LMFDB - Embedded newform 1080.4.q.b.361.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1080,4,Mod(361,1080)] 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("1080.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1080, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,40,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$63.7220628062$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 16 x^{14} + 14 x^{13} - 284 x^{12} + 764 x^{11} + 19770 x^{10} + 55106 x^{9} + \cdots + 193472540143 $ x^16 - 6x^15 + 16x^14 + 14x^13 - 284x^12 + 764x^11 + 19770x^10 + 55106x^9 + 266991x^8 + 335444x^7 - 4573302x^6 + 2211506x^5 + 251845642x^4 + 1438820210x^3 + 12022873630x^2 + 75245140836*x + 193472540143
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$2.04823 + 5.08932i$ of defining polynomial
Character	$\chi$	$=$	1080.361
Dual form			1080.4.q.b.721.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.50000 - 4.33013i) q^{5} +(-17.3532 - 30.0566i) q^{7} +(-13.7375 - 23.7941i) q^{11} +(33.0855 - 57.3058i) q^{13} -58.2319 q^{17} +106.853 q^{19} +(10.3464 - 17.9205i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-95.7760 - 165.889i) q^{29} +(117.666 - 203.804i) q^{31} -173.532 q^{35} -63.4389 q^{37} +(115.337 - 199.769i) q^{41} +(202.677 + 351.048i) q^{43} +(252.504 + 437.350i) q^{47} +(-430.766 + 746.109i) q^{49} -693.794 q^{53} -137.375 q^{55} +(-218.818 + 379.004i) q^{59} +(-287.707 - 498.323i) q^{61} +(-165.428 - 286.529i) q^{65} +(351.378 - 608.604i) q^{67} -238.922 q^{71} +661.982 q^{73} +(-476.780 + 825.808i) q^{77} +(-288.259 - 499.279i) q^{79} +(625.483 + 1083.37i) q^{83} +(-145.580 + 252.152i) q^{85} -402.160 q^{89} -2296.56 q^{91} +(267.133 - 462.689i) q^{95} +(409.278 + 708.891i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) $ 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	−17.3532	−	30.0566i	−0.936984	−	1.62290i	−0.771057	−	0.636766i	$-0.780272\pi$
$7$	−17.3532	−	30.0566i	−0.936984	−	1.62290i	−0.165927	−	0.986138i	$-0.553062\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.7375	−	23.7941i	−0.376548	−	0.652200i	0.614010	−	0.789298i	$-0.289556\pi$
$11$	−13.7375	−	23.7941i	−0.376548	−	0.652200i	−0.990557	+	0.137099i	$0.956222\pi$
$12$		0			0
$13$	33.0855	−	57.3058i	0.705867	−	1.22260i	−0.260511	−	0.965471i	$-0.583891\pi$
$13$	33.0855	−	57.3058i	0.705867	−	1.22260i	0.966378	−	0.257126i	$-0.0827756\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−58.2319			−0.830783			−0.415392	−	0.909643i	$-0.636355\pi$
$17$	−58.2319			−0.830783			−0.415392	+	0.909643i	$0.636355\pi$
$18$		0			0
$19$	106.853			1.29020			0.645101	−	0.764097i	$-0.276815\pi$
$19$	106.853			1.29020			0.645101	+	0.764097i	$0.276815\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	10.3464	−	17.9205i	0.0937990	−	0.162465i	−0.815308	−	0.579028i	$-0.803432\pi$
$23$	10.3464	−	17.9205i	0.0937990	−	0.162465i	0.909107	+	0.416563i	$0.136766\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−95.7760	−	165.889i	−0.613281	−	1.06223i	−0.990683	−	0.136185i	$-0.956516\pi$
$29$	−95.7760	−	165.889i	−0.613281	−	1.06223i	0.377402	−	0.926049i	$-0.376817\pi$
$30$		0			0
$31$	117.666	−	203.804i	0.681727	−	1.18079i	−0.292727	−	0.956196i	$-0.594563\pi$
$31$	117.666	−	203.804i	0.681727	−	1.18079i	0.974454	−	0.224589i	$-0.0721040\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−173.532			−0.838064
$36$		0			0
$37$	−63.4389			−0.281873			−0.140936	−	0.990019i	$-0.545011\pi$
$37$	−63.4389			−0.281873			−0.140936	+	0.990019i	$0.545011\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	115.337	−	199.769i	0.439330	−	0.760942i	−0.558308	−	0.829634i	$-0.688549\pi$
$41$	115.337	−	199.769i	0.439330	−	0.760942i	0.997638	+	0.0686916i	$0.0218824\pi$
$42$		0			0
$43$	202.677	+	351.048i	0.718791	+	1.24498i	0.961479	+	0.274879i	$0.0886377\pi$
$43$	202.677	+	351.048i	0.718791	+	1.24498i	−0.242688	+	0.970104i	$0.578029\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	252.504	+	437.350i	0.783650	+	1.35732i	0.929803	+	0.368059i	$0.119977\pi$
$47$	252.504	+	437.350i	0.783650	+	1.35732i	−0.146153	+	0.989262i	$0.546689\pi$
$48$		0			0
$49$	−430.766	+	746.109i	−1.25588	+	2.17525i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−693.794			−1.79811			−0.899056	−	0.437833i	$-0.855746\pi$
$53$	−693.794			−1.79811			−0.899056	+	0.437833i	$0.855746\pi$
$54$		0			0
$55$	−137.375			−0.336794
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−218.818	+	379.004i	−0.482843	+	0.836308i	−0.999806	−	0.0196995i	$-0.993729\pi$
$59$	−218.818	+	379.004i	−0.482843	+	0.836308i	0.516963	+	0.856008i	$0.327062\pi$
$60$		0			0
$61$	−287.707	−	498.323i	−0.603886	−	1.04596i	−0.992226	−	0.124446i	$-0.960285\pi$
$61$	−287.707	−	498.323i	−0.603886	−	1.04596i	0.388340	−	0.921516i	$-0.373049\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−165.428	−	286.529i	−0.315673	−	0.546762i
$66$		0			0
$67$	351.378	−	608.604i	0.640711	−	1.10974i	−0.344563	−	0.938763i	$-0.611973\pi$
$67$	351.378	−	608.604i	0.640711	−	1.10974i	0.985274	−	0.170981i	$-0.0546937\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−238.922			−0.399363			−0.199682	−	0.979861i	$-0.563991\pi$
$71$	−238.922			−0.399363			−0.199682	+	0.979861i	$0.563991\pi$
$72$		0			0
$73$	661.982			1.06136			0.530679	−	0.847573i	$-0.321937\pi$
$73$	661.982			1.06136			0.530679	+	0.847573i	$0.321937\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−476.780	+	825.808i	−0.705638	+	1.22220i
$78$		0			0
$79$	−288.259	−	499.279i	−0.410527	−	0.711054i	0.584420	−	0.811451i	$-0.301322\pi$
$79$	−288.259	−	499.279i	−0.410527	−	0.711054i	−0.994947	+	0.100397i	$0.967989\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	625.483	+	1083.37i	0.827177	+	1.43271i	0.900244	+	0.435386i	$0.143388\pi$
