LMFDB - Embedded newform 480.4.f.b.289.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,4,Mod(289,480)] 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("480.289"); 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, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-20,0,0,0,-18,0,60] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$28.3209168028$
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			289.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	480.289
Dual form			480.4.f.b.289.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} -18.0000i q^{7} -9.00000 q^{9} +30.0000 q^{11} -38.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} -70.0000i q^{17} -12.0000 q^{19} -54.0000 q^{21} -72.0000i q^{23} +(75.0000 + 100.000i) q^{25} +27.0000i q^{27} +64.0000 q^{29} -312.000 q^{31} -90.0000i q^{33} +(-90.0000 + 180.000i) q^{35} +138.000i q^{37} -114.000 q^{39} -374.000 q^{41} +468.000i q^{43} +(90.0000 + 45.0000i) q^{45} +132.000i q^{47} +19.0000 q^{49} -210.000 q^{51} -446.000i q^{53} +(-300.000 - 150.000i) q^{55} +36.0000i q^{57} -510.000 q^{59} +754.000 q^{61} +162.000i q^{63} +(-190.000 + 380.000i) q^{65} +384.000i q^{67} -216.000 q^{69} -924.000 q^{71} +340.000i q^{73} +(300.000 - 225.000i) q^{75} -540.000i q^{77} -72.0000 q^{79} +81.0000 q^{81} +156.000i q^{83} +(-350.000 + 700.000i) q^{85} -192.000i q^{87} +290.000 q^{89} -684.000 q^{91} +936.000i q^{93} +(120.000 + 60.0000i) q^{95} -376.000i q^{97} -270.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 20 q^{5} - 18 q^{9} + 60 q^{11} - 30 q^{15} - 24 q^{19} - 108 q^{21} + 150 q^{25} + 128 q^{29} - 624 q^{31} - 180 q^{35} - 228 q^{39} - 748 q^{41} + 180 q^{45} + 38 q^{49} - 420 q^{51} - 600 q^{55}+ \cdots - 540 q^{99}+O(q^{100}) $ 2 * q - 20 * q^5 - 18 * q^9 + 60 * q^11 - 30 * q^15 - 24 * q^19 - 108 * q^21 + 150 * q^25 + 128 * q^29 - 624 * q^31 - 180 * q^35 - 228 * q^39 - 748 * q^41 + 180 * q^45 + 38 * q^49 - 420 * q^51 - 600 * q^55 - 1020 * q^59 + 1508 * q^61 - 380 * q^65 - 432 * q^69 - 1848 * q^71 + 600 * q^75 - 144 * q^79 + 162 * q^81 - 700 * q^85 + 580 * q^89 - 1368 * q^91 + 240 * q^95 - 540 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/480\mathbb{Z}\right)^\times$.

$n$	$31$	$97$	$161$	$421$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$7$		−	18.0000i		−	0.971909i	−0.873984	−	0.485954i	$-0.838472\pi$
$7$		−	18.0000i		−	0.971909i	0.873984	−	0.485954i	$-0.161528\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	30.0000			0.822304			0.411152	−	0.911567i	$-0.365127\pi$
$11$	30.0000			0.822304			0.411152	+	0.911567i	$0.365127\pi$
$12$		0			0
$13$		−	38.0000i		−	0.810716i	−0.914158	−	0.405358i	$-0.867147\pi$
$13$		−	38.0000i		−	0.810716i	0.914158	−	0.405358i	$-0.132853\pi$
$14$		0			0
$15$	−15.0000	+	30.0000i	−0.258199	+	0.516398i
$16$		0			0
$17$		−	70.0000i		−	0.998676i	−0.866407	−	0.499338i	$-0.833577\pi$
$17$		−	70.0000i		−	0.998676i	0.866407	−	0.499338i	$-0.166423\pi$
$18$		0			0
$19$	−12.0000			−0.144894			−0.0724471	−	0.997372i	$-0.523081\pi$
$19$	−12.0000			−0.144894			−0.0724471	+	0.997372i	$0.523081\pi$
$20$		0			0
$21$	−54.0000			−0.561132
$22$		0			0
$23$		−	72.0000i		−	0.652741i	−0.945242	−	0.326370i	$-0.894174\pi$
$23$		−	72.0000i		−	0.652741i	0.945242	−	0.326370i	$-0.105826\pi$
$24$		0			0
$25$	75.0000	+	100.000i	0.600000	+	0.800000i
$26$		0			0
$27$			27.0000i			0.192450i
$28$		0			0
$29$	64.0000			0.409810			0.204905	−	0.978782i	$-0.434311\pi$
$29$	64.0000			0.409810			0.204905	+	0.978782i	$0.434311\pi$
$30$		0			0
$31$	−312.000			−1.80764			−0.903820	−	0.427912i	$-0.859249\pi$
$31$	−312.000			−1.80764			−0.903820	+	0.427912i	$0.859249\pi$
$32$		0			0
$33$		−	90.0000i		−	0.474757i
$34$		0			0
$35$	−90.0000	+	180.000i	−0.434651	+	0.869302i
$36$		0			0
$37$			138.000i			0.613164i	0.951844	+	0.306582i	$0.0991853\pi$
$37$			138.000i			0.613164i	−0.951844	+	0.306582i	$0.900815\pi$
$38$		0			0
$39$	−114.000			−0.468067
$40$		0			0
$41$	−374.000			−1.42461			−0.712305	−	0.701870i	$-0.752349\pi$
$41$	−374.000			−1.42461			−0.712305	+	0.701870i	$0.752349\pi$
$42$		0			0
$43$			468.000i			1.65975i	0.557948	+	0.829876i	$0.311589\pi$
$43$			468.000i			1.65975i	−0.557948	+	0.829876i	$0.688411\pi$
$44$		0			0
$45$	90.0000	+	45.0000i	0.298142	+	0.149071i
$46$		0			0
$47$			132.000i			0.409663i	0.978797	+	0.204832i	$0.0656647\pi$
$47$			132.000i			0.409663i	−0.978797	+	0.204832i	$0.934335\pi$
$48$		0			0
$49$	19.0000			0.0553936
$50$		0			0
$51$	−210.000			−0.576586
$52$		0			0
$53$		−	446.000i		−	1.15590i	−0.816071	−	0.577951i	$-0.803852\pi$
$53$		−	446.000i		−	1.15590i	0.816071	−	0.577951i	$-0.196148\pi$
$54$		0			0
$55$	−300.000	−	150.000i	−0.735491	−	0.367745i
$56$		0			0
$57$			36.0000i			0.0836547i
$58$		0			0
$59$	−510.000			−1.12536			−0.562681	−	0.826674i	$-0.690230\pi$
$59$	−510.000			−1.12536			−0.562681	+	0.826674i	$0.690230\pi$
$60$		0			0
$61$	754.000			1.58262			0.791310	−	0.611415i	$-0.209399\pi$
$61$	754.000			1.58262			0.791310	+	0.611415i	$0.209399\pi$
$62$		0			0
$63$			162.000i			0.323970i
$64$		0			0
$65$	−190.000	+	380.000i	−0.362563	+	0.725126i
$66$		0			0
$67$			384.000i			0.700195i	0.936713	+	0.350098i	$0.113852\pi$
$67$			384.000i			0.700195i	−0.936713	+	0.350098i	$0.886148\pi$
$68$		0			0
$69$	−216.000			−0.376860
$70$		0			0
$71$	−924.000			−1.54449			−0.772244	−	0.635326i	$-0.780866\pi$
$71$	−924.000			−1.54449			−0.772244	+	0.635326i	$0.780866\pi$
$72$		0			0
$73$			340.000i			0.545123i	0.962138	+	0.272562i	$0.0878708\pi$
$73$			340.000i			0.545123i	−0.962138	+	0.272562i	$0.912129\pi$
