LMFDB - Embedded newform 60.4.d.a.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [60,4,Mod(49,60)] 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("60.49"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	60.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$3.54011460034$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	60.49
Dual form			60.4.d.a.49.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} -22.0000i q^{7} -9.00000 q^{9} -14.0000 q^{11} -30.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} -62.0000i q^{17} +120.000 q^{19} -66.0000 q^{21} +188.000i q^{23} +(75.0000 + 100.000i) q^{25} +27.0000i q^{27} -96.0000 q^{29} +184.000 q^{31} +42.0000i q^{33} +(-110.000 + 220.000i) q^{35} -406.000i q^{37} -90.0000 q^{39} +130.000 q^{41} +148.000i q^{43} +(90.0000 + 45.0000i) q^{45} -448.000i q^{47} -141.000 q^{49} -186.000 q^{51} -414.000i q^{53} +(140.000 + 70.0000i) q^{55} -360.000i q^{57} -266.000 q^{59} -838.000 q^{61} +198.000i q^{63} +(-150.000 + 300.000i) q^{65} -248.000i q^{67} +564.000 q^{69} +1020.00 q^{71} +484.000i q^{73} +(300.000 - 225.000i) q^{75} +308.000i q^{77} +48.0000 q^{79} +81.0000 q^{81} +548.000i q^{83} +(-310.000 + 620.000i) q^{85} +288.000i q^{87} +650.000 q^{89} -660.000 q^{91} -552.000i q^{93} +(-1200.00 - 600.000i) q^{95} +1816.00i q^{97} +126.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{5} - 18 q^{9} - 28 q^{11} - 30 q^{15} + 240 q^{19} - 132 q^{21} + 150 q^{25} - 192 q^{29} + 368 q^{31} - 220 q^{35} - 180 q^{39} + 260 q^{41} + 180 q^{45} - 282 q^{49} - 372 q^{51} + 280 q^{55}+ \cdots + 252 q^{99}+O(q^{100}) $ 2 * q - 20 * q^5 - 18 * q^9 - 28 * q^11 - 30 * q^15 + 240 * q^19 - 132 * q^21 + 150 * q^25 - 192 * q^29 + 368 * q^31 - 220 * q^35 - 180 * q^39 + 260 * q^41 + 180 * q^45 - 282 * q^49 - 372 * q^51 + 280 * q^55 - 532 * q^59 - 1676 * q^61 - 300 * q^65 + 1128 * q^69 + 2040 * q^71 + 600 * q^75 + 96 * q^79 + 162 * q^81 - 620 * q^85 + 1300 * q^89 - 1320 * q^91 - 2400 * q^95 + 252 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$.

$n$	$31$	$37$	$41$
$\chi(n)$	$1$	$-1$	$1$

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
$3$		−	3.00000i		−	0.577350i
$3$		−	3.00000i		−	0.577350i
$4$		0			0
$5$	−10.0000	−	5.00000i	−0.894427	−	0.447214i
$5$	−10.0000	−	5.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$		−	22.0000i		−	1.18789i	−0.804506	−	0.593944i	$-0.797570\pi$
$7$		−	22.0000i		−	1.18789i	0.804506	−	0.593944i	$-0.202430\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	−14.0000			−0.383742			−0.191871	−	0.981420i	$-0.561455\pi$
$11$	−14.0000			−0.383742			−0.191871	+	0.981420i	$0.561455\pi$
$12$		0			0
$13$		−	30.0000i		−	0.640039i	−0.947411	−	0.320019i	$-0.896311\pi$
$13$		−	30.0000i		−	0.640039i	0.947411	−	0.320019i	$-0.103689\pi$
$14$		0			0
$15$	−15.0000	+	30.0000i	−0.258199	+	0.516398i
$16$		0			0
$17$		−	62.0000i		−	0.884542i	−0.896882	−	0.442271i	$-0.854173\pi$
$17$		−	62.0000i		−	0.884542i	0.896882	−	0.442271i	$-0.145827\pi$
$18$		0			0
$19$	120.000			1.44894			0.724471	−	0.689306i	$-0.242084\pi$
$19$	120.000			1.44894			0.724471	+	0.689306i	$0.242084\pi$
$20$		0			0
$21$	−66.0000			−0.685828
$22$		0			0
$23$			188.000i			1.70438i	0.523234	+	0.852189i	$0.324726\pi$
$23$			188.000i			1.70438i	−0.523234	+	0.852189i	$0.675274\pi$
$24$		0			0
$25$	75.0000	+	100.000i	0.600000	+	0.800000i
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$	−96.0000			−0.614716			−0.307358	−	0.951594i	$-0.599445\pi$
$29$	−96.0000			−0.614716			−0.307358	+	0.951594i	$0.599445\pi$
$30$		0			0
$31$	184.000			1.06604			0.533022	−	0.846101i	$-0.321056\pi$
$31$	184.000			1.06604			0.533022	+	0.846101i	$0.321056\pi$
$32$		0			0
$33$			42.0000i			0.221553i
$34$		0			0
$35$	−110.000	+	220.000i	−0.531240	+	1.06248i
$36$		0			0
$37$		−	406.000i		−	1.80395i	−0.431793	−	0.901973i	$-0.642119\pi$
$37$		−	406.000i		−	1.80395i	0.431793	−	0.901973i	$-0.357881\pi$
$38$		0			0
$39$	−90.0000			−0.369527
$40$		0			0
$41$	130.000			0.495185			0.247593	−	0.968864i	$-0.420361\pi$
$41$	130.000			0.495185			0.247593	+	0.968864i	$0.420361\pi$
$42$		0			0
$43$			148.000i			0.524879i	0.964948	+	0.262439i	$0.0845270\pi$
$43$			148.000i			0.524879i	−0.964948	+	0.262439i	$0.915473\pi$
$44$		0			0
$45$	90.0000	+	45.0000i	0.298142	+	0.149071i
$46$		0			0
$47$		−	448.000i		−	1.39037i	−0.718830	−	0.695186i	$-0.755322\pi$
$47$		−	448.000i		−	1.39037i	0.718830	−	0.695186i	$-0.244678\pi$
$48$		0			0
$49$	−141.000			−0.411079
$50$		0			0
$51$	−186.000			−0.510690
$52$		0			0
$53$		−	414.000i		−	1.07297i	−0.843911	−	0.536484i	$-0.819752\pi$
$53$		−	414.000i		−	1.07297i	0.843911	−	0.536484i	$-0.180248\pi$
$54$		0			0
$55$	140.000	+	70.0000i	0.343229	+	0.171615i
$56$		0			0
$57$		−	360.000i		−	0.836547i
$58$		0			0
$59$	−266.000			−0.586953			−0.293477	−	0.955966i	$-0.594812\pi$
$59$	−266.000			−0.586953			−0.293477	+	0.955966i	$0.594812\pi$
$60$		0			0
$61$	−838.000			−1.75893			−0.879466	−	0.475961i	$-0.842100\pi$
$61$	−838.000			−1.75893			−0.879466	+	0.475961i	$0.842100\pi$
$62$		0			0
$63$			198.000i			0.395963i
$64$		0			0
$65$	−150.000	+	300.000i	−0.286234	+	0.572468i
$66$		0			0
$67$		−	248.000i		−	0.452209i	−0.974103	−	0.226105i	$-0.927401\pi$
$67$		−	248.000i		−	0.452209i	0.974103	−	0.226105i	$-0.0725991\pi$
$68$		0			0
$69$	564.000			0.984023
$70$		0			0
$71$	1020.00			1.70495			0.852477	−	0.522765i	$-0.175099\pi$
$71$	1020.00			1.70495			0.852477	+	0.522765i	$0.175099\pi$
$72$		0			0
$73$			484.000i			0.775999i	0.921660	+	0.387999i	$0.126834\pi$
$73$			484.000i			0.775999i	−0.921660	+	0.387999i	$0.873166\pi$
