00	−0.168901
$741$	0	0
$742$	0	0
$743$	−13400.0	−0.661640	−0.330820	−	0.943694i	$-0.607325\pi$
$743$	−13400.0	−0.661640	−0.330820	+	0.943694i	$0.607325\pi$
$744$	0	0
$745$	−350.000	−0.0172121
$746$	280.000	0.0137420
$747$	0	0
$748$	−280.000	−0.0136869
$749$	0	0
$750$	0	0
$751$	−23219.0	−1.12819	−0.564097	−	0.825709i	$-0.690776\pi$
$751$	−23219.0	−1.12819	−0.564097	+	0.825709i	$0.690776\pi$
$752$	−2560.00	−0.124140
$753$	0	0
$754$	−35200.0	−1.70014
$755$	11240.0	0.541809
$756$	0	0
$757$	19630.0	0.942489	0.471245	−	0.882003i	$-0.343805\pi$
$757$	19630.0	0.942489	0.471245	+	0.882003i	$0.343805\pi$
$758$	−12434.0	−0.595809
$759$	0	0
$760$	−13560.0	−0.647201
$761$	2940.00	0.140046	0.0700229	−	0.997545i	$-0.477693\pi$
$761$	2940.00	0.140046	0.0700229	+	0.997545i	$0.477693\pi$
$762$	0	0
$763$	0	0
$764$	19960.0	0.945193
$765$	0	0
$766$	−9102.00	−0.429332
$767$	44800.0	2.10904
$768$	0	0
$769$	−13987.0	−0.655896	−0.327948	−	0.944696i	$-0.606357\pi$
$769$	−13987.0	−0.655896	−0.327948	+	0.944696i	$0.606357\pi$
$770$	0	0
$771$	0	0
$772$	9040.00	0.421447
$773$	19839.0	0.923104	0.461552	−	0.887113i	$-0.347293\pi$
$773$	19839.0	0.923104	0.461552	+	0.887113i	$0.347293\pi$
$774$	0	0
$775$	−4725.00	−0.219003
$776$	−35520.0	−1.64316
$777$	0	0
$778$	4620.00	0.212898
$779$	14690.0	0.675640
$780$	0	0
$781$	8900.00	0.407768
$782$	1134.00	0.0518565
$783$	0	0
$784$	5488.00	0.250000
$785$	5050.00	0.229608
$786$	0	0
$787$	−38390.0	−1.73883	−0.869413	−	0.494086i	$-0.835503\pi$
$787$	−38390.0	−1.73883	−0.869413	+	0.494086i	$0.835503\pi$
$788$	8988.00	0.406325
$789$	0	0
$790$	270.000	0.0121597
$791$	0	0
$792$	0	0
$793$	−18320.0	−0.820381
$794$	5800.00	0.259237
$795$	0	0
$796$	−18256.0	−0.812898
$797$	−28027.0	−1.24563	−0.622815	−	0.782369i	$-0.714011\pi$
$797$	−28027.0	−1.24563	−0.622815	+	0.782369i	$0.714011\pi$
$798$	0	0
$799$	1120.00	0.0495904
$800$	−4000.00	−0.176777
$801$	0	0
$802$	−4500.00	−0.198130
$803$	−8900.00	−0.391126
$804$	0	0
$805$	0	0
$806$	−30240.0	−1.32154
$807$	0	0
$808$	−36000.0	−1.56742
$809$	8630.00	0.375049	0.187525	−	0.982260i	$-0.439954\pi$
$809$	8630.00	0.375049	0.187525	+	0.982260i	$0.439954\pi$
$810$	0	0
$811$	−1932.00	−0.0836519	−0.0418260	−	0.999125i	$-0.513317\pi$
$811$	−1932.00	−0.0836519	−0.0418260	+	0.999125i	$0.513317\pi$
$812$	0	0
$813$	0	0
$814$	−3400.00	−0.146400
$815$	2950.00	0.126790
$816$	0	0
$817$	−1130.00	−0.0483889
$818$	22526.0	0.962840
$819$	0	0
$820$	2600.00	0.110727
$821$	−18090.0	−0.768996	−0.384498	−	0.923126i	$-0.625625\pi$
$821$	−18090.0	−0.768996	−0.384498	+	0.923126i	$0.625625\pi$
$822$	0	0
$823$	12890.0	0.545950	0.272975	−	0.962021i	$-0.411992\pi$
$823$	12890.0	0.545950	0.272975	+	0.962021i	$0.411992\pi$
$824$	−11040.0	−0.466743
$825$	0	0
$826$	0	0
$827$	14887.0	0.625963	0.312982	−	0.949759i	$-0.398672\pi$
$827$	14887.0	0.625963	0.312982	+	0.949759i	$0.398672\pi$
$828$	0	0
$829$	12666.0	0.530649	0.265325	−	0.964159i	$-0.414521\pi$
$829$	12666.0	0.530649	0.265325	+	0.964159i	$0.414521\pi$
