LMFDB - Embedded newform 1404.2.k.a.1225.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1404,2,Mod(1153,1404)] 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("1404.1153"); 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, 4])) N = Newforms(chi, 2, names="a")

Level:	$N$	$=$	$1404 = 2^{2} \cdot 3^{3} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1404.k (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			1225.1
Character	$\chi$	$=$	1404.1225
Dual form			1404.2.k.a.1153.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.03195 + 3.51944i) q^{5} -1.23400 q^{7} +(-3.15860 + 5.47085i) q^{11} +(0.199477 + 3.60003i) q^{13} +(2.45192 - 4.24685i) q^{17} +(0.346591 - 0.600313i) q^{19} -1.41399 q^{23} +(-5.75764 - 9.97252i) q^{25} +(3.35150 - 5.80497i) q^{29} +(1.88555 - 3.26586i) q^{31} +(2.50742 - 4.34298i) q^{35} +(-1.48705 - 2.57565i) q^{37} -7.12257 q^{41} -1.13355 q^{43} +(3.76496 + 6.52111i) q^{47} -5.47725 q^{49} -13.1961 q^{53} +(-12.8362 - 22.2330i) q^{55} +(5.22399 + 9.04821i) q^{59} +5.76307 q^{61} +(-13.0754 - 6.61303i) q^{65} +7.40886 q^{67} +(1.36182 - 2.35874i) q^{71} -1.45266 q^{73} +(3.89770 - 6.75102i) q^{77} +(3.73863 + 6.47550i) q^{79} +(-4.02165 - 6.96570i) q^{83} +(9.96436 + 17.2588i) q^{85} +(-6.75612 - 11.7020i) q^{89} +(-0.246154 - 4.44243i) q^{91} +(1.40851 + 2.43961i) q^{95} +3.06393 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$28 q - 4 q^{7} + 4 q^{11} + q^{13} + 8 q^{17} - q^{19} - 8 q^{23} - 14 q^{25} + 13 q^{29} + 2 q^{31} - 3 q^{35} - q^{37} + 8 q^{41} - 4 q^{43} - 11 q^{47} + 24 q^{49} - 52 q^{53} + 8 q^{59} + 14 q^{61}+ \cdots + 50 q^{97}+O(q^{100})$ 28 * q - 4 * q^7 + 4 * q^11 + q^13 + 8 * q^17 - q^19 - 8 * q^23 - 14 * q^25 + 13 * q^29 + 2 * q^31 - 3 * q^35 - q^37 + 8 * q^41 - 4 * q^43 - 11 * q^47 + 24 * q^49 - 52 * q^53 + 8 * q^59 + 14 * q^61 - 38 * q^65 + 14 * q^67 + 12 * q^71 + 14 * q^73 + 28 * q^77 + 5 * q^79 - 9 * q^83 + 18 * q^85 + 11 * q^89 - q^91 - 28 * q^95 + 50 * q^97

Character values

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

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

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	−0.0928649	+	0.995679i	$0.529602\pi$
$5$	−2.03195	+	3.51944i	−0.908716	+	1.57394i	−0.815851	+	0.578263i	$0.803731\pi$
$6$		0			0
$7$	−1.23400			−0.466408			−0.233204	−	0.972428i	$-0.574921\pi$
$7$	−1.23400			−0.466408			−0.233204	+	0.972428i	$0.574921\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.15860	+	5.47085i	−0.952353	+	1.64952i	−0.212039	+	0.977261i	$0.568010\pi$
$11$	−3.15860	+	5.47085i	−0.952353	+	1.64952i	−0.740314	+	0.672262i	$0.765323\pi$
$12$		0			0
$13$	0.199477	+	3.60003i	0.0553249	+	0.998468i
$13$	0.199477	+	3.60003i	0.0553249	+	0.998468i
$14$		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
$23$	−1.41399			−0.294837			−0.147419	−	0.989074i	$-0.547096\pi$
$23$	−1.41399			−0.294837			−0.147419	+	0.989074i	$0.547096\pi$
$24$		0			0
$25$	−5.75764	−	9.97252i	−1.15153	−	1.99450i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.35150	−	5.80497i	0.622358	−	1.07796i	−0.366687	−	0.930344i	$-0.619508\pi$
$29$	3.35150	−	5.80497i	0.622358	−	1.07796i	0.989045	−	0.147612i	$-0.0471586\pi$
$30$		0			0
$31$	1.88555	−	3.26586i	0.338654	−	0.586566i	−0.645526	−	0.763738i	$-0.723362\pi$
$31$	1.88555	−	3.26586i	0.338654	−	0.586566i	0.984180	+	0.177173i	$0.0566951\pi$
$32$		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
$41$	−7.12257			−1.11236			−0.556179	−	0.831062i	$-0.687733\pi$
$41$	−7.12257			−1.11236			−0.556179	+	0.831062i	$0.687733\pi$
$42$		0			0
$43$	−1.13355			−0.172864			−0.0864320	−	0.996258i	$-0.527547\pi$
