LMFDB - Embedded newform 270.4.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [270,4,Mod(1,270)] 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("270.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	270.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,0,4,5,0,-34] 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:	$15.9305157015$
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	$\chi$	$=$	270.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+2.00000 q^{2} +4.00000 q^{4} +5.00000 q^{5} -34.0000 q^{7} +8.00000 q^{8} +10.0000 q^{10} -48.0000 q^{11} -70.0000 q^{13} -68.0000 q^{14} +16.0000 q^{16} -27.0000 q^{17} +119.000 q^{19} +20.0000 q^{20} -96.0000 q^{22} -51.0000 q^{23} +25.0000 q^{25} -140.000 q^{26} -136.000 q^{28} -30.0000 q^{29} -133.000 q^{31} +32.0000 q^{32} -54.0000 q^{34} -170.000 q^{35} +218.000 q^{37} +238.000 q^{38} +40.0000 q^{40} +156.000 q^{41} -88.0000 q^{43} -192.000 q^{44} -102.000 q^{46} -516.000 q^{47} +813.000 q^{49} +50.0000 q^{50} -280.000 q^{52} -639.000 q^{53} -240.000 q^{55} -272.000 q^{56} -60.0000 q^{58} -654.000 q^{59} +461.000 q^{61} -266.000 q^{62} +64.0000 q^{64} -350.000 q^{65} +182.000 q^{67} -108.000 q^{68} -340.000 q^{70} +900.000 q^{71} +704.000 q^{73} +436.000 q^{74} +476.000 q^{76} +1632.00 q^{77} -1375.00 q^{79} +80.0000 q^{80} +312.000 q^{82} +915.000 q^{83} -135.000 q^{85} -176.000 q^{86} -384.000 q^{88} -1116.00 q^{89} +2380.00 q^{91} -204.000 q^{92} -1032.00 q^{94} +595.000 q^{95} -16.0000 q^{97} +1626.00 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−34.0000	−1.83583	−0.917914	−	0.396780i	$-0.870128\pi$
$7$	−34.0000	−1.83583	−0.917914	+	0.396780i	$0.870128\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	10.0000	0.316228
$11$	−48.0000	−1.31569	−0.657843	−	0.753155i	$-0.728531\pi$
$11$	−48.0000	−1.31569	−0.657843	+	0.753155i	$0.728531\pi$
$12$	0	0
$13$	−70.0000	−1.49342	−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−70.0000	−1.49342	−0.746712	+	0.665148i	$0.768369\pi$
$14$	−68.0000	−1.29813
$15$	0	0
$16$	16.0000	0.250000
$17$	−27.0000	−0.385204	−0.192602	−	0.981277i	$-0.561693\pi$
$17$	−27.0000	−0.385204	−0.192602	+	0.981277i	$0.561693\pi$
$18$	0	0
$19$	119.000	1.43687	0.718433	−	0.695596i	$-0.244859\pi$
$19$	119.000	1.43687	0.718433	+	0.695596i	$0.244859\pi$
$20$	20.0000	0.223607
$21$	0	0
$22$	−96.0000	−0.930330
$23$	−51.0000	−0.462358	−0.231179	−	0.972911i	$-0.574258\pi$
$23$	−51.0000	−0.462358	−0.231179	+	0.972911i	$0.574258\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	−140.000	−1.05601
$27$	0	0
$28$	−136.000	−0.917914
$29$	−30.0000	−0.192099	−0.0960493	−	0.995377i	$-0.530621\pi$
$29$	−30.0000	−0.192099	−0.0960493	+	0.995377i	$0.530621\pi$
$30$	0	0
$31$	−133.000	−0.770565	−0.385282	−	0.922799i	$-0.625896\pi$
$31$	−133.000	−0.770565	−0.385282	+	0.922799i	$0.625896\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−54.0000	−0.272380
$35$	−170.000	−0.821007
$36$	0	0
$37$	218.000	0.968621	0.484311	−	0.874896i	$-0.339070\pi$
$37$	218.000	0.968621	0.484311	+	0.874896i	$0.339070\pi$
$38$	238.000	1.01602
$39$	0	0
$40$	40.0000	0.158114
$41$	156.000	0.594222	0.297111	−	0.954843i	$-0.403977\pi$
$41$	156.000	0.594222	0.297111	+	0.954843i	$0.403977\pi$
$42$	0	0
$43$	−88.0000	−0.312090	−0.156045	−	0.987750i	$-0.549875\pi$
$43$	−88.0000	−0.312090	−0.156045	+	0.987750i	$0.549875\pi$
$44$	−192.000	−0.657843
$45$	0	0
$46$	−102.000	−0.326937
$47$	−516.000	−1.60141	−0.800706	−	0.599058i	$-0.795542\pi$
$47$	−516.000	−1.60141	−0.800706	+	0.599058i	$0.795542\pi$
$48$	0	0
$49$	813.000	2.37026
$50$	50.0000	0.141421
$51$	0	0
$52$	−280.000	−0.746712
$53$	−639.000	−1.65610	−0.828051	−	0.560653i	$-0.810550\pi$
$53$	−639.000	−1.65610	−0.828051	+	0.560653i	$0.810550\pi$
$54$	0	0
$55$	−240.000	−0.588393
$56$	−272.000	−0.649063
$57$	0	0
$58$	−60.0000	−0.135834
$59$	−654.000	−1.44311	−0.721555	−	0.692357i	$-0.756573\pi$
$59$	−654.000	−1.44311	−0.721555	+	0.692357i	$0.756573\pi$
$60$	0	0
$61$	461.000	0.967623	0.483811	−	0.875172i	$-0.339252\pi$
$61$	461.000	0.967623	0.483811	+	0.875172i	$0.339252\pi$
$62$	−266.000	−0.544872
$63$	0	0
$64$	64.0000	0.125000
$65$	−350.000	−0.667879
$66$	0	0
$67$	182.000	0.331863	0.165932	−	0.986137i	$-0.446937\pi$
$67$	182.000	0.331863	0.165932	+	0.986137i	$0.446937\pi$
$68$	−108.000	−0.192602
$69$	0	0
$70$	−340.000	−0.580540
$71$	900.000	1.50437	0.752186	−	0.658951i	$-0.229000\pi$
$71$	900.000	1.50437	0.752186	+	0.658951i	$0.229000\pi$
$72$	0	0
$73$	704.000	1.12873	0.564363	−	0.825527i	$-0.309122\pi$
$73$	704.000	1.12873	0.564363	+	0.825527i	$0.309122\pi$
$74$	436.000	0.684919
$75$	0	0
$76$	476.000	0.718433
$77$	1632.00	2.41537
$78$	0	0
$79$	−1375.00	−1.95822	−0.979111	−	0.203325i	$-0.934825\pi$
