LMFDB - Embedded newform 480.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,4,Mod(1,480)] 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("480.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,5,0,-32] 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:	$28.3209168028$
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	$\chi$	$=$	480.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} +5.00000 q^{5} -32.0000 q^{7} +9.00000 q^{9} -64.0000 q^{11} -6.00000 q^{13} -15.0000 q^{15} +38.0000 q^{17} +116.000 q^{19} +96.0000 q^{21} +120.000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -122.000 q^{29} -164.000 q^{31} +192.000 q^{33} -160.000 q^{35} +146.000 q^{37} +18.0000 q^{39} -238.000 q^{41} +148.000 q^{43} +45.0000 q^{45} +184.000 q^{47} +681.000 q^{49} -114.000 q^{51} +470.000 q^{53} -320.000 q^{55} -348.000 q^{57} +216.000 q^{59} +806.000 q^{61} -288.000 q^{63} -30.0000 q^{65} +732.000 q^{67} -360.000 q^{69} -264.000 q^{71} -638.000 q^{73} -75.0000 q^{75} +2048.00 q^{77} -596.000 q^{79} +81.0000 q^{81} +884.000 q^{83} +190.000 q^{85} +366.000 q^{87} +930.000 q^{89} +192.000 q^{91} +492.000 q^{93} +580.000 q^{95} +322.000 q^{97} -576.000 q^{99} +O(q^{100})$

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
$7$	−32.0000	−1.72784	−0.863919	−	0.503631i	$-0.831997\pi$
$7$	−32.0000	−1.72784	−0.863919	+	0.503631i	$0.831997\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−64.0000	−1.75425	−0.877124	−	0.480264i	$-0.840541\pi$
$11$	−64.0000	−1.75425	−0.877124	+	0.480264i	$0.840541\pi$
$12$	0	0
$13$	−6.00000	−0.128008	−0.0640039	−	0.997950i	$-0.520387\pi$
$13$	−6.00000	−0.128008	−0.0640039	+	0.997950i	$0.520387\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	38.0000	0.542138	0.271069	−	0.962560i	$-0.412623\pi$
$17$	38.0000	0.542138	0.271069	+	0.962560i	$0.412623\pi$
$18$	0	0
$19$	116.000	1.40064	0.700322	−	0.713827i	$-0.253040\pi$
$19$	116.000	1.40064	0.700322	+	0.713827i	$0.253040\pi$
$20$	0	0
$21$	96.0000	0.997567
$22$	0	0
$23$	120.000	1.08790	0.543951	−	0.839117i	$-0.316928\pi$
$23$	120.000	1.08790	0.543951	+	0.839117i	$0.316928\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−122.000	−0.781201	−0.390601	−	0.920560i	$-0.627733\pi$
$29$	−122.000	−0.781201	−0.390601	+	0.920560i	$0.627733\pi$
$30$	0	0
$31$	−164.000	−0.950170	−0.475085	−	0.879940i	$-0.657583\pi$
$31$	−164.000	−0.950170	−0.475085	+	0.879940i	$0.657583\pi$
$32$	0	0
$33$	192.000	1.01282
$34$	0	0
$35$	−160.000	−0.772712
$36$	0	0
$37$	146.000	0.648710	0.324355	−	0.945936i	$-0.394853\pi$
$37$	146.000	0.648710	0.324355	+	0.945936i	$0.394853\pi$
$38$	0	0
$39$	18.0000	0.0739053
$40$	0	0
$41$	−238.000	−0.906570	−0.453285	−	0.891366i	$-0.649748\pi$
$41$	−238.000	−0.906570	−0.453285	+	0.891366i	$0.649748\pi$
$42$	0	0
$43$	148.000	0.524879	0.262439	−	0.964948i	$-0.415473\pi$
$43$	148.000	0.524879	0.262439	+	0.964948i	$0.415473\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	184.000	0.571046	0.285523	−	0.958372i	$-0.407833\pi$
$47$	184.000	0.571046	0.285523	+	0.958372i	$0.407833\pi$
$48$	0	0
$49$	681.000	1.98542
$50$	0	0
$51$	−114.000	−0.313004
$52$	0	0
$53$	470.000	1.21810	0.609052	−	0.793131i	$-0.291550\pi$
$53$	470.000	1.21810	0.609052	+	0.793131i	$0.291550\pi$
$54$	0	0
$55$	−320.000	−0.784523
$56$	0	0
$57$	−348.000	−0.808662
$58$	0	0
$59$	216.000	0.476624	0.238312	−	0.971189i	$-0.423406\pi$
$59$	216.000	0.476624	0.238312	+	0.971189i	$0.423406\pi$
$60$	0	0
$61$	806.000	1.69177	0.845883	−	0.533369i	$-0.179074\pi$
$61$	806.000	1.69177	0.845883	+	0.533369i	$0.179074\pi$
$62$	0	0
$63$	−288.000	−0.575946
$64$	0	0
$65$	−30.0000	−0.0572468
$66$	0	0
$67$	732.000	1.33475	0.667373	−	0.744723i	$-0.267419\pi$
$67$	732.000	1.33475	0.667373	+	0.744723i	$0.267419\pi$
$68$	0	0
$69$	−360.000	−0.628100
$70$	0	0
$71$	−264.000	−0.441282	−0.220641	−	0.975355i	$-0.570815\pi$
$71$	−264.000	−0.441282	−0.220641	+	0.975355i	$0.570815\pi$
$72$	0	0
$73$	−638.000	−1.02291	−0.511454	−	0.859311i	$-0.670893\pi$
$73$	−638.000	−1.02291	−0.511454	+	0.859311i	$0.670893\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	2048.00	3.03106
$78$	0	0
$79$	−596.000	−0.848800	−0.424400	−	0.905475i	$-0.639515\pi$
$79$	−596.000	−0.848800	−0.424400	+	0.905475i	$0.639515\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	884.000	1.16906	0.584528	−	0.811374i	$-0.301280\pi$