$43$	−1.13355			−0.172864			−0.0864320	+	0.996258i	$0.527547\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.76496	+	6.52111i	0.549176	+	0.951201i	0.998331	+	0.0577473i	$0.0183918\pi$
$47$	3.76496	+	6.52111i	0.549176	+	0.951201i	−0.449155	+	0.893454i	$0.648275\pi$
$48$		0			0
$49$	−5.47725			−0.782464
$50$		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
$59$	5.22399	+	9.04821i	0.680105	+	1.17798i	0.974948	+	0.222431i	$0.0713991\pi$
$59$	5.22399	+	9.04821i	0.680105	+	1.17798i	−0.294844	+	0.955546i	$0.595268\pi$
$60$		0			0
$61$	5.76307			0.737886			0.368943	−	0.929452i	$-0.379720\pi$
$61$	5.76307			0.737886			0.368943	+	0.929452i	$0.379720\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−13.0754	−	6.61303i	−1.62181	−	0.820246i
$66$		0			0
$67$	7.40886			0.905136			0.452568	−	0.891730i	$-0.350508\pi$
$67$	7.40886			0.905136			0.452568	+	0.891730i	$0.350508\pi$
$68$		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	0.996001	−	0.0893413i	$-0.0284762\pi$
$79$	3.73863	+	6.47550i	0.420629	+	0.728550i	−0.575372	+	0.817892i	$0.695143\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−4.02165	−	6.96570i	−0.441433	−	0.764585i	0.556363	−	0.830939i	$-0.312196\pi$
$83$	−4.02165	−	6.96570i	−0.441433	−	0.764585i	−0.997796	+	0.0663546i	$0.978863\pi$
$84$		0			0
$85$	9.96436	+	17.2588i	1.08079	+	1.87198i
$86$		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
$95$	1.40851	+	2.43961i	0.144510	+	0.250299i
$96$		0			0
$97$	3.06393			0.311095			0.155547	−	0.987828i	$-0.450286\pi$
$97$	3.06393			0.311095			0.155547	+	0.987828i	$0.450286\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.05797	+	5.29656i	−0.304279	+	0.527027i	−0.977101	−	0.212778i	$-0.931749\pi$
$101$	−3.05797	+	5.29656i	−0.304279	+	0.527027i	0.672821	+	0.739805i	$0.265082\pi$
$102$		0			0
$103$	−2.17185	+	3.76176i	−0.213999	+	0.370657i	−0.952962	−	0.303089i	$-0.901982\pi$
$103$	−2.17185	+	3.76176i	−0.213999	+	0.370657i	0.738964	+	0.673745i	$0.235316\pi$
$104$		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
$113$	−3.05239	−	5.28690i	−0.287145	−	0.497350i	0.685982	−	0.727618i	$-0.259373\pi$
$113$	−3.05239	−	5.28690i	−0.287145	−	0.497350i	−0.973127	+	0.230269i	$0.926040\pi$
$114$		0			0
$115$	2.87315	−	4.97645i	0.267923	−	0.464056i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−3.02567	+	5.24061i	−0.277363	+	0.480406i
$120$		0			0
$121$	−14.4535	−	25.0341i	−1.31395	−	2.27583i
$122$		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	−1.00000		0.000776096i	$-0.999753\pi$
$131$	−5.71507	+	9.89879i	−0.499328	+	0.864861i	0.500672	+	0.865637i	$0.333086\pi$
$132$		0			0
$133$	−0.427692	+	0.740785i	−0.0370856	+	0.0642342i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−11.7040			−0.999939			−0.499969	−	0.866043i	$-0.666655\pi$
$137$	−11.7040			−0.999939			−0.499969	+	0.866043i	$0.666655\pi$
$138$		0			0
$139$	−3.49733	−	6.05755i	−0.296639	−	0.513794i	0.678726	−	0.734392i	$-0.262533\pi$
$139$	−3.49733	−	6.05755i	−0.296639	−	0.513794i	−0.975365	+	0.220598i	$0.929199\pi$
$140$		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
$149$	4.55253	+	7.88521i	0.372957	+	0.645981i	0.990019	−	0.140934i	$-0.0450104\pi$
$149$	4.55253	+	7.88521i	0.372957	+	0.645981i	−0.617062	+	0.786915i	$0.711677\pi$
$150$		0			0
$151$	0.185604	+	0.321476i	0.0151043	+	0.0261614i	0.873479	−	0.486862i	$-0.161859\pi$
$151$	0.185604	+	0.321476i	0.0151043	+	0.0261614i	−0.858374	+	0.513024i	$0.828525\pi$
$152$		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	−0.892117	−	0.451804i	$-0.850781\pi$
$157$	−0.686457	+	1.18898i	−0.0547852	+	0.0948908i	0.837332	+	0.546694i	$0.184114\pi$
$158$		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
$167$	−3.14950			−0.243716			−0.121858	−	0.992548i	$-0.538885\pi$
$167$	−3.14950			−0.243716			−0.121858	+	0.992548i	$0.538885\pi$
