LMFDB - Embedded newform 1080.4.q.e.721.6

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 = [20,0,0,0,-50,0,22] 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:	$20$
Relative dimension:	$10$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} + 1542 x^{18} + 976429 x^{16} + 327887620 x^{14} + 62946909772 x^{12} + 6953278937404 x^{10} + \cdots + 45\!\cdots\!76 $ x^20 + 1542x^18 + 976429x^16 + 327887620x^14 + 62946909772x^12 + 6953278937404x^10 + 426056944129744x^8 + 13516703885670016x^6 + 186027361852684288x^4 + 550048054374998016*x^2 + 452028941289112576
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{25} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.6
Root			$-1.23574i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.4.q.e.361.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.50000 - 4.33013i) q^{5} +(1.57018 - 2.71963i) q^{7} +(7.30996 - 12.6612i) q^{11} +(-27.3440 - 47.3612i) q^{13} -45.7952 q^{17} -50.3457 q^{19} +(-9.94951 - 17.2331i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-82.1830 + 142.345i) q^{29} +(123.317 + 213.592i) q^{31} -15.7018 q^{35} +268.484 q^{37} +(-166.114 - 287.718i) q^{41} +(55.4147 - 95.9811i) q^{43} +(17.8310 - 30.8843i) q^{47} +(166.569 + 288.506i) q^{49} -482.657 q^{53} -73.0996 q^{55} +(-164.766 - 285.383i) q^{59} +(-80.0901 + 138.720i) q^{61} +(-136.720 + 236.806i) q^{65} +(145.738 + 252.425i) q^{67} -191.669 q^{71} +866.440 q^{73} +(-22.9559 - 39.7608i) q^{77} +(-369.794 + 640.503i) q^{79} +(-244.041 + 422.691i) q^{83} +(114.488 + 198.299i) q^{85} -173.938 q^{89} -171.740 q^{91} +(125.864 + 218.003i) q^{95} +(-744.116 + 1288.85i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 50 q^{5} + 22 q^{7} - 50 q^{11} - 16 q^{13} + 68 q^{17} + 212 q^{19} - 50 q^{23} - 250 q^{25} + 64 q^{29} - 22 q^{31} - 220 q^{35} + 600 q^{37} + 198 q^{41} - 382 q^{43} - 128 q^{47} - 1230 q^{49}+ \cdots - 1354 q^{97}+O(q^{100}) $ 20 * q - 50 * q^5 + 22 * q^7 - 50 * q^11 - 16 * q^13 + 68 * q^17 + 212 * q^19 - 50 * q^23 - 250 * q^25 + 64 * q^29 - 22 * q^31 - 220 * q^35 + 600 * q^37 + 198 * q^41 - 382 * q^43 - 128 * q^47 - 1230 * q^49 + 1128 * q^53 + 500 * q^55 + 200 * q^59 - 232 * q^61 - 80 * q^65 - 1316 * q^67 + 228 * q^71 + 828 * q^73 - 616 * q^77 - 300 * q^79 + 1082 * q^83 - 170 * q^85 + 768 * q^89 + 1972 * q^91 - 530 * q^95 - 1354 * q^97

Character values

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

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$7$	1.57018	−	2.71963i	0.0847818	−	0.146846i	−0.820516	−	0.571623i	$-0.806314\pi$
$7$	1.57018	−	2.71963i	0.0847818	−	0.146846i	0.905298	+	0.424777i	$0.139647\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	7.30996	−	12.6612i	0.200367	−	0.347046i	−0.748280	−	0.663383i	$-0.769120\pi$
$11$	7.30996	−	12.6612i	0.200367	−	0.347046i	0.948647	+	0.316338i	$0.102453\pi$
$12$		0			0
$13$	−27.3440	−	47.3612i	−0.583374	−	1.01043i	−0.995076	−	0.0991143i	$-0.968399\pi$
$13$	−27.3440	−	47.3612i	−0.583374	−	1.01043i	0.411703	−	0.911318i	$-0.364934\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−45.7952			−0.653351			−0.326676	−	0.945137i	$-0.605928\pi$
$17$	−45.7952			−0.653351			−0.326676	+	0.945137i	$0.605928\pi$
$18$		0			0
$19$	−50.3457			−0.607900			−0.303950	−	0.952688i	$-0.598306\pi$
$19$	−50.3457			−0.607900			−0.303950	+	0.952688i	$0.598306\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.94951	−	17.2331i	−0.0902007	−	0.156232i	0.817395	−	0.576078i	$-0.195418\pi$
$23$	−9.94951	−	17.2331i	−0.0902007	−	0.156232i	−0.907595	+	0.419846i	$0.862084\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−82.1830	+	142.345i	−0.526242	+	0.911477i	0.473291	+	0.880906i	$0.343066\pi$
$29$	−82.1830	+	142.345i	−0.526242	+	0.911477i	−0.999533	+	0.0305711i	$0.990267\pi$
$30$		0			0
$31$	123.317	+	213.592i	0.714466	+	1.23749i	0.963165	+	0.268911i	$0.0866636\pi$
$31$	123.317	+	213.592i	0.714466	+	1.23749i	−0.248699	+	0.968581i	$0.580003\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−15.7018			−0.0758312
$36$		0			0
$37$	268.484			1.19293			0.596467	−	0.802637i	$-0.296571\pi$
$37$	268.484			1.19293			0.596467	+	0.802637i	$0.296571\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−166.114	−	287.718i	−0.632747	−	1.09595i	−0.986988	−	0.160795i	$-0.948594\pi$
$41$	−166.114	−	287.718i	−0.632747	−	1.09595i	0.354241	−	0.935154i	$-0.384739\pi$
$42$		0			0
$43$	55.4147	−	95.9811i	0.196527	−	0.340395i	−0.750873	−	0.660447i	$-0.770367\pi$
$43$	55.4147	−	95.9811i	0.196527	−	0.340395i	0.947400	+	0.320052i	$0.103700\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	17.8310	−	30.8843i	0.0553388	−	0.0958496i	−0.837029	−	0.547159i	$-0.815709\pi$
$47$	17.8310	−	30.8843i	0.0553388	−	0.0958496i	0.892368	+	0.451309i	$0.149043\pi$
$48$		0			0
$49$	166.569	+	288.506i	0.485624	+	0.841126i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−482.657			−1.25091			−0.625453	−	0.780262i	$-0.715086\pi$
$53$	−482.657			−1.25091			−0.625453	+	0.780262i	$0.715086\pi$
$54$		0			0
$55$	−73.0996			−0.179214
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−164.766	−	285.383i	−0.363572	−	0.629724i	0.624974	−	0.780645i	$-0.285109\pi$
$59$	−164.766	−	285.383i	−0.363572	−	0.629724i	−0.988546	+	0.150921i	$0.951776\pi$
$60$		0			0
$61$	−80.0901	+	138.720i	−0.168106	+	0.291169i	−0.937754	−	0.347300i	$-0.887099\pi$
$61$	−80.0901	+	138.720i	−0.168106	+	0.291169i	0.769648	+	0.638469i	$0.220432\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−136.720	+	236.806i	−0.260893	+	0.451879i
$66$		0			0
$67$	145.738	+	252.425i	0.265742	+	0.460278i	0.967758	−	0.251883i	$-0.0810498\pi$
$67$	145.738	+	252.425i	0.265742	+	0.460278i	−0.702016	+	0.712161i	$0.747716\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−191.669			−0.320380			−0.160190	−	0.987086i	$-0.551211\pi$
