↑

Properties

Label 3871.1.j.a
Level $3871$
Weight $1$
Character orbit 3871.j
Analytic conductor $1.932$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -7
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3871,1,Mod(214,3871)] 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("3871.214"); S:= CuspForms(chi, 1); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 5])) B = ModularForms(chi, 1).cuspidal_submodule().basis() N = [B[i] for i in range(len(B))]
 
Level: \( N \) \(=\) \( 3871 = 7^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3871.j (of order \(6\), degree \(2\), 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: \(1.93188066390\)
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: \(D_{6}\)
Projective field: Galois closure of 6.0.150775763551.1

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + q^{8} + q^{9} + \zeta_{6}^{2} q^{11} + \zeta_{6} q^{16} + \zeta_{6} q^{18} - q^{22} + q^{23} - q^{25} + \zeta_{6} q^{46} - \zeta_{6} q^{50} + q^{64} - \zeta_{6} q^{67} + q^{72} + \cdots + \zeta_{6}^{2} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 2 q^{8} + 2 q^{9} - q^{11} + q^{16} + q^{18} - 2 q^{22} + 2 q^{23} - 2 q^{25} + q^{46} - q^{50} + 2 q^{64} - q^{67} + 2 q^{72} - q^{79} + 2 q^{81} - q^{88} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3871\mathbb{Z}\right)^\times\).

\(n\) \(1030\) \(2845\)
\(\chi(n)\) \(\zeta_{6}\) \(\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
214.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 − 0.866025i 0 0 0 0 0 1.00000 1.00000 0
814.1 0.500000 + 0.866025i 0 0 0 0 0 1.00000 1.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
553.j odd 6 1 inner
553.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3871.1.j.a 2
7.b odd 2 1 CM 3871.1.j.a 2
7.c even 3 1 3871.1.n.a ✓ 2
7.c even 3 1 3871.1.s.a 2
7.d odd 6 1 3871.1.n.a ✓ 2
7.d odd 6 1 3871.1.s.a 2
79.d odd 6 1 3871.1.s.a 2
553.j odd 6 1 inner 3871.1.j.a 2
553.k even 6 1 3871.1.s.a 2
553.q even 6 1 inner 3871.1.j.a 2
553.r even 6 1 3871.1.n.a ✓ 2
553.s odd 6 1 3871.1.n.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3871.1.j.a 2 1.a even 1 1 trivial
3871.1.j.a 2 7.b odd 2 1 CM
3871.1.j.a 2 553.j odd 6 1 inner
3871.1.j.a 2 553.q even 6 1 inner
3871.1.n.a ✓ 2 7.c even 3 1
3871.1.n.a ✓ 2 7.d odd 6 1
3871.1.n.a ✓ 2 553.r even 6 1
3871.1.n.a ✓ 2 553.s odd 6 1
3871.1.s.a 2 7.c even 3 1
3871.1.s.a 2 7.d odd 6 1
3871.1.s.a 2 79.d odd 6 1
3871.1.s.a 2 553.k even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3871, [\chi])\).

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