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Properties

Label 112.12.i.e
Level $112$
Weight $12$
Character orbit 112.i
Analytic conductor $86.054$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [112,12,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,-463] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(86.0544362227\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 56)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 463 q^{3} - 1277 q^{5} + 20812 q^{7} - 683418 q^{9} - 42963 q^{11} - 4788164 q^{13} - 451934 q^{15} + 308351 q^{17} - 9407833 q^{19} + 13339137 q^{21} + 14617071 q^{23} - 98205820 q^{25} + 23955914 q^{27}+ \cdots - 422825405532 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1 0 −382.344 − 662.239i 0 2309.34 − 3999.90i 0 −44457.2 − 937.995i 0 −203800. + 352992.i 0
65.2 0 −296.161 − 512.966i 0 −660.420 + 1143.88i 0 44354.6 + 3162.21i 0 −86849.6 + 150428.i 0
65.3 0 −243.403 − 421.586i 0 −5334.80 + 9240.15i 0 11449.5 + 42967.9i 0 −29916.5 + 51817.0i 0
65.4 0 −201.532 − 349.063i 0 6257.04 − 10837.5i 0 37201.2 − 24359.8i 0 7343.58 − 12719.4i 0
65.5 0 −177.531 − 307.492i 0 −2756.02 + 4773.57i 0 −33285.7 − 29485.4i 0 25539.2 − 44235.2i 0
65.6 0 16.3265 + 28.2783i 0 3033.50 − 5254.17i 0 −13812.1 + 42267.6i 0 88040.4 − 152490.i 0
65.7 0 47.0112 + 81.4257i 0 37.2961 − 64.5987i 0 −7256.54 − 43871.1i 0 84153.4 − 145758.i 0
65.8 0 87.2788 + 151.171i 0 −4044.30 + 7004.93i 0 41808.6 − 15144.9i 0 73338.3 − 127026.i 0
65.9 0 278.915 + 483.094i 0 −5115.99 + 8861.15i 0 −40990.2 + 17237.4i 0 −67013.2 + 116070.i 0
65.10 0 313.471 + 542.948i 0 555.329 − 961.858i 0 42749.4 + 12240.1i 0 −107955. + 186984.i 0
65.11 0 326.468 + 565.459i 0 5080.53 − 8799.74i 0 −27355.3 − 35057.3i 0 −124589. + 215795.i 0
81.1 0 −382.344 + 662.239i 0 2309.34 + 3999.90i 0 −44457.2 + 937.995i 0 −203800. − 352992.i 0
81.2 0 −296.161 + 512.966i 0 −660.420 − 1143.88i 0 44354.6 − 3162.21i 0 −86849.6 − 150428.i 0
81.3 0 −243.403 + 421.586i 0 −5334.80 − 9240.15i 0 11449.5 − 42967.9i 0 −29916.5 − 51817.0i 0
81.4 0 −201.532 + 349.063i 0 6257.04 + 10837.5i 0 37201.2 + 24359.8i 0 7343.58 + 12719.4i 0
81.5 0 −177.531 + 307.492i 0 −2756.02 − 4773.57i 0 −33285.7 + 29485.4i 0 25539.2 + 44235.2i 0
81.6 0 16.3265 − 28.2783i 0 3033.50 + 5254.17i 0 −13812.1 − 42267.6i 0 88040.4 + 152490.i 0
81.7 0 47.0112 − 81.4257i 0 37.2961 + 64.5987i 0 −7256.54 + 43871.1i 0 84153.4 + 145758.i 0
81.8 0 87.2788 − 151.171i 0 −4044.30 − 7004.93i 0 41808.6 + 15144.9i 0 73338.3 + 127026.i 0
81.9 0 278.915 − 483.094i 0 −5115.99 − 8861.15i 0 −40990.2 − 17237.4i 0 −67013.2 − 116070.i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 65.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 112.12.i.e 22
4.b odd 2 1 56.12.i.b ✓ 22
7.c even 3 1 inner 112.12.i.e 22
28.g odd 6 1 56.12.i.b ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.12.i.b ✓ 22 4.b odd 2 1
56.12.i.b ✓ 22 28.g odd 6 1
112.12.i.e 22 1.a even 1 1 trivial
112.12.i.e 22 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{22} + 463 T_{3}^{21} + 1423202 T_{3}^{20} + 508514191 T_{3}^{19} + 1203980330744 T_{3}^{18} + \cdots + 14\!\cdots\!81 \) acting on \(S_{12}^{\mathrm{new}}(112, [\chi])\). Copy content Toggle raw display