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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [380,2,Mod(13,380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("380.13"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(380, base_ring=CyclotomicField(36)) chi = DirichletCharacter(H, H._module([0, 27, 10])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 380.bh (of order \(36\), degree \(12\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.03431527681\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(10\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

Embedding invariants

Embedding label 13.6
Character \(\chi\) \(=\) 380.13
Dual form 380.2.bh.a.117.6

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.224420 - 0.104649i) q^{3} +(-2.15088 + 0.611320i) q^{5} +(0.357605 + 1.33460i) q^{7} +(-1.88895 + 2.25116i) q^{9} +(0.00188012 + 0.00325647i) q^{11} +(-1.47530 + 3.16379i) q^{13} +(-0.418728 + 0.362280i) q^{15} +(0.0221536 - 0.253216i) q^{17} +(0.384763 + 4.34188i) q^{19} +(0.219918 + 0.262088i) q^{21} +(-3.63066 + 2.54221i) q^{23} +(4.25258 - 2.62975i) q^{25} +(-0.380604 + 1.42043i) q^{27} +(1.17897 + 0.989275i) q^{29} +(0.377920 + 0.218192i) q^{31} +(0.000762724 + 0.000534065i) q^{33} +(-1.58503 - 2.65195i) q^{35} +(0.496355 + 0.496355i) q^{37} +0.864406i q^{39} +(2.34618 - 6.44608i) q^{41} +(-3.96540 + 5.66318i) q^{43} +(2.68673 - 5.99673i) q^{45} +(-2.70864 + 0.236975i) q^{47} +(4.40890 - 2.54548i) q^{49} +(-0.0215271 - 0.0591453i) q^{51} +(-3.00243 - 4.28792i) q^{53} +(-0.00603466 - 0.00585492i) q^{55} +(0.540722 + 0.934142i) q^{57} +(7.08129 - 5.94191i) q^{59} +(-0.899822 + 5.10314i) q^{61} +(-3.67990 - 1.71597i) q^{63} +(1.23910 - 7.70680i) q^{65} +(1.01014 + 11.5460i) q^{67} +(-0.548754 + 0.950469i) q^{69} +(-1.85753 + 0.327532i) q^{71} +(-4.92742 - 10.5669i) q^{73} +(0.679164 - 1.03520i) q^{75} +(-0.00367374 + 0.00367374i) q^{77} +(10.2851 + 3.74347i) q^{79} +(-1.46766 - 8.32351i) q^{81} +(-1.18134 + 0.316538i) q^{83} +(0.107147 + 0.558181i) q^{85} +(0.368112 + 0.0986353i) q^{87} +(4.35576 - 1.58537i) q^{89} +(-4.74996 - 0.837546i) q^{91} +(0.107647 + 0.00941786i) q^{93} +(-3.48186 - 9.10366i) q^{95} +(17.5189 + 1.53271i) q^{97} +(-0.0108823 - 0.00191884i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 6 q^{7} + 18 q^{15} - 18 q^{17} + 48 q^{21} - 36 q^{23} - 24 q^{25} - 60 q^{33} - 18 q^{35} - 12 q^{41} - 36 q^{43} + 18 q^{45} - 24 q^{47} + 96 q^{51} - 18 q^{53} + 72 q^{55} - 6 q^{57} - 24 q^{61}+ \cdots - 60 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/380\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(77\) \(191\)