$168$		0			0
$169$	−12.9204	+	1.43624i	−0.993878	+	0.110480i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−8.80265			−0.669253			−0.334626	−	0.942351i	$-0.608610\pi$
$173$	−8.80265			−0.669253			−0.334626	+	0.942351i	$0.608610\pi$
$174$		0			0
$175$	7.10492	+	12.3061i	0.537081	+	0.930252i
$176$		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
$185$	12.0864			0.888613
$186$		0			0
$187$	15.4893	+	26.8282i	1.13269	+	1.96187i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.23950			−0.162044			−0.0810221	−	0.996712i	$-0.525818\pi$
$191$	−2.23950			−0.162044			−0.0810221	+	0.996712i	$0.525818\pi$
$192$		0			0
$193$	−18.9028			−1.36065			−0.680326	−	0.732909i	$-0.738162\pi$
$193$	−18.9028			−1.36065			−0.680326	+	0.732909i	$0.738162\pi$
$194$		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
$205$	14.4727	−	25.0675i	1.01082	−	1.75079i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	2.18948	+	3.79229i	0.151449	+	0.262318i
$210$		0			0
$211$	−18.3932			−1.26624			−0.633119	−	0.774055i	$-0.718225\pi$
$211$	−18.3932			−1.26624			−0.633119	+	0.774055i	$0.718225\pi$
$212$		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
$221$	15.7779	+	7.97984i	1.06134	+	0.536782i
$222$		0			0
$223$	10.5975	−	18.3553i	0.709658	−	1.22916i	−0.255326	−	0.966855i	$-0.582183\pi$
$223$	10.5975	−	18.3553i	0.709658	−	1.22916i	0.964984	−	0.262309i	$-0.0844840\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−19.0279			−1.26293			−0.631464	−	0.775405i	$-0.717546\pi$
$227$	−19.0279			−1.26293			−0.631464	+	0.775405i	$0.717546\pi$
$228$		0			0
$229$	7.57460	−	13.1196i	0.500544	−	0.866967i	−0.499456	−	0.866339i	$-0.666467\pi$
$229$	7.57460	−	13.1196i	0.500544	−	0.866967i	1.00000		0.000627856i	$-0.000199853\pi$
$230$		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	−0.826302	−	0.563228i	$-0.809559\pi$
$239$	1.15357	−	1.99805i	0.0746186	−	0.129243i	0.900920	+	0.433985i	$0.142893\pi$
$240$		0			0
$241$	−11.7563			−0.757288			−0.378644	−	0.925542i	$-0.623610\pi$
$241$	−11.7563			−0.757288			−0.378644	+	0.925542i	$0.623610\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	11.1295	−	19.2768i	0.711037	−	1.23155i
$246$		0			0
$247$	2.23028	+	1.12799i	0.141909	+	0.0717722i
$248$		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
$257$	3.59795			0.224434			0.112217	−	0.993684i	$-0.464205\pi$
$257$	3.59795			0.224434			0.112217	+	0.993684i	$0.464205\pi$
$258$		0			0
$259$	1.83502	+	3.17834i	0.114022	+	0.197493i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	3.18175	−	5.51095i	0.196195	−	0.339819i	−0.751097	−	0.660192i	$-0.770475\pi$
$263$	3.18175	−	5.51095i	0.196195	−	0.339819i	0.947292	+	0.320373i	$0.103808\pi$
$264$		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
$275$	72.7442			4.38664
$276$		0			0
$277$	15.0872			0.906503			0.453252	−	0.891383i	$-0.350264\pi$
$277$	15.0872			0.906503			0.453252	+	0.891383i	$0.350264\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	9.06790	+	15.7061i	0.540945	+	0.936945i	0.998850	+	0.0479435i	$0.0152668\pi$
$281$	9.06790	+	15.7061i	0.540945	+	0.936945i	−0.457905	+	0.889001i	$0.651400\pi$
$282$		0			0
$283$	−9.52842			−0.566406			−0.283203	−	0.959060i	$-0.591397\pi$
$283$	−9.52842			−0.566406			−0.283203	+	0.959060i	$0.591397\pi$
$284$		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
$293$	−0.154098	−	0.266906i	−0.00900252	−	0.0155928i	0.861489	−	0.507776i	$-0.169532\pi$
$293$	−0.154098	−	0.266906i	−0.00900252	−	0.0155928i	−0.870492	+	0.492183i	$0.836199\pi$
$294$		0			0
$295$	−42.4595			−2.47209
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.282058	−	5.09040i	−0.0163118	−	0.294386i
$300$		0			0
$301$	1.39879			0.0806251
