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Properties

Label 208.10.f.d
Level $208$
Weight $10$
Character orbit 208.f
Analytic conductor $107.127$
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] = [208,10,Mod(129,208)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, 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("208.129"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,162] 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: \(107.127453922\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 104)
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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1 0 −263.861 0 − 382.881i 0 − 2900.70i 0 49939.8 0
129.2 0 −263.861 0 382.881i 0 2900.70i 0 49939.8 0
129.3 0 −196.614 0 − 2713.78i 0 8420.63i 0 18973.9 0
129.4 0 −196.614 0 2713.78i 0 − 8420.63i 0 18973.9 0
129.5 0 −195.153 0 2044.04i 0 11607.1i 0 18401.8 0
129.6 0 −195.153 0 − 2044.04i 0 − 11607.1i 0 18401.8 0
129.7 0 −161.892 0 1806.71i 0 − 1769.55i 0 6526.18 0
129.8 0 −161.892 0 − 1806.71i 0 1769.55i 0 6526.18 0
129.9 0 −132.434 0 − 362.362i 0 − 3483.24i 0 −2144.23 0
129.10 0 −132.434 0 362.362i 0 3483.24i 0 −2144.23 0
129.11 0 −117.861 0 287.112i 0 10358.2i 0 −5791.89 0
129.12 0 −117.861 0 − 287.112i 0 − 10358.2i 0 −5791.89 0
129.13 0 −35.4013 0 405.839i 0 − 7970.77i 0 −18429.7 0
129.14 0 −35.4013 0 − 405.839i 0 7970.77i 0 −18429.7 0
129.15 0 12.4770 0 − 1341.60i 0 − 1173.85i 0 −19527.3 0
129.16 0 12.4770 0 1341.60i 0 1173.85i 0 −19527.3 0
129.17 0 33.7913 0 − 2480.68i 0 − 3601.42i 0 −18541.1 0
129.18 0 33.7913 0 2480.68i 0 3601.42i 0 −18541.1 0
129.19 0 56.8673 0 1609.31i 0 − 9634.72i 0 −16449.1 0
129.20 0 56.8673 0 − 1609.31i 0 9634.72i 0 −16449.1 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.32
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 208.10.f.d 32
4.b odd 2 1 104.10.f.a ✓ 32
13.b even 2 1 inner 208.10.f.d 32
52.b odd 2 1 104.10.f.a ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.10.f.a ✓ 32 4.b odd 2 1
104.10.f.a ✓ 32 52.b odd 2 1
208.10.f.d 32 1.a even 1 1 trivial
208.10.f.d 32 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 81 T_{3}^{15} - 209952 T_{3}^{14} + 15504548 T_{3}^{13} + 17318953958 T_{3}^{12} + \cdots - 48\!\cdots\!04 \) acting on \(S_{10}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display