$83$	884.000	1.16906	0.584528	+	0.811374i	$0.301280\pi$
$84$	0	0
$85$	190.000	0.242452
$86$	0	0
$87$	366.000	0.451027
$88$	0	0
$89$	930.000	1.10764	0.553819	−	0.832637i	$-0.313170\pi$
$89$	930.000	1.10764	0.553819	+	0.832637i	$0.313170\pi$
$90$	0	0
$91$	192.000	0.221177
$92$	0	0
$93$	492.000	0.548581
$94$	0	0
$95$	580.000	0.626387
$96$	0	0
$97$	322.000	0.337053	0.168527	−	0.985697i	$-0.446099\pi$
$97$	322.000	0.337053	0.168527	+	0.985697i	$0.446099\pi$
$98$	0	0
$99$	−576.000	−0.584749
$100$	0	0
$101$	−946.000	−0.931985	−0.465993	−	0.884789i	$-0.654303\pi$
$101$	−946.000	−0.931985	−0.465993	+	0.884789i	$0.654303\pi$
$102$	0	0
$103$	−424.000	−0.405611	−0.202806	−	0.979219i	$-0.565006\pi$
$103$	−424.000	−0.405611	−0.202806	+	0.979219i	$0.565006\pi$
$104$	0	0
$105$	480.000	0.446126
$106$	0	0
$107$	1668.00	1.50702	0.753512	−	0.657434i	$-0.228358\pi$
$107$	1668.00	1.50702	0.753512	+	0.657434i	$0.228358\pi$
$108$	0	0
$109$	−1634.00	−1.43586	−0.717930	−	0.696115i	$-0.754910\pi$
$109$	−1634.00	−1.43586	−0.717930	+	0.696115i	$0.754910\pi$
$110$	0	0
$111$	−438.000	−0.374533
$112$	0	0
$113$	1686.00	1.40359	0.701794	−	0.712380i	$-0.252383\pi$
$113$	1686.00	1.40359	0.701794	+	0.712380i	$0.252383\pi$
$114$	0	0
$115$	600.000	0.486524
$116$	0	0
$117$	−54.0000	−0.0426692
$118$	0	0
$119$	−1216.00	−0.936727
$120$	0	0
$121$	2765.00	2.07739
$122$	0	0
$123$	714.000	0.523408
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−2640.00	−1.84458	−0.922292	−	0.386494i	$-0.873686\pi$
$127$	−2640.00	−1.84458	−0.922292	+	0.386494i	$0.873686\pi$
$128$	0	0
$129$	−444.000	−0.303039
$130$	0	0
$131$	2536.00	1.69138	0.845692	−	0.533671i	$-0.179188\pi$
$131$	2536.00	1.69138	0.845692	+	0.533671i	$0.179188\pi$
$132$	0	0
$133$	−3712.00	−2.42008
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−634.000	−0.395374	−0.197687	−	0.980265i	$-0.563343\pi$
$137$	−634.000	−0.395374	−0.197687	+	0.980265i	$0.563343\pi$
$138$	0	0
$139$	−2980.00	−1.81842	−0.909210	−	0.416338i	$-0.863313\pi$
$139$	−2980.00	−1.81842	−0.909210	+	0.416338i	$0.863313\pi$
$140$	0	0
$141$	−552.000	−0.329694
$142$	0	0
$143$	384.000	0.224557
$144$	0	0
$145$	−610.000	−0.349364
$146$	0	0
$147$	−2043.00	−1.14628
$148$	0	0
$149$	−338.000	−0.185839	−0.0929196	−	0.995674i	$-0.529620\pi$
$149$	−338.000	−0.185839	−0.0929196	+	0.995674i	$0.529620\pi$
$150$	0	0
$151$	428.000	0.230663	0.115332	−	0.993327i	$-0.463207\pi$
$151$	428.000	0.230663	0.115332	+	0.993327i	$0.463207\pi$
$152$	0	0
$153$	342.000	0.180713
$154$	0	0
$155$	−820.000	−0.424929
$156$	0	0
$157$	3010.00	1.53009	0.765045	−	0.643977i	$-0.222717\pi$
$157$	3010.00	1.53009	0.765045	+	0.643977i	$0.222717\pi$
$158$	0	0
$159$	−1410.00	−0.703272
$160$	0	0
$161$	−3840.00	−1.87972
$162$	0	0
$163$	132.000	0.0634297	0.0317148	−	0.999497i	$-0.489903\pi$
$163$	132.000	0.0634297	0.0317148	+	0.999497i	$0.489903\pi$
$164$	0	0
$165$	960.000	0.452945
$166$	0	0
$167$	−2176.00	−1.00829	−0.504144	−	0.863620i	$-0.668192\pi$
$167$	−2176.00	−1.00829	−0.504144	+	0.863620i	$0.668192\pi$
$168$	0	0
$169$	−2161.00	−0.983614
$170$	0	0
$171$	1044.00	0.466881
$172$	0	0
$173$	−2778.00	−1.22085	−0.610426	−	0.792073i	$-0.709002\pi$
$173$	−2778.00	−1.22085	−0.610426	+	0.792073i	$0.709002\pi$
$174$	0	0
$175$	−800.000	−0.345568
$176$	0	0
$177$	−648.000	−0.275179
$178$	0	0
$179$	3888.00	1.62348	0.811740	−	0.584020i	$-0.198521\pi$
$179$	3888.00	1.62348	0.811740	+	0.584020i	$0.198521\pi$
$180$	0	0
$181$	1350.00	0.554391	0.277195	−	0.960814i	$-0.410595\pi$
$181$	1350.00	0.554391	0.277195	+	0.960814i	$0.410595\pi$
$182$	0	0
$183$	−2418.00	−0.976742
$184$	0	0
$185$	730.000	0.290112
$186$	0	0
$187$	−2432.00	−0.951045
$188$	0	0
$189$	864.000	0.332522
$190$	0	0
$191$	4760.00	1.80325	0.901627	−	0.432514i	$-0.142374\pi$
$191$	4760.00	1.80325	0.901627	+	0.432514i	$0.142374\pi$
$192$	0	0
$193$	1034.00	0.385642	0.192821	−	0.981234i	$-0.438236\pi$
$193$	1034.00	0.385642	0.192821	+	0.981234i	$0.438236\pi$
$194$	0	0
$195$	90.0000	0.0330515
$196$	0	0
$197$	−1354.00	−0.489688	−0.244844	−	0.969563i	$-0.578737\pi$
$197$	−1354.00	−0.489688	−0.244844	+	0.969563i	$0.578737\pi$
$198$	0	0
$199$	2324.00	0.827859	0.413930	−	0.910309i	$-0.364156\pi$
$199$	2324.00	0.827859	0.413930	+	0.910309i	$0.364156\pi$
$200$	0	0
$201$	−2196.00	−0.770616
$202$	0	0
$203$	3904.00	1.34979
$204$	0	0
$205$	−1190.00	−0.405430
$206$	0	0
