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Properties

Label 3871.1.m.d
Level $3871$
Weight $1$
Character orbit 3871.m
Analytic conductor $1.932$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -79
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] = [3871,1,Mod(1500,3871)] 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("3871.1500"); 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([2, 3])) 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.m (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,4,0,-6,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.93188066390\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.2140663.1
Artin image: $C_3\times D_8$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} + ( - 3 \beta_{2} - 3) q^{4} + ( - \beta_{3} - \beta_1) q^{5} - 4 q^{8} + \beta_{2} q^{9} - 2 \beta_1 q^{10} + \beta_{3} q^{13} + 5 \beta_{2} q^{16} + (2 \beta_{2} + 2) q^{18} + ( - \beta_{3} - \beta_1) q^{19}+ \cdots - \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 6 q^{4} - 16 q^{8} - 2 q^{9} - 10 q^{16} + 4 q^{18} - 2 q^{25} + 12 q^{32} + 12 q^{36} - 8 q^{50} + 28 q^{64} - 4 q^{65} + 8 q^{72} + 2 q^{79} - 2 q^{81} - 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(1030\) \(2845\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1500.1
−0.707107 − 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 − 1.22474i
1.00000 − 1.73205i 0 −1.50000 − 2.59808i −0.707107 + 1.22474i 0 0 −4.00000 −0.500000 + 0.866025i 1.41421 + 2.44949i
1500.2 1.00000 − 1.73205i 0 −1.50000 − 2.59808i 0.707107 − 1.22474i 0 0 −4.00000 −0.500000 + 0.866025i −1.41421 − 2.44949i
3791.1 1.00000 + 1.73205i 0 −1.50000 + 2.59808i −0.707107 − 1.22474i 0 0 −4.00000 −0.500000 − 0.866025i 1.41421 − 2.44949i
3791.2 1.00000 + 1.73205i 0 −1.50000 + 2.59808i 0.707107 + 1.22474i 0 0 −4.00000 −0.500000 − 0.866025i −1.41421 + 2.44949i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
79.b odd 2 1 CM by \(\Q(\sqrt{-79}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
553.d even 2 1 inner
553.l even 6 1 inner
553.m odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3871.1.m.d 4
7.b odd 2 1 inner 3871.1.m.d 4
7.c even 3 1 3871.1.c.b ✓ 2
7.c even 3 1 inner 3871.1.m.d 4
7.d odd 6 1 3871.1.c.b ✓ 2
7.d odd 6 1 inner 3871.1.m.d 4
79.b odd 2 1 CM 3871.1.m.d 4
553.d even 2 1 inner 3871.1.m.d 4
553.l even 6 1 3871.1.c.b ✓ 2
553.l even 6 1 inner 3871.1.m.d 4
553.m odd 6 1 3871.1.c.b ✓ 2
553.m odd 6 1 inner 3871.1.m.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3871.1.c.b ✓ 2 7.c even 3 1
3871.1.c.b ✓ 2 7.d odd 6 1
3871.1.c.b ✓ 2 553.l even 6 1
3871.1.c.b ✓ 2 553.m odd 6 1
3871.1.m.d 4 1.a even 1 1 trivial
3871.1.m.d 4 7.b odd 2 1 inner
3871.1.m.d 4 7.c even 3 1 inner
3871.1.m.d 4 7.d odd 6 1 inner
3871.1.m.d 4 79.b odd 2 1 CM
3871.1.m.d 4 553.d even 2 1 inner
3871.1.m.d 4 553.l even 6 1 inner
3871.1.m.d 4 553.m odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3871, [\chi])\):

\( T_{2}^{2} - 2T_{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{4} + 2T_{5}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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