$83$	625.483	+	1083.37i	0.827177	+	1.43271i	−0.0730664	+	0.997327i	$0.523279\pi$
$84$		0			0
$85$	−145.580	+	252.152i	−0.185769	+	0.321761i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−402.160			−0.478976			−0.239488	−	0.970899i	$-0.576980\pi$
$89$	−402.160			−0.478976			−0.239488	+	0.970899i	$0.576980\pi$
$90$		0			0
$91$	−2296.56			−2.64554
$92$		0			0
$93$		0			0
$94$		0			0
$95$	267.133	−	462.689i	0.288498	−	0.499693i
$96$		0			0
$97$	409.278	+	708.891i	0.428411	+	0.742030i	0.996732	−	0.0807768i	$-0.0257401\pi$
$97$	409.278	+	708.891i	0.428411	+	0.742030i	−0.568321	+	0.822807i	$0.692407\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−222.832	−	385.956i	−0.219530	−	0.380238i	0.735134	−	0.677922i	$-0.237119\pi$
$101$	−222.832	−	385.956i	−0.219530	−	0.380238i	−0.954665	+	0.297684i	$0.903786\pi$
$102$		0			0
$103$	−9.19756	+	15.9306i	−0.00879867	+	0.0152397i	−0.870391	−	0.492361i	$-0.836134\pi$
$103$	−9.19756	+	15.9306i	−0.00879867	+	0.0152397i	0.861593	+	0.507601i	$0.169467\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1649.12			−1.48997			−0.744985	−	0.667082i	$-0.767543\pi$
$107$	−1649.12			−1.48997			−0.744985	+	0.667082i	$0.767543\pi$
$108$		0			0
$109$	588.802			0.517404			0.258702	−	0.965957i	$-0.416705\pi$
$109$	588.802			0.517404			0.258702	+	0.965957i	$0.416705\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	248.491	−	430.400i	0.206868	−	0.358306i	−0.743858	−	0.668337i	$-0.767006\pi$
$113$	248.491	−	430.400i	0.206868	−	0.358306i	0.950726	+	0.310031i	$0.100340\pi$
$114$		0			0
$115$	−51.7321	−	89.6026i	−0.0419482	−	0.0726564i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1010.51	+	1750.25i	0.778431	+	1.34828i
$120$		0			0
$121$	288.060	−	498.935i	0.216424	−	0.374857i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−125.000			−0.0894427
$126$		0			0
$127$	666.725			0.465844			0.232922	−	0.972495i	$-0.425171\pi$
$127$	666.725			0.465844			0.232922	+	0.972495i	$0.425171\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−433.896	+	751.530i	−0.289387	+	0.501232i	−0.973663	−	0.227990i	$-0.926785\pi$
$131$	−433.896	+	751.530i	−0.289387	+	0.501232i	0.684277	+	0.729222i	$0.260118\pi$
$132$		0			0
$133$	−1854.25	−	3211.65i	−1.20890	−	2.09388i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	697.909	+	1208.81i	0.435229	+	0.753839i	0.997314	−	0.0732406i	$-0.0233341\pi$
$137$	697.909	+	1208.81i	0.435229	+	0.753839i	−0.562085	+	0.827079i	$0.690001\pi$
$138$		0			0
$139$	−536.221	+	928.762i	−0.327206	+	0.566738i	−0.981956	−	0.189107i	$-0.939441\pi$
$139$	−536.221	+	928.762i	−0.327206	+	0.566738i	0.654750	+	0.755845i	$0.272774\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1818.05			−1.06317
$144$		0			0
$145$	−957.760			−0.548535
$146$		0			0
$147$		0			0
$148$		0			0
$149$	1075.01	−	1861.98i	0.591064	−	1.02375i	−0.403026	−	0.915189i	$-0.632042\pi$
$149$	1075.01	−	1861.98i	0.591064	−	1.02375i	0.994090	−	0.108564i	$-0.0346251\pi$
$150$		0			0
$151$	−936.168	−	1621.49i	−0.504532	−	0.873874i	−0.999986	−	0.00524048i	$-0.998332\pi$
$151$	−936.168	−	1621.49i	−0.504532	−	0.873874i	0.495455	−	0.868634i	$-0.335001\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−588.332	−	1019.02i	−0.304877	−	0.528063i
$156$		0			0
$157$	−692.805	+	1199.97i	−0.352178	+	0.609989i	−0.986631	−	0.162972i	$-0.947892\pi$
$157$	−692.805	+	1199.97i	−0.352178	+	0.609989i	0.634453	+	0.772961i	$0.281225\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−718.174			−0.351553
$162$		0			0
$163$	141.709			0.0680950			0.0340475	−	0.999420i	$-0.489160\pi$
$163$	141.709			0.0680950			0.0340475	+	0.999420i	$0.489160\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−405.399	+	702.171i	−0.187849	+	0.325363i	−0.944533	−	0.328417i	$-0.893485\pi$
$167$	−405.399	+	702.171i	−0.187849	+	0.325363i	0.756684	+	0.653781i	$0.226818\pi$
$168$		0			0
$169$	−1090.80	−	1889.32i	−0.496496	−	0.859955i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−670.032	−	1160.53i	−0.294460	−	0.510020i	0.680399	−	0.732842i	$-0.261807\pi$
$173$	−670.032	−	1160.53i	−0.294460	−	0.510020i	−0.974859	+	0.222822i	$0.928473\pi$
$174$		0			0
$175$	−433.830	+	751.415i	−0.187397	+	0.324581i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	2401.28			1.00268			0.501341	−	0.865250i	$-0.332840\pi$
$179$	2401.28			1.00268			0.501341	+	0.865250i	$0.332840\pi$
$180$		0			0
$181$	−3779.56			−1.55211			−0.776056	−	0.630664i	$-0.782783\pi$
$181$	−3779.56			−1.55211			−0.776056	+	0.630664i	$0.782783\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−158.597	+	274.699i	−0.0630287	+	0.109169i
$186$		0			0
$187$	799.963	+	1385.58i	0.312830	+	0.541837i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−790.438	−	1369.08i	−0.299445	−	0.518655i	0.676564	−	0.736384i	$-0.263468\pi$
$191$	−790.438	−	1369.08i	−0.299445	−	0.518655i	−0.976009	+	0.217729i	$0.930135\pi$
$192$		0			0
$193$	−686.848	+	1189.65i	−0.256168	+	0.443695i	−0.965212	−	0.261468i	$-0.915793\pi$
$193$	−686.848	+	1189.65i	−0.256168	+	0.443695i	0.709044	+	0.705164i	$0.249127\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	719.881			0.260352			0.130176	−	0.991491i	$-0.458446\pi$
$197$	719.881			0.260352			0.130176	+	0.991491i	$0.458446\pi$
$198$		0			0
$199$	−3171.51			−1.12976			−0.564880	−	0.825173i	$-0.691077\pi$
$199$	−3171.51			−1.12976			−0.564880	+	0.825173i	$0.691077\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−3324.04	+	5757.40i	−1.14927	+	1.99059i
$204$		0			0
$205$	−576.683	−	998.844i	−0.196474	−	0.340304i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−1467.90	−	2542.48i	−0.485823	−	0.841470i
$210$		0			0