$74$		0			0
$75$	300.000	−	225.000i	0.461880	−	0.346410i
$76$		0			0
$77$		−	540.000i		−	0.799204i
$78$		0			0
$79$	−72.0000			−0.102540			−0.0512698	−	0.998685i	$-0.516327\pi$
$79$	−72.0000			−0.102540			−0.0512698	+	0.998685i	$0.516327\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			156.000i			0.206304i	0.994666	+	0.103152i	$0.0328928\pi$
$83$			156.000i			0.206304i	−0.994666	+	0.103152i	$0.967107\pi$
$84$		0			0
$85$	−350.000	+	700.000i	−0.446622	+	0.893243i
$86$		0			0
$87$		−	192.000i		−	0.236604i
$88$		0			0
$89$	290.000			0.345393			0.172696	−	0.984975i	$-0.444752\pi$
$89$	290.000			0.345393			0.172696	+	0.984975i	$0.444752\pi$
$90$		0			0
$91$	−684.000			−0.787942
$92$		0			0
$93$			936.000i			1.04364i
$94$		0			0
$95$	120.000	+	60.0000i	0.129597	+	0.0647986i
$96$		0			0
$97$		−	376.000i		−	0.393577i	−0.980446	−	0.196789i	$-0.936949\pi$
$97$		−	376.000i		−	0.393577i	0.980446	−	0.196789i	$-0.0630513\pi$
$98$		0			0
$99$	−270.000			−0.274101
$100$		0			0
$101$	104.000			0.102459			0.0512296	−	0.998687i	$-0.483686\pi$
$101$	104.000			0.102459			0.0512296	+	0.998687i	$0.483686\pi$
$102$		0			0
$103$		−	1014.00i		−	0.970023i	−0.874508	−	0.485012i	$-0.838815\pi$
$103$		−	1014.00i		−	0.970023i	0.874508	−	0.485012i	$-0.161185\pi$
$104$		0			0
$105$	540.000	+	270.000i	0.501891	+	0.250946i
$106$		0			0
$107$			1284.00i			1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$			1284.00i			1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$		0			0
$109$	−570.000			−0.500882			−0.250441	−	0.968132i	$-0.580576\pi$
$109$	−570.000			−0.500882			−0.250441	+	0.968132i	$0.580576\pi$
$110$		0			0
$111$	414.000			0.354010
$112$		0			0
$113$		−	254.000i		−	0.211454i	−0.994395	−	0.105727i	$-0.966283\pi$
$113$		−	254.000i		−	0.211454i	0.994395	−	0.105727i	$-0.0337170\pi$
$114$		0			0
$115$	−360.000	+	720.000i	−0.291915	+	0.583829i
$116$		0			0
$117$			342.000i			0.270239i
$118$		0			0
$119$	−1260.00			−0.970622
$120$		0			0
$121$	−431.000			−0.323817
$122$		0			0
$123$			1122.00i			0.822499i
$124$		0			0
$125$	−250.000	−	1375.00i	−0.178885	−	0.983870i
$126$		0			0
$127$		−	138.000i		−	0.0964214i	−0.998837	−	0.0482107i	$-0.984648\pi$
$127$		−	138.000i		−	0.0964214i	0.998837	−	0.0482107i	$-0.0153519\pi$
$128$		0			0
$129$	1404.00			0.958258
$130$		0			0
$131$	−894.000			−0.596253			−0.298126	−	0.954526i	$-0.596362\pi$
$131$	−894.000			−0.596253			−0.298126	+	0.954526i	$0.596362\pi$
$132$		0			0
$133$			216.000i			0.140824i
$134$		0			0
$135$	135.000	−	270.000i	0.0860663	−	0.172133i
$136$		0			0
$137$		−	3050.00i		−	1.90204i	−0.309134	−	0.951019i	$-0.600039\pi$
$137$		−	3050.00i		−	1.90204i	0.309134	−	0.951019i	$-0.399961\pi$
$138$		0			0
$139$	2352.00			1.43521			0.717604	−	0.696451i	$-0.245239\pi$
$139$	2352.00			1.43521			0.717604	+	0.696451i	$0.245239\pi$
$140$		0			0
$141$	396.000			0.236519
$142$		0			0
$143$		−	1140.00i		−	0.666654i
$144$		0			0
$145$	−640.000	−	320.000i	−0.366546	−	0.183273i
$146$		0			0
$147$		−	57.0000i		−	0.0319815i
$148$		0			0
$149$	−2476.00			−1.36135			−0.680677	−	0.732583i	$-0.738314\pi$
$149$	−2476.00			−1.36135			−0.680677	+	0.732583i	$0.738314\pi$
$150$		0			0
$151$	−3192.00			−1.72027			−0.860137	−	0.510064i	$-0.829622\pi$
$151$	−3192.00			−1.72027			−0.860137	+	0.510064i	$0.829622\pi$
$152$		0			0
$153$			630.000i			0.332892i
$154$		0			0
$155$	3120.00	+	1560.00i	1.61680	+	0.808401i
$156$		0			0
$157$		−	1218.00i		−	0.619153i	−0.950875	−	0.309576i	$-0.899813\pi$
$157$		−	1218.00i		−	0.619153i	0.950875	−	0.309576i	$-0.100187\pi$
$158$		0			0
$159$	−1338.00			−0.667360
$160$		0			0
$161$	−1296.00			−0.634404
$162$		0			0
$163$			2328.00i			1.11867i	0.828942	+	0.559334i	$0.188943\pi$
$163$			2328.00i			1.11867i	−0.828942	+	0.559334i	$0.811057\pi$
$164$		0			0
$165$	−450.000	+	900.000i	−0.212318	+	0.424636i
$166$		0			0
$167$		−	2928.00i		−	1.35674i	−0.734721	−	0.678370i	$-0.762687\pi$
$167$		−	2928.00i		−	1.35674i	0.734721	−	0.678370i	$-0.237313\pi$
$168$		0			0
$169$	753.000			0.342740
$170$		0			0
$171$	108.000			0.0482980
$172$		0			0
$173$			502.000i			0.220615i	0.993898	+	0.110307i	$0.0351835\pi$
$173$			502.000i			0.220615i	−0.993898	+	0.110307i	$0.964816\pi$
$174$		0			0
$175$	1800.00	−	1350.00i	0.777527	−	0.583145i
$176$		0			0
$177$			1530.00i			0.649728i
$178$		0			0
$179$	4026.00			1.68110			0.840551	−	0.541732i	$-0.182231\pi$
$179$	4026.00			1.68110			0.840551	+	0.541732i	$0.182231\pi$
$180$		0			0
$181$	594.000			0.243932			0.121966	−	0.992534i	$-0.461080\pi$
$181$	594.000			0.243932			0.121966	+	0.992534i	$0.461080\pi$
$182$		0			0
$183$		−	2262.00i		−	0.913726i
$184$		0			0
$185$	690.000	−	1380.00i	0.274215	−	0.548430i
$186$		0			0
$187$		−	2100.00i		−	0.821215i
$188$		0			0
$189$	486.000			0.187044
$190$		0			0
$191$	1044.00			0.395504			0.197752	−	0.980252i	$-0.436636\pi$
$191$	1044.00			0.395504			0.197752	+	0.980252i	$0.436636\pi$
$192$		0			0
$193$			3252.00i			1.21287i	0.795133	+	0.606435i	$0.207401\pi$
$193$			3252.00i			1.21287i	−0.795133	+	0.606435i	$0.792599\pi$
$194$		0			0
$195$	1140.00	+	570.000i	0.418652	+	0.209326i
$196$		0			0
$197$		−	2914.00i		−	1.05388i	−0.849903	−	0.526939i	$-0.823340\pi$
$197$		−	2914.00i		−	1.05388i	0.849903	−	0.526939i	$-0.176660\pi$
$198$		0			0
$199$	4872.00			1.73551			0.867756	−	0.496990i	$-0.165562\pi$
$199$	4872.00			1.73551			0.867756	+	0.496990i	$0.165562\pi$
$200$		0			0
$201$	1152.00			0.404258
$202$		0			0
$203$		−	1152.00i		−	0.398298i
$204$		0			0
$205$	3740.00	+	1870.00i	1.27421	+	0.637105i
$206$		0			0