$79$	−1375.00	−1.95822	−0.979111	+	0.203325i	$0.934825\pi$
$80$	80.0000	0.111803
$81$	0	0
$82$	312.000	0.420178
$83$	915.000	1.21005	0.605026	−	0.796206i	$-0.293163\pi$
$83$	915.000	1.21005	0.605026	+	0.796206i	$0.293163\pi$
$84$	0	0
$85$	−135.000	−0.172268
$86$	−176.000	−0.220681
$87$	0	0
$88$	−384.000	−0.465165
$89$	−1116.00	−1.32917	−0.664583	−	0.747215i	$-0.731391\pi$
$89$	−1116.00	−1.32917	−0.664583	+	0.747215i	$0.731391\pi$
$90$	0	0
$91$	2380.00	2.74167
$92$	−204.000	−0.231179
$93$	0	0
$94$	−1032.00	−1.13237
$95$	595.000	0.642586
$96$	0	0
$97$	−16.0000	−0.0167480	−0.00837399	−	0.999965i	$-0.502666\pi$
$97$	−16.0000	−0.0167480	−0.00837399	+	0.999965i	$0.502666\pi$
$98$	1626.00	1.67603
$99$	0	0
$100$	100.000	0.100000
$101$	348.000	0.342844	0.171422	−	0.985198i	$-0.445164\pi$
$101$	348.000	0.342844	0.171422	+	0.985198i	$0.445164\pi$
$102$	0	0
$103$	−412.000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−412.000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−560.000	−0.528005
$105$	0	0
$106$	−1278.00	−1.17104
$107$	−900.000	−0.813143	−0.406571	−	0.913619i	$-0.633276\pi$
$107$	−900.000	−0.813143	−0.406571	+	0.913619i	$0.633276\pi$
$108$	0	0
$109$	−115.000	−0.101055	−0.0505275	−	0.998723i	$-0.516090\pi$
$109$	−115.000	−0.101055	−0.0505275	+	0.998723i	$0.516090\pi$
$110$	−480.000	−0.416056
$111$	0	0
$112$	−544.000	−0.458957
$113$	966.000	0.804191	0.402096	−	0.915598i	$-0.368282\pi$
$113$	966.000	0.804191	0.402096	+	0.915598i	$0.368282\pi$
$114$	0	0
$115$	−255.000	−0.206773
$116$	−120.000	−0.0960493
$117$	0	0
$118$	−1308.00	−1.02043
$119$	918.000	0.707167
$120$	0	0
$121$	973.000	0.731029
$122$	922.000	0.684213
$123$	0	0
$124$	−532.000	−0.385282
$125$	125.000	0.0894427
$126$	0	0
$127$	1406.00	0.982381	0.491190	−	0.871052i	$-0.336562\pi$
$127$	1406.00	0.982381	0.491190	+	0.871052i	$0.336562\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	−700.000	−0.472262
$131$	246.000	0.164070	0.0820348	−	0.996629i	$-0.473858\pi$
$131$	246.000	0.164070	0.0820348	+	0.996629i	$0.473858\pi$
$132$	0	0
$133$	−4046.00	−2.63784
$134$	364.000	0.234663
$135$	0	0
$136$	−216.000	−0.136190
$137$	519.000	0.323658	0.161829	−	0.986819i	$-0.448261\pi$
$137$	519.000	0.323658	0.161829	+	0.986819i	$0.448261\pi$
$138$	0	0
$139$	1316.00	0.803034	0.401517	−	0.915852i	$-0.368483\pi$
$139$	1316.00	0.803034	0.401517	+	0.915852i	$0.368483\pi$
$140$	−680.000	−0.410503
$141$	0	0
$142$	1800.00	1.06375
$143$	3360.00	1.96488
$144$	0	0
$145$	−150.000	−0.0859091
$146$	1408.00	0.798130
$147$	0	0
$148$	872.000	0.484311
$149$	372.000	0.204533	0.102267	−	0.994757i	$-0.467391\pi$
$149$	372.000	0.204533	0.102267	+	0.994757i	$0.467391\pi$
$150$	0	0
$151$	−1456.00	−0.784686	−0.392343	−	0.919819i	$-0.628335\pi$
$151$	−1456.00	−0.784686	−0.392343	+	0.919819i	$0.628335\pi$
$152$	952.000	0.508009
$153$	0	0
$154$	3264.00	1.70793
$155$	−665.000	−0.344607
$156$	0	0
$157$	956.000	0.485969	0.242984	−	0.970030i	$-0.421874\pi$
$157$	956.000	0.485969	0.242984	+	0.970030i	$0.421874\pi$
$158$	−2750.00	−1.38467
$159$	0	0
$160$	160.000	0.0790569
$161$	1734.00	0.848810
$162$	0	0
$163$	−2446.00	−1.17537	−0.587686	−	0.809089i	$-0.699961\pi$
$163$	−2446.00	−1.17537	−0.587686	+	0.809089i	$0.699961\pi$
$164$	624.000	0.297111
$165$	0	0
$166$	1830.00	0.855636
$167$	−3111.00	−1.44154	−0.720768	−	0.693177i	$-0.756211\pi$
$167$	−3111.00	−1.44154	−0.720768	+	0.693177i	$0.756211\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	−270.000	−0.121812
$171$	0	0
$172$	−352.000	−0.156045
$173$	−2397.00	−1.05341	−0.526707	−	0.850047i	$-0.676573\pi$
$173$	−2397.00	−1.05341	−0.526707	+	0.850047i	$0.676573\pi$
$174$	0	0
$175$	−850.000	−0.367165
$176$	−768.000	−0.328921
$177$	0	0
$178$	−2232.00	−0.939862
$179$	−540.000	−0.225483	−0.112742	−	0.993624i	$-0.535963\pi$
$179$	−540.000	−0.225483	−0.112742	+	0.993624i	$0.535963\pi$
$180$	0	0
$181$	2333.00	0.958069	0.479035	−	0.877796i	$-0.340987\pi$
$181$	2333.00	0.958069	0.479035	+	0.877796i	$0.340987\pi$
$182$	4760.00	1.93865
$183$	0	0
$184$	−408.000	−0.163468
$185$	1090.00	0.433181
$186$	0	0
$187$	1296.00	0.506807
$188$	−2064.00	−0.800706
$189$	0	0
$190$	1190.00	0.454377
$191$	−2730.00	−1.03422	−0.517110	−	0.855919i	$-0.672992\pi$
$191$	−2730.00	−1.03422	−0.517110	+	0.855919i	$0.672992\pi$
$192$	0	0
$193$	−4570.00	−1.70443	−0.852217	−	0.523188i	$-0.824742\pi$
$193$	−4570.00	−1.70443	−0.852217	+	0.523188i	$0.824742\pi$
$194$	−32.0000	−0.0118426
$195$	0	0
$196$	3252.00	1.18513
$197$	675.000	0.244121	0.122060	−	0.992523i	$-0.461050\pi$
$197$	675.000	0.244121	0.122060	+	0.992523i	$0.461050\pi$
$198$	0	0
$199$	−3112.00	−1.10856	−0.554281	−	0.832330i	$-0.687007\pi$
$199$	−3112.00	−1.10856	−0.554281	+	0.832330i	$0.687007\pi$
$200$	200.000	0.0707107