$74$		0			0
$75$	300.000	−	225.000i	0.461880	−	0.346410i
$76$		0			0
$77$			308.000i			0.455842i
$78$		0			0
$79$	48.0000			0.0683598			0.0341799	−	0.999416i	$-0.489118\pi$
$79$	48.0000			0.0683598			0.0341799	+	0.999416i	$0.489118\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			548.000i			0.724709i	0.932040	+	0.362354i	$0.118027\pi$
$83$			548.000i			0.724709i	−0.932040	+	0.362354i	$0.881973\pi$
$84$		0			0
$85$	−310.000	+	620.000i	−0.395579	+	0.791158i
$86$		0			0
$87$			288.000i			0.354906i
$88$		0			0
$89$	650.000			0.774156			0.387078	−	0.922047i	$-0.373484\pi$
$89$	650.000			0.774156			0.387078	+	0.922047i	$0.373484\pi$
$90$		0			0
$91$	−660.000			−0.760294
$92$		0			0
$93$		−	552.000i		−	0.615481i
$94$		0			0
$95$	−1200.00	−	600.000i	−1.29597	−	0.647986i
$96$		0			0
$97$			1816.00i			1.90090i	0.310884	+	0.950448i	$0.399375\pi$
$97$			1816.00i			1.90090i	−0.310884	+	0.950448i	$0.600625\pi$
$98$		0			0
$99$	126.000			0.127914
$100$		0			0
$101$	1688.00			1.66299			0.831496	−	0.555530i	$-0.187485\pi$
$101$	1688.00			1.66299			0.831496	+	0.555530i	$0.187485\pi$
$102$		0			0
$103$		−	298.000i		−	0.285076i	−0.989789	−	0.142538i	$-0.954474\pi$
$103$		−	298.000i		−	0.285076i	0.989789	−	0.142538i	$-0.0455263\pi$
$104$		0			0
$105$	660.000	+	330.000i	0.613423	+	0.306711i
$106$		0			0
$107$		−	276.000i		−	0.249364i	−0.992197	−	0.124682i	$-0.960209\pi$
$107$		−	276.000i		−	0.249364i	0.992197	−	0.124682i	$-0.0397910\pi$
$108$		0			0
$109$	−322.000			−0.282954			−0.141477	−	0.989942i	$-0.545185\pi$
$109$	−322.000			−0.282954			−0.141477	+	0.989942i	$0.545185\pi$
$110$		0			0
$111$	−1218.00			−1.04151
$112$		0			0
$113$		−	486.000i		−	0.404593i	−0.979324	−	0.202297i	$-0.935159\pi$
$113$		−	486.000i		−	0.404593i	0.979324	−	0.202297i	$-0.0648405\pi$
$114$		0			0
$115$	940.000	−	1880.00i	0.762221	−	1.52444i
$116$		0			0
$117$			270.000i			0.213346i
$118$		0			0
$119$	−1364.00			−1.05074
$120$		0			0
$121$	−1135.00			−0.852742
$122$		0			0
$123$		−	390.000i		−	0.285895i
$124$		0			0
$125$	−250.000	−	1375.00i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	502.000i		−	0.350750i	−0.984502	−	0.175375i	$-0.943886\pi$
$127$		−	502.000i		−	0.350750i	0.984502	−	0.175375i	$-0.0561138\pi$
$128$		0			0
$129$	444.000			0.303039
$130$		0			0
$131$	−18.0000			−0.0120051			−0.00600255	−	0.999982i	$-0.501911\pi$
$131$	−18.0000			−0.0120051			−0.00600255	+	0.999982i	$0.501911\pi$
$132$		0			0
$133$		−	2640.00i		−	1.72118i
$134$		0			0
$135$	135.000	−	270.000i	0.0860663	−	0.172133i
$136$		0			0
$137$			2334.00i			1.45553i	0.685829	+	0.727763i	$0.259440\pi$
$137$			2334.00i			1.45553i	−0.685829	+	0.727763i	$0.740560\pi$
$138$		0			0
$139$	1676.00			1.02271			0.511354	−	0.859370i	$-0.329144\pi$
$139$	1676.00			1.02271			0.511354	+	0.859370i	$0.329144\pi$
$140$		0			0
$141$	−1344.00			−0.802732
$142$		0			0
$143$			420.000i			0.245610i
$144$		0			0
$145$	960.000	+	480.000i	0.549818	+	0.274909i
$146$		0			0
$147$			423.000i			0.237336i
$148$		0			0
$149$	−3452.00			−1.89798			−0.948989	−	0.315308i	$-0.897892\pi$
$149$	−3452.00			−1.89798			−0.948989	+	0.315308i	$0.897892\pi$
$150$		0			0
$151$	1720.00			0.926964			0.463482	−	0.886106i	$-0.346600\pi$
$151$	1720.00			0.926964			0.463482	+	0.886106i	$0.346600\pi$
$152$		0			0
$153$			558.000i			0.294847i
$154$		0			0
$155$	−1840.00	−	920.000i	−0.953499	−	0.476750i
$156$		0			0
$157$			1246.00i			0.633386i	0.948528	+	0.316693i	$0.102573\pi$
$157$			1246.00i			0.633386i	−0.948528	+	0.316693i	$0.897427\pi$
$158$		0			0
$159$	−1242.00			−0.619478
$160$		0			0
$161$	4136.00			2.02461
$162$		0			0
$163$			1760.00i			0.845729i	0.906193	+	0.422865i	$0.138975\pi$
$163$			1760.00i			0.845729i	−0.906193	+	0.422865i	$0.861025\pi$
$164$		0			0
$165$	210.000	−	420.000i	0.0990817	−	0.198163i
$166$		0			0
$167$			2724.00i			1.26221i	0.775696	+	0.631106i	$0.217399\pi$
$167$			2724.00i			1.26221i	−0.775696	+	0.631106i	$0.782601\pi$
$168$		0			0
$169$	1297.00			0.590350
$170$		0			0
$171$	−1080.00			−0.482980
$172$		0			0
$173$		−	2378.00i		−	1.04506i	−0.852620	−	0.522532i	$-0.824988\pi$
$173$		−	2378.00i		−	1.04506i	0.852620	−	0.522532i	$-0.175012\pi$
$174$		0			0
$175$	2200.00	−	1650.00i	0.950311	−	0.712733i
$176$		0			0
$177$			798.000i			0.338878i
$178$		0			0
$179$	3094.00			1.29194			0.645968	−	0.763365i	$-0.276454\pi$
$179$	3094.00			1.29194			0.645968	+	0.763365i	$0.276454\pi$
$180$		0			0
$181$	−310.000			−0.127305			−0.0636523	−	0.997972i	$-0.520275\pi$
$181$	−310.000			−0.127305			−0.0636523	+	0.997972i	$0.520275\pi$
$182$		0			0
$183$			2514.00i			1.01552i
$184$		0			0
$185$	−2030.00	+	4060.00i	−0.806749	+	1.61350i
$186$		0			0
$187$			868.000i			0.339436i
$188$		0			0
$189$	594.000			0.228609
$190$		0			0
$191$	−516.000			−0.195479			−0.0977394	−	0.995212i	$-0.531161\pi$
$191$	−516.000			−0.195479			−0.0977394	+	0.995212i	$0.531161\pi$
$192$		0			0
$193$		−	188.000i		−	0.0701168i	−0.999385	−	0.0350584i	$-0.988838\pi$
$193$		−	188.000i		−	0.0701168i	0.999385	−	0.0350584i	$-0.0111617\pi$
$194$		0			0
$195$	900.000	+	450.000i	0.330515	+	0.165257i
$196$		0			0
$197$		−	2578.00i		−	0.932360i	−0.884690	−	0.466180i	$-0.845630\pi$
$197$		−	2578.00i		−	0.932360i	0.884690	−	0.466180i	$-0.154370\pi$
$198$		0			0
$199$	528.000			0.188085			0.0940425	−	0.995568i	$-0.470021\pi$
$199$	528.000			0.188085			0.0940425	+	0.995568i	$0.470021\pi$
$200$		0			0
$201$	−744.000			−0.261083
$202$		0			0
$203$			2112.00i			0.730213i
$204$		0			0
$205$	−1300.00	−	650.000i	−0.442907	−	0.221454i