$302$		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	0.321833	+	0.946796i	$0.395701\pi$
$311$	−11.6222	+	20.1302i	−0.659033	+	1.14148i	−0.980866	+	0.194682i	$0.937632\pi$
$312$		0			0
$313$	3.47080	+	6.01160i	0.196181	+	0.339796i	0.947287	−	0.320386i	$-0.103813\pi$
$313$	3.47080	+	6.01160i	0.196181	+	0.339796i	−0.751106	+	0.660182i	$0.770479\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.08590	−	10.5411i	−0.341818	−	0.592046i	0.642952	−	0.765906i	$-0.277709\pi$
$317$	−6.08590	−	10.5411i	−0.341818	−	0.592046i	−0.984770	+	0.173860i	$0.944376\pi$
$318$		0			0
$319$	21.1721	+	36.6711i	1.18541	+	2.05319i
$320$		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
$329$	−4.64596	−	8.04704i	−0.256140	−	0.443648i
$330$		0			0
$331$	−18.6570			−1.02548			−0.512741	−	0.858543i	$-0.671370\pi$
$331$	−18.6570			−1.02548			−0.512741	+	0.858543i	$0.671370\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.0544	+	26.0750i	−0.822511	+	1.42463i
$336$		0			0
$337$	−10.0206	+	17.3562i	−0.545858	+	0.945454i	0.452694	+	0.891666i	$0.350463\pi$
$337$	−10.0206	+	17.3562i	−0.545858	+	0.945454i	−0.998552	+	0.0537879i	$0.982871\pi$
$338$		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
$347$	−10.1229	−	17.5333i	−0.543424	−	0.941239i	−0.998704	−	0.0508902i	$-0.983794\pi$
$347$	−10.1229	−	17.5333i	−0.543424	−	0.941239i	0.455280	−	0.890348i	$-0.349539\pi$
$348$		0			0
$349$	−17.9009	+	31.0052i	−0.958212	+	1.65967i	−0.231372	+	0.972865i	$0.574322\pi$
$349$	−17.9009	+	31.0052i	−0.958212	+	1.65967i	−0.726840	+	0.686807i	$0.759012\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.2650	−	22.9756i	0.706022	−	1.22287i	−0.260299	−	0.965528i	$-0.583821\pi$
$353$	13.2650	−	22.9756i	0.706022	−	1.22287i	0.966321	−	0.257339i	$-0.0828456\pi$
$354$		0			0
$355$	5.53429	+	9.58566i	0.293729	+	0.508754i
$356$		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
$367$	−10.0026	+	17.3250i	−0.522130	+	0.904356i	0.477538	+	0.878611i	$0.341529\pi$
$367$	−10.0026	+	17.3250i	−0.522130	+	0.904356i	−0.999669	+	0.0257451i	$0.991804\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	16.2840			0.845424
$372$		0			0
$373$	9.86899	+	17.0936i	0.510997	+	0.885073i	0.999919	+	0.0127450i	$0.00405696\pi$
$373$	9.86899	+	17.0936i	0.510997	+	0.885073i	−0.488922	+	0.872328i	$0.662610\pi$
$374$		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
$383$	1.75166	+	3.03397i	0.0895058	+	0.155029i	0.907302	−	0.420479i	$-0.138138\pi$
$383$	1.75166	+	3.03397i	0.0895058	+	0.155029i	−0.817797	+	0.575507i	$0.804805\pi$
$384$		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	0.605612	−	0.795760i	$-0.292929\pi$
$389$	−7.61988	−	13.1980i	−0.386343	−	0.669166i	−0.991955	+	0.126595i	$0.959595\pi$
$390$		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
$401$	−14.1903			−0.708632			−0.354316	−	0.935126i	$-0.615286\pi$
$401$	−14.1903			−0.708632			−0.354316	+	0.935126i	$0.615286\pi$
$402$		0			0
$403$	12.1333	+	6.13655i	0.604403	+	0.305684i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	18.7880			0.931285
$408$		0			0
$409$	−7.54929	−	13.0758i	−0.373289	−	0.646555i	0.616781	−	0.787135i	$-0.288437\pi$
$409$	−7.54929	−	13.0758i	−0.373289	−	0.646555i	−0.990069	+	0.140580i	$0.955103\pi$
$410$		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
$419$	−14.9144			−0.728617			−0.364309	−	0.931278i	$-0.618695\pi$
$419$	−14.9144			−0.728617			−0.364309	+	0.931278i	$0.618695\pi$
$420$		0			0
$421$	5.12630	+	8.87902i	0.249841	+	0.432737i	0.963481	−	0.267775i	$-0.0862884\pi$
$421$	5.12630	+	8.87902i	0.249841	+	0.432737i	−0.713641	+	0.700512i	$0.752955\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−56.4691			−2.73915
$426$		0			0
$427$	−7.11162			−0.344156