$71$	−191.669			−0.320380			−0.160190	+	0.987086i	$0.551211\pi$
$72$		0			0
$73$	866.440			1.38917			0.694583	−	0.719412i	$-0.255589\pi$
$73$	866.440			1.38917			0.694583	+	0.719412i	$0.255589\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−22.9559	−	39.7608i	−0.0339749	−	0.0588463i
$78$		0			0
$79$	−369.794	+	640.503i	−0.526647	+	0.912180i	0.472871	+	0.881132i	$0.343218\pi$
$79$	−369.794	+	640.503i	−0.526647	+	0.912180i	−0.999518	+	0.0310478i	$0.990116\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−244.041	+	422.691i	−0.322734	+	0.558992i	−0.981051	−	0.193749i	$-0.937935\pi$
$83$	−244.041	+	422.691i	−0.322734	+	0.558992i	0.658317	+	0.752741i	$0.271269\pi$
$84$		0			0
$85$	114.488	+	198.299i	0.146094	+	0.253042i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−173.938			−0.207161			−0.103581	−	0.994621i	$-0.533030\pi$
$89$	−173.938			−0.207161			−0.103581	+	0.994621i	$0.533030\pi$
$90$		0			0
$91$	−171.740			−0.197838
$92$		0			0
$93$		0			0
$94$		0			0
$95$	125.864	+	218.003i	0.135931	+	0.235439i
$96$		0			0
$97$	−744.116	+	1288.85i	−0.778902	+	1.34910i	0.153673	+	0.988122i	$0.450890\pi$
$97$	−744.116	+	1288.85i	−0.778902	+	1.34910i	−0.932575	+	0.360976i	$0.882444\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−197.683	+	342.397i	−0.194754	+	0.337324i	−0.946820	−	0.321764i	$-0.895724\pi$
$101$	−197.683	+	342.397i	−0.194754	+	0.337324i	0.752066	+	0.659088i	$0.229058\pi$
$102$		0			0
$103$	−428.044	−	741.394i	−0.409480	−	0.709240i	0.585352	−	0.810779i	$-0.300956\pi$
$103$	−428.044	−	741.394i	−0.409480	−	0.709240i	−0.994831	+	0.101540i	$0.967623\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−423.634			−0.382750			−0.191375	−	0.981517i	$-0.561295\pi$
$107$	−423.634			−0.382750			−0.191375	+	0.981517i	$0.561295\pi$
$108$		0			0
$109$	−840.730			−0.738783			−0.369391	−	0.929274i	$-0.620434\pi$
$109$	−840.730			−0.738783			−0.369391	+	0.929274i	$0.620434\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	853.149	+	1477.70i	0.710243	+	1.23018i	0.964766	+	0.263111i	$0.0847484\pi$
$113$	853.149	+	1477.70i	0.710243	+	1.23018i	−0.254522	+	0.967067i	$0.581918\pi$
$114$		0			0
$115$	−49.7475	+	86.1653i	−0.0403390	+	0.0698692i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−71.9068	+	124.546i	−0.0553923	+	0.0959423i
$120$		0			0
$121$	558.629	+	967.574i	0.419706	+	0.726953i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			0.0894427
$126$		0			0
$127$	1075.14			0.751205			0.375602	−	0.926781i	$-0.377436\pi$
$127$	1075.14			0.751205			0.375602	+	0.926781i	$0.377436\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	443.630	+	768.390i	0.295879	+	0.512477i	0.975189	−	0.221374i	$-0.0710542\pi$
$131$	443.630	+	768.390i	0.295879	+	0.512477i	−0.679310	+	0.733851i	$0.737721\pi$
$132$		0			0
$133$	−79.0519	+	136.922i	−0.0515389	+	0.0892679i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−988.251	+	1711.70i	−0.616292	+	1.06745i	0.373865	+	0.927483i	$0.378032\pi$
$137$	−988.251	+	1711.70i	−0.616292	+	1.06745i	−0.990156	+	0.139965i	$0.955301\pi$
$138$		0			0
$139$	906.632	+	1570.33i	0.553234	+	0.958229i	0.998039	+	0.0626017i	$0.0199398\pi$
$139$	906.632	+	1570.33i	0.553234	+	0.958229i	−0.444805	+	0.895628i	$0.646727\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−799.534			−0.467555
$144$		0			0
$145$	821.830			0.470685
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−505.197	−	875.027i	−0.277767	−	0.481107i	0.693062	−	0.720878i	$-0.256261\pi$
$149$	−505.197	−	875.027i	−0.277767	−	0.481107i	−0.970830	+	0.239771i	$0.922928\pi$
$150$		0			0
$151$	−577.277	+	999.874i	−0.311114	+	0.538865i	−0.978604	−	0.205754i	$-0.934035\pi$
$151$	−577.277	+	999.874i	−0.311114	+	0.538865i	0.667490	+	0.744619i	$0.267369\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	616.587	−	1067.96i	0.319519	−	0.553423i
$156$		0			0
$157$	−569.678	−	986.711i	−0.289587	−	0.501580i	0.684124	−	0.729366i	$-0.260185\pi$
$157$	−569.678	−	986.711i	−0.289587	−	0.501580i	−0.973711	+	0.227786i	$0.926851\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−62.4901			−0.0305895
$162$		0			0
$163$	−3320.87			−1.59577			−0.797886	−	0.602809i	$-0.794048\pi$
$163$	−3320.87			−1.59577			−0.797886	+	0.602809i	$0.794048\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−1076.63	−	1864.78i	−0.498876	−	0.864078i	0.501123	−	0.865376i	$-0.332920\pi$
$167$	−1076.63	−	1864.78i	−0.498876	−	0.864078i	−0.999999	+	0.00129774i	$0.999587\pi$
$168$		0			0
$169$	−396.887	+	687.428i	−0.180649	+	0.312894i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1898.21	+	3287.80i	−0.834211	+	1.44490i	0.0604599	+	0.998171i	$0.480743\pi$
$173$	−1898.21	+	3287.80i	−0.834211	+	1.44490i	−0.894671	+	0.446725i	$0.852590\pi$
$174$		0			0
$175$	39.2545	+	67.9908i	0.0169564	+	0.0293693i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1763.35			−0.736305			−0.368153	−	0.929765i	$-0.620010\pi$
$179$	−1763.35			−0.736305			−0.368153	+	0.929765i	$0.620010\pi$
$180$		0			0
$181$	194.975			0.0800683			0.0400342	−	0.999198i	$-0.487253\pi$
$181$	194.975			0.0800683			0.0400342	+	0.999198i	$0.487253\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−671.211	−	1162.57i	−0.266748	−	0.462022i
$186$		0			0
$187$	−334.761	+	579.823i	−0.130910	+	0.226743i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1718.62	+	2976.74i	−0.651074	+	1.12769i	0.331789	+	0.943354i	$0.392348\pi$
$191$	−1718.62	+	2976.74i	−0.651074	+	1.12769i	−0.982863	+	0.184339i	$0.940985\pi$
$192$		0			0
$193$	1272.75	+	2204.47i	0.474686	+	0.822181i	0.999580	−	0.0289870i	$-0.00922813\pi$
$193$	1272.75	+	2204.47i	0.474686	+	0.822181i	−0.524893	+	0.851168i	$0.675895\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3772.65			1.36442			0.682209	−	0.731158i	$-0.261020\pi$