$207$			648.000i			0.217580i
$208$		0			0
$209$	−360.000			−0.119147
$210$		0			0
$211$	1872.00			0.610776			0.305388	−	0.952228i	$-0.401214\pi$
$211$	1872.00			0.610776			0.305388	+	0.952228i	$0.401214\pi$
$212$		0			0
$213$			2772.00i			0.891710i
$214$		0			0
$215$	2340.00	−	4680.00i	0.742264	−	1.48453i
$216$		0			0
$217$			5616.00i			1.75686i
$218$		0			0
$219$	1020.00			0.314727
$220$		0			0
$221$	−2660.00			−0.809642
$222$		0			0
$223$		−	1302.00i		−	0.390979i	−0.980706	−	0.195490i	$-0.937370\pi$
$223$		−	1302.00i		−	0.390979i	0.980706	−	0.195490i	$-0.0626296\pi$
$224$		0			0
$225$	−675.000	−	900.000i	−0.200000	−	0.266667i
$226$		0			0
$227$		−	708.000i		−	0.207011i	−0.994629	−	0.103506i	$-0.966994\pi$
$227$		−	708.000i		−	0.207011i	0.994629	−	0.103506i	$-0.0330060\pi$
$228$		0			0
$229$	5082.00			1.46650			0.733249	−	0.679960i	$-0.238003\pi$
$229$	5082.00			1.46650			0.733249	+	0.679960i	$0.238003\pi$
$230$		0			0
$231$	−1620.00			−0.461421
$232$		0			0
$233$		−	5786.00i		−	1.62684i	−0.581678	−	0.813419i	$-0.697603\pi$
$233$		−	5786.00i		−	1.62684i	0.581678	−	0.813419i	$-0.302397\pi$
$234$		0			0
$235$	660.000	−	1320.00i	0.183207	−	0.366414i
$236$		0			0
$237$			216.000i			0.0592013i
$238$		0			0
$239$	3156.00			0.854162			0.427081	−	0.904213i	$-0.359542\pi$
$239$	3156.00			0.854162			0.427081	+	0.904213i	$0.359542\pi$
$240$		0			0
$241$	−3306.00			−0.883644			−0.441822	−	0.897103i	$-0.645668\pi$
$241$	−3306.00			−0.883644			−0.441822	+	0.897103i	$0.645668\pi$
$242$		0			0
$243$		−	243.000i		−	0.0641500i
$244$		0			0
$245$	−190.000	−	95.0000i	−0.0495455	−	0.0247728i
$246$		0			0
$247$			456.000i			0.117468i
$248$		0			0
$249$	468.000			0.119110
$250$		0			0
$251$	−4134.00			−1.03958			−0.519792	−	0.854293i	$-0.673991\pi$
$251$	−4134.00			−1.03958			−0.519792	+	0.854293i	$0.673991\pi$
$252$		0			0
$253$		−	2160.00i		−	0.536751i
$254$		0			0
$255$	2100.00	+	1050.00i	0.515714	+	0.257857i
$256$		0			0
$257$		−	3482.00i		−	0.845141i	−0.906330	−	0.422570i	$-0.861128\pi$
$257$		−	3482.00i		−	0.845141i	0.906330	−	0.422570i	$-0.138872\pi$
$258$		0			0
$259$	2484.00			0.595939
$260$		0			0
$261$	−576.000			−0.136603
$262$		0			0
$263$			2856.00i			0.669614i	0.942287	+	0.334807i	$0.108671\pi$
$263$			2856.00i			0.669614i	−0.942287	+	0.334807i	$0.891329\pi$
$264$		0			0
$265$	−2230.00	+	4460.00i	−0.516935	+	1.03387i
$266$		0			0
$267$		−	870.000i		−	0.199412i
$268$		0			0
$269$	−6464.00			−1.46512			−0.732560	−	0.680703i	$-0.761674\pi$
$269$	−6464.00			−1.46512			−0.732560	+	0.680703i	$0.761674\pi$
$270$		0			0
$271$	−5184.00			−1.16201			−0.581007	−	0.813899i	$-0.697341\pi$
$271$	−5184.00			−1.16201			−0.581007	+	0.813899i	$0.697341\pi$
$272$		0			0
$273$			2052.00i			0.454918i
$274$		0			0
$275$	2250.00	+	3000.00i	0.493382	+	0.657843i
$276$		0			0
$277$			3818.00i			0.828164i	0.910240	+	0.414082i	$0.135897\pi$
$277$			3818.00i			0.828164i	−0.910240	+	0.414082i	$0.864103\pi$
$278$		0			0
$279$	2808.00			0.602547
$280$		0			0
$281$	6094.00			1.29373			0.646864	−	0.762605i	$-0.276080\pi$
$281$	6094.00			1.29373			0.646864	+	0.762605i	$0.276080\pi$
$282$		0			0
$283$		−	4440.00i		−	0.932617i	−0.884622	−	0.466308i	$-0.845584\pi$
$283$		−	4440.00i		−	0.932617i	0.884622	−	0.466308i	$-0.154416\pi$
$284$		0			0
$285$	180.000	−	360.000i	0.0374115	−	0.0748230i
$286$		0			0
$287$			6732.00i			1.38459i
$288$		0			0
$289$	13.0000			0.00264604
$290$		0			0
$291$	−1128.00			−0.227232
$292$		0			0
$293$		−	5950.00i		−	1.18636i	−0.805071	−	0.593179i	$-0.797873\pi$
$293$		−	5950.00i		−	1.18636i	0.805071	−	0.593179i	$-0.202127\pi$
$294$		0			0
$295$	5100.00	+	2550.00i	1.00655	+	0.503277i
$296$		0			0
$297$			810.000i			0.158252i
$298$		0			0
$299$	−2736.00			−0.529187
$300$		0			0
$301$	8424.00			1.61313
$302$		0			0
$303$		−	312.000i		−	0.0591549i
$304$		0			0
$305$	−7540.00	−	3770.00i	−1.41554	−	0.707769i
$306$		0			0
$307$			924.000i			0.171777i	0.996305	+	0.0858884i	$0.0273728\pi$
$307$			924.000i			0.171777i	−0.996305	+	0.0858884i	$0.972627\pi$
$308$		0			0
$309$	−3042.00			−0.560043
$310$		0			0
$311$	−7020.00			−1.27996			−0.639980	−	0.768391i	$-0.721057\pi$
$311$	−7020.00			−1.27996			−0.639980	+	0.768391i	$0.721057\pi$
$312$		0			0
$313$		−	5244.00i		−	0.946992i	−0.880796	−	0.473496i	$-0.842992\pi$
$313$		−	5244.00i		−	0.946992i	0.880796	−	0.473496i	$-0.157008\pi$
$314$		0			0
$315$	810.000	−	1620.00i	0.144884	−	0.289767i
$316$		0			0
$317$			5194.00i			0.920265i	0.887850	+	0.460133i	$0.152198\pi$
$317$			5194.00i			0.920265i	−0.887850	+	0.460133i	$0.847802\pi$
$318$		0			0
$319$	1920.00			0.336989
$320$		0			0
$321$	3852.00			0.669775
$322$		0			0
$323$			840.000i			0.144702i
$324$		0			0
$325$	3800.00	−	2850.00i	0.648573	−	0.486429i
$326$		0			0
$327$			1710.00i			0.289184i
$328$		0			0
$329$	2376.00			0.398155
$330$		0			0
$331$	−11628.0			−1.93091			−0.965457	−	0.260562i	$-0.916092\pi$
$331$	−11628.0			−1.93091			−0.965457	+	0.260562i	$0.916092\pi$
$332$		0			0
$333$		−	1242.00i		−	0.204388i
$334$		0			0
$335$	1920.00	−	3840.00i	0.313137	−	0.626273i
$336$		0			0
$337$			4372.00i			0.706700i	0.935491	+	0.353350i	$0.114958\pi$
$337$			4372.00i			0.706700i	−0.935491	+	0.353350i	$0.885042\pi$
$338$		0			0
$339$	−762.000			−0.122083
$340$		0			0
$341$	−9360.00			−1.48643
$342$		0			0
$343$		−	6516.00i		−	1.02575i
$344$		0			0
$345$	2160.00	+	1080.00i	0.337074	+	0.168537i
$346$		0			0
$347$		−	12564.0i		−	1.94372i	−0.235559	−	0.971860i	$-0.575692\pi$
$347$		−	12564.0i		−	1.94372i	0.235559	−	0.971860i	$-0.424308\pi$
$348$		0			0