$428$		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
$439$	−4.44439	+	7.69790i	−0.212119	+	0.367401i	−0.952377	−	0.304922i	$-0.901370\pi$
$439$	−4.44439	+	7.69790i	−0.212119	+	0.367401i	0.740259	+	0.672322i	$0.234703\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	16.7207	+	28.9612i	0.794426	+	1.37599i	0.923203	+	0.384313i	$0.125562\pi$
$443$	16.7207	+	28.9612i	0.794426	+	1.37599i	−0.128777	+	0.991674i	$0.541105\pi$
$444$		0			0
$445$	54.9124			2.60310
$446$		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
$455$	16.1350	+	8.16047i	0.756422	+	0.382569i
$456$		0			0
$457$	−2.40357	+	4.16311i	−0.112434	+	0.194742i	−0.916751	−	0.399459i	$-0.869198\pi$
$457$	−2.40357	+	4.16311i	−0.112434	+	0.194742i	0.804317	+	0.594201i	$0.202531\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−27.9065			−1.29973			−0.649867	−	0.760048i	$-0.725175\pi$
$461$	−27.9065			−1.29973			−0.649867	+	0.760048i	$0.725175\pi$
$462$		0			0
$463$	5.16665	−	8.94891i	0.240115	−	0.415891i	−0.720632	−	0.693318i	$-0.756148\pi$
$463$	5.16665	−	8.94891i	0.240115	−	0.415891i	0.960747	+	0.277427i	$0.0894816\pi$
$464$		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
$475$	−7.98218			−0.366247
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−12.2295	+	21.1821i	−0.558780	+	0.967836i	0.438818	+	0.898576i	$0.355397\pi$
$479$	−12.2295	+	21.1821i	−0.558780	+	0.967836i	−0.997599	+	0.0692603i	$0.977936\pi$
$480$		0			0
$481$	8.97577	−	5.86721i	0.409260	−	0.267522i
$482$		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
$491$	20.0941			0.906834			0.453417	−	0.891299i	$-0.350205\pi$
$491$	20.0941			0.906834			0.453417	+	0.891299i	$0.350205\pi$
$492$		0			0
$493$	−16.4352	−	28.4667i	−0.740206	−	1.28207i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1.68048	+	2.91068i	−0.0753798	+	0.130562i
$498$		0			0
$499$	−13.0843	+	22.6626i	−0.585733	+	1.01452i	0.409051	+	0.912512i	$0.365860\pi$
$499$	−13.0843	+	22.6626i	−0.585733	+	1.01452i	−0.994784	+	0.102007i	$0.967473\pi$
$500$		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
$509$	−22.6648			−1.00460			−0.502299	−	0.864694i	$-0.667513\pi$
$509$	−22.6648			−1.00460			−0.502299	+	0.864694i	$0.667513\pi$
$510$		0			0
$511$	1.79258			0.0792990
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−8.82618	−	15.2874i	−0.388928	−	0.673643i
$516$		0			0
$517$	−47.5680			−2.09204
$518$		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
$527$	−9.24642	−	16.0153i	−0.402780	−	0.697636i
$528$		0			0
$529$	−21.0006			−0.913071
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.42079	−	25.6415i	−0.0615412	−	1.11066i
$534$		0			0
$535$	24.0830			1.04120
$536$		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	0.881792	−	0.471638i	$-0.156337\pi$
$547$	0.758838	+	1.31435i	0.0324456	+	0.0561974i	−0.849347	+	0.527835i	$0.823004\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.32320	−	4.02390i	−0.0989716	−	0.171424i
$552$		0			0
$553$	−4.61346	−	7.99075i	−0.196184	−	0.339801i
$554$		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
$563$	−3.71374	−	6.43238i	−0.156515	−	0.271093i	0.777094	−	0.629384i	$-0.216693\pi$
$563$	−3.71374	−	6.43238i	−0.156515	−	0.271093i	−0.933610	+	0.358291i	$0.883359\pi$
$564$		0			0
$565$	24.8092			1.04373
$566$		0			0
$567$		0			0
$568$		0			0
$569$	12.7578	−	22.0972i	0.534836	−	0.926362i	−0.464336	−	0.885659i	$-0.653707\pi$
$569$	12.7578	−	22.0972i	0.534836	−	0.926362i	0.999171	−	0.0407030i	$-0.0129598\pi$
$570$		0			0
$571$	−13.2087	+	22.8782i	−0.552769	+	0.957423i	0.445305	+	0.895379i	$0.353095\pi$
$571$	−13.2087	+	22.8782i	−0.552769	+	0.957423i	−0.998073	+	0.0620443i	$0.980238\pi$
$572$		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
$581$	4.96271	+	8.59566i	0.205888	+	0.356608i