$207$	1080.00	0.362634
$208$	0	0
$209$	−7424.00	−2.45708
$210$	0	0
$211$	3220.00	1.05059	0.525294	−	0.850921i	$-0.323955\pi$
$211$	3220.00	1.05059	0.525294	+	0.850921i	$0.323955\pi$
$212$	0	0
$213$	792.000	0.254774
$214$	0	0
$215$	740.000	0.234733
$216$	0	0
$217$	5248.00	1.64174
$218$	0	0
$219$	1914.00	0.590576
$220$	0	0
$221$	−228.000	−0.0693979
$222$	0	0
$223$	32.0000	0.00960932	0.00480466	−	0.999988i	$-0.498471\pi$
$223$	32.0000	0.00960932	0.00480466	+	0.999988i	$0.498471\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3996.00	1.16839	0.584193	−	0.811614i	$-0.301411\pi$
$227$	3996.00	1.16839	0.584193	+	0.811614i	$0.301411\pi$
$228$	0	0
$229$	−3010.00	−0.868587	−0.434293	−	0.900771i	$-0.643002\pi$
$229$	−3010.00	−0.868587	−0.434293	+	0.900771i	$0.643002\pi$
$230$	0	0
$231$	−6144.00	−1.74998
$232$	0	0
$233$	−2698.00	−0.758592	−0.379296	−	0.925275i	$-0.623834\pi$
$233$	−2698.00	−0.758592	−0.379296	+	0.925275i	$0.623834\pi$
$234$	0	0
$235$	920.000	0.255380
$236$	0	0
$237$	1788.00	0.490055
$238$	0	0
$239$	2200.00	0.595423	0.297712	−	0.954656i	$-0.403777\pi$
$239$	2200.00	0.595423	0.297712	+	0.954656i	$0.403777\pi$
$240$	0	0
$241$	−6174.00	−1.65022	−0.825109	−	0.564974i	$-0.808886\pi$
$241$	−6174.00	−1.65022	−0.825109	+	0.564974i	$0.808886\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	3405.00	0.887908
$246$	0	0
$247$	−696.000	−0.179293
$248$	0	0
$249$	−2652.00	−0.674955
$250$	0	0
$251$	−4808.00	−1.20908	−0.604538	−	0.796576i	$-0.706642\pi$
$251$	−4808.00	−1.20908	−0.604538	+	0.796576i	$0.706642\pi$
$252$	0	0
$253$	−7680.00	−1.90845
$254$	0	0
$255$	−570.000	−0.139980
$256$	0	0
$257$	3974.00	0.964558	0.482279	−	0.876018i	$-0.339809\pi$
$257$	3974.00	0.964558	0.482279	+	0.876018i	$0.339809\pi$
$258$	0	0
$259$	−4672.00	−1.12086
$260$	0	0
$261$	−1098.00	−0.260400
$262$	0	0
$263$	−1408.00	−0.330118	−0.165059	−	0.986284i	$-0.552781\pi$
$263$	−1408.00	−0.330118	−0.165059	+	0.986284i	$0.552781\pi$
$264$	0	0
$265$	2350.00	0.544752
$266$	0	0
$267$	−2790.00	−0.639495
$268$	0	0
$269$	−3258.00	−0.738453	−0.369226	−	0.929340i	$-0.620377\pi$
$269$	−3258.00	−0.738453	−0.369226	+	0.929340i	$0.620377\pi$
$270$	0	0
$271$	−8612.00	−1.93041	−0.965206	−	0.261490i	$-0.915786\pi$
$271$	−8612.00	−1.93041	−0.965206	+	0.261490i	$0.915786\pi$
$272$	0	0
$273$	−576.000	−0.127696
$274$	0	0
$275$	−1600.00	−0.350850
$276$	0	0
$277$	−4006.00	−0.868943	−0.434472	−	0.900686i	$-0.643065\pi$
$277$	−4006.00	−0.868943	−0.434472	+	0.900686i	$0.643065\pi$
$278$	0	0
$279$	−1476.00	−0.316723
$280$	0	0
$281$	6194.00	1.31496	0.657479	−	0.753473i	$-0.271623\pi$
$281$	6194.00	1.31496	0.657479	+	0.753473i	$0.271623\pi$
$282$	0	0
$283$	−1724.00	−0.362124	−0.181062	−	0.983472i	$-0.557954\pi$
$283$	−1724.00	−0.362124	−0.181062	+	0.983472i	$0.557954\pi$
$284$	0	0
$285$	−1740.00	−0.361645
$286$	0	0
$287$	7616.00	1.56641
$288$	0	0
$289$	−3469.00	−0.706086
$290$	0	0
$291$	−966.000	−0.194598
$292$	0	0
$293$	2502.00	0.498868	0.249434	−	0.968392i	$-0.419755\pi$
$293$	2502.00	0.498868	0.249434	+	0.968392i	$0.419755\pi$
$294$	0	0
$295$	1080.00	0.213153
$296$	0	0
$297$	1728.00	0.337605
$298$	0	0
$299$	−720.000	−0.139260
$300$	0	0
$301$	−4736.00	−0.906905
$302$	0	0
$303$	2838.00	0.538082
$304$	0	0
$305$	4030.00	0.756581
$306$	0	0
$307$	6404.00	1.19054	0.595270	−	0.803526i	$-0.297045\pi$
$307$	6404.00	1.19054	0.595270	+	0.803526i	$0.297045\pi$
$308$	0	0
$309$	1272.00	0.234180
$310$	0	0
$311$	−896.000	−0.163368	−0.0816841	−	0.996658i	$-0.526030\pi$
$311$	−896.000	−0.163368	−0.0816841	+	0.996658i	$0.526030\pi$
$312$	0	0
$313$	−4110.00	−0.742207	−0.371104	−	0.928591i	$-0.621021\pi$
$313$	−4110.00	−0.742207	−0.371104	+	0.928591i	$0.621021\pi$
$314$	0	0
$315$	−1440.00	−0.257571
$316$	0	0
$317$	6926.00	1.22714	0.613569	−	0.789641i	$-0.289733\pi$
$317$	6926.00	1.22714	0.613569	+	0.789641i	$0.289733\pi$
$318$	0	0
$319$	7808.00	1.37042
$320$	0	0
$321$	−5004.00	−0.870081
$322$	0	0
$323$	4408.00	0.759343
$324$	0	0
$325$	−150.000	−0.0256015
$326$	0	0
$327$	4902.00	0.828995
$328$	0	0
$329$	−5888.00	−0.986675
$330$	0	0
$331$	2692.00	0.447026	0.223513	−	0.974701i	$-0.428247\pi$
$331$	2692.00	0.447026	0.223513	+	0.974701i	$0.428247\pi$
$332$	0	0
$333$	1314.00	0.216237
$334$	0	0
$335$	3660.00	0.596917
$336$	0	0
$337$	11914.0	1.92581	0.962903	−	0.269846i	$-0.0869728\pi$
$337$	11914.0	1.92581	0.962903	+	0.269846i	$0.0869728\pi$