$206$		0			0
$207$		−	1692.00i		−	0.568126i
$208$		0			0
$209$	−1680.00			−0.556019
$210$		0			0
$211$	−3660.00			−1.19415			−0.597073	−	0.802187i	$-0.703670\pi$
$211$	−3660.00			−1.19415			−0.597073	+	0.802187i	$0.703670\pi$
$212$		0			0
$213$		−	3060.00i		−	0.984356i
$214$		0			0
$215$	740.000	−	1480.00i	0.234733	−	0.469466i
$216$		0			0
$217$		−	4048.00i		−	1.26634i
$218$		0			0
$219$	1452.00			0.448023
$220$		0			0
$221$	−1860.00			−0.566141
$222$		0			0
$223$			2350.00i			0.705684i	0.935683	+	0.352842i	$0.114785\pi$
$223$			2350.00i			0.705684i	−0.935683	+	0.352842i	$0.885215\pi$
$224$		0			0
$225$	−675.000	−	900.000i	−0.200000	−	0.266667i
$226$		0			0
$227$		−	3260.00i		−	0.953189i	−0.879123	−	0.476594i	$-0.841871\pi$
$227$		−	3260.00i		−	0.953189i	0.879123	−	0.476594i	$-0.158129\pi$
$228$		0			0
$229$	466.000			0.134472			0.0672361	−	0.997737i	$-0.478582\pi$
$229$	466.000			0.134472			0.0672361	+	0.997737i	$0.478582\pi$
$230$		0			0
$231$	924.000			0.263181
$232$		0			0
$233$		−	3170.00i		−	0.891303i	−0.895207	−	0.445652i	$-0.852972\pi$
$233$		−	3170.00i		−	0.891303i	0.895207	−	0.445652i	$-0.147028\pi$
$234$		0			0
$235$	−2240.00	+	4480.00i	−0.621794	+	1.24359i
$236$		0			0
$237$		−	144.000i		−	0.0394675i
$238$		0			0
$239$	−292.000			−0.0790289			−0.0395145	−	0.999219i	$-0.512581\pi$
$239$	−292.000			−0.0790289			−0.0395145	+	0.999219i	$0.512581\pi$
$240$		0			0
$241$	−842.000			−0.225054			−0.112527	−	0.993649i	$-0.535894\pi$
$241$	−842.000			−0.225054			−0.112527	+	0.993649i	$0.535894\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	1410.00	+	705.000i	0.367680	+	0.183840i
$246$		0			0
$247$		−	3600.00i		−	0.927379i
$248$		0			0
$249$	1644.00			0.418411
$250$		0			0
$251$	5838.00			1.46809			0.734046	−	0.679099i	$-0.237629\pi$
$251$	5838.00			1.46809			0.734046	+	0.679099i	$0.237629\pi$
$252$		0			0
$253$		−	2632.00i		−	0.654041i
$254$		0			0
$255$	1860.00	+	930.000i	0.456775	+	0.228388i
$256$		0			0
$257$		−	4626.00i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	4626.00i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$	−8932.00			−2.14289
$260$		0			0
$261$	864.000			0.204905
$262$		0			0
$263$			5468.00i			1.28202i	0.767532	+	0.641010i	$0.221484\pi$
$263$			5468.00i			1.28202i	−0.767532	+	0.641010i	$0.778516\pi$
$264$		0			0
$265$	−2070.00	+	4140.00i	−0.479846	+	0.959691i
$266$		0			0
$267$		−	1950.00i		−	0.446959i
$268$		0			0
$269$	2976.00			0.674535			0.337268	−	0.941409i	$-0.390497\pi$
$269$	2976.00			0.674535			0.337268	+	0.941409i	$0.390497\pi$
$270$		0			0
$271$	−56.0000			−0.0125526			−0.00627631	−	0.999980i	$-0.501998\pi$
$271$	−56.0000			−0.0125526			−0.00627631	+	0.999980i	$0.501998\pi$
$272$		0			0
$273$			1980.00i			0.438956i
$274$		0			0
$275$	−1050.00	−	1400.00i	−0.230245	−	0.306993i
$276$		0			0
$277$			4106.00i			0.890634i	0.895373	+	0.445317i	$0.146909\pi$
$277$			4106.00i			0.890634i	−0.895373	+	0.445317i	$0.853091\pi$
$278$		0			0
$279$	−1656.00			−0.355348
$280$		0			0
$281$	−2274.00			−0.482760			−0.241380	−	0.970431i	$-0.577600\pi$
$281$	−2274.00			−0.482760			−0.241380	+	0.970431i	$0.577600\pi$
$282$		0			0
$283$			5504.00i			1.15611i	0.815998	+	0.578054i	$0.196188\pi$
$283$			5504.00i			1.15611i	−0.815998	+	0.578054i	$0.803812\pi$
$284$		0			0
$285$	−1800.00	+	3600.00i	−0.374115	+	0.748230i
$286$		0			0
$287$		−	2860.00i		−	0.588225i
$288$		0			0
$289$	1069.00			0.217586
$290$		0			0
$291$	5448.00			1.09748
$292$		0			0
$293$			8034.00i			1.60188i	0.598744	+	0.800941i	$0.295667\pi$
$293$			8034.00i			1.60188i	−0.598744	+	0.800941i	$0.704333\pi$
$294$		0			0
$295$	2660.00	+	1330.00i	0.524987	+	0.262494i
$296$		0			0
$297$		−	378.000i		−	0.0738511i
$298$		0			0
$299$	5640.00			1.09087
$300$		0			0
$301$	3256.00			0.623497
$302$		0			0
$303$		−	5064.00i		−	0.960129i
$304$		0			0
$305$	8380.00	+	4190.00i	1.57324	+	0.786619i
$306$		0			0
$307$		−	996.000i		−	0.185162i	−0.995705	−	0.0925810i	$-0.970488\pi$
$307$		−	996.000i		−	0.185162i	0.995705	−	0.0925810i	$-0.0295117\pi$
$308$		0			0
$309$	−894.000			−0.164589
$310$		0			0
$311$	−8676.00			−1.58190			−0.790950	−	0.611881i	$-0.790413\pi$
$311$	−8676.00			−1.58190			−0.790950	+	0.611881i	$0.790413\pi$
$312$		0			0
$313$			1732.00i			0.312775i	0.987696	+	0.156387i	$0.0499848\pi$
$313$			1732.00i			0.312775i	−0.987696	+	0.156387i	$0.950015\pi$
$314$		0			0
$315$	990.000	−	1980.00i	0.177080	−	0.354160i
$316$		0			0
$317$			2938.00i			0.520551i	0.965534	+	0.260275i	$0.0838133\pi$
$317$			2938.00i			0.520551i	−0.965534	+	0.260275i	$0.916187\pi$
$318$		0			0
$319$	1344.00			0.235892
$320$		0			0
$321$	−828.000			−0.143970
$322$		0			0
$323$		−	7440.00i		−	1.28165i
$324$		0			0
$325$	3000.00	−	2250.00i	0.512031	−	0.384023i
$326$		0			0
$327$			966.000i			0.163364i
$328$		0			0
$329$	−9856.00			−1.65161
$330$		0			0
$331$	−128.000			−0.0212553			−0.0106277	−	0.999944i	$-0.503383\pi$
$331$	−128.000			−0.0212553			−0.0106277	+	0.999944i	$0.503383\pi$
$332$		0			0
$333$			3654.00i			0.601315i
$334$		0			0
$335$	−1240.00	+	2480.00i	−0.202234	+	0.404468i
$336$		0			0
$337$		−	5596.00i		−	0.904551i	−0.891878	−	0.452275i	$-0.850612\pi$
$337$		−	5596.00i		−	0.904551i	0.891878	−	0.452275i	$-0.149388\pi$
$338$		0			0
$339$	−1458.00			−0.233592
$340$		0			0
$341$	−2576.00			−0.409086
$342$		0			0
$343$		−	4444.00i		−	0.699573i
$344$		0			0
$345$	−5640.00	−	2820.00i	−0.880137	−	0.440069i
$346$		0			0
$347$			924.000i			0.142948i	0.997442	+	0.0714739i	$0.0227703\pi$
$347$			924.000i			0.142948i	−0.997442	+	0.0714739i	$0.977230\pi$
