Properties

Label 432.8.a.g.1.1
Level $432$
Weight $8$
Character 432.1
Self dual yes
Analytic conductor $134.950$
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] = [432,8,Mod(1,432)] 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("432.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,120,0,-377] 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: \(134.950331009\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 54)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 432.1

$q$-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)
 
\(f(q)\) \(=\) \(q+120.000 q^{5} -377.000 q^{7} -600.000 q^{11} +5369.00 q^{13} +12168.0 q^{17} -16211.0 q^{19} -106392. q^{23} -63725.0 q^{25} +177216. q^{29} +268060. q^{31} -45240.0 q^{35} +114959. q^{37} -112128. q^{41} +115048. q^{43} -561336. q^{47} -681414. q^{49} -1.78776e6 q^{53} -72000.0 q^{55} +1.78634e6 q^{59} -1.30684e6 q^{61} +644280. q^{65} +2.01382e6 q^{67} +4.06094e6 q^{71} -3.85064e6 q^{73} +226200. q^{77} -1.03723e6 q^{79} -9.20357e6 q^{83} +1.46016e6 q^{85} -1.28930e6 q^{89} -2.02411e6 q^{91} -1.94532e6 q^{95} +8.55588e6 q^{97} +O(q^{100})\)

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
\(5\) 120.000 0.429325 0.214663 − 0.976688i \(-0.431135\pi\)
0.214663 + 0.976688i \(0.431135\pi\)
\(6\) 0 0
\(7\) −377.000 −0.415430 −0.207715 − 0.978189i \(-0.566603\pi\)
−0.207715 + 0.978189i \(0.566603\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −600.000 −0.135918 −0.0679590 − 0.997688i \(-0.521649\pi\)
−0.0679590 + 0.997688i \(0.521649\pi\)
\(12\) 0 0
\(13\) 5369.00 0.677785 0.338892 − 0.940825i \(-0.389948\pi\)
0.338892 + 0.940825i \(0.389948\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 12168.0 0.600687 0.300343 − 0.953831i \(-0.402899\pi\)
0.300343 + 0.953831i \(0.402899\pi\)
\(18\) 0 0
\(19\) −16211.0 −0.542216 −0.271108 − 0.962549i \(-0.587390\pi\)
−0.271108 + 0.962549i \(0.587390\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −106392. −1.82331 −0.911657 − 0.410952i \(-0.865197\pi\)
−0.911657 + 0.410952i \(0.865197\pi\)
\(24\) 0 0
\(25\) −63725.0 −0.815680
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 177216. 1.34930 0.674652 − 0.738136i \(-0.264294\pi\)
0.674652 + 0.738136i \(0.264294\pi\)
\(30\) 0 0
\(31\) 268060. 1.61609 0.808046 − 0.589119i \(-0.200525\pi\)
0.808046 + 0.589119i \(0.200525\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −45240.0 −0.178355
\(36\) 0 0
\(37\) 114959. 0.373110 0.186555 − 0.982445i \(-0.440268\pi\)
0.186555 + 0.982445i \(0.440268\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −112128. −0.254080 −0.127040 − 0.991898i \(-0.540548\pi\)
−0.127040 + 0.991898i \(0.540548\pi\)
\(42\) 0 0
\(43\) 115048. 0.220668 0.110334 − 0.993895i \(-0.464808\pi\)
0.110334 + 0.993895i \(0.464808\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −561336. −0.788643 −0.394321 − 0.918973i \(-0.629020\pi\)
−0.394321 + 0.918973i \(0.629020\pi\)
\(48\) 0 0
\(49\) −681414. −0.827418
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.78776e6 −1.64947 −0.824734 − 0.565521i \(-0.808675\pi\)
−0.824734 + 0.565521i \(0.808675\pi\)
\(54\) 0 0
\(55\) −72000.0 −0.0583530
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.78634e6 1.13236 0.566178 − 0.824283i \(-0.308421\pi\)
0.566178 + 0.824283i \(0.308421\pi\)
\(60\) 0 0
\(61\) −1.30684e6 −0.737169 −0.368584 − 0.929594i \(-0.620157\pi\)
−0.368584 + 0.929594i \(0.620157\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 644280. 0.290990
\(66\) 0 0
\(67\) 2.01382e6 0.818009 0.409005 − 0.912532i \(-0.365876\pi\)
0.409005 + 0.912532i \(0.365876\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.06094e6 1.34655 0.673275 − 0.739392i \(-0.264887\pi\)
0.673275 + 0.739392i \(0.264887\pi\)
\(72\) 0 0
\(73\) −3.85064e6 −1.15852 −0.579259 − 0.815144i \(-0.696658\pi\)
−0.579259 + 0.815144i \(0.696658\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 226200. 0.0564644
\(78\) 0 0
\(79\) −1.03723e6 −0.236690 −0.118345 − 0.992973i \(-0.537759\pi\)
−0.118345 + 0.992973i \(0.537759\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.20357e6 −1.76678 −0.883391 − 0.468637i \(-0.844745\pi\)
−0.883391 + 0.468637i \(0.844745\pi\)
\(84\) 0 0
\(85\) 1.46016e6 0.257890
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.28930e6 −0.193861 −0.0969305 − 0.995291i \(-0.530902\pi\)
−0.0969305 + 0.995291i \(0.530902\pi\)
\(90\) 0 0
\(91\) −2.02411e6 −0.281572
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.94532e6 −0.232787
\(96\) 0 0
\(97\) 8.55588e6 0.951840 0.475920 − 0.879489i \(-0.342115\pi\)
0.475920 + 0.879489i \(0.342115\pi\)
\(98\) 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 432.8.a.g.1.1 1
3.2 odd 2 432.8.a.b.1.1 1
4.3 odd 2 54.8.a.f.1.1 yes 1
12.11 even 2 54.8.a.a.1.1 ✓ 1
36.7 odd 6 162.8.c.b.109.1 2
36.11 even 6 162.8.c.k.109.1 2
36.23 even 6 162.8.c.k.55.1 2
36.31 odd 6 162.8.c.b.55.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
54.8.a.a.1.1 ✓ 1 12.11 even 2
54.8.a.f.1.1 yes 1 4.3 odd 2
162.8.c.b.55.1 2 36.31 odd 6
162.8.c.b.109.1 2 36.7 odd 6
162.8.c.k.55.1 2 36.23 even 6
162.8.c.k.109.1 2 36.11 even 6
432.8.a.b.1.1 1 3.2 odd 2
432.8.a.g.1.1 1 1.1 even 1 trivial