Properties

Label 144.14.a.k.1.1
Level $144$
Weight $14$
Character 144.1
Self dual yes
Analytic conductor $154.413$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,14,Mod(1,144)] 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("144.1"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,30210,0,-235088] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(154.412537691\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 144.1

$q$-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)
 
\(f(q)\) \(=\) \(q+30210.0 q^{5} -235088. q^{7} -1.11829e7 q^{11} +8.04961e6 q^{13} +1.17495e8 q^{17} +2.14061e8 q^{19} +8.30556e8 q^{23} -3.08059e8 q^{25} +1.25240e9 q^{29} -6.15935e9 q^{31} -7.10201e9 q^{35} -5.49819e9 q^{37} +4.67869e9 q^{41} -7.11501e9 q^{43} -2.95288e10 q^{47} -4.16226e10 q^{49} +2.04125e11 q^{53} -3.37836e11 q^{55} -2.99098e10 q^{59} -1.34392e11 q^{61} +2.43179e11 q^{65} -3.48519e11 q^{67} +1.31434e12 q^{71} -1.17888e12 q^{73} +2.62897e12 q^{77} +1.07242e12 q^{79} +1.12403e12 q^{83} +3.54951e12 q^{85} -2.23561e12 q^{89} -1.89237e12 q^{91} +6.46679e12 q^{95} -1.42153e13 q^{97} +O(q^{100})\)

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 0
\(4\) 0 0
\(5\) 30210.0 0.864661 0.432330 − 0.901715i \(-0.357691\pi\)
0.432330 + 0.901715i \(0.357691\pi\)
\(6\) 0 0
\(7\) −235088. −0.755254 −0.377627 − 0.925958i \(-0.623260\pi\)
−0.377627 + 0.925958i \(0.623260\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.11829e7 −1.90328 −0.951639 − 0.307218i \(-0.900602\pi\)
−0.951639 + 0.307218i \(0.900602\pi\)
\(12\) 0 0
\(13\) 8.04961e6 0.462534 0.231267 − 0.972890i \(-0.425713\pi\)
0.231267 + 0.972890i \(0.425713\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.17495e8 1.18059 0.590296 − 0.807187i \(-0.299011\pi\)
0.590296 + 0.807187i \(0.299011\pi\)
\(18\) 0 0
\(19\) 2.14061e8 1.04385 0.521927 − 0.852990i \(-0.325213\pi\)
0.521927 + 0.852990i \(0.325213\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.30556e8 1.16987 0.584935 − 0.811080i \(-0.301120\pi\)
0.584935 + 0.811080i \(0.301120\pi\)
\(24\) 0 0
\(25\) −3.08059e8 −0.252362
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.25240e9 0.390981 0.195491 − 0.980706i \(-0.437370\pi\)
0.195491 + 0.980706i \(0.437370\pi\)
\(30\) 0 0
\(31\) −6.15935e9 −1.24648 −0.623238 − 0.782032i \(-0.714183\pi\)
−0.623238 + 0.782032i \(0.714183\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.10201e9 −0.653039
\(36\) 0 0
\(37\) −5.49819e9 −0.352297 −0.176148 − 0.984364i \(-0.556364\pi\)
−0.176148 + 0.984364i \(0.556364\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.67869e9 0.153826 0.0769129 − 0.997038i \(-0.475494\pi\)
0.0769129 + 0.997038i \(0.475494\pi\)
\(42\) 0 0
\(43\) −7.11501e9 −0.171645 −0.0858224 − 0.996310i \(-0.527352\pi\)
−0.0858224 + 0.996310i \(0.527352\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.95288e10 −0.399585 −0.199793 − 0.979838i \(-0.564027\pi\)
−0.199793 + 0.979838i \(0.564027\pi\)
\(48\) 0 0
\(49\) −4.16226e10 −0.429591
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.04125e11 1.26504 0.632518 − 0.774545i \(-0.282021\pi\)
0.632518 + 0.774545i \(0.282021\pi\)
\(54\) 0 0
\(55\) −3.37836e11 −1.64569
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.99098e10 −0.0923157 −0.0461579 − 0.998934i \(-0.514698\pi\)
−0.0461579 + 0.998934i \(0.514698\pi\)
\(60\) 0 0
\(61\) −1.34392e11 −0.333987 −0.166993 − 0.985958i \(-0.553406\pi\)
−0.166993 + 0.985958i \(0.553406\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.43179e11 0.399935
\(66\) 0 0
\(67\) −3.48519e11 −0.470695 −0.235348 − 0.971911i \(-0.575623\pi\)
−0.235348 + 0.971911i \(0.575623\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.31434e12 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(72\) 0 0
\(73\) −1.17888e12 −0.911737 −0.455868 − 0.890047i \(-0.650671\pi\)
−0.455868 + 0.890047i \(0.650671\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.62897e12 1.43746
\(78\) 0 0
\(79\) 1.07242e12 0.496351 0.248176 − 0.968715i \(-0.420169\pi\)
0.248176 + 0.968715i \(0.420169\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.12403e12 0.377371 0.188685 − 0.982038i \(-0.439577\pi\)
0.188685 + 0.982038i \(0.439577\pi\)
\(84\) 0 0
\(85\) 3.54951e12 1.02081
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.23561e12 −0.476827 −0.238414 − 0.971164i \(-0.576627\pi\)
−0.238414 + 0.971164i \(0.576627\pi\)
\(90\) 0 0
\(91\) −1.89237e12 −0.349330
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.46679e12 0.902580
\(96\) 0 0
\(97\) −1.42153e13 −1.73276 −0.866380 − 0.499385i \(-0.833559\pi\)
−0.866380 + 0.499385i \(0.833559\pi\)
\(98\) 0 0
\(99\) 0 0
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 144.14.a.k.1.1 1
3.2 odd 2 48.14.a.c.1.1 1
4.3 odd 2 9.14.a.a.1.1 1
12.11 even 2 3.14.a.a.1.1 ✓ 1
24.5 odd 2 192.14.a.e.1.1 1
24.11 even 2 192.14.a.j.1.1 1
60.23 odd 4 75.14.b.b.49.2 2
60.47 odd 4 75.14.b.b.49.1 2
60.59 even 2 75.14.a.a.1.1 1
84.83 odd 2 147.14.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.14.a.a.1.1 ✓ 1 12.11 even 2
9.14.a.a.1.1 1 4.3 odd 2
48.14.a.c.1.1 1 3.2 odd 2
75.14.a.a.1.1 1 60.59 even 2
75.14.b.b.49.1 2 60.47 odd 4
75.14.b.b.49.2 2 60.23 odd 4
144.14.a.k.1.1 1 1.1 even 1 trivial
147.14.a.a.1.1 1 84.83 odd 2
192.14.a.e.1.1 1 24.5 odd 2
192.14.a.j.1.1 1 24.11 even 2