LMFDB - Embedded newform 120.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [120,4,Mod(1,120)] 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("120.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(120, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 120 = 2^{3} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	120.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,5,0,-16] 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:	$7.08022920069$
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	$\chi$	$=$	120.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +5.00000 q^{5} -16.0000 q^{7} +9.00000 q^{9} -28.0000 q^{11} -26.0000 q^{13} -15.0000 q^{15} -62.0000 q^{17} -68.0000 q^{19} +48.0000 q^{21} -208.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -58.0000 q^{29} +160.000 q^{31} +84.0000 q^{33} -80.0000 q^{35} +270.000 q^{37} +78.0000 q^{39} +282.000 q^{41} +76.0000 q^{43} +45.0000 q^{45} -280.000 q^{47} -87.0000 q^{49} +186.000 q^{51} -210.000 q^{53} -140.000 q^{55} +204.000 q^{57} +196.000 q^{59} +742.000 q^{61} -144.000 q^{63} -130.000 q^{65} +836.000 q^{67} +624.000 q^{69} -504.000 q^{71} -1062.00 q^{73} -75.0000 q^{75} +448.000 q^{77} +768.000 q^{79} +81.0000 q^{81} -1052.00 q^{83} -310.000 q^{85} +174.000 q^{87} -726.000 q^{89} +416.000 q^{91} -480.000 q^{93} -340.000 q^{95} -1406.00 q^{97} -252.000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−16.0000	−0.863919	−0.431959	−	0.901893i	$-0.642178\pi$
$7$	−16.0000	−0.863919	−0.431959	+	0.901893i	$0.642178\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−28.0000	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$11$	−28.0000	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$12$	0	0
$13$	−26.0000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−26.0000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	−62.0000	−0.884542	−0.442271	−	0.896882i	$-0.645827\pi$
$17$	−62.0000	−0.884542	−0.442271	+	0.896882i	$0.645827\pi$
$18$	0	0
$19$	−68.0000	−0.821067	−0.410533	−	0.911846i	$-0.634657\pi$
$19$	−68.0000	−0.821067	−0.410533	+	0.911846i	$0.634657\pi$
$20$	0	0
$21$	48.0000	0.498784
$22$	0	0
$23$	−208.000	−1.88570	−0.942848	−	0.333224i	$-0.891864\pi$
$23$	−208.000	−1.88570	−0.942848	+	0.333224i	$0.891864\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−58.0000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−58.0000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	160.000	0.926995	0.463498	−	0.886098i	$-0.346594\pi$
$31$	160.000	0.926995	0.463498	+	0.886098i	$0.346594\pi$
$32$	0	0
$33$	84.0000	0.443107
$34$	0	0
$35$	−80.0000	−0.386356
$36$	0	0
$37$	270.000	1.19967	0.599834	−	0.800124i	$-0.295233\pi$
$37$	270.000	1.19967	0.599834	+	0.800124i	$0.295233\pi$
$38$	0	0
$39$	78.0000	0.320256
$40$	0	0
$41$	282.000	1.07417	0.537085	−	0.843528i	$-0.319525\pi$
$41$	282.000	1.07417	0.537085	+	0.843528i	$0.319525\pi$
$42$	0	0
$43$	76.0000	0.269532	0.134766	−	0.990877i	$-0.456972\pi$
$43$	76.0000	0.269532	0.134766	+	0.990877i	$0.456972\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−280.000	−0.868983	−0.434491	−	0.900676i	$-0.643072\pi$
$47$	−280.000	−0.868983	−0.434491	+	0.900676i	$0.643072\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	186.000	0.510690
$52$	0	0
$53$	−210.000	−0.544259	−0.272129	−	0.962261i	$-0.587728\pi$
$53$	−210.000	−0.544259	−0.272129	+	0.962261i	$0.587728\pi$
$54$	0	0
$55$	−140.000	−0.343229
$56$	0	0
$57$	204.000	0.474043
$58$	0	0
$59$	196.000	0.432492	0.216246	−	0.976339i	$-0.430619\pi$
$59$	196.000	0.432492	0.216246	+	0.976339i	$0.430619\pi$
$60$	0	0
$61$	742.000	1.55743	0.778716	−	0.627376i	$-0.215871\pi$
$61$	742.000	1.55743	0.778716	+	0.627376i	$0.215871\pi$
$62$	0	0
$63$	−144.000	−0.287973
$64$	0	0
$65$	−130.000	−0.248069
$66$	0	0
$67$	836.000	1.52438	0.762191	−	0.647352i	$-0.224123\pi$
$67$	836.000	1.52438	0.762191	+	0.647352i	$0.224123\pi$
$68$	0	0
$69$	624.000	1.08871
$70$	0	0
$71$	−504.000	−0.842448	−0.421224	−	0.906957i	$-0.638399\pi$
$71$	−504.000	−0.842448	−0.421224	+	0.906957i	$0.638399\pi$
$72$	0	0
$73$	−1062.00	−1.70271	−0.851354	−	0.524591i	$-0.824218\pi$
$73$	−1062.00	−1.70271	−0.851354	+	0.524591i	$0.824218\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	448.000	0.663043
$78$	0	0
$79$	768.000	1.09376	0.546878	−	0.837212i	$-0.315816\pi$
$79$	768.000	1.09376	0.546878	+	0.837212i	$0.315816\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−1052.00	−1.39123	−0.695614	−	0.718415i	$-0.744868\pi$
