LMFDB - Embedded newform 1404.2.j.a.289.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1404,2,Mod(289,1404)] 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("1404.289"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1404, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1404 = 2^{2} \cdot 3^{3} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1404.j (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$11.2109964438$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(\zeta_{3})$
Twist minimal:	no (minimal twist has level 468)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.1
Character	$\chi$	$=$	1404.289
Dual form			1404.2.j.a.685.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.03195 - 3.51944i) q^{5} +(0.616999 + 1.06867i) q^{7} +6.31719 q^{11} +(-3.21746 - 1.62726i) q^{13} +(2.45192 - 4.24685i) q^{17} +(0.346591 - 0.600313i) q^{19} +(0.706995 - 1.22455i) q^{23} +(-5.75764 + 9.97252i) q^{25} -6.70300 q^{29} +(1.88555 + 3.26586i) q^{31} +(2.50742 - 4.34298i) q^{35} +(-1.48705 - 2.57565i) q^{37} +(3.56129 - 6.16833i) q^{41} +(0.566773 + 0.981679i) q^{43} +(3.76496 - 6.52111i) q^{47} +(2.73862 - 4.74344i) q^{49} -13.1961 q^{53} +(-12.8362 - 22.2330i) q^{55} -10.4480 q^{59} +(-2.88154 - 4.99097i) q^{61} +(0.810654 + 14.6302i) q^{65} +(-3.70443 + 6.41626i) q^{67} +(1.36182 - 2.35874i) q^{71} -1.45266 q^{73} +(3.89770 + 6.75102i) q^{77} +(3.73863 - 6.47550i) q^{79} +(-4.02165 + 6.96570i) q^{83} -19.9287 q^{85} +(-6.75612 - 11.7020i) q^{89} +(-0.246154 - 4.44243i) q^{91} -2.81702 q^{95} +(-1.53197 - 2.65344i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 2 q^{7} - 8 q^{11} + q^{13} + 8 q^{17} - q^{19} + 4 q^{23} - 14 q^{25} - 26 q^{29} + 2 q^{31} - 3 q^{35} - q^{37} - 4 q^{41} + 2 q^{43} - 11 q^{47} - 12 q^{49} - 52 q^{53} - 16 q^{59} - 7 q^{61}+ \cdots - 25 q^{97}+O(q^{100}) $ 28 * q + 2 * q^7 - 8 * q^11 + q^13 + 8 * q^17 - q^19 + 4 * q^23 - 14 * q^25 - 26 * q^29 + 2 * q^31 - 3 * q^35 - q^37 - 4 * q^41 + 2 * q^43 - 11 * q^47 - 12 * q^49 - 52 * q^53 - 16 * q^59 - 7 * q^61 + 31 * q^65 - 7 * q^67 + 12 * q^71 + 14 * q^73 + 28 * q^77 + 5 * q^79 - 9 * q^83 - 36 * q^85 + 11 * q^89 - q^91 + 56 * q^95 - 25 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1404\mathbb{Z}\right)^\times$.

$n$	$677$	$703$	$1081$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−2.03195	−	3.51944i	−0.908716	−	1.57394i	−0.815851	−	0.578263i	$-0.803731\pi$
$5$	−2.03195	−	3.51944i	−0.908716	−	1.57394i	−0.0928649	−	0.995679i	$-0.529602\pi$
$6$		0			0
$7$	0.616999	+	1.06867i	0.233204	+	0.403921i	0.958749	−	0.284253i	$-0.0917457\pi$
$7$	0.616999	+	1.06867i	0.233204	+	0.403921i	−0.725545	+	0.688174i	$0.758412\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	6.31719			1.90471			0.952353	−	0.304999i	$-0.0986563\pi$
$11$	6.31719			1.90471			0.952353	+	0.304999i	$0.0986563\pi$
$12$		0			0
$13$	−3.21746	−	1.62726i	−0.892361	−	0.451321i
$13$	−3.21746	−	1.62726i	−0.892361	−	0.451321i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.45192	−	4.24685i	0.594678	−	1.03001i	−0.398914	−	0.916988i	$-0.630613\pi$
$17$	2.45192	−	4.24685i	0.594678	−	1.03001i	0.993592	−	0.113025i	$-0.0360540\pi$
$18$		0			0
$19$	0.346591	−	0.600313i	0.0795134	−	0.137721i	−0.823527	−	0.567277i	$-0.807997\pi$
$19$	0.346591	−	0.600313i	0.0795134	−	0.137721i	0.903040	+	0.429556i	$0.141330\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.706995	−	1.22455i	0.147419	−	0.255336i	−0.782854	−	0.622205i	$-0.786237\pi$
$23$	0.706995	−	1.22455i	0.147419	−	0.255336i	0.930273	+	0.366869i	$0.119570\pi$
$24$		0			0
$25$	−5.75764	+	9.97252i	−1.15153	+	1.99450i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.70300			−1.24472			−0.622358	−	0.782733i	$-0.713825\pi$
$29$	−6.70300			−1.24472			−0.622358	+	0.782733i	$0.713825\pi$
$30$		0			0
$31$	1.88555	+	3.26586i	0.338654	+	0.586566i	0.984180	−	0.177173i	$-0.0566951\pi$
$31$	1.88555	+	3.26586i	0.338654	+	0.586566i	−0.645526	+	0.763738i	$0.723362\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.50742	−	4.34298i	0.423832	−	0.734098i
$36$		0			0
$37$	−1.48705	−	2.57565i	−0.244470	−	0.423434i	0.717513	−	0.696545i	$-0.245280\pi$
$37$	−1.48705	−	2.57565i	−0.244470	−	0.423434i	−0.961982	+	0.273112i	$0.911947\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.56129	−	6.16833i	0.556179	−	0.963331i	−0.441631	−	0.897197i	$-0.645600\pi$
$41$	3.56129	−	6.16833i	0.556179	−	0.963331i	0.997811	−	0.0661343i	$-0.0210666\pi$
$42$		0			0
$43$	0.566773	+	0.981679i	0.0864320	+	0.149705i	0.906001	−	0.423277i	$-0.139120\pi$
$43$	0.566773	+	0.981679i	0.0864320	+	0.149705i	−0.819569	+	0.572981i	$0.805787\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.76496	−	6.52111i	0.549176	−	0.951201i	−0.449155	−	0.893454i	$-0.648275\pi$
$47$	3.76496	−	6.52111i	0.549176	−	0.951201i	0.998331	−	0.0577473i	$-0.0183918\pi$
$48$		0			0
$49$	2.73862	−	4.74344i	0.391232	−	0.677634i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13.1961			−1.81263			−0.906315	−	0.422603i	$-0.861116\pi$
$53$	−13.1961			−1.81263			−0.906315	+	0.422603i	$0.861116\pi$
$54$		0			0
$55$	−12.8362	−	22.2330i	−1.73084	−	2.99789i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−10.4480			−1.36021			−0.680105	−	0.733115i	$-0.738066\pi$
$59$	−10.4480			−1.36021			−0.680105	+	0.733115i	$0.738066\pi$
$60$		0			0
$61$	−2.88154	−	4.99097i	−0.368943	−	0.639028i	0.620458	−	0.784240i	$-0.286947\pi$
$61$	−2.88154	−	4.99097i	−0.368943	−	0.639028i	−0.989401	+	0.145212i	$0.953614\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.810654	+	14.6302i	0.100549	+	1.81465i
$66$		0			0
$67$	−3.70443	+	6.41626i	−0.452568	+	0.783871i	−0.998545	−	0.0539295i	$-0.982825\pi$
$67$	−3.70443	+	6.41626i	−0.452568	+	0.783871i	0.545977	+	0.837800i	$0.316159\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	1.36182	−	2.35874i	0.161618	−	0.279930i	−0.773831	−	0.633392i	$-0.781662\pi$