$338$	0	0
$339$	−5058.00	−0.810362
$340$	0	0
$341$	10496.0	1.66683
$342$	0	0
$343$	−10816.0	−1.70265
$344$	0	0
$345$	−1800.00	−0.280895
$346$	0	0
$347$	6660.00	1.03034	0.515169	−	0.857088i	$-0.327729\pi$
$347$	6660.00	1.03034	0.515169	+	0.857088i	$0.327729\pi$
$348$	0	0
$349$	3046.00	0.467188	0.233594	−	0.972334i	$-0.424951\pi$
$349$	3046.00	0.467188	0.233594	+	0.972334i	$0.424951\pi$
$350$	0	0
$351$	162.000	0.0246351
$352$	0	0
$353$	−3522.00	−0.531040	−0.265520	−	0.964105i	$-0.585544\pi$
$353$	−3522.00	−0.531040	−0.265520	+	0.964105i	$0.585544\pi$
$354$	0	0
$355$	−1320.00	−0.197347
$356$	0	0
$357$	3648.00	0.540820
$358$	0	0
$359$	8656.00	1.27255	0.636276	−	0.771461i	$-0.280474\pi$
$359$	8656.00	1.27255	0.636276	+	0.771461i	$0.280474\pi$
$360$	0	0
$361$	6597.00	0.961802
$362$	0	0
$363$	−8295.00	−1.19938
$364$	0	0
$365$	−3190.00	−0.457458
$366$	0	0
$367$	−936.000	−0.133130	−0.0665651	−	0.997782i	$-0.521204\pi$
$367$	−936.000	−0.133130	−0.0665651	+	0.997782i	$0.521204\pi$
$368$	0	0
$369$	−2142.00	−0.302190
$370$	0	0
$371$	−15040.0	−2.10468
$372$	0	0
$373$	11578.0	1.60720	0.803601	−	0.595169i	$-0.202915\pi$
$373$	11578.0	1.60720	0.803601	+	0.595169i	$0.202915\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	732.000	0.0999998
$378$	0	0
$379$	−9948.00	−1.34827	−0.674135	−	0.738608i	$-0.735483\pi$
$379$	−9948.00	−1.34827	−0.674135	+	0.738608i	$0.735483\pi$
$380$	0	0
$381$	7920.00	1.06497
$382$	0	0
$383$	−8336.00	−1.11214	−0.556070	−	0.831135i	$-0.687691\pi$
$383$	−8336.00	−1.11214	−0.556070	+	0.831135i	$0.687691\pi$
$384$	0	0
$385$	10240.0	1.35553
$386$	0	0
$387$	1332.00	0.174960
$388$	0	0
$389$	−6370.00	−0.830262	−0.415131	−	0.909762i	$-0.636264\pi$
$389$	−6370.00	−0.830262	−0.415131	+	0.909762i	$0.636264\pi$
$390$	0	0
$391$	4560.00	0.589793
$392$	0	0
$393$	−7608.00	−0.976521
$394$	0	0
$395$	−2980.00	−0.379595
$396$	0	0
$397$	10394.0	1.31400	0.657002	−	0.753888i	$-0.271824\pi$
$397$	10394.0	1.31400	0.657002	+	0.753888i	$0.271824\pi$
$398$	0	0
$399$	11136.0	1.39724
$400$	0	0
$401$	−7470.00	−0.930259	−0.465130	−	0.885243i	$-0.653992\pi$
$401$	−7470.00	−0.930259	−0.465130	+	0.885243i	$0.653992\pi$
$402$	0	0
$403$	984.000	0.121629
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−9344.00	−1.13800
$408$	0	0
$409$	2810.00	0.339720	0.169860	−	0.985468i	$-0.445668\pi$
$409$	2810.00	0.339720	0.169860	+	0.985468i	$0.445668\pi$
$410$	0	0
$411$	1902.00	0.228269
$412$	0	0
$413$	−6912.00	−0.823529
$414$	0	0
$415$	4420.00	0.522818
$416$	0	0
$417$	8940.00	1.04986
$418$	0	0
$419$	4320.00	0.503689	0.251845	−	0.967768i	$-0.418963\pi$
$419$	4320.00	0.503689	0.251845	+	0.967768i	$0.418963\pi$
$420$	0	0
$421$	−15122.0	−1.75060	−0.875298	−	0.483583i	$-0.839335\pi$
$421$	−15122.0	−1.75060	−0.875298	+	0.483583i	$0.839335\pi$
$422$	0	0
$423$	1656.00	0.190349
$424$	0	0
$425$	950.000	0.108428
$426$	0	0
$427$	−25792.0	−2.92310
$428$	0	0
$429$	−1152.00	−0.129648
$430$	0	0
$431$	12616.0	1.40996	0.704978	−	0.709229i	$-0.250957\pi$
$431$	12616.0	1.40996	0.704978	+	0.709229i	$0.250957\pi$
$432$	0	0
$433$	15098.0	1.67567	0.837833	−	0.545926i	$-0.183822\pi$
$433$	15098.0	1.67567	0.837833	+	0.545926i	$0.183822\pi$
$434$	0	0
$435$	1830.00	0.201705
$436$	0	0
$437$	13920.0	1.52376
$438$	0	0
$439$	2372.00	0.257880	0.128940	−	0.991652i	$-0.458843\pi$
$439$	2372.00	0.257880	0.128940	+	0.991652i	$0.458843\pi$
$440$	0	0
$441$	6129.00	0.661808
$442$	0	0
$443$	36.0000	0.00386097	0.00193049	−	0.999998i	$-0.499386\pi$
$443$	36.0000	0.00386097	0.00193049	+	0.999998i	$0.499386\pi$
$444$	0	0
$445$	4650.00	0.495351
$446$	0	0
$447$	1014.00	0.107294
$448$	0	0
$449$	7330.00	0.770432	0.385216	−	0.922826i	$-0.374127\pi$
$449$	7330.00	0.770432	0.385216	+	0.922826i	$0.374127\pi$
$450$	0	0
$451$	15232.0	1.59035
$452$	0	0
$453$	−1284.00	−0.133173
$454$	0	0
$455$	960.000	0.0989132
$456$	0	0
$457$	9642.00	0.986945	0.493472	−	0.869761i	$-0.335727\pi$
$457$	9642.00	0.986945	0.493472	+	0.869761i	$0.335727\pi$
$458$	0	0
$459$	−1026.00	−0.104335
$460$	0	0
$461$	9654.00	0.975340	0.487670	−	0.873028i	$-0.337847\pi$
$461$	9654.00	0.975340	0.487670	+	0.873028i	$0.337847\pi$
$462$	0	0
$463$	13960.0	1.40124	0.700622	−	0.713532i	$-0.252906\pi$
$463$	13960.0	1.40124	0.700622	+	0.713532i	$0.252906\pi$
$464$	0	0
$465$	2460.00	0.245333
$466$	0	0
$467$	−9996.00	−0.990492	−0.495246	−	0.868753i	$-0.664922\pi$