$582$		0			0
$583$	41.6813	−	72.1941i	1.72626	−	2.98997i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.07874	−	13.9928i	0.333445	−	0.577544i	−0.649740	−	0.760157i	$-0.725122\pi$
$587$	8.07874	−	13.9928i	0.333445	−	0.577544i	0.983185	+	0.182612i	$0.0584554\pi$
$588$		0			0
$589$	−1.30702	−	2.26383i	−0.0538550	−	0.0932796i
$590$		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	−0.585975	−	0.810329i	$-0.699288\pi$
$599$	10.0046	−	17.3285i	0.408778	−	0.708024i	0.994753	+	0.102305i	$0.0326218\pi$
$600$		0			0
$601$	4.50392	−	7.80102i	0.183719	−	0.318210i	−0.759425	−	0.650595i	$-0.774520\pi$
$601$	4.50392	−	7.80102i	0.183719	−	0.318210i	0.943144	+	0.332384i	$0.107853\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	117.475			4.77603
$606$		0			0
$607$	−14.3741	−	24.8967i	−0.583427	−	1.01053i	−0.995070	−	0.0991800i	$-0.968378\pi$
$607$	−14.3741	−	24.8967i	−0.583427	−	1.01053i	0.411642	−	0.911345i	$-0.364955\pi$
$608$		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
$617$	9.34148	+	16.1799i	0.376074	+	0.651379i	0.990487	−	0.137605i	$-0.0439406\pi$
$617$	9.34148	+	16.1799i	0.376074	+	0.651379i	−0.614413	+	0.788984i	$0.710607\pi$
$618$		0			0
$619$	9.16531	+	15.8748i	0.368385	+	0.638061i	0.989313	−	0.145806i	$-0.0465776\pi$
$619$	9.16531	+	15.8748i	0.368385	+	0.638061i	−0.620928	+	0.783867i	$0.713244\pi$
$620$		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
$635$	−35.7975			−1.42058
$636$		0			0
$637$	−1.09258	−	19.7182i	−0.0432898	−	0.781265i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−17.6912			−0.698760			−0.349380	−	0.936981i	$-0.613608\pi$
$641$	−17.6912			−0.698760			−0.349380	+	0.936981i	$0.613608\pi$
$642$		0			0
$643$	−22.9864	−	39.8137i	−0.906497	−	1.57010i	−0.818895	−	0.573943i	$-0.805413\pi$
$643$	−22.9864	−	39.8137i	−0.906497	−	1.57010i	−0.0876021	−	0.996156i	$-0.527920\pi$
$644$		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
$653$	−25.8466			−1.01146			−0.505728	−	0.862693i	$-0.668776\pi$
$653$	−25.8466			−1.01146			−0.505728	+	0.862693i	$0.668776\pi$
$654$		0			0
$655$	−23.2255	−	40.2277i	−0.907494	−	1.57183i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	0.943057			0.0367363			0.0183681	−	0.999831i	$-0.494153\pi$
$659$	0.943057			0.0367363			0.0183681	+	0.999831i	$0.494153\pi$
$660$		0			0
$661$	9.86572			0.383732			0.191866	−	0.981421i	$-0.438546\pi$
$661$	9.86572			0.383732			0.191866	+	0.981421i	$0.438546\pi$
$662$		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
$673$	12.6742	−	21.9524i	0.488555	−	0.846202i	−0.511358	−	0.859368i	$-0.670858\pi$
$673$	12.6742	−	21.9524i	0.488555	−	0.846202i	0.999913	+	0.0131657i	$0.00419088\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.2261	+	35.0326i	0.777352	+	1.34641i	0.933463	+	0.358674i	$0.116771\pi$
$677$	20.2261	+	35.0326i	0.777352	+	1.34641i	−0.156111	+	0.987740i	$0.549896\pi$
$678$		0			0
$679$	−3.78089			−0.145097
$680$		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
$689$	−2.63233	−	47.5065i	−0.100284	−	1.80985i
$690$		0			0
$691$	1.21656	−	2.10714i	0.0462801	−	0.0801595i	−0.841957	−	0.539544i	$-0.818597\pi$
$691$	1.21656	−	2.10714i	0.0462801	−	0.0801595i	0.888238	+	0.459384i	$0.151930\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	28.4256			1.07824
$696$		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
$709$	12.1334			0.455681			0.227840	−	0.973698i	$-0.426834\pi$
$709$	12.1334			0.455681			0.227840	+	0.973698i	$0.426834\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−2.66614	+	4.61789i	−0.0998478	+	0.172941i
$714$		0			0
$715$	77.4788	−	50.6457i	2.89754	−	1.89404i
$716$		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
$725$	−77.1869			−2.86665
$726$		0			0
