LMFDB - Embedded newform 270.4.a.a.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:	$0$
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	$-0.271469\pi$
$11$	48.0000	1.31569	0.657843	+	0.753155i	$0.271469\pi$
$12$	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	$-0.438307\pi$
$17$	27.0000	0.385204	0.192602	+	0.981277i	$0.438307\pi$
$18$	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	$-0.425742\pi$
$23$	51.0000	0.462358	0.231179	+	0.972911i	$0.425742\pi$
$24$	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	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	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	$-0.596023\pi$
$41$	−156.000	−0.594222	−0.297111	+	0.954843i	$0.596023\pi$
$42$	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	$-0.204458\pi$
$47$	516.000	1.60141	0.800706	+	0.599058i	$0.204458\pi$
$48$	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	$-0.189450\pi$
$53$	639.000	1.65610	0.828051	+	0.560653i	$0.189450\pi$
$54$	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	$-0.243427\pi$
$59$	654.000	1.44311	0.721555	+	0.692357i	$0.243427\pi$
$60$	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	$-0.771000\pi$
$71$	−900.000	−1.50437	−0.752186	+	0.658951i	$0.771000\pi$
$72$	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	$-0.706837\pi$
$83$	−915.000	−1.21005	−0.605026	+	0.796206i	$0.706837\pi$
$84$	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	$-0.268609\pi$
$89$	1116.00	1.32917	0.664583	+	0.747215i	$0.268609\pi$
$90$	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	$-0.554836\pi$
$101$	−348.000	−0.342844	−0.171422	+	0.985198i	$0.554836\pi$
$102$	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	$-0.366724\pi$
$107$	900.000	0.813143	0.406571	+	0.913619i	$0.366724\pi$
$108$	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	$-0.631718\pi$
$113$	−966.000	−0.804191	−0.402096	+	0.915598i	$0.631718\pi$
$114$	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	$-0.526142\pi$
$131$	−246.000	−0.164070	−0.0820348	+	0.996629i	$0.526142\pi$
$132$	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	$-0.551739\pi$
$137$	−519.000	−0.323658	−0.161829	+	0.986819i	$0.551739\pi$
$138$	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	$-0.532609\pi$
$149$	−372.000	−0.204533	−0.102267	+	0.994757i	$0.532609\pi$
$150$	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	$-0.243789\pi$
$167$	3111.00	1.44154	0.720768	+	0.693177i	$0.243789\pi$
$168$	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	$-0.323427\pi$
$173$	2397.00	1.05341	0.526707	+	0.850047i	$0.323427\pi$
$174$	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	$-0.464037\pi$
$179$	540.000	0.225483	0.112742	+	0.993624i	$0.464037\pi$
$180$	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	$-0.327008\pi$
$191$	2730.00	1.03422	0.517110	+	0.855919i	$0.327008\pi$
$192$	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	$-0.538950\pi$
$197$	−675.000	−0.244121	−0.122060	+	0.992523i	$0.538950\pi$
$198$	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	$-0.278978\pi$
$227$	4377.00	1.27979	0.639894	+	0.768464i	$0.278978\pi$
$228$	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	$-0.450668\pi$
$233$	1098.00	0.308723	0.154361	+	0.988014i	$0.450668\pi$
$234$	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	$-0.839853\pi$
$239$	−6474.00	−1.75217	−0.876084	+	0.482158i	$0.839853\pi$
$240$	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	$-0.569716\pi$
$251$	−1728.00	−0.434543	−0.217272	+	0.976111i	$0.569716\pi$
$252$	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	$-0.268981\pi$
$257$	5469.00	1.32742	0.663710	+	0.747990i	$0.268981\pi$
$258$	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	$-0.376952\pi$
$263$	3216.00	0.754019	0.377010	+	0.926209i	$0.376952\pi$
$264$	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	$-0.137786\pi$
$269$	8010.00	1.81553	0.907766	+	0.419476i	$0.137786\pi$
$270$	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	$-0.340329\pi$
$281$	4530.00	0.961698	0.480849	+	0.876803i	$0.340329\pi$
$282$	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	$-0.208588\pi$
$293$	7953.00	1.58573	0.792866	+	0.609397i	$0.208588\pi$