$348$		0			0
$349$	6206.00			0.951861			0.475931	−	0.879483i	$-0.342111\pi$
$349$	6206.00			0.951861			0.475931	+	0.879483i	$0.342111\pi$
$350$		0			0
$351$	810.000			0.123176
$352$		0			0
$353$		−	3506.00i		−	0.528628i	−0.964437	−	0.264314i	$-0.914855\pi$
$353$		−	3506.00i		−	0.528628i	0.964437	−	0.264314i	$-0.0851455\pi$
$354$		0			0
$355$	−10200.0	−	5100.00i	−1.52496	−	0.762479i
$356$		0			0
$357$			4092.00i			0.606643i
$358$		0			0
$359$	7104.00			1.04439			0.522193	−	0.852827i	$-0.325114\pi$
$359$	7104.00			1.04439			0.522193	+	0.852827i	$0.325114\pi$
$360$		0			0
$361$	7541.00			1.09943
$362$		0			0
$363$			3405.00i			0.492331i
$364$		0			0
$365$	2420.00	−	4840.00i	0.347037	−	0.694074i
$366$		0			0
$367$		−	2902.00i		−	0.412761i	−0.978472	−	0.206380i	$-0.933832\pi$
$367$		−	2902.00i		−	0.412761i	0.978472	−	0.206380i	$-0.0661684\pi$
$368$		0			0
$369$	−1170.00			−0.165062
$370$		0			0
$371$	−9108.00			−1.27457
$372$		0			0
$373$		−	4542.00i		−	0.630498i	−0.949009	−	0.315249i	$-0.897912\pi$
$373$		−	4542.00i		−	0.630498i	0.949009	−	0.315249i	$-0.102088\pi$
$374$		0			0
$375$	−4125.00	+	750.000i	−0.568038	+	0.103280i
$376$		0			0
$377$			2880.00i			0.393442i
$378$		0			0
$379$	−5852.00			−0.793132			−0.396566	−	0.918006i	$-0.629798\pi$
$379$	−5852.00			−0.793132			−0.396566	+	0.918006i	$0.629798\pi$
$380$		0			0
$381$	−1506.00			−0.202506
$382$		0			0
$383$		−	8936.00i		−	1.19219i	−0.802914	−	0.596094i	$-0.796718\pi$
$383$		−	8936.00i		−	1.19219i	0.802914	−	0.596094i	$-0.203282\pi$
$384$		0			0
$385$	1540.00	−	3080.00i	0.203859	−	0.407718i
$386$		0			0
$387$		−	1332.00i		−	0.174960i
$388$		0			0
$389$	−1924.00			−0.250773			−0.125386	−	0.992108i	$-0.540017\pi$
$389$	−1924.00			−0.250773			−0.125386	+	0.992108i	$0.540017\pi$
$390$		0			0
$391$	11656.0			1.50759
$392$		0			0
$393$			54.0000i			0.00693114i
$394$		0			0
$395$	−480.000	−	240.000i	−0.0611428	−	0.0305714i
$396$		0			0
$397$			9194.00i			1.16230i	0.813796	+	0.581151i	$0.197397\pi$
$397$			9194.00i			1.16230i	−0.813796	+	0.581151i	$0.802603\pi$
$398$		0			0
$399$	−7920.00			−0.993724
$400$		0			0
$401$	−10714.0			−1.33424			−0.667122	−	0.744949i	$-0.732474\pi$
$401$	−10714.0			−1.33424			−0.667122	+	0.744949i	$0.732474\pi$
$402$		0			0
$403$		−	5520.00i		−	0.682310i
$404$		0			0
$405$	−810.000	−	405.000i	−0.0993808	−	0.0496904i
$406$		0			0
$407$			5684.00i			0.692249i
$408$		0			0
$409$	12346.0			1.49259			0.746296	−	0.665614i	$-0.231830\pi$
$409$	12346.0			1.49259			0.746296	+	0.665614i	$0.231830\pi$
$410$		0			0
$411$	7002.00			0.840348
$412$		0			0
$413$			5852.00i			0.697235i
$414$		0			0
$415$	2740.00	−	5480.00i	0.324100	−	0.648199i
$416$		0			0
$417$		−	5028.00i		−	0.590461i
$418$		0			0
$419$	13562.0			1.58126			0.790629	−	0.612296i	$-0.209754\pi$
$419$	13562.0			1.58126			0.790629	+	0.612296i	$0.209754\pi$
$420$		0			0
$421$	5230.00			0.605450			0.302725	−	0.953078i	$-0.402104\pi$
$421$	5230.00			0.605450			0.302725	+	0.953078i	$0.402104\pi$
$422$		0			0
$423$			4032.00i			0.463458i
$424$		0			0
$425$	6200.00	−	4650.00i	0.707633	−	0.530725i
$426$		0			0
$427$			18436.0i			2.08942i
$428$		0			0
$429$	1260.00			0.141803
$430$		0			0
$431$	−11552.0			−1.29104			−0.645522	−	0.763741i	$-0.723360\pi$
$431$	−11552.0			−1.29104			−0.645522	+	0.763741i	$0.723360\pi$
$432$		0			0
$433$			7972.00i			0.884780i	0.896823	+	0.442390i	$0.145869\pi$
$433$			7972.00i			0.884780i	−0.896823	+	0.442390i	$0.854131\pi$
$434$		0			0
$435$	1440.00	−	2880.00i	0.158719	−	0.317438i
$436$		0			0
$437$			22560.0i			2.46954i
$438$		0			0
$439$	−7152.00			−0.777554			−0.388777	−	0.921332i	$-0.627102\pi$
$439$	−7152.00			−0.777554			−0.388777	+	0.921332i	$0.627102\pi$
$440$		0			0
$441$	1269.00			0.137026
$442$		0			0
$443$			2652.00i			0.284425i	0.989836	+	0.142213i	$0.0454217\pi$
$443$			2652.00i			0.284425i	−0.989836	+	0.142213i	$0.954578\pi$
$444$		0			0
$445$	−6500.00	−	3250.00i	−0.692426	−	0.346213i
$446$		0			0
$447$			10356.0i			1.09580i
$448$		0			0
$449$	−4542.00			−0.477395			−0.238697	−	0.971094i	$-0.576720\pi$
$449$	−4542.00			−0.477395			−0.238697	+	0.971094i	$0.576720\pi$
$450$		0			0
$451$	−1820.00			−0.190023
$452$		0			0
$453$		−	5160.00i		−	0.535183i
$454$		0			0
$455$	6600.00	+	3300.00i	0.680028	+	0.340014i
$456$		0			0
$457$		−	16000.0i		−	1.63774i	−0.573977	−	0.818871i	$-0.694600\pi$
$457$		−	16000.0i		−	1.63774i	0.573977	−	0.818871i	$-0.305400\pi$
$458$		0			0
$459$	1674.00			0.170230
$460$		0			0
$461$	2076.00			0.209737			0.104869	−	0.994486i	$-0.466558\pi$
$461$	2076.00			0.209737			0.104869	+	0.994486i	$0.466558\pi$
$462$		0			0
$463$		−	5406.00i		−	0.542631i	−0.962490	−	0.271315i	$-0.912541\pi$
$463$		−	5406.00i		−	0.542631i	0.962490	−	0.271315i	$-0.0874587\pi$
$464$		0			0
$465$	−2760.00	+	5520.00i	−0.275251	+	0.550503i
$466$		0			0
$467$			444.000i			0.0439954i	0.999758	+	0.0219977i	$0.00700266\pi$
$467$			444.000i			0.0439954i	−0.999758	+	0.0219977i	$0.992997\pi$
$468$		0			0
$469$	−5456.00			−0.537174
$470$		0			0
$471$	3738.00			0.365686
$472$		0			0
$473$		−	2072.00i		−	0.201418i
$474$		0			0
$475$	9000.00	+	12000.0i	0.869365	+	1.15915i
$476$		0			0
$477$			3726.00i			0.357656i
$478$		0			0
$479$	1724.00			0.164450			0.0822250	−	0.996614i	$-0.473797\pi$
$479$	1724.00			0.164450			0.0822250	+	0.996614i	$0.473797\pi$
$480$		0			0
$481$	−12180.0			−1.15460
$482$		0			0
$483$		−	12408.0i		−	1.16891i
$484$		0			0
$485$	9080.00	−	18160.0i	0.850106	−	1.70021i
$486$		0			0
$487$			17046.0i			1.58609i	0.609160	+	0.793047i	$0.291507\pi$
