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Properties

Label 3888.1.o.b
Level $3888$
Weight $1$
Character orbit 3888.o
Analytic conductor $1.940$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3888,1,Mod(1135,3888)] 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("3888.1135"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3888, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 4])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3888 = 2^{4} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3888.o (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-3,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0, 3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.94036476912\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.45349632.2

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{6}^{2} - 1) q^{7} - 2 \zeta_{6}^{2} q^{13} + \zeta_{6} q^{25} + (\zeta_{6} + 1) q^{31} - q^{37} + ( - \zeta_{6}^{2} + 1) q^{43} + ( - \zeta_{6}^{2} - \zeta_{6} + 1) q^{49} + \zeta_{6} q^{61} + \cdots + 2 \zeta_{6} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{7} + 2 q^{13} + q^{25} + 3 q^{31} - 2 q^{37} + 3 q^{43} + 2 q^{49} + q^{61} + 3 q^{67} + 2 q^{73} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3888\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(2431\) \(2917\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1135.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 −1.50000 − 0.866025i 0 0 0
3727.1 0 0 0 0 0 −1.50000 + 0.866025i 0 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3888.1.o.b 2
3.b odd 2 1 CM 3888.1.o.b 2
4.b odd 2 1 3888.1.o.f 2
9.c even 3 1 3888.1.g.a ✓ 2
9.c even 3 1 3888.1.o.f 2
9.d odd 6 1 3888.1.g.a ✓ 2
9.d odd 6 1 3888.1.o.f 2
12.b even 2 1 3888.1.o.f 2
36.f odd 6 1 3888.1.g.a ✓ 2
36.f odd 6 1 inner 3888.1.o.b 2
36.h even 6 1 3888.1.g.a ✓ 2
36.h even 6 1 inner 3888.1.o.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3888.1.g.a ✓ 2 9.c even 3 1
3888.1.g.a ✓ 2 9.d odd 6 1
3888.1.g.a ✓ 2 36.f odd 6 1
3888.1.g.a ✓ 2 36.h even 6 1
3888.1.o.b 2 1.a even 1 1 trivial
3888.1.o.b 2 3.b odd 2 1 CM
3888.1.o.b 2 36.f odd 6 1 inner
3888.1.o.b 2 36.h even 6 1 inner
3888.1.o.f 2 4.b odd 2 1
3888.1.o.f 2 9.c even 3 1
3888.1.o.f 2 9.d odd 6 1
3888.1.o.f 2 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3888, [\chi])\):

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