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Properties

Label 126.3.q.a
Level $126$
Weight $3$
Character orbit 126.q
Analytic conductor $3.433$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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] = [126,3,Mod(29,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.29"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 126.q (of order \(6\), degree \(2\), 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: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 24 q^{4} + 36 q^{5} + 8 q^{6} - 32 q^{9} - 24 q^{12} - 44 q^{15} - 48 q^{16} + 48 q^{18} + 24 q^{19} + 72 q^{20} + 28 q^{21} + 24 q^{22} - 72 q^{23} - 16 q^{24} + 72 q^{25} - 108 q^{29} - 56 q^{30}+ \cdots - 440 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −1.22474 − 0.707107i −2.92142 + 0.682149i 1.00000 + 1.73205i 2.76866 − 1.59849i 4.06034 + 1.23029i −1.32288 + 2.29129i − 2.82843i 8.06934 − 3.98568i −4.52120
29.2 −1.22474 − 0.707107i −1.47915 − 2.61001i 1.00000 + 1.73205i −3.47352 + 2.00544i −0.0339773 + 4.24250i −1.32288 + 2.29129i − 2.82843i −4.62426 + 7.72115i 5.67224
29.3 −1.22474 − 0.707107i −1.06149 − 2.80593i 1.00000 + 1.73205i 6.70848 − 3.87314i −0.684038 + 4.18713i 1.32288 − 2.29129i − 2.82843i −6.74648 + 5.95693i −10.9549
29.4 −1.22474 − 0.707107i −0.392589 + 2.97420i 1.00000 + 1.73205i −4.50944 + 2.60353i 2.58390 − 3.36504i 1.32288 − 2.29129i − 2.82843i −8.69175 − 2.33528i 7.36389
29.5 −1.22474 − 0.707107i 0.530065 + 2.95280i 1.00000 + 1.73205i 5.20486 − 3.00503i 1.43875 − 3.99124i −1.32288 + 2.29129i − 2.82843i −8.43806 + 3.13035i −8.49950
29.6 −1.22474 − 0.707107i 2.87508 + 0.856672i 1.00000 + 1.73205i 2.30097 − 1.32846i −2.91549 − 3.08220i 1.32288 − 2.29129i − 2.82843i 7.53223 + 4.92601i −3.75747
29.7 1.22474 + 0.707107i −2.99983 + 0.0321734i 1.00000 + 1.73205i 7.44329 − 4.29739i −3.69677 − 2.08179i −1.32288 + 2.29129i 2.82843i 8.99793 − 0.193030i 12.1548
29.8 1.22474 + 0.707107i −1.41155 + 2.64717i 1.00000 + 1.73205i −3.60855 + 2.08340i −3.60062 + 2.24400i −1.32288 + 2.29129i 2.82843i −5.01506 − 7.47323i −5.89274
29.9 1.22474 + 0.707107i 0.880705 + 2.86781i 1.00000 + 1.73205i 7.45939 − 4.30668i −0.949211 + 4.13509i 1.32288 − 2.29129i 2.82843i −7.44872 + 5.05140i 12.1811
29.10 1.22474 + 0.707107i 1.12701 − 2.78026i 1.00000 + 1.73205i 2.56117 − 1.47869i 3.34624 − 2.60819i 1.32288 − 2.29129i 2.82843i −6.45970 − 6.26676i 4.18237
29.11 1.22474 + 0.707107i 1.86278 + 2.35160i 1.00000 + 1.73205i −5.52056 + 3.18729i 0.618597 + 4.19730i 1.32288 − 2.29129i 2.82843i −2.06009 + 8.76105i −9.01503
29.12 1.22474 + 0.707107i 2.99037 − 0.240190i 1.00000 + 1.73205i 0.665254 − 0.384085i 3.83228 + 1.82034i −1.32288 + 2.29129i 2.82843i 8.88462 − 1.43651i 1.08636
113.1 −1.22474 + 0.707107i −2.92142 − 0.682149i 1.00000 − 1.73205i 2.76866 + 1.59849i 4.06034 − 1.23029i −1.32288 − 2.29129i 2.82843i 8.06934 + 3.98568i −4.52120
113.2 −1.22474 + 0.707107i −1.47915 + 2.61001i 1.00000 − 1.73205i −3.47352 − 2.00544i −0.0339773 − 4.24250i −1.32288 − 2.29129i 2.82843i −4.62426 − 7.72115i 5.67224
113.3 −1.22474 + 0.707107i −1.06149 + 2.80593i 1.00000 − 1.73205i 6.70848 + 3.87314i −0.684038 − 4.18713i 1.32288 + 2.29129i 2.82843i −6.74648 − 5.95693i −10.9549
113.4 −1.22474 + 0.707107i −0.392589 − 2.97420i 1.00000 − 1.73205i −4.50944 − 2.60353i 2.58390 + 3.36504i 1.32288 + 2.29129i 2.82843i −8.69175 + 2.33528i 7.36389
113.5 −1.22474 + 0.707107i 0.530065 − 2.95280i 1.00000 − 1.73205i 5.20486 + 3.00503i 1.43875 + 3.99124i −1.32288 − 2.29129i 2.82843i −8.43806 − 3.13035i −8.49950
113.6 −1.22474 + 0.707107i 2.87508 − 0.856672i 1.00000 − 1.73205i 2.30097 + 1.32846i −2.91549 + 3.08220i 1.32288 + 2.29129i 2.82843i 7.53223 − 4.92601i −3.75747
113.7 1.22474 − 0.707107i −2.99983 − 0.0321734i 1.00000 − 1.73205i 7.44329 + 4.29739i −3.69677 + 2.08179i −1.32288 − 2.29129i − 2.82843i 8.99793 + 0.193030i 12.1548
113.8 1.22474 − 0.707107i −1.41155 − 2.64717i 1.00000 − 1.73205i −3.60855 − 2.08340i −3.60062 − 2.24400i −1.32288 − 2.29129i − 2.82843i −5.01506 + 7.47323i −5.89274
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.3.q.a ✓ 24
3.b odd 2 1 378.3.q.a 24
9.c even 3 1 378.3.q.a 24
9.c even 3 1 1134.3.b.c 24
9.d odd 6 1 inner 126.3.q.a ✓ 24
9.d odd 6 1 1134.3.b.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.3.q.a ✓ 24 1.a even 1 1 trivial
126.3.q.a ✓ 24 9.d odd 6 1 inner
378.3.q.a 24 3.b odd 2 1
378.3.q.a 24 9.c even 3 1
1134.3.b.c 24 9.c even 3 1
1134.3.b.c 24 9.d odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(126, [\chi])\).