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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,8,Mod(25,72)] 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("72.25"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 72.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.4917218349\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 44 q^{3} - 125 q^{5} - 1245 q^{7} - 3766 q^{9} + 1699 q^{11} - 4937 q^{13} - 19349 q^{15} + 26540 q^{17} + 28976 q^{19} - 75315 q^{21} - 18239 q^{23} - 109168 q^{25} - 87680 q^{27} - 3525 q^{29}+ \cdots + 24043955 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1 0 −43.7152 − 16.6126i 0 42.6088 + 73.8006i 0 328.506 − 568.988i 0 1635.04 + 1452.45i 0
25.2 0 −43.1020 − 18.1443i 0 −19.0488 − 32.9935i 0 −722.808 + 1251.94i 0 1528.57 + 1564.11i 0
25.3 0 −34.8262 + 31.2112i 0 125.246 + 216.932i 0 413.999 − 717.067i 0 238.728 − 2173.93i 0
25.4 0 −32.5814 + 33.5478i 0 −214.094 − 370.822i 0 −343.228 + 594.488i 0 −63.9074 − 2186.07i 0
25.5 0 −9.85077 − 45.7161i 0 −186.216 − 322.535i 0 726.432 − 1258.22i 0 −1992.92 + 900.678i 0
25.6 0 −0.287929 − 46.7645i 0 193.063 + 334.394i 0 −111.981 + 193.956i 0 −2186.83 + 26.9297i 0
25.7 0 14.6863 + 44.3995i 0 225.055 + 389.807i 0 −817.403 + 1415.78i 0 −1755.63 + 1304.12i 0
25.8 0 14.8033 + 44.3606i 0 −53.7582 − 93.1119i 0 322.045 − 557.798i 0 −1748.73 + 1313.36i 0
25.9 0 24.3366 − 39.9340i 0 −160.855 − 278.610i 0 −649.814 + 1125.51i 0 −1002.46 − 1943.72i 0
25.10 0 44.0088 + 15.8184i 0 −123.366 − 213.675i 0 5.73097 − 9.92633i 0 1686.56 + 1392.30i 0
25.11 0 44.5285 − 14.2902i 0 108.865 + 188.560i 0 226.022 − 391.481i 0 1778.58 − 1272.65i 0
49.1 0 −43.7152 + 16.6126i 0 42.6088 − 73.8006i 0 328.506 + 568.988i 0 1635.04 − 1452.45i 0
49.2 0 −43.1020 + 18.1443i 0 −19.0488 + 32.9935i 0 −722.808 − 1251.94i 0 1528.57 − 1564.11i 0
49.3 0 −34.8262 − 31.2112i 0 125.246 − 216.932i 0 413.999 + 717.067i 0 238.728 + 2173.93i 0
49.4 0 −32.5814 − 33.5478i 0 −214.094 + 370.822i 0 −343.228 − 594.488i 0 −63.9074 + 2186.07i 0
49.5 0 −9.85077 + 45.7161i 0 −186.216 + 322.535i 0 726.432 + 1258.22i 0 −1992.92 − 900.678i 0
49.6 0 −0.287929 + 46.7645i 0 193.063 − 334.394i 0 −111.981 − 193.956i 0 −2186.83 − 26.9297i 0
49.7 0 14.6863 − 44.3995i 0 225.055 − 389.807i 0 −817.403 − 1415.78i 0 −1755.63 − 1304.12i 0
49.8 0 14.8033 − 44.3606i 0 −53.7582 + 93.1119i 0 322.045 + 557.798i 0 −1748.73 − 1313.36i 0
49.9 0 24.3366 + 39.9340i 0 −160.855 + 278.610i 0 −649.814 − 1125.51i 0 −1002.46 + 1943.72i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 25.11
Significant digits:
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Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.8.i.b ✓ 22
3.b odd 2 1 216.8.i.b 22
4.b odd 2 1 144.8.i.f 22
9.c even 3 1 inner 72.8.i.b ✓ 22
9.d odd 6 1 216.8.i.b 22
12.b even 2 1 432.8.i.f 22
36.f odd 6 1 144.8.i.f 22
36.h even 6 1 432.8.i.f 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.i.b ✓ 22 1.a even 1 1 trivial
72.8.i.b ✓ 22 9.c even 3 1 inner
144.8.i.f 22 4.b odd 2 1
144.8.i.f 22 36.f odd 6 1
216.8.i.b 22 3.b odd 2 1
216.8.i.b 22 9.d odd 6 1
432.8.i.f 22 12.b even 2 1
432.8.i.f 22 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{22} + 125 T_{5}^{21} + 492084 T_{5}^{20} + 57952559 T_{5}^{19} + 154919763673 T_{5}^{18} + \cdots + 17\!\cdots\!00 \) acting on \(S_{8}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display