$71$	1.36182	−	2.35874i	0.161618	−	0.279930i	0.935449	+	0.353461i	$0.114995\pi$
$72$		0			0
$73$	−1.45266			−0.170021			−0.0850104	−	0.996380i	$-0.527092\pi$
$73$	−1.45266			−0.170021			−0.0850104	+	0.996380i	$0.527092\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	3.89770	+	6.75102i	0.444185	+	0.769350i
$78$		0			0
$79$	3.73863	−	6.47550i	0.420629	−	0.728550i	−0.575372	−	0.817892i	$-0.695143\pi$
$79$	3.73863	−	6.47550i	0.420629	−	0.728550i	0.996001	+	0.0893413i	$0.0284762\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.02165	+	6.96570i	−0.441433	+	0.764585i	−0.997796	−	0.0663546i	$-0.978863\pi$
$83$	−4.02165	+	6.96570i	−0.441433	+	0.764585i	0.556363	+	0.830939i	$0.312196\pi$
$84$		0			0
$85$	−19.9287			−2.16157
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.75612	−	11.7020i	−0.716148	−	1.24040i	−0.962515	−	0.271228i	$-0.912570\pi$
$89$	−6.75612	−	11.7020i	−0.716148	−	1.24040i	0.246367	−	0.969177i	$-0.420763\pi$
$90$		0			0
$91$	−0.246154	−	4.44243i	−0.0258040	−	0.465693i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.81702			−0.289020
$96$		0			0
$97$	−1.53197	−	2.65344i	−0.155547	−	0.269416i	0.777711	−	0.628622i	$-0.216381\pi$
$97$	−1.53197	−	2.65344i	−0.155547	−	0.269416i	−0.933258	+	0.359206i	$0.883047\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	6.11594			0.608558			0.304279	−	0.952583i	$-0.401584\pi$
$101$	6.11594			0.608558			0.304279	+	0.952583i	$0.401584\pi$
$102$		0			0
$103$	−2.17185	−	3.76176i	−0.213999	−	0.370657i	0.738964	−	0.673745i	$-0.235316\pi$
$103$	−2.17185	−	3.76176i	−0.213999	−	0.370657i	−0.952962	+	0.303089i	$0.901982\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.96304	−	5.13214i	−0.286448	−	0.496143i	0.686511	−	0.727119i	$-0.259141\pi$
$107$	−2.96304	−	5.13214i	−0.286448	−	0.496143i	−0.972959	+	0.230976i	$0.925808\pi$
$108$		0			0
$109$	−0.553911			−0.0530550			−0.0265275	−	0.999648i	$-0.508445\pi$
$109$	−0.553911			−0.0530550			−0.0265275	+	0.999648i	$0.508445\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	6.10479			0.574290			0.287145	−	0.957887i	$-0.407294\pi$
$113$	6.10479			0.574290			0.287145	+	0.957887i	$0.407294\pi$
$114$		0			0
$115$	−5.74631			−0.535846
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.05134			0.554725
$120$		0			0
$121$	28.9069			2.62790
$122$		0			0
$123$		0			0
$124$		0			0
$125$	26.4774			2.36821
$126$		0			0
$127$	4.40433	+	7.62852i	0.390821	+	0.676922i	0.992558	−	0.121773i	$-0.0388579\pi$
$127$	4.40433	+	7.62852i	0.390821	+	0.676922i	−0.601737	+	0.798694i	$0.705525\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.71507	−	9.89879i	−0.499328	−	0.864861i	0.500672	−	0.865637i	$-0.333086\pi$
$131$	−5.71507	−	9.89879i	−0.499328	−	0.864861i	−1.00000		0.000776096i	$0.999753\pi$
$132$		0			0
$133$	0.855385			0.0741713
$134$		0			0
$135$		0			0
$136$		0			0
$137$	5.85199	+	10.1359i	0.499969	+	0.865972i	1.00000		3.54390e-5i	$-1.12806e-5\pi$
$137$	5.85199	+	10.1359i	0.499969	+	0.865972i	−0.500031	+	0.866008i	$0.666678\pi$
$138$		0			0
$139$	6.99465			0.593279			0.296639	−	0.954990i	$-0.404134\pi$
$139$	6.99465			0.593279			0.296639	+	0.954990i	$0.404134\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−20.3253	−	10.2797i	−1.69969	−	0.859634i
$144$		0			0
$145$	13.6202	+	23.5908i	1.13109	+	1.95911i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−9.10505			−0.745915			−0.372957	−	0.927848i	$-0.621656\pi$
$149$	−9.10505			−0.745915			−0.372957	+	0.927848i	$0.621656\pi$
$150$		0			0
$151$	0.185604	−	0.321476i	0.0151043	−	0.0261614i	−0.858374	−	0.513024i	$-0.828525\pi$
$151$	0.185604	−	0.321476i	0.0151043	−	0.0261614i	0.873479	+	0.486862i	$0.161859\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.66267	−	13.2721i	0.615480	−	1.06604i
$156$		0			0
$157$	−0.686457	−	1.18898i	−0.0547852	−	0.0948908i	0.837332	−	0.546694i	$-0.184114\pi$
$157$	−0.686457	−	1.18898i	−0.0547852	−	0.0948908i	−0.892117	+	0.451804i	$0.850781\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.74486			0.137514
$162$		0			0
$163$	−8.53586	+	14.7845i	−0.668580	+	1.15802i	0.309721	+	0.950828i	$0.399764\pi$
$163$	−8.53586	+	14.7845i	−0.668580	+	1.15802i	−0.978301	+	0.207188i	$0.933569\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.57475	−	2.72755i	0.121858	−	0.211064i	−0.798643	−	0.601806i	$-0.794448\pi$
$167$	1.57475	−	2.72755i	0.121858	−	0.211064i	0.920500	+	0.390742i	$0.127781\pi$
$168$		0			0
$169$	7.70403	+	10.4713i	0.592618	+	0.805484i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.40132	+	7.62332i	0.334626	+	0.579590i	0.983413	−	0.181380i	$-0.0580565\pi$
$173$	4.40132	+	7.62332i	0.334626	+	0.579590i	−0.648787	+	0.760970i	$0.724723\pi$
$174$		0			0
$175$	−14.2098			−1.07416
$176$		0			0
$177$		0			0
$178$		0			0
$179$	10.8390	+	18.7736i	0.810142	+	1.40321i	0.912764	+	0.408487i	$0.133944\pi$
$179$	10.8390	+	18.7736i	0.810142	+	1.40321i	−0.102622	+	0.994720i	$0.532723\pi$
$180$		0			0
$181$	17.3637			1.29063			0.645315	−	0.763916i	$-0.276726\pi$
$181$	17.3637			1.29063			0.645315	+	0.763916i	$0.276726\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.04322	+	10.4672i	−0.444307	+	0.769562i
$186$		0			0
$187$	15.4893	−	26.8282i	1.13269	−	1.96187i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.11975	+	1.93946i	0.0810221	+	0.140334i	0.903689	−	0.428189i	$-0.140848\pi$
$191$	1.11975	+	1.93946i	0.0810221	+	0.140334i	−0.822667	+	0.568523i	$0.807515\pi$
$192$		0			0
$193$	9.45140	−	16.3703i	0.680326	−	1.17836i	−0.294555	−	0.955635i	$-0.595171\pi$
$193$	9.45140	−	16.3703i	0.680326	−	1.17836i	0.974881	−	0.222725i	$-0.0714953\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−4.09745	−	7.09699i	−0.291931	−	0.505640i	0.682335	−	0.731039i	$-0.260964\pi$
$197$	−4.09745	−	7.09699i	−0.291931	−	0.505640i	−0.974266	+	0.225400i	$0.927631\pi$
