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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [338,3,Mod(19,338)] 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("338.19"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(338, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 338.f (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,-12,8,0,-16,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.20983293538\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{2} + 8 q^{3} - 16 q^{5} + 8 q^{6} + 24 q^{7} - 48 q^{8} - 28 q^{9} + 16 q^{11} - 96 q^{14} + 116 q^{15} + 48 q^{16} + 56 q^{18} + 92 q^{19} - 16 q^{20} + 256 q^{21} + 32 q^{22} - 16 q^{24}+ \cdots - 224 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 0.366025 + 1.36603i −2.20695 + 3.82256i −1.73205 + 1.00000i −4.82072 + 4.82072i −6.02951 − 1.61560i 2.96642 − 11.0708i −2.00000 − 2.00000i −5.24129 − 9.07818i −8.34973 − 4.82072i
19.2 0.366025 + 1.36603i −1.16612 + 2.01978i −1.73205 + 1.00000i −6.63447 + 6.63447i −3.18590 − 0.853660i −2.33791 + 8.72520i −2.00000 − 2.00000i 1.78032 + 3.08361i −11.4912 − 6.63447i
19.3 0.366025 + 1.36603i 0.364183 − 0.630784i −1.73205 + 1.00000i −0.573283 + 0.573283i 0.994967 + 0.266601i 0.628480 − 2.34552i −2.00000 − 2.00000i 4.23474 + 7.33479i −0.992956 − 0.573283i
19.4 0.366025 + 1.36603i 0.631417 − 1.09365i −1.73205 + 1.00000i 5.71353 − 5.71353i 1.72506 + 0.462230i −2.13989 + 7.98619i −2.00000 − 2.00000i 3.70262 + 6.41313i 9.89612 + 5.71353i
19.5 0.366025 + 1.36603i 1.61556 − 2.79824i −1.73205 + 1.00000i −0.725610 + 0.725610i 4.41380 + 1.18267i −2.54949 + 9.51482i −2.00000 − 2.00000i −0.720083 − 1.24722i −1.25679 − 0.725610i
19.6 0.366025 + 1.36603i 2.76191 − 4.78377i −1.73205 + 1.00000i 3.04055 − 3.04055i 7.54568 + 2.02186i −0.959916 + 3.58245i −2.00000 − 2.00000i −10.7563 − 18.6305i 5.26639 + 3.04055i
89.1 0.366025 − 1.36603i −2.20695 − 3.82256i −1.73205 − 1.00000i −4.82072 − 4.82072i −6.02951 + 1.61560i 2.96642 + 11.0708i −2.00000 + 2.00000i −5.24129 + 9.07818i −8.34973 + 4.82072i
89.2 0.366025 − 1.36603i −1.16612 − 2.01978i −1.73205 − 1.00000i −6.63447 − 6.63447i −3.18590 + 0.853660i −2.33791 − 8.72520i −2.00000 + 2.00000i 1.78032 − 3.08361i −11.4912 + 6.63447i
89.3 0.366025 − 1.36603i 0.364183 + 0.630784i −1.73205 − 1.00000i −0.573283 − 0.573283i 0.994967 − 0.266601i 0.628480 + 2.34552i −2.00000 + 2.00000i 4.23474 − 7.33479i −0.992956 + 0.573283i
89.4 0.366025 − 1.36603i 0.631417 + 1.09365i −1.73205 − 1.00000i 5.71353 + 5.71353i 1.72506 − 0.462230i −2.13989 − 7.98619i −2.00000 + 2.00000i 3.70262 − 6.41313i 9.89612 − 5.71353i
89.5 0.366025 − 1.36603i 1.61556 + 2.79824i −1.73205 − 1.00000i −0.725610 − 0.725610i 4.41380 − 1.18267i −2.54949 − 9.51482i −2.00000 + 2.00000i −0.720083 + 1.24722i −1.25679 + 0.725610i
89.6 0.366025 − 1.36603i 2.76191 + 4.78377i −1.73205 − 1.00000i 3.04055 + 3.04055i 7.54568 − 2.02186i −0.959916 − 3.58245i −2.00000 + 2.00000i −10.7563 + 18.6305i 5.26639 − 3.04055i