$201$	0	0
$202$	696.000	0.242428
$203$	1020.00	0.352660
$204$	0	0
$205$	780.000	0.265744
$206$	−824.000	−0.278693
$207$	0	0
$208$	−1120.00	−0.373356
$209$	−5712.00	−1.89047
$210$	0	0
$211$	2441.00	0.796424	0.398212	−	0.917294i	$-0.369631\pi$
$211$	2441.00	0.796424	0.398212	+	0.917294i	$0.369631\pi$
$212$	−2556.00	−0.828051
$213$	0	0
$214$	−1800.00	−0.574979
$215$	−440.000	−0.139571
$216$	0	0
$217$	4522.00	1.41462
$218$	−230.000	−0.0714567
$219$	0	0
$220$	−960.000	−0.294196
$221$	1890.00	0.575272
$222$	0	0
$223$	−3418.00	−1.02640	−0.513198	−	0.858270i	$-0.671539\pi$
$223$	−3418.00	−1.02640	−0.513198	+	0.858270i	$0.671539\pi$
$224$	−1088.00	−0.324532
$225$	0	0
$226$	1932.00	0.568649
$227$	−4377.00	−1.27979	−0.639894	−	0.768464i	$-0.721022\pi$
$227$	−4377.00	−1.27979	−0.639894	+	0.768464i	$0.721022\pi$
$228$	0	0
$229$	4187.00	1.20823	0.604115	−	0.796897i	$-0.293527\pi$
$229$	4187.00	1.20823	0.604115	+	0.796897i	$0.293527\pi$
$230$	−510.000	−0.146210
$231$	0	0
$232$	−240.000	−0.0679171
$233$	−1098.00	−0.308723	−0.154361	−	0.988014i	$-0.549332\pi$
$233$	−1098.00	−0.308723	−0.154361	+	0.988014i	$0.549332\pi$
$234$	0	0
$235$	−2580.00	−0.716173
$236$	−2616.00	−0.721555
$237$	0	0
$238$	1836.00	0.500043
$239$	6474.00	1.75217	0.876084	−	0.482158i	$-0.160147\pi$
$239$	6474.00	1.75217	0.876084	+	0.482158i	$0.160147\pi$
$240$	0	0
$241$	3251.00	0.868943	0.434472	−	0.900686i	$-0.356935\pi$
$241$	3251.00	0.868943	0.434472	+	0.900686i	$0.356935\pi$
$242$	1946.00	0.516916
$243$	0	0
$244$	1844.00	0.483811
$245$	4065.00	1.06001
$246$	0	0
$247$	−8330.00	−2.14585
$248$	−1064.00	−0.272436
$249$	0	0
$250$	250.000	0.0632456
$251$	1728.00	0.434543	0.217272	−	0.976111i	$-0.430284\pi$
$251$	1728.00	0.434543	0.217272	+	0.976111i	$0.430284\pi$
$252$	0	0
$253$	2448.00	0.608318
$254$	2812.00	0.694648
$255$	0	0
$256$	256.000	0.0625000
$257$	−5469.00	−1.32742	−0.663710	−	0.747990i	$-0.731019\pi$
$257$	−5469.00	−1.32742	−0.663710	+	0.747990i	$0.731019\pi$
$258$	0	0
$259$	−7412.00	−1.77822
$260$	−1400.00	−0.333940
$261$	0	0
$262$	492.000	0.116015
$263$	−3216.00	−0.754019	−0.377010	−	0.926209i	$-0.623048\pi$
$263$	−3216.00	−0.754019	−0.377010	+	0.926209i	$0.623048\pi$
$264$	0	0
$265$	−3195.00	−0.740631
$266$	−8092.00	−1.86523
$267$	0	0
$268$	728.000	0.165932
$269$	−8010.00	−1.81553	−0.907766	−	0.419476i	$-0.862214\pi$
$269$	−8010.00	−1.81553	−0.907766	+	0.419476i	$0.862214\pi$
$270$	0	0
$271$	−3805.00	−0.852905	−0.426453	−	0.904510i	$-0.640237\pi$
$271$	−3805.00	−0.852905	−0.426453	+	0.904510i	$0.640237\pi$
$272$	−432.000	−0.0963009
$273$	0	0
$274$	1038.00	0.228861
$275$	−1200.00	−0.263137
$276$	0	0
$277$	3224.00	0.699319	0.349660	−	0.936877i	$-0.386297\pi$
$277$	3224.00	0.699319	0.349660	+	0.936877i	$0.386297\pi$
$278$	2632.00	0.567830
$279$	0	0
$280$	−1360.00	−0.290270
$281$	−4530.00	−0.961698	−0.480849	−	0.876803i	$-0.659671\pi$
$281$	−4530.00	−0.961698	−0.480849	+	0.876803i	$0.659671\pi$
$282$	0	0
$283$	−3292.00	−0.691481	−0.345740	−	0.938330i	$-0.612372\pi$
$283$	−3292.00	−0.691481	−0.345740	+	0.938330i	$0.612372\pi$
$284$	3600.00	0.752186
$285$	0	0
$286$	6720.00	1.38938
$287$	−5304.00	−1.09089
$288$	0	0
$289$	−4184.00	−0.851618
$290$	−300.000	−0.0607469
$291$	0	0
$292$	2816.00	0.564363
$293$	−7953.00	−1.58573	−0.792866	−	0.609397i	$-0.791412\pi$
$293$	−7953.00	−1.58573	−0.792866	+	0.609397i	$0.791412\pi$
$294$	0	0
$295$	−3270.00	−0.645379
$296$	1744.00	0.342459
$297$	0	0
$298$	744.000	0.144627
$299$	3570.00	0.690496
$300$	0	0
$301$	2992.00	0.572944
$302$	−2912.00	−0.554857
$303$	0	0
$304$	1904.00	0.359217
$305$	2305.00	0.432734
$306$	0	0
$307$	−5290.00	−0.983441	−0.491720	−	0.870753i	$-0.663632\pi$
$307$	−5290.00	−0.983441	−0.491720	+	0.870753i	$0.663632\pi$
$308$	6528.00	1.20769
$309$	0	0
$310$	−1330.00	−0.243674
$311$	5358.00	0.976927	0.488464	−	0.872584i	$-0.337558\pi$
$311$	5358.00	0.976927	0.488464	+	0.872584i	$0.337558\pi$
$312$	0	0
$313$	5600.00	1.01128	0.505640	−	0.862744i	$-0.331256\pi$
$313$	5600.00	1.01128	0.505640	+	0.862744i	$0.331256\pi$
$314$	1912.00	0.343632
$315$	0	0
$316$	−5500.00	−0.979111
$317$	7341.00	1.30067	0.650334	−	0.759649i	$-0.274629\pi$
$317$	7341.00	1.30067	0.650334	+	0.759649i	$0.274629\pi$
$318$	0	0
$319$	1440.00	0.252741
$320$	320.000	0.0559017
$321$	0	0
$322$	3468.00	0.600199
$323$	−3213.00	−0.553486
$324$	0	0
$325$	−1750.00	−0.298685
$326$	−4892.00	−0.831113
$327$	0	0
$328$	1248.00	0.210089
$329$	17544.0	2.93991
$330$	0	0
$331$	380.000	0.0631018	0.0315509	−	0.999502i	$-0.489955\pi$
$331$	380.000	0.0631018	0.0315509	+	0.999502i	$0.489955\pi$
$332$	3660.00	0.605026
$333$	0	0
$334$	−6222.00	−1.01932
$335$	910.000	0.148414
$336$	0	0
$337$	434.000	0.0701528	0.0350764	−	0.999385i	$-0.488833\pi$