$349$	−970.000			−0.148776			−0.0743881	−	0.997229i	$-0.523700\pi$
$349$	−970.000			−0.148776			−0.0743881	+	0.997229i	$0.523700\pi$
$350$		0			0
$351$	1026.00			0.156022
$352$		0			0
$353$			1958.00i			0.295223i	0.989045	+	0.147612i	$0.0471586\pi$
$353$			1958.00i			0.295223i	−0.989045	+	0.147612i	$0.952841\pi$
$354$		0			0
$355$	9240.00	+	4620.00i	1.38143	+	0.690716i
$356$		0			0
$357$			3780.00i			0.560389i
$358$		0			0
$359$	1920.00			0.282267			0.141133	−	0.989991i	$-0.454925\pi$
$359$	1920.00			0.282267			0.141133	+	0.989991i	$0.454925\pi$
$360$		0			0
$361$	−6715.00			−0.979006
$362$		0			0
$363$			1293.00i			0.186956i
$364$		0			0
$365$	1700.00	−	3400.00i	0.243786	−	0.487573i
$366$		0			0
$367$			8886.00i			1.26388i	0.775016	+	0.631942i	$0.217742\pi$
$367$			8886.00i			1.26388i	−0.775016	+	0.631942i	$0.782258\pi$
$368$		0			0
$369$	3366.00			0.474870
$370$		0			0
$371$	−8028.00			−1.12343
$372$		0			0
$373$		−	7470.00i		−	1.03695i	−0.855093	−	0.518474i	$-0.826500\pi$
$373$		−	7470.00i		−	1.03695i	0.855093	−	0.518474i	$-0.173500\pi$
$374$		0			0
$375$	−4125.00	+	750.000i	−0.568038	+	0.103280i
$376$		0			0
$377$		−	2432.00i		−	0.332240i
$378$		0			0
$379$	−6312.00			−0.855477			−0.427738	−	0.903903i	$-0.640690\pi$
$379$	−6312.00			−0.855477			−0.427738	+	0.903903i	$0.640690\pi$
$380$		0			0
$381$	−414.000			−0.0556689
$382$		0			0
$383$			12876.0i			1.71784i	0.512109	+	0.858920i	$0.328864\pi$
$383$			12876.0i			1.71784i	−0.512109	+	0.858920i	$0.671136\pi$
$384$		0			0
$385$	−2700.00	+	5400.00i	−0.357415	+	0.714830i
$386$		0			0
$387$		−	4212.00i		−	0.553251i
$388$		0			0
$389$	8780.00			1.14438			0.572190	−	0.820121i	$-0.306094\pi$
$389$	8780.00			1.14438			0.572190	+	0.820121i	$0.306094\pi$
$390$		0			0
$391$	−5040.00			−0.651877
$392$		0			0
$393$			2682.00i			0.344247i
$394$		0			0
$395$	720.000	+	360.000i	0.0917143	+	0.0458571i
$396$		0			0
$397$		−	3246.00i		−	0.410358i	−0.978724	−	0.205179i	$-0.934222\pi$
$397$		−	3246.00i		−	0.410358i	0.978724	−	0.205179i	$-0.0657776\pi$
$398$		0			0
$399$	648.000			0.0813047
$400$		0			0
$401$	1870.00			0.232876			0.116438	−	0.993198i	$-0.462852\pi$
$401$	1870.00			0.232876			0.116438	+	0.993198i	$0.462852\pi$
$402$		0			0
$403$			11856.0i			1.46548i
$404$		0			0
$405$	−810.000	−	405.000i	−0.0993808	−	0.0496904i
$406$		0			0
$407$			4140.00i			0.504207i
$408$		0			0
$409$	−15990.0			−1.93314			−0.966570	−	0.256401i	$-0.917463\pi$
$409$	−15990.0			−1.93314			−0.966570	+	0.256401i	$0.917463\pi$
$410$		0			0
$411$	−9150.00			−1.09814
$412$		0			0
$413$			9180.00i			1.09375i
$414$		0			0
$415$	780.000	−	1560.00i	0.0922619	−	0.184524i
$416$		0			0
$417$		−	7056.00i		−	0.828618i
$418$		0			0
$419$	3054.00			0.356080			0.178040	−	0.984023i	$-0.443024\pi$
$419$	3054.00			0.356080			0.178040	+	0.984023i	$0.443024\pi$
$420$		0			0
$421$	−1194.00			−0.138223			−0.0691116	−	0.997609i	$-0.522016\pi$
$421$	−1194.00			−0.138223			−0.0691116	+	0.997609i	$0.522016\pi$
$422$		0			0
$423$		−	1188.00i		−	0.136554i
$424$		0			0
$425$	7000.00	−	5250.00i	0.798941	−	0.599206i
$426$		0			0
$427$		−	13572.0i		−	1.53816i
$428$		0			0
$429$	−3420.00			−0.384893
$430$		0			0
$431$	−15264.0			−1.70590			−0.852948	−	0.521996i	$-0.825188\pi$
$431$	−15264.0			−1.70590			−0.852948	+	0.521996i	$0.825188\pi$
$432$		0			0
$433$		−	8220.00i		−	0.912305i	−0.889902	−	0.456152i	$-0.849227\pi$
$433$		−	8220.00i		−	0.912305i	0.889902	−	0.456152i	$-0.150773\pi$
$434$		0			0
$435$	−960.000	+	1920.00i	−0.105813	+	0.211625i
$436$		0			0
$437$			864.000i			0.0945783i
$438$		0			0
$439$	−16248.0			−1.76646			−0.883229	−	0.468943i	$-0.844635\pi$
$439$	−16248.0			−1.76646			−0.883229	+	0.468943i	$0.844635\pi$
$440$		0			0
$441$	−171.000			−0.0184645
$442$		0			0
$443$		−	8220.00i		−	0.881589i	−0.897608	−	0.440795i	$-0.854697\pi$
$443$		−	8220.00i		−	0.881589i	0.897608	−	0.440795i	$-0.145303\pi$
$444$		0			0
$445$	−2900.00	−	1450.00i	−0.308929	−	0.154464i
$446$		0			0
$447$			7428.00i			0.785978i
$448$		0			0
$449$	9490.00			0.997463			0.498731	−	0.866757i	$-0.333799\pi$
$449$	9490.00			0.997463			0.498731	+	0.866757i	$0.333799\pi$
$450$		0			0
$451$	−11220.0			−1.17146
$452$		0			0
$453$			9576.00i			0.993200i
$454$		0			0
$455$	6840.00	+	3420.00i	0.704756	+	0.352378i
$456$		0			0
$457$			5776.00i			0.591225i	0.955308	+	0.295613i	$0.0955237\pi$
$457$			5776.00i			0.591225i	−0.955308	+	0.295613i	$0.904476\pi$
$458$		0			0
$459$	1890.00			0.192195
$460$		0			0
$461$	2812.00			0.284095			0.142048	−	0.989860i	$-0.454631\pi$
$461$	2812.00			0.284095			0.142048	+	0.989860i	$0.454631\pi$
$462$		0			0
$463$			13710.0i			1.37615i	0.725639	+	0.688075i	$0.241544\pi$
$463$			13710.0i			1.37615i	−0.725639	+	0.688075i	$0.758456\pi$
$464$		0			0
$465$	4680.00	−	9360.00i	0.466731	−	0.933462i
$466$		0			0
$467$			6252.00i			0.619503i	0.950817	+	0.309752i	$0.100246\pi$
$467$			6252.00i			0.619503i	−0.950817	+	0.309752i	$0.899754\pi$
$468$		0			0
$469$	6912.00			0.680526
$470$		0			0
$471$	−3654.00			−0.357468
$472$		0			0
$473$			14040.0i			1.36482i
$474$		0			0
$475$	−900.000	−	1200.00i	−0.0869365	−	0.115915i
$476$		0			0
$477$			4014.00i			0.385301i
$478$		0			0
$479$	7908.00			0.754333			0.377167	−	0.926145i	$-0.376898\pi$
$479$	7908.00			0.754333			0.377167	+	0.926145i	$0.376898\pi$
$480$		0			0
$481$	5244.00			0.497101
$482$		0			0
$483$			3888.00i			0.366274i
$484$		0			0
$485$	−1880.00	+	3760.00i	−0.176013	+	0.352026i
$486$		0			0
$487$			17562.0i			1.63411i	0.576562	+	0.817054i	$0.304394\pi$