\(\chi(n)\) \(e\left(\frac{5}{18}\right)\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.224420 − 0.104649i 0.129569 − 0.0604191i −0.356753 − 0.934199i \(-0.616116\pi\)
0.486322 + 0.873780i \(0.338338\pi\)
\(4\) 0 0
\(5\) −2.15088 + 0.611320i −0.961903 + 0.273391i
\(6\) 0 0
\(7\) 0.357605 + 1.33460i 0.135162 + 0.504431i 0.999997 + 0.00237552i \(0.000756153\pi\)
−0.864835 + 0.502056i \(0.832577\pi\)
\(8\) 0 0
\(9\) −1.88895 + 2.25116i −0.629650 + 0.750388i
\(10\) 0 0
\(11\) 0.00188012 + 0.00325647i 0.000566878 + 0.000981862i 0.866309 − 0.499509i \(-0.166486\pi\)
−0.865742 + 0.500491i \(0.833153\pi\)
\(12\) 0 0
\(13\) −1.47530 + 3.16379i −0.409174 + 0.877476i 0.588447 + 0.808536i \(0.299740\pi\)
−0.997621 + 0.0689404i \(0.978038\pi\)
\(14\) 0 0
\(15\) −0.418728 + 0.362280i −0.108115 + 0.0935403i
\(16\) 0 0
\(17\) 0.0221536 − 0.253216i 0.00537303 − 0.0614140i −0.993035 − 0.117821i \(-0.962409\pi\)
0.998408 + 0.0564073i \(0.0179645\pi\)
\(18\) 0 0
\(19\) 0.384763 + 4.34188i 0.0882708 + 0.996097i
\(20\) 0 0
\(21\) 0.219918 + 0.262088i 0.0479901 + 0.0571924i
\(22\) 0 0
\(23\) −3.63066 + 2.54221i −0.757045 + 0.530088i −0.887189 − 0.461407i \(-0.847345\pi\)
0.130144 + 0.991495i \(0.458456\pi\)
\(24\) 0 0
\(25\) 4.25258 − 2.62975i 0.850515 − 0.525950i
\(26\) 0 0
\(27\) −0.380604 + 1.42043i −0.0732472 + 0.273362i
\(28\) 0 0
\(29\) 1.17897 + 0.989275i 0.218930 + 0.183704i 0.745656 − 0.666331i \(-0.232136\pi\)
−0.526726 + 0.850035i \(0.676581\pi\)
\(30\) 0 0
\(31\) 0.377920 + 0.218192i 0.0678765 + 0.0391885i 0.533554 − 0.845766i \(-0.320856\pi\)
−0.465678 + 0.884954i \(0.654189\pi\)
\(32\) 0 0
\(33\) 0.000762724 0 0.000534065i 0.000132773 0 9.29688e-5i
\(34\) 0 0
\(35\) −1.58503 − 2.65195i −0.267919 − 0.448262i
\(36\) 0 0
\(37\) 0.496355 + 0.496355i 0.0816002 + 0.0816002i 0.746729 − 0.665129i \(-0.231623\pi\)
−0.665129 + 0.746729i \(0.731623\pi\)
\(38\) 0 0
\(39\) 0.864406i 0.138416i
\(40\) 0 0
\(41\) 2.34618 − 6.44608i 0.366412 − 1.00671i −0.610303 − 0.792168i \(-0.708952\pi\)
0.976715 − 0.214541i \(-0.0688255\pi\)
\(42\) 0 0
\(43\) −3.96540 + 5.66318i −0.604718 + 0.863627i −0.998453 − 0.0555970i \(-0.982294\pi\)
0.393735 + 0.919224i \(0.371183\pi\)
\(44\) 0 0
\(45\) 2.68673 − 5.99673i 0.400513 − 0.893940i
\(46\) 0 0
\(47\) −2.70864 + 0.236975i −0.395096 + 0.0345664i −0.282973 − 0.959128i \(-0.591321\pi\)
−0.112123 + 0.993694i \(0.535765\pi\)
\(48\) 0 0
\(49\) 4.40890 − 2.54548i 0.629843 − 0.363640i
\(50\) 0 0
\(51\) −0.0215271 − 0.0591453i −0.00301440 − 0.00828200i
\(52\) 0 0
\(53\) −3.00243 − 4.28792i −0.412416 − 0.588991i 0.558176 − 0.829723i \(-0.311502\pi\)