$727$	−1.58322	−	2.74222i	−0.0587184	−	0.101703i	0.835172	−	0.549989i	$-0.185368\pi$
$727$	−1.58322	−	2.74222i	−0.0587184	−	0.101703i	−0.893890	+	0.448286i	$0.852035\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.77936	+	4.81400i	−0.102799	+	0.178052i
$732$		0			0
$733$	11.4180	−	19.7766i	0.421734	−	0.730466i	−0.574375	−	0.818592i	$-0.694755\pi$
$733$	11.4180	−	19.7766i	0.421734	−	0.730466i	0.996109	+	0.0881269i	$0.0280881\pi$
$734$		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
$743$	10.9092			0.400218			0.200109	−	0.979774i	$-0.435870\pi$
$743$	10.9092			0.400218			0.200109	+	0.979774i	$0.435870\pi$
$744$		0			0
$745$	−37.0020			−1.35565
$746$		0			0
$747$		0			0
$748$		0			0
$749$	3.65639	+	6.33306i	0.133602	+	0.231405i
$750$		0			0
$751$	−29.2500			−1.06735			−0.533674	−	0.845690i	$-0.679189\pi$
$751$	−29.2500			−1.06735			−0.533674	+	0.845690i	$0.679189\pi$
$752$		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
$761$	11.6645	+	20.2035i	0.422838	+	0.732377i	0.996216	−	0.0869143i	$-0.0277006\pi$
$761$	11.6645	+	20.2035i	0.422838	+	0.732377i	−0.573378	+	0.819291i	$0.694367\pi$
$762$		0			0
$763$	0.683525			0.0247453
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−31.5317	+	20.6114i	−1.13855	+	0.744235i
$768$		0			0
$769$	8.21952			0.296404			0.148202	−	0.988957i	$-0.452651\pi$
$769$	8.21952			0.296404			0.148202	+	0.988957i	$0.452651\pi$
$770$		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
$787$	−21.0076	−	36.3863i	−0.748841	−	1.29703i	−0.948379	−	0.317140i	$-0.897277\pi$
$787$	−21.0076	−	36.3863i	−0.748841	−	1.29703i	0.199538	−	0.979890i	$-0.436056\pi$
$788$		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
$797$	−14.2345	−	24.6549i	−0.504212	−	0.873321i	−0.999988	−	0.00487082i	$-0.998450\pi$
$797$	−14.2345	−	24.6549i	−0.504212	−	0.873321i	0.495776	−	0.868451i	$-0.334884\pi$
$798$		0			0
$799$	36.9256			1.30633
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4.58836	−	7.94728i	0.161920	−	0.280453i
$804$		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
$815$	−34.6889	−	60.0829i	−1.21510	−	2.10461i
$816$		0			0
$817$	−0.392876	+	0.680482i	−0.0137450	+	0.0238070i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11.3074	+	19.5850i	−0.394632	+	0.683522i	−0.993054	−	0.117659i	$-0.962461\pi$
$821$	−11.3074	+	19.5850i	−0.394632	+	0.683522i	0.598422	+	0.801181i	$0.295794\pi$
$822$		0			0
$823$	2.12463	+	3.67997i	0.0740600	+	0.128276i	0.900677	−	0.434489i	$-0.143071\pi$
$823$	2.12463	+	3.67997i	0.0740600	+	0.128276i	−0.826617	+	0.562765i	$0.809738\pi$
$824$		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
$835$	6.39962	−	11.0845i	0.221468	−	0.383594i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−9.27534			−0.320220			−0.160110	−	0.987099i	$-0.551185\pi$
$839$	−9.27534			−0.320220			−0.160110	+	0.987099i	$0.551185\pi$
$840$		0			0
$841$	−7.96513	−	13.7960i	−0.274660	−	0.475724i
$842$		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
$851$	2.10267	+	3.64194i	0.0720787	+	0.124844i
$852$		0			0
$853$	8.60320	+	14.9012i	0.294568	+	0.510207i	0.974884	−	0.222712i	$-0.0714909\pi$
$853$	8.60320	+	14.9012i	0.294568	+	0.510207i	−0.680316	+	0.732919i	$0.738158\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.2451	−	17.7450i	−0.349964	−	0.606156i	0.636278	−	0.771460i	$-0.280473\pi$
$857$	−10.2451	−	17.7450i	−0.349964	−	0.606156i	−0.986243	+	0.165303i	$0.947140\pi$
$858$		0			0
$859$	8.09018	−	14.0126i	0.276034	−	0.478104i	−0.694362	−	0.719626i	$-0.744313\pi$
$859$	8.09018	−	14.0126i	0.276034	−	0.478104i	0.970395	+	0.241522i	$0.0776466\pi$
$860$		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
$869$	−47.2353			−1.60235
$870$		0			0