$197$	3772.65			1.36442			0.682209	+	0.731158i	$0.261020\pi$
$198$		0			0
$199$	−1011.51			−0.360321			−0.180160	−	0.983637i	$-0.557662\pi$
$199$	−1011.51			−0.360321			−0.180160	+	0.983637i	$0.557662\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	258.085	+	447.016i	0.0892314	+	0.154553i
$204$		0			0
$205$	−830.569	+	1438.59i	−0.282973	+	0.490124i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−368.025	+	637.438i	−0.121803	+	0.210969i
$210$		0			0
$211$	−2104.05	−	3644.33i	−0.686488	−	1.18903i	−0.972967	−	0.230946i	$-0.925818\pi$
$211$	−2104.05	−	3644.33i	−0.686488	−	1.18903i	0.286478	−	0.958087i	$-0.407515\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−554.147			−0.175779
$216$		0			0
$217$	774.522			0.242295
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1252.22	+	2168.91i	0.381148	+	0.660167i
$222$		0			0
$223$	1576.10	−	2729.89i	0.473290	−	0.819762i	−0.526243	−	0.850335i	$-0.676400\pi$
$223$	1576.10	−	2729.89i	0.473290	−	0.819762i	0.999533	+	0.0305721i	$0.00973293\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1027.79	+	1780.18i	−0.300515	+	0.520507i	−0.976253	−	0.216635i	$-0.930492\pi$
$227$	−1027.79	+	1780.18i	−0.300515	+	0.520507i	0.675738	+	0.737142i	$0.263825\pi$
$228$		0			0
$229$	2339.71	+	4052.50i	0.675164	+	1.16942i	0.976421	+	0.215875i	$0.0692604\pi$
$229$	2339.71	+	4052.50i	0.675164	+	1.16942i	−0.301257	+	0.953543i	$0.597406\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	707.658			0.198971			0.0994854	−	0.995039i	$-0.468280\pi$
$233$	707.658			0.198971			0.0994854	+	0.995039i	$0.468280\pi$
$234$		0			0
$235$	−178.310			−0.0494965
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−3046.20	−	5276.18i	−0.824446	−	1.42798i	−0.902342	−	0.431021i	$-0.858154\pi$
$239$	−3046.20	−	5276.18i	−0.824446	−	1.42798i	0.0778960	−	0.996961i	$-0.475180\pi$
$240$		0			0
$241$	−826.336	+	1431.26i	−0.220867	+	0.382553i	−0.955072	−	0.296375i	$-0.904222\pi$
$241$	−826.336	+	1431.26i	−0.220867	+	0.382553i	0.734204	+	0.678929i	$0.237555\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	832.845	−	1442.53i	0.217178	−	0.376163i
$246$		0			0
$247$	1376.65	+	2384.43i	0.354633	+	0.614242i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1497.98			−0.376700			−0.188350	−	0.982102i	$-0.560314\pi$
$251$	−1497.98			−0.376700			−0.188350	+	0.982102i	$0.560314\pi$
$252$		0			0
$253$	−290.922			−0.0722929
$254$		0			0
$255$		0			0
$256$		0			0
$257$	826.195	+	1431.01i	0.200532	+	0.347331i	0.948700	−	0.316178i	$-0.102400\pi$
$257$	826.195	+	1431.01i	0.200532	+	0.347331i	−0.748168	+	0.663509i	$0.769066\pi$
$258$		0			0
$259$	421.569	−	730.179i	0.101139	−	0.175178i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−2877.51	+	4983.99i	−0.674657	+	1.16854i	0.301912	+	0.953336i	$0.402375\pi$
$263$	−2877.51	+	4983.99i	−0.674657	+	1.16854i	−0.976569	+	0.215205i	$0.930958\pi$
$264$		0			0
$265$	1206.64	+	2089.97i	0.279711	+	0.484474i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4242.00			0.961484			0.480742	−	0.876862i	$-0.340367\pi$
$269$	4242.00			0.961484			0.480742	+	0.876862i	$0.340367\pi$
$270$		0			0
$271$	4037.67			0.905059			0.452529	−	0.891750i	$-0.350522\pi$
$271$	4037.67			0.905059			0.452529	+	0.891750i	$0.350522\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	182.749	+	316.531i	0.0400734	+	0.0694091i
$276$		0			0
$277$	−425.127	+	736.341i	−0.0922144	+	0.159720i	−0.908443	−	0.418010i	$-0.862728\pi$
$277$	−425.127	+	736.341i	−0.0922144	+	0.159720i	0.816228	+	0.577730i	$0.196061\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2110.97	+	3656.30i	−0.448149	+	0.776216i	−0.998266	−	0.0588713i	$-0.981250\pi$
$281$	−2110.97	+	3656.30i	−0.448149	+	0.776216i	0.550117	+	0.835088i	$0.314583\pi$
$282$		0			0
$283$	−1225.85	−	2123.23i	−0.257488	−	0.445982i	0.708081	−	0.706132i	$-0.249561\pi$
$283$	−1225.85	−	2123.23i	−0.257488	−	0.445982i	−0.965568	+	0.260150i	$0.916228\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−1043.32			−0.214582
$288$		0			0
$289$	−2815.80			−0.573132
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1912.62	−	3312.75i	−0.381353	−	0.660523i	0.609903	−	0.792476i	$-0.291208\pi$
$293$	−1912.62	−	3312.75i	−0.381353	−	0.660523i	−0.991256	+	0.131953i	$0.957875\pi$
$294$		0			0
$295$	−823.831	+	1426.92i	−0.162594	+	0.281621i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−544.118	+	942.441i	−0.105241	+	0.182283i
$300$		0			0
$301$	−174.022	−	301.416i	−0.0333239	−	0.0577186i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	800.901			0.150359
$306$		0			0
$307$	2771.63			0.515262			0.257631	−	0.966243i	$-0.417058\pi$
$307$	2771.63			0.515262			0.257631	+	0.966243i	$0.417058\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4336.74	−	7511.46i	−0.790721	−	1.36957i	−0.925521	−	0.378696i	$-0.876373\pi$
$311$	−4336.74	−	7511.46i	−0.790721	−	1.36957i	0.134801	−	0.990873i	$-0.456961\pi$
$312$		0			0
$313$	−1772.78	+	3070.55i	−0.320139	+	0.554497i	−0.980516	−	0.196437i	$-0.937063\pi$
$313$	−1772.78	+	3070.55i	−0.320139	+	0.554497i	0.660378	+	0.750934i	$0.270396\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	3081.76	−	5337.77i	0.546022	−	0.945738i	−0.452520	−	0.891754i	$-0.649475\pi$
$317$	3081.76	−	5337.77i	0.546022	−	0.945738i	0.998542	−	0.0539836i	$-0.0171919\pi$
$318$		0			0
$319$	1201.51	+	2081.08i	0.210883	+	0.365260i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	2305.59			0.397172
$324$		0			0
$325$	1367.20			0.233349
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−55.9959	−	96.9878i	−0.00938345	−	0.0162526i
$330$		0			0
$331$	4358.97	−	7549.96i	0.723839	−	1.25373i	−0.235611	−	0.971847i	$-0.575709\pi$
$331$	4358.97	−	7549.96i	0.723839	−	1.25373i	0.959450	−	0.281878i	$-0.0909574\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	728.689	−	1262.13i	0.118843	−	0.205843i