$467$	−9996.00	−0.990492	−0.495246	+	0.868753i	$0.664922\pi$
$468$	0	0
$469$	−23424.0	−2.30623
$470$	0	0
$471$	−9030.00	−0.883398
$472$	0	0
$473$	−9472.00	−0.920767
$474$	0	0
$475$	2900.00	0.280129
$476$	0	0
$477$	4230.00	0.406034
$478$	0	0
$479$	8736.00	0.833315	0.416658	−	0.909063i	$-0.363201\pi$
$479$	8736.00	0.833315	0.416658	+	0.909063i	$0.363201\pi$
$480$	0	0
$481$	−876.000	−0.0830398
$482$	0	0
$483$	11520.0	1.08525
$484$	0	0
$485$	1610.00	0.150735
$486$	0	0
$487$	−6712.00	−0.624537	−0.312269	−	0.949994i	$-0.601089\pi$
$487$	−6712.00	−0.624537	−0.312269	+	0.949994i	$0.601089\pi$
$488$	0	0
$489$	−396.000	−0.0366211
$490$	0	0
$491$	2512.00	0.230886	0.115443	−	0.993314i	$-0.463171\pi$
$491$	2512.00	0.230886	0.115443	+	0.993314i	$0.463171\pi$
$492$	0	0
$493$	−4636.00	−0.423519
$494$	0	0
$495$	−2880.00	−0.261508
$496$	0	0
$497$	8448.00	0.762464
$498$	0	0
$499$	5708.00	0.512074	0.256037	−	0.966667i	$-0.417583\pi$
$499$	5708.00	0.512074	0.256037	+	0.966667i	$0.417583\pi$
$500$	0	0
$501$	6528.00	0.582135
$502$	0	0
$503$	5440.00	0.482222	0.241111	−	0.970498i	$-0.422488\pi$
$503$	5440.00	0.482222	0.241111	+	0.970498i	$0.422488\pi$
$504$	0	0
$505$	−4730.00	−0.416797
$506$	0	0
$507$	6483.00	0.567890
$508$	0	0
$509$	3942.00	0.343273	0.171637	−	0.985160i	$-0.445095\pi$
$509$	3942.00	0.343273	0.171637	+	0.985160i	$0.445095\pi$
$510$	0	0
$511$	20416.0	1.76742
$512$	0	0
$513$	−3132.00	−0.269554
$514$	0	0
$515$	−2120.00	−0.181395
$516$	0	0
$517$	−11776.0	−1.00176
$518$	0	0
$519$	8334.00	0.704859
$520$	0	0
$521$	−2310.00	−0.194247	−0.0971237	−	0.995272i	$-0.530964\pi$
$521$	−2310.00	−0.194247	−0.0971237	+	0.995272i	$0.530964\pi$
$522$	0	0
$523$	2956.00	0.247145	0.123573	−	0.992336i	$-0.460565\pi$
$523$	2956.00	0.247145	0.123573	+	0.992336i	$0.460565\pi$
$524$	0	0
$525$	2400.00	0.199513
$526$	0	0
$527$	−6232.00	−0.515124
$528$	0	0
$529$	2233.00	0.183529
$530$	0	0
$531$	1944.00	0.158875
$532$	0	0
$533$	1428.00	0.116048
$534$	0	0
$535$	8340.00	0.673962
$536$	0	0
$537$	−11664.0	−0.937316
$538$	0	0
$539$	−43584.0	−3.48292
$540$	0	0
$541$	2078.00	0.165139	0.0825695	−	0.996585i	$-0.473687\pi$
$541$	2078.00	0.165139	0.0825695	+	0.996585i	$0.473687\pi$
$542$	0	0
$543$	−4050.00	−0.320078
$544$	0	0
$545$	−8170.00	−0.642136
$546$	0	0
$547$	−6164.00	−0.481816	−0.240908	−	0.970548i	$-0.577445\pi$
$547$	−6164.00	−0.481816	−0.240908	+	0.970548i	$0.577445\pi$
$548$	0	0
$549$	7254.00	0.563922
$550$	0	0
$551$	−14152.0	−1.09418
$552$	0	0
$553$	19072.0	1.46659
$554$	0	0
$555$	−2190.00	−0.167496
$556$	0	0
$557$	13526.0	1.02893	0.514466	−	0.857511i	$-0.327990\pi$
$557$	13526.0	1.02893	0.514466	+	0.857511i	$0.327990\pi$
$558$	0	0
$559$	−888.000	−0.0671885
$560$	0	0
$561$	7296.00	0.549086
$562$	0	0
$563$	−276.000	−0.0206608	−0.0103304	−	0.999947i	$-0.503288\pi$
$563$	−276.000	−0.0206608	−0.0103304	+	0.999947i	$0.503288\pi$
$564$	0	0
$565$	8430.00	0.627704
$566$	0	0
$567$	−2592.00	−0.191982
$568$	0	0
$569$	−774.000	−0.0570260	−0.0285130	−	0.999593i	$-0.509077\pi$
$569$	−774.000	−0.0570260	−0.0285130	+	0.999593i	$0.509077\pi$
$570$	0	0
$571$	6676.00	0.489285	0.244643	−	0.969613i	$-0.421329\pi$
$571$	6676.00	0.489285	0.244643	+	0.969613i	$0.421329\pi$
$572$	0	0
$573$	−14280.0	−1.04111
$574$	0	0
$575$	3000.00	0.217580
$576$	0	0
$577$	−11774.0	−0.849494	−0.424747	−	0.905312i	$-0.639637\pi$
$577$	−11774.0	−0.849494	−0.424747	+	0.905312i	$0.639637\pi$
$578$	0	0
$579$	−3102.00	−0.222651
$580$	0	0
$581$	−28288.0	−2.01994
$582$	0	0
$583$	−30080.0	−2.13685
$584$	0	0
$585$	−270.000	−0.0190823
$586$	0	0
$587$	12292.0	0.864302	0.432151	−	0.901801i	$-0.357755\pi$
$587$	12292.0	0.864302	0.432151	+	0.901801i	$0.357755\pi$
$588$	0	0
$589$	−19024.0	−1.33085
$590$	0	0
$591$	4062.00	0.282721
$592$	0	0
$593$	7110.00	0.492365	0.246183	−	0.969223i	$-0.420824\pi$
$593$	7110.00	0.492365	0.246183	+	0.969223i	$0.420824\pi$
$594$	0	0
$595$	−6080.00	−0.418917
$596$	0	0
$597$	−6972.00	−0.477965
$598$	0	0
$599$	−4416.00	−0.301223	−0.150612	−	0.988593i	$-0.548124\pi$
$599$	−4416.00	−0.301223	−0.150612	+	0.988593i	$0.548124\pi$
$600$	0	0
$601$	3850.00	0.261306	0.130653	−	0.991428i	$-0.458293\pi$
$601$	3850.00	0.261306	0.130653	+	0.991428i	$0.458293\pi$
$602$	0	0
$603$	6588.00	0.444916
$604$	0	0
$605$	13825.0	0.929035
$606$	0	0
$607$	21880.0	1.46307	0.731534	−	0.681805i	$-0.238805\pi$