$211$	−559.827	+	969.648i	−0.182654	+	0.316367i	−0.942784	−	0.333405i	$-0.891802\pi$
$211$	−559.827	+	969.648i	−0.182654	+	0.316367i	0.760129	+	0.649772i	$0.225136\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2026.77			0.642907
$216$		0			0
$217$	−8167.56			−2.55507
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1926.63	+	3337.03i	−0.586422	+	1.01571i
$222$		0			0
$223$	−1104.78	−	1913.53i	−0.331754	−	0.574616i	0.651101	−	0.758991i	$-0.274307\pi$
$223$	−1104.78	−	1913.53i	−0.331754	−	0.574616i	−0.982856	+	0.184375i	$0.940974\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2005.00	+	3472.76i	0.586240	+	1.01540i	0.994720	+	0.102631i	$0.0327260\pi$
$227$	2005.00	+	3472.76i	0.586240	+	1.01540i	−0.408479	+	0.912768i	$0.633941\pi$
$228$		0			0
$229$	−89.0792	+	154.290i	−0.0257053	+	0.0445229i	−0.878592	−	0.477573i	$-0.841516\pi$
$229$	−89.0792	+	154.290i	−0.0257053	+	0.0445229i	0.852887	+	0.522096i	$0.174850\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−573.057			−0.161125			−0.0805627	−	0.996750i	$-0.525672\pi$
$233$	−573.057			−0.161125			−0.0805627	+	0.996750i	$0.525672\pi$
$234$		0			0
$235$	2525.04			0.700917
$236$		0			0
$237$		0			0
$238$		0			0
$239$	775.518	−	1343.24i	0.209892	−	0.363543i	−0.741789	−	0.670634i	$-0.766022\pi$
$239$	775.518	−	1343.24i	0.209892	−	0.363543i	0.951680	+	0.307091i	$0.0993555\pi$
$240$		0			0
$241$	687.731	+	1191.18i	0.183820	+	0.318386i	0.943178	−	0.332287i	$-0.107820\pi$
$241$	687.731	+	1191.18i	0.183820	+	0.318386i	−0.759358	+	0.650673i	$0.774487\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2153.83	+	3730.55i	0.561646	+	0.972800i
$246$		0			0
$247$	3535.30	−	6123.32i	0.910711	−	1.57740i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1043.09			0.262309			0.131155	−	0.991362i	$-0.458132\pi$
$251$	1043.09			0.262309			0.131155	+	0.991362i	$0.458132\pi$
$252$		0			0
$253$	−568.537			−0.141279
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2096.54	+	3631.32i	−0.508866	+	0.881382i	0.491081	+	0.871114i	$0.336602\pi$
$257$	−2096.54	+	3631.32i	−0.508866	+	0.881382i	−0.999947	+	0.0102684i	$0.996731\pi$
$258$		0			0
$259$	1100.87	+	1906.76i	0.264110	+	0.457453i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	2172.64	+	3763.12i	0.509393	+	0.882295i	0.999941	+	0.0108807i	$0.00346351\pi$
$263$	2172.64	+	3763.12i	0.509393	+	0.882295i	−0.490547	+	0.871415i	$0.663203\pi$
$264$		0			0
$265$	−1734.49	+	3004.22i	−0.402070	+	0.696406i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2993.79			−0.678567			−0.339284	−	0.940684i	$-0.610185\pi$
$269$	−2993.79			−0.678567			−0.339284	+	0.940684i	$0.610185\pi$
$270$		0			0
$271$	7757.79			1.73894			0.869469	−	0.493987i	$-0.164461\pi$
$271$	7757.79			1.73894			0.869469	+	0.493987i	$0.164461\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−343.438	+	594.853i	−0.0753095	+	0.130440i
$276$		0			0
$277$	3698.00	+	6405.12i	0.802134	+	1.38934i	0.918209	+	0.396096i	$0.129635\pi$
$277$	3698.00	+	6405.12i	0.802134	+	1.38934i	−0.116075	+	0.993240i	$0.537031\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1964.10	−	3401.92i	−0.416969	−	0.722212i	0.578664	−	0.815566i	$-0.303574\pi$
$281$	−1964.10	−	3401.92i	−0.416969	−	0.722212i	−0.995633	+	0.0933546i	$0.970241\pi$
$282$		0			0
$283$	479.285	−	830.146i	0.100673	−	0.174371i	−0.811289	−	0.584645i	$-0.801234\pi$
$283$	479.285	−	830.146i	0.100673	−	0.174371i	0.911962	+	0.410274i	$0.134567\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−8005.83			−1.64658
$288$		0			0
$289$	−1522.04			−0.309799
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1994.24	−	3454.12i	0.397627	−	0.688709i	−0.595806	−	0.803128i	$-0.703167\pi$
$293$	1994.24	−	3454.12i	0.397627	−	0.688709i	0.993433	+	0.114419i	$0.0365006\pi$
$294$		0			0
$295$	1094.09	+	1895.02i	0.215934	+	0.374008i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−684.633	−	1185.82i	−0.132419	−	0.229357i
$300$		0			0
$301$	7034.20	−	12183.6i	1.34699	−	2.33306i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2877.07			−0.540132
$306$		0			0
$307$	−778.464			−0.144721			−0.0723604	−	0.997379i	$-0.523053\pi$
$307$	−778.464			−0.144721			−0.0723604	+	0.997379i	$0.523053\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−601.630	+	1042.05i	−0.109695	+	0.189998i	−0.915647	−	0.401984i	$-0.868321\pi$
$311$	−601.630	+	1042.05i	−0.109695	+	0.189998i	0.805951	+	0.591982i	$0.201654\pi$
$312$		0			0
$313$	−4766.23	−	8255.35i	−0.860713	−	1.49080i	−0.871242	−	0.490853i	$-0.836685\pi$
$313$	−4766.23	−	8255.35i	−0.860713	−	1.49080i	0.0105296	−	0.999945i	$-0.496648\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3766.10	−	6523.08i	−0.667272	−	1.15575i	−0.978664	−	0.205468i	$-0.934128\pi$
$317$	−3766.10	−	6523.08i	−0.667272	−	1.15575i	0.311392	−	0.950282i	$-0.399205\pi$
$318$		0			0
$319$	−2631.45	+	4557.81i	−0.461859	+	0.799964i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6222.28			−1.07188
$324$		0			0
$325$	−1654.28			−0.282347
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8763.51	−	15178.8i	1.46853	−	2.54358i
$330$		0			0
$331$	5814.22	+	10070.5i	0.965494	+	1.67228i	0.708282	+	0.705930i	$0.249471\pi$
$331$	5814.22	+	10070.5i	0.965494	+	1.67228i	0.257212	+	0.966355i	$0.417196\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1756.89	−	3043.02i	−0.286535	−	0.496293i
$336$		0			0
$337$	−972.879	+	1685.08i	−0.157258	+	0.272380i	−0.933879	−	0.357589i	$-0.883599\pi$
$337$	−972.879	+	1685.08i	−0.157258	+	0.272380i	0.776621	+	0.629969i	$0.216932\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6465.79			−1.02681
$342$		0			0
$343$	17996.4			2.83299
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6029.45	−	10443.3i	0.932790	−	1.61564i	0.154261	−	0.988030i	$-0.450700\pi$