$487$			17562.0i			1.63411i	−0.576562	+	0.817054i	$0.695606\pi$
$488$		0			0
$489$	6984.00			0.645864
$490$		0			0
$491$	−13062.0			−1.20057			−0.600285	−	0.799786i	$-0.704946\pi$
$491$	−13062.0			−1.20057			−0.600285	+	0.799786i	$0.704946\pi$
$492$		0			0
$493$		−	4480.00i		−	0.409268i
$494$		0			0
$495$	2700.00	+	1350.00i	0.245164	+	0.122582i
$496$		0			0
$497$			16632.0i			1.50110i
$498$		0			0
$499$	−8556.00			−0.767573			−0.383787	−	0.923422i	$-0.625380\pi$
$499$	−8556.00			−0.767573			−0.383787	+	0.923422i	$0.625380\pi$
$500$		0			0
$501$	−8784.00			−0.783314
$502$		0			0
$503$		−	14436.0i		−	1.27966i	−0.768516	−	0.639830i	$-0.779005\pi$
$503$		−	14436.0i		−	1.27966i	0.768516	−	0.639830i	$-0.220995\pi$
$504$		0			0
$505$	−1040.00	−	520.000i	−0.0916424	−	0.0458212i
$506$		0			0
$507$		−	2259.00i		−	0.197881i
$508$		0			0
$509$	−9536.00			−0.830404			−0.415202	−	0.909729i	$-0.636289\pi$
$509$	−9536.00			−0.830404			−0.415202	+	0.909729i	$0.636289\pi$
$510$		0			0
$511$	6120.00			0.529810
$512$		0			0
$513$		−	324.000i		−	0.0278849i
$514$		0			0
$515$	−5070.00	+	10140.0i	−0.433808	+	0.867615i
$516$		0			0
$517$			3960.00i			0.336868i
$518$		0			0
$519$	1506.00			0.127372
$520$		0			0
$521$	−1130.00			−0.0950215			−0.0475107	−	0.998871i	$-0.515129\pi$
$521$	−1130.00			−0.0950215			−0.0475107	+	0.998871i	$0.515129\pi$
$522$		0			0
$523$		−	4104.00i		−	0.343127i	−0.985173	−	0.171563i	$-0.945118\pi$
$523$		−	4104.00i		−	0.343127i	0.985173	−	0.171563i	$-0.0548819\pi$
$524$		0			0
$525$	−4050.00	−	5400.00i	−0.336679	−	0.448905i
$526$		0			0
$527$			21840.0i			1.80525i
$528$		0			0
$529$	6983.00			0.573929
$530$		0			0
$531$	4590.00			0.375121
$532$		0			0
$533$			14212.0i			1.15495i
$534$		0			0
$535$	6420.00	−	12840.0i	0.518805	−	1.03761i
$536$		0			0
$537$		−	12078.0i		−	0.970585i
$538$		0			0
$539$	570.000			0.0455503
$540$		0			0
$541$	6810.00			0.541192			0.270596	−	0.962693i	$-0.412779\pi$
$541$	6810.00			0.541192			0.270596	+	0.962693i	$0.412779\pi$
$542$		0			0
$543$		−	1782.00i		−	0.140834i
$544$		0			0
$545$	5700.00	+	2850.00i	0.448002	+	0.224001i
$546$		0			0
$547$		−	23772.0i		−	1.85817i	−0.369870	−	0.929083i	$-0.620598\pi$
$547$		−	23772.0i		−	1.85817i	0.369870	−	0.929083i	$-0.379402\pi$
$548$		0			0
$549$	−6786.00			−0.527540
$550$		0			0
$551$	−768.000			−0.0593791
$552$		0			0
$553$			1296.00i			0.0996592i
$554$		0			0
$555$	−4140.00	−	2070.00i	−0.316636	−	0.158318i
$556$		0			0
$557$		−	10762.0i		−	0.818672i	−0.912384	−	0.409336i	$-0.865760\pi$
$557$		−	10762.0i		−	0.818672i	0.912384	−	0.409336i	$-0.134240\pi$
$558$		0			0
$559$	17784.0			1.34559
$560$		0			0
$561$	−6300.00			−0.474129
$562$		0			0
$563$		−	1716.00i		−	0.128456i	−0.997935	−	0.0642280i	$-0.979541\pi$
$563$		−	1716.00i		−	0.128456i	0.997935	−	0.0642280i	$-0.0204585\pi$
$564$		0			0
$565$	−1270.00	+	2540.00i	−0.0945651	+	0.189130i
$566$		0			0
$567$		−	1458.00i		−	0.107990i
$568$		0			0
$569$	−6670.00			−0.491425			−0.245713	−	0.969343i	$-0.579022\pi$
$569$	−6670.00			−0.491425			−0.245713	+	0.969343i	$0.579022\pi$
$570$		0			0
$571$	−5580.00			−0.408959			−0.204480	−	0.978871i	$-0.565550\pi$
$571$	−5580.00			−0.408959			−0.204480	+	0.978871i	$0.565550\pi$
$572$		0			0
$573$		−	3132.00i		−	0.228344i
$574$		0			0
$575$	7200.00	−	5400.00i	0.522193	−	0.391644i
$576$		0			0
$577$		−	17808.0i		−	1.28485i	−0.766350	−	0.642424i	$-0.777929\pi$
$577$		−	17808.0i		−	1.28485i	0.766350	−	0.642424i	$-0.222071\pi$
$578$		0			0
$579$	9756.00			0.700251
$580$		0			0
$581$	2808.00			0.200509
$582$		0			0
$583$		−	13380.0i		−	0.950503i
$584$		0			0
$585$	1710.00	−	3420.00i	0.120854	−	0.241709i
$586$		0			0
$587$		−	23004.0i		−	1.61751i	−0.588148	−	0.808754i	$-0.700143\pi$
$587$		−	23004.0i		−	1.61751i	0.588148	−	0.808754i	$-0.299857\pi$
$588$		0			0
$589$	3744.00			0.261917
$590$		0			0
$591$	−8742.00			−0.608457
$592$		0			0
$593$			3266.00i			0.226170i	0.993585	+	0.113085i	$0.0360732\pi$
$593$			3266.00i			0.226170i	−0.993585	+	0.113085i	$0.963927\pi$
$594$		0			0
$595$	12600.0	+	6300.00i	0.868151	+	0.434075i
$596$		0			0
$597$		−	14616.0i		−	1.00200i
$598$		0			0
$599$	2208.00			0.150612			0.0753059	−	0.997160i	$-0.476007\pi$
$599$	2208.00			0.150612			0.0753059	+	0.997160i	$0.476007\pi$
$600$		0			0
$601$	−21166.0			−1.43657			−0.718285	−	0.695748i	$-0.755073\pi$
$601$	−21166.0			−1.43657			−0.718285	+	0.695748i	$0.755073\pi$
$602$		0			0
$603$		−	3456.00i		−	0.233398i
$604$		0			0
$605$	4310.00	+	2155.00i	0.289630	+	0.144815i
$606$		0			0
$607$			17610.0i			1.17754i	0.808300	+	0.588771i	$0.200388\pi$
$607$			17610.0i			1.17754i	−0.808300	+	0.588771i	$0.799612\pi$
$608$		0			0
$609$	−3456.00			−0.229958
$610$		0			0
$611$	5016.00			0.332121
$612$		0			0
$613$		−	29610.0i		−	1.95096i	−0.220095	−	0.975478i	$-0.570637\pi$
$613$		−	29610.0i		−	1.95096i	0.220095	−	0.975478i	$-0.429363\pi$
$614$		0			0
$615$	5610.00	−	11220.0i	0.367833	−	0.735665i
$616$		0			0
$617$		−	2294.00i		−	0.149681i	−0.997196	−	0.0748403i	$-0.976155\pi$
$617$		−	2294.00i		−	0.149681i	0.997196	−	0.0748403i	$-0.0238447\pi$
$618$		0			0
$619$	19704.0			1.27944			0.639718	−	0.768610i	$-0.279051\pi$
$619$	19704.0			1.27944			0.639718	+	0.768610i	$0.279051\pi$
$620$		0			0
$621$	1944.00			0.125620
$622$		0			0
$623$		−	5220.00i		−	0.335690i
$624$		0			0
$625$	−4375.00	+	15000.0i	−0.280000	+	0.960000i
$626$		0			0
$627$			1080.00i			0.0687895i
$628$		0			0
$629$	9660.00			0.612352
$630$		0			0
$631$	4392.00			0.277088			0.138544	−	0.990356i	$-0.455758\pi$
$631$	4392.00			0.277088			0.138544	+	0.990356i	$0.455758\pi$