$607$	21880.0	1.46307	0.731534	+	0.681805i	$0.238805\pi$
$608$	0	0
$609$	−11712.0	−0.779301
$610$	0	0
$611$	−1104.00	−0.0730983
$612$	0	0
$613$	2786.00	0.183565	0.0917826	−	0.995779i	$-0.470744\pi$
$613$	2786.00	0.183565	0.0917826	+	0.995779i	$0.470744\pi$
$614$	0	0
$615$	3570.00	0.234075
$616$	0	0
$617$	1014.00	0.0661622	0.0330811	−	0.999453i	$-0.489468\pi$
$617$	1014.00	0.0661622	0.0330811	+	0.999453i	$0.489468\pi$
$618$	0	0
$619$	−10708.0	−0.695300	−0.347650	−	0.937624i	$-0.613020\pi$
$619$	−10708.0	−0.695300	−0.347650	+	0.937624i	$0.613020\pi$
$620$	0	0
$621$	−3240.00	−0.209367
$622$	0	0
$623$	−29760.0	−1.91382
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	22272.0	1.41859
$628$	0	0
$629$	5548.00	0.351690
$630$	0	0
$631$	6660.00	0.420175	0.210087	−	0.977683i	$-0.432625\pi$
$631$	6660.00	0.420175	0.210087	+	0.977683i	$0.432625\pi$
$632$	0	0
$633$	−9660.00	−0.606557
$634$	0	0
$635$	−13200.0	−0.824923
$636$	0	0
$637$	−4086.00	−0.254149
$638$	0	0
$639$	−2376.00	−0.147094
$640$	0	0
$641$	90.0000	0.00554569	0.00277284	−	0.999996i	$-0.499117\pi$
$641$	90.0000	0.00554569	0.00277284	+	0.999996i	$0.499117\pi$
$642$	0	0
$643$	21684.0	1.32991	0.664956	−	0.746882i	$-0.268450\pi$
$643$	21684.0	1.32991	0.664956	+	0.746882i	$0.268450\pi$
$644$	0	0
$645$	−2220.00	−0.135523
$646$	0	0
$647$	1344.00	0.0816663	0.0408331	−	0.999166i	$-0.486999\pi$
$647$	1344.00	0.0816663	0.0408331	+	0.999166i	$0.486999\pi$
$648$	0	0
$649$	−13824.0	−0.836116
$650$	0	0
$651$	−15744.0	−0.947859
$652$	0	0
$653$	−9210.00	−0.551937	−0.275969	−	0.961167i	$-0.588999\pi$
$653$	−9210.00	−0.551937	−0.275969	+	0.961167i	$0.588999\pi$
$654$	0	0
$655$	12680.0	0.756410
$656$	0	0
$657$	−5742.00	−0.340969
$658$	0	0
$659$	−184.000	−0.0108765	−0.00543826	−	0.999985i	$-0.501731\pi$
$659$	−184.000	−0.0108765	−0.00543826	+	0.999985i	$0.501731\pi$
$660$	0	0
$661$	−12866.0	−0.757079	−0.378540	−	0.925585i	$-0.623574\pi$
$661$	−12866.0	−0.757079	−0.378540	+	0.925585i	$0.623574\pi$
$662$	0	0
$663$	684.000	0.0400669
$664$	0	0
$665$	−18560.0	−1.08229
$666$	0	0
$667$	−14640.0	−0.849870
$668$	0	0
$669$	−96.0000	−0.00554794
$670$	0	0
$671$	−51584.0	−2.96778
$672$	0	0
$673$	−26534.0	−1.51978	−0.759889	−	0.650053i	$-0.774747\pi$
$673$	−26534.0	−1.51978	−0.759889	+	0.650053i	$0.774747\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	30062.0	1.70661	0.853306	−	0.521410i	$-0.174594\pi$
$677$	30062.0	1.70661	0.853306	+	0.521410i	$0.174594\pi$
$678$	0	0
$679$	−10304.0	−0.582373
$680$	0	0
$681$	−11988.0	−0.674569
$682$	0	0
$683$	−12724.0	−0.712841	−0.356420	−	0.934326i	$-0.616003\pi$
$683$	−12724.0	−0.712841	−0.356420	+	0.934326i	$0.616003\pi$
$684$	0	0
$685$	−3170.00	−0.176817
$686$	0	0
$687$	9030.00	0.501479
$688$	0	0
$689$	−2820.00	−0.155927
$690$	0	0
$691$	21972.0	1.20963	0.604815	−	0.796366i	$-0.293247\pi$
$691$	21972.0	1.20963	0.604815	+	0.796366i	$0.293247\pi$
$692$	0	0
$693$	18432.0	1.01035
$694$	0	0
$695$	−14900.0	−0.813222
$696$	0	0
$697$	−9044.00	−0.491486
$698$	0	0
$699$	8094.00	0.437973
$700$	0	0
$701$	−15642.0	−0.842782	−0.421391	−	0.906879i	$-0.638458\pi$
$701$	−15642.0	−0.842782	−0.421391	+	0.906879i	$0.638458\pi$
$702$	0	0
$703$	16936.0	0.908611
$704$	0	0
$705$	−2760.00	−0.147443
$706$	0	0
$707$	30272.0	1.61032
$708$	0	0
$709$	36398.0	1.92801	0.964003	−	0.265893i	$-0.0856668\pi$
$709$	36398.0	1.92801	0.964003	+	0.265893i	$0.0856668\pi$
$710$	0	0
$711$	−5364.00	−0.282933
$712$	0	0
$713$	−19680.0	−1.03369
$714$	0	0
$715$	1920.00	0.100425
$716$	0	0
$717$	−6600.00	−0.343768
$718$	0	0
$719$	36248.0	1.88014	0.940071	−	0.340978i	$-0.110758\pi$
$719$	36248.0	1.88014	0.940071	+	0.340978i	$0.110758\pi$
$720$	0	0
$721$	13568.0	0.700830
$722$	0	0
$723$	18522.0	0.952753
$724$	0	0
$725$	−3050.00	−0.156240
$726$	0	0
$727$	4936.00	0.251810	0.125905	−	0.992042i	$-0.459816\pi$
$727$	4936.00	0.251810	0.125905	+	0.992042i	$0.459816\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	5624.00	0.284557
$732$	0	0
$733$	−23838.0	−1.20120	−0.600598	−	0.799551i	$-0.705071\pi$
$733$	−23838.0	−1.20120	−0.600598	+	0.799551i	$0.705071\pi$
$734$	0	0
$735$	−10215.0	−0.512634
$736$	0	0
$737$	−46848.0	−2.34148
$738$	0	0
$739$	−31700.0	−1.57795	−0.788974	−	0.614427i	$-0.789387\pi$
$739$	−31700.0	−1.57795	−0.788974	+	0.614427i	$0.789387\pi$
$740$	0	0
$741$	2088.00	0.103515
$742$	0	0
$743$	13128.0	0.648209	0.324105	−	0.946021i	$-0.394937\pi$