$337$	434.000	0.0701528	0.0350764	+	0.999385i	$0.488833\pi$
$338$	5406.00	0.869963
$339$	0	0
$340$	−540.000	−0.0861342
$341$	6384.00	1.01382
$342$	0	0
$343$	−15980.0	−2.51557
$344$	−704.000	−0.110341
$345$	0	0
$346$	−4794.00	−0.744876
$347$	8004.00	1.23826	0.619131	−	0.785287i	$-0.287485\pi$
$347$	8004.00	1.23826	0.619131	+	0.785287i	$0.287485\pi$
$348$	0	0
$349$	1109.00	0.170096	0.0850479	−	0.996377i	$-0.472896\pi$
$349$	1109.00	0.170096	0.0850479	+	0.996377i	$0.472896\pi$
$350$	−1700.00	−0.259625
$351$	0	0
$352$	−1536.00	−0.232583
$353$	−7662.00	−1.15526	−0.577630	−	0.816298i	$-0.696023\pi$
$353$	−7662.00	−1.15526	−0.577630	+	0.816298i	$0.696023\pi$
$354$	0	0
$355$	4500.00	0.672775
$356$	−4464.00	−0.664583
$357$	0	0
$358$	−1080.00	−0.159441
$359$	−8478.00	−1.24638	−0.623192	−	0.782069i	$-0.714164\pi$
$359$	−8478.00	−1.24638	−0.623192	+	0.782069i	$0.714164\pi$
$360$	0	0
$361$	7302.00	1.06459
$362$	4666.00	0.677457
$363$	0	0
$364$	9520.00	1.37083
$365$	3520.00	0.504781
$366$	0	0
$367$	13286.0	1.88971	0.944855	−	0.327489i	$-0.106202\pi$
$367$	13286.0	1.88971	0.944855	+	0.327489i	$0.106202\pi$
$368$	−816.000	−0.115590
$369$	0	0
$370$	2180.00	0.306305
$371$	21726.0	3.04032
$372$	0	0
$373$	3080.00	0.427551	0.213775	−	0.976883i	$-0.431424\pi$
$373$	3080.00	0.427551	0.213775	+	0.976883i	$0.431424\pi$
$374$	2592.00	0.358367
$375$	0	0
$376$	−4128.00	−0.566184
$377$	2100.00	0.286885
$378$	0	0
$379$	10109.0	1.37009	0.685045	−	0.728500i	$-0.259782\pi$
$379$	10109.0	1.37009	0.685045	+	0.728500i	$0.259782\pi$
$380$	2380.00	0.321293
$381$	0	0
$382$	−5460.00	−0.731303
$383$	−8727.00	−1.16431	−0.582153	−	0.813080i	$-0.697789\pi$
$383$	−8727.00	−1.16431	−0.582153	+	0.813080i	$0.697789\pi$
$384$	0	0
$385$	8160.00	1.08019
$386$	−9140.00	−1.20522
$387$	0	0
$388$	−64.0000	−0.00837399
$389$	−2712.00	−0.353480	−0.176740	−	0.984258i	$-0.556555\pi$
$389$	−2712.00	−0.353480	−0.176740	+	0.984258i	$0.556555\pi$
$390$	0	0
$391$	1377.00	0.178102
$392$	6504.00	0.838014
$393$	0	0
$394$	1350.00	0.172619
$395$	−6875.00	−0.875744
$396$	0	0
$397$	−8818.00	−1.11477	−0.557384	−	0.830255i	$-0.688195\pi$
$397$	−8818.00	−1.11477	−0.557384	+	0.830255i	$0.688195\pi$
$398$	−6224.00	−0.783872
$399$	0	0
$400$	400.000	0.0500000
$401$	3306.00	0.411705	0.205853	−	0.978583i	$-0.434003\pi$
$401$	3306.00	0.411705	0.205853	+	0.978583i	$0.434003\pi$
$402$	0	0
$403$	9310.00	1.15078
$404$	1392.00	0.171422
$405$	0	0
$406$	2040.00	0.249368
$407$	−10464.0	−1.27440
$408$	0	0
$409$	6401.00	0.773861	0.386930	−	0.922109i	$-0.373535\pi$
$409$	6401.00	0.773861	0.386930	+	0.922109i	$0.373535\pi$
$410$	1560.00	0.187910
$411$	0	0
$412$	−1648.00	−0.197066
$413$	22236.0	2.64930
$414$	0	0
$415$	4575.00	0.541152
$416$	−2240.00	−0.264002
$417$	0	0
$418$	−11424.0	−1.33676
$419$	−2256.00	−0.263038	−0.131519	−	0.991314i	$-0.541985\pi$
$419$	−2256.00	−0.263038	−0.131519	+	0.991314i	$0.541985\pi$
$420$	0	0
$421$	1811.00	0.209650	0.104825	−	0.994491i	$-0.466572\pi$
$421$	1811.00	0.209650	0.104825	+	0.994491i	$0.466572\pi$
$422$	4882.00	0.563156
$423$	0	0
$424$	−5112.00	−0.585520
$425$	−675.000	−0.0770407
$426$	0	0
$427$	−15674.0	−1.77639
$428$	−3600.00	−0.406571
$429$	0	0
$430$	−880.000	−0.0986916
$431$	−5454.00	−0.609536	−0.304768	−	0.952427i	$-0.598579\pi$
$431$	−5454.00	−0.609536	−0.304768	+	0.952427i	$0.598579\pi$
$432$	0	0
$433$	2990.00	0.331848	0.165924	−	0.986139i	$-0.446939\pi$
$433$	2990.00	0.331848	0.165924	+	0.986139i	$0.446939\pi$
$434$	9044.00	1.00029
$435$	0	0
$436$	−460.000	−0.0505275
$437$	−6069.00	−0.664347
$438$	0	0
$439$	9371.00	1.01880	0.509400	−	0.860530i	$-0.329867\pi$
$439$	9371.00	1.01880	0.509400	+	0.860530i	$0.329867\pi$
$440$	−1920.00	−0.208028
$441$	0	0
$442$	3780.00	0.406779
$443$	6171.00	0.661835	0.330918	−	0.943660i	$-0.392642\pi$
$443$	6171.00	0.661835	0.330918	+	0.943660i	$0.392642\pi$
$444$	0	0
$445$	−5580.00	−0.594421
$446$	−6836.00	−0.725771
$447$	0	0
$448$	−2176.00	−0.229478
$449$	4122.00	0.433250	0.216625	−	0.976255i	$-0.430495\pi$
$449$	4122.00	0.433250	0.216625	+	0.976255i	$0.430495\pi$
$450$	0	0
$451$	−7488.00	−0.781810
$452$	3864.00	0.402096
$453$	0	0
$454$	−8754.00	−0.904946
$455$	11900.0	1.22611
$456$	0	0
$457$	7076.00	0.724292	0.362146	−	0.932121i	$-0.382044\pi$
$457$	7076.00	0.724292	0.362146	+	0.932121i	$0.382044\pi$
$458$	8374.00	0.854348
$459$	0	0
$460$	−1020.00	−0.103386
$461$	762.000	0.0769846	0.0384923	−	0.999259i	$-0.487745\pi$
$461$	762.000	0.0769846	0.0384923	+	0.999259i	$0.487745\pi$
$462$	0	0
$463$	8822.00	0.885514	0.442757	−	0.896642i	$-0.354000\pi$
$463$	8822.00	0.885514	0.442757	+	0.896642i	$0.354000\pi$
$464$	−480.000	−0.0480247
$465$	0	0
$466$	−2196.00	−0.218300
$467$	−4977.00	−0.493165	−0.246583	−	0.969122i	$-0.579308\pi$
