Properties

Label 144.16.a.o.1.1
Level $144$
Weight $16$
Character 144.1
Self dual yes
Analytic conductor $205.479$
Analytic rank $0$
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] = [144,16,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, 16); 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, 16, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,314490,0,-2025056] 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: \(205.478647344\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
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)
 
\(f(q)\) \(=\) \(q+314490. q^{5} -2.02506e6 q^{7} +1.10255e8 q^{11} +5.60479e7 q^{13} +1.93010e9 q^{17} -2.16319e9 q^{19} +6.22897e9 q^{23} +6.83864e10 q^{25} -6.47437e10 q^{29} +2.02376e10 q^{31} -6.36860e11 q^{35} +4.88968e11 q^{37} +7.72359e11 q^{41} -1.30677e12 q^{43} +3.35182e12 q^{47} -6.46710e11 q^{49} -9.38781e12 q^{53} +3.46741e13 q^{55} +2.89304e13 q^{59} +4.23931e13 q^{61} +1.76265e13 q^{65} +5.22472e13 q^{67} -2.71945e13 q^{71} -9.16042e13 q^{73} -2.23273e14 q^{77} -6.28821e13 q^{79} -2.23567e14 q^{83} +6.06999e14 q^{85} -5.54199e14 q^{89} -1.13500e14 q^{91} -6.80301e14 q^{95} -1.38887e15 q^{97} +O(q^{100})\)

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
\(5\) 314490. 1.80025 0.900123 − 0.435636i \(-0.143477\pi\)
0.900123 + 0.435636i \(0.143477\pi\)
\(6\) 0 0
\(7\) −2.02506e6 −0.929398 −0.464699 − 0.885469i \(-0.653837\pi\)
−0.464699 + 0.885469i \(0.653837\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.10255e8 1.70590 0.852950 − 0.521993i \(-0.174811\pi\)
0.852950 + 0.521993i \(0.174811\pi\)
\(12\) 0 0
\(13\) 5.60479e7 0.247733 0.123867 − 0.992299i \(-0.460471\pi\)
0.123867 + 0.992299i \(0.460471\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.93010e9 1.14081 0.570406 − 0.821363i \(-0.306786\pi\)
0.570406 + 0.821363i \(0.306786\pi\)
\(18\) 0 0
\(19\) −2.16319e9 −0.555191 −0.277595 − 0.960698i \(-0.589537\pi\)
−0.277595 + 0.960698i \(0.589537\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.22897e9 0.381468 0.190734 − 0.981642i \(-0.438913\pi\)
0.190734 + 0.981642i \(0.438913\pi\)
\(24\) 0 0
\(25\) 6.83864e10 2.24088
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −6.47437e10 −0.696968 −0.348484 − 0.937315i \(-0.613303\pi\)
−0.348484 + 0.937315i \(0.613303\pi\)
\(30\) 0 0
\(31\) 2.02376e10 0.132113 0.0660567 − 0.997816i \(-0.478958\pi\)
0.0660567 + 0.997816i \(0.478958\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −6.36860e11 −1.67314
\(36\) 0 0
\(37\) 4.88968e11 0.846773 0.423387 − 0.905949i \(-0.360841\pi\)
0.423387 + 0.905949i \(0.360841\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.72359e11 0.619356 0.309678 − 0.950841i \(-0.399779\pi\)
0.309678 + 0.950841i \(0.399779\pi\)
\(42\) 0 0
\(43\) −1.30677e12 −0.733136 −0.366568 − 0.930391i \(-0.619467\pi\)
−0.366568 + 0.930391i \(0.619467\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.35182e12 0.965044 0.482522 − 0.875884i \(-0.339721\pi\)
0.482522 + 0.875884i \(0.339721\pi\)
\(48\) 0 0
\(49\) −6.46710e11 −0.136219
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.38781e12 −1.09773 −0.548865 − 0.835911i \(-0.684940\pi\)
−0.548865 + 0.835911i \(0.684940\pi\)
\(54\) 0 0
\(55\) 3.46741e13 3.07104
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.89304e13 1.51343 0.756717 − 0.653742i \(-0.226802\pi\)
0.756717 + 0.653742i \(0.226802\pi\)
\(60\) 0 0
\(61\) 4.23931e13 1.72711 0.863557 − 0.504251i \(-0.168231\pi\)
0.863557 + 0.504251i \(0.168231\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.76265e13 0.445980
\(66\) 0 0
\(67\) 5.22472e13 1.05318 0.526590 − 0.850120i \(-0.323470\pi\)
0.526590 + 0.850120i \(0.323470\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.71945e13 −0.354849 −0.177425 − 0.984134i \(-0.556777\pi\)
−0.177425 + 0.984134i \(0.556777\pi\)
\(72\) 0 0
\(73\) −9.16042e13 −0.970496 −0.485248 − 0.874376i \(-0.661271\pi\)
−0.485248 + 0.874376i \(0.661271\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.23273e14 −1.58546
\(78\) 0 0
\(79\) −6.28821e13 −0.368404 −0.184202 − 0.982888i \(-0.558970\pi\)
−0.184202 + 0.982888i \(0.558970\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.23567e14 −0.904321 −0.452161 − 0.891937i \(-0.649347\pi\)
−0.452161 + 0.891937i \(0.649347\pi\)
\(84\) 0 0
\(85\) 6.06999e14 2.05374
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.54199e14 −1.32813 −0.664065 − 0.747675i \(-0.731170\pi\)
−0.664065 + 0.747675i \(0.731170\pi\)
\(90\) 0 0
\(91\) −1.13500e14 −0.230243
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −6.80301e14 −0.999480
\(96\) 0 0
\(97\) −1.38887e15 −1.74531 −0.872657 − 0.488333i \(-0.837605\pi\)
−0.872657 + 0.488333i \(0.837605\pi\)
\(98\) 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 144.16.a.o.1.1 1
3.2 odd 2 48.16.a.c.1.1 1
4.3 odd 2 18.16.a.f.1.1 1
12.11 even 2 6.16.a.a.1.1 ✓ 1
60.23 odd 4 150.16.c.i.49.2 2
60.47 odd 4 150.16.c.i.49.1 2
60.59 even 2 150.16.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.16.a.a.1.1 ✓ 1 12.11 even 2
18.16.a.f.1.1 1 4.3 odd 2
48.16.a.c.1.1 1 3.2 odd 2
144.16.a.o.1.1 1 1.1 even 1 trivial
150.16.a.h.1.1 1 60.59 even 2
150.16.c.i.49.1 2 60.47 odd 4
150.16.c.i.49.2 2 60.23 odd 4