$487$			17046.0i			1.58609i	−0.609160	+	0.793047i	$0.708493\pi$
$488$		0			0
$489$	5280.00			0.488282
$490$		0			0
$491$	8814.00			0.810123			0.405061	−	0.914290i	$-0.367250\pi$
$491$	8814.00			0.810123			0.405061	+	0.914290i	$0.367250\pi$
$492$		0			0
$493$			5952.00i			0.543742i
$494$		0			0
$495$	−1260.00	−	630.000i	−0.114410	−	0.0572048i
$496$		0			0
$497$		−	22440.0i		−	2.02529i
$498$		0			0
$499$	−5256.00			−0.471525			−0.235762	−	0.971811i	$-0.575759\pi$
$499$	−5256.00			−0.471525			−0.235762	+	0.971811i	$0.575759\pi$
$500$		0			0
$501$	8172.00			0.728739
$502$		0			0
$503$			5232.00i			0.463784i	0.972742	+	0.231892i	$0.0744916\pi$
$503$			5232.00i			0.463784i	−0.972742	+	0.231892i	$0.925508\pi$
$504$		0			0
$505$	−16880.0	−	8440.00i	−1.48743	−	0.743713i
$506$		0			0
$507$		−	3891.00i		−	0.340839i
$508$		0			0
$509$	−4128.00			−0.359470			−0.179735	−	0.983715i	$-0.557524\pi$
$509$	−4128.00			−0.359470			−0.179735	+	0.983715i	$0.557524\pi$
$510$		0			0
$511$	10648.0			0.921800
$512$		0			0
$513$			3240.00i			0.278849i
$514$		0			0
$515$	−1490.00	+	2980.00i	−0.127490	+	0.254980i
$516$		0			0
$517$			6272.00i			0.533544i
$518$		0			0
$519$	−7134.00			−0.603368
$520$		0			0
$521$	−538.000			−0.0452403			−0.0226202	−	0.999744i	$-0.507201\pi$
$521$	−538.000			−0.0452403			−0.0226202	+	0.999744i	$0.507201\pi$
$522$		0			0
$523$		−	10336.0i		−	0.864172i	−0.901833	−	0.432086i	$-0.857778\pi$
$523$		−	10336.0i		−	0.864172i	0.901833	−	0.432086i	$-0.142222\pi$
$524$		0			0
$525$	−4950.00	−	6600.00i	−0.411497	−	0.548662i
$526$		0			0
$527$		−	11408.0i		−	0.942961i
$528$		0			0
$529$	−23177.0			−1.90491
$530$		0			0
$531$	2394.00			0.195651
$532$		0			0
$533$		−	3900.00i		−	0.316938i
$534$		0			0
$535$	−1380.00	+	2760.00i	−0.111519	+	0.223038i
$536$		0			0
$537$		−	9282.00i		−	0.745899i
$538$		0			0
$539$	1974.00			0.157748
$540$		0			0
$541$	−8942.00			−0.710622			−0.355311	−	0.934748i	$-0.615625\pi$
$541$	−8942.00			−0.710622			−0.355311	+	0.934748i	$0.615625\pi$
$542$		0			0
$543$			930.000i			0.0734993i
$544$		0			0
$545$	3220.00	+	1610.00i	0.253082	+	0.126541i
$546$		0			0
$547$		−	11404.0i		−	0.891407i	−0.895181	−	0.445704i	$-0.852954\pi$
$547$		−	11404.0i		−	0.891407i	0.895181	−	0.445704i	$-0.147046\pi$
$548$		0			0
$549$	7542.00			0.586311
$550$		0			0
$551$	−11520.0			−0.890687
$552$		0			0
$553$		−	1056.00i		−	0.0812038i
$554$		0			0
$555$	12180.0	+	6090.00i	0.931554	+	0.465777i
$556$		0			0
$557$		−	3114.00i		−	0.236884i	−0.992961	−	0.118442i	$-0.962210\pi$
$557$		−	3114.00i		−	0.236884i	0.992961	−	0.118442i	$-0.0377900\pi$
$558$		0			0
$559$	4440.00			0.335943
$560$		0			0
$561$	2604.00			0.195973
$562$		0			0
$563$		−	3620.00i		−	0.270985i	−0.990778	−	0.135493i	$-0.956738\pi$
$563$		−	3620.00i		−	0.270985i	0.990778	−	0.135493i	$-0.0432617\pi$
$564$		0			0
$565$	−2430.00	+	4860.00i	−0.180940	+	0.361879i
$566$		0			0
$567$		−	1782.00i		−	0.131988i
$568$		0			0
$569$	6506.00			0.479342			0.239671	−	0.970854i	$-0.422960\pi$
$569$	6506.00			0.479342			0.239671	+	0.970854i	$0.422960\pi$
$570$		0			0
$571$	−17600.0			−1.28991			−0.644954	−	0.764222i	$-0.723123\pi$
$571$	−17600.0			−1.28991			−0.644954	+	0.764222i	$0.723123\pi$
$572$		0			0
$573$			1548.00i			0.112860i
$574$		0			0
$575$	−18800.0	+	14100.0i	−1.36350	+	1.02263i
$576$		0			0
$577$		−	4864.00i		−	0.350938i	−0.984485	−	0.175469i	$-0.943856\pi$
$577$		−	4864.00i		−	0.350938i	0.984485	−	0.175469i	$-0.0561441\pi$
$578$		0			0
$579$	−564.000			−0.0404819
$580$		0			0
$581$	12056.0			0.860873
$582$		0			0
$583$			5796.00i			0.411742i
$584$		0			0
$585$	1350.00	−	2700.00i	0.0954113	−	0.190823i
$586$		0			0
$587$			18348.0i			1.29012i	0.764130	+	0.645062i	$0.223169\pi$
$587$			18348.0i			1.29012i	−0.764130	+	0.645062i	$0.776831\pi$
$588$		0			0
$589$	22080.0			1.54464
$590$		0			0
$591$	−7734.00			−0.538298
$592$		0			0
$593$			16218.0i			1.12309i	0.827446	+	0.561546i	$0.189793\pi$
$593$			16218.0i			1.12309i	−0.827446	+	0.561546i	$0.810207\pi$
$594$		0			0
$595$	13640.0	+	6820.00i	0.939808	+	0.469904i
$596$		0			0
$597$		−	1584.00i		−	0.108591i
$598$		0			0
$599$	24576.0			1.67637			0.838187	−	0.545383i	$-0.183616\pi$
$599$	24576.0			1.67637			0.838187	+	0.545383i	$0.183616\pi$
$600$		0			0
$601$	6578.00			0.446460			0.223230	−	0.974766i	$-0.428340\pi$
$601$	6578.00			0.446460			0.223230	+	0.974766i	$0.428340\pi$
$602$		0			0
$603$			2232.00i			0.150736i
$604$		0			0
$605$	11350.0	+	5675.00i	0.762716	+	0.381358i
$606$		0			0
$607$		−	25346.0i		−	1.69483i	−0.530930	−	0.847415i	$-0.678157\pi$
$607$		−	25346.0i		−	1.69483i	0.530930	−	0.847415i	$-0.321843\pi$
$608$		0			0
$609$	6336.00			0.421589
$610$		0			0
$611$	−13440.0			−0.889892
$612$		0			0
$613$			27214.0i			1.79309i	0.442954	+	0.896544i	$0.353930\pi$
$613$			27214.0i			1.79309i	−0.442954	+	0.896544i	$0.646070\pi$
$614$		0			0
$615$	−1950.00	+	3900.00i	−0.127856	+	0.255712i
$616$		0			0
$617$		−	22478.0i		−	1.46666i	−0.679872	−	0.733331i	$-0.737965\pi$
$617$		−	22478.0i		−	1.46666i	0.679872	−	0.733331i	$-0.262035\pi$
$618$		0			0
$619$	−3356.00			−0.217914			−0.108957	−	0.994046i	$-0.534751\pi$
$619$	−3356.00			−0.217914			−0.108957	+	0.994046i	$0.534751\pi$
$620$		0			0
$621$	−5076.00			−0.328008
$622$		0			0
$623$		−	14300.0i		−	0.919611i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$		0			0
$627$			5040.00i			0.321018i
$628$		0			0
$629$	−25172.0			−1.59567
$630$		0			0
$631$	19952.0			1.25876			0.629379	−	0.777098i	$-0.283309\pi$
$631$	19952.0			1.25876			0.629379	+	0.777098i	$0.283309\pi$
$632$		0			0
$633$			10980.0i			0.689440i