$336$		0			0
$337$	3254.65	+	5637.22i	0.526090	+	0.911214i	0.999538	+	0.0303925i	$0.00967573\pi$
$337$	3254.65	+	5637.22i	0.526090	+	0.911214i	−0.473448	+	0.880822i	$0.656991\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	3605.78			0.572621
$342$		0			0
$343$	2123.32			0.334252
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1852.56	−	3208.72i	−0.286601	−	0.496407i	0.686395	−	0.727228i	$-0.259192\pi$
$347$	−1852.56	−	3208.72i	−0.286601	−	0.496407i	−0.972996	+	0.230822i	$0.925859\pi$
$348$		0			0
$349$	−5113.11	+	8856.17i	−0.784237	+	1.35834i	0.145217	+	0.989400i	$0.453612\pi$
$349$	−5113.11	+	8856.17i	−0.784237	+	1.35834i	−0.929454	+	0.368938i	$0.879721\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1156.50	−	2003.11i	0.174374	−	0.302025i	−0.765570	−	0.643352i	$-0.777543\pi$
$353$	1156.50	−	2003.11i	0.174374	−	0.302025i	0.939944	+	0.341327i	$0.110876\pi$
$354$		0			0
$355$	479.173	+	829.953i	0.0716391	+	0.124083i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−11018.9			−1.61993			−0.809967	−	0.586475i	$-0.800515\pi$
$359$	−11018.9			−1.61993			−0.809967	+	0.586475i	$0.800515\pi$
$360$		0			0
$361$	−4324.31			−0.630458
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2166.10	−	3751.80i	−0.310627	−	0.538022i
$366$		0			0
$367$	6462.97	−	11194.2i	0.919248	−	1.59218i	0.118689	−	0.992932i	$-0.462131\pi$
$367$	6462.97	−	11194.2i	0.919248	−	1.59218i	0.800560	−	0.599253i	$-0.204536\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−757.859	+	1312.65i	−0.106054	+	0.183691i
$372$		0			0
$373$	−1003.06	−	1737.36i	−0.139240	−	0.241171i	0.787969	−	0.615715i	$-0.211133\pi$
$373$	−1003.06	−	1737.36i	−0.139240	−	0.241171i	−0.927209	+	0.374544i	$0.877799\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	8988.85			1.22798
$378$		0			0
$379$	−6167.89			−0.835945			−0.417972	−	0.908460i	$-0.637259\pi$
$379$	−6167.89			−0.835945			−0.417972	+	0.908460i	$0.637259\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1344.31	−	2328.41i	−0.179350	−	0.310643i	0.762308	−	0.647214i	$-0.224066\pi$
$383$	−1344.31	−	2328.41i	−0.179350	−	0.310643i	−0.941658	+	0.336571i	$0.890733\pi$
$384$		0			0
$385$	−114.780	+	198.804i	−0.0151941	+	0.0263169i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−857.500	+	1485.23i	−0.111766	+	0.193584i	−0.916482	−	0.400075i	$-0.868984\pi$
$389$	−857.500	+	1485.23i	−0.111766	+	0.193584i	0.804716	+	0.593660i	$0.202317\pi$
$390$		0			0
$391$	455.640	+	789.191i	0.0589327	+	0.102074i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3697.94			0.471048
$396$		0			0
$397$	8048.37			1.01747			0.508735	−	0.860923i	$-0.330113\pi$
$397$	8048.37			1.01747			0.508735	+	0.860923i	$0.330113\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	4821.71	+	8351.45i	0.600461	+	1.04003i	0.992751	+	0.120188i	$0.0383496\pi$
$401$	4821.71	+	8351.45i	0.600461	+	1.04003i	−0.392290	+	0.919842i	$0.628317\pi$
$402$		0			0
$403$	6743.97	−	11680.9i	0.833601	−	1.44384i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1962.61	−	3399.34i	0.239025	−	0.414003i
$408$		0			0
$409$	−7105.36	−	12306.8i	−0.859016	−	1.48786i	−0.872869	−	0.487955i	$-0.837743\pi$
$409$	−7105.36	−	12306.8i	−0.859016	−	1.48786i	0.0138525	−	0.999904i	$-0.495590\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1034.85			−0.123297
$414$		0			0
$415$	2440.41			0.288662
$416$		0			0
$417$		0			0
$418$		0			0
$419$	1114.29	+	1930.01i	0.129921	+	0.225029i	0.923646	−	0.383248i	$-0.125194\pi$
$419$	1114.29	+	1930.01i	0.129921	+	0.225029i	−0.793725	+	0.608277i	$0.791861\pi$
$420$		0			0
$421$	776.902	−	1345.63i	0.0899380	−	0.155777i	−0.817547	−	0.575862i	$-0.804666\pi$
$421$	776.902	−	1345.63i	0.0899380	−	0.155777i	0.907485	+	0.420085i	$0.138000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	572.440	−	991.495i	0.0653351	−	0.113164i
$426$		0			0
$427$	251.512	+	435.632i	0.0285047	+	0.0493716i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−462.976			−0.0517419			−0.0258710	−	0.999665i	$-0.508236\pi$
$431$	−462.976			−0.0517419			−0.0258710	+	0.999665i	$0.508236\pi$
$432$		0			0
$433$	−580.010			−0.0643729			−0.0321865	−	0.999482i	$-0.510247\pi$
$433$	−580.010			−0.0643729			−0.0321865	+	0.999482i	$0.510247\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	500.915	+	867.610i	0.0548330	+	0.0949735i
$438$		0			0
$439$	−1096.63	+	1899.41i	−0.119223	+	0.206501i	−0.919460	−	0.393183i	$-0.871374\pi$
$439$	−1096.63	+	1899.41i	−0.119223	+	0.206501i	0.800237	+	0.599684i	$0.204707\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1710.70	+	2963.02i	−0.183471	+	0.317781i	−0.943060	−	0.332622i	$-0.892067\pi$
$443$	−1710.70	+	2963.02i	−0.183471	+	0.317781i	0.759589	+	0.650403i	$0.225400\pi$
$444$		0			0
$445$	434.844	+	753.172i	0.0463227	+	0.0802332i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	13914.9			1.46255			0.731273	−	0.682085i	$-0.238926\pi$
$449$	13914.9			1.46255			0.731273	+	0.682085i	$0.238926\pi$
$450$		0			0
$451$	−4857.14			−0.507126
$452$		0			0
$453$		0			0
$454$		0			0
$455$	429.350	+	743.656i	0.0442379	+	0.0766223i
$456$		0			0
$457$	−3864.00	+	6692.65i	−0.395515	+	0.685052i	−0.993167	−	0.116703i	$-0.962767\pi$
$457$	−3864.00	+	6692.65i	−0.395515	+	0.685052i	0.597652	+	0.801756i	$0.296101\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3464.26	−	6000.28i	0.349993	−	0.606206i	−0.636255	−	0.771479i	$-0.719517\pi$
$461$	3464.26	−	6000.28i	0.349993	−	0.606206i	0.986248	+	0.165273i	$0.0528506\pi$
$462$		0			0
$463$	−9738.83	−	16868.1i	−0.977541	−	1.69315i	−0.671280	−	0.741204i	$-0.734255\pi$
$463$	−9738.83	−	16868.1i	−0.977541	−	1.69315i	−0.306261	−	0.951948i	$-0.599078\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	18244.0			1.80777			0.903887	−	0.427772i	$-0.140701\pi$
