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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,1,Mod(55,324)] 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("324.55"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.161697064093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.324.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.419904.3

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} + \zeta_{6} q^{5} + q^{8} - q^{10} + \zeta_{6} q^{13} + \zeta_{6}^{2} q^{16} - q^{17} - \zeta_{6}^{2} q^{20} - q^{26} - \zeta_{6}^{2} q^{29} - \zeta_{6} q^{32} + \cdots + q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} + q^{5} + 2 q^{8} - 2 q^{10} + q^{13} - q^{16} - 2 q^{17} + q^{20} - 2 q^{26} + q^{29} - q^{32} + q^{34} - 2 q^{37} + q^{40} - 2 q^{41} - q^{49} + q^{52} + 4 q^{53} + q^{58}+ \cdots + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(-1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.500000 + 0.866025i 0 0 1.00000 0 −1.00000
271.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.500000 − 0.866025i 0 0 1.00000 0 −1.00000
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 324.1.f.a 2
3.b odd 2 1 324.1.f.b 2
4.b odd 2 1 CM 324.1.f.a 2
9.c even 3 1 324.1.d.b yes 1
9.c even 3 1 inner 324.1.f.a 2
9.d odd 6 1 324.1.d.a ✓ 1
9.d odd 6 1 324.1.f.b 2
12.b even 2 1 324.1.f.b 2
27.e even 9 6 2916.1.j.d 6
27.f odd 18 6 2916.1.j.e 6
36.f odd 6 1 324.1.d.b yes 1
36.f odd 6 1 inner 324.1.f.a 2
36.h even 6 1 324.1.d.a ✓ 1
36.h even 6 1 324.1.f.b 2
108.j odd 18 6 2916.1.j.d 6
108.l even 18 6 2916.1.j.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
324.1.d.a ✓ 1 9.d odd 6 1
324.1.d.a ✓ 1 36.h even 6 1
324.1.d.b yes 1 9.c even 3 1
324.1.d.b yes 1 36.f odd 6 1
324.1.f.a 2 1.a even 1 1 trivial
324.1.f.a 2 4.b odd 2 1 CM
324.1.f.a 2 9.c even 3 1 inner
324.1.f.a 2 36.f odd 6 1 inner
324.1.f.b 2 3.b odd 2 1
324.1.f.b 2 9.d odd 6 1
324.1.f.b 2 12.b even 2 1
324.1.f.b 2 36.h even 6 1
2916.1.j.d 6 27.e even 9 6
2916.1.j.d 6 108.j odd 18 6
2916.1.j.e 6 27.f odd 18 6
2916.1.j.e 6 108.l even 18 6

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(324, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( (T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T + 1)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
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