$198$		0			0
$199$	−0.765637	+	1.32612i	−0.0542745	+	0.0940063i	−0.891886	−	0.452260i	$-0.850618\pi$
$199$	−0.765637	+	1.32612i	−0.0542745	+	0.0940063i	0.837612	+	0.546266i	$0.183951\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.13575	−	7.16333i	−0.290273	−	0.502767i
$204$		0			0
$205$	−28.9454			−2.02164
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.18948	−	3.79229i	0.151449	−	0.262318i
$210$		0			0
$211$	9.19658	−	15.9289i	0.633119	−	1.09659i	−0.353792	−	0.935324i	$-0.615108\pi$
$211$	9.19658	−	15.9289i	0.633119	−	1.09659i	0.986910	−	0.161269i	$-0.0515588\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.30331	−	3.98944i	0.157084	−	0.272078i
$216$		0			0
$217$	−2.32676	+	4.03007i	−0.157951	+	0.273579i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−14.7997	+	9.67414i	−0.995535	+	0.650753i
$222$		0			0
$223$	−21.1949			−1.41932			−0.709658	−	0.704546i	$-0.751151\pi$
$223$	−21.1949			−1.41932			−0.709658	+	0.704546i	$0.751151\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.51397	+	16.4787i	0.631464	+	1.09373i	0.987253	+	0.159162i	$0.0508791\pi$
$227$	9.51397	+	16.4787i	0.631464	+	1.09373i	−0.355788	+	0.934567i	$0.615788\pi$
$228$		0			0
$229$	7.57460	+	13.1196i	0.500544	+	0.866967i	1.00000		0.000627856i	$0.000199853\pi$
$229$	7.57460	+	13.1196i	0.500544	+	0.866967i	−0.499456	+	0.866339i	$0.666467\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.1509			1.32013			0.660064	−	0.751209i	$-0.270529\pi$
$233$	20.1509			1.32013			0.660064	+	0.751209i	$0.270529\pi$
$234$		0			0
$235$	−30.6009			−1.99618
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.15357	+	1.99805i	0.0746186	+	0.129243i	0.900920	−	0.433985i	$-0.142893\pi$
$239$	1.15357	+	1.99805i	0.0746186	+	0.129243i	−0.826302	+	0.563228i	$0.809559\pi$
$240$		0			0
$241$	5.87814	+	10.1812i	0.378644	+	0.655831i	0.990865	−	0.134856i	$-0.0430571\pi$
$241$	5.87814	+	10.1812i	0.378644	+	0.655831i	−0.612221	+	0.790687i	$0.709724\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−22.2590			−1.42207
$246$		0			0
$247$	−2.09201	+	1.36749i	−0.133111	+	0.0870110i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	9.40736	−	16.2940i	0.593787	−	1.02847i	−0.399929	−	0.916546i	$-0.630965\pi$
$251$	9.40736	−	16.2940i	0.593787	−	1.02847i	0.993717	−	0.111924i	$-0.0357013\pi$
$252$		0			0
$253$	4.46622	−	7.73572i	0.280789	−	0.486341i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.79897	+	3.11591i	−0.112217	+	0.194365i	−0.916664	−	0.399659i	$-0.869128\pi$
$257$	−1.79897	+	3.11591i	−0.112217	+	0.194365i	0.804447	+	0.594025i	$0.202462\pi$
$258$		0			0
$259$	1.83502	−	3.17834i	0.114022	−	0.197493i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.36349			−0.392390			−0.196195	−	0.980565i	$-0.562859\pi$
$263$	−6.36349			−0.392390			−0.196195	+	0.980565i	$0.562859\pi$
$264$		0			0
$265$	26.8139	+	46.4430i	1.64716	+	2.85297i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.24582	−	7.35398i	0.258872	−	0.448380i	−0.707068	−	0.707146i	$-0.749982\pi$
$269$	4.24582	−	7.35398i	0.258872	−	0.448380i	0.965940	+	0.258766i	$0.0833157\pi$
$270$		0			0
$271$	15.8863	+	27.5159i	0.965023	+	1.67147i	0.709553	+	0.704652i	$0.248897\pi$
$271$	15.8863	+	27.5159i	0.965023	+	1.67147i	0.255470	+	0.966817i	$0.417770\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−36.3721	+	62.9983i	−2.19332	+	3.79894i
$276$		0			0
$277$	−7.54361	−	13.0659i	−0.453252	−	0.785055i	0.545334	−	0.838219i	$-0.316403\pi$
$277$	−7.54361	−	13.0659i	−0.453252	−	0.785055i	−0.998586	+	0.0531638i	$0.983069\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.06790	−	15.7061i	0.540945	−	0.936945i	−0.457905	−	0.889001i	$-0.651400\pi$
$281$	9.06790	−	15.7061i	0.540945	−	0.936945i	0.998850	−	0.0479435i	$-0.0152668\pi$
$282$		0			0
$283$	4.76421	−	8.25186i	0.283203	−	0.490522i	−0.688969	−	0.724791i	$-0.741936\pi$
$283$	4.76421	−	8.25186i	0.283203	−	0.490522i	0.972172	+	0.234269i	$0.0752696\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.78924			0.518813
$288$		0			0
$289$	−3.52384	−	6.10348i	−0.207285	−	0.359028i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.308197			0.0180050			0.00900252	−	0.999959i	$-0.497134\pi$
$293$	0.308197			0.0180050			0.00900252	+	0.999959i	$0.497134\pi$
$294$		0			0
$295$	21.2298	+	36.7710i	1.23604	+	2.14089i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−4.26739	+	2.78947i	−0.246789	+	0.161319i
$300$		0			0
$301$	−0.699397	+	1.21139i	−0.0403125	+	0.0698234i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−11.7103	+	20.2828i	−0.670528	+	1.16139i
$306$		0			0
$307$	−30.6770			−1.75083			−0.875415	−	0.483372i	$-0.839412\pi$
$307$	−30.6770			−1.75083			−0.875415	+	0.483372i	$0.839412\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−11.6222	−	20.1302i	−0.659033	−	1.14148i	−0.980866	−	0.194682i	$-0.937632\pi$
$311$	−11.6222	−	20.1302i	−0.659033	−	1.14148i	0.321833	−	0.946796i	$-0.395701\pi$
$312$		0			0
$313$	3.47080	−	6.01160i	0.196181	−	0.339796i	−0.751106	−	0.660182i	$-0.770479\pi$
$313$	3.47080	−	6.01160i	0.196181	−	0.339796i	0.947287	+	0.320386i	$0.103813\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.08590	+	10.5411i	−0.341818	+	0.592046i	−0.984770	−	0.173860i	$-0.944376\pi$
$317$	−6.08590	+	10.5411i	−0.341818	+	0.592046i	0.642952	+	0.765906i	$0.277709\pi$
$318$		0			0
$319$	−42.3442			−2.37082
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−1.69963	−	2.94384i	−0.0945698	−	0.163800i
$324$		0			0
$325$	34.7529	−	22.7170i	1.92774	−	1.26011i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.29192			0.512280
$330$		0			0
$331$	9.32850	+	16.1574i	0.512741	+	0.888093i	0.999891	+	0.0147750i	$0.00470321\pi$
$331$	9.32850	+	16.1574i	0.512741	+	0.888093i	−0.487150	+	0.873318i	$0.661963\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	30.1089			1.64502
$336$		0			0