$467$	−4977.00	−0.493165	−0.246583	+	0.969122i	$0.579308\pi$
$468$	0	0
$469$	−6188.00	−0.609244
$470$	−5160.00	−0.506411
$471$	0	0
$472$	−5232.00	−0.510217
$473$	4224.00	0.410613
$474$	0	0
$475$	2975.00	0.287373
$476$	3672.00	0.353584
$477$	0	0
$478$	12948.0	1.23897
$479$	10104.0	0.963807	0.481903	−	0.876224i	$-0.339946\pi$
$479$	10104.0	0.963807	0.481903	+	0.876224i	$0.339946\pi$
$480$	0	0
$481$	−15260.0	−1.44656
$482$	6502.00	0.614436
$483$	0	0
$484$	3892.00	0.365515
$485$	−80.0000	−0.00748992
$486$	0	0
$487$	14924.0	1.38865	0.694323	−	0.719663i	$-0.255704\pi$
$487$	14924.0	1.38865	0.694323	+	0.719663i	$0.255704\pi$
$488$	3688.00	0.342106
$489$	0	0
$490$	8130.00	0.749543
$491$	1146.00	0.105332	0.0526662	−	0.998612i	$-0.483228\pi$
$491$	1146.00	0.105332	0.0526662	+	0.998612i	$0.483228\pi$
$492$	0	0
$493$	810.000	0.0739971
$494$	−16660.0	−1.51735
$495$	0	0
$496$	−2128.00	−0.192641
$497$	−30600.0	−2.76177
$498$	0	0
$499$	−14965.0	−1.34254	−0.671268	−	0.741215i	$-0.734250\pi$
$499$	−14965.0	−1.34254	−0.671268	+	0.741215i	$0.734250\pi$
$500$	500.000	0.0447214
$501$	0	0
$502$	3456.00	0.307269
$503$	15525.0	1.37619	0.688097	−	0.725619i	$-0.258446\pi$
$503$	15525.0	1.37619	0.688097	+	0.725619i	$0.258446\pi$
$504$	0	0
$505$	1740.00	0.153325
$506$	4896.00	0.430146
$507$	0	0
$508$	5624.00	0.491190
$509$	−8196.00	−0.713716	−0.356858	−	0.934159i	$-0.616152\pi$
$509$	−8196.00	−0.713716	−0.356858	+	0.934159i	$0.616152\pi$
$510$	0	0
$511$	−23936.0	−2.07215
$512$	512.000	0.0441942
$513$	0	0
$514$	−10938.0	−0.938627
$515$	−2060.00	−0.176261
$516$	0	0
$517$	24768.0	2.10695
$518$	−14824.0	−1.25739
$519$	0	0
$520$	−2800.00	−0.236131
$521$	4932.00	0.414731	0.207365	−	0.978264i	$-0.433511\pi$
$521$	4932.00	0.414731	0.207365	+	0.978264i	$0.433511\pi$
$522$	0	0
$523$	−5938.00	−0.496464	−0.248232	−	0.968701i	$-0.579850\pi$
$523$	−5938.00	−0.496464	−0.248232	+	0.968701i	$0.579850\pi$
$524$	984.000	0.0820348
$525$	0	0
$526$	−6432.00	−0.533172
$527$	3591.00	0.296824
$528$	0	0
$529$	−9566.00	−0.786225
$530$	−6390.00	−0.523705
$531$	0	0
$532$	−16184.0	−1.31892
$533$	−10920.0	−0.887425
$534$	0	0
$535$	−4500.00	−0.363649
$536$	1456.00	0.117331
$537$	0	0
$538$	−16020.0	−1.28378
$539$	−39024.0	−3.11852
$540$	0	0
$541$	−6730.00	−0.534834	−0.267417	−	0.963581i	$-0.586170\pi$
$541$	−6730.00	−0.534834	−0.267417	+	0.963581i	$0.586170\pi$
$542$	−7610.00	−0.603095
$543$	0	0
$544$	−864.000	−0.0680950
$545$	−575.000	−0.0451932
$546$	0	0
$547$	−17656.0	−1.38010	−0.690051	−	0.723761i	$-0.742412\pi$
$547$	−17656.0	−1.38010	−0.690051	+	0.723761i	$0.742412\pi$
$548$	2076.00	0.161829
$549$	0	0
$550$	−2400.00	−0.186066
$551$	−3570.00	−0.276020
$552$	0	0
$553$	46750.0	3.59496
$554$	6448.00	0.494493
$555$	0	0
$556$	5264.00	0.401517
$557$	7974.00	0.606587	0.303294	−	0.952897i	$-0.401914\pi$
$557$	7974.00	0.606587	0.303294	+	0.952897i	$0.401914\pi$
$558$	0	0
$559$	6160.00	0.466083
$560$	−2720.00	−0.205252
$561$	0	0
$562$	−9060.00	−0.680023
$563$	−25332.0	−1.89630	−0.948150	−	0.317824i	$-0.897048\pi$
$563$	−25332.0	−1.89630	−0.948150	+	0.317824i	$0.897048\pi$
$564$	0	0
$565$	4830.00	0.359645
$566$	−6584.00	−0.488951
$567$	0	0
$568$	7200.00	0.531876
$569$	−1038.00	−0.0764767	−0.0382383	−	0.999269i	$-0.512175\pi$
$569$	−1038.00	−0.0764767	−0.0382383	+	0.999269i	$0.512175\pi$
$570$	0	0
$571$	15671.0	1.14853	0.574265	−	0.818669i	$-0.305288\pi$
$571$	15671.0	1.14853	0.574265	+	0.818669i	$0.305288\pi$
$572$	13440.0	0.982438
$573$	0	0
$574$	−10608.0	−0.771375
$575$	−1275.00	−0.0924716
$576$	0	0
$577$	−916.000	−0.0660894	−0.0330447	−	0.999454i	$-0.510520\pi$
$577$	−916.000	−0.0660894	−0.0330447	+	0.999454i	$0.510520\pi$
$578$	−8368.00	−0.602185
$579$	0	0
$580$	−600.000	−0.0429546
$581$	−31110.0	−2.22145
$582$	0	0
$583$	30672.0	2.17891
$584$	5632.00	0.399065
$585$	0	0
$586$	−15906.0	−1.12128
$587$	9141.00	0.642742	0.321371	−	0.946953i	$-0.395856\pi$
$587$	9141.00	0.642742	0.321371	+	0.946953i	$0.395856\pi$
$588$	0	0
$589$	−15827.0	−1.10720
$590$	−6540.00	−0.456352
$591$	0	0
$592$	3488.00	0.242155
$593$	5247.00	0.363353	0.181677	−	0.983358i	$-0.441848\pi$
$593$	5247.00	0.363353	0.181677	+	0.983358i	$0.441848\pi$
$594$	0	0
$595$	4590.00	0.316255
$596$	1488.00	0.102267
$597$	0	0
$598$	7140.00	0.488255
$599$	−24162.0	−1.64813	−0.824067	−	0.566492i	$-0.808300\pi$
$599$	−24162.0	−1.64813	−0.824067	+	0.566492i	$0.808300\pi$
$600$	0	0
$601$	14357.0	0.974433	0.487217	−	0.873281i	$-0.338012\pi$
$601$	14357.0	0.974433	0.487217	+	0.873281i	$0.338012\pi$
$602$	5984.00	0.405132
$603$	0	0
$604$	−5824.00	−0.392343
$605$	4865.00	0.326926
$606$	0	0
$607$	3152.00	0.210767	0.105384	−	0.994432i	$-0.466393\pi$
$607$	3152.00	0.210767	0.105384	+	0.994432i	$0.466393\pi$
$608$	3808.00	0.254005