$632$		0			0
$633$		−	5616.00i		−	0.352632i
$634$		0			0
$635$	−690.000	+	1380.00i	−0.0431210	+	0.0862419i
$636$		0			0
$637$		−	722.000i		−	0.0449084i
$638$		0			0
$639$	8316.00			0.514829
$640$		0			0
$641$	19586.0			1.20687			0.603433	−	0.797414i	$-0.293799\pi$
$641$	19586.0			1.20687			0.603433	+	0.797414i	$0.293799\pi$
$642$		0			0
$643$		−	624.000i		−	0.0382709i	−0.999817	−	0.0191354i	$-0.993909\pi$
$643$		−	624.000i		−	0.0382709i	0.999817	−	0.0191354i	$-0.00609137\pi$
$644$		0			0
$645$	−14040.0	−	7020.00i	−0.857092	−	0.428546i
$646$		0			0
$647$		−	8556.00i		−	0.519893i	−0.965623	−	0.259947i	$-0.916295\pi$
$647$		−	8556.00i		−	0.519893i	0.965623	−	0.259947i	$-0.0837050\pi$
$648$		0			0
$649$	−15300.0			−0.925389
$650$		0			0
$651$	16848.0			1.01432
$652$		0			0
$653$			22910.0i			1.37295i	0.727153	+	0.686476i	$0.240843\pi$
$653$			22910.0i			1.37295i	−0.727153	+	0.686476i	$0.759157\pi$
$654$		0			0
$655$	8940.00	+	4470.00i	0.533305	+	0.266652i
$656$		0			0
$657$		−	3060.00i		−	0.181708i
$658$		0			0
$659$	7590.00			0.448656			0.224328	−	0.974514i	$-0.427981\pi$
$659$	7590.00			0.448656			0.224328	+	0.974514i	$0.427981\pi$
$660$		0			0
$661$	31970.0			1.88122			0.940612	−	0.339484i	$-0.110252\pi$
$661$	31970.0			1.88122			0.940612	+	0.339484i	$0.110252\pi$
$662$		0			0
$663$			7980.00i			0.467447i
$664$		0			0
$665$	1080.00	−	2160.00i	0.0629784	−	0.125957i
$666$		0			0
$667$		−	4608.00i		−	0.267500i
$668$		0			0
$669$	−3906.00			−0.225732
$670$		0			0
$671$	22620.0			1.30139
$672$		0			0
$673$		−	24996.0i		−	1.43169i	−0.698261	−	0.715843i	$-0.746043\pi$
$673$		−	24996.0i		−	1.43169i	0.698261	−	0.715843i	$-0.253957\pi$
$674$		0			0
$675$	−2700.00	+	2025.00i	−0.153960	+	0.115470i
$676$		0			0
$677$			3226.00i			0.183139i	0.995799	+	0.0915696i	$0.0291884\pi$
$677$			3226.00i			0.183139i	−0.995799	+	0.0915696i	$0.970812\pi$
$678$		0			0
$679$	−6768.00			−0.382521
$680$		0			0
$681$	−2124.00			−0.119518
$682$		0			0
$683$			23412.0i			1.31162i	0.754927	+	0.655809i	$0.227672\pi$
$683$			23412.0i			1.31162i	−0.754927	+	0.655809i	$0.772328\pi$
$684$		0			0
$685$	−15250.0	+	30500.0i	−0.850617	+	1.70123i
$686$		0			0
$687$		−	15246.0i		−	0.846683i
$688$		0			0
$689$	−16948.0			−0.937108
$690$		0			0
$691$	16572.0			0.912342			0.456171	−	0.889892i	$-0.349220\pi$
$691$	16572.0			0.912342			0.456171	+	0.889892i	$0.349220\pi$
$692$		0			0
$693$			4860.00i			0.266401i
$694$		0			0
$695$	−23520.0	−	11760.0i	−1.28369	−	0.641845i
$696$		0			0
$697$			26180.0i			1.42272i
$698$		0			0
$699$	−17358.0			−0.939256
$700$		0			0
$701$	−448.000			−0.0241380			−0.0120690	−	0.999927i	$-0.503842\pi$
$701$	−448.000			−0.0241380			−0.0120690	+	0.999927i	$0.503842\pi$
$702$		0			0
$703$		−	1656.00i		−	0.0888438i
$704$		0			0
$705$	−3960.00	−	1980.00i	−0.211549	−	0.105775i
$706$		0			0
$707$		−	1872.00i		−	0.0995811i
$708$		0			0
$709$	11922.0			0.631509			0.315755	−	0.948841i	$-0.397742\pi$
$709$	11922.0			0.631509			0.315755	+	0.948841i	$0.397742\pi$
$710$		0			0
$711$	648.000			0.0341799
$712$		0			0
$713$			22464.0i			1.17992i
$714$		0			0
$715$	−5700.00	+	11400.0i	−0.298137	+	0.596274i
$716$		0			0
$717$		−	9468.00i		−	0.493151i
$718$		0			0
$719$	−4344.00			−0.225318			−0.112659	−	0.993634i	$-0.535937\pi$
$719$	−4344.00			−0.225318			−0.112659	+	0.993634i	$0.535937\pi$
$720$		0			0
$721$	−18252.0			−0.942774
$722$		0			0
$723$			9918.00i			0.510172i
$724$		0			0
$725$	4800.00	+	6400.00i	0.245886	+	0.327848i
$726$		0			0
$727$		−	26934.0i		−	1.37404i	−0.726639	−	0.687020i	$-0.758919\pi$
$727$		−	26934.0i		−	1.37404i	0.726639	−	0.687020i	$-0.241081\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$	32760.0			1.65755
$732$		0			0
$733$		−	35962.0i		−	1.81212i	−0.423145	−	0.906062i	$-0.639074\pi$
$733$		−	35962.0i		−	1.81212i	0.423145	−	0.906062i	$-0.360926\pi$
$734$		0			0
$735$	−285.000	+	570.000i	−0.0143026	+	0.0286051i
$736$		0			0
$737$			11520.0i			0.575773i
$738$		0			0
$739$	4860.00			0.241919			0.120959	−	0.992657i	$-0.461403\pi$
$739$	4860.00			0.241919			0.120959	+	0.992657i	$0.461403\pi$
$740$		0			0
$741$	1368.00			0.0678202
$742$		0			0
$743$		−	29844.0i		−	1.47358i	−0.676121	−	0.736790i	$-0.736341\pi$
$743$		−	29844.0i		−	1.47358i	0.676121	−	0.736790i	$-0.263659\pi$
$744$		0			0
$745$	24760.0	+	12380.0i	1.21763	+	0.608816i
$746$		0			0
$747$		−	1404.00i		−	0.0687680i
$748$		0			0
$749$	23112.0			1.12750
$750$		0			0
$751$	1248.00			0.0606394			0.0303197	−	0.999540i	$-0.490347\pi$
$751$	1248.00			0.0606394			0.0303197	+	0.999540i	$0.490347\pi$
$752$		0			0
$753$			12402.0i			0.600205i
$754$		0			0
$755$	31920.0	+	15960.0i	1.53866	+	0.769330i
$756$		0			0
$757$		−	25094.0i		−	1.20483i	−0.798183	−	0.602415i	$-0.794205\pi$
$757$		−	25094.0i		−	1.20483i	0.798183	−	0.602415i	$-0.205795\pi$
$758$		0			0
$759$	−6480.00			−0.309893
$760$		0			0
$761$	23294.0			1.10960			0.554801	−	0.831983i	$-0.312794\pi$
$761$	23294.0			1.10960			0.554801	+	0.831983i	$0.312794\pi$
$762$		0			0
$763$			10260.0i			0.486811i
$764$		0			0
$765$	3150.00	−	6300.00i	0.148874	−	0.297748i
$766$		0			0
$767$			19380.0i			0.912348i
$768$		0			0
$769$	254.000			0.0119109			0.00595544	−	0.999982i	$-0.498104\pi$
$769$	254.000			0.0119109			0.00595544	+	0.999982i	$0.498104\pi$
$770$		0			0
$771$	−10446.0			−0.487942
$772$		0			0
$773$			21886.0i			1.01835i	0.860663	+	0.509175i	$0.170049\pi$
$773$			21886.0i			1.01835i	−0.860663	+	0.509175i	$0.829951\pi$
$774$		0			0
$775$	−23400.0	−	31200.0i	−1.08458	−	1.44611i
$776$		0			0
$777$		−	7452.00i		−	0.344066i