$830$	−4290.00	−0.179407
$831$	0	0
$832$	−35840.0	−1.49342
$833$	−2401.00	−0.0998676
$834$	0	0
$835$	12015.0	0.497960
$836$	4520.00	0.186995
$837$	0	0
$838$	13820.0	0.569694
$839$	43820.0	1.80314	0.901570	−	0.432633i	$-0.142416\pi$
$839$	43820.0	1.80314	0.901570	+	0.432633i	$0.142416\pi$
$840$	0	0
$841$	24011.0	0.984501
$842$	10498.0	0.429673
$843$	0	0
$844$	−19796.0	−0.807354
$845$	21015.0	0.855548
$846$	0	0
$847$	0	0
$848$	−10096.0	−0.408842
$849$	0	0
$850$	−350.000	−0.0141234
$851$	−13770.0	−0.554676
$852$	0	0
$853$	19320.0	0.775503	0.387752	−	0.921764i	$-0.373252\pi$
$853$	19320.0	0.775503	0.387752	+	0.921764i	$0.373252\pi$
$854$	0	0
$855$	0	0
$856$	−10080.0	−0.402485
$857$	−3653.00	−0.145606	−0.0728029	−	0.997346i	$-0.523194\pi$
$857$	−3653.00	−0.145606	−0.0728029	+	0.997346i	$0.523194\pi$
$858$	0	0
$859$	24373.0	0.968098	0.484049	−	0.875041i	$-0.339166\pi$
$859$	24373.0	0.968098	0.484049	+	0.875041i	$0.339166\pi$
$860$	−200.000	−0.00793017
$861$	0	0
$862$	23760.0	0.938827
$863$	−17629.0	−0.695363	−0.347681	−	0.937613i	$-0.613031\pi$
$863$	−17629.0	−0.695363	−0.347681	+	0.937613i	$0.613031\pi$
$864$	0	0
$865$	4005.00	0.157427
$866$	8560.00	0.335890
$867$	0	0
$868$	0	0
$869$	−270.000	−0.0105398
$870$	0	0
$871$	−60000.0	−2.33412
$872$	−14568.0	−0.565751
$873$	0	0
$874$	−18306.0	−0.708478
$875$	0	0
$876$	0	0
$877$	−21210.0	−0.816660	−0.408330	−	0.912834i	$-0.633889\pi$
$877$	−21210.0	−0.816660	−0.408330	+	0.912834i	$0.633889\pi$
$878$	−12926.0	−0.496847
$879$	0	0
$880$	−800.000	−0.0306454
$881$	39340.0	1.50442	0.752212	−	0.658921i	$-0.228987\pi$
$881$	39340.0	1.50442	0.752212	+	0.658921i	$0.228987\pi$
$882$	0	0
$883$	−4240.00	−0.161594	−0.0807969	−	0.996731i	$-0.525747\pi$
$883$	−4240.00	−0.161594	−0.0807969	+	0.996731i	$0.525747\pi$
$884$	2240.00	0.0852255
$885$	0	0
$886$	−23442.0	−0.888882
$887$	−15933.0	−0.603132	−0.301566	−	0.953445i	$-0.597509\pi$
$887$	−15933.0	−0.603132	−0.301566	+	0.953445i	$0.597509\pi$
$888$	0	0
$889$	0	0
$890$	7500.00	0.282473
$891$	0	0
$892$	−15560.0	−0.584067
$893$	−18080.0	−0.677519
$894$	0	0
$895$	−11800.0	−0.440704
$896$	0	0
$897$	0	0
$898$	4360.00	0.162021
$899$	41580.0	1.54257
$900$	0	0
$901$	4417.00	0.163320
$902$	2600.00	0.0959762
$903$	0	0
$904$	52080.0	1.91610
$905$	6205.00	0.227913
$906$	0	0
$907$	−6780.00	−0.248210	−0.124105	−	0.992269i	$-0.539606\pi$
$907$	−6780.00	−0.248210	−0.124105	+	0.992269i	$0.539606\pi$
$908$	−9812.00	−0.358615
$909$	0	0
$910$	0	0
$911$	−24740.0	−0.899751	−0.449875	−	0.893091i	$-0.648532\pi$
$911$	−24740.0	−0.899751	−0.449875	+	0.893091i	$0.648532\pi$
$912$	0	0
$913$	4290.00	0.155507
$914$	35680.0	1.29124
$915$	0	0
$916$	24852.0	0.896434
$917$	0	0
$918$	0	0
$919$	−48344.0	−1.73528	−0.867640	−	0.497194i	$-0.834364\pi$
$919$	−48344.0	−1.73528	−0.867640	+	0.497194i	$0.834364\pi$
$920$	−9720.00	−0.348325
$921$	0	0
$922$	4500.00	0.160737
$923$	−71200.0	−2.53909
$924$	0	0
$925$	4250.00	0.151069
$926$	−2460.00	−0.0873009
$927$	0	0
$928$	35200.0	1.24515