249.1 −1.36603 − 0.366025i −2.20695 − 3.82256i 1.73205 + 1.00000i −4.82072 + 4.82072i 1.61560 + 6.02951i −11.0708 + 2.96642i −2.00000 − 2.00000i −5.24129 + 9.07818i 8.34973 − 4.82072i
249.2 −1.36603 − 0.366025i −1.16612 − 2.01978i 1.73205 + 1.00000i −6.63447 + 6.63447i 0.853660 + 3.18590i 8.72520 − 2.33791i −2.00000 − 2.00000i 1.78032 − 3.08361i 11.4912 − 6.63447i
249.3 −1.36603 − 0.366025i 0.364183 + 0.630784i 1.73205 + 1.00000i −0.573283 + 0.573283i −0.266601 − 0.994967i −2.34552 + 0.628480i −2.00000 − 2.00000i 4.23474 − 7.33479i 0.992956 − 0.573283i
249.4 −1.36603 − 0.366025i 0.631417 + 1.09365i 1.73205 + 1.00000i 5.71353 − 5.71353i −0.462230 − 1.72506i 7.98619 − 2.13989i −2.00000 − 2.00000i 3.70262 − 6.41313i −9.89612 + 5.71353i
249.5 −1.36603 − 0.366025i 1.61556 + 2.79824i 1.73205 + 1.00000i −0.725610 + 0.725610i −1.18267 − 4.41380i 9.51482 − 2.54949i −2.00000 − 2.00000i −0.720083 + 1.24722i 1.25679 − 0.725610i
249.6 −1.36603 − 0.366025i 2.76191 + 4.78377i 1.73205 + 1.00000i 3.04055 − 3.04055i −2.02186 − 7.54568i 3.58245 − 0.959916i −2.00000 − 2.00000i −10.7563 + 18.6305i −5.26639 + 3.04055i
319.1 −1.36603 + 0.366025i −2.20695 + 3.82256i 1.73205 − 1.00000i −4.82072 − 4.82072i 1.61560 − 6.02951i −11.0708 − 2.96642i −2.00000 + 2.00000i −5.24129 − 9.07818i 8.34973 + 4.82072i
319.2 −1.36603 + 0.366025i −1.16612 + 2.01978i 1.73205 − 1.00000i −6.63447 − 6.63447i 0.853660 − 3.18590i 8.72520 + 2.33791i −2.00000 + 2.00000i 1.78032 + 3.08361i 11.4912 + 6.63447i
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 19.6
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner
13.d odd 4 1 inner
13.f odd 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 338.3.f.k 24
13.b even 2 1 338.3.f.l 24
13.c even 3 1 338.3.d.i yes 12
13.c even 3 1 inner 338.3.f.k 24
13.d odd 4 1 inner 338.3.f.k 24
13.d odd 4 1 338.3.f.l 24
13.e even 6 1 338.3.d.h ✓ 12
13.e even 6 1 338.3.f.l 24
13.f odd 12 1 338.3.d.h ✓ 12
13.f odd 12 1 338.3.d.i yes 12
13.f odd 12 1 inner 338.3.f.k 24
13.f odd 12 1 338.3.f.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
338.3.d.h ✓ 12 13.e even 6 1
338.3.d.h ✓ 12 13.f odd 12 1
338.3.d.i yes 12 13.c even 3 1
338.3.d.i yes 12 13.f odd 12 1
338.3.f.k 24 1.a even 1 1 trivial
338.3.f.k 24 13.c even 3 1 inner
338.3.f.k 24 13.d odd 4 1 inner
338.3.f.k 24 13.f odd 12 1 inner
338.3.f.l 24 13.b even 2 1
338.3.f.l 24 13.d odd 4 1
338.3.f.l 24 13.e even 6 1
338.3.f.l 24 13.f odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(338, [\chi])\):

\( T_{3}^{12} - 4 T_{3}^{11} + 42 T_{3}^{10} - 76 T_{3}^{9} + 941 T_{3}^{8} - 1918 T_{3}^{7} + 9556 T_{3}^{6} + \cdots + 28561 \) Copy content Toggle raw display
\( T_{5}^{12} + 8 T_{5}^{11} + 32 T_{5}^{10} - 168 T_{5}^{9} + 5402 T_{5}^{8} + 35656 T_{5}^{7} + \cdots + 3418801 \) Copy content Toggle raw display
\( T_{7}^{24} - 24 T_{7}^{23} + 288 T_{7}^{22} - 3048 T_{7}^{21} + 12567 T_{7}^{20} + 176488 T_{7}^{19} + \cdots + 33\!\cdots\!61 \) Copy content Toggle raw display