$778$		0			0
$779$	4488.00			0.206418
$780$		0			0
$781$	−27720.0			−1.27004
$782$		0			0
$783$			1728.00i			0.0788680i
$784$		0			0
$785$	−6090.00	+	12180.0i	−0.276894	+	0.553787i
$786$		0			0
$787$			14904.0i			0.675057i	0.941315	+	0.337529i	$0.109591\pi$
$787$			14904.0i			0.675057i	−0.941315	+	0.337529i	$0.890409\pi$
$788$		0			0
$789$	8568.00			0.386602
$790$		0			0
$791$	−4572.00			−0.205514
$792$		0			0
$793$		−	28652.0i		−	1.28305i
$794$		0			0
$795$	13380.0	+	6690.00i	0.596905	+	0.298453i
$796$		0			0
$797$			23834.0i			1.05928i	0.848224	+	0.529638i	$0.177672\pi$
$797$			23834.0i			1.05928i	−0.848224	+	0.529638i	$0.822328\pi$
$798$		0			0
$799$	9240.00			0.409121
$800$		0			0
$801$	−2610.00			−0.115131
$802$		0			0
$803$			10200.0i			0.448257i
$804$		0			0
$805$	12960.0	+	6480.00i	0.567429	+	0.283714i
$806$		0			0
$807$			19392.0i			0.845887i
$808$		0			0
$809$	31502.0			1.36904			0.684519	−	0.728995i	$-0.260012\pi$
$809$	31502.0			1.36904			0.684519	+	0.728995i	$0.260012\pi$
$810$		0			0
$811$	21072.0			0.912377			0.456189	−	0.889883i	$-0.349214\pi$
$811$	21072.0			0.912377			0.456189	+	0.889883i	$0.349214\pi$
$812$		0			0
$813$			15552.0i			0.670889i
$814$		0			0
$815$	11640.0	−	23280.0i	0.500284	−	1.00057i
$816$		0			0
$817$		−	5616.00i		−	0.240488i
$818$		0			0
$819$	6156.00			0.262647
$820$		0			0
$821$	−38072.0			−1.61842			−0.809209	−	0.587520i	$-0.800104\pi$
$821$	−38072.0			−1.61842			−0.809209	+	0.587520i	$0.800104\pi$
$822$		0			0
$823$		−	31578.0i		−	1.33747i	−0.743500	−	0.668736i	$-0.766836\pi$
$823$		−	31578.0i		−	1.33747i	0.743500	−	0.668736i	$-0.233164\pi$
$824$		0			0
$825$	9000.00	−	6750.00i	0.379806	−	0.284854i
$826$		0			0
$827$		−	41532.0i		−	1.74632i	−0.487431	−	0.873162i	$-0.662066\pi$
$827$		−	41532.0i		−	1.74632i	0.487431	−	0.873162i	$-0.337934\pi$
$828$		0			0
$829$	−24786.0			−1.03842			−0.519212	−	0.854646i	$-0.673774\pi$
$829$	−24786.0			−1.03842			−0.519212	+	0.854646i	$0.673774\pi$
$830$		0			0
$831$	11454.0			0.478141
$832$		0			0
$833$		−	1330.00i		−	0.0553203i
$834$		0			0
$835$	−14640.0	+	29280.0i	−0.606752	+	1.21350i
$836$		0			0
$837$		−	8424.00i		−	0.347881i
$838$		0			0
$839$	−1608.00			−0.0661673			−0.0330836	−	0.999453i	$-0.510533\pi$
$839$	−1608.00			−0.0661673			−0.0330836	+	0.999453i	$0.510533\pi$
$840$		0			0
$841$	−20293.0			−0.832055
$842$		0			0
$843$		−	18282.0i		−	0.746934i
$844$		0			0
$845$	−7530.00	−	3765.00i	−0.306556	−	0.153278i
$846$		0			0
$847$			7758.00i			0.314720i
$848$		0			0
$849$	−13320.0			−0.538447
$850$		0			0
$851$	9936.00			0.400237
$852$		0			0
$853$		−	8514.00i		−	0.341751i	−0.985293	−	0.170876i	$-0.945340\pi$
$853$		−	8514.00i		−	0.341751i	0.985293	−	0.170876i	$-0.0546596\pi$
$854$		0			0
$855$	−1080.00	−	540.000i	−0.0431991	−	0.0215995i
$856$		0			0
$857$		−	29726.0i		−	1.18486i	−0.805624	−	0.592428i	$-0.798170\pi$
$857$		−	29726.0i		−	1.18486i	0.805624	−	0.592428i	$-0.201830\pi$
$858$		0			0
$859$	−2136.00			−0.0848421			−0.0424211	−	0.999100i	$-0.513507\pi$
$859$	−2136.00			−0.0848421			−0.0424211	+	0.999100i	$0.513507\pi$
$860$		0			0
$861$	20196.0			0.799394
$862$		0			0
$863$		−	11928.0i		−	0.470491i	−0.971936	−	0.235246i	$-0.924411\pi$
$863$		−	11928.0i		−	0.470491i	0.971936	−	0.235246i	$-0.0755894\pi$
$864$		0			0
$865$	2510.00	−	5020.00i	0.0986619	−	0.197324i
$866$		0			0
$867$		−	39.0000i		−	0.00152769i
$868$		0			0
$869$	−2160.00			−0.0843187
$870$		0			0
$871$	14592.0			0.567659
$872$		0			0
$873$			3384.00i			0.131192i
$874$		0			0
$875$	−24750.0	+	4500.00i	−0.956232	+	0.173860i
$876$		0			0
$877$			27006.0i			1.03983i	0.854219	+	0.519913i	$0.174036\pi$
$877$			27006.0i			1.03983i	−0.854219	+	0.519913i	$0.825964\pi$
$878$		0			0
$879$	−17850.0			−0.684944
$880$		0			0
$881$	29318.0			1.12117			0.560584	−	0.828098i	$-0.310577\pi$
$881$	29318.0			1.12117			0.560584	+	0.828098i	$0.310577\pi$
$882$		0			0
$883$			7008.00i			0.267087i	0.991043	+	0.133544i	$0.0426356\pi$
$883$			7008.00i			0.267087i	−0.991043	+	0.133544i	$0.957364\pi$
$884$		0			0
$885$	7650.00	−	15300.0i	0.290567	−	0.581134i
$886$		0			0
$887$			15024.0i			0.568722i	0.958717	+	0.284361i	$0.0917814\pi$
$887$			15024.0i			0.568722i	−0.958717	+	0.284361i	$0.908219\pi$
$888$		0			0
$889$	−2484.00			−0.0937128
$890$		0			0
$891$	2430.00			0.0913671
$892$		0			0
$893$		−	1584.00i		−	0.0593578i
$894$		0			0
$895$	−40260.0	−	20130.0i	−1.50362	−	0.751812i
$896$		0			0
$897$			8208.00i			0.305526i
$898$		0			0
$899$	−19968.0			−0.740790
$900$		0			0
$901$	−31220.0			−1.15437
$902$		0			0
$903$		−	25272.0i		−	0.931339i
$904$		0			0
$905$	−5940.00	−	2970.00i	−0.218179	−	0.109090i
$906$		0			0
$907$		−	1380.00i		−	0.0505206i	−0.999681	−	0.0252603i	$-0.991959\pi$
$907$		−	1380.00i		−	0.0505206i	0.999681	−	0.0252603i	$-0.00804145\pi$
$908$		0			0
$909$	−936.000			−0.0341531
$910$		0			0
$911$	17952.0			0.652883			0.326441	−	0.945217i	$-0.394150\pi$
$911$	17952.0			0.652883			0.326441	+	0.945217i	$0.394150\pi$
$912$		0			0
$913$			4680.00i			0.169644i
$914$		0			0
$915$	−11310.0	+	22620.0i	−0.408631	+	0.817261i
$916$		0			0
$917$			16092.0i			0.579503i
$918$		0			0
$919$	−19056.0			−0.684004			−0.342002	−	0.939699i	$-0.611105\pi$
$919$	−19056.0			−0.684004			−0.342002	+	0.939699i	$0.611105\pi$
$920$		0			0
$921$	2772.00			0.0991754
$922$		0			0
$923$			35112.0i			1.25214i
$924$		0			0
$925$	−13800.0	+	10350.0i	−0.490531	+	0.367898i
$926$		0			0
$927$			9126.00i			0.323341i
$928$		0			0
$929$	−26398.0			−0.932282			−0.466141	−	0.884710i	$-0.654356\pi$