$467$	18244.0			1.80777			0.903887	+	0.427772i	$0.140701\pi$
$468$		0			0
$469$	915.339			0.0901203
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−810.159	−	1403.24i	−0.0787551	−	0.136408i
$474$		0			0
$475$	629.321	−	1090.02i	0.0607900	−	0.105291i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2243.72	+	3886.24i	−0.214025	+	0.370703i	−0.952971	−	0.303063i	$-0.901991\pi$
$479$	−2243.72	+	3886.24i	−0.214025	+	0.370703i	0.738945	+	0.673765i	$0.235324\pi$
$480$		0			0
$481$	−7341.43	−	12715.7i	−0.695926	−	1.20538i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7441.16			0.696671
$486$		0			0
$487$	−2136.17			−0.198766			−0.0993832	−	0.995049i	$-0.531687\pi$
$487$	−2136.17			−0.198766			−0.0993832	+	0.995049i	$0.531687\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7048.39	−	12208.2i	−0.647840	−	1.12209i	−0.983638	−	0.180158i	$-0.942339\pi$
$491$	−7048.39	−	12208.2i	−0.647840	−	1.12209i	0.335797	−	0.941934i	$-0.390994\pi$
$492$		0			0
$493$	3763.59	−	6518.73i	0.343821	−	0.595515i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−300.956	+	521.270i	−0.0271624	+	0.0470466i
$498$		0			0
$499$	−10083.5	−	17465.2i	−0.904612	−	1.56683i	−0.821437	−	0.570299i	$-0.806827\pi$
$499$	−10083.5	−	17465.2i	−0.904612	−	1.56683i	−0.0831753	−	0.996535i	$-0.526506\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	19912.3			1.76510			0.882551	−	0.470217i	$-0.155825\pi$
$503$	19912.3			1.76510			0.882551	+	0.470217i	$0.155825\pi$
$504$		0			0
$505$	1976.83			0.174193
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−9789.42	−	16955.8i	−0.852472	−	1.47652i	−0.878971	−	0.476876i	$-0.841769\pi$
$509$	−9789.42	−	16955.8i	−0.852472	−	1.47652i	0.0264988	−	0.999649i	$-0.491564\pi$
$510$		0			0
$511$	1360.47	−	2356.40i	0.117776	−	0.203994i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2140.22	+	3706.97i	−0.183125	+	0.317182i
$516$		0			0
$517$	−260.688	−	451.526i	−0.0221761	−	0.0384102i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18389.9			1.54640			0.773200	−	0.634163i	$-0.218655\pi$
$521$	18389.9			1.54640			0.773200	+	0.634163i	$0.218655\pi$
$522$		0			0
$523$	−20546.3			−1.71783			−0.858916	−	0.512116i	$-0.828862\pi$
$523$	−20546.3			−1.71783			−0.858916	+	0.512116i	$0.828862\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−5647.34	−	9781.49i	−0.466797	−	0.808517i
$528$		0			0
$529$	5885.51	−	10194.0i	0.483728	−	0.837841i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−9084.42	+	15734.7i	−0.738255	+	1.27870i
$534$		0			0
$535$	1059.09	+	1834.39i	0.0855856	+	0.148239i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	4870.45			0.389212
$540$		0			0
$541$	4660.74			0.370390			0.185195	−	0.982702i	$-0.440708\pi$
$541$	4660.74			0.370390			0.185195	+	0.982702i	$0.440708\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	2101.83	+	3640.47i	0.165197	+	0.286129i
$546$		0			0
$547$	1458.99	−	2527.04i	0.114044	−	0.197529i	−0.803353	−	0.595503i	$-0.796953\pi$
$547$	1458.99	−	2527.04i	0.114044	−	0.197529i	0.917397	+	0.397973i	$0.130286\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4137.56	−	7166.47i	0.319902	−	0.554087i
$552$		0			0
$553$	1161.29	+	2011.41i	0.0893002	+	0.154672i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	14120.6			1.07417			0.537084	−	0.843529i	$-0.319526\pi$
$557$	14120.6			1.07417			0.537084	+	0.843529i	$0.319526\pi$
$558$		0			0
$559$	−6061.04			−0.458595
$560$		0			0
$561$		0			0
$562$		0			0
$563$	6642.62	+	11505.4i	0.497252	+	0.861266i	0.999995	−	0.00316990i	$-0.00100901\pi$
$563$	6642.62	+	11505.4i	0.497252	+	0.861266i	−0.502743	+	0.864436i	$0.667676\pi$
$564$		0			0
$565$	4265.74	−	7388.49i	0.317630	−	0.550152i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−6058.65	+	10493.9i	−0.446383	+	0.773158i	−0.998147	−	0.0608422i	$-0.980621\pi$
$569$	−6058.65	+	10493.9i	−0.446383	+	0.773158i	0.551765	+	0.834000i	$0.313955\pi$
$570$		0			0
$571$	3443.57	+	5964.44i	0.252380	+	0.437135i	0.964181	−	0.265247i	$-0.0854534\pi$
$571$	3443.57	+	5964.44i	0.252380	+	0.437135i	−0.711801	+	0.702381i	$0.752120\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	497.475			0.0360803
$576$		0			0
$577$	6389.22			0.460982			0.230491	−	0.973074i	$-0.425967\pi$
$577$	6389.22			0.460982			0.230491	+	0.973074i	$0.425967\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	766.376	+	1327.40i	0.0547240	+	0.0947848i
$582$		0			0
$583$	−3528.20	+	6111.03i	−0.250640	+	0.434121i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	12313.1	−	21327.0i	0.865787	−	1.49959i	−0.000476796	−	1.00000i	$-0.500152\pi$
$587$	12313.1	−	21327.0i	0.865787	−	1.49959i	0.866264	−	0.499587i	$-0.166515\pi$
$588$		0			0
$589$	−6208.50	−	10753.4i	−0.434324	−	0.752271i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−18703.3			−1.29520			−0.647598	−	0.761982i	$-0.724226\pi$
$593$	−18703.3			−1.29520			−0.647598	+	0.761982i	$0.724226\pi$
$594$		0			0
$595$	719.068			0.0495444
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1258.93	−	2180.54i	−0.0858741	−	0.148738i	0.819889	−	0.572522i	$-0.194035\pi$
$599$	−1258.93	−	2180.54i	−0.0858741	−	0.148738i	−0.905763	+	0.423784i	$0.860702\pi$
$600$		0			0
$601$	−10579.5	+	18324.2i	−0.718048	+	1.24369i	0.243725	+	0.969844i	$0.421631\pi$
$601$	−10579.5	+	18324.2i	−0.718048	+	1.24369i	−0.961772	+	0.273851i	$0.911703\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2793.14	−	4837.87i	0.187698	−	0.325103i
$606$		0			0
$607$	14378.7	+	24904.7i	0.961474	+	1.66532i	0.718804	+	0.695213i	$0.244690\pi$
$607$	14378.7	+	24904.7i	0.961474	+	1.66532i	0.242670	+	0.970109i	$0.421977\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1950.29			−0.129133
$612$		0			0
$613$	12342.1			0.813205			0.406602	−	0.913605i	$-0.366713\pi$