$337$	−10.0206	−	17.3562i	−0.545858	−	0.945454i	−0.998552	−	0.0537879i	$-0.982871\pi$
$337$	−10.0206	−	17.3562i	−0.545858	−	0.945454i	0.452694	−	0.891666i	$-0.350463\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	11.9114	+	20.6311i	0.645036	+	1.11723i
$342$		0			0
$343$	15.3969			0.831355
$344$		0			0
$345$		0			0
$346$		0			0
$347$	20.2457			1.08685			0.543424	−	0.839458i	$-0.317127\pi$
$347$	20.2457			1.08685			0.543424	+	0.839458i	$0.317127\pi$
$348$		0			0
$349$	35.8018			1.91642			0.958212	−	0.286058i	$-0.0923452\pi$
$349$	35.8018			1.91642			0.958212	+	0.286058i	$0.0923452\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−26.5299			−1.41204			−0.706022	−	0.708189i	$-0.749512\pi$
$353$	−26.5299			−1.41204			−0.706022	+	0.708189i	$0.749512\pi$
$354$		0			0
$355$	−11.0686			−0.587459
$356$		0			0
$357$		0			0
$358$		0			0
$359$	5.94247			0.313632			0.156816	−	0.987628i	$-0.449877\pi$
$359$	5.94247			0.313632			0.156816	+	0.987628i	$0.449877\pi$
$360$		0			0
$361$	9.25975	+	16.0384i	0.487355	+	0.844124i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.95173	+	5.11255i	0.154501	+	0.267603i
$366$		0			0
$367$	20.0052			1.04426			0.522130	−	0.852866i	$-0.325137\pi$
$367$	20.0052			1.04426			0.522130	+	0.852866i	$0.325137\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−8.14201	−	14.1024i	−0.422712	−	0.732159i
$372$		0			0
$373$	−19.7380			−1.02199			−0.510997	−	0.859583i	$-0.670724\pi$
$373$	−19.7380			−1.02199			−0.510997	+	0.859583i	$0.670724\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	21.5666	+	10.9075i	1.11074	+	0.561767i
$378$		0			0
$379$	−1.58794	−	2.75040i	−0.0815672	−	0.141278i	0.822356	−	0.568973i	$-0.192659\pi$
$379$	−1.58794	−	2.75040i	−0.0815672	−	0.141278i	−0.903923	+	0.427695i	$0.859326\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.50333			−0.179012			−0.0895058	−	0.995986i	$-0.528529\pi$
$383$	−3.50333			−0.179012			−0.0895058	+	0.995986i	$0.528529\pi$
$384$		0			0
$385$	15.8399	−	27.4355i	0.807275	−	1.39824i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.61988	+	13.1980i	−0.386343	+	0.669166i	−0.991955	−	0.126595i	$-0.959595\pi$
$389$	−7.61988	+	13.1980i	−0.386343	+	0.669166i	0.605612	+	0.795760i	$0.292929\pi$
$390$		0			0
$391$	−3.46699	−	6.00501i	−0.175333	−	0.303686i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−30.3868			−1.52893
$396$		0			0
$397$	−6.12766	+	10.6134i	−0.307538	+	0.532672i	−0.977823	−	0.209432i	$-0.932839\pi$
$397$	−6.12766	+	10.6134i	−0.307538	+	0.532672i	0.670285	+	0.742104i	$0.266172\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	7.09517	−	12.2892i	0.354316	−	0.613693i	−0.632685	−	0.774409i	$-0.718047\pi$
$401$	7.09517	−	12.2892i	0.354316	−	0.613693i	0.987001	+	0.160716i	$0.0513804\pi$
$402$		0			0
$403$	−0.752245	−	13.5760i	−0.0374720	−	0.676270i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−9.39398	−	16.2709i	−0.465642	−	0.806516i
$408$		0			0
$409$	15.0986			0.746577			0.373289	−	0.927715i	$-0.378230\pi$
$409$	15.0986			0.746577			0.373289	+	0.927715i	$0.378230\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−6.44639	−	11.1655i	−0.317206	−	0.549417i
$414$		0			0
$415$	32.6872			1.60455
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.45721	−	12.9163i	0.364309	−	0.631001i	−0.624356	−	0.781140i	$-0.714639\pi$
$419$	7.45721	−	12.9163i	0.364309	−	0.631001i	0.988665	+	0.150139i	$0.0479720\pi$
$420$		0			0
$421$	5.12630	−	8.87902i	0.249841	−	0.432737i	−0.713641	−	0.700512i	$-0.752955\pi$
$421$	5.12630	−	8.87902i	0.249841	−	0.432737i	0.963481	+	0.267775i	$0.0862884\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	28.2346	+	48.9037i	1.36958	+	2.37218i
$426$		0			0
$427$	3.55581	−	6.15885i	0.172078	−	0.298047i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−4.95272	−	8.57837i	−0.238564	−	0.413206i	0.721738	−	0.692166i	$-0.243343\pi$
$431$	−4.95272	−	8.57837i	−0.238564	−	0.413206i	−0.960303	+	0.278961i	$0.910010\pi$
$432$		0			0
$433$	4.64659	−	8.04813i	0.223301	−	0.386768i	−0.732507	−	0.680759i	$-0.761650\pi$
$433$	4.64659	−	8.04813i	0.223301	−	0.386768i	0.955808	+	0.293991i	$0.0949834\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.490076	−	0.848836i	−0.0234435	−	0.0406053i
$438$		0			0
$439$	8.88877			0.424238			0.212119	−	0.977244i	$-0.431964\pi$
$439$	8.88877			0.424238			0.212119	+	0.977244i	$0.431964\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.7207	−	28.9612i	0.794426	−	1.37599i	−0.128777	−	0.991674i	$-0.541105\pi$
$443$	16.7207	−	28.9612i	0.794426	−	1.37599i	0.923203	−	0.384313i	$-0.125562\pi$
$444$		0			0
$445$	−27.4562	+	47.5556i	−1.30155	+	2.25435i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−6.81689	+	11.8072i	−0.321709	+	0.557216i	−0.980841	−	0.194811i	$-0.937591\pi$
$449$	−6.81689	+	11.8072i	−0.321709	+	0.557216i	0.659132	+	0.752027i	$0.270924\pi$
$450$		0			0
$451$	22.4973	−	38.9665i	1.05936	−	1.83486i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−15.1347	+	9.89312i	−0.709526	+	0.463797i
$456$		0			0
$457$	4.80714			0.224869			0.112434	−	0.993659i	$-0.464135\pi$
$457$	4.80714			0.224869			0.112434	+	0.993659i	$0.464135\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13.9532	+	24.1677i	0.649867	+	1.12560i	0.983154	+	0.182777i	$0.0585087\pi$
$461$	13.9532	+	24.1677i	0.649867	+	1.12560i	−0.333288	+	0.942825i	$0.608158\pi$
$462$		0			0
$463$	5.16665	+	8.94891i	0.240115	+	0.415891i	0.960747	−	0.277427i	$-0.0894816\pi$
$463$	5.16665	+	8.94891i	0.240115	+	0.415891i	−0.720632	+	0.693318i	$0.756148\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	10.3158			0.477359			0.238679	−	0.971098i	$-0.423286\pi$
$467$	10.3158			0.477359			0.238679	+	0.971098i	$0.423286\pi$
$468$		0			0
$469$	−9.14252			−0.422162
$470$		0			0
$471$		0			0
$472$		0			0
$473$	3.58041	+	6.20145i	0.164627	+	0.285143i
$474$		0			0