$347$	6029.45	−	10443.3i	0.932790	−	1.61564i	0.778529	−	0.627609i	$-0.215966\pi$
$348$		0			0
$349$	−3570.73	−	6184.69i	−0.547670	−	0.948593i	−0.998434	−	0.0559492i	$-0.982182\pi$
$349$	−3570.73	−	6184.69i	−0.547670	−	0.948593i	0.450763	−	0.892643i	$-0.351152\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4966.00	+	8601.36i	0.748763	+	1.29690i	0.948416	+	0.317029i	$0.102685\pi$
$353$	4966.00	+	8601.36i	0.748763	+	1.29690i	−0.199653	+	0.979867i	$0.563982\pi$
$354$		0			0
$355$	−597.304	+	1034.56i	−0.0893003	+	0.154673i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−603.288			−0.0886918			−0.0443459	−	0.999016i	$-0.514120\pi$
$359$	−603.288			−0.0886918			−0.0443459	+	0.999016i	$0.514120\pi$
$360$		0			0
$361$	4558.65			0.664622
$362$		0			0
$363$		0			0
$364$		0			0
$365$	1654.95	−	2866.46i	0.237327	−	0.411062i
$366$		0			0
$367$	3530.10	+	6114.32i	0.502098	+	0.869659i	0.999997	+	0.00242403i	$0.000771594\pi$
$367$	3530.10	+	6114.32i	0.502098	+	0.869659i	−0.497899	+	0.867235i	$0.665895\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	12039.5	+	20853.1i	1.68480	+	2.91816i
$372$		0			0
$373$	2535.97	−	4392.42i	0.352030	−	0.609734i	−0.634575	−	0.772862i	$-0.718825\pi$
$373$	2535.97	−	4392.42i	0.352030	−	0.609734i	0.986605	+	0.163127i	$0.0521580\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12675.2			−1.73158
$378$		0			0
$379$	−2562.11			−0.347247			−0.173624	−	0.984812i	$-0.555548\pi$
$379$	−2562.11			−0.347247			−0.173624	+	0.984812i	$0.555548\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2984.06	−	5168.54i	0.398116	−	0.689556i	−0.595378	−	0.803446i	$-0.702998\pi$
$383$	2984.06	−	5168.54i	0.398116	−	0.689556i	0.993493	+	0.113889i	$0.0363310\pi$
$384$		0			0
$385$	2383.90	+	4129.04i	0.315571	+	0.546585i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1273.43	+	2205.64i	0.165978	+	0.287482i	0.937002	−	0.349324i	$-0.113589\pi$
$389$	1273.43	+	2205.64i	0.165978	+	0.287482i	−0.771024	+	0.636806i	$0.780255\pi$
$390$		0			0
$391$	−602.492	+	1043.55i	−0.0779267	+	0.134973i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2882.59			−0.367187
$396$		0			0
$397$	426.226			0.0538833			0.0269416	−	0.999637i	$-0.491423\pi$
$397$	426.226			0.0538833			0.0269416	+	0.999637i	$0.491423\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3109.37	−	5385.59i	0.387219	−	0.670682i	−0.604856	−	0.796335i	$-0.706769\pi$
$401$	3109.37	−	5385.59i	0.387219	−	0.670682i	0.992074	+	0.125653i	$0.0401025\pi$
$402$		0			0
$403$	−7786.11	−	13485.9i	−0.962416	−	1.66695i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	871.495	+	1509.47i	0.106139	+	0.183837i
$408$		0			0
$409$	3063.37	−	5305.91i	0.370352	−	0.641468i	−0.619268	−	0.785180i	$-0.712571\pi$
$409$	3063.37	−	5305.91i	0.370352	−	0.641468i	0.989620	+	0.143712i	$0.0459039\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	15188.8			1.80966
$414$		0			0
$415$	6254.83			0.739850
$416$		0			0
$417$		0			0
$418$		0			0
$419$	6463.81	−	11195.7i	0.753647	−	1.30535i	−0.192398	−	0.981317i	$-0.561626\pi$
$419$	6463.81	−	11195.7i	0.753647	−	1.30535i	0.946044	−	0.324037i	$-0.105040\pi$
$420$		0			0
$421$	6959.45	+	12054.1i	0.805660	+	1.39544i	0.915845	+	0.401532i	$0.131522\pi$
$421$	6959.45	+	12054.1i	0.805660	+	1.39544i	−0.110185	+	0.993911i	$0.535144\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	727.899	+	1260.76i	0.0830783	+	0.143896i
$426$		0			0
$427$	−9985.26	+	17295.0i	−1.13166	+	1.96010i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	262.768			0.0293668			0.0146834	−	0.999892i	$-0.495326\pi$
$431$	262.768			0.0293668			0.0146834	+	0.999892i	$0.495326\pi$
$432$		0			0
$433$	4163.66			0.462108			0.231054	−	0.972941i	$-0.425783\pi$
$433$	4163.66			0.462108			0.231054	+	0.972941i	$0.425783\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1105.55	−	1914.87i	0.121020	−	0.209612i
$438$		0			0
$439$	5439.92	+	9422.22i	0.591419	+	1.02437i	0.994042	+	0.109002i	$0.0347656\pi$
$439$	5439.92	+	9422.22i	0.591419	+	1.02437i	−0.402622	+	0.915366i	$0.631901\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2561.71	−	4437.02i	−0.274742	−	0.475867i	0.695328	−	0.718693i	$-0.255259\pi$
$443$	−2561.71	−	4437.02i	−0.274742	−	0.475867i	−0.970070	+	0.242825i	$0.921926\pi$
$444$		0			0
$445$	−1005.40	+	1741.40i	−0.107102	+	0.185507i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	6313.23			0.663562			0.331781	−	0.943356i	$-0.392350\pi$
$449$	6313.23			0.663562			0.331781	+	0.943356i	$0.392350\pi$
$450$		0			0
$451$	−6337.76			−0.661715
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−5741.39	+	9944.38i	−0.591562	+	1.02461i
$456$		0			0
$457$	−4900.19	−	8487.37i	−0.501578	−	0.868758i	−0.999998	−	0.00182282i	$-0.999420\pi$
$457$	−4900.19	−	8487.37i	−0.501578	−	0.868758i	0.498421	−	0.866935i	$-0.333914\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	8163.20	+	14139.1i	0.824724	+	1.42846i	0.902130	+	0.431465i	$0.142003\pi$
$461$	8163.20	+	14139.1i	0.824724	+	1.42846i	−0.0774052	+	0.997000i	$0.524664\pi$
$462$		0			0
$463$	−4251.04	+	7363.01i	−0.426701	+	0.739067i	−0.996578	−	0.0826628i	$-0.973658\pi$
$463$	−4251.04	+	7363.01i	−0.426701	+	0.739067i	0.569877	+	0.821730i	$0.306991\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−705.493			−0.0699065			−0.0349532	−	0.999389i	$-0.511128\pi$
$467$	−705.493			−0.0699065			−0.0349532	+	0.999389i	$0.511128\pi$
$468$		0			0
$469$	−24390.1			−2.40134
$470$		0			0
$471$		0			0
$472$		0			0
$473$	5568.58	−	9645.06i	0.541318	−	0.937591i
$474$		0			0
$475$	−1335.67	−	2313.44i	−0.129020	−	0.223470i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3412.92	+	5911.35i	0.325554	+	0.563876i	0.981624	−	0.190824i	$-0.0611159\pi$