$743$	13128.0	0.648209	0.324105	+	0.946021i	$0.394937\pi$
$744$	0	0
$745$	−1690.00	−0.0831098
$746$	0	0
$747$	7956.00	0.389685
$748$	0	0
$749$	−53376.0	−2.60389
$750$	0	0
$751$	15676.0	0.761685	0.380842	−	0.924640i	$-0.375634\pi$
$751$	15676.0	0.761685	0.380842	+	0.924640i	$0.375634\pi$
$752$	0	0
$753$	14424.0	0.698061
$754$	0	0
$755$	2140.00	0.103156
$756$	0	0
$757$	−13238.0	−0.635592	−0.317796	−	0.948159i	$-0.602943\pi$
$757$	−13238.0	−0.635592	−0.317796	+	0.948159i	$0.602943\pi$
$758$	0	0
$759$	23040.0	1.10184
$760$	0	0
$761$	−5270.00	−0.251035	−0.125517	−	0.992091i	$-0.540059\pi$
$761$	−5270.00	−0.251035	−0.125517	+	0.992091i	$0.540059\pi$
$762$	0	0
$763$	52288.0	2.48093
$764$	0	0
$765$	1710.00	0.0808172
$766$	0	0
$767$	−1296.00	−0.0610115
$768$	0	0
$769$	−8526.00	−0.399812	−0.199906	−	0.979815i	$-0.564064\pi$
$769$	−8526.00	−0.399812	−0.199906	+	0.979815i	$0.564064\pi$
$770$	0	0
$771$	−11922.0	−0.556888
$772$	0	0
$773$	13606.0	0.633084	0.316542	−	0.948579i	$-0.397478\pi$
$773$	13606.0	0.633084	0.316542	+	0.948579i	$0.397478\pi$
$774$	0	0
$775$	−4100.00	−0.190034
$776$	0	0
$777$	14016.0	0.647132
$778$	0	0
$779$	−27608.0	−1.26978
$780$	0	0
$781$	16896.0	0.774118
$782$	0	0
$783$	3294.00	0.150342
$784$	0	0
$785$	15050.0	0.684277
$786$	0	0
$787$	8836.00	0.400215	0.200108	−	0.979774i	$-0.435871\pi$
$787$	8836.00	0.400215	0.200108	+	0.979774i	$0.435871\pi$
$788$	0	0
$789$	4224.00	0.190594
$790$	0	0
$791$	−53952.0	−2.42517
$792$	0	0
$793$	−4836.00	−0.216559
$794$	0	0
$795$	−7050.00	−0.314513
$796$	0	0
$797$	−29082.0	−1.29252	−0.646259	−	0.763118i	$-0.723668\pi$
$797$	−29082.0	−1.29252	−0.646259	+	0.763118i	$0.723668\pi$
$798$	0	0
$799$	6992.00	0.309586
$800$	0	0
$801$	8370.00	0.369213
$802$	0	0
$803$	40832.0	1.79443
$804$	0	0
$805$	−19200.0	−0.840635
$806$	0	0
$807$	9774.00	0.426346
$808$	0	0
$809$	26994.0	1.17313	0.586563	−	0.809904i	$-0.300481\pi$
$809$	26994.0	1.17313	0.586563	+	0.809904i	$0.300481\pi$
$810$	0	0
$811$	9268.00	0.401287	0.200643	−	0.979664i	$-0.435697\pi$
$811$	9268.00	0.401287	0.200643	+	0.979664i	$0.435697\pi$
$812$	0	0
$813$	25836.0	1.11452
$814$	0	0
$815$	660.000	0.0283666
$816$	0	0
$817$	17168.0	0.735168
$818$	0	0
$819$	1728.00	0.0737255
$820$	0	0
$821$	6286.00	0.267214	0.133607	−	0.991034i	$-0.457344\pi$
$821$	6286.00	0.267214	0.133607	+	0.991034i	$0.457344\pi$
$822$	0	0
$823$	−44088.0	−1.86733	−0.933664	−	0.358150i	$-0.883408\pi$
$823$	−44088.0	−1.86733	−0.933664	+	0.358150i	$0.883408\pi$
$824$	0	0
$825$	4800.00	0.202563
$826$	0	0
$827$	−30100.0	−1.26563	−0.632817	−	0.774301i	$-0.718102\pi$
$827$	−30100.0	−1.26563	−0.632817	+	0.774301i	$0.718102\pi$
$828$	0	0
$829$	18254.0	0.764762	0.382381	−	0.924005i	$-0.375104\pi$
$829$	18254.0	0.764762	0.382381	+	0.924005i	$0.375104\pi$
$830$	0	0
$831$	12018.0	0.501684
$832$	0	0
$833$	25878.0	1.07637
$834$	0	0
$835$	−10880.0	−0.450920
$836$	0	0
$837$	4428.00	0.182860
$838$	0	0
$839$	−13008.0	−0.535263	−0.267632	−	0.963521i	$-0.586241\pi$
$839$	−13008.0	−0.535263	−0.267632	+	0.963521i	$0.586241\pi$
$840$	0	0
$841$	−9505.00	−0.389725
$842$	0	0
$843$	−18582.0	−0.759191
$844$	0	0
$845$	−10805.0	−0.439886
$846$	0	0
$847$	−88480.0	−3.58938
$848$	0	0
$849$	5172.00	0.209073
$850$	0	0
$851$	17520.0	0.705732
$852$	0	0
$853$	−16094.0	−0.646012	−0.323006	−	0.946397i	$-0.604693\pi$
$853$	−16094.0	−0.646012	−0.323006	+	0.946397i	$0.604693\pi$
$854$	0	0
$855$	5220.00	0.208796
$856$	0	0
$857$	−25938.0	−1.03387	−0.516934	−	0.856025i	$-0.672927\pi$
$857$	−25938.0	−1.03387	−0.516934	+	0.856025i	$0.672927\pi$
$858$	0	0
$859$	38564.0	1.53177	0.765883	−	0.642980i	$-0.222302\pi$
$859$	38564.0	1.53177	0.765883	+	0.642980i	$0.222302\pi$
$860$	0	0
$861$	−22848.0	−0.904364
$862$	0	0
$863$	−10216.0	−0.402963	−0.201481	−	0.979492i	$-0.564576\pi$
$863$	−10216.0	−0.402963	−0.201481	+	0.979492i	$0.564576\pi$
$864$	0	0
$865$	−13890.0	−0.545982
$866$	0	0
$867$	10407.0	0.407659
$868$	0	0
$869$	38144.0	1.48901
$870$	0	0
$871$	−4392.00	−0.170858
$872$	0	0
$873$	2898.00	0.112351
$874$	0	0
$875$	−4000.00	−0.154542
$876$	0	0
$877$	−17006.0	−0.654791	−0.327396	−	0.944887i	$-0.606171\pi$
$877$	−17006.0	−0.654791	−0.327396	+	0.944887i	$0.606171\pi$
$878$	0	0
$879$	−7506.00	−0.288022
$880$	0	0
$881$	43898.0	1.67873	0.839365	−	0.543568i	$-0.182927\pi$
$881$	43898.0	1.67873	0.839365	+	0.543568i	$0.182927\pi$