$475$	3.99109	+	6.91277i	0.183124	+	0.317180i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	24.4590			1.11756			0.558780	−	0.829316i	$-0.311269\pi$
$479$	24.4590			1.11756			0.558780	+	0.829316i	$0.311269\pi$
$480$		0			0
$481$	0.593264	+	10.7068i	0.0270505	+	0.488190i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−6.22575	+	10.7833i	−0.282697	+	0.489645i
$486$		0			0
$487$	−2.64195	+	4.57600i	−0.119718	+	0.207358i	−0.919656	−	0.392725i	$-0.871532\pi$
$487$	−2.64195	+	4.57600i	−0.119718	+	0.207358i	0.799938	+	0.600083i	$0.204866\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−10.0470	+	17.4020i	−0.453417	+	0.785341i	−0.998596	−	0.0529788i	$-0.983128\pi$
$491$	−10.0470	+	17.4020i	−0.453417	+	0.785341i	0.545179	+	0.838320i	$0.316462\pi$
$492$		0			0
$493$	−16.4352	+	28.4667i	−0.740206	+	1.28207i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	3.36096			0.150760
$498$		0			0
$499$	−13.0843	−	22.6626i	−0.585733	−	1.01452i	−0.994784	−	0.102007i	$-0.967473\pi$
$499$	−13.0843	−	22.6626i	−0.585733	−	1.01452i	0.409051	−	0.912512i	$-0.365860\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	10.6471	−	18.4414i	0.474732	−	0.822260i	−0.524849	−	0.851195i	$-0.675878\pi$
$503$	10.6471	−	18.4414i	0.474732	−	0.822260i	0.999581	+	0.0289349i	$0.00921154\pi$
$504$		0			0
$505$	−12.4273	−	21.5247i	−0.553006	−	0.957835i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	11.3324	−	19.6283i	0.502299	−	0.870008i	−0.497697	−	0.867351i	$-0.665821\pi$
$509$	11.3324	−	19.6283i	0.502299	−	0.870008i	0.999996	−	0.00265726i	$-0.000845834\pi$
$510$		0			0
$511$	−0.896290	−	1.55242i	−0.0396495	−	0.0686750i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.82618	+	15.2874i	−0.388928	+	0.673643i
$516$		0			0
$517$	23.7840	−	41.1951i	1.04602	−	1.81176i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.1834			−0.533765			−0.266883	−	0.963729i	$-0.585994\pi$
$521$	−12.1834			−0.533765			−0.266883	+	0.963729i	$0.585994\pi$
$522$		0			0
$523$	17.2158	+	29.8186i	0.752794	+	1.30388i	0.946464	+	0.322810i	$0.104628\pi$
$523$	17.2158	+	29.8186i	0.752794	+	1.30388i	−0.193670	+	0.981067i	$0.562039\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	18.4928			0.805561
$528$		0			0
$529$	10.5003	+	18.1871i	0.456536	+	0.790743i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−21.4958	+	14.0512i	−0.931085	+	0.608624i
$534$		0			0
$535$	−12.0415	+	20.8565i	−0.520600	+	0.901705i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	17.3004	−	29.9652i	0.745181	−	1.29069i
$540$		0			0
$541$	−44.8888			−1.92992			−0.964960	−	0.262398i	$-0.915487\pi$
$541$	−44.8888			−1.92992			−0.964960	+	0.262398i	$0.915487\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.12552	+	1.94946i	0.0482119	+	0.0835055i
$546$		0			0
$547$	0.758838	−	1.31435i	0.0324456	−	0.0561974i	−0.849347	−	0.527835i	$-0.823004\pi$
$547$	0.758838	−	1.31435i	0.0324456	−	0.0561974i	0.881792	+	0.471638i	$0.156337\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.32320	+	4.02390i	−0.0989716	+	0.171424i
$552$		0			0
$553$	9.22693			0.392369
$554$		0			0
$555$		0			0
$556$		0			0
$557$	11.0013	+	19.0548i	0.466139	+	0.807376i	0.999252	−	0.0386680i	$-0.0123115\pi$
$557$	11.0013	+	19.0548i	0.466139	+	0.807376i	−0.533114	+	0.846044i	$0.678978\pi$
$558$		0			0
$559$	−0.226116	−	4.08080i	−0.00956369	−	0.172599i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	7.42747			0.313031			0.156515	−	0.987676i	$-0.449974\pi$
$563$	7.42747			0.313031			0.156515	+	0.987676i	$0.449974\pi$
$564$		0			0
$565$	−12.4046	−	21.4854i	−0.521866	−	0.903899i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−25.5156			−1.06967			−0.534836	−	0.844956i	$-0.679626\pi$
$569$	−25.5156			−1.06967			−0.534836	+	0.844956i	$0.679626\pi$
$570$		0			0
$571$	−13.2087	−	22.8782i	−0.552769	−	0.957423i	−0.998073	−	0.0620443i	$-0.980238\pi$
$571$	−13.2087	−	22.8782i	−0.552769	−	0.957423i	0.445305	−	0.895379i	$-0.353095\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.14124	+	14.1010i	0.339513	+	0.588054i
$576$		0			0
$577$	8.20970			0.341774			0.170887	−	0.985291i	$-0.445337\pi$
$577$	8.20970			0.341774			0.170887	+	0.985291i	$0.445337\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−9.92542			−0.411776
$582$		0			0
$583$	−83.3626			−3.45252
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.1575			−0.666891			−0.333445	−	0.942769i	$-0.608211\pi$
$587$	−16.1575			−0.666891			−0.333445	+	0.942769i	$0.608211\pi$
$588$		0			0
$589$	2.61405			0.107710
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−38.1696			−1.56744			−0.783718	−	0.621117i	$-0.786679\pi$
$593$	−38.1696			−1.56744			−0.783718	+	0.621117i	$0.786679\pi$
$594$		0			0
$595$	−12.2960	−	21.2973i	−0.504087	−	0.873105i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	10.0046	+	17.3285i	0.408778	+	0.708024i	0.994753	−	0.102305i	$-0.0326218\pi$
$599$	10.0046	+	17.3285i	0.408778	+	0.708024i	−0.585975	+	0.810329i	$0.699288\pi$
$600$		0			0
$601$	−9.00784			−0.367438			−0.183719	−	0.982979i	$-0.558814\pi$
$601$	−9.00784			−0.367438			−0.183719	+	0.982979i	$0.558814\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−58.7374	−	101.736i	−2.38801	−	4.13616i
$606$		0			0
$607$	28.7482			1.16685			0.583427	−	0.812165i	$-0.301711\pi$
$607$	28.7482			1.16685			0.583427	+	0.812165i	$0.301711\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−22.7252	+	14.8548i	−0.919361	+	0.600960i
$612$		0			0
$613$	4.46519	+	7.73394i	0.180347	+	0.312371i	0.941999	−	0.335616i	$-0.108944\pi$
$613$	4.46519	+	7.73394i	0.180347	+	0.312371i	−0.761651	+	0.647987i	$0.775611\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−18.6830			−0.752148			−0.376074	−	0.926590i	$-0.622726\pi$
$617$	−18.6830			−0.752148			−0.376074	+	0.926590i	$0.622726\pi$
$618$		0			0
$619$	9.16531	−	15.8748i	0.368385	−	0.638061i	−0.620928	−	0.783867i	$-0.713244\pi$