$609$	0	0
$610$	4610.00	0.305989
$611$	36120.0	2.39159
$612$	0	0
$613$	4592.00	0.302560	0.151280	−	0.988491i	$-0.451661\pi$
$613$	4592.00	0.302560	0.151280	+	0.988491i	$0.451661\pi$
$614$	−10580.0	−0.695397
$615$	0	0
$616$	13056.0	0.853963
$617$	−7359.00	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$617$	−7359.00	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$618$	0	0
$619$	−15712.0	−1.02022	−0.510112	−	0.860108i	$-0.670396\pi$
$619$	−15712.0	−1.02022	−0.510112	+	0.860108i	$0.670396\pi$
$620$	−2660.00	−0.172304
$621$	0	0
$622$	10716.0	0.690792
$623$	37944.0	2.44012
$624$	0	0
$625$	625.000	0.0400000
$626$	11200.0	0.715083
$627$	0	0
$628$	3824.00	0.242984
$629$	−5886.00	−0.373116
$630$	0	0
$631$	−3175.00	−0.200309	−0.100154	−	0.994972i	$-0.531934\pi$
$631$	−3175.00	−0.200309	−0.100154	+	0.994972i	$0.531934\pi$
$632$	−11000.0	−0.692336
$633$	0	0
$634$	14682.0	0.919711
$635$	7030.00	0.439334
$636$	0	0
$637$	−56910.0	−3.53981
$638$	2880.00	0.178715
$639$	0	0
$640$	640.000	0.0395285
$641$	96.0000	0.00591540	0.00295770	−	0.999996i	$-0.499059\pi$
$641$	96.0000	0.00591540	0.00295770	+	0.999996i	$0.499059\pi$
$642$	0	0
$643$	−18070.0	−1.10826	−0.554130	−	0.832430i	$-0.686949\pi$
$643$	−18070.0	−1.10826	−0.554130	+	0.832430i	$0.686949\pi$
$644$	6936.00	0.424405
$645$	0	0
$646$	−6426.00	−0.391374
$647$	1341.00	0.0814840	0.0407420	−	0.999170i	$-0.487028\pi$
$647$	1341.00	0.0814840	0.0407420	+	0.999170i	$0.487028\pi$
$648$	0	0
$649$	31392.0	1.89868
$650$	−3500.00	−0.211202
$651$	0	0
$652$	−9784.00	−0.587686
$653$	−24495.0	−1.46794	−0.733969	−	0.679183i	$-0.762334\pi$
$653$	−24495.0	−1.46794	−0.733969	+	0.679183i	$0.762334\pi$
$654$	0	0
$655$	1230.00	0.0733742
$656$	2496.00	0.148556
$657$	0	0
$658$	35088.0	2.07883
$659$	−12378.0	−0.731682	−0.365841	−	0.930677i	$-0.619219\pi$
$659$	−12378.0	−0.731682	−0.365841	+	0.930677i	$0.619219\pi$
$660$	0	0
$661$	−24442.0	−1.43825	−0.719125	−	0.694880i	$-0.755457\pi$
$661$	−24442.0	−1.43825	−0.719125	+	0.694880i	$0.755457\pi$
$662$	760.000	0.0446197
$663$	0	0
$664$	7320.00	0.427818
$665$	−20230.0	−1.17968
$666$	0	0
$667$	1530.00	0.0888183
$668$	−12444.0	−0.720768
$669$	0	0
$670$	1820.00	0.104944
$671$	−22128.0	−1.27309
$672$	0	0
$673$	2378.00	0.136204	0.0681019	−	0.997678i	$-0.478306\pi$
$673$	2378.00	0.136204	0.0681019	+	0.997678i	$0.478306\pi$
$674$	868.000	0.0496055
$675$	0	0
$676$	10812.0	0.615157
$677$	5478.00	0.310985	0.155492	−	0.987837i	$-0.450304\pi$
$677$	5478.00	0.310985	0.155492	+	0.987837i	$0.450304\pi$
$678$	0	0
$679$	544.000	0.0307464
$680$	−1080.00	−0.0609060
$681$	0	0
$682$	12768.0	0.716880
$683$	8595.00	0.481521	0.240760	−	0.970585i	$-0.422603\pi$
$683$	8595.00	0.481521	0.240760	+	0.970585i	$0.422603\pi$
$684$	0	0
$685$	2595.00	0.144744
$686$	−31960.0	−1.77877
$687$	0	0
$688$	−1408.00	−0.0780225
$689$	44730.0	2.47326
$690$	0	0
$691$	−31615.0	−1.74051	−0.870254	−	0.492603i	$-0.836046\pi$
$691$	−31615.0	−1.74051	−0.870254	+	0.492603i	$0.836046\pi$
$692$	−9588.00	−0.526707
$693$	0	0
$694$	16008.0	0.875584
$695$	6580.00	0.359128
$696$	0	0
$697$	−4212.00	−0.228897
$698$	2218.00	0.120276
$699$	0	0
$700$	−3400.00	−0.183583
$701$	29790.0	1.60507	0.802534	−	0.596606i	$-0.203485\pi$
$701$	29790.0	1.60507	0.802534	+	0.596606i	$0.203485\pi$
$702$	0	0
$703$	25942.0	1.39178
$704$	−3072.00	−0.164461
$705$	0	0
$706$	−15324.0	−0.816893
$707$	−11832.0	−0.629403
$708$	0	0
$709$	3818.00	0.202240	0.101120	−	0.994874i	$-0.467757\pi$
$709$	3818.00	0.202240	0.101120	+	0.994874i	$0.467757\pi$
$710$	9000.00	0.475724
$711$	0	0
$712$	−8928.00	−0.469931
$713$	6783.00	0.356277
$714$	0	0
$715$	16800.0	0.878719
$716$	−2160.00	−0.112742
$717$	0	0
$718$	−16956.0	−0.881326
$719$	−28314.0	−1.46861	−0.734307	−	0.678817i	$-0.762493\pi$
$719$	−28314.0	−1.46861	−0.734307	+	0.678817i	$0.762493\pi$
$720$	0	0
$721$	14008.0	0.723558
$722$	14604.0	0.752776
$723$	0	0
$724$	9332.00	0.479035
$725$	−750.000	−0.0384197
$726$	0	0
$727$	56.0000	0.00285684	0.00142842	−	0.999999i	$-0.499545\pi$
$727$	56.0000	0.00285684	0.00142842	+	0.999999i	$0.499545\pi$
$728$	19040.0	0.969326
$729$	0	0
$730$	7040.00	0.356934
$731$	2376.00	0.120218
$732$	0	0
$733$	−34432.0	−1.73503	−0.867514	−	0.497413i	$-0.834283\pi$
$733$	−34432.0	−1.73503	−0.867514	+	0.497413i	$0.834283\pi$
$734$	26572.0	1.33623
$735$	0	0
$736$	−1632.00	−0.0817341
$737$	−8736.00	−0.436628
$738$	0	0
$739$	−1051.00	−0.0523162	−0.0261581	−	0.999658i	$-0.508327\pi$
$739$	−1051.00	−0.0523162	−0.0261581	+	0.999658i	$0.508327\pi$
$740$	4360.00	0.216590
$741$	0	0
$742$	43452.0	2.14983
$743$	39144.0	1.93278	0.966389	−	0.257084i	$-0.0827618\pi$
$743$	39144.0	1.93278	0.966389	+	0.257084i	$0.0827618\pi$
$744$	0	0
$745$	1860.00	0.0914700
$746$	6160.00	0.302324
