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Properties

Label 2664.1.cy.a
Level $2664$
Weight $1$
Character orbit 2664.cy
Analytic conductor $1.330$
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] = [2664,1,Mod(1099,2664)] 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("2664.1099"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2664, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 3, 0, 2])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 2664 = 2^{3} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2664.cy (of order \(6\), degree \(2\), 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: \(1.32950919365\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.262848.3

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{24}^{7} q^{2} - \zeta_{24}^{2} q^{4} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{5} + \zeta_{24}^{2} q^{7} + \zeta_{24}^{9} q^{8} + (\zeta_{24}^{6} - 1) q^{10} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{11} + \cdots - q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{10} + 4 q^{16} - 8 q^{19} - 4 q^{22} + 4 q^{25} - 4 q^{28} + 4 q^{34} + 4 q^{40} + 8 q^{43} - 4 q^{52} - 4 q^{58} + 4 q^{67} - 4 q^{70} + 8 q^{73} - 8 q^{88} + 4 q^{91} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1333\) \(1999\) \(2369\)
\(\chi(n)\) \(\zeta_{24}^{8}\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1099.1
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i 0 0.866025 + 0.500000i 1.22474 + 0.707107i 0 −0.866025 − 0.500000i −0.707107 − 0.707107i 0 −1.00000 − 1.00000i
1099.2 −0.258819 + 0.965926i 0 −0.866025 − 0.500000i 1.22474 + 0.707107i 0 0.866025 + 0.500000i 0.707107 − 0.707107i 0 −1.00000 + 1.00000i
1099.3 0.258819 − 0.965926i 0 −0.866025 − 0.500000i −1.22474 − 0.707107i 0 0.866025 + 0.500000i −0.707107 + 0.707107i 0 −1.00000 + 1.00000i
1099.4 0.965926 + 0.258819i 0 0.866025 + 0.500000i −1.22474 − 0.707107i 0 −0.866025 − 0.500000i 0.707107 + 0.707107i 0 −1.00000 − 1.00000i
1675.1 −0.965926 + 0.258819i 0 0.866025 − 0.500000i 1.22474 − 0.707107i 0 −0.866025 + 0.500000i −0.707107 + 0.707107i 0 −1.00000 + 1.00000i
1675.2 −0.258819 − 0.965926i 0 −0.866025 + 0.500000i 1.22474 − 0.707107i 0 0.866025 − 0.500000i 0.707107 + 0.707107i 0 −1.00000 − 1.00000i
1675.3 0.258819 + 0.965926i 0 −0.866025 + 0.500000i −1.22474 + 0.707107i 0 0.866025 − 0.500000i −0.707107 − 0.707107i 0 −1.00000 − 1.00000i
1675.4 0.965926 − 0.258819i 0 0.866025 − 0.500000i −1.22474 + 0.707107i 0 −0.866025 + 0.500000i 0.707107 − 0.707107i 0 −1.00000 + 1.00000i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1099.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner
37.c even 3 1 inner
111.i odd 6 1 inner
296.p odd 6 1 inner
888.bd even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2664.1.cy.a ✓ 8
3.b odd 2 1 inner 2664.1.cy.a ✓ 8
8.d odd 2 1 inner 2664.1.cy.a ✓ 8
24.f even 2 1 inner 2664.1.cy.a ✓ 8
37.c even 3 1 inner 2664.1.cy.a ✓ 8
111.i odd 6 1 inner 2664.1.cy.a ✓ 8
296.p odd 6 1 inner 2664.1.cy.a ✓ 8
888.bd even 6 1 inner 2664.1.cy.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2664.1.cy.a ✓ 8 1.a even 1 1 trivial
2664.1.cy.a ✓ 8 3.b odd 2 1 inner
2664.1.cy.a ✓ 8 8.d odd 2 1 inner
2664.1.cy.a ✓ 8 24.f even 2 1 inner
2664.1.cy.a ✓ 8 37.c even 3 1 inner
2664.1.cy.a ✓ 8 111.i odd 6 1 inner
2664.1.cy.a ✓ 8 296.p odd 6 1 inner
2664.1.cy.a ✓ 8 888.bd even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(2664, [\chi])\).

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