$83$	−1052.00	−1.39123	−0.695614	+	0.718415i	$0.744868\pi$
$84$	0	0
$85$	−310.000	−0.395579
$86$	0	0
$87$	174.000	0.214423
$88$	0	0
$89$	−726.000	−0.864672	−0.432336	−	0.901712i	$-0.642311\pi$
$89$	−726.000	−0.864672	−0.432336	+	0.901712i	$0.642311\pi$
$90$	0	0
$91$	416.000	0.479216
$92$	0	0
$93$	−480.000	−0.535201
$94$	0	0
$95$	−340.000	−0.367192
$96$	0	0
$97$	−1406.00	−1.47173	−0.735864	−	0.677129i	$-0.763224\pi$
$97$	−1406.00	−1.47173	−0.735864	+	0.677129i	$0.763224\pi$
$98$	0	0
$99$	−252.000	−0.255828
$100$	0	0
$101$	990.000	0.975333	0.487667	−	0.873030i	$-0.337848\pi$
$101$	990.000	0.975333	0.487667	+	0.873030i	$0.337848\pi$
$102$	0	0
$103$	736.000	0.704080	0.352040	−	0.935985i	$-0.385488\pi$
$103$	736.000	0.704080	0.352040	+	0.935985i	$0.385488\pi$
$104$	0	0
$105$	240.000	0.223063
$106$	0	0
$107$	1212.00	1.09503	0.547516	−	0.836795i	$-0.315573\pi$
$107$	1212.00	1.09503	0.547516	+	0.836795i	$0.315573\pi$
$108$	0	0
$109$	−1834.00	−1.61161	−0.805804	−	0.592182i	$-0.798267\pi$
$109$	−1834.00	−1.61161	−0.805804	+	0.592182i	$0.798267\pi$
$110$	0	0
$111$	−810.000	−0.692629
$112$	0	0
$113$	−2046.00	−1.70329	−0.851644	−	0.524121i	$-0.824394\pi$
$113$	−2046.00	−1.70329	−0.851644	+	0.524121i	$0.824394\pi$
$114$	0	0
$115$	−1040.00	−0.843309
$116$	0	0
$117$	−234.000	−0.184900
$118$	0	0
$119$	992.000	0.764172
$120$	0	0
$121$	−547.000	−0.410969
$122$	0	0
$123$	−846.000	−0.620173
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1176.00	0.821678	0.410839	−	0.911708i	$-0.365236\pi$
$127$	1176.00	0.821678	0.410839	+	0.911708i	$0.365236\pi$
$128$	0	0
$129$	−228.000	−0.155615
$130$	0	0
$131$	12.0000	0.00800340	0.00400170	−	0.999992i	$-0.498726\pi$
$131$	12.0000	0.00800340	0.00400170	+	0.999992i	$0.498726\pi$
$132$	0	0
$133$	1088.00	0.709335
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−790.000	−0.492659	−0.246329	−	0.969186i	$-0.579225\pi$
$137$	−790.000	−0.492659	−0.246329	+	0.969186i	$0.579225\pi$
$138$	0	0
$139$	−924.000	−0.563832	−0.281916	−	0.959439i	$-0.590970\pi$
$139$	−924.000	−0.563832	−0.281916	+	0.959439i	$0.590970\pi$
$140$	0	0
$141$	840.000	0.501708
$142$	0	0
$143$	728.000	0.425723
$144$	0	0
$145$	−290.000	−0.166091
$146$	0	0
$147$	261.000	0.146442
$148$	0	0
$149$	3022.00	1.66156	0.830778	−	0.556604i	$-0.187896\pi$
$149$	3022.00	1.66156	0.830778	+	0.556604i	$0.187896\pi$
$150$	0	0
$151$	1736.00	0.935587	0.467794	−	0.883838i	$-0.345049\pi$
$151$	1736.00	0.935587	0.467794	+	0.883838i	$0.345049\pi$
$152$	0	0
$153$	−558.000	−0.294847
$154$	0	0
$155$	800.000	0.414565
$156$	0	0
$157$	−1322.00	−0.672020	−0.336010	−	0.941858i	$-0.609078\pi$
$157$	−1322.00	−0.672020	−0.336010	+	0.941858i	$0.609078\pi$
$158$	0	0
$159$	630.000	0.314228
$160$	0	0
$161$	3328.00	1.62909
$162$	0	0
$163$	−908.000	−0.436319	−0.218160	−	0.975913i	$-0.570005\pi$
$163$	−908.000	−0.436319	−0.218160	+	0.975913i	$0.570005\pi$
$164$	0	0
$165$	420.000	0.198163
$166$	0	0
$167$	1296.00	0.600524	0.300262	−	0.953857i	$-0.402926\pi$
$167$	1296.00	0.600524	0.300262	+	0.953857i	$0.402926\pi$
$168$	0	0
$169$	−1521.00	−0.692308
$170$	0	0
$171$	−612.000	−0.273689
$172$	0	0
$173$	2134.00	0.937832	0.468916	−	0.883243i	$-0.344645\pi$
$173$	2134.00	0.937832	0.468916	+	0.883243i	$0.344645\pi$
$174$	0	0
$175$	−400.000	−0.172784
$176$	0	0
$177$	−588.000	−0.249699
$178$	0	0
$179$	1612.00	0.673109	0.336555	−	0.941664i	$-0.390738\pi$
$179$	1612.00	0.673109	0.336555	+	0.941664i	$0.390738\pi$
$180$	0	0
$181$	3086.00	1.26730	0.633648	−	0.773621i	$-0.281557\pi$
$181$	3086.00	1.26730	0.633648	+	0.773621i	$0.281557\pi$
$182$	0	0
$183$	−2226.00	−0.899184
$184$	0	0
$185$	1350.00	0.536508
$186$	0	0
$187$	1736.00	0.678871
$188$	0	0
$189$	432.000	0.166261
$190$	0	0
$191$	−4208.00	−1.59414	−0.797069	−	0.603889i	$-0.793617\pi$
$191$	−4208.00	−1.59414	−0.797069	+	0.603889i	$0.793617\pi$
$192$	0	0
$193$	2818.00	1.05101	0.525503	−	0.850792i	$-0.323877\pi$
$193$	2818.00	1.05101	0.525503	+	0.850792i	$0.323877\pi$
$194$	0	0
$195$	390.000	0.143223
$196$	0	0
$197$	−418.000	−0.151174	−0.0755870	−	0.997139i	$-0.524083\pi$
$197$	−418.000	−0.151174	−0.0755870	+	0.997139i	$0.524083\pi$
$198$	0	0
$199$	−3352.00	−1.19406	−0.597028	−	0.802221i	$-0.703652\pi$
$199$	−3352.00	−1.19406	−0.597028	+	0.802221i	$0.703652\pi$
$200$	0	0
$201$	−2508.00	−0.880103
$202$	0	0
$203$	928.000	0.320851
$204$	0	0
$205$	1410.00	0.480384
$206$	0	0
$207$	−1872.00	−0.628565
