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Properties

Label 3871.2.a.i
Level $3871$
Weight $2$
Character orbit 3871.a
Self dual yes
Analytic conductor $30.910$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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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] = [3871,2,Mod(1,3871)] 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("3871.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3871, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3871 = 7^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3871.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24,4,0,28,0] 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: yes
Analytic conductor: \(30.9100906224\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{2} + 28 q^{4} + 18 q^{8} + 50 q^{9} + 14 q^{11} + 18 q^{15} + 36 q^{16} + 10 q^{18} - 12 q^{22} + 22 q^{23} + 56 q^{25} + 36 q^{29} + 14 q^{30} + 30 q^{32} + 8 q^{36} + 24 q^{37} + 56 q^{39}+ \cdots + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.41231 −2.98272 3.81923 −3.55917 7.19525 0 −4.38854 5.89665 8.58580
1.2 −2.41231 2.98272 3.81923 3.55917 −7.19525 0 −4.38854 5.89665 −8.58580
1.3 −2.35106 −0.333714 3.52747 −1.89309 0.784581 0 −3.59117 −2.88863 4.45076
1.4 −2.35106 0.333714 3.52747 1.89309 −0.784581 0 −3.59117 −2.88863 −4.45076
1.5 −1.75669 −0.436287 1.08596 3.17234 0.766421 0 1.60568 −2.80965 −5.57282
1.6 −1.75669 0.436287 1.08596 −3.17234 −0.766421 0 1.60568 −2.80965 5.57282
1.7 −1.16173 −3.26388 −0.650393 3.42778 3.79173 0 3.07903 7.65292 −3.98215
1.8 −1.16173 3.26388 −0.650393 −3.42778 −3.79173 0 3.07903 7.65292 3.98215
1.9 −0.380222 −1.62042 −1.85543 3.79610 0.616119 0 1.46592 −0.374245 −1.44336
1.10 −0.380222 1.62042 −1.85543 −3.79610 −0.616119 0 1.46592 −0.374245 1.44336
1.11 −0.0847728 −2.25088 −1.99281 −0.792214 0.190813 0 0.338482 2.06645 0.0671582
1.12 −0.0847728 2.25088 −1.99281 0.792214 −0.190813 0 0.338482 2.06645 −0.0671582
1.13 0.113614 −3.40956 −1.98709 −2.24850 −0.387373 0 −0.452989 8.62510 −0.255460
1.14 0.113614 3.40956 −1.98709 2.24850 0.387373 0 −0.452989 8.62510 0.255460
1.15 0.899639 −1.39029 −1.19065 1.12263 −1.25076 0 −2.87043 −1.06708 1.00996
1.16 0.899639 1.39029 −1.19065 −1.12263 1.25076 0 −2.87043 −1.06708 −1.00996
1.17 1.96843 −3.07428 1.87472 −1.53880 −6.05151 0 −0.246606 6.45121 −3.02903
1.18 1.96843 3.07428 1.87472 1.53880 6.05151 0 −0.246606 6.45121 3.02903
1.19 2.05025 −2.04954 2.20353 −3.37907 −4.20207 0 0.417285 1.20062 −6.92795
1.20 2.05025 2.04954 2.20353 3.37907 4.20207 0 0.417285 1.20062 6.92795
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(79\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3871.2.a.i ✓ 24
7.b odd 2 1 inner 3871.2.a.i ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3871.2.a.i ✓ 24 1.a even 1 1 trivial
3871.2.a.i ✓ 24 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3871))\):

\( T_{2}^{12} - 2 T_{2}^{11} - 17 T_{2}^{10} + 31 T_{2}^{9} + 105 T_{2}^{8} - 166 T_{2}^{7} - 285 T_{2}^{6} + \cdots + 1 \) Copy content Toggle raw display
\( T_{3}^{24} - 61 T_{3}^{22} + 1613 T_{3}^{20} - 24286 T_{3}^{18} + 230099 T_{3}^{16} - 1431843 T_{3}^{14} + \cdots + 180224 \) Copy content Toggle raw display