$747$	0	0
$748$	5184.00	0.253403
$749$	30600.0	1.49279
$750$	0	0
$751$	−1735.00	−0.0843023	−0.0421512	−	0.999111i	$-0.513421\pi$
$751$	−1735.00	−0.0843023	−0.0421512	+	0.999111i	$0.513421\pi$
$752$	−8256.00	−0.400353
$753$	0	0
$754$	4200.00	0.202858
$755$	−7280.00	−0.350922
$756$	0	0
$757$	6698.00	0.321589	0.160795	−	0.986988i	$-0.448594\pi$
$757$	6698.00	0.321589	0.160795	+	0.986988i	$0.448594\pi$
$758$	20218.0	0.968801
$759$	0	0
$760$	4760.00	0.227189
$761$	38766.0	1.84660	0.923302	−	0.384074i	$-0.125479\pi$
$761$	38766.0	1.84660	0.923302	+	0.384074i	$0.125479\pi$
$762$	0	0
$763$	3910.00	0.185520
$764$	−10920.0	−0.517110
$765$	0	0
$766$	−17454.0	−0.823288
$767$	45780.0	2.15518
$768$	0	0
$769$	23501.0	1.10204	0.551019	−	0.834492i	$-0.314239\pi$
$769$	23501.0	1.10204	0.551019	+	0.834492i	$0.314239\pi$
$770$	16320.0	0.763808
$771$	0	0
$772$	−18280.0	−0.852217
$773$	−3591.00	−0.167088	−0.0835442	−	0.996504i	$-0.526624\pi$
$773$	−3591.00	−0.167088	−0.0835442	+	0.996504i	$0.526624\pi$
$774$	0	0
$775$	−3325.00	−0.154113
$776$	−128.000	−0.00592130
$777$	0	0
$778$	−5424.00	−0.249948
$779$	18564.0	0.853818
$780$	0	0
$781$	−43200.0	−1.97928
$782$	2754.00	0.125937
$783$	0	0
$784$	13008.0	0.592566
$785$	4780.00	0.217332
$786$	0	0
$787$	−20716.0	−0.938305	−0.469152	−	0.883117i	$-0.655440\pi$
$787$	−20716.0	−0.938305	−0.469152	+	0.883117i	$0.655440\pi$
$788$	2700.00	0.122060
$789$	0	0
$790$	−13750.0	−0.619244
$791$	−32844.0	−1.47636
$792$	0	0
$793$	−32270.0	−1.44507
$794$	−17636.0	−0.788260
$795$	0	0
$796$	−12448.0	−0.554281
$797$	−42981.0	−1.91024	−0.955122	−	0.296211i	$-0.904277\pi$
$797$	−42981.0	−1.91024	−0.955122	+	0.296211i	$0.904277\pi$
$798$	0	0
$799$	13932.0	0.616869
$800$	800.000	0.0353553
$801$	0	0
$802$	6612.00	0.291119
$803$	−33792.0	−1.48505
$804$	0	0
$805$	8670.00	0.379599
$806$	18620.0	0.813724
$807$	0	0
$808$	2784.00	0.121214
$809$	−2268.00	−0.0985644	−0.0492822	−	0.998785i	$-0.515693\pi$
$809$	−2268.00	−0.0985644	−0.0492822	+	0.998785i	$0.515693\pi$
$810$	0	0
$811$	11756.0	0.509012	0.254506	−	0.967071i	$-0.418087\pi$
$811$	11756.0	0.509012	0.254506	+	0.967071i	$0.418087\pi$
$812$	4080.00	0.176330
$813$	0	0
$814$	−20928.0	−0.901138
$815$	−12230.0	−0.525642
$816$	0	0
$817$	−10472.0	−0.448432
$818$	12802.0	0.547202
$819$	0	0
$820$	3120.00	0.132872
$821$	8646.00	0.367537	0.183768	−	0.982970i	$-0.441170\pi$
$821$	8646.00	0.367537	0.183768	+	0.982970i	$0.441170\pi$
$822$	0	0
$823$	10784.0	0.456752	0.228376	−	0.973573i	$-0.426659\pi$
$823$	10784.0	0.456752	0.228376	+	0.973573i	$0.426659\pi$
$824$	−3296.00	−0.139347
$825$	0	0
$826$	44472.0	1.87334
$827$	42597.0	1.79110	0.895552	−	0.444957i	$-0.146781\pi$
$827$	42597.0	1.79110	0.895552	+	0.444957i	$0.146781\pi$
$828$	0	0
$829$	−26458.0	−1.10847	−0.554237	−	0.832359i	$-0.686990\pi$
$829$	−26458.0	−1.10847	−0.554237	+	0.832359i	$0.686990\pi$
$830$	9150.00	0.382652
$831$	0	0
$832$	−4480.00	−0.186678
$833$	−21951.0	−0.913034
$834$	0	0
$835$	−15555.0	−0.644674
$836$	−22848.0	−0.945233
$837$	0	0
$838$	−4512.00	−0.185996
$839$	−11496.0	−0.473046	−0.236523	−	0.971626i	$-0.576008\pi$
$839$	−11496.0	−0.473046	−0.236523	+	0.971626i	$0.576008\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	3622.00	0.148245
$843$	0	0
$844$	9764.00	0.398212
$845$	13515.0	0.550213
$846$	0	0
$847$	−33082.0	−1.34204
$848$	−10224.0	−0.414025
$849$	0	0
$850$	−1350.00	−0.0544760
$851$	−11118.0	−0.447850
$852$	0	0
$853$	21548.0	0.864935	0.432467	−	0.901650i	$-0.357643\pi$
$853$	21548.0	0.864935	0.432467	+	0.901650i	$0.357643\pi$
$854$	−31348.0	−1.25610
$855$	0	0
$856$	−7200.00	−0.287489
$857$	6261.00	0.249559	0.124779	−	0.992185i	$-0.460178\pi$
$857$	6261.00	0.249559	0.124779	+	0.992185i	$0.460178\pi$
$858$	0	0
$859$	−3355.00	−0.133261	−0.0666305	−	0.997778i	$-0.521225\pi$
$859$	−3355.00	−0.133261	−0.0666305	+	0.997778i	$0.521225\pi$
$860$	−1760.00	−0.0697855
$861$	0	0
$862$	−10908.0	−0.431007
$863$	19701.0	0.777091	0.388546	−	0.921429i	$-0.372978\pi$
$863$	19701.0	0.777091	0.388546	+	0.921429i	$0.372978\pi$
$864$	0	0
$865$	−11985.0	−0.471101
$866$	5980.00	0.234652
$867$	0	0
$868$	18088.0	0.707312
$869$	66000.0	2.57641
$870$	0	0
$871$	−12740.0	−0.495612
$872$	−920.000	−0.0357284
$873$	0	0
$874$	−12138.0	−0.469764
$875$	−4250.00	−0.164201
$876$	0	0
$877$	16292.0	0.627300	0.313650	−	0.949539i	$-0.398448\pi$
$877$	16292.0	0.627300	0.313650	+	0.949539i	$0.398448\pi$
$878$	18742.0	0.720401
$879$	0	0
$880$	−3840.00	−0.147098
$881$	9270.00	0.354500	0.177250	−	0.984166i	$-0.443280\pi$
$881$	9270.00	0.354500	0.177250	+	0.984166i	$0.443280\pi$
$882$	0	0
$883$	38486.0	1.46677	0.733384	−	0.679814i	$-0.237940\pi$
$883$	38486.0	1.46677	0.733384	+	0.679814i	$0.237940\pi$
$884$	7560.00	0.287636
$885$	0	0