$619$	9.16531	−	15.8748i	0.368385	−	0.638061i	0.989313	+	0.145806i	$0.0465776\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8.33705	−	14.4402i	0.334017	−	0.578534i
$624$		0			0
$625$	−25.0126	−	43.3231i	−1.00050	−	1.73292i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14.5845			−0.581523
$630$		0			0
$631$	−4.52993	+	7.84607i	−0.180334	+	0.312347i	−0.941994	−	0.335629i	$-0.891051\pi$
$631$	−4.52993	+	7.84607i	−0.180334	+	0.312347i	0.761660	+	0.647976i	$0.224384\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.8987	−	31.0015i	0.710290	−	1.23026i
$636$		0			0
$637$	−16.5302	+	10.8053i	−0.654951	+	0.428123i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	8.84560	+	15.3210i	0.349380	+	0.605144i	0.986139	−	0.165919i	$-0.0530589\pi$
$641$	8.84560	+	15.3210i	0.349380	+	0.605144i	−0.636759	+	0.771063i	$0.719726\pi$
$642$		0			0
$643$	45.9729			1.81299			0.906497	−	0.422212i	$-0.138746\pi$
$643$	45.9729			1.81299			0.906497	+	0.422212i	$0.138746\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.5411	−	32.1142i	−0.728927	−	1.26254i	−0.957337	−	0.288973i	$-0.906686\pi$
$647$	−18.5411	−	32.1142i	−0.728927	−	1.26254i	0.228411	−	0.973565i	$-0.426647\pi$
$648$		0			0
$649$	−66.0018			−2.59080
$650$		0			0
$651$		0			0
$652$		0			0
$653$	12.9233	−	22.3838i	0.505728	−	0.875946i	−0.494250	−	0.869320i	$-0.664557\pi$
$653$	12.9233	−	22.3838i	0.505728	−	0.875946i	0.999978	−	0.00662658i	$-0.00210932\pi$
$654$		0			0
$655$	−23.2255	+	40.2277i	−0.907494	+	1.57183i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.471529	−	0.816712i	−0.0183681	−	0.0318146i	0.856695	−	0.515823i	$-0.172514\pi$
$659$	−0.471529	−	0.816712i	−0.0183681	−	0.0318146i	−0.875063	+	0.484008i	$0.839180\pi$
$660$		0			0
$661$	−4.93286	+	8.54396i	−0.191866	+	0.332322i	−0.945869	−	0.324550i	$-0.894787\pi$
$661$	−4.93286	+	8.54396i	−0.191866	+	0.332322i	0.754003	+	0.656871i	$0.228121\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.73810	−	3.01048i	−0.0674006	−	0.116741i
$666$		0			0
$667$	−4.73899	+	8.20817i	−0.183494	+	0.317821i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.2032	−	31.5289i	−0.702727	−	1.21716i
$672$		0			0
$673$	−25.3484			−0.977110			−0.488555	−	0.872533i	$-0.662476\pi$
$673$	−25.3484			−0.977110			−0.488555	+	0.872533i	$0.662476\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.2261	−	35.0326i	0.777352	−	1.34641i	−0.156111	−	0.987740i	$-0.549896\pi$
$677$	20.2261	−	35.0326i	0.777352	−	1.34641i	0.933463	−	0.358674i	$-0.116771\pi$
$678$		0			0
$679$	1.89044	−	3.27434i	0.0725485	−	0.125658i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.6007	−	20.0931i	0.443890	−	0.768840i	−0.554084	−	0.832461i	$-0.686931\pi$
$683$	11.6007	−	20.0931i	0.443890	−	0.768840i	0.997974	+	0.0636208i	$0.0202648\pi$
$684$		0			0
$685$	23.7819	−	41.1915i	0.908660	−	1.57384i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	42.4580	+	21.4736i	1.61752	+	0.818079i
$690$		0			0
$691$	−2.43312			−0.0925602			−0.0462801	−	0.998929i	$-0.514737\pi$
$691$	−2.43312			−0.0925602			−0.0462801	+	0.998929i	$0.514737\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−14.2128	−	24.6173i	−0.539122	−	0.933786i
$696$		0			0
$697$	−17.4640	−	30.2485i	−0.661496	−	1.14574i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.254890			−0.00962706			−0.00481353	−	0.999988i	$-0.501532\pi$
$701$	−0.254890			−0.00962706			−0.00481353	+	0.999988i	$0.501532\pi$
$702$		0			0
$703$	−2.06159			−0.0777544
$704$		0			0
$705$		0			0
$706$		0			0
$707$	3.77353	+	6.53594i	0.141918	+	0.245809i
$708$		0			0
$709$	−6.06672	−	10.5079i	−0.227840	−	0.394631i	0.729327	−	0.684165i	$-0.239833\pi$
$709$	−6.06672	−	10.5079i	−0.227840	−	0.394631i	−0.957168	+	0.289534i	$0.906500\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.33228			0.199696
$714$		0			0
$715$	5.12106	+	92.4215i	0.191517	+	3.45637i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−26.3387	+	45.6200i	−0.982268	+	1.70134i	−0.328771	+	0.944410i	$0.606635\pi$
$719$	−26.3387	+	45.6200i	−0.982268	+	1.70134i	−0.653497	+	0.756929i	$0.726699\pi$
$720$		0			0
$721$	2.68006	−	4.64200i	0.0998107	−	0.172877i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	38.5935	−	66.8459i	1.43333	−	2.48259i
$726$		0			0
$727$	−1.58322	+	2.74222i	−0.0587184	+	0.101703i	−0.893890	−	0.448286i	$-0.852035\pi$
$727$	−1.58322	+	2.74222i	−0.0587184	+	0.101703i	0.835172	+	0.549989i	$0.185368\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.55873			0.205597
$732$		0			0
$733$	11.4180	+	19.7766i	0.421734	+	0.730466i	0.996109	−	0.0881269i	$-0.0280881\pi$
$733$	11.4180	+	19.7766i	0.421734	+	0.730466i	−0.574375	+	0.818592i	$0.694755\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−23.4016	+	40.5327i	−0.862009	+	1.49304i
$738$		0			0
$739$	10.9208	+	18.9154i	0.401728	+	0.695813i	0.993935	−	0.109973i	$-0.0350765\pi$
$739$	10.9208	+	18.9154i	0.401728	+	0.695813i	−0.592207	+	0.805786i	$0.701743\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−5.45458	+	9.44761i	−0.200109	+	0.346599i	−0.948563	−	0.316587i	$-0.897463\pi$
$743$	−5.45458	+	9.44761i	−0.200109	+	0.346599i	0.748454	+	0.663186i	$0.230796\pi$
$744$		0			0
$745$	18.5010	+	32.0447i	0.677824	+	1.17403i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.65639	−	6.33306i	0.133602	−	0.231405i
$750$		0			0
$751$	14.6250	−	25.3313i	0.533674	−	0.924351i	−0.465552	−	0.885020i	$-0.654144\pi$
$751$	14.6250	−	25.3313i	0.533674	−	0.924351i	0.999226	−	0.0393303i	$-0.0125224\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−1.50856			−0.0549020
$756$		0			0
$757$	−15.6746	−	27.1492i	−0.569703	−	0.986755i	−0.996595	−	0.0824524i	$-0.973725\pi$
$757$	−15.6746	−	27.1492i	−0.569703	−	0.986755i	0.426892	−	0.904303i	$-0.359609\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−23.3290			−0.845676			−0.422838	−	0.906205i	$-0.638966\pi$