$634$		0			0
$635$	−2510.00	+	5020.00i	−0.156860	+	0.313721i
$636$		0			0
$637$			4230.00i			0.263106i
$638$		0			0
$639$	−9180.00			−0.568318
$640$		0			0
$641$	26434.0			1.62883			0.814415	−	0.580283i	$-0.197058\pi$
$641$	26434.0			1.62883			0.814415	+	0.580283i	$0.197058\pi$
$642$		0			0
$643$			10632.0i			0.652076i	0.945357	+	0.326038i	$0.105714\pi$
$643$			10632.0i			0.652076i	−0.945357	+	0.326038i	$0.894286\pi$
$644$		0			0
$645$	−4440.00	−	2220.00i	−0.271046	−	0.135523i
$646$		0			0
$647$			9384.00i			0.570206i	0.958497	+	0.285103i	$0.0920278\pi$
$647$			9384.00i			0.570206i	−0.958497	+	0.285103i	$0.907972\pi$
$648$		0			0
$649$	3724.00			0.225239
$650$		0			0
$651$	−12144.0			−0.731123
$652$		0			0
$653$		−	3922.00i		−	0.235038i	−0.993071	−	0.117519i	$-0.962506\pi$
$653$		−	3922.00i		−	0.235038i	0.993071	−	0.117519i	$-0.0374941\pi$
$654$		0			0
$655$	180.000	+	90.0000i	0.0107377	+	0.00536884i
$656$		0			0
$657$		−	4356.00i		−	0.258666i
$658$		0			0
$659$	−12542.0			−0.741376			−0.370688	−	0.928757i	$-0.620878\pi$
$659$	−12542.0			−0.741376			−0.370688	+	0.928757i	$0.620878\pi$
$660$		0			0
$661$	−8662.00			−0.509702			−0.254851	−	0.966980i	$-0.582026\pi$
$661$	−8662.00			−0.509702			−0.254851	+	0.966980i	$0.582026\pi$
$662$		0			0
$663$			5580.00i			0.326862i
$664$		0			0
$665$	−13200.0	+	26400.0i	−0.769735	+	1.53947i
$666$		0			0
$667$		−	18048.0i		−	1.04771i
$668$		0			0
$669$	7050.00			0.407427
$670$		0			0
$671$	11732.0			0.674976
$672$		0			0
$673$			5820.00i			0.333350i	0.986012	+	0.166675i	$0.0533031\pi$
$673$			5820.00i			0.333350i	−0.986012	+	0.166675i	$0.946697\pi$
$674$		0			0
$675$	−2700.00	+	2025.00i	−0.153960	+	0.115470i
$676$		0			0
$677$			778.000i			0.0441669i	0.999756	+	0.0220834i	$0.00702995\pi$
$677$			778.000i			0.0441669i	−0.999756	+	0.0220834i	$0.992970\pi$
$678$		0			0
$679$	39952.0			2.25805
$680$		0			0
$681$	−9780.00			−0.550324
$682$		0			0
$683$		−	5548.00i		−	0.310817i	−0.987850	−	0.155409i	$-0.950331\pi$
$683$		−	5548.00i		−	0.310817i	0.987850	−	0.155409i	$-0.0496695\pi$
$684$		0			0
$685$	11670.0	−	23340.0i	0.650931	−	1.30186i
$686$		0			0
$687$		−	1398.00i		−	0.0776376i
$688$		0			0
$689$	−12420.0			−0.686741
$690$		0			0
$691$	5488.00			0.302132			0.151066	−	0.988524i	$-0.451729\pi$
$691$	5488.00			0.302132			0.151066	+	0.988524i	$0.451729\pi$
$692$		0			0
$693$		−	2772.00i		−	0.151947i
$694$		0			0
$695$	−16760.0	−	8380.00i	−0.914738	−	0.457369i
$696$		0			0
$697$		−	8060.00i		−	0.438012i
$698$		0			0
$699$	−9510.00			−0.514594
$700$		0			0
$701$	1216.00			0.0655174			0.0327587	−	0.999463i	$-0.489571\pi$
$701$	1216.00			0.0655174			0.0327587	+	0.999463i	$0.489571\pi$
$702$		0			0
$703$		−	48720.0i		−	2.61381i
$704$		0			0
$705$	13440.0	+	6720.00i	0.717985	+	0.358993i
$706$		0			0
$707$		−	37136.0i		−	1.97545i
$708$		0			0
$709$	−20406.0			−1.08091			−0.540454	−	0.841374i	$-0.681747\pi$
$709$	−20406.0			−1.08091			−0.540454	+	0.841374i	$0.681747\pi$
$710$		0			0
$711$	−432.000			−0.0227866
$712$		0			0
$713$			34592.0i			1.81694i
$714$		0			0
$715$	2100.00	−	4200.00i	0.109840	−	0.219680i
$716$		0			0
$717$			876.000i			0.0456274i
$718$		0			0
$719$	−19672.0			−1.02036			−0.510182	−	0.860066i	$-0.670422\pi$
$719$	−19672.0			−1.02036			−0.510182	+	0.860066i	$0.670422\pi$
$720$		0			0
$721$	−6556.00			−0.338638
$722$		0			0
$723$			2526.00i			0.129935i
$724$		0			0
$725$	−7200.00	−	9600.00i	−0.368829	−	0.491772i
$726$		0			0
$727$		−	1194.00i		−	0.0609120i	−0.999536	−	0.0304560i	$-0.990304\pi$
$727$		−	1194.00i		−	0.0609120i	0.999536	−	0.0304560i	$-0.00969594\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	9176.00			0.464277
$732$		0			0
$733$		−	21802.0i		−	1.09860i	−0.835625	−	0.549301i	$-0.814894\pi$
$733$		−	21802.0i		−	1.09860i	0.835625	−	0.549301i	$-0.185106\pi$
$734$		0			0
$735$	2115.00	−	4230.00i	0.106140	−	0.212280i
$736$		0			0
$737$			3472.00i			0.173532i
$738$		0			0
$739$	−15280.0			−0.760601			−0.380300	−	0.924863i	$-0.624179\pi$
$739$	−15280.0			−0.760601			−0.380300	+	0.924863i	$0.624179\pi$
$740$		0			0
$741$	−10800.0			−0.535422
$742$		0			0
$743$			6672.00i			0.329437i	0.986341	+	0.164719i	$0.0526716\pi$
$743$			6672.00i			0.329437i	−0.986341	+	0.164719i	$0.947328\pi$
$744$		0			0
$745$	34520.0	+	17260.0i	1.69760	+	0.848802i
$746$		0			0
$747$		−	4932.00i		−	0.241570i
$748$		0			0
$749$	−6072.00			−0.296216
$750$		0			0
$751$	−11008.0			−0.534870			−0.267435	−	0.963576i	$-0.586176\pi$
$751$	−11008.0			−0.534870			−0.267435	+	0.963576i	$0.586176\pi$
$752$		0			0
$753$		−	17514.0i		−	0.847604i
$754$		0			0
$755$	−17200.0	−	8600.00i	−0.829102	−	0.414551i
$756$		0			0
$757$			3242.00i			0.155657i	0.996967	+	0.0778286i	$0.0247987\pi$
$757$			3242.00i			0.155657i	−0.996967	+	0.0778286i	$0.975201\pi$
$758$		0			0
$759$	−7896.00			−0.377611
$760$		0			0
$761$	17982.0			0.856566			0.428283	−	0.903645i	$-0.359119\pi$
$761$	17982.0			0.856566			0.428283	+	0.903645i	$0.359119\pi$
$762$		0			0
$763$			7084.00i			0.336118i
$764$		0			0
$765$	2790.00	−	5580.00i	0.131860	−	0.263719i
$766$		0			0
$767$			7980.00i			0.375673i
$768$		0			0
$769$	30462.0			1.42846			0.714231	−	0.699910i	$-0.246776\pi$
$769$	30462.0			1.42846			0.714231	+	0.699910i	$0.246776\pi$
$770$		0			0
$771$	−13878.0			−0.648254
$772$		0			0
$773$		−	17394.0i		−	0.809339i	−0.914463	−	0.404669i	$-0.867387\pi$
$773$		−	17394.0i		−	0.809339i	0.914463	−	0.404669i	$-0.132613\pi$
$774$		0			0
$775$	13800.0	+	18400.0i	0.639627	+	0.852835i
$776$		0			0
$777$			26796.0i			1.23720i
$778$		0			0