$613$	12342.1			0.813205			0.406602	+	0.913605i	$0.366713\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−666.521	−	1154.45i	−0.0434897	−	0.0753263i	0.843461	−	0.537190i	$-0.180514\pi$
$617$	−666.521	−	1154.45i	−0.0434897	−	0.0753263i	−0.886951	+	0.461864i	$0.847181\pi$
$618$		0			0
$619$	1037.12	−	1796.34i	0.0673429	−	0.116641i	−0.830388	−	0.557186i	$-0.811881\pi$
$619$	1037.12	−	1796.34i	0.0673429	−	0.116641i	0.897731	+	0.440544i	$0.145215\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−273.114	+	473.047i	−0.0175635	+	0.0304209i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12295.3			−0.779405
$630$		0			0
$631$	9839.77			0.620785			0.310392	−	0.950609i	$-0.399540\pi$
$631$	9839.77			0.620785			0.310392	+	0.950609i	$0.399540\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2687.84	−	4655.48i	−0.167975	−	0.290940i
$636$		0			0
$637$	9109.32	−	15777.8i	0.566600	−	0.981381i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−7909.90	+	13700.3i	−0.487398	+	0.844198i	−0.999895	−	0.0144910i	$-0.995387\pi$
$641$	−7909.90	+	13700.3i	−0.487398	+	0.844198i	0.512497	+	0.858689i	$0.328721\pi$
$642$		0			0
$643$	−7355.81	−	12740.6i	−0.451143	−	0.781402i	0.547315	−	0.836927i	$-0.315650\pi$
$643$	−7355.81	−	12740.6i	−0.451143	−	0.781402i	−0.998457	+	0.0555249i	$0.982317\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	16466.5			1.00056			0.500281	−	0.865863i	$-0.333230\pi$
$647$	16466.5			1.00056			0.500281	+	0.865863i	$0.333230\pi$
$648$		0			0
$649$	−4817.73			−0.291391
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−4076.89	−	7061.38i	−0.244320	−	0.423175i	0.717620	−	0.696435i	$-0.245231\pi$
$653$	−4076.89	−	7061.38i	−0.244320	−	0.423175i	−0.961940	+	0.273260i	$0.911898\pi$
$654$		0			0
$655$	2218.15	−	3841.95i	0.132321	−	0.229187i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−15003.6	+	25987.0i	−0.886886	+	1.53613i	−0.0433494	+	0.999060i	$0.513803\pi$
$659$	−15003.6	+	25987.0i	−0.886886	+	1.53613i	−0.843537	+	0.537072i	$0.819530\pi$
$660$		0			0
$661$	−10689.5	−	18514.8i	−0.629008	−	1.08947i	−0.987751	−	0.156037i	$-0.950128\pi$
$661$	−10689.5	−	18514.8i	−0.629008	−	1.08947i	0.358744	−	0.933436i	$-0.383205\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	790.519			0.0460978
$666$		0			0
$667$	3270.72			0.189869
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1170.91	+	2028.08i	0.0673659	+	0.116681i
$672$		0			0
$673$	−12310.3	+	21322.0i	−0.705091	+	1.22125i	0.261568	+	0.965185i	$0.415760\pi$
$673$	−12310.3	+	21322.0i	−0.705091	+	1.22125i	−0.966659	+	0.256068i	$0.917573\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	13782.0	−	23871.1i	0.782399	−	1.35515i	−0.148142	−	0.988966i	$-0.547329\pi$
$677$	13782.0	−	23871.1i	0.782399	−	1.35515i	0.930541	−	0.366188i	$-0.119337\pi$
$678$		0			0
$679$	2336.79	+	4047.44i	0.132073	+	0.228758i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9068.36			0.508040			0.254020	−	0.967199i	$-0.418247\pi$
$683$	9068.36			0.508040			0.254020	+	0.967199i	$0.418247\pi$
$684$		0			0
$685$	9882.51			0.551228
$686$		0			0
$687$		0			0
$688$		0			0
$689$	13197.8	+	22859.2i	0.729746	+	1.26396i
$690$		0			0
$691$	−11432.8	+	19802.2i	−0.629413	+	1.09018i	0.358256	+	0.933623i	$0.383371\pi$
$691$	−11432.8	+	19802.2i	−0.629413	+	1.09018i	−0.987670	+	0.156553i	$0.949962\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	4533.16	−	7851.66i	0.247414	−	0.428533i
$696$		0			0
$697$	7607.22	+	13176.1i	0.413406	+	0.716040i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2080.10			−0.112075			−0.0560373	−	0.998429i	$-0.517847\pi$
$701$	−2080.10			−0.112075			−0.0560373	+	0.998429i	$0.517847\pi$
$702$		0			0
$703$	−13517.0			−0.725185
$704$		0			0
$705$		0			0
$706$		0			0
$707$	620.796	+	1075.25i	0.0330232	+	0.0571979i
$708$		0			0
$709$	11853.8	−	20531.4i	0.627898	−	1.08755i	−0.360074	−	0.932924i	$-0.617249\pi$
$709$	11853.8	−	20531.4i	0.627898	−	1.08755i	0.987973	−	0.154628i	$-0.0494180\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2453.89	−	4250.27i	0.128891	−	0.223245i
$714$		0			0
$715$	1998.83	+	3462.08i	0.104548	+	0.181083i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−12675.0			−0.657438			−0.328719	−	0.944428i	$-0.606617\pi$
$719$	−12675.0			−0.657438			−0.328719	+	0.944428i	$0.606617\pi$
$720$		0			0
$721$	−2688.43			−0.138866
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−2054.58	−	3558.63i	−0.105248	−	0.182295i
$726$		0			0
$727$	9534.79	−	16514.7i	0.486418	−	0.842500i	−0.513460	−	0.858113i	$-0.671637\pi$
$727$	9534.79	−	16514.7i	0.486418	−	0.842500i	0.999878	+	0.0156129i	$0.00496995\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2537.73	+	4395.48i	−0.128401	+	0.222397i
$732$		0			0
$733$	11079.3	+	19189.9i	0.558285	+	0.966978i	0.997640	+	0.0686641i	$0.0218737\pi$
$733$	11079.3	+	19189.9i	0.558285	+	0.966978i	−0.439355	+	0.898313i	$0.644793\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4261.35			0.212983
$738$		0			0
$739$	−15425.3			−0.767834			−0.383917	−	0.923368i	$-0.625425\pi$
$739$	−15425.3			−0.767834			−0.383917	+	0.923368i	$0.625425\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17721.2	−	30694.0i	−0.875004	−	1.51555i	−0.856759	−	0.515717i	$-0.827525\pi$
$743$	−17721.2	−	30694.0i	−0.875004	−	1.51555i	−0.0182450	−	0.999834i	$-0.505808\pi$
$744$		0			0
$745$	−2525.98	+	4375.13i	−0.124221	+	0.215158i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−665.183	+	1152.13i	−0.0324503	+	0.0562055i
$750$		0			0
$751$	8106.23	+	14040.4i	0.393875	+	0.682212i	0.992957	−	0.118476i	$-0.0378008\pi$
$751$	8106.23	+	14040.4i	0.393875	+	0.682212i	−0.599082	+	0.800688i	$0.704468\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	5772.77			0.278269
$756$		0			0
$757$	20286.0			0.973984			0.486992	−	0.873407i	$-0.338094\pi$
