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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [432,8,Mod(145,432)] 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("432.145"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 432 = 2^{4} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 432.i (of order \(3\), degree \(2\), not 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: \(134.950331009\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: no (minimal twist has level 72)
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q + 125 q^{5} + 1245 q^{7} + 1699 q^{11} - 4937 q^{13} - 26540 q^{17} - 28976 q^{19} - 18239 q^{23} - 109168 q^{25} + 3525 q^{29} - 23753 q^{31} - 613122 q^{35} - 108420 q^{37} - 75063 q^{41} + 604385 q^{43}+ \cdots + 1704887 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1 0 0 0 −225.055 + 389.807i 0 817.403 + 1415.78i 0 0 0
145.2 0 0 0 −193.063 + 334.394i 0 111.981 + 193.956i 0 0 0
145.3 0 0 0 −125.246 + 216.932i 0 −413.999 − 717.067i 0 0 0
145.4 0 0 0 −108.865 + 188.560i 0 −226.022 − 391.481i 0 0 0
145.5 0 0 0 −42.6088 + 73.8006i 0 −328.506 − 568.988i 0 0 0
145.6 0 0 0 19.0488 − 32.9935i 0 722.808 + 1251.94i 0 0 0
145.7 0 0 0 53.7582 − 93.1119i 0 −322.045 − 557.798i 0 0 0
145.8 0 0 0 123.366 − 213.675i 0 −5.73097 − 9.92633i 0 0 0
145.9 0 0 0 160.855 − 278.610i 0 649.814 + 1125.51i 0 0 0
145.10 0 0 0 186.216 − 322.535i 0 −726.432 − 1258.22i 0 0 0
145.11 0 0 0 214.094 − 370.822i 0 343.228 + 594.488i 0 0 0
289.1 0 0 0 −225.055 − 389.807i 0 817.403 − 1415.78i 0 0 0
289.2 0 0 0 −193.063 − 334.394i 0 111.981 − 193.956i 0 0 0
289.3 0 0 0 −125.246 − 216.932i 0 −413.999 + 717.067i 0 0 0
289.4 0 0 0 −108.865 − 188.560i 0 −226.022 + 391.481i 0 0 0
289.5 0 0 0 −42.6088 − 73.8006i 0 −328.506 + 568.988i 0 0 0
289.6 0 0 0 19.0488 + 32.9935i 0 722.808 − 1251.94i 0 0 0
289.7 0 0 0 53.7582 + 93.1119i 0 −322.045 + 557.798i 0 0 0
289.8 0 0 0 123.366 + 213.675i 0 −5.73097 + 9.92633i 0 0 0
289.9 0 0 0 160.855 + 278.610i 0 649.814 − 1125.51i 0 0 0
See all 22 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 145.11
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 432.8.i.f 22
3.b odd 2 1 144.8.i.f 22
4.b odd 2 1 216.8.i.b 22
9.c even 3 1 inner 432.8.i.f 22
9.d odd 6 1 144.8.i.f 22
12.b even 2 1 72.8.i.b ✓ 22
36.f odd 6 1 216.8.i.b 22
36.h even 6 1 72.8.i.b ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.8.i.b ✓ 22 12.b even 2 1
72.8.i.b ✓ 22 36.h even 6 1
144.8.i.f 22 3.b odd 2 1
144.8.i.f 22 9.d odd 6 1
216.8.i.b 22 4.b odd 2 1
216.8.i.b 22 36.f odd 6 1
432.8.i.f 22 1.a even 1 1 trivial
432.8.i.f 22 9.c even 3 1 inner

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 \(S_{8}^{\mathrm{new}}(432, [\chi])\). Copy content Toggle raw display