$761$	−23.3290			−0.845676			−0.422838	+	0.906205i	$0.638966\pi$
$762$		0			0
$763$	−0.341763	−	0.591950i	−0.0123726	−	0.0214300i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	33.6159	+	17.0016i	1.21380	+	0.613892i
$768$		0			0
$769$	−4.10976	+	7.11832i	−0.148202	+	0.256693i	−0.930563	−	0.366132i	$-0.880682\pi$
$769$	−4.10976	+	7.11832i	−0.148202	+	0.256693i	0.782361	+	0.622825i	$0.214015\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.9461	−	34.5476i	0.717411	−	1.24259i	−0.244611	−	0.969621i	$-0.578660\pi$
$773$	19.9461	−	34.5476i	0.717411	−	1.24259i	0.962022	−	0.272971i	$-0.0880064\pi$
$774$		0			0
$775$	−43.4252			−1.55988
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−2.46862	−	4.27577i	−0.0884474	−	0.153195i
$780$		0			0
$781$	8.60286	−	14.9006i	0.307834	−	0.533185i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.78969	+	4.83189i	−0.0995683	+	0.172457i
$786$		0			0
$787$	42.0152			1.49768			0.748841	−	0.662750i	$-0.230611\pi$
$787$	42.0152			1.49768			0.748841	+	0.662750i	$0.230611\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3.76665	+	6.52403i	0.133927	+	0.231968i
$792$		0			0
$793$	1.14960	+	20.7472i	0.0408235	+	0.736755i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	28.4690			1.00842			0.504212	−	0.863580i	$-0.331783\pi$
$797$	28.4690			1.00842			0.504212	+	0.863580i	$0.331783\pi$
$798$		0			0
$799$	−18.4628	−	31.9785i	−0.653167	−	1.13132i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−9.17673			−0.323840
$804$		0			0
$805$	−3.54547	−	6.14093i	−0.124961	−	0.216439i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.8229	+	44.7266i	0.907885	+	1.57250i	0.816997	+	0.576643i	$0.195637\pi$
$809$	25.8229	+	44.7266i	0.907885	+	1.57250i	0.0908889	+	0.995861i	$0.471029\pi$
$810$		0			0
$811$	−14.1240			−0.495959			−0.247980	−	0.968765i	$-0.579767\pi$
$811$	−14.1240			−0.495959			−0.247980	+	0.968765i	$0.579767\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	69.3778			2.43020
$816$		0			0
$817$	0.785752			0.0274900
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.6148			0.789263			0.394632	−	0.918839i	$-0.370872\pi$
$821$	22.6148			0.789263			0.394632	+	0.918839i	$0.370872\pi$
$822$		0			0
$823$	−4.24926			−0.148120			−0.0740600	−	0.997254i	$-0.523596\pi$
$823$	−4.24926			−0.148120			−0.0740600	+	0.997254i	$0.523596\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	4.50665			0.156711			0.0783557	−	0.996925i	$-0.475033\pi$
$827$	4.50665			0.156711			0.0783557	+	0.996925i	$0.475033\pi$
$828$		0			0
$829$	4.99777	+	8.65638i	0.173580	+	0.300649i	0.939669	−	0.342086i	$-0.111133\pi$
$829$	4.99777	+	8.65638i	0.173580	+	0.300649i	−0.766089	+	0.642734i	$0.777800\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−13.4298	−	23.2611i	−0.465314	−	0.805948i
$834$		0			0
$835$	−12.7992			−0.442936
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.63767	+	8.03268i	0.160110	+	0.277319i	0.934908	−	0.354890i	$-0.115482\pi$
$839$	4.63767	+	8.03268i	0.160110	+	0.277319i	−0.774798	+	0.632209i	$0.782148\pi$
$840$		0			0
$841$	15.9303			0.549319
$842$		0			0
$843$		0			0
$844$		0			0
$845$	21.1989	−	48.3910i	0.729263	−	1.66470i
$846$		0			0
$847$	17.8355	+	30.8921i	0.612837	+	1.06146i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4.20535			−0.144157
$852$		0			0
$853$	8.60320	−	14.9012i	0.294568	−	0.510207i	−0.680316	−	0.732919i	$-0.738158\pi$
$853$	8.60320	−	14.9012i	0.294568	−	0.510207i	0.974884	+	0.222712i	$0.0714909\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.2451	+	17.7450i	−0.349964	+	0.606156i	−0.986243	−	0.165303i	$-0.947140\pi$
$857$	−10.2451	+	17.7450i	−0.349964	+	0.606156i	0.636278	+	0.771460i	$0.280473\pi$
$858$		0			0
$859$	8.09018	+	14.0126i	0.276034	+	0.478104i	0.970395	−	0.241522i	$-0.0776466\pi$
$859$	8.09018	+	14.0126i	0.276034	+	0.478104i	−0.694362	+	0.719626i	$0.744313\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26.3268			0.896176			0.448088	−	0.893989i	$-0.352105\pi$
$863$	26.3268			0.896176			0.448088	+	0.893989i	$0.352105\pi$
$864$		0			0
$865$	17.8865	−	30.9804i	0.608160	−	1.05336i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	23.6176	−	40.9070i	0.801174	−	1.38767i
$870$		0			0
$871$	22.3598	−	14.6159i	0.757632	−	0.495242i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	16.3366	+	28.2957i	0.552276	+	0.956571i
$876$		0			0
$877$	−33.4379			−1.12912			−0.564559	−	0.825393i	$-0.690954\pi$
$877$	−33.4379			−1.12912			−0.564559	+	0.825393i	$0.690954\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.74235	+	3.01783i	0.0587011	+	0.101673i	0.893883	−	0.448301i	$-0.147971\pi$
$881$	1.74235	+	3.01783i	0.0587011	+	0.101673i	−0.835181	+	0.549974i	$0.814637\pi$
$882$		0			0
$883$	−21.7609			−0.732313			−0.366156	−	0.930553i	$-0.619326\pi$
$883$	−21.7609			−0.732313			−0.366156	+	0.930553i	$0.619326\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−17.4467	+	30.2185i	−0.585802	+	1.01464i	0.408973	+	0.912546i	$0.365887\pi$
$887$	−17.4467	+	30.2185i	−0.585802	+	1.01464i	−0.994775	+	0.102092i	$0.967446\pi$
$888$		0			0
$889$	−5.43493	+	9.41358i	−0.182282	+	0.315721i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−2.60980	−	4.52031i	−0.0873337	−	0.151266i
$894$		0			0
$895$	44.0485	−	76.2942i	1.47238	−	2.55023i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−12.6388	−	21.8911i	−0.421528	−	0.730108i
$900$		0			0
$901$	−32.3559	+	56.0421i	−1.07793	+	1.86703i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−35.2821	−	61.1104i	−1.17282	−	2.03138i
$906$		0			0
$907$	18.0350			0.598841			0.299420	−	0.954121i	$-0.403207\pi$
$907$	18.0350			0.598841			0.299420	+	0.954121i	$0.403207\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.8853	−	44.8346i	0.857617	−	1.48544i	−0.0165778	−	0.999863i	$-0.505277\pi$
$911$	25.8853	−	44.8346i	0.857617	−	1.48544i	0.874195	−	0.485574i	$-0.161390\pi$
$912$		0			0
