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Properties

Label 3344.1.bc.a
Level $3344$
Weight $1$
Character orbit 3344.bc
Analytic conductor $1.669$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
RM discriminant 44
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3344,1,Mod(2463,3344)] 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("3344.2463"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 3, 5])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3344.bc (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,-1,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.66887340224\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.4793727664.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{5} - q^{7} + \zeta_{6} q^{9} + q^{11} - \zeta_{6}^{2} q^{19} - \zeta_{6}^{2} q^{35} + (\zeta_{6}^{2} + \zeta_{6}) q^{37} + 2 \zeta_{6}^{2} q^{43} - q^{45} + (\zeta_{6}^{2} - 1) q^{53} + \cdots + \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} - 2 q^{7} + q^{9} + 2 q^{11} + q^{19} + q^{35} - 2 q^{43} - 2 q^{45} - 3 q^{53} - q^{55} - q^{63} - 2 q^{77} + 3 q^{79} - q^{81} - 2 q^{83} + q^{95} + 3 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3344\mathbb{Z}\right)^\times\).

\(n\) \(705\) \(837\) \(2433\) \(2927\)
\(\chi(n)\) \(\zeta_{6}\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2463.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 −0.500000 − 0.866025i 0 −1.00000 0 0.500000 − 0.866025i 0
2991.1 0 0 0 −0.500000 + 0.866025i 0 −1.00000 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
44.c even 2 1 RM by \(\Q(\sqrt{11}) \)
19.d odd 6 1 inner
836.m odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3344.1.bc.a ✓ 2
4.b odd 2 1 3344.1.bc.b yes 2
11.b odd 2 1 3344.1.bc.b yes 2
19.d odd 6 1 inner 3344.1.bc.a ✓ 2
44.c even 2 1 RM 3344.1.bc.a ✓ 2
76.f even 6 1 3344.1.bc.b yes 2
209.g even 6 1 3344.1.bc.b yes 2
836.m odd 6 1 inner 3344.1.bc.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3344.1.bc.a ✓ 2 1.a even 1 1 trivial
3344.1.bc.a ✓ 2 19.d odd 6 1 inner
3344.1.bc.a ✓ 2 44.c even 2 1 RM
3344.1.bc.a ✓ 2 836.m odd 6 1 inner
3344.1.bc.b yes 2 4.b odd 2 1
3344.1.bc.b yes 2 11.b odd 2 1
3344.1.bc.b yes 2 76.f even 6 1
3344.1.bc.b yes 2 209.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 1 \) acting on \(S_{1}^{\mathrm{new}}(3344, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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