$479$	3412.92	+	5911.35i	0.325554	+	0.563876i	−0.656071	+	0.754700i	$0.727783\pi$
$480$		0			0
$481$	−2098.91	+	3635.42i	−0.198965	+	0.344617i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4092.78			0.383183
$486$		0			0
$487$	11052.2			1.02839			0.514193	−	0.857675i	$-0.328092\pi$
$487$	11052.2			1.02839			0.514193	+	0.857675i	$0.328092\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	3093.91	−	5358.80i	0.284371	−	0.492545i	−0.688086	−	0.725630i	$-0.741549\pi$
$491$	3093.91	−	5358.80i	0.284371	−	0.492545i	0.972456	+	0.233085i	$0.0748820\pi$
$492$		0			0
$493$	5577.22	+	9660.03i	0.509504	+	0.882487i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4146.05	+	7181.17i	0.374197	+	0.648128i
$498$		0			0
$499$	8156.64	−	14127.7i	0.731746	−	1.26742i	−0.224390	−	0.974499i	$-0.572039\pi$
$499$	8156.64	−	14127.7i	0.731746	−	1.26742i	0.956136	−	0.292922i	$-0.0946278\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−8648.39			−0.766625			−0.383313	−	0.923619i	$-0.625217\pi$
$503$	−8648.39			−0.766625			−0.383313	+	0.923619i	$0.625217\pi$
$504$		0			0
$505$	−2228.32			−0.196354
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2937.25	−	5087.47i	0.255779	−	0.443022i	−0.709328	−	0.704879i	$-0.751001\pi$
$509$	2937.25	−	5087.47i	0.255779	−	0.443022i	0.965107	+	0.261857i	$0.0843348\pi$
$510$		0			0
$511$	−11487.5	−	19896.9i	−0.994475	−	1.72248i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	45.9878	+	79.6532i	0.00393488	+	0.00681542i
$516$		0			0
$517$	6937.57	−	12016.2i	0.590163	−	1.02219i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8851.73			−0.744340			−0.372170	−	0.928164i	$-0.621386\pi$
$521$	−8851.73			−0.744340			−0.372170	+	0.928164i	$0.621386\pi$
$522$		0			0
$523$	6699.71			0.560149			0.280075	−	0.959978i	$-0.409641\pi$
$523$	6699.71			0.560149			0.280075	+	0.959978i	$0.409641\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−6851.95	+	11867.9i	−0.566367	+	0.980977i
$528$		0			0
$529$	5869.40	+	10166.1i	0.482403	+	0.835547i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7631.93	−	13218.9i	−0.620217	−	1.07425i
$534$		0			0
$535$	−4122.81	+	7140.91i	−0.333167	+	0.577063i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	23670.7			1.89159
$540$		0			0
$541$	2595.30			0.206249			0.103124	−	0.994668i	$-0.467116\pi$
$541$	2595.30			0.206249			0.103124	+	0.994668i	$0.467116\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1472.01	−	2549.59i	0.115695	−	0.200390i
$546$		0			0
$547$	279.236	+	483.650i	0.0218268	+	0.0378051i	0.876732	−	0.480978i	$-0.159718\pi$
$547$	279.236	+	483.650i	0.0218268	+	0.0378051i	−0.854906	+	0.518783i	$0.826385\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−10234.0	−	17725.8i	−0.791257	−	1.37050i
$552$		0			0
$553$	−10004.4	+	17328.2i	−0.769315	+	1.33249i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	5151.52			0.391880			0.195940	−	0.980616i	$-0.437224\pi$
$557$	5151.52			0.391880			0.195940	+	0.980616i	$0.437224\pi$
$558$		0			0
$559$	26822.7			2.02948
$560$		0			0
$561$		0			0
$562$		0			0
$563$	8901.70	−	15418.2i	0.666362	−	1.15417i	−0.312552	−	0.949901i	$-0.601184\pi$
$563$	8901.70	−	15418.2i	0.666362	−	1.15417i	0.978914	−	0.204272i	$-0.0654827\pi$
$564$		0			0
$565$	−1242.46	−	2152.00i	−0.0925142	−	0.160239i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−4694.29	−	8130.75i	−0.345861	−	0.599049i	0.639649	−	0.768667i	$-0.279080\pi$
$569$	−4694.29	−	8130.75i	−0.345861	−	0.599049i	−0.985510	+	0.169618i	$0.945747\pi$
$570$		0			0
$571$	4928.34	−	8536.13i	0.361199	−	0.625614i	−0.626960	−	0.779052i	$-0.715701\pi$
$571$	4928.34	−	8536.13i	0.361199	−	0.625614i	0.988158	+	0.153437i	$0.0490343\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−517.321			−0.0375196
$576$		0			0
$577$	12115.4			0.874126			0.437063	−	0.899431i	$-0.356019\pi$
$577$	12115.4			0.874126			0.437063	+	0.899431i	$0.356019\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	21708.3	−	37599.8i	1.55010	−	2.68486i
$582$		0			0
$583$	9531.02	+	16508.2i	0.677075	+	1.17273i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12384.1	+	21449.8i	0.870775	+	1.50823i	0.861197	+	0.508272i	$0.169716\pi$
$587$	12384.1	+	21449.8i	0.870775	+	1.50823i	0.00957831	+	0.999954i	$0.496951\pi$
$588$		0			0
$589$	12573.1	−	21777.2i	0.879565	−	1.52345i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6341.37			0.439138			0.219569	−	0.975597i	$-0.429535\pi$
$593$	6341.37			0.439138			0.219569	+	0.975597i	$0.429535\pi$
$594$		0			0
$595$	10105.1			0.696250
$596$		0			0
$597$		0			0
$598$		0			0
$599$	9280.18	−	16073.7i	0.633018	−	1.09642i	−0.353914	−	0.935278i	$-0.615149\pi$
$599$	9280.18	−	16073.7i	0.633018	−	1.09642i	0.986931	−	0.161141i	$-0.0515174\pi$
$600$		0			0
$601$	−3194.27	−	5532.63i	−0.216800	−	0.375509i	0.737028	−	0.675862i	$-0.236229\pi$
$601$	−3194.27	−	5532.63i	−0.216800	−	0.375509i	−0.953828	+	0.300354i	$0.902895\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−1440.30	−	2494.67i	−0.0967877	−	0.167641i
$606$		0			0
$607$	−11744.6	+	20342.2i	−0.785334	+	1.36024i	0.143466	+	0.989655i	$0.454175\pi$
$607$	−11744.6	+	20342.2i	−0.785334	+	1.36024i	−0.928800	+	0.370582i	$0.879158\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	33416.9			2.21261
$612$		0			0
$613$	−13134.2			−0.865389			−0.432694	−	0.901541i	$-0.642437\pi$
$613$	−13134.2			−0.865389			−0.432694	+	0.901541i	$0.642437\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−7625.20	+	13207.2i	−0.497535	+	0.861756i	−0.999996	−	0.00284411i	$-0.999095\pi$
$617$	−7625.20	+	13207.2i	−0.497535	+	0.861756i	0.502461	+	0.864600i	$0.332428\pi$
$618$		0			0