$208$	0	0
$209$	1904.00	0.630155
$210$	0	0
$211$	−4276.00	−1.39513	−0.697564	−	0.716523i	$-0.745733\pi$
$211$	−4276.00	−1.39513	−0.697564	+	0.716523i	$0.745733\pi$
$212$	0	0
$213$	1512.00	0.486387
$214$	0	0
$215$	380.000	0.120539
$216$	0	0
$217$	−2560.00	−0.800848
$218$	0	0
$219$	3186.00	0.983059
$220$	0	0
$221$	1612.00	0.490655
$222$	0	0
$223$	4712.00	1.41497	0.707486	−	0.706727i	$-0.249829\pi$
$223$	4712.00	1.41497	0.707486	+	0.706727i	$0.249829\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−732.000	−0.214029	−0.107014	−	0.994257i	$-0.534129\pi$
$227$	−732.000	−0.214029	−0.107014	+	0.994257i	$0.534129\pi$
$228$	0	0
$229$	−5186.00	−1.49651	−0.748254	−	0.663412i	$-0.769108\pi$
$229$	−5186.00	−1.49651	−0.748254	+	0.663412i	$0.769108\pi$
$230$	0	0
$231$	−1344.00	−0.382808
$232$	0	0
$233$	−3798.00	−1.06788	−0.533938	−	0.845523i	$-0.679289\pi$
$233$	−3798.00	−1.06788	−0.533938	+	0.845523i	$0.679289\pi$
$234$	0	0
$235$	−1400.00	−0.388621
$236$	0	0
$237$	−2304.00	−0.631481
$238$	0	0
$239$	−3120.00	−0.844419	−0.422209	−	0.906498i	$-0.638745\pi$
$239$	−3120.00	−0.844419	−0.422209	+	0.906498i	$0.638745\pi$
$240$	0	0
$241$	1490.00	0.398255	0.199127	−	0.979974i	$-0.436189\pi$
$241$	1490.00	0.398255	0.199127	+	0.979974i	$0.436189\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−435.000	−0.113433
$246$	0	0
$247$	1768.00	0.455446
$248$	0	0
$249$	3156.00	0.803226
$250$	0	0
$251$	−5292.00	−1.33079	−0.665395	−	0.746492i	$-0.731737\pi$
$251$	−5292.00	−1.33079	−0.665395	+	0.746492i	$0.731737\pi$
$252$	0	0
$253$	5824.00	1.44724
$254$	0	0
$255$	930.000	0.228388
$256$	0	0
$257$	−3918.00	−0.950965	−0.475483	−	0.879725i	$-0.657727\pi$
$257$	−3918.00	−0.950965	−0.475483	+	0.879725i	$0.657727\pi$
$258$	0	0
$259$	−4320.00	−1.03642
$260$	0	0
$261$	−522.000	−0.123797
$262$	0	0
$263$	6624.00	1.55305	0.776527	−	0.630084i	$-0.216979\pi$
$263$	6624.00	1.55305	0.776527	+	0.630084i	$0.216979\pi$
$264$	0	0
$265$	−1050.00	−0.243400
$266$	0	0
$267$	2178.00	0.499219
$268$	0	0
$269$	−2954.00	−0.669549	−0.334774	−	0.942298i	$-0.608660\pi$
$269$	−2954.00	−0.669549	−0.334774	+	0.942298i	$0.608660\pi$
$270$	0	0
$271$	−6576.00	−1.47404	−0.737018	−	0.675874i	$-0.763767\pi$
$271$	−6576.00	−1.47404	−0.737018	+	0.675874i	$0.763767\pi$
$272$	0	0
$273$	−1248.00	−0.276675
$274$	0	0
$275$	−700.000	−0.153497
$276$	0	0
$277$	4478.00	0.971325	0.485662	−	0.874146i	$-0.338578\pi$
$277$	4478.00	0.971325	0.485662	+	0.874146i	$0.338578\pi$
$278$	0	0
$279$	1440.00	0.308998
$280$	0	0
$281$	−6358.00	−1.34977	−0.674887	−	0.737921i	$-0.735808\pi$
$281$	−6358.00	−1.34977	−0.674887	+	0.737921i	$0.735808\pi$
$282$	0	0
$283$	860.000	0.180642	0.0903210	−	0.995913i	$-0.471211\pi$
$283$	860.000	0.180642	0.0903210	+	0.995913i	$0.471211\pi$
$284$	0	0
$285$	1020.00	0.211999
$286$	0	0
$287$	−4512.00	−0.927996
$288$	0	0
$289$	−1069.00	−0.217586
$290$	0	0
$291$	4218.00	0.849703
$292$	0	0
$293$	−5794.00	−1.15525	−0.577626	−	0.816301i	$-0.696021\pi$
$293$	−5794.00	−1.15525	−0.577626	+	0.816301i	$0.696021\pi$
$294$	0	0
$295$	980.000	0.193416
$296$	0	0
$297$	756.000	0.147702
$298$	0	0
$299$	5408.00	1.04600
$300$	0	0
$301$	−1216.00	−0.232854
$302$	0	0
$303$	−2970.00	−0.563109
$304$	0	0
$305$	3710.00	0.696505
$306$	0	0
$307$	−6860.00	−1.27531	−0.637656	−	0.770321i	$-0.720096\pi$
$307$	−6860.00	−1.27531	−0.637656	+	0.770321i	$0.720096\pi$
$308$	0	0
$309$	−2208.00	−0.406501
$310$	0	0
$311$	−6248.00	−1.13920	−0.569601	−	0.821922i	$-0.692902\pi$
$311$	−6248.00	−1.13920	−0.569601	+	0.821922i	$0.692902\pi$
$312$	0	0
$313$	11018.0	1.98969	0.994847	−	0.101388i	$-0.0323284\pi$
$313$	11018.0	1.98969	0.994847	+	0.101388i	$0.0323284\pi$
$314$	0	0
$315$	−720.000	−0.128785
$316$	0	0
$317$	−954.000	−0.169028	−0.0845142	−	0.996422i	$-0.526934\pi$
$317$	−954.000	−0.169028	−0.0845142	+	0.996422i	$0.526934\pi$
$318$	0	0
$319$	1624.00	0.285036
$320$	0	0
$321$	−3636.00	−0.632217
$322$	0	0
$323$	4216.00	0.726268
$324$	0	0
$325$	−650.000	−0.110940
$326$	0	0
$327$	5502.00	0.930463
$328$	0	0
$329$	4480.00	0.750731
$330$	0	0
$331$	9396.00	1.56027	0.780137	−	0.625608i	$-0.215149\pi$
$331$	9396.00	1.56027	0.780137	+	0.625608i	$0.215149\pi$
$332$	0	0
$333$	2430.00	0.399889
$334$	0	0
$335$	4180.00	0.681725
$336$	0	0
$337$	5074.00	0.820173	0.410087	−	0.912047i	$-0.365498\pi$
$337$	5074.00	0.820173	0.410087	+	0.912047i	$0.365498\pi$
$338$	0	0
$339$	6138.00	0.983394
$340$	0	0
$341$	−4480.00	−0.711453
$342$	0	0
$343$	6880.00	1.08305
