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Properties

Label 1296.1.x.a
Level $1296$
Weight $1$
Character orbit 1296.x
Analytic conductor $0.647$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1296,1,Mod(53,1296)] 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("1296.53"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1296, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 10])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1296.x (of order \(12\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.646788256372\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 432)
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.55296.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24} q^{2} + \zeta_{24}^{2} q^{4} + \zeta_{24}^{5} q^{5} - \zeta_{24}^{10} q^{7} - \zeta_{24}^{3} q^{8} - \zeta_{24}^{6} q^{10} + \zeta_{24} q^{11} + ( - \zeta_{24}^{8} - \zeta_{24}^{2}) q^{13} + \cdots - \zeta_{24}^{4} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{13} + 4 q^{16} + 8 q^{28} - 4 q^{31} + 4 q^{34} + 4 q^{40} + 4 q^{43} - 4 q^{52} - 4 q^{67} - 4 q^{70} + 4 q^{85} - 4 q^{88} - 8 q^{91} - 4 q^{94} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(\zeta_{24}^{6}\) \(1\) \(-\zeta_{24}^{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.258819 + 0.965926i
−0.258819 − 0.965926i
−0.965926 − 0.258819i 0 0.866025 + 0.500000i 0.258819 + 0.965926i 0 0.866025 − 0.500000i −0.707107 − 0.707107i 0 − 1.00000i
53.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i −0.258819 − 0.965926i 0 0.866025 − 0.500000i 0.707107 + 0.707107i 0 − 1.00000i
269.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 0.258819 − 0.965926i 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 1.00000i
269.2 0.965926 − 0.258819i 0 0.866025 − 0.500000i −0.258819 + 0.965926i 0 0.866025 + 0.500000i 0.707107 − 0.707107i 0 1.00000i
701.1 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 0.965926 − 0.258819i 0 −0.866025 + 0.500000i 0.707107 − 0.707107i 0 1.00000i
701.2 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −0.965926 + 0.258819i 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 1.00000i
917.1 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 0.965926 + 0.258819i 0 −0.866025 − 0.500000i 0.707107 + 0.707107i 0 − 1.00000i
917.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −0.965926 − 0.258819i 0 −0.866025 − 0.500000i −0.707107 − 0.707107i 0 − 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
16.e even 4 1 inner
48.i odd 4 1 inner
144.w odd 12 1 inner
144.x even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.1.x.a 8
3.b odd 2 1 inner 1296.1.x.a 8
9.c even 3 1 432.1.j.a ✓ 4
9.c even 3 1 inner 1296.1.x.a 8
9.d odd 6 1 432.1.j.a ✓ 4
9.d odd 6 1 inner 1296.1.x.a 8
16.e even 4 1 inner 1296.1.x.a 8
36.f odd 6 1 1728.1.j.a 4
36.h even 6 1 1728.1.j.a 4
48.i odd 4 1 inner 1296.1.x.a 8
72.j odd 6 1 3456.1.j.b 4
72.l even 6 1 3456.1.j.a 4
72.n even 6 1 3456.1.j.b 4
72.p odd 6 1 3456.1.j.a 4
144.u even 12 1 1728.1.j.a 4
144.u even 12 1 3456.1.j.a 4
144.v odd 12 1 1728.1.j.a 4
144.v odd 12 1 3456.1.j.a 4
144.w odd 12 1 432.1.j.a ✓ 4
144.w odd 12 1 inner 1296.1.x.a 8
144.w odd 12 1 3456.1.j.b 4
144.x even 12 1 432.1.j.a ✓ 4
144.x even 12 1 inner 1296.1.x.a 8
144.x even 12 1 3456.1.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.1.j.a ✓ 4 9.c even 3 1
432.1.j.a ✓ 4 9.d odd 6 1
432.1.j.a ✓ 4 144.w odd 12 1
432.1.j.a ✓ 4 144.x even 12 1
1296.1.x.a 8 1.a even 1 1 trivial
1296.1.x.a 8 3.b odd 2 1 inner
1296.1.x.a 8 9.c even 3 1 inner
1296.1.x.a 8 9.d odd 6 1 inner
1296.1.x.a 8 16.e even 4 1 inner
1296.1.x.a 8 48.i odd 4 1 inner
1296.1.x.a 8 144.w odd 12 1 inner
1296.1.x.a 8 144.x even 12 1 inner
1728.1.j.a 4 36.f odd 6 1
1728.1.j.a 4 36.h even 6 1
1728.1.j.a 4 144.u even 12 1
1728.1.j.a 4 144.v odd 12 1
3456.1.j.a 4 72.l even 6 1
3456.1.j.a 4 72.p odd 6 1
3456.1.j.a 4 144.u even 12 1
3456.1.j.a 4 144.v odd 12 1
3456.1.j.b 4 72.j odd 6 1
3456.1.j.b 4 72.n even 6 1
3456.1.j.b 4 144.w odd 12 1
3456.1.j.b 4 144.x even 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1296, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$13$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$89$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
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