$619$	−11117.1	−	19255.4i	−0.721865	−	1.25031i	−0.960252	−	0.279136i	$-0.909952\pi$
$619$	−11117.1	−	19255.4i	−0.721865	−	1.25031i	0.238387	−	0.971170i	$-0.423381\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6978.76	+	12087.6i	0.448793	+	0.777332i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	3694.17			0.234175
$630$		0			0
$631$	−24006.1			−1.51453			−0.757265	−	0.653108i	$-0.773465\pi$
$631$	−24006.1			−1.51453			−0.757265	+	0.653108i	$0.773465\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	1666.81	−	2887.00i	0.104166	−	0.180421i
$636$		0			0
$637$	28504.2	+	49370.8i	1.77297	+	3.07087i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5888.32	−	10198.9i	−0.362831	−	0.628442i	0.625594	−	0.780148i	$-0.284856\pi$
$641$	−5888.32	−	10198.9i	−0.362831	−	0.628442i	−0.988426	+	0.151707i	$0.951523\pi$
$642$		0			0
$643$	11862.2	−	20545.9i	0.727525	−	1.26011i	−0.230401	−	0.973096i	$-0.574004\pi$
$643$	11862.2	−	20545.9i	0.727525	−	1.26011i	0.957926	−	0.287015i	$-0.0926629\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−14545.5			−0.883839			−0.441919	−	0.897055i	$-0.645702\pi$
$647$	−14545.5			−0.883839			−0.441919	+	0.897055i	$0.645702\pi$
$648$		0			0
$649$	12024.1			0.727253
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−686.969	+	1189.86i	−0.0411687	+	0.0713063i	−0.885876	−	0.463923i	$-0.846441\pi$
$653$	−686.969	+	1189.86i	−0.0411687	+	0.0713063i	0.844707	+	0.535229i	$0.179775\pi$
$654$		0			0
$655$	2169.48	+	3757.65i	0.129418	+	0.224158i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−296.323	−	513.246i	−0.0175161	−	0.0303387i	0.857135	−	0.515093i	$-0.172242\pi$
$659$	−296.323	−	513.246i	−0.0175161	−	0.0303387i	−0.874651	+	0.484754i	$0.838909\pi$
$660$		0			0
$661$	−7955.53	+	13779.4i	−0.468130	+	0.810826i	−0.999337	−	0.0364168i	$-0.988406\pi$
$661$	−7955.53	+	13779.4i	−0.468130	+	0.810826i	0.531206	+	0.847243i	$0.321739\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−18542.5			−1.08127
$666$		0			0
$667$	−3963.75			−0.230101
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−7904.76	+	13691.5i	−0.454784	+	0.787709i
$672$		0			0
$673$	499.218	+	864.670i	0.0285935	+	0.0495254i	0.879968	−	0.475033i	$-0.157564\pi$
$673$	499.218	+	864.670i	0.0285935	+	0.0495254i	−0.851375	+	0.524558i	$0.824231\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4873.92	−	8441.88i	−0.276691	−	0.479243i	0.693869	−	0.720101i	$-0.255905\pi$
$677$	−4873.92	−	8441.88i	−0.276691	−	0.479243i	−0.970560	+	0.240858i	$0.922571\pi$
$678$		0			0
$679$	14204.6	−	24603.0i	0.802829	−	1.39054i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−18314.6			−1.02605			−0.513023	−	0.858375i	$-0.671474\pi$
$683$	−18314.6			−1.02605			−0.513023	+	0.858375i	$0.671474\pi$
$684$		0			0
$685$	6979.09			0.389281
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−22954.5	+	39758.4i	−1.26923	+	2.19837i
$690$		0			0
$691$	−5800.22	−	10046.3i	−0.319321	−	0.553080i	0.661026	−	0.750363i	$-0.270121\pi$
$691$	−5800.22	−	10046.3i	−0.319321	−	0.553080i	−0.980347	+	0.197283i	$0.936788\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	2681.11	+	4643.81i	0.146331	+	0.253453i
$696$		0			0
$697$	−6716.27	+	11632.9i	−0.364988	+	0.632178i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−28849.1			−1.55438			−0.777188	−	0.629269i	$-0.783354\pi$
$701$	−28849.1			−1.55438			−0.777188	+	0.629269i	$0.783354\pi$
$702$		0			0
$703$	−6778.67			−0.363673
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7733.68	+	13395.1i	−0.411393	+	0.712554i
$708$		0			0
$709$	−15505.1	−	26855.6i	−0.821304	−	1.42254i	−0.904711	−	0.426025i	$-0.859913\pi$
$709$	−15505.1	−	26855.6i	−0.821304	−	1.42254i	0.0834070	−	0.996516i	$-0.473420\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2434.85	−	4217.29i	−0.127891	−	0.221513i
$714$		0			0
$715$	−4545.13	+	7872.40i	−0.237732	+	0.411764i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	3437.02			0.178275			0.0891373	−	0.996019i	$-0.471589\pi$
$719$	3437.02			0.178275			0.0891373	+	0.996019i	$0.471589\pi$
$720$		0			0
$721$	638.428			0.0329768
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2394.40	+	4147.22i	−0.122656	+	0.212447i
$726$		0			0
$727$	8217.94	+	14233.9i	0.419239	+	0.726143i	0.995863	−	0.0908665i	$-0.0289637\pi$
$727$	8217.94	+	14233.9i	0.419239	+	0.726143i	−0.576624	+	0.817009i	$0.695630\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−11802.3	−	20442.2i	−0.597160	−	1.03431i
$732$		0			0
$733$	13548.4	−	23466.6i	0.682705	−	1.18248i	−0.291447	−	0.956587i	$-0.594137\pi$
$733$	13548.4	−	23466.6i	0.682705	−	1.18248i	0.974152	−	0.225893i	$-0.0725298\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−19308.3			−0.965033
$738$		0			0
$739$	9125.36			0.454238			0.227119	−	0.973867i	$-0.427069\pi$
$739$	9125.36			0.454238			0.227119	+	0.973867i	$0.427069\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1754.29	+	3038.52i	−0.0866201	+	0.150030i	−0.906080	−	0.423106i	$-0.860940\pi$
$743$	−1754.29	+	3038.52i	−0.0866201	+	0.150030i	0.819460	+	0.573136i	$0.194273\pi$
$744$		0			0
$745$	−5375.06	−	9309.88i	−0.264332	−	0.457836i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	28617.5	+	49567.0i	1.39608	+	2.41808i
$750$		0			0
$751$	−1514.80	+	2623.71i	−0.0736029	+	0.127484i	−0.900478	−	0.434902i	$-0.856783\pi$
$751$	−1514.80	+	2623.71i	−0.0736029	+	0.127484i	0.826875	+	0.562386i	$0.190116\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−9361.68			−0.451267
$756$		0			0
$757$	22804.0			1.09488			0.547441	−	0.836845i	$-0.315602\pi$
$757$	22804.0			1.09488			0.547441	+	0.836845i	$0.315602\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7377.18	−	12777.6i	0.351409	−	0.608659i	−0.635087	−	0.772440i	$-0.719036\pi$