$344$	0	0
$345$	3120.00	0.486885
$346$	0	0
$347$	3916.00	0.605827	0.302913	−	0.953018i	$-0.402041\pi$
$347$	3916.00	0.605827	0.302913	+	0.953018i	$0.402041\pi$
$348$	0	0
$349$	−1818.00	−0.278840	−0.139420	−	0.990233i	$-0.544524\pi$
$349$	−1818.00	−0.278840	−0.139420	+	0.990233i	$0.544524\pi$
$350$	0	0
$351$	702.000	0.106752
$352$	0	0
$353$	−7118.00	−1.07324	−0.536619	−	0.843825i	$-0.680299\pi$
$353$	−7118.00	−1.07324	−0.536619	+	0.843825i	$0.680299\pi$
$354$	0	0
$355$	−2520.00	−0.376754
$356$	0	0
$357$	−2976.00	−0.441195
$358$	0	0
$359$	5304.00	0.779762	0.389881	−	0.920865i	$-0.372516\pi$
$359$	5304.00	0.779762	0.389881	+	0.920865i	$0.372516\pi$
$360$	0	0
$361$	−2235.00	−0.325849
$362$	0	0
$363$	1641.00	0.237273
$364$	0	0
$365$	−5310.00	−0.761474
$366$	0	0
$367$	5672.00	0.806747	0.403373	−	0.915036i	$-0.367838\pi$
$367$	5672.00	0.806747	0.403373	+	0.915036i	$0.367838\pi$
$368$	0	0
$369$	2538.00	0.358057
$370$	0	0
$371$	3360.00	0.470195
$372$	0	0
$373$	7774.00	1.07915	0.539574	−	0.841938i	$-0.318585\pi$
$373$	7774.00	1.07915	0.539574	+	0.841938i	$0.318585\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	1508.00	0.206010
$378$	0	0
$379$	−5516.00	−0.747593	−0.373797	−	0.927511i	$-0.621944\pi$
$379$	−5516.00	−0.747593	−0.373797	+	0.927511i	$0.621944\pi$
$380$	0	0
$381$	−3528.00	−0.474396
$382$	0	0
$383$	−7128.00	−0.950976	−0.475488	−	0.879722i	$-0.657728\pi$
$383$	−7128.00	−0.950976	−0.475488	+	0.879722i	$0.657728\pi$
$384$	0	0
$385$	2240.00	0.296522
$386$	0	0
$387$	684.000	0.0898441
$388$	0	0
$389$	−10722.0	−1.39750	−0.698749	−	0.715367i	$-0.746260\pi$
$389$	−10722.0	−1.39750	−0.698749	+	0.715367i	$0.746260\pi$
$390$	0	0
$391$	12896.0	1.66798
$392$	0	0
$393$	−36.0000	−0.00462076
$394$	0	0
$395$	3840.00	0.489143
$396$	0	0
$397$	−12122.0	−1.53246	−0.766229	−	0.642568i	$-0.777869\pi$
$397$	−12122.0	−1.53246	−0.766229	+	0.642568i	$0.777869\pi$
$398$	0	0
$399$	−3264.00	−0.409535
$400$	0	0
$401$	10482.0	1.30535	0.652676	−	0.757637i	$-0.273646\pi$
$401$	10482.0	1.30535	0.652676	+	0.757637i	$0.273646\pi$
$402$	0	0
$403$	−4160.00	−0.514204
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−7560.00	−0.920726
$408$	0	0
$409$	3850.00	0.465453	0.232726	−	0.972542i	$-0.425235\pi$
$409$	3850.00	0.465453	0.232726	+	0.972542i	$0.425235\pi$
$410$	0	0
$411$	2370.00	0.284437
$412$	0	0
$413$	−3136.00	−0.373638
$414$	0	0
$415$	−5260.00	−0.622176
$416$	0	0
$417$	2772.00	0.325529
$418$	0	0
$419$	−5796.00	−0.675783	−0.337892	−	0.941185i	$-0.609714\pi$
$419$	−5796.00	−0.675783	−0.337892	+	0.941185i	$0.609714\pi$
$420$	0	0
$421$	3294.00	0.381330	0.190665	−	0.981655i	$-0.438936\pi$
$421$	3294.00	0.381330	0.190665	+	0.981655i	$0.438936\pi$
$422$	0	0
$423$	−2520.00	−0.289661
$424$	0	0
$425$	−1550.00	−0.176908
$426$	0	0
$427$	−11872.0	−1.34549
$428$	0	0
$429$	−2184.00	−0.245791
$430$	0	0
$431$	1696.00	0.189544	0.0947720	−	0.995499i	$-0.469788\pi$
$431$	1696.00	0.189544	0.0947720	+	0.995499i	$0.469788\pi$
$432$	0	0
$433$	−12334.0	−1.36890	−0.684451	−	0.729059i	$-0.739958\pi$
$433$	−12334.0	−1.36890	−0.684451	+	0.729059i	$0.739958\pi$
$434$	0	0
$435$	870.000	0.0958927
$436$	0	0
$437$	14144.0	1.54828
$438$	0	0
$439$	376.000	0.0408781	0.0204391	−	0.999791i	$-0.493494\pi$
$439$	376.000	0.0408781	0.0204391	+	0.999791i	$0.493494\pi$
$440$	0	0
$441$	−783.000	−0.0845481
$442$	0	0
$443$	8028.00	0.860997	0.430499	−	0.902591i	$-0.358338\pi$
$443$	8028.00	0.860997	0.430499	+	0.902591i	$0.358338\pi$
$444$	0	0
$445$	−3630.00	−0.386693
$446$	0	0
$447$	−9066.00	−0.959300
$448$	0	0
$449$	8898.00	0.935240	0.467620	−	0.883930i	$-0.345112\pi$
$449$	8898.00	0.935240	0.467620	+	0.883930i	$0.345112\pi$
$450$	0	0
$451$	−7896.00	−0.824408
$452$	0	0
$453$	−5208.00	−0.540162
$454$	0	0
$455$	2080.00	0.214312
$456$	0	0
$457$	10330.0	1.05737	0.528684	−	0.848819i	$-0.322686\pi$
$457$	10330.0	1.05737	0.528684	+	0.848819i	$0.322686\pi$
$458$	0	0
$459$	1674.00	0.170230
$460$	0	0
$461$	1878.00	0.189734	0.0948668	−	0.995490i	$-0.469757\pi$
$461$	1878.00	0.189734	0.0948668	+	0.995490i	$0.469757\pi$
$462$	0	0
$463$	−13224.0	−1.32737	−0.663684	−	0.748013i	$-0.731008\pi$
$463$	−13224.0	−1.32737	−0.663684	+	0.748013i	$0.731008\pi$
$464$	0	0
$465$	−2400.00	−0.239349
$466$	0	0
$467$	−8012.00	−0.793900	−0.396950	−	0.917840i	$-0.629931\pi$
$467$	−8012.00	−0.793900	−0.396950	+	0.917840i	$0.629931\pi$
$468$	0	0
$469$	−13376.0	−1.31694
$470$	0	0
$471$	3966.00	0.387991
$472$	0	0