$757$	20286.0			0.973984			0.486992	+	0.873407i	$0.338094\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−17748.5	−	30741.2i	−0.845442	−	1.46435i	−0.885237	−	0.465140i	$-0.846004\pi$
$761$	−17748.5	−	30741.2i	−0.845442	−	1.46435i	0.0397950	−	0.999208i	$-0.487330\pi$
$762$		0			0
$763$	−1320.10	+	2286.48i	−0.0626354	+	0.108488i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9010.72	+	15607.0i	−0.424196	+	0.734729i
$768$		0			0
$769$	−17945.3	−	31082.3i	−0.841516	−	1.45755i	−0.888613	−	0.458658i	$-0.848330\pi$
$769$	−17945.3	−	31082.3i	−0.841516	−	1.45755i	0.0470967	−	0.998890i	$-0.485003\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−22725.0			−1.05739			−0.528694	−	0.848812i	$-0.677318\pi$
$773$	−22725.0			−1.05739			−0.528694	+	0.848812i	$0.677318\pi$
$774$		0			0
$775$	−6165.87			−0.285786
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8363.12	+	14485.3i	0.384647	+	0.666228i
$780$		0			0
$781$	−1401.10	+	2426.77i	−0.0641935	+	0.111186i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2848.39	+	4933.55i	−0.129507	+	0.224313i
$786$		0			0
$787$	16475.1	+	28535.6i	0.746217	+	1.29249i	0.949624	+	0.313391i	$0.101465\pi$
$787$	16475.1	+	28535.6i	0.746217	+	1.29249i	−0.203407	+	0.979094i	$0.565202\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	5358.39			0.240863
$792$		0			0
$793$	8759.93			0.392275
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9869.56	−	17094.6i	−0.438642	−	0.759751i	0.558943	−	0.829206i	$-0.311207\pi$
$797$	−9869.56	−	17094.6i	−0.438642	−	0.759751i	−0.997585	+	0.0694555i	$0.977874\pi$
$798$		0			0
$799$	−816.576	+	1414.35i	−0.0361557	+	0.0626235i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6333.64	−	10970.2i	0.278343	−	0.482104i
$804$		0			0
$805$	156.225	+	270.590i	0.00684002	+	0.0118473i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−32852.8			−1.42774			−0.713870	−	0.700278i	$-0.753059\pi$
$809$	−32852.8			−1.42774			−0.713870	+	0.700278i	$0.753059\pi$
$810$		0			0
$811$	20122.0			0.871245			0.435623	−	0.900129i	$-0.356528\pi$
$811$	20122.0			0.871245			0.435623	+	0.900129i	$0.356528\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	8302.18	+	14379.8i	0.356825	+	0.618040i
$816$		0			0
$817$	−2789.89	+	4832.24i	−0.119469	+	0.206926i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−12567.8	+	21768.0i	−0.534249	+	0.925347i	0.464950	+	0.885337i	$0.346072\pi$
$821$	−12567.8	+	21768.0i	−0.534249	+	0.925347i	−0.999199	+	0.0400102i	$0.987261\pi$
$822$		0			0
$823$	11558.9	+	20020.6i	0.489572	+	0.847963i	0.999928	−	0.0119999i	$-0.00381978\pi$
$823$	11558.9	+	20020.6i	0.489572	+	0.847963i	−0.510356	+	0.859963i	$0.670486\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6362.04			−0.267509			−0.133754	−	0.991015i	$-0.542703\pi$
$827$	−6362.04			−0.267509			−0.133754	+	0.991015i	$0.542703\pi$
$828$		0			0
$829$	−21021.5			−0.880706			−0.440353	−	0.897825i	$-0.645147\pi$
$829$	−21021.5			−0.880706			−0.440353	+	0.897825i	$0.645147\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−7628.07	−	13212.2i	−0.317283	−	0.549550i
$834$		0			0
$835$	−5383.16	+	9323.90i	−0.223104	+	0.386427i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−23031.0	+	39890.9i	−0.947698	+	1.64146i	−0.197441	+	0.980315i	$0.563263\pi$
$839$	−23031.0	+	39890.9i	−0.947698	+	1.64146i	−0.750257	+	0.661146i	$0.770070\pi$
$840$		0			0
$841$	−1313.60	−	2275.23i	−0.0538605	−	0.0932891i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	3968.87			0.161578
$846$		0			0
$847$	3508.60			0.142334
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−2671.29	−	4626.81i	−0.107604	−	0.186375i
$852$		0			0
$853$	−8792.39	+	15228.9i	−0.352926	+	0.611285i	−0.986761	−	0.162183i	$-0.948147\pi$
$853$	−8792.39	+	15228.9i	−0.352926	+	0.611285i	0.633835	+	0.773468i	$0.281480\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5156.94	+	8932.09i	−0.205552	+	0.356026i	−0.950308	−	0.311310i	$-0.899232\pi$
$857$	−5156.94	+	8932.09i	−0.205552	+	0.356026i	0.744757	+	0.667336i	$0.232566\pi$
$858$		0			0
$859$	15293.9	+	26489.9i	0.607476	+	1.05218i	0.991655	+	0.128921i	$0.0411514\pi$
$859$	15293.9	+	26489.9i	0.607476	+	1.05218i	−0.384178	+	0.923259i	$0.625515\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	3387.21			0.133606			0.0668029	−	0.997766i	$-0.478720\pi$
$863$	3387.21			0.133606			0.0668029	+	0.997766i	$0.478720\pi$
$864$		0			0
$865$	18982.1			0.746141
$866$		0			0
$867$		0			0
$868$		0			0
$869$	5406.36	+	9364.10i	0.211045	+	0.365541i
$870$		0			0
$871$	7970.10	−	13804.6i	0.310053	−	0.537028i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	196.273	−	339.954i	0.00758312	−	0.0131343i
$876$		0			0
$877$	9232.83	+	15991.7i	0.355497	+	0.615738i	0.987203	−	0.159470i	$-0.0509784\pi$
$877$	9232.83	+	15991.7i	0.355497	+	0.615738i	−0.631706	+	0.775208i	$0.717645\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−20662.0			−0.790148			−0.395074	−	0.918649i	$-0.629281\pi$
$881$	−20662.0			−0.790148			−0.395074	+	0.918649i	$0.629281\pi$
$882$		0			0
$883$	386.519			0.0147309			0.00736545	−	0.999973i	$-0.497655\pi$
$883$	386.519			0.0147309			0.00736545	+	0.999973i	$0.497655\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	6678.06	+	11566.7i	0.252793	+	0.437850i	0.964294	−	0.264835i	$-0.0853175\pi$
$887$	6678.06	+	11566.7i	0.252793	+	0.437850i	−0.711501	+	0.702685i	$0.751984\pi$
$888$		0			0
$889$	1688.16	−	2923.98i	0.0636885	−	0.110312i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−897.717	+	1554.89i	−0.0336405	+	0.0582670i
$894$		0			0
$895$	4408.36	+	7635.51i	0.164643	+	0.285170i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−40538.4			−1.50393
$900$		0			0
$901$	22103.4			0.817281
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−487.437	−	844.266i	−0.0179038	−	0.0310103i
