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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,13,Mod(74,75)] 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("75.74"); S:= CuspForms(chi, 13); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 13, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,64612,0,-106948] 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: \(68.5495362957\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 15)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 64612 q^{4} - 106948 q^{6} + 3055600 q^{9} + 272933636 q^{16} + 440507360 q^{19} + 268559472 q^{21} + 93284964 q^{24} - 4111882976 q^{31} - 2956627176 q^{34} + 9354564244 q^{36} - 26816664352 q^{39}+ \cdots + 3214626713600 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1 −124.574 −478.686 + 549.819i 11422.6 0 59631.7 − 68493.1i 124156.i −912706. −73161.3 − 526381.i 0
74.2 −124.574 −478.686 − 549.819i 11422.6 0 59631.7 + 68493.1i − 124156.i −912706. −73161.3 + 526381.i 0
74.3 −118.018 675.345 − 274.499i 9832.29 0 −79703.0 + 32395.8i − 80183.4i −676986. 380742. − 370763.i 0
74.4 −118.018 675.345 + 274.499i 9832.29 0 −79703.0 − 32395.8i 80183.4i −676986. 380742. + 370763.i 0
74.5 −77.3702 11.6559 − 728.907i 1890.15 0 −901.823 + 56395.7i 140164.i 170667. −531169. − 16992.2i 0
74.6 −77.3702 11.6559 + 728.907i 1890.15 0 −901.823 − 56395.7i − 140164.i 170667. −531169. + 16992.2i 0
74.7 −75.0256 641.888 − 345.574i 1532.85 0 −48158.0 + 25926.9i − 47419.9i 192302. 292599. − 443639.i 0
74.8 −75.0256 641.888 + 345.574i 1532.85 0 −48158.0 − 25926.9i 47419.9i 192302. 292599. + 443639.i 0
74.9 −73.7688 −728.809 + 16.6838i 1345.83 0 53763.3 − 1230.74i − 82132.6i 202877. 530884. − 24318.6i 0
74.10 −73.7688 −728.809 − 16.6838i 1345.83 0 53763.3 + 1230.74i 82132.6i 202877. 530884. + 24318.6i 0
74.11 −41.3682 451.804 − 572.114i −2384.67 0 −18690.3 + 23667.3i 204687.i 268094. −123187. − 516967.i 0
74.12 −41.3682 451.804 + 572.114i −2384.67 0 −18690.3 − 23667.3i − 204687.i 268094. −123187. + 516967.i 0
74.13 −21.4312 −679.771 + 263.348i −3636.71 0 14568.3 − 5643.86i − 90117.1i 165721. 392737. − 358033.i 0
74.14 −21.4312 −679.771 − 263.348i −3636.71 0 14568.3 + 5643.86i 90117.1i 165721. 392737. + 358033.i 0
74.15 −15.7045 461.463 − 564.352i −3849.37 0 −7247.06 + 8862.87i − 125051.i 124778. −105544. − 520855.i 0
74.16 −15.7045 461.463 + 564.352i −3849.37 0 −7247.06 − 8862.87i 125051.i 124778. −105544. + 520855.i 0
74.17 15.7045 −461.463 − 564.352i −3849.37 0 −7247.06 − 8862.87i − 125051.i −124778. −105544. + 520855.i 0
74.18 15.7045 −461.463 + 564.352i −3849.37 0 −7247.06 + 8862.87i 125051.i −124778. −105544. − 520855.i 0
74.19 21.4312 679.771 + 263.348i −3636.71 0 14568.3 + 5643.86i − 90117.1i −165721. 392737. + 358033.i 0
74.20 21.4312 679.771 − 263.348i −3636.71 0 14568.3 − 5643.86i 90117.1i −165721. 392737. − 358033.i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 74.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.13.d.c 32
3.b odd 2 1 inner 75.13.d.c 32
5.b even 2 1 inner 75.13.d.c 32
5.c odd 4 1 15.13.c.a ✓ 16
5.c odd 4 1 75.13.c.d 16
15.d odd 2 1 inner 75.13.d.c 32
15.e even 4 1 15.13.c.a ✓ 16
15.e even 4 1 75.13.c.d 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.13.c.a ✓ 16 5.c odd 4 1
15.13.c.a ✓ 16 15.e even 4 1
75.13.c.d 16 5.c odd 4 1
75.13.c.d 16 15.e even 4 1
75.13.d.c 32 1.a even 1 1 trivial
75.13.d.c 32 3.b odd 2 1 inner
75.13.d.c 32 5.b even 2 1 inner
75.13.d.c 32 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} - 48921 T_{2}^{14} + 929053656 T_{2}^{12} - 8756105588596 T_{2}^{10} + \cdots + 76\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display