$473$	−2128.00	−0.206862
$474$	0	0
$475$	−1700.00	−0.164213
$476$	0	0
$477$	−1890.00	−0.181420
$478$	0	0
$479$	−1792.00	−0.170936	−0.0854682	−	0.996341i	$-0.527239\pi$
$479$	−1792.00	−0.170936	−0.0854682	+	0.996341i	$0.527239\pi$
$480$	0	0
$481$	−7020.00	−0.665456
$482$	0	0
$483$	−9984.00	−0.940554
$484$	0	0
$485$	−7030.00	−0.658177
$486$	0	0
$487$	8272.00	0.769692	0.384846	−	0.922981i	$-0.374255\pi$
$487$	8272.00	0.769692	0.384846	+	0.922981i	$0.374255\pi$
$488$	0	0
$489$	2724.00	0.251909
$490$	0	0
$491$	516.000	0.0474272	0.0237136	−	0.999719i	$-0.492451\pi$
$491$	516.000	0.0474272	0.0237136	+	0.999719i	$0.492451\pi$
$492$	0	0
$493$	3596.00	0.328511
$494$	0	0
$495$	−1260.00	−0.114410
$496$	0	0
$497$	8064.00	0.727807
$498$	0	0
$499$	−14020.0	−1.25776	−0.628879	−	0.777503i	$-0.716486\pi$
$499$	−14020.0	−1.25776	−0.628879	+	0.777503i	$0.716486\pi$
$500$	0	0
$501$	−3888.00	−0.346713
$502$	0	0
$503$	1872.00	0.165941	0.0829705	−	0.996552i	$-0.473559\pi$
$503$	1872.00	0.165941	0.0829705	+	0.996552i	$0.473559\pi$
$504$	0	0
$505$	4950.00	0.436182
$506$	0	0
$507$	4563.00	0.399704
$508$	0	0
$509$	8678.00	0.755689	0.377844	−	0.925869i	$-0.376665\pi$
$509$	8678.00	0.755689	0.377844	+	0.925869i	$0.376665\pi$
$510$	0	0
$511$	16992.0	1.47100
$512$	0	0
$513$	1836.00	0.158014
$514$	0	0
$515$	3680.00	0.314874
$516$	0	0
$517$	7840.00	0.666930
$518$	0	0
$519$	−6402.00	−0.541458
$520$	0	0
$521$	18074.0	1.51984	0.759920	−	0.650017i	$-0.225238\pi$
$521$	18074.0	1.51984	0.759920	+	0.650017i	$0.225238\pi$
$522$	0	0
$523$	−20852.0	−1.74339	−0.871696	−	0.490047i	$-0.836980\pi$
$523$	−20852.0	−1.74339	−0.871696	+	0.490047i	$0.836980\pi$
$524$	0	0
$525$	1200.00	0.0997567
$526$	0	0
$527$	−9920.00	−0.819966
$528$	0	0
$529$	31097.0	2.55585
$530$	0	0
$531$	1764.00	0.144164
$532$	0	0
$533$	−7332.00	−0.595843
$534$	0	0
$535$	6060.00	0.489713
$536$	0	0
$537$	−4836.00	−0.388620
$538$	0	0
$539$	2436.00	0.194668
$540$	0	0
$541$	−12410.0	−0.986225	−0.493112	−	0.869966i	$-0.664141\pi$
$541$	−12410.0	−0.986225	−0.493112	+	0.869966i	$0.664141\pi$
$542$	0	0
$543$	−9258.00	−0.731674
$544$	0	0
$545$	−9170.00	−0.720733
$546$	0	0
$547$	3620.00	0.282962	0.141481	−	0.989941i	$-0.454814\pi$
$547$	3620.00	0.282962	0.141481	+	0.989941i	$0.454814\pi$
$548$	0	0
$549$	6678.00	0.519144
$550$	0	0
$551$	3944.00	0.304937
$552$	0	0
$553$	−12288.0	−0.944917
$554$	0	0
$555$	−4050.00	−0.309753
$556$	0	0
$557$	11734.0	0.892613	0.446307	−	0.894880i	$-0.352739\pi$
$557$	11734.0	0.892613	0.446307	+	0.894880i	$0.352739\pi$
$558$	0	0
$559$	−1976.00	−0.149510
$560$	0	0
$561$	−5208.00	−0.391946
$562$	0	0
$563$	−1372.00	−0.102705	−0.0513525	−	0.998681i	$-0.516353\pi$
$563$	−1372.00	−0.102705	−0.0513525	+	0.998681i	$0.516353\pi$
$564$	0	0
$565$	−10230.0	−0.761733
$566$	0	0
$567$	−1296.00	−0.0959910
$568$	0	0
$569$	18922.0	1.39412	0.697058	−	0.717015i	$-0.254492\pi$
$569$	18922.0	1.39412	0.697058	+	0.717015i	$0.254492\pi$
$570$	0	0
$571$	14596.0	1.06974	0.534872	−	0.844933i	$-0.320360\pi$
$571$	14596.0	1.06974	0.534872	+	0.844933i	$0.320360\pi$
$572$	0	0
$573$	12624.0	0.920376
$574$	0	0
$575$	−5200.00	−0.377139
$576$	0	0
$577$	−2302.00	−0.166089	−0.0830446	−	0.996546i	$-0.526464\pi$
$577$	−2302.00	−0.166089	−0.0830446	+	0.996546i	$0.526464\pi$
$578$	0	0
$579$	−8454.00	−0.606798
$580$	0	0
$581$	16832.0	1.20191
$582$	0	0
$583$	5880.00	0.417710
$584$	0	0
$585$	−1170.00	−0.0826898
$586$	0	0
$587$	23292.0	1.63776	0.818879	−	0.573966i	$-0.194596\pi$
$587$	23292.0	1.63776	0.818879	+	0.573966i	$0.194596\pi$
$588$	0	0
$589$	−10880.0	−0.761125
$590$	0	0
$591$	1254.00	0.0872803
$592$	0	0
$593$	−16542.0	−1.14553	−0.572764	−	0.819720i	$-0.694129\pi$
$593$	−16542.0	−1.14553	−0.572764	+	0.819720i	$0.694129\pi$
$594$	0	0
$595$	4960.00	0.341748
$596$	0	0
$597$	10056.0	0.689388
$598$	0	0
$599$	7464.00	0.509133	0.254567	−	0.967055i	$-0.418067\pi$
$599$	7464.00	0.509133	0.254567	+	0.967055i	$0.418067\pi$
$600$	0	0
$601$	−17270.0	−1.17214	−0.586072	−	0.810259i	$-0.699326\pi$
$601$	−17270.0	−1.17214	−0.586072	+	0.810259i	$0.699326\pi$
$602$	0	0
$603$	7524.00	0.508128
$604$	0	0
$605$	−2735.00	−0.183791
$606$	0	0
$607$	984.000	0.0657979	0.0328990	−	0.999459i	$-0.489526\pi$
$607$	984.000	0.0657979	0.0328990	+	0.999459i	$0.489526\pi$
$608$	0	0
$609$	−2784.00	−0.185244
$610$	0	0
$611$	7280.00	0.482025
$612$	0	0
$613$	7278.00	0.479536	0.239768	−	0.970830i	$-0.422929\pi$
$613$	7278.00	0.479536	0.239768	+	0.970830i	$0.422929\pi$
