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Properties

Label 104.10.f.a
Level $104$
Weight $10$
Character orbit 104.f
Analytic conductor $53.564$
Analytic rank $0$
Dimension $32$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [104,10,Mod(25,104)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(104, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("104.25"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 104 = 2^{3} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 104.f (of order \(2\), degree \(1\), 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: \(53.5637269610\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 162 q^{3} + 223074 q^{9} + 66270 q^{13} - 487902 q^{17} + 3171556 q^{23} - 13526722 q^{25} - 3694974 q^{27} + 8833508 q^{29} - 8281126 q^{35} - 12056860 q^{39} + 89959038 q^{43} - 172344874 q^{49}+ \cdots + 1741143356 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −258.209 0 − 2361.50i 0 − 3336.28i 0 46988.7 0
25.2 0 −258.209 0 2361.50i 0 3336.28i 0 46988.7 0
25.3 0 −241.055 0 999.605i 0 3393.35i 0 38424.4 0
25.4 0 −241.055 0 − 999.605i 0 − 3393.35i 0 38424.4 0
25.5 0 −182.615 0 909.303i 0 3898.48i 0 13665.4 0
25.6 0 −182.615 0 − 909.303i 0 − 3898.48i 0 13665.4 0
25.7 0 −178.824 0 − 872.510i 0 12429.2i 0 12295.0 0
25.8 0 −178.824 0 872.510i 0 − 12429.2i 0 12295.0 0
25.9 0 −144.011 0 − 1019.43i 0 5657.65i 0 1056.24 0
25.10 0 −144.011 0 1019.43i 0 − 5657.65i 0 1056.24 0
25.11 0 −76.3670 0 − 1713.06i 0 − 425.676i 0 −13851.1 0
25.12 0 −76.3670 0 1713.06i 0 425.676i 0 −13851.1 0
25.13 0 −56.8673 0 − 1609.31i 0 − 9634.72i 0 −16449.1 0
25.14 0 −56.8673 0 1609.31i 0 9634.72i 0 −16449.1 0
25.15 0 −33.7913 0 2480.68i 0 − 3601.42i 0 −18541.1 0
25.16 0 −33.7913 0 − 2480.68i 0 3601.42i 0 −18541.1 0
25.17 0 −12.4770 0 1341.60i 0 − 1173.85i 0 −19527.3 0
25.18 0 −12.4770 0 − 1341.60i 0 1173.85i 0 −19527.3 0
25.19 0 35.4013 0 − 405.839i 0 − 7970.77i 0 −18429.7 0
25.20 0 35.4013 0 405.839i 0 7970.77i 0 −18429.7 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 25.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 104.10.f.a ✓ 32
4.b odd 2 1 208.10.f.d 32
13.b even 2 1 inner 104.10.f.a ✓ 32
52.b odd 2 1 208.10.f.d 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.10.f.a ✓ 32 1.a even 1 1 trivial
104.10.f.a ✓ 32 13.b even 2 1 inner
208.10.f.d 32 4.b odd 2 1
208.10.f.d 32 52.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(104, [\chi])\).