$779$	15600.0			0.717494
$780$		0			0
$781$	−14280.0			−0.654262
$782$		0			0
$783$		−	2592.00i		−	0.118302i
$784$		0			0
$785$	6230.00	−	12460.0i	0.283259	−	0.566518i
$786$		0			0
$787$			20336.0i			0.921093i	0.887636	+	0.460546i	$0.152347\pi$
$787$			20336.0i			0.921093i	−0.887636	+	0.460546i	$0.847653\pi$
$788$		0			0
$789$	16404.0			0.740175
$790$		0			0
$791$	−10692.0			−0.480612
$792$		0			0
$793$			25140.0i			1.12579i
$794$		0			0
$795$	12420.0	+	6210.00i	0.554078	+	0.277039i
$796$		0			0
$797$			19546.0i			0.868701i	0.900744	+	0.434351i	$0.143022\pi$
$797$			19546.0i			0.868701i	−0.900744	+	0.434351i	$0.856978\pi$
$798$		0			0
$799$	−27776.0			−1.22984
$800$		0			0
$801$	−5850.00			−0.258052
$802$		0			0
$803$		−	6776.00i		−	0.297783i
$804$		0			0
$805$	−41360.0	−	20680.0i	−1.81087	−	0.905434i
$806$		0			0
$807$		−	8928.00i		−	0.389443i
$808$		0			0
$809$	−3954.00			−0.171836			−0.0859179	−	0.996302i	$-0.527382\pi$
$809$	−3954.00			−0.171836			−0.0859179	+	0.996302i	$0.527382\pi$
$810$		0			0
$811$	−3860.00			−0.167131			−0.0835653	−	0.996502i	$-0.526631\pi$
$811$	−3860.00			−0.167131			−0.0835653	+	0.996502i	$0.526631\pi$
$812$		0			0
$813$			168.000i			0.00724725i
$814$		0			0
$815$	8800.00	−	17600.0i	0.378222	−	0.756443i
$816$		0			0
$817$			17760.0i			0.760519i
$818$		0			0
$819$	5940.00			0.253431
$820$		0			0
$821$	37016.0			1.57353			0.786764	−	0.617253i	$-0.211755\pi$
$821$	37016.0			1.57353			0.786764	+	0.617253i	$0.211755\pi$
$822$		0			0
$823$			25794.0i			1.09249i	0.837624	+	0.546247i	$0.183944\pi$
$823$			25794.0i			1.09249i	−0.837624	+	0.546247i	$0.816056\pi$
$824$		0			0
$825$	−4200.00	+	3150.00i	−0.177243	+	0.132932i
$826$		0			0
$827$			44116.0i			1.85497i	0.373855	+	0.927487i	$0.378036\pi$
$827$			44116.0i			1.85497i	−0.373855	+	0.927487i	$0.621964\pi$
$828$		0			0
$829$	11622.0			0.486910			0.243455	−	0.969912i	$-0.421719\pi$
$829$	11622.0			0.486910			0.243455	+	0.969912i	$0.421719\pi$
$830$		0			0
$831$	12318.0			0.514208
$832$		0			0
$833$			8742.00i			0.363616i
$834$		0			0
$835$	13620.0	−	27240.0i	0.564478	−	1.12896i
$836$		0			0
$837$			4968.00i			0.205160i
$838$		0			0
$839$	2584.00			0.106328			0.0531642	−	0.998586i	$-0.483069\pi$
$839$	2584.00			0.106328			0.0531642	+	0.998586i	$0.483069\pi$
$840$		0			0
$841$	−15173.0			−0.622125
$842$		0			0
$843$			6822.00i			0.278721i
$844$		0			0
$845$	−12970.0	−	6485.00i	−0.528026	−	0.264013i
$846$		0			0
$847$			24970.0i			1.01296i
$848$		0			0
$849$	16512.0			0.667480
$850$		0			0
$851$	76328.0			3.07461
$852$		0			0
$853$		−	39754.0i		−	1.59572i	−0.602841	−	0.797861i	$-0.705965\pi$
$853$		−	39754.0i		−	1.59572i	0.602841	−	0.797861i	$-0.294035\pi$
$854$		0			0
$855$	10800.0	+	5400.00i	0.431991	+	0.215995i
$856$		0			0
$857$		−	18534.0i		−	0.738751i	−0.929280	−	0.369375i	$-0.879572\pi$
$857$		−	18534.0i		−	0.738751i	0.929280	−	0.369375i	$-0.120428\pi$
$858$		0			0
$859$	11140.0			0.442482			0.221241	−	0.975219i	$-0.428989\pi$
$859$	11140.0			0.442482			0.221241	+	0.975219i	$0.428989\pi$
$860$		0			0
$861$	−8580.00			−0.339612
$862$		0			0
$863$			28356.0i			1.11848i	0.829005	+	0.559241i	$0.188907\pi$
$863$			28356.0i			1.11848i	−0.829005	+	0.559241i	$0.811093\pi$
$864$		0			0
$865$	−11890.0	+	23780.0i	−0.467367	+	0.934733i
$866$		0			0
$867$		−	3207.00i		−	0.125623i
$868$		0			0
$869$	−672.000			−0.0262325
$870$		0			0
$871$	−7440.00			−0.289431
$872$		0			0
$873$		−	16344.0i		−	0.633632i
$874$		0			0
$875$	−30250.0	+	5500.00i	−1.16873	+	0.212496i
$876$		0			0
$877$			34942.0i			1.34539i	0.739920	+	0.672695i	$0.234863\pi$
$877$			34942.0i			1.34539i	−0.739920	+	0.672695i	$0.765137\pi$
$878$		0			0
$879$	24102.0			0.924847
$880$		0			0
$881$	−13890.0			−0.531176			−0.265588	−	0.964087i	$-0.585566\pi$
$881$	−13890.0			−0.531176			−0.265588	+	0.964087i	$0.585566\pi$
$882$		0			0
$883$			11496.0i			0.438133i	0.975710	+	0.219066i	$0.0703011\pi$
$883$			11496.0i			0.438133i	−0.975710	+	0.219066i	$0.929699\pi$
$884$		0			0
$885$	3990.00	−	7980.00i	0.151551	−	0.303101i
$886$		0			0
$887$			29988.0i			1.13517i	0.823314	+	0.567587i	$0.192123\pi$
$887$			29988.0i			1.13517i	−0.823314	+	0.567587i	$0.807877\pi$
$888$		0			0
$889$	−11044.0			−0.416652
$890$		0			0
$891$	−1134.00			−0.0426380
$892$		0			0
$893$		−	53760.0i		−	2.01457i
$894$		0			0
$895$	−30940.0	−	15470.0i	−1.15554	−	0.577771i
$896$		0			0
$897$		−	16920.0i		−	0.629813i
$898$		0			0
$899$	−17664.0			−0.655314
$900$		0			0
$901$	−25668.0			−0.949084
$902$		0			0
$903$		−	9768.00i		−	0.359976i
$904$		0			0
$905$	3100.00	+	1550.00i	0.113865	+	0.0569323i
$906$		0			0
$907$			16764.0i			0.613715i	0.951755	+	0.306857i	$0.0992775\pi$
$907$			16764.0i			0.613715i	−0.951755	+	0.306857i	$0.900723\pi$
$908$		0			0
$909$	−15192.0			−0.554331
$910$		0			0
$911$	−19152.0			−0.696525			−0.348262	−	0.937397i	$-0.613228\pi$
$911$	−19152.0			−0.696525			−0.348262	+	0.937397i	$0.613228\pi$
$912$		0			0
$913$		−	7672.00i		−	0.278101i
$914$		0			0
$915$	12570.0	−	25140.0i	0.454155	−	0.908309i
$916$		0			0
$917$			396.000i			0.0142607i
$918$		0			0
$919$	−35712.0			−1.28186			−0.640930	−	0.767599i	$-0.721451\pi$
$919$	−35712.0			−1.28186			−0.640930	+	0.767599i	$0.721451\pi$
$920$		0			0
$921$	−2988.00			−0.106903
$922$		0			0
$923$		−	30600.0i		−	1.09124i
$924$		0			0
$925$	40600.0	−	30450.0i	1.44316	−	1.08237i
$926$		0			0
$927$			2682.00i			0.0950253i
$928$		0			0
$929$	16098.0			0.568523			0.284262	−	0.958747i	$-0.408252\pi$
$929$	16098.0			0.568523			0.284262	+	0.958747i	$0.408252\pi$
$930$		0			0