$614$	0	0
$615$	−4230.00	−0.277350
$616$	0	0
$617$	18090.0	1.18035	0.590175	−	0.807275i	$-0.299059\pi$
$617$	18090.0	1.18035	0.590175	+	0.807275i	$0.299059\pi$
$618$	0	0
$619$	24740.0	1.60644	0.803219	−	0.595684i	$-0.203119\pi$
$619$	24740.0	1.60644	0.803219	+	0.595684i	$0.203119\pi$
$620$	0	0
$621$	5616.00	0.362902
$622$	0	0
$623$	11616.0	0.747007
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−5712.00	−0.363820
$628$	0	0
$629$	−16740.0	−1.06116
$630$	0	0
$631$	19720.0	1.24412	0.622061	−	0.782969i	$-0.286296\pi$
$631$	19720.0	1.24412	0.622061	+	0.782969i	$0.286296\pi$
$632$	0	0
$633$	12828.0	0.805477
$634$	0	0
$635$	5880.00	0.367466
$636$	0	0
$637$	2262.00	0.140697
$638$	0	0
$639$	−4536.00	−0.280816
$640$	0	0
$641$	−16542.0	−1.01930	−0.509649	−	0.860383i	$-0.670225\pi$
$641$	−16542.0	−1.01930	−0.509649	+	0.860383i	$0.670225\pi$
$642$	0	0
$643$	−10092.0	−0.618957	−0.309479	−	0.950906i	$-0.600155\pi$
$643$	−10092.0	−0.618957	−0.309479	+	0.950906i	$0.600155\pi$
$644$	0	0
$645$	−1140.00	−0.0695930
$646$	0	0
$647$	−14544.0	−0.883746	−0.441873	−	0.897078i	$-0.645686\pi$
$647$	−14544.0	−0.883746	−0.441873	+	0.897078i	$0.645686\pi$
$648$	0	0
$649$	−5488.00	−0.331930
$650$	0	0
$651$	7680.00	0.462370
$652$	0	0
$653$	23062.0	1.38206	0.691030	−	0.722826i	$-0.257157\pi$
$653$	23062.0	1.38206	0.691030	+	0.722826i	$0.257157\pi$
$654$	0	0
$655$	60.0000	0.00357923
$656$	0	0
$657$	−9558.00	−0.567569
$658$	0	0
$659$	−28020.0	−1.65630	−0.828152	−	0.560504i	$-0.810608\pi$
$659$	−28020.0	−1.65630	−0.828152	+	0.560504i	$0.810608\pi$
$660$	0	0
$661$	−6738.00	−0.396487	−0.198243	−	0.980153i	$-0.563524\pi$
$661$	−6738.00	−0.396487	−0.198243	+	0.980153i	$0.563524\pi$
$662$	0	0
$663$	−4836.00	−0.283280
$664$	0	0
$665$	5440.00	0.317224
$666$	0	0
$667$	12064.0	0.700330
$668$	0	0
$669$	−14136.0	−0.816935
$670$	0	0
$671$	−20776.0	−1.19530
$672$	0	0
$673$	−14430.0	−0.826502	−0.413251	−	0.910617i	$-0.635607\pi$
$673$	−14430.0	−0.826502	−0.413251	+	0.910617i	$0.635607\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−17890.0	−1.01561	−0.507805	−	0.861472i	$-0.669543\pi$
$677$	−17890.0	−1.01561	−0.507805	+	0.861472i	$0.669543\pi$
$678$	0	0
$679$	22496.0	1.27145
$680$	0	0
$681$	2196.00	0.123570
$682$	0	0
$683$	10860.0	0.608413	0.304207	−	0.952606i	$-0.401609\pi$
$683$	10860.0	0.608413	0.304207	+	0.952606i	$0.401609\pi$
$684$	0	0
$685$	−3950.00	−0.220324
$686$	0	0
$687$	15558.0	0.864010
$688$	0	0
$689$	5460.00	0.301900
$690$	0	0
$691$	−8692.00	−0.478523	−0.239261	−	0.970955i	$-0.576905\pi$
$691$	−8692.00	−0.478523	−0.239261	+	0.970955i	$0.576905\pi$
$692$	0	0
$693$	4032.00	0.221014
$694$	0	0
$695$	−4620.00	−0.252153
$696$	0	0
$697$	−17484.0	−0.950149
$698$	0	0
$699$	11394.0	0.616539
$700$	0	0
$701$	−698.000	−0.0376078	−0.0188039	−	0.999823i	$-0.505986\pi$
$701$	−698.000	−0.0376078	−0.0188039	+	0.999823i	$0.505986\pi$
$702$	0	0
$703$	−18360.0	−0.985008
$704$	0	0
$705$	4200.00	0.224370
$706$	0	0
$707$	−15840.0	−0.842609
$708$	0	0
$709$	2654.00	0.140583	0.0702913	−	0.997527i	$-0.477607\pi$
$709$	2654.00	0.140583	0.0702913	+	0.997527i	$0.477607\pi$
$710$	0	0
$711$	6912.00	0.364585
$712$	0	0
$713$	−33280.0	−1.74803
$714$	0	0
$715$	3640.00	0.190389
$716$	0	0
$717$	9360.00	0.487525
$718$	0	0
$719$	−28240.0	−1.46478	−0.732388	−	0.680887i	$-0.761594\pi$
$719$	−28240.0	−1.46478	−0.732388	+	0.680887i	$0.761594\pi$
$720$	0	0
$721$	−11776.0	−0.608268
$722$	0	0
$723$	−4470.00	−0.229932
$724$	0	0
$725$	−1450.00	−0.0742781
$726$	0	0
$727$	−8320.00	−0.424445	−0.212223	−	0.977221i	$-0.568070\pi$
$727$	−8320.00	−0.424445	−0.212223	+	0.977221i	$0.568070\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−4712.00	−0.238413
$732$	0	0
$733$	−2154.00	−0.108540	−0.0542700	−	0.998526i	$-0.517283\pi$
$733$	−2154.00	−0.108540	−0.0542700	+	0.998526i	$0.517283\pi$
$734$	0	0
$735$	1305.00	0.0654907
$736$	0	0
$737$	−23408.0	−1.16994
$738$	0	0
$739$	22380.0	1.11402	0.557011	−	0.830505i	$-0.311948\pi$
$739$	22380.0	1.11402	0.557011	+	0.830505i	$0.311948\pi$
$740$	0	0
$741$	−5304.00	−0.262952
$742$	0	0
$743$	−5760.00	−0.284406	−0.142203	−	0.989837i	$-0.545419\pi$
$743$	−5760.00	−0.284406	−0.142203	+	0.989837i	$0.545419\pi$
$744$	0	0
$745$	15110.0	0.743071
$746$	0	0
$747$	−9468.00	−0.463743
$748$	0	0
$749$	−19392.0	−0.946019
$750$	0	0
$751$	−6192.00	−0.300865	−0.150432	−	0.988620i	$-0.548067\pi$
$751$	−6192.00	−0.300865	−0.150432	+	0.988620i	$0.548067\pi$
$752$	0	0