$294$	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	$-0.662442\pi$
$311$	−5358.00	−0.976927	−0.488464	+	0.872584i	$0.662442\pi$
$312$	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	$-0.725371\pi$
$317$	−7341.00	−1.30067	−0.650334	+	0.759649i	$0.725371\pi$
$318$	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	$-0.712515\pi$
$347$	−8004.00	−1.23826	−0.619131	+	0.785287i	$0.712515\pi$
$348$	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	$-0.303977\pi$
$353$	7662.00	1.15526	0.577630	+	0.816298i	$0.303977\pi$
$354$	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	$-0.285836\pi$
$359$	8478.00	1.24638	0.623192	+	0.782069i	$0.285836\pi$
$360$	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	$-0.302211\pi$
$383$	8727.00	1.16431	0.582153	+	0.813080i	$0.302211\pi$
$384$	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	$-0.443445\pi$
$389$	2712.00	0.353480	0.176740	+	0.984258i	$0.443445\pi$
$390$	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	$-0.565997\pi$
$401$	−3306.00	−0.411705	−0.205853	+	0.978583i	$0.565997\pi$
$402$	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	$-0.458015\pi$
$419$	2256.00	0.263038	0.131519	+	0.991314i	$0.458015\pi$
$420$	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	$-0.401421\pi$
$431$	5454.00	0.609536	0.304768	+	0.952427i	$0.401421\pi$
$432$	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	$-0.607358\pi$
$443$	−6171.00	−0.661835	−0.330918	+	0.943660i	$0.607358\pi$
$444$	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	$-0.569505\pi$
$449$	−4122.00	−0.433250	−0.216625	+	0.976255i	$0.569505\pi$
$450$	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	$-0.512255\pi$
$461$	−762.000	−0.0769846	−0.0384923	+	0.999259i	$0.512255\pi$
$462$	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	$-0.420692\pi$
$467$	4977.00	0.493165	0.246583	+	0.969122i	$0.420692\pi$
$468$	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	$-0.660054\pi$
$479$	−10104.0	−0.963807	−0.481903	+	0.876224i	$0.660054\pi$
$480$	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	$-0.516772\pi$
$491$	−1146.00	−0.105332	−0.0526662	+	0.998612i	$0.516772\pi$
$492$	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	$-0.741554\pi$
$503$	−15525.0	−1.37619	−0.688097	+	0.725619i	$0.741554\pi$
$504$	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	$-0.383848\pi$
$509$	8196.00	0.713716	0.356858	+	0.934159i	$0.383848\pi$
$510$	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	$-0.566489\pi$
$521$	−4932.00	−0.414731	−0.207365	+	0.978264i	$0.566489\pi$
$522$	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	$-0.598086\pi$
$557$	−7974.00	−0.606587	−0.303294	+	0.952897i	$0.598086\pi$
$558$	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	$-0.102952\pi$
$563$	25332.0	1.89630	0.948150	+	0.317824i	$0.102952\pi$
$564$	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	$-0.487825\pi$
$569$	1038.00	0.0764767	0.0382383	+	0.999269i	$0.487825\pi$
$570$	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	$-0.604144\pi$
$587$	−9141.00	−0.642742	−0.321371	+	0.946953i	$0.604144\pi$
$588$	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	$-0.558152\pi$
$593$	−5247.00	−0.363353	−0.181677	+	0.983358i	$0.558152\pi$
$594$	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	$-0.191700\pi$
$599$	24162.0	1.64813	0.824067	+	0.566492i	$0.191700\pi$
$600$	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	$-0.422825\pi$
$617$	7359.00	0.480166	0.240083	+	0.970752i	$0.422825\pi$
$618$	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	$-0.500941\pi$
$641$	−96.0000	−0.00591540	−0.00295770	+	0.999996i	$0.500941\pi$
$642$	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	$-0.512972\pi$
$647$	−1341.00	−0.0814840	−0.0407420	+	0.999170i	$0.512972\pi$
$648$	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	$-0.237666\pi$
$653$	24495.0	1.46794	0.733969	+	0.679183i	$0.237666\pi$
$654$	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	$-0.380781\pi$
$659$	12378.0	0.731682	0.365841	+	0.930677i	$0.380781\pi$
$660$	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	$-0.549696\pi$
$677$	−5478.00	−0.310985	−0.155492	+	0.987837i	$0.549696\pi$