$761$	7377.18	−	12777.6i	0.351409	−	0.608659i	0.986497	+	0.163782i	$0.0523693\pi$
$762$		0			0
$763$	−10217.6	−	17697.4i	−0.484799	−	0.839697i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	14479.4	+	25079.1i	0.681645	+	1.18064i
$768$		0			0
$769$	−9006.49	+	15599.7i	−0.422344	+	0.731520i	−0.996168	−	0.0874577i	$-0.972126\pi$
$769$	−9006.49	+	15599.7i	−0.422344	+	0.731520i	0.573825	+	0.818978i	$0.305459\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	24820.4			1.15489			0.577445	−	0.816430i	$-0.304050\pi$
$773$	24820.4			1.15489			0.577445	+	0.816430i	$0.304050\pi$
$774$		0			0
$775$	−5883.32			−0.272691
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12324.1	−	21346.0i	0.566825	−	0.981770i
$780$		0			0
$781$	3282.19	+	5684.93i	0.150379	+	0.260464i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	3464.03	+	5999.87i	0.157499	+	0.272796i
$786$		0			0
$787$	−18283.1	+	31667.2i	−0.828109	+	1.43433i	0.0714116	+	0.997447i	$0.477250\pi$
$787$	−18283.1	+	31667.2i	−0.828109	+	1.43433i	−0.899520	+	0.436879i	$0.856084\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−17248.5			−0.775329
$792$		0			0
$793$	−38075.7			−1.70505
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−21278.7	+	36855.7i	−0.945707	+	1.63801i	−0.191378	+	0.981516i	$0.561296\pi$
$797$	−21278.7	+	36855.7i	−0.945707	+	1.63801i	−0.754329	+	0.656497i	$0.772038\pi$
$798$		0			0
$799$	−14703.8	−	25467.7i	−0.651043	−	1.12764i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9094.00	−	15751.3i	−0.399652	−	0.692217i
$804$		0			0
$805$	−1795.43	+	3109.78i	−0.0786096	+	0.136156i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−25817.9			−1.12201			−0.561007	−	0.827811i	$-0.689586\pi$
$809$	−25817.9			−1.12201			−0.561007	+	0.827811i	$0.689586\pi$
$810$		0			0
$811$	19818.9			0.858120			0.429060	−	0.903276i	$-0.358845\pi$
$811$	19818.9			0.858120			0.429060	+	0.903276i	$0.358845\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	354.272	−	613.617i	0.0152265	−	0.0263731i
$816$		0			0
$817$	21656.8	+	37510.6i	0.927386	+	1.60628i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	11389.6	+	19727.4i	0.484166	+	0.838600i	0.999835	−	0.0181878i	$-0.00578968\pi$
$821$	11389.6	+	19727.4i	0.484166	+	0.838600i	−0.515668	+	0.856788i	$0.672456\pi$
$822$		0			0
$823$	−7234.48	+	12530.5i	−0.306413	+	0.530723i	−0.977575	−	0.210588i	$-0.932462\pi$
$823$	−7234.48	+	12530.5i	−0.306413	+	0.530723i	0.671162	+	0.741311i	$0.265796\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−21163.1			−0.889858			−0.444929	−	0.895566i	$-0.646771\pi$
$827$	−21163.1			−0.889858			−0.444929	+	0.895566i	$0.646771\pi$
$828$		0			0
$829$	−1919.89			−0.0804348			−0.0402174	−	0.999191i	$-0.512805\pi$
$829$	−1919.89			−0.0804348			−0.0402174	+	0.999191i	$0.512805\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	25084.4	−	43447.4i	1.04336	−	1.80716i
$834$		0			0
$835$	2026.99	+	3510.86i	0.0840084	+	0.145507i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−17590.9	−	30468.4i	−0.723845	−	1.25374i	−0.959448	−	0.281887i	$-0.909040\pi$
$839$	−17590.9	−	30468.4i	−0.723845	−	1.25374i	0.235603	−	0.971849i	$-0.424294\pi$
$840$		0			0
$841$	−6151.58	+	10654.9i	−0.252228	+	0.436871i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−10908.0			−0.444079
$846$		0			0
$847$	−19995.0			−0.811143
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−656.366	+	1136.86i	−0.0264394	+	0.0457944i
$852$		0			0
$853$	3310.95	+	5734.73i	0.132901	+	0.230192i	0.924794	−	0.380469i	$-0.124237\pi$
$853$	3310.95	+	5734.73i	0.132901	+	0.230192i	−0.791893	+	0.610660i	$0.790904\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21840.3	−	37828.5i	−0.870538	−	1.50782i	−0.861441	−	0.507857i	$-0.830438\pi$
$857$	−21840.3	−	37828.5i	−0.870538	−	1.50782i	−0.00909673	−	0.999959i	$-0.502896\pi$
$858$		0			0
$859$	7842.56	−	13583.7i	0.311507	−	0.539546i	−0.667182	−	0.744895i	$-0.732500\pi$
$859$	7842.56	−	13583.7i	0.311507	−	0.539546i	0.978689	+	0.205349i	$0.0658329\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	38263.8			1.50929			0.754643	−	0.656135i	$-0.227810\pi$
$863$	38263.8			1.50929			0.754643	+	0.656135i	$0.227810\pi$
$864$		0			0
$865$	−6700.32			−0.263373
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−7919.94	+	13717.7i	−0.309166	+	0.535492i
$870$		0			0
$871$	−23251.0	−	40272.0i	−0.904513	−	1.56666i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	2169.15	+	3757.08i	0.0838064	+	0.145157i
$876$		0			0
$877$	18834.7	−	32622.6i	0.725201	−	1.25609i	−0.233690	−	0.972311i	$-0.575080\pi$
$877$	18834.7	−	32622.6i	0.725201	−	1.25609i	0.958891	−	0.283774i	$-0.0915867\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1945.38			0.0743945			0.0371972	−	0.999308i	$-0.488157\pi$
$881$	1945.38			0.0743945			0.0371972	+	0.999308i	$0.488157\pi$
$882$		0			0
$883$	−48739.4			−1.85754			−0.928772	−	0.370653i	$-0.879134\pi$
$883$	−48739.4			−1.85754			−0.928772	+	0.370653i	$0.879134\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12886.8	+	22320.6i	−0.487820	+	0.844928i	−0.999902	−	0.0140080i	$-0.995541\pi$
$887$	−12886.8	+	22320.6i	−0.487820	+	0.844928i	0.512082	+	0.858936i	$0.328874\pi$
$888$		0			0
$889$	−11569.8	−	20039.5i	−0.436489	−	0.756021i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	26980.9	+	46732.3i	1.01107	+	1.75122i
$894$		0			0
$895$	6003.20	−	10397.8i	0.224206	−	0.388337i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−45078.5			−1.67236
$900$		0			0
$901$	40401.0			1.49384
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−9448.89	+	16366.0i	−0.347063	+	0.601130i
$906$		0			0
$907$	2842.92	+	4924.08i	0.104077	+	0.180266i	0.913361	−	0.407152i	$-0.133478\pi$