$753$	15876.0	0.768331
$754$	0	0
$755$	8680.00	0.418407
$756$	0	0
$757$	−13666.0	−0.656142	−0.328071	−	0.944653i	$-0.606398\pi$
$757$	−13666.0	−0.656142	−0.328071	+	0.944653i	$0.606398\pi$
$758$	0	0
$759$	−17472.0	−0.835564
$760$	0	0
$761$	−32022.0	−1.52536	−0.762678	−	0.646778i	$-0.776116\pi$
$761$	−32022.0	−1.52536	−0.762678	+	0.646778i	$0.776116\pi$
$762$	0	0
$763$	29344.0	1.39230
$764$	0	0
$765$	−2790.00	−0.131860
$766$	0	0
$767$	−5096.00	−0.239903
$768$	0	0
$769$	22786.0	1.06851	0.534255	−	0.845323i	$-0.320592\pi$
$769$	22786.0	1.06851	0.534255	+	0.845323i	$0.320592\pi$
$770$	0	0
$771$	11754.0	0.549040
$772$	0	0
$773$	8286.00	0.385546	0.192773	−	0.981243i	$-0.438252\pi$
$773$	8286.00	0.385546	0.192773	+	0.981243i	$0.438252\pi$
$774$	0	0
$775$	4000.00	0.185399
$776$	0	0
$777$	12960.0	0.598375
$778$	0	0
$779$	−19176.0	−0.881966
$780$	0	0
$781$	14112.0	0.646565
$782$	0	0
$783$	1566.00	0.0714742
$784$	0	0
$785$	−6610.00	−0.300536
$786$	0	0
$787$	−25804.0	−1.16876	−0.584379	−	0.811481i	$-0.698662\pi$
$787$	−25804.0	−1.16876	−0.584379	+	0.811481i	$0.698662\pi$
$788$	0	0
$789$	−19872.0	−0.896656
$790$	0	0
$791$	32736.0	1.47150
$792$	0	0
$793$	−19292.0	−0.863908
$794$	0	0
$795$	3150.00	0.140527
$796$	0	0
$797$	17670.0	0.785324	0.392662	−	0.919683i	$-0.371554\pi$
$797$	17670.0	0.785324	0.392662	+	0.919683i	$0.371554\pi$
$798$	0	0
$799$	17360.0	0.768652
$800$	0	0
$801$	−6534.00	−0.288224
$802$	0	0
$803$	29736.0	1.30680
$804$	0	0
$805$	16640.0	0.728550
$806$	0	0
$807$	8862.00	0.386564
$808$	0	0
$809$	−7398.00	−0.321508	−0.160754	−	0.986995i	$-0.551393\pi$
$809$	−7398.00	−0.321508	−0.160754	+	0.986995i	$0.551393\pi$
$810$	0	0
$811$	−28108.0	−1.21702	−0.608511	−	0.793545i	$-0.708233\pi$
$811$	−28108.0	−1.21702	−0.608511	+	0.793545i	$0.708233\pi$
$812$	0	0
$813$	19728.0	0.851035
$814$	0	0
$815$	−4540.00	−0.195128
$816$	0	0
$817$	−5168.00	−0.221304
$818$	0	0
$819$	3744.00	0.159739
$820$	0	0
$821$	30830.0	1.31057	0.655283	−	0.755384i	$-0.272549\pi$
$821$	30830.0	1.31057	0.655283	+	0.755384i	$0.272549\pi$
$822$	0	0
$823$	5872.00	0.248706	0.124353	−	0.992238i	$-0.460314\pi$
$823$	5872.00	0.248706	0.124353	+	0.992238i	$0.460314\pi$
$824$	0	0
$825$	2100.00	0.0886214
$826$	0	0
$827$	−16308.0	−0.685713	−0.342857	−	0.939388i	$-0.611394\pi$
$827$	−16308.0	−0.685713	−0.342857	+	0.939388i	$0.611394\pi$
$828$	0	0
$829$	28294.0	1.18539	0.592697	−	0.805426i	$-0.298063\pi$
$829$	28294.0	1.18539	0.592697	+	0.805426i	$0.298063\pi$
$830$	0	0
$831$	−13434.0	−0.560795
$832$	0	0
$833$	5394.00	0.224359
$834$	0	0
$835$	6480.00	0.268562
$836$	0	0
$837$	−4320.00	−0.178400
$838$	0	0
$839$	20536.0	0.845032	0.422516	−	0.906356i	$-0.361147\pi$
$839$	20536.0	0.845032	0.422516	+	0.906356i	$0.361147\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	0	0
$843$	19074.0	0.779292
$844$	0	0
$845$	−7605.00	−0.309609
$846$	0	0
$847$	8752.00	0.355044
$848$	0	0
$849$	−2580.00	−0.104294
$850$	0	0
$851$	−56160.0	−2.26221
$852$	0	0
$853$	27710.0	1.11228	0.556139	−	0.831090i	$-0.312282\pi$
$853$	27710.0	1.11228	0.556139	+	0.831090i	$0.312282\pi$
$854$	0	0
$855$	−3060.00	−0.122397
$856$	0	0
$857$	12858.0	0.512510	0.256255	−	0.966609i	$-0.417511\pi$
$857$	12858.0	0.512510	0.256255	+	0.966609i	$0.417511\pi$
$858$	0	0
$859$	−3148.00	−0.125039	−0.0625194	−	0.998044i	$-0.519914\pi$
$859$	−3148.00	−0.125039	−0.0625194	+	0.998044i	$0.519914\pi$
$860$	0	0
$861$	13536.0	0.535779
$862$	0	0
$863$	48456.0	1.91131	0.955656	−	0.294487i	$-0.0951487\pi$
$863$	48456.0	1.91131	0.955656	+	0.294487i	$0.0951487\pi$
$864$	0	0
$865$	10670.0	0.419411
$866$	0	0
$867$	3207.00	0.125623
$868$	0	0
$869$	−21504.0	−0.839440
$870$	0	0
$871$	−21736.0	−0.845576
$872$	0	0
$873$	−12654.0	−0.490576
$874$	0	0
$875$	−2000.00	−0.0772712
$876$	0	0
$877$	9478.00	0.364937	0.182468	−	0.983212i	$-0.441591\pi$
$877$	9478.00	0.364937	0.182468	+	0.983212i	$0.441591\pi$
$878$	0	0
$879$	17382.0	0.666986
$880$	0	0
$881$	8178.00	0.312740	0.156370	−	0.987699i	$-0.450021\pi$
$881$	8178.00	0.312740	0.156370	+	0.987699i	$0.450021\pi$
$882$	0	0
$883$	−316.000	−0.0120433	−0.00602166	−	0.999982i	$-0.501917\pi$
$883$	−316.000	−0.0120433	−0.00602166	+	0.999982i	$0.501917\pi$
$884$	0	0
$885$	−2940.00	−0.111669
$886$	0	0
$887$	−6304.00	−0.238633	−0.119317	−	0.992856i	$-0.538070\pi$
$887$	−6304.00	−0.238633	−0.119317	+	0.992856i	$0.538070\pi$
$888$	0	0
$889$	−18816.0	−0.709863
$890$	0	0
$891$	−2268.00	−0.0852759
$892$	0	0