$929$	−29650.0	−1.04713	−0.523566	−	0.851985i	$-0.675399\pi$
$929$	−29650.0	−1.04713	−0.523566	+	0.851985i	$0.675399\pi$
$930$	0	0
$931$	38759.0	1.36442
$932$	13800.0	0.485015
$933$	0	0
$934$	−11626.0	−0.407296
$935$	350.000	0.0122420
$936$	0	0
$937$	10260.0	0.357716	0.178858	−	0.983875i	$-0.442760\pi$
$937$	10260.0	0.357716	0.178858	+	0.983875i	$0.442760\pi$
$938$	0	0
$939$	0	0
$940$	−3200.00	−0.111035
$941$	−16270.0	−0.563642	−0.281821	−	0.959467i	$-0.590938\pi$
$941$	−16270.0	−0.563642	−0.281821	+	0.959467i	$0.590938\pi$
$942$	0	0
$943$	10530.0	0.363631
$944$	8960.00	0.308923
$945$	0	0
$946$	−200.000	−0.00687374
$947$	−23103.0	−0.792763	−0.396382	−	0.918086i	$-0.629734\pi$
$947$	−23103.0	−0.792763	−0.396382	+	0.918086i	$0.629734\pi$
$948$	0	0
$949$	71200.0	2.43546
$950$	5650.00	0.192958
$951$	0	0
$952$	0	0
$953$	−32090.0	−1.09076	−0.545381	−	0.838188i	$-0.683615\pi$
$953$	−32090.0	−1.09076	−0.545381	+	0.838188i	$0.683615\pi$
$954$	0	0
$955$	−24950.0	−0.845406
$956$	−25960.0	−0.878249
$957$	0	0
$958$	13500.0	0.455287
$959$	0	0
$960$	0	0
$961$	5930.00	0.199053
$962$	27200.0	0.911604
$963$	0	0
$964$	13604.0	0.454518
$965$	−11300.0	−0.376953
$966$	0	0
$967$	−42010.0	−1.39705	−0.698527	−	0.715584i	$-0.746161\pi$
$967$	−42010.0	−1.39705	−0.698527	+	0.715584i	$0.746161\pi$
$968$	−29544.0	−0.980971
$969$	0	0
$970$	14800.0	0.489896
$971$	17490.0	0.578044	0.289022	−	0.957322i	$-0.406670\pi$
$971$	17490.0	0.578044	0.289022	+	0.957322i	$0.406670\pi$
$972$	0	0
$973$	0	0
$974$	13220.0	0.434904
$975$	0	0
$976$	−3664.00	−0.120166
$977$	−22130.0	−0.724669	−0.362334	−	0.932048i	$-0.618020\pi$
$977$	−22130.0	−0.724669	−0.362334	+	0.932048i	$0.618020\pi$
$978$	0	0
$979$	−7500.00	−0.244843
$980$	6860.00	0.223607
$981$	0	0
$982$	9980.00	0.324312
$983$	40959.0	1.32898	0.664491	−	0.747296i	$-0.268648\pi$
$983$	40959.0	1.32898	0.664491	+	0.747296i	$0.268648\pi$
$984$	0	0
$985$	−11235.0	−0.363428
$986$	3080.00	0.0994799
$987$	0	0
$988$	−36160.0	−1.16438
$989$	−810.000	−0.0260430
$990$	0	0
$991$	61169.0	1.96074	0.980372	−	0.197157i	$-0.0631707\pi$
$991$	61169.0	1.96074	0.980372	+	0.197157i	$0.0631707\pi$
$992$	30240.0	0.967864
$993$	0	0
$994$	0	0
$995$	22820.0	0.727078
$996$	0	0
$997$	26190.0	0.831941	0.415971	−	0.909378i	$-0.363442\pi$
$997$	26190.0	0.831941	0.415971	+	0.909378i	$0.363442\pi$
$998$	−2966.00	−0.0940752
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.4.a.a.1.1	✓	1	1.1	even	1	trivial
135.4.a.d.1.1	yes	1	3.2	odd	2
405.4.e.e.136.1		2	9.5	odd	6
405.4.e.e.271.1		2	9.2	odd	6
405.4.e.j.136.1		2	9.4	even	3
405.4.e.j.271.1		2	9.7	even	3
675.4.a.b.1.1		1	15.14	odd	2
675.4.a.i.1.1		1	5.4	even	2
675.4.b.c.649.1		2	15.8	even	4
675.4.b.c.649.2		2	15.2	even	4
675.4.b.d.649.1		2	5.2	odd	4
675.4.b.d.649.2		2	5.3	odd	4
2160.4.a.d.1.1		1	12.11	even	2
2160.4.a.n.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 $\F_{q}$

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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