$907$	2842.92	+	4924.08i	0.104077	+	0.180266i	−0.809284	+	0.587418i	$0.800145\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19342.8	−	33502.7i	−0.703463	−	1.21843i	−0.967243	−	0.253852i	$-0.918303\pi$
$911$	−19342.8	−	33502.7i	−0.703463	−	1.21843i	0.263780	−	0.964583i	$-0.415031\pi$
$912$		0			0
$913$	17185.2	−	29765.6i	0.622943	−	1.07897i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	30117.9			1.08460
$918$		0			0
$919$	−22903.7			−0.822115			−0.411057	−	0.911609i	$-0.634840\pi$
$919$	−22903.7			−0.822115			−0.411057	+	0.911609i	$0.634840\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7904.84	+	13691.6i	−0.281897	+	0.488260i
$924$		0			0
$925$	792.987	+	1373.49i	0.0281873	+	0.0488218i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19361.3	−	33534.8i	−0.683773	−	1.18433i	−0.973821	−	0.227317i	$-0.927005\pi$
$929$	−19361.3	−	33534.8i	−0.683773	−	1.18433i	0.290048	−	0.957012i	$-0.406329\pi$
$930$		0			0
$931$	−46028.9	+	79724.3i	−1.62034	+	2.80651i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7999.63			0.279803
$936$		0			0
$937$	−24777.8			−0.863879			−0.431940	−	0.901903i	$-0.642171\pi$
$937$	−24777.8			−0.863879			−0.431940	+	0.901903i	$0.642171\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−12059.3	+	20887.4i	−0.417772	+	0.723602i	−0.995715	−	0.0924747i	$-0.970522\pi$
$941$	−12059.3	+	20887.4i	−0.417772	+	0.723602i	0.577943	+	0.816077i	$0.303856\pi$
$942$		0			0
$943$	−2386.64	−	4133.78i	−0.0824175	−	0.142751i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13793.3	−	23890.7i	−0.473307	−	0.819791i	0.526226	−	0.850345i	$-0.323606\pi$
$947$	−13793.3	−	23890.7i	−0.473307	−	0.819791i	−0.999533	+	0.0305531i	$0.990273\pi$
$948$		0			0
$949$	21902.0	−	37935.4i	0.749177	−	1.29761i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	17641.7			0.599654			0.299827	−	0.953994i	$-0.403071\pi$
$953$	17641.7			0.599654			0.299827	+	0.953994i	$0.403071\pi$
$954$		0			0
$955$	−7904.38			−0.267832
$956$		0			0
$957$		0			0
$958$		0			0
$959$	24221.9	−	41953.5i	0.815605	−	1.41267i
$960$		0			0
$961$	−12795.3	−	22162.1i	−0.429503	−	0.743920i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	3434.24	+	5948.27i	0.114562	+	0.198427i
$966$		0			0
$967$	13075.9	−	22648.1i	0.434843	−	0.753169i	−0.562440	−	0.826838i	$-0.690137\pi$
$967$	13075.9	−	22648.1i	0.434843	−	0.753169i	0.997283	+	0.0736685i	$0.0234707\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−6455.27			−0.213347			−0.106673	−	0.994294i	$-0.534020\pi$
$971$	−6455.27			−0.213347			−0.106673	+	0.994294i	$0.534020\pi$
$972$		0			0
$973$	37220.6			1.22635
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−4569.07	+	7913.86i	−0.149619	+	0.259147i	−0.931087	−	0.364798i	$-0.881138\pi$
$977$	−4569.07	+	7913.86i	−0.149619	+	0.259147i	0.781468	+	0.623945i	$0.214471\pi$
$978$		0			0
$979$	5524.69	+	9569.04i	0.180357	+	0.312388i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7207.09	−	12483.0i	−0.233846	−	0.405033i	0.725091	−	0.688653i	$-0.241798\pi$
$983$	−7207.09	−	12483.0i	−0.233846	−	0.405033i	−0.958937	+	0.283620i	$0.908464\pi$
$984$		0			0
$985$	1799.70	−	3117.18i	0.0582165	−	0.100834i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8387.95			0.269688
$990$		0			0
$991$	6347.11			0.203454			0.101727	−	0.994812i	$-0.467563\pi$
$991$	6347.11			0.203454			0.101727	+	0.994812i	$0.467563\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−7928.77	+	13733.0i	−0.252622	+	0.437554i
$996$		0			0
$997$	−18183.0	−	31493.9i	−0.577595	−	1.00042i	−0.995754	−	0.0920507i	$-0.970658\pi$
$997$	−18183.0	−	31493.9i	−0.577595	−	1.00042i	0.418159	−	0.908374i	$-0.362676\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.q.b.361.1		16
3.2	odd	2		360.4.q.b.121.6	✓	16
9.2	odd	6		360.4.q.b.241.6	yes	16
9.7	even	3	inner	1080.4.q.b.721.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.q.b.121.6	✓	16	3.2	odd	2
360.4.q.b.241.6	yes	16	9.2	odd	6
1080.4.q.b.361.1		16	1.1	even	1	trivial
1080.4.q.b.721.1		16	9.7	even	3	inner

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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.50000 - 4.33013i) q^{5} +(-17.3532 - 30.0566i) q^{7} +(-13.7375 - 23.7941i) q^{11} +(33.0855 - 57.3058i) q^{13} -58.2319 q^{17} +106.853 q^{19} +(10.3464 - 17.9205i) q^{23} +(-12.5000 - 21.6506i) q^{25} +(-95.7760 - 165.889i) q^{29} +(117.666 - 203.804i) q^{31} -173.532 q^{35} -63.4389 q^{37} +(115.337 - 199.769i) q^{41} +(202.677 + 351.048i) q^{43} +(252.504 + 437.350i) q^{47} +(-430.766 + 746.109i) q^{49} -693.794 q^{53} -137.375 q^{55} +(-218.818 + 379.004i) q^{59} +(-287.707 - 498.323i) q^{61} +(-165.428 - 286.529i) q^{65} +(351.378 - 608.604i) q^{67} -238.922 q^{71} +661.982 q^{73} +(-476.780 + 825.808i) q^{77} +(-288.259 - 499.279i) q^{79} +(625.483 + 1083.37i) q^{83} +(-145.580 + 252.152i) q^{85} -402.160 q^{89} -2296.56 q^{91} +(267.133 - 462.689i) q^{95} +(409.278 + 708.891i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 40 q^{5} - 34 q^{7} + 64 q^{11} - 48 q^{13} - 212 q^{17} + 456 q^{19} + 166 q^{23} - 200 q^{25} + 110 q^{29} - 160 q^{31} - 340 q^{35} + 104 q^{37} + 280 q^{41} + 136 q^{43} + 594 q^{47} - 1094 q^{49}+ \cdots + 182 q^{97}+O(q^{100}) \) 16 * q + 40 * q^5 - 34 * q^7 + 64 * q^11 - 48 * q^13 - 212 * q^17 + 456 * q^19 + 166 * q^23 - 200 * q^25 + 110 * q^29 - 160 * q^31 - 340 * q^35 + 104 * q^37 + 280 * q^41 + 136 * q^43 + 594 * q^47 - 1094 * q^49 - 752 * q^53 + 640 * q^55 + 24 * q^59 - 950 * q^61 + 240 * q^65 - 426 * q^67 + 136 * q^71 + 308 * q^73 - 696 * q^77 - 764 * q^79 + 1362 * q^83 - 530 * q^85 + 172 * q^89 - 1224 * q^91 + 1140 * q^95 + 182 * q^97

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