$893$	19040.0	0.713493
$894$	0	0
$895$	8060.00	0.301024
$896$	0	0
$897$	−16224.0	−0.603906
$898$	0	0
$899$	−9280.00	−0.344277
$900$	0	0
$901$	13020.0	0.481420
$902$	0	0
$903$	3648.00	0.134438
$904$	0	0
$905$	15430.0	0.566752
$906$	0	0
$907$	1596.00	0.0584281	0.0292141	−	0.999573i	$-0.490700\pi$
$907$	1596.00	0.0584281	0.0292141	+	0.999573i	$0.490700\pi$
$908$	0	0
$909$	8910.00	0.325111
$910$	0	0
$911$	−25792.0	−0.938010	−0.469005	−	0.883196i	$-0.655387\pi$
$911$	−25792.0	−0.938010	−0.469005	+	0.883196i	$0.655387\pi$
$912$	0	0
$913$	29456.0	1.06775
$914$	0	0
$915$	−11130.0	−0.402127
$916$	0	0
$917$	−192.000	−0.00691428
$918$	0	0
$919$	−9736.00	−0.349468	−0.174734	−	0.984616i	$-0.555907\pi$
$919$	−9736.00	−0.349468	−0.174734	+	0.984616i	$0.555907\pi$
$920$	0	0
$921$	20580.0	0.736302
$922$	0	0
$923$	13104.0	0.467306
$924$	0	0
$925$	6750.00	0.239934
$926$	0	0
$927$	6624.00	0.234693
$928$	0	0
$929$	−94.0000	−0.00331974	−0.00165987	−	0.999999i	$-0.500528\pi$
$929$	−94.0000	−0.00331974	−0.00165987	+	0.999999i	$0.500528\pi$
$930$	0	0
$931$	5916.00	0.208259
$932$	0	0
$933$	18744.0	0.657718
$934$	0	0
$935$	8680.00	0.303600
$936$	0	0
$937$	−8678.00	−0.302559	−0.151280	−	0.988491i	$-0.548339\pi$
$937$	−8678.00	−0.302559	−0.151280	+	0.988491i	$0.548339\pi$
$938$	0	0
$939$	−33054.0	−1.14875
$940$	0	0
$941$	28406.0	0.984069	0.492035	−	0.870576i	$-0.336253\pi$
$941$	28406.0	0.984069	0.492035	+	0.870576i	$0.336253\pi$
$942$	0	0
$943$	−58656.0	−2.02556
$944$	0	0
$945$	2160.00	0.0743543
$946$	0	0
$947$	31988.0	1.09765	0.548823	−	0.835939i	$-0.315076\pi$
$947$	31988.0	1.09765	0.548823	+	0.835939i	$0.315076\pi$
$948$	0	0
$949$	27612.0	0.944493
$950$	0	0
$951$	2862.00	0.0975885
$952$	0	0
$953$	6714.00	0.228214	0.114107	−	0.993468i	$-0.463599\pi$
$953$	6714.00	0.228214	0.114107	+	0.993468i	$0.463599\pi$
$954$	0	0
$955$	−21040.0	−0.712920
$956$	0	0
$957$	−4872.00	−0.164566
$958$	0	0
$959$	12640.0	0.425617
$960$	0	0
$961$	−4191.00	−0.140680
$962$	0	0
$963$	10908.0	0.365011
$964$	0	0
$965$	14090.0	0.470024
$966$	0	0
$967$	15312.0	0.509204	0.254602	−	0.967046i	$-0.418055\pi$
$967$	15312.0	0.509204	0.254602	+	0.967046i	$0.418055\pi$
$968$	0	0
$969$	−12648.0	−0.419311
$970$	0	0
$971$	−8540.00	−0.282247	−0.141123	−	0.989992i	$-0.545071\pi$
$971$	−8540.00	−0.282247	−0.141123	+	0.989992i	$0.545071\pi$
$972$	0	0
$973$	14784.0	0.487105
$974$	0	0
$975$	1950.00	0.0640513
$976$	0	0
$977$	−8126.00	−0.266094	−0.133047	−	0.991110i	$-0.542476\pi$
$977$	−8126.00	−0.266094	−0.133047	+	0.991110i	$0.542476\pi$
$978$	0	0
$979$	20328.0	0.663622
$980$	0	0
$981$	−16506.0	−0.537203
$982$	0	0
$983$	1392.00	0.0451657	0.0225829	−	0.999745i	$-0.492811\pi$
$983$	1392.00	0.0451657	0.0225829	+	0.999745i	$0.492811\pi$
$984$	0	0
$985$	−2090.00	−0.0676070
$986$	0	0
$987$	−13440.0	−0.433435
$988$	0	0
$989$	−15808.0	−0.508256
$990$	0	0
$991$	−48832.0	−1.56529	−0.782644	−	0.622470i	$-0.786129\pi$
$991$	−48832.0	−1.56529	−0.782644	+	0.622470i	$0.786129\pi$
$992$	0	0
$993$	−28188.0	−0.900825
$994$	0	0
$995$	−16760.0	−0.533998
$996$	0	0
$997$	46926.0	1.49063	0.745317	−	0.666711i	$-0.232298\pi$
$997$	46926.0	1.49063	0.745317	+	0.666711i	$0.232298\pi$
$998$	0	0
$999$	−7290.00	−0.230876

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	120.4.a.c.1.1	✓	1
3.2	odd	2		360.4.a.b.1.1		1
4.3	odd	2		240.4.a.l.1.1		1
5.2	odd	4		600.4.f.c.49.2		2
5.3	odd	4		600.4.f.c.49.1		2
5.4	even	2		600.4.a.q.1.1		1
8.3	odd	2		960.4.a.h.1.1		1
8.5	even	2		960.4.a.u.1.1		1
12.11	even	2		720.4.a.l.1.1		1
15.2	even	4		1800.4.f.r.649.1		2
15.8	even	4		1800.4.f.r.649.2		2
15.14	odd	2		1800.4.a.bb.1.1		1
20.3	even	4		1200.4.f.o.49.2		2
20.7	even	4		1200.4.f.o.49.1		2
20.19	odd	2		1200.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.c.1.1	✓	1	1.1	even	1	trivial
240.4.a.l.1.1		1	4.3	odd	2
360.4.a.b.1.1		1	3.2	odd	2
600.4.a.q.1.1		1	5.4	even	2
600.4.f.c.49.1		2	5.3	odd	4
600.4.f.c.49.2		2	5.2	odd	4
720.4.a.l.1.1		1	12.11	even	2
960.4.a.h.1.1		1	8.3	odd	2
960.4.a.u.1.1		1	8.5	even	2
1200.4.a.c.1.1		1	20.19	odd	2
1200.4.f.o.49.1		2	20.7	even	4
1200.4.f.o.49.2		2	20.3	even	4
1800.4.a.bb.1.1		1	15.14	odd	2
1800.4.f.r.649.1		2	15.2	even	4
1800.4.f.r.649.2		2	15.8	even	4

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:	\( N \)	\(=\)	\( 120 = 2^{3} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	120.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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