$678$	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	$-0.577397\pi$
$683$	−8595.00	−0.481521	−0.240760	+	0.970585i	$0.577397\pi$
$684$	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	$-0.796515\pi$
$701$	−29790.0	−1.60507	−0.802534	+	0.596606i	$0.796515\pi$
$702$	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	$-0.237507\pi$
$719$	28314.0	1.46861	0.734307	+	0.678817i	$0.237507\pi$
$720$	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	$-0.917238\pi$
$743$	−39144.0	−1.93278	−0.966389	+	0.257084i	$0.917238\pi$
$744$	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	$-0.874521\pi$
$761$	−38766.0	−1.84660	−0.923302	+	0.384074i	$0.874521\pi$
$762$	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	$-0.473376\pi$
$773$	3591.00	0.167088	0.0835442	+	0.996504i	$0.473376\pi$
$774$	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	$-0.0957233\pi$
$797$	42981.0	1.91024	0.955122	+	0.296211i	$0.0957233\pi$
$798$	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	$-0.484307\pi$
$809$	2268.00	0.0985644	0.0492822	+	0.998785i	$0.484307\pi$
$810$	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	$-0.558830\pi$
$821$	−8646.00	−0.367537	−0.183768	+	0.982970i	$0.558830\pi$
$822$	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	$-0.853219\pi$
$827$	−42597.0	−1.79110	−0.895552	+	0.444957i	$0.853219\pi$
$828$	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	$-0.423992\pi$
$839$	11496.0	0.473046	0.236523	+	0.971626i	$0.423992\pi$
$840$	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	$-0.539822\pi$
$857$	−6261.00	−0.249559	−0.124779	+	0.992185i	$0.539822\pi$
$858$	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	$-0.627022\pi$
$863$	−19701.0	−0.777091	−0.388546	+	0.921429i	$0.627022\pi$
$864$	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	$-0.556720\pi$
$881$	−9270.00	−0.354500	−0.177250	+	0.984166i	$0.556720\pi$
$882$	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	$-0.511407\pi$
$887$	−1893.00	−0.0716581	−0.0358290	+	0.999358i	$0.511407\pi$
$888$	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	$-0.642481\pi$
$911$	−23802.0	−0.865637	−0.432819	+	0.901481i	$0.642481\pi$
$912$	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	$-0.551144\pi$
$929$	−9060.00	−0.319967	−0.159983	+	0.987120i	$0.551144\pi$
$930$	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	$-0.476922\pi$
$941$	4182.00	0.144877	0.0724385	+	0.997373i	$0.476922\pi$
$942$	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	$-0.772980\pi$
$947$	−44079.0	−1.51254	−0.756270	+	0.654260i	$0.772980\pi$
$948$	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	$-0.569393\pi$
$953$	−12726.0	−0.432566	−0.216283	+	0.976331i	$0.569393\pi$
$954$	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	$-0.479060\pi$
$971$	3978.00	0.131473	0.0657364	+	0.997837i	$0.479060\pi$
$972$	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	$-0.625526\pi$
$977$	−23466.0	−0.768417	−0.384209	+	0.923246i	$0.625526\pi$
$978$	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	$-0.783415\pi$
$983$	−47913.0	−1.55462	−0.777308	+	0.629120i	$0.783415\pi$
$984$	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.a.1.1	✓	1
3.2	odd	2		270.4.a.k.1.1	yes	1
4.3	odd	2		2160.4.a.j.1.1		1
5.2	odd	4		1350.4.c.r.649.1		2
5.3	odd	4		1350.4.c.r.649.2		2
5.4	even	2		1350.4.a.bb.1.1		1
9.2	odd	6		810.4.e.d.271.1		2
9.4	even	3		810.4.e.x.541.1		2
9.5	odd	6		810.4.e.d.541.1		2
9.7	even	3		810.4.e.x.271.1		2
12.11	even	2		2160.4.a.t.1.1		1
15.2	even	4		1350.4.c.c.649.2		2
15.8	even	4		1350.4.c.c.649.1		2
15.14	odd	2		1350.4.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.a.1.1	✓	1	1.1	even	1	trivial
270.4.a.k.1.1	yes	1	3.2	odd	2
810.4.e.d.271.1		2	9.2	odd	6
810.4.e.d.541.1		2	9.5	odd	6
810.4.e.x.271.1		2	9.7	even	3
810.4.e.x.541.1		2	9.4	even	3
1350.4.a.n.1.1		1	15.14	odd	2
1350.4.a.bb.1.1		1	5.4	even	2
1350.4.c.c.649.1		2	15.8	even	4
1350.4.c.c.649.2		2	15.2	even	4
1350.4.c.r.649.1		2	5.2	odd	4
1350.4.c.r.649.2		2	5.3	odd	4
2160.4.a.j.1.1		1	4.3	odd	2
2160.4.a.t.1.1		1	12.11	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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more