$882$	0	0
$883$	−29180.0	−1.11210	−0.556051	−	0.831149i	$-0.687684\pi$
$883$	−29180.0	−1.11210	−0.556051	+	0.831149i	$0.687684\pi$
$884$	0	0
$885$	−3240.00	−0.123064
$886$	0	0
$887$	14752.0	0.558426	0.279213	−	0.960229i	$-0.409926\pi$
$887$	14752.0	0.558426	0.279213	+	0.960229i	$0.409926\pi$
$888$	0	0
$889$	84480.0	3.18714
$890$	0	0
$891$	−5184.00	−0.194916
$892$	0	0
$893$	21344.0	0.799832
$894$	0	0
$895$	19440.0	0.726042
$896$	0	0
$897$	2160.00	0.0804017
$898$	0	0
$899$	20008.0	0.742274
$900$	0	0
$901$	17860.0	0.660381
$902$	0	0
$903$	14208.0	0.523602
$904$	0	0
$905$	6750.00	0.247931
$906$	0	0
$907$	12020.0	0.440041	0.220021	−	0.975495i	$-0.429388\pi$
$907$	12020.0	0.440041	0.220021	+	0.975495i	$0.429388\pi$
$908$	0	0
$909$	−8514.00	−0.310662
$910$	0	0
$911$	18560.0	0.674995	0.337497	−	0.941326i	$-0.390420\pi$
$911$	18560.0	0.674995	0.337497	+	0.941326i	$0.390420\pi$
$912$	0	0
$913$	−56576.0	−2.05081
$914$	0	0
$915$	−12090.0	−0.436812
$916$	0	0
$917$	−81152.0	−2.92244
$918$	0	0
$919$	10244.0	0.367702	0.183851	−	0.982954i	$-0.441144\pi$
$919$	10244.0	0.367702	0.183851	+	0.982954i	$0.441144\pi$
$920$	0	0
$921$	−19212.0	−0.687358
$922$	0	0
$923$	1584.00	0.0564875
$924$	0	0
$925$	3650.00	0.129742
$926$	0	0
$927$	−3816.00	−0.135204
$928$	0	0
$929$	1266.00	0.0447106	0.0223553	−	0.999750i	$-0.492884\pi$
$929$	1266.00	0.0447106	0.0223553	+	0.999750i	$0.492884\pi$
$930$	0	0
$931$	78996.0	2.78087
$932$	0	0
$933$	2688.00	0.0943207
$934$	0	0
$935$	−12160.0	−0.425320
$936$	0	0
$937$	47626.0	1.66048	0.830242	−	0.557403i	$-0.188202\pi$
$937$	47626.0	1.66048	0.830242	+	0.557403i	$0.188202\pi$
$938$	0	0
$939$	12330.0	0.428514
$940$	0	0
$941$	31958.0	1.10712	0.553561	−	0.832809i	$-0.313269\pi$
$941$	31958.0	1.10712	0.553561	+	0.832809i	$0.313269\pi$
$942$	0	0
$943$	−28560.0	−0.986258
$944$	0	0
$945$	4320.00	0.148709
$946$	0	0
$947$	−25196.0	−0.864583	−0.432291	−	0.901734i	$-0.642295\pi$
$947$	−25196.0	−0.864583	−0.432291	+	0.901734i	$0.642295\pi$
$948$	0	0
$949$	3828.00	0.130940
$950$	0	0
$951$	−20778.0	−0.708489
$952$	0	0
$953$	51574.0	1.75304	0.876519	−	0.481367i	$-0.159859\pi$
$953$	51574.0	1.75304	0.876519	+	0.481367i	$0.159859\pi$
$954$	0	0
$955$	23800.0	0.806440
$956$	0	0
$957$	−23424.0	−0.791213
$958$	0	0
$959$	20288.0	0.683143
$960$	0	0
$961$	−2895.00	−0.0971770
$962$	0	0
$963$	15012.0	0.502342
$964$	0	0
$965$	5170.00	0.172464
$966$	0	0
$967$	−48296.0	−1.60610	−0.803048	−	0.595914i	$-0.796790\pi$
$967$	−48296.0	−1.60610	−0.803048	+	0.595914i	$0.796790\pi$
$968$	0	0
$969$	−13224.0	−0.438407
$970$	0	0
$971$	27288.0	0.901868	0.450934	−	0.892557i	$-0.351091\pi$
$971$	27288.0	0.901868	0.450934	+	0.892557i	$0.351091\pi$
$972$	0	0
$973$	95360.0	3.14193
$974$	0	0
$975$	450.000	0.0147811
$976$	0	0
$977$	14406.0	0.471739	0.235869	−	0.971785i	$-0.424206\pi$
$977$	14406.0	0.471739	0.235869	+	0.971785i	$0.424206\pi$
$978$	0	0
$979$	−59520.0	−1.94307
$980$	0	0
$981$	−14706.0	−0.478620
$982$	0	0
$983$	−18952.0	−0.614929	−0.307464	−	0.951560i	$-0.599480\pi$
$983$	−18952.0	−0.614929	−0.307464	+	0.951560i	$0.599480\pi$
$984$	0	0
$985$	−6770.00	−0.218995
$986$	0	0
$987$	17664.0	0.569657
$988$	0	0
$989$	17760.0	0.571016
$990$	0	0
$991$	44308.0	1.42027	0.710136	−	0.704064i	$-0.248633\pi$
$991$	44308.0	1.42027	0.710136	+	0.704064i	$0.248633\pi$
$992$	0	0
$993$	−8076.00	−0.258091
$994$	0	0
$995$	11620.0	0.370230
$996$	0	0
$997$	954.000	0.0303044	0.0151522	−	0.999885i	$-0.495177\pi$
$997$	954.000	0.0303044	0.0151522	+	0.999885i	$0.495177\pi$
$998$	0	0
$999$	−3942.00	−0.124844

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.c.1.1	✓	1
3.2	odd	2		1440.4.a.a.1.1		1
4.3	odd	2		480.4.a.l.1.1	yes	1
5.4	even	2		2400.4.a.v.1.1		1
8.3	odd	2		960.4.a.i.1.1		1
8.5	even	2		960.4.a.t.1.1		1
12.11	even	2		1440.4.a.j.1.1		1
20.19	odd	2		2400.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.c.1.1	✓	1	1.1	even	1	trivial
480.4.a.l.1.1	yes	1	4.3	odd	2
960.4.a.i.1.1		1	8.3	odd	2
960.4.a.t.1.1		1	8.5	even	2
1440.4.a.a.1.1		1	3.2	odd	2
1440.4.a.j.1.1		1	12.11	even	2
2400.4.a.a.1.1		1	20.19	odd	2
2400.4.a.v.1.1		1	5.4	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)

Introduction

L-functions

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Fields

Representations

Groups

Database

Properties

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