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Properties

Label 3871.2.a.e
Level $3871$
Weight $2$
Character orbit 3871.a
Self dual yes
Analytic conductor $30.910$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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 = [8,3,-3,7,-4] 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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 7x^{6} + 24x^{5} + 6x^{4} - 40x^{3} + 6x^{2} + 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 553)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{4} - \beta_{5} q^{5} + (\beta_{3} - \beta_1) q^{6} + (\beta_{6} - \beta_{5} + \beta_{2} + \beta_1) q^{8} + ( - \beta_{4} - \beta_{2}) q^{9}+ \cdots + (\beta_{7} + 2 \beta_{6} - \beta_{5} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{2} - 3 q^{3} + 7 q^{4} - 4 q^{5} - 6 q^{6} + 6 q^{8} - q^{9} - 10 q^{10} + 3 q^{11} - 10 q^{12} - 3 q^{13} + 11 q^{15} + 13 q^{16} - 14 q^{17} + 4 q^{18} - 12 q^{19} - 12 q^{20} - 15 q^{22}+ \cdots - 25 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 7x^{6} + 24x^{5} + 6x^{4} - 40x^{3} + 6x^{2} + 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 6\nu^{5} + 33\nu^{4} - 58\nu^{2} + 10\nu + 10 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{7} + 5\nu^{6} + 15\nu^{5} - 39\nu^{4} - 18\nu^{3} + 59\nu^{2} - 8\nu - 5 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{7} + 3\nu^{6} + 7\nu^{5} - 24\nu^{4} - 6\nu^{3} + 40\nu^{2} - 7\nu - 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -4\nu^{7} + 13\nu^{6} + 24\nu^{5} - 102\nu^{4} + 6\nu^{3} + 160\nu^{2} - 64\nu - 19 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 17\nu^{6} + 30\nu^{5} - 135\nu^{4} + 9\nu^{3} + 218\nu^{2} - 89\nu - 29 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -8\nu^{7} + 26\nu^{6} + 51\nu^{5} - 204\nu^{4} - 9\nu^{3} + 317\nu^{2} - 110\nu - 35 ) / 3 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 7\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 7\beta_{6} - 9\beta_{5} + \beta_{4} - \beta_{3} + 8\beta_{2} + 30\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{7} - 8\beta_{6} - 3\beta_{5} + 36\beta_{4} - 46\beta_{3} + 44\beta_{2} + 48\beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 13\beta_{7} + 43\beta_{6} - 66\beta_{5} + 10\beta_{4} - 17\beta_{3} + 54\beta_{2} + 189\beta _1 + 11 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.40260
−1.39220
−0.250631
−0.174832
0.719508
1.55105
2.36846
2.58125
−2.40260 0.382380 3.77249 −1.34101 −0.918706 0 −4.25859 −2.85379 3.22191
1.2 −1.39220 −1.18707 −0.0617699 3.15859 1.65264 0 2.87040 −1.59087 −4.39739
1.3 −0.250631 −2.10019 −1.93718 −2.19126 0.526374 0 0.986782 1.41081 0.549199
1.4 −0.174832 2.31835 −1.96943 1.01700 −0.405323 0 0.693986 2.37476 −0.177804
1.5 0.719508 1.78774 −1.48231 0.429026 1.28629 0 −2.50555 0.196014 0.308688
1.6 1.55105 −2.56317 0.405768 −2.92345 −3.97562 0 −2.47274 3.56985 −4.53442
1.7 2.36846 −0.293835 3.60960 −2.70993 −0.695936 0 3.81226 −2.91366 −6.41837
1.8 2.58125 −1.34420 4.66284 0.561042 −3.46973 0 6.87345 −1.19311 1.44819
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3871.2.a.e 8
7.b odd 2 1 553.2.a.b ✓ 8
21.c even 2 1 4977.2.a.j 8
28.d even 2 1 8848.2.a.s 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
553.2.a.b ✓ 8 7.b odd 2 1
3871.2.a.e 8 1.a even 1 1 trivial
4977.2.a.j 8 21.c even 2 1
8848.2.a.s 8 28.d even 2 1

Hecke kernels

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

\( T_{2}^{8} - 3T_{2}^{7} - 7T_{2}^{6} + 24T_{2}^{5} + 6T_{2}^{4} - 40T_{2}^{3} + 6T_{2}^{2} + 8T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{8} + 3T_{3}^{7} - 7T_{3}^{6} - 26T_{3}^{5} + 3T_{3}^{4} + 55T_{3}^{3} + 31T_{3}^{2} - 9T_{3} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 3 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{8} + 4 T^{7} + \cdots + 18 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{7} + \cdots - 3566 \) Copy content Toggle raw display
$17$ \( T^{8} + 14 T^{7} + \cdots - 1864 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + \cdots + 1782 \) Copy content Toggle raw display
$23$ \( T^{8} - 17 T^{7} + \cdots + 1108 \) Copy content Toggle raw display
$29$ \( T^{8} + 6 T^{7} + \cdots - 4534 \) Copy content Toggle raw display
$31$ \( T^{8} + 15 T^{7} + \cdots + 235834 \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + \cdots - 3726 \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots - 17944 \) Copy content Toggle raw display
$43$ \( T^{8} - 9 T^{7} + \cdots - 2708 \) Copy content Toggle raw display
$47$ \( T^{8} + 34 T^{7} + \cdots - 591912 \) Copy content Toggle raw display
$53$ \( T^{8} - 2 T^{7} + \cdots - 4669922 \) Copy content Toggle raw display
$59$ \( T^{8} + 21 T^{7} + \cdots + 973944 \) Copy content Toggle raw display
$61$ \( T^{8} + 11 T^{7} + \cdots - 1340704 \) Copy content Toggle raw display
$67$ \( T^{8} - 14 T^{7} + \cdots + 19248472 \) Copy content Toggle raw display
$71$ \( T^{8} + 2 T^{7} + \cdots - 1296 \) Copy content Toggle raw display
$73$ \( T^{8} + 7 T^{7} + \cdots + 2229818 \) Copy content Toggle raw display
$79$ \( (T - 1)^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 32 T^{7} + \cdots + 435826 \) Copy content Toggle raw display
$89$ \( T^{8} + 6 T^{7} + \cdots - 7218 \) Copy content Toggle raw display
$97$ \( T^{8} - 39 T^{7} + \cdots - 139626 \) Copy content Toggle raw display
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