$886$	12342.0	0.467988
$887$	1893.00	0.0716581	0.0358290	−	0.999358i	$-0.488593\pi$
$887$	1893.00	0.0716581	0.0358290	+	0.999358i	$0.488593\pi$
$888$	0	0
$889$	−47804.0	−1.80348
$890$	−11160.0	−0.420319
$891$	0	0
$892$	−13672.0	−0.513198
$893$	−61404.0	−2.30102
$894$	0	0
$895$	−2700.00	−0.100839
$896$	−4352.00	−0.162266
$897$	0	0
$898$	8244.00	0.306354
$899$	3990.00	0.148024
$900$	0	0
$901$	17253.0	0.637936
$902$	−14976.0	−0.552823
$903$	0	0
$904$	7728.00	0.284325
$905$	11665.0	0.428462
$906$	0	0
$907$	44876.0	1.64287	0.821435	−	0.570302i	$-0.193174\pi$
$907$	44876.0	1.64287	0.821435	+	0.570302i	$0.193174\pi$
$908$	−17508.0	−0.639894
$909$	0	0
$910$	23800.0	0.866992
$911$	23802.0	0.865637	0.432819	−	0.901481i	$-0.357519\pi$
$911$	23802.0	0.865637	0.432819	+	0.901481i	$0.357519\pi$
$912$	0	0
$913$	−43920.0	−1.59205
$914$	14152.0	0.512152
$915$	0	0
$916$	16748.0	0.604115
$917$	−8364.00	−0.301204
$918$	0	0
$919$	−24784.0	−0.889607	−0.444803	−	0.895628i	$-0.646726\pi$
$919$	−24784.0	−0.889607	−0.444803	+	0.895628i	$0.646726\pi$
$920$	−2040.00	−0.0731052
$921$	0	0
$922$	1524.00	0.0544363
$923$	−63000.0	−2.24666
$924$	0	0
$925$	5450.00	0.193724
$926$	17644.0	0.626153
$927$	0	0
$928$	−960.000	−0.0339586
$929$	9060.00	0.319967	0.159983	−	0.987120i	$-0.448856\pi$
$929$	9060.00	0.319967	0.159983	+	0.987120i	$0.448856\pi$
$930$	0	0
$931$	96747.0	3.40575
$932$	−4392.00	−0.154361
$933$	0	0
$934$	−9954.00	−0.348720
$935$	6480.00	0.226651
$936$	0	0
$937$	6176.00	0.215327	0.107663	−	0.994187i	$-0.465663\pi$
$937$	6176.00	0.215327	0.107663	+	0.994187i	$0.465663\pi$
$938$	−12376.0	−0.430800
$939$	0	0
$940$	−10320.0	−0.358086
$941$	−4182.00	−0.144877	−0.0724385	−	0.997373i	$-0.523078\pi$
$941$	−4182.00	−0.144877	−0.0724385	+	0.997373i	$0.523078\pi$
$942$	0	0
$943$	−7956.00	−0.274743
$944$	−10464.0	−0.360778
$945$	0	0
$946$	8448.00	0.290347
$947$	44079.0	1.51254	0.756270	−	0.654260i	$-0.227020\pi$
$947$	44079.0	1.51254	0.756270	+	0.654260i	$0.227020\pi$
$948$	0	0
$949$	−49280.0	−1.68567
$950$	5950.00	0.203204
$951$	0	0
$952$	7344.00	0.250021
$953$	12726.0	0.432566	0.216283	−	0.976331i	$-0.430607\pi$
$953$	12726.0	0.432566	0.216283	+	0.976331i	$0.430607\pi$
$954$	0	0
$955$	−13650.0	−0.462517
$956$	25896.0	0.876084
$957$	0	0
$958$	20208.0	0.681514
$959$	−17646.0	−0.594180
$960$	0	0
$961$	−12102.0	−0.406230
$962$	−30520.0	−1.02287
$963$	0	0
$964$	13004.0	0.434472
$965$	−22850.0	−0.762246
$966$	0	0
$967$	45218.0	1.50374	0.751868	−	0.659314i	$-0.229153\pi$
$967$	45218.0	1.50374	0.751868	+	0.659314i	$0.229153\pi$
$968$	7784.00	0.258458
$969$	0	0
$970$	−160.000	−0.00529618
$971$	−3978.00	−0.131473	−0.0657364	−	0.997837i	$-0.520940\pi$
$971$	−3978.00	−0.131473	−0.0657364	+	0.997837i	$0.520940\pi$
$972$	0	0
$973$	−44744.0	−1.47423
$974$	29848.0	0.981922
$975$	0	0
$976$	7376.00	0.241906
$977$	23466.0	0.768417	0.384209	−	0.923246i	$-0.374474\pi$
$977$	23466.0	0.768417	0.384209	+	0.923246i	$0.374474\pi$
$978$	0	0
$979$	53568.0	1.74876
$980$	16260.0	0.530007
$981$	0	0
$982$	2292.00	0.0744813
$983$	47913.0	1.55462	0.777308	−	0.629120i	$-0.216585\pi$
$983$	47913.0	1.55462	0.777308	+	0.629120i	$0.216585\pi$
$984$	0	0
$985$	3375.00	0.109174
$986$	1620.00	0.0523238
$987$	0	0
$988$	−33320.0	−1.07293
$989$	4488.00	0.144297
$990$	0	0
$991$	31997.0	1.02565	0.512825	−	0.858493i	$-0.328599\pi$
$991$	31997.0	1.02565	0.512825	+	0.858493i	$0.328599\pi$
$992$	−4256.00	−0.136218
$993$	0	0
$994$	−61200.0	−1.95286
$995$	−15560.0	−0.495764
$996$	0	0
$997$	−45628.0	−1.44940	−0.724701	−	0.689064i	$-0.758022\pi$
$997$	−45628.0	−1.44940	−0.724701	+	0.689064i	$0.758022\pi$
$998$	−29930.0	−0.949316
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.4.a.k.1.1	yes	1
3.2	odd	2		270.4.a.a.1.1	✓	1
4.3	odd	2		2160.4.a.t.1.1		1
5.2	odd	4		1350.4.c.c.649.2		2
5.3	odd	4		1350.4.c.c.649.1		2
5.4	even	2		1350.4.a.n.1.1		1
9.2	odd	6		810.4.e.x.271.1		2
9.4	even	3		810.4.e.d.541.1		2
9.5	odd	6		810.4.e.x.541.1		2
9.7	even	3		810.4.e.d.271.1		2
12.11	even	2		2160.4.a.j.1.1		1
15.2	even	4		1350.4.c.r.649.1		2
15.8	even	4		1350.4.c.r.649.2		2
15.14	odd	2		1350.4.a.bb.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.a.1.1	✓	1	3.2	odd	2
270.4.a.k.1.1	yes	1	1.1	even	1	trivial
810.4.e.d.271.1		2	9.7	even	3
810.4.e.d.541.1		2	9.4	even	3
810.4.e.x.271.1		2	9.2	odd	6
810.4.e.x.541.1		2	9.5	odd	6
1350.4.a.n.1.1		1	5.4	even	2
1350.4.a.bb.1.1		1	15.14	odd	2
1350.4.c.c.649.1		2	5.3	odd	4
1350.4.c.c.649.2		2	5.2	odd	4
1350.4.c.r.649.1		2	15.2	even	4
1350.4.c.r.649.2		2	15.8	even	4
2160.4.a.j.1.1		1	12.11	even	2
2160.4.a.t.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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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