$929$	−26398.0			−0.932282			−0.466141	+	0.884710i	$0.654356\pi$
$930$		0			0
$931$	−228.000			−0.00802621
$932$		0			0
$933$			21060.0i			0.738985i
$934$		0			0
$935$	−10500.0	+	21000.0i	−0.367259	+	0.734517i
$936$		0			0
$937$		−	21192.0i		−	0.738861i	−0.929258	−	0.369430i	$-0.879553\pi$
$937$		−	21192.0i		−	0.738861i	0.929258	−	0.369430i	$-0.120447\pi$
$938$		0			0
$939$	−15732.0			−0.546746
$940$		0			0
$941$	7300.00			0.252894			0.126447	−	0.991973i	$-0.459643\pi$
$941$	7300.00			0.252894			0.126447	+	0.991973i	$0.459643\pi$
$942$		0			0
$943$			26928.0i			0.929901i
$944$		0			0
$945$	−4860.00	−	2430.00i	−0.167297	−	0.0836486i
$946$		0			0
$947$			33540.0i			1.15090i	0.817836	+	0.575451i	$0.195173\pi$
$947$			33540.0i			1.15090i	−0.817836	+	0.575451i	$0.804827\pi$
$948$		0			0
$949$	12920.0			0.441940
$950$		0			0
$951$	15582.0			0.531315
$952$		0			0
$953$			32714.0i			1.11197i	0.831191	+	0.555987i	$0.187659\pi$
$953$			32714.0i			1.11197i	−0.831191	+	0.555987i	$0.812341\pi$
$954$		0			0
$955$	−10440.0	−	5220.00i	−0.353749	−	0.176875i
$956$		0			0
$957$		−	5760.00i		−	0.194560i
$958$		0			0
$959$	−54900.0			−1.84861
$960$		0			0
$961$	67553.0			2.26756
$962$		0			0
$963$		−	11556.0i		−	0.386695i
$964$		0			0
$965$	16260.0	−	32520.0i	0.542412	−	1.08482i
$966$		0			0
$967$			35334.0i			1.17504i	0.809209	+	0.587521i	$0.199896\pi$
$967$			35334.0i			1.17504i	−0.809209	+	0.587521i	$0.800104\pi$
$968$		0			0
$969$	2520.00			0.0835439
$970$		0			0
$971$	−17094.0			−0.564956			−0.282478	−	0.959274i	$-0.591156\pi$
$971$	−17094.0			−0.564956			−0.282478	+	0.959274i	$0.591156\pi$
$972$		0			0
$973$		−	42336.0i		−	1.39489i
$974$		0			0
$975$	−8550.00	−	11400.0i	−0.280840	−	0.374454i
$976$		0			0
$977$		−	46814.0i		−	1.53297i	−0.642262	−	0.766485i	$-0.722004\pi$
$977$		−	46814.0i		−	1.53297i	0.642262	−	0.766485i	$-0.277996\pi$
$978$		0			0
$979$	8700.00			0.284018
$980$		0			0
$981$	5130.00			0.166961
$982$		0			0
$983$			5076.00i			0.164699i	0.996604	+	0.0823496i	$0.0262424\pi$
$983$			5076.00i			0.164699i	−0.996604	+	0.0823496i	$0.973758\pi$
$984$		0			0
$985$	−14570.0	+	29140.0i	−0.471308	+	0.942617i
$986$		0			0
$987$		−	7128.00i		−	0.229875i
$988$		0			0
$989$	33696.0			1.08339
$990$		0			0
$991$	−9456.00			−0.303108			−0.151554	−	0.988449i	$-0.548428\pi$
$991$	−9456.00			−0.303108			−0.151554	+	0.988449i	$0.548428\pi$
$992$		0			0
$993$			34884.0i			1.11481i
$994$		0			0
$995$	−48720.0	−	24360.0i	−1.55229	−	0.776145i
$996$		0			0
$997$			33486.0i			1.06370i	0.846838	+	0.531852i	$0.178504\pi$
$997$			33486.0i			1.06370i	−0.846838	+	0.531852i	$0.821496\pi$
$998$		0			0
$999$	−3726.00			−0.118003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.f.b.289.1	yes	2
3.2	odd	2		1440.4.f.c.289.2		2
4.3	odd	2		480.4.f.a.289.2	yes	2
5.2	odd	4		2400.4.a.j.1.1		1
5.3	odd	4		2400.4.a.n.1.1		1
5.4	even	2	inner	480.4.f.b.289.2	yes	2
8.3	odd	2		960.4.f.k.769.1		2
8.5	even	2		960.4.f.h.769.2		2
12.11	even	2		1440.4.f.d.289.2		2
15.14	odd	2		1440.4.f.c.289.1		2
20.3	even	4		2400.4.a.i.1.1		1
20.7	even	4		2400.4.a.m.1.1		1
20.19	odd	2		480.4.f.a.289.1	✓	2
40.19	odd	2		960.4.f.k.769.2		2
40.29	even	2		960.4.f.h.769.1		2
60.59	even	2		1440.4.f.d.289.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.f.a.289.1	✓	2	20.19	odd	2
480.4.f.a.289.2	yes	2	4.3	odd	2
480.4.f.b.289.1	yes	2	1.1	even	1	trivial
480.4.f.b.289.2	yes	2	5.4	even	2	inner
960.4.f.h.769.1		2	40.29	even	2
960.4.f.h.769.2		2	8.5	even	2
960.4.f.k.769.1		2	8.3	odd	2
960.4.f.k.769.2		2	40.19	odd	2
1440.4.f.c.289.1		2	15.14	odd	2
1440.4.f.c.289.2		2	3.2	odd	2
1440.4.f.d.289.1		2	60.59	even	2
1440.4.f.d.289.2		2	12.11	even	2
2400.4.a.i.1.1		1	20.3	even	4
2400.4.a.j.1.1		1	5.2	odd	4
2400.4.a.m.1.1		1	20.7	even	4
2400.4.a.n.1.1		1	5.3	odd	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:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.00000i q^{3} +(-10.0000 - 5.00000i) q^{5} -18.0000i q^{7} -9.00000 q^{9} +30.0000 q^{11} -38.0000i q^{13} +(-15.0000 + 30.0000i) q^{15} -70.0000i q^{17} -12.0000 q^{19} -54.0000 q^{21} -72.0000i q^{23} +(75.0000 + 100.000i) q^{25} +27.0000i q^{27} +64.0000 q^{29} -312.000 q^{31} -90.0000i q^{33} +(-90.0000 + 180.000i) q^{35} +138.000i q^{37} -114.000 q^{39} -374.000 q^{41} +468.000i q^{43} +(90.0000 + 45.0000i) q^{45} +132.000i q^{47} +19.0000 q^{49} -210.000 q^{51} -446.000i q^{53} +(-300.000 - 150.000i) q^{55} +36.0000i q^{57} -510.000 q^{59} +754.000 q^{61} +162.000i q^{63} +(-190.000 + 380.000i) q^{65} +384.000i q^{67} -216.000 q^{69} -924.000 q^{71} +340.000i q^{73} +(300.000 - 225.000i) q^{75} -540.000i q^{77} -72.0000 q^{79} +81.0000 q^{81} +156.000i q^{83} +(-350.000 + 700.000i) q^{85} -192.000i q^{87} +290.000 q^{89} -684.000 q^{91} +936.000i q^{93} +(120.000 + 60.0000i) q^{95} -376.000i q^{97} -270.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 20 q^{5} - 18 q^{9} + 60 q^{11} - 30 q^{15} - 24 q^{19} - 108 q^{21} + 150 q^{25} + 128 q^{29} - 624 q^{31} - 180 q^{35} - 228 q^{39} - 748 q^{41} + 180 q^{45} + 38 q^{49} - 420 q^{51} - 600 q^{55}+ \cdots - 540 q^{99}+O(q^{100}) \) 2 * q - 20 * q^5 - 18 * q^9 + 60 * q^11 - 30 * q^15 - 24 * q^19 - 108 * q^21 + 150 * q^25 + 128 * q^29 - 624 * q^31 - 180 * q^35 - 228 * q^39 - 748 * q^41 + 180 * q^45 + 38 * q^49 - 420 * q^51 - 600 * q^55 - 1020 * q^59 + 1508 * q^61 - 380 * q^65 - 432 * q^69 - 1848 * q^71 + 600 * q^75 - 144 * q^79 + 162 * q^81 - 700 * q^85 + 580 * q^89 - 1368 * q^91 + 240 * q^95 - 540 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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