$931$	−16920.0			−0.595629
$932$		0			0
$933$			26028.0i			0.913310i
$934$		0			0
$935$	4340.00	−	8680.00i	0.151800	−	0.303600i
$936$		0			0
$937$		−	24776.0i		−	0.863817i	−0.901917	−	0.431909i	$-0.857840\pi$
$937$		−	24776.0i		−	0.863817i	0.901917	−	0.431909i	$-0.142160\pi$
$938$		0			0
$939$	5196.00			0.180580
$940$		0			0
$941$	−39500.0			−1.36840			−0.684199	−	0.729295i	$-0.739848\pi$
$941$	−39500.0			−1.36840			−0.684199	+	0.729295i	$0.739848\pi$
$942$		0			0
$943$			24440.0i			0.843983i
$944$		0			0
$945$	−5940.00	−	2970.00i	−0.204474	−	0.102237i
$946$		0			0
$947$		−	22836.0i		−	0.783601i	−0.920050	−	0.391801i	$-0.871852\pi$
$947$		−	22836.0i		−	0.783601i	0.920050	−	0.391801i	$-0.128148\pi$
$948$		0			0
$949$	14520.0			0.496669
$950$		0			0
$951$	8814.00			0.300540
$952$		0			0
$953$		−	28478.0i		−	0.967988i	−0.875071	−	0.483994i	$-0.839186\pi$
$953$		−	28478.0i		−	0.967988i	0.875071	−	0.483994i	$-0.160814\pi$
$954$		0			0
$955$	5160.00	+	2580.00i	0.174842	+	0.0874208i
$956$		0			0
$957$		−	4032.00i		−	0.136192i
$958$		0			0
$959$	51348.0			1.72900
$960$		0			0
$961$	4065.00			0.136451
$962$		0			0
$963$			2484.00i			0.0831213i
$964$		0			0
$965$	−940.000	+	1880.00i	−0.0313572	+	0.0627143i
$966$		0			0
$967$		−	24830.0i		−	0.825728i	−0.910793	−	0.412864i	$-0.864528\pi$
$967$		−	24830.0i		−	0.825728i	0.910793	−	0.412864i	$-0.135472\pi$
$968$		0			0
$969$	−22320.0			−0.739960
$970$		0			0
$971$	37038.0			1.22411			0.612053	−	0.790817i	$-0.290344\pi$
$971$	37038.0			1.22411			0.612053	+	0.790817i	$0.290344\pi$
$972$		0			0
$973$		−	36872.0i		−	1.21486i
$974$		0			0
$975$	−6750.00	−	9000.00i	−0.221716	−	0.295621i
$976$		0			0
$977$			26346.0i			0.862726i	0.902178	+	0.431363i	$0.141967\pi$
$977$			26346.0i			0.862726i	−0.902178	+	0.431363i	$0.858033\pi$
$978$		0			0
$979$	−9100.00			−0.297076
$980$		0			0
$981$	2898.00			0.0943181
$982$		0			0
$983$			11464.0i			0.371968i	0.982553	+	0.185984i	$0.0595473\pi$
$983$			11464.0i			0.371968i	−0.982553	+	0.185984i	$0.940453\pi$
$984$		0			0
$985$	−12890.0	+	25780.0i	−0.416964	+	0.833928i
$986$		0			0
$987$			29568.0i			0.953556i
$988$		0			0
$989$	−27824.0			−0.894592
$990$		0			0
$991$	−28952.0			−0.928043			−0.464021	−	0.885824i	$-0.653594\pi$
$991$	−28952.0			−0.928043			−0.464021	+	0.885824i	$0.653594\pi$
$992$		0			0
$993$			384.000i			0.0122718i
$994$		0			0
$995$	−5280.00	−	2640.00i	−0.168228	−	0.0841142i
$996$		0			0
$997$			8022.00i			0.254824i	0.991850	+	0.127412i	$0.0406670\pi$
$997$			8022.00i			0.254824i	−0.991850	+	0.127412i	$0.959333\pi$
$998$		0			0
$999$	10962.0			0.347170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.d.a.49.1	✓	2
3.2	odd	2		180.4.d.b.109.2		2
4.3	odd	2		240.4.f.a.49.2		2
5.2	odd	4		300.4.a.d.1.1		1
5.3	odd	4		300.4.a.f.1.1		1
5.4	even	2	inner	60.4.d.a.49.2	yes	2
8.3	odd	2		960.4.f.i.769.1		2
8.5	even	2		960.4.f.j.769.2		2
12.11	even	2		720.4.f.h.289.2		2
15.2	even	4		900.4.a.o.1.1		1
15.8	even	4		900.4.a.d.1.1		1
15.14	odd	2		180.4.d.b.109.1		2
20.3	even	4		1200.4.a.q.1.1		1
20.7	even	4		1200.4.a.w.1.1		1
20.19	odd	2		240.4.f.a.49.1		2
40.19	odd	2		960.4.f.i.769.2		2
40.29	even	2		960.4.f.j.769.1		2
60.59	even	2		720.4.f.h.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.d.a.49.1	✓	2	1.1	even	1	trivial
60.4.d.a.49.2	yes	2	5.4	even	2	inner
180.4.d.b.109.1		2	15.14	odd	2
180.4.d.b.109.2		2	3.2	odd	2
240.4.f.a.49.1		2	20.19	odd	2
240.4.f.a.49.2		2	4.3	odd	2
300.4.a.d.1.1		1	5.2	odd	4
300.4.a.f.1.1		1	5.3	odd	4
720.4.f.h.289.1		2	60.59	even	2
720.4.f.h.289.2		2	12.11	even	2
900.4.a.d.1.1		1	15.8	even	4
900.4.a.o.1.1		1	15.2	even	4
960.4.f.i.769.1		2	8.3	odd	2
960.4.f.i.769.2		2	40.19	odd	2
960.4.f.j.769.1		2	40.29	even	2
960.4.f.j.769.2		2	8.5	even	2
1200.4.a.q.1.1		1	20.3	even	4
1200.4.a.w.1.1		1	20.7	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:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	60.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} -22.0000i q^{7} -9.00000 q^{9} -14.0000 q^{11} -30.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} -62.0000i q^{17} +120.000 q^{19} -66.0000 q^{21} +188.000i q^{23} +(75.0000 + 100.000i) q^{25} +27.0000i q^{27} -96.0000 q^{29} +184.000 q^{31} +42.0000i q^{33} +(-110.000 + 220.000i) q^{35} -406.000i q^{37} -90.0000 q^{39} +130.000 q^{41} +148.000i q^{43} +(90.0000 + 45.0000i) q^{45} -448.000i q^{47} -141.000 q^{49} -186.000 q^{51} -414.000i q^{53} +(140.000 + 70.0000i) q^{55} -360.000i q^{57} -266.000 q^{59} -838.000 q^{61} +198.000i q^{63} +(-150.000 + 300.000i) q^{65} -248.000i q^{67} +564.000 q^{69} +1020.00 q^{71} +484.000i q^{73} +(300.000 - 225.000i) q^{75} +308.000i q^{77} +48.0000 q^{79} +81.0000 q^{81} +548.000i q^{83} +(-310.000 + 620.000i) q^{85} +288.000i q^{87} +650.000 q^{89} -660.000 q^{91} -552.000i q^{93} +(-1200.00 - 600.000i) q^{95} +1816.00i q^{97} +126.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{5} - 18 q^{9} - 28 q^{11} - 30 q^{15} + 240 q^{19} - 132 q^{21} + 150 q^{25} - 192 q^{29} + 368 q^{31} - 220 q^{35} - 180 q^{39} + 260 q^{41} + 180 q^{45} - 282 q^{49} - 372 q^{51} + 280 q^{55}+ \cdots + 252 q^{99}+O(q^{100}) \) 2 * q - 20 * q^5 - 18 * q^9 - 28 * q^11 - 30 * q^15 + 240 * q^19 - 132 * q^21 + 150 * q^25 - 192 * q^29 + 368 * q^31 - 220 * q^35 - 180 * q^39 + 260 * q^41 + 180 * q^45 - 282 * q^49 - 372 * q^51 + 280 * q^55 - 532 * q^59 - 1676 * q^61 - 300 * q^65 + 1128 * q^69 + 2040 * q^71 + 600 * q^75 + 96 * q^79 + 162 * q^81 - 620 * q^85 + 1300 * q^89 - 1320 * q^91 - 2400 * q^95 + 252 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more