$871$	1.47790	+	26.6721i	0.0500766	+	0.903750i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−32.6731			−1.10455
$876$		0			0
$877$	16.7190	+	28.9581i	0.564559	+	0.977845i	0.997091	+	0.0762261i	$0.0242871\pi$
$877$	16.7190	+	28.9581i	0.564559	+	0.977845i	−0.432532	+	0.901619i	$0.642380\pi$
$878$		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
$887$	34.8933			1.17160			0.585802	−	0.810454i	$-0.300780\pi$
$887$	34.8933			1.17160			0.585802	+	0.810454i	$0.300780\pi$
$888$		0			0
$889$	−5.43493	−	9.41358i	−0.182282	−	0.315721i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.21960			0.174667
$894$		0			0
$895$	−88.0969			−2.94476
$896$		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
$907$	−9.01748	+	15.6187i	−0.299420	+	0.518611i	−0.976003	−	0.217755i	$-0.930127\pi$
$907$	−9.01748	+	15.6187i	−0.299420	+	0.518611i	0.676583	+	0.736366i	$0.263460\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	25.8853	+	44.8346i	0.857617	+	1.48544i	0.874195	+	0.485574i	$0.161390\pi$
$911$	25.8853	+	44.8346i	0.857617	+	1.48544i	−0.0165778	+	0.999863i	$0.505277\pi$
$912$		0			0
$913$	50.8111			1.68160
$914$		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
$923$	8.76317	+	4.43207i	0.288443	+	0.145883i
$924$		0			0
$925$	−17.1238	+	29.6593i	−0.563027	+	0.975191i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	41.6437			1.36629			0.683144	−	0.730284i	$-0.260612\pi$
$929$	41.6437			1.36629			0.683144	+	0.730284i	$0.260612\pi$
$930$		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	−0.871410	−	0.490555i	$-0.836794\pi$
$941$	−0.333504	+	0.577647i	−0.0108719	+	0.0188307i	0.860538	+	0.509386i	$0.170127\pi$
$942$		0			0
$943$	10.0712			0.327965
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−6.27850	+	10.8747i	−0.204024	+	0.353380i	−0.949821	−	0.312793i	$-0.898735\pi$
$947$	−6.27850	+	10.8747i	−0.204024	+	0.353380i	0.745797	+	0.666173i	$0.232069\pi$
$948$		0			0
$949$	−0.289772	−	5.22961i	−0.00940639	−	0.169760i
$950$		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
$959$	14.4427			0.466379
$960$		0			0
$961$	8.38944	+	14.5309i	0.270627	+	0.468740i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	38.4095	−	66.5272i	1.23645	−	2.14159i
$966$		0			0
$967$	−22.6585	+	39.2457i	−0.728648	+	1.26206i	0.228807	+	0.973472i	$0.426518\pi$
$967$	−22.6585	+	39.2457i	−0.728648	+	1.26206i	−0.957455	+	0.288583i	$0.906816\pi$
$968$		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
$977$	19.6241			0.627830			0.313915	−	0.949451i	$-0.398359\pi$
$977$	19.6241			0.627830			0.313915	+	0.949451i	$0.398359\pi$
$978$		0			0
$979$	85.3595			2.72810
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.4131	−	37.0886i	−0.682972	−	1.18294i	−0.974070	−	0.226249i	$-0.927354\pi$
$983$	−21.4131	−	37.0886i	−0.682972	−	1.18294i	0.291098	−	0.956693i	$-0.405980\pi$
$984$		0			0
$985$	33.3032			1.06113
$986$		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
$995$	−3.11147	−	5.38922i	−0.0986402	−	0.170850i
$996$		0			0
$997$	17.0256			0.539208			0.269604	−	0.962971i	$-0.413107\pi$
$997$	17.0256			0.539208			0.269604	+	0.962971i	$0.413107\pi$
$998$		0			0
$999$		0			0

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

)

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
468.2.j.a.133.3	✓	28	9.2	odd	6
468.2.j.a.373.3	yes	28	39.35	odd	6
468.2.k.a.61.7	yes	28	117.74	odd	6
468.2.k.a.445.7	yes	28	3.2	odd	2
1404.2.j.a.289.1		28	9.7	even	3
1404.2.j.a.685.1		28	13.9	even	3
1404.2.k.a.1153.1		28	117.61	even	3	inner
1404.2.k.a.1225.1		28	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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1404.2.k.a.1225.1

Level	$1404$
Weight	$2$
Character	1404.1225
Analytic conductor	$11.211$
Analytic rank	$0$
Dimension	$28$
Inner twists	$2$