−0.970592 + 0.240732i \(0.922613\pi\)
\(54\) 0 0
\(55\) −0.00603466 − 0.00585492i −0.000813714 − 0.000789477i
\(56\) 0 0
\(57\) 0.540722 + 0.934142i 0.0716204 + 0.123730i
\(58\) 0 0
\(59\) 7.08129 − 5.94191i 0.921906 − 0.773571i −0.0524407 − 0.998624i \(-0.516700\pi\)
0.974346 + 0.225053i \(0.0722556\pi\)
\(60\) 0 0
\(61\) −0.899822 + 5.10314i −0.115210 + 0.653391i 0.871435 + 0.490510i \(0.163190\pi\)
−0.986646 + 0.162880i \(0.947922\pi\)
\(62\) 0 0
\(63\) −3.67990 − 1.71597i −0.463624 − 0.216191i
\(64\) 0 0
\(65\) 1.23910 − 7.70680i 0.153692 − 0.955911i
\(66\) 0 0
\(67\) 1.01014 + 11.5460i 0.123409 + 1.41057i 0.765251 + 0.643732i \(0.222615\pi\)
−0.641843 + 0.766836i \(0.721830\pi\)
\(68\) 0 0
\(69\) −0.548754 + 0.950469i −0.0660622 + 0.114423i
\(70\) 0 0
\(71\) −1.85753 + 0.327532i −0.220448 + 0.0388709i −0.282781 − 0.959184i \(-0.591257\pi\)
0.0623331 + 0.998055i \(0.480146\pi\)
\(72\) 0 0
\(73\) −4.92742 − 10.5669i −0.576711 − 1.23676i −0.950926 − 0.309419i \(-0.899866\pi\)
0.374215 − 0.927342i \(-0.377912\pi\)
\(74\) 0 0
\(75\) 0.679164 − 1.03520i 0.0784231 − 0.119534i
\(76\) 0 0
\(77\) −0.00367374 + 0.00367374i −0.000418662 + 0.000418662i
\(78\) 0 0
\(79\) 10.2851 + 3.74347i 1.15716 + 0.421173i 0.848084 − 0.529863i \(-0.177757\pi\)
0.309081 + 0.951036i \(0.399979\pi\)
\(80\) 0 0
\(81\) −1.46766 − 8.32351i −0.163073 − 0.924835i
\(82\) 0 0
\(83\) −1.18134 + 0.316538i −0.129669 + 0.0347446i −0.323070 − 0.946375i \(-0.604715\pi\)
0.193401 + 0.981120i \(0.438048\pi\)
\(84\) 0 0
\(85\) 0.107147 + 0.558181i 0.0116217 + 0.0605433i
\(86\) 0 0
\(87\) 0.368112 + 0.0986353i 0.0394657 + 0.0105748i
\(88\) 0 0
\(89\) 4.35576 − 1.58537i 0.461710 − 0.168049i −0.100683 − 0.994919i \(-0.532103\pi\)
0.562393 + 0.826870i \(0.309881\pi\)
\(90\) 0 0
\(91\) −4.74996 − 0.837546i −0.497931 − 0.0877987i
\(92\) 0 0
\(93\) 0.107647 + 0.00941786i 0.0111624 + 0.000976587i
\(94\) 0 0
\(95\) −3.48186 − 9.10366i −0.357231 − 0.934016i
\(96\) 0 0
\(97\) 17.5189 + 1.53271i 1.77878 + 0.155623i 0.928650 − 0.370956i \(-0.120970\pi\)
0.850126 + 0.526579i \(0.176526\pi\)
\(98\) 0 0
\(99\) −0.0108823 − 0.00191884i −0.00109371 − 0.000192851i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 380.2.bh.a.13.6 ✓ 120
5.2 odd 4 inner 380.2.bh.a.317.5 yes 120
19.3 odd 18 inner 380.2.bh.a.193.5 yes 120
95.22 even 36 inner 380.2.bh.a.117.6 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.bh.a.13.6 ✓ 120 1.1 even 1 trivial
380.2.bh.a.117.6 yes 120 95.22 even 36 inner
380.2.bh.a.193.5 yes 120 19.3 odd 18 inner
380.2.bh.a.317.5 yes 120 5.2 odd 4 inner