$906$		0			0
$907$	−14680.6	+	25427.6i	−0.537445	+	0.930882i	0.461596	+	0.887090i	$0.347277\pi$
$907$	−14680.6	+	25427.6i	−0.537445	+	0.930882i	−0.999041	+	0.0437913i	$0.986056\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−3836.01	+	6644.16i	−0.139509	+	0.241636i	−0.927311	−	0.374292i	$-0.877886\pi$
$911$	−3836.01	+	6644.16i	−0.139509	+	0.241636i	0.787802	+	0.615929i	$0.211219\pi$
$912$		0			0
$913$	3567.86	+	6179.71i	0.129331	+	0.224007i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	2786.32			0.100341
$918$		0			0
$919$	−25818.6			−0.926742			−0.463371	−	0.886164i	$-0.653360\pi$
$919$	−25818.6			−0.926742			−0.463371	+	0.886164i	$0.653360\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	5241.00	+	9077.68i	0.186901	+	0.323722i
$924$		0			0
$925$	−3356.06	+	5812.86i	−0.119293	+	0.206622i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13689.5	+	23710.9i	−0.483463	+	0.837383i	−0.999820	−	0.0189908i	$-0.993955\pi$
$929$	−13689.5	+	23710.9i	−0.483463	+	0.837383i	0.516356	+	0.856374i	$0.327288\pi$
$930$		0			0
$931$	−8386.04	−	14525.0i	−0.295211	−	0.511320i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3347.61			0.117089
$936$		0			0
$937$	−4696.47			−0.163743			−0.0818713	−	0.996643i	$-0.526090\pi$
$937$	−4696.47			−0.163743			−0.0818713	+	0.996643i	$0.526090\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1419.08	+	2457.92i	0.0491612	+	0.0851497i	0.889559	−	0.456821i	$-0.151012\pi$
$941$	1419.08	+	2457.92i	0.0491612	+	0.0851497i	−0.840398	+	0.541970i	$0.817679\pi$
$942$		0			0
$943$	−3305.50	+	5725.30i	−0.114148	+	0.197711i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	23449.9	−	40616.5i	0.804668	−	1.39373i	−0.111847	−	0.993725i	$-0.535677\pi$
$947$	23449.9	−	40616.5i	0.804668	−	1.39373i	0.916515	−	0.400000i	$-0.130990\pi$
$948$		0			0
$949$	−23691.9	−	41035.6i	−0.810403	−	1.40366i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−32713.6			−1.11196			−0.555980	−	0.831196i	$-0.687657\pi$
$953$	−32713.6			−1.11196			−0.555980	+	0.831196i	$0.687657\pi$
$954$		0			0
$955$	17186.2			0.582338
$956$		0			0
$957$		0			0
$958$		0			0
$959$	3103.47	+	5375.36i	0.104501	+	0.181001i
$960$		0			0
$961$	−15518.8	+	26879.4i	−0.520923	+	0.902266i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	6363.75	−	11022.3i	0.212286	−	0.367691i
$966$		0			0
$967$	12065.4	+	20897.8i	0.401236	+	0.694962i	0.993875	−	0.110506i	$-0.0352473\pi$
$967$	12065.4	+	20897.8i	0.401236	+	0.694962i	−0.592639	+	0.805468i	$0.701914\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	21198.9			0.700623			0.350312	−	0.936633i	$-0.386076\pi$
$971$	21198.9			0.700623			0.350312	+	0.936633i	$0.386076\pi$
$972$		0			0
$973$	5694.31			0.187617
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−2070.13	−	3585.57i	−0.0677885	−	0.117413i	0.830139	−	0.557556i	$-0.188261\pi$
$977$	−2070.13	−	3585.57i	−0.0677885	−	0.117413i	−0.897928	+	0.440143i	$0.854928\pi$
$978$		0			0
$979$	−1271.48	+	2202.26i	−0.0415082	+	0.0718944i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11969.4	−	20731.6i	0.388367	−	0.672672i	−0.603863	−	0.797088i	$-0.706373\pi$
$983$	11969.4	−	20731.6i	0.388367	−	0.672672i	0.992230	+	0.124417i	$0.0397059\pi$
$984$		0			0
$985$	−9431.62	−	16336.1i	−0.305093	−	0.528436i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−2205.40			−0.0709075
$990$		0			0
$991$	−40688.9			−1.30426			−0.652131	−	0.758106i	$-0.726125\pi$
$991$	−40688.9			−1.30426			−0.652131	+	0.758106i	$0.726125\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	2528.77	+	4379.95i	0.0805702	+	0.139552i
$996$		0			0
$997$	1689.21	−	2925.80i	0.0536589	−	0.0929400i	−0.837948	−	0.545750i	$-0.816245\pi$
$997$	1689.21	−	2925.80i	0.0536589	−	0.0929400i	0.891607	+	0.452810i	$0.149578\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.4.q.e.721.6		20
3.2	odd	2		360.4.q.e.241.1	yes	20
9.4	even	3	inner	1080.4.q.e.361.6		20
9.5	odd	6		360.4.q.e.121.1	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.4.q.e.121.1	✓	20	9.5	odd	6
360.4.q.e.241.1	yes	20	3.2	odd	2
1080.4.q.e.361.6		20	9.4	even	3	inner
1080.4.q.e.721.6		20	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+(-2.50000 - 4.33013i) q^{5} +(1.57018 - 2.71963i) q^{7} +(7.30996 - 12.6612i) q^{11} +(-27.3440 - 47.3612i) q^{13} -45.7952 q^{17} -50.3457 q^{19} +(-9.94951 - 17.2331i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(-82.1830 + 142.345i) q^{29} +(123.317 + 213.592i) q^{31} -15.7018 q^{35} +268.484 q^{37} +(-166.114 - 287.718i) q^{41} +(55.4147 - 95.9811i) q^{43} +(17.8310 - 30.8843i) q^{47} +(166.569 + 288.506i) q^{49} -482.657 q^{53} -73.0996 q^{55} +(-164.766 - 285.383i) q^{59} +(-80.0901 + 138.720i) q^{61} +(-136.720 + 236.806i) q^{65} +(145.738 + 252.425i) q^{67} -191.669 q^{71} +866.440 q^{73} +(-22.9559 - 39.7608i) q^{77} +(-369.794 + 640.503i) q^{79} +(-244.041 + 422.691i) q^{83} +(114.488 + 198.299i) q^{85} -173.938 q^{89} -171.740 q^{91} +(125.864 + 218.003i) q^{95} +(-744.116 + 1288.85i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 50 q^{5} + 22 q^{7} - 50 q^{11} - 16 q^{13} + 68 q^{17} + 212 q^{19} - 50 q^{23} - 250 q^{25} + 64 q^{29} - 22 q^{31} - 220 q^{35} + 600 q^{37} + 198 q^{41} - 382 q^{43} - 128 q^{47} - 1230 q^{49}+ \cdots - 1354 q^{97}+O(q^{100}) \) 20 * q - 50 * q^5 + 22 * q^7 - 50 * q^11 - 16 * q^13 + 68 * q^17 + 212 * q^19 - 50 * q^23 - 250 * q^25 + 64 * q^29 - 22 * q^31 - 220 * q^35 + 600 * q^37 + 198 * q^41 - 382 * q^43 - 128 * q^47 - 1230 * q^49 + 1128 * q^53 + 500 * q^55 + 200 * q^59 - 232 * q^61 - 80 * q^65 - 1316 * q^67 + 228 * q^71 + 828 * q^73 - 616 * q^77 - 300 * q^79 + 1082 * q^83 - 170 * q^85 + 768 * q^89 + 1972 * q^91 - 530 * q^95 - 1354 * q^97

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