$913$	−25.4055	+	44.0037i	−0.840800	+	1.45631i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7.05239	−	12.2151i	0.232890	−	0.403378i
$918$		0			0
$919$	25.9608	−	44.9655i	0.856369	−	1.48327i	−0.0190005	−	0.999819i	$-0.506048\pi$
$919$	25.9608	−	44.9655i	0.856369	−	1.48327i	0.875369	−	0.483455i	$-0.160618\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−8.21987	+	5.37309i	−0.270560	+	0.176858i
$924$		0			0
$925$	34.2476			1.12605
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.8219	−	36.0645i	−0.683144	−	1.18324i	−0.974016	−	0.226478i	$-0.927279\pi$
$929$	−20.8219	−	36.0645i	−0.683144	−	1.18324i	0.290873	−	0.956762i	$-0.406054\pi$
$930$		0			0
$931$	−1.89836	−	3.28806i	−0.0622163	−	0.107762i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−125.894			−4.11716
$936$		0			0
$937$	24.2358			0.791749			0.395875	−	0.918305i	$-0.370441\pi$
$937$	24.2358			0.791749			0.395875	+	0.918305i	$0.370441\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−0.333504	−	0.577647i	−0.0108719	−	0.0188307i	0.860538	−	0.509386i	$-0.170127\pi$
$941$	−0.333504	−	0.577647i	−0.0108719	−	0.0188307i	−0.871410	+	0.490555i	$0.836794\pi$
$942$		0			0
$943$	−5.03562	−	8.72195i	−0.163982	−	0.284026i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	12.5570			0.408048			0.204024	−	0.978966i	$-0.434598\pi$
$947$	12.5570			0.408048			0.204024	+	0.978966i	$0.434598\pi$
$948$		0			0
$949$	4.67387	+	2.36386i	0.151720	+	0.0767341i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.43257	+	5.94539i	−0.111192	+	0.192590i	−0.916251	−	0.400604i	$-0.868800\pi$
$953$	−3.43257	+	5.94539i	−0.111192	+	0.192590i	0.805059	+	0.593194i	$0.202133\pi$
$954$		0			0
$955$	4.55054	−	7.88177i	0.147252	−	0.255048i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−7.22135	+	12.5077i	−0.233190	+	0.403896i
$960$		0			0
$961$	8.38944	−	14.5309i	0.270627	−	0.468740i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−76.8190			−2.47289
$966$		0			0
$967$	−22.6585	−	39.2457i	−0.728648	−	1.26206i	−0.957455	−	0.288583i	$-0.906816\pi$
$967$	−22.6585	−	39.2457i	−0.728648	−	1.26206i	0.228807	−	0.973472i	$-0.426518\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−22.8736	+	39.6183i	−0.734049	+	1.27141i	0.221090	+	0.975253i	$0.429039\pi$
$971$	−22.8736	+	39.6183i	−0.734049	+	1.27141i	−0.955139	+	0.296157i	$0.904295\pi$
$972$		0			0
$973$	4.31570	+	7.47501i	0.138355	+	0.239638i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.81204	+	16.9950i	−0.313915	+	0.543717i	−0.979206	−	0.202867i	$-0.934974\pi$
$977$	−9.81204	+	16.9950i	−0.313915	+	0.543717i	0.665291	+	0.746584i	$0.268307\pi$
$978$		0			0
$979$	−42.6797	−	73.9235i	−1.36405	−	2.36260i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.4131	+	37.0886i	−0.682972	+	1.18294i	0.291098	+	0.956693i	$0.405980\pi$
$983$	−21.4131	+	37.0886i	−0.682972	+	1.18294i	−0.974070	+	0.226249i	$0.927354\pi$
$984$		0			0
$985$	−16.6516	+	28.8415i	−0.530565	+	0.918965i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.60282			0.0509667
$990$		0			0
$991$	24.4276	+	42.3099i	0.775969	+	1.34402i	0.934248	+	0.356623i	$0.116072\pi$
$991$	24.4276	+	42.3099i	0.775969	+	1.34402i	−0.158279	+	0.987394i	$0.550595\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.22294			0.197280
$996$		0			0
$997$	−8.51282	−	14.7446i	−0.269604	−	0.466968i	0.699156	−	0.714969i	$-0.253559\pi$
$997$	−8.51282	−	14.7446i	−0.269604	−	0.466968i	−0.968760	+	0.248002i	$0.920226\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1404.2.j.a.289.1		28
3.2	odd	2		468.2.j.a.133.3	✓	28
9.4	even	3		1404.2.k.a.1225.1		28
9.5	odd	6		468.2.k.a.445.7	yes	28
13.9	even	3		1404.2.k.a.1153.1		28
39.35	odd	6		468.2.k.a.61.7	yes	28
117.22	even	3	inner	1404.2.j.a.685.1		28
117.113	odd	6		468.2.j.a.373.3	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.j.a.133.3	✓	28	3.2	odd	2
468.2.j.a.373.3	yes	28	117.113	odd	6
468.2.k.a.61.7	yes	28	39.35	odd	6
468.2.k.a.445.7	yes	28	9.5	odd	6
1404.2.j.a.289.1		28	1.1	even	1	trivial
1404.2.j.a.685.1		28	117.22	even	3	inner
1404.2.k.a.1153.1		28	13.9	even	3
1404.2.k.a.1225.1		28	9.4	even	3

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:	\( N \)	\(=\)	\( 1404 = 2^{2} \cdot 3^{3} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1404.j (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-2.03195 - 3.51944i) q^{5} +(0.616999 + 1.06867i) q^{7} +6.31719 q^{11} +(-3.21746 - 1.62726i) q^{13} +(2.45192 - 4.24685i) q^{17} +(0.346591 - 0.600313i) q^{19} +(0.706995 - 1.22455i) q^{23} +(-5.75764 + 9.97252i) q^{25} -6.70300 q^{29} +(1.88555 + 3.26586i) q^{31} +(2.50742 - 4.34298i) q^{35} +(-1.48705 - 2.57565i) q^{37} +(3.56129 - 6.16833i) q^{41} +(0.566773 + 0.981679i) q^{43} +(3.76496 - 6.52111i) q^{47} +(2.73862 - 4.74344i) q^{49} -13.1961 q^{53} +(-12.8362 - 22.2330i) q^{55} -10.4480 q^{59} +(-2.88154 - 4.99097i) q^{61} +(0.810654 + 14.6302i) q^{65} +(-3.70443 + 6.41626i) q^{67} +(1.36182 - 2.35874i) q^{71} -1.45266 q^{73} +(3.89770 + 6.75102i) q^{77} +(3.73863 - 6.47550i) q^{79} +(-4.02165 + 6.96570i) q^{83} -19.9287 q^{85} +(-6.75612 - 11.7020i) q^{89} +(-0.246154 - 4.44243i) q^{91} -2.81702 q^{95} +(-1.53197 - 2.65344i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 2 q^{7} - 8 q^{11} + q^{13} + 8 q^{17} - q^{19} + 4 q^{23} - 14 q^{25} - 26 q^{29} + 2 q^{31} - 3 q^{35} - q^{37} - 4 q^{41} + 2 q^{43} - 11 q^{47} - 12 q^{49} - 52 q^{53} - 16 q^{59} - 7 q^{61}+ \cdots - 25 q^{97}+O(q^{100}) \) 28 * q + 2 * q^7 - 8 * q^11 + q^13 + 8 * q^17 - q^19 + 4 * q^23 - 14 * q^25 - 26 * q^29 + 2 * q^31 - 3 * q^35 - q^37 - 4 * q^41 + 2 * q^43 - 11 * q^47 - 12 * q^49 - 52 * q^53 - 16 * q^59 - 7 * q^61 + 31 * q^65 - 7 * q^67 + 12 * q^71 + 14 * q^73 + 28 * q^77 + 5 * q^79 - 9 * q^83 - 36 * q^85 + 11 * q^89 - q^91 + 56 * q^95 - 25 * q^97

Introduction

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