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Properties

Label 3872.2.a.bo
Level $3872$
Weight $2$
Character orbit 3872.a
Self dual yes
Analytic conductor $30.918$
Analytic rank $1$
Dimension $6$
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] = [3872,2,Mod(1,3872)] 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("3872.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3872, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3872 = 2^{5} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3872.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-5,0,0,0,2,0,7,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(30.9180756626\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.19898000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 352)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} - \beta_{5} q^{5} + (\beta_{5} + \beta_{2} - \beta_1 + 1) q^{7} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{9} + ( - \beta_{5} - 2 \beta_{4}) q^{13} + (\beta_{5} - \beta_{4} - \beta_{3} - 1) q^{15}+ \cdots + ( - 2 \beta_{5} - \beta_{2} - 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 5 q^{3} + 2 q^{7} + 7 q^{9} - 4 q^{13} - 8 q^{15} + 9 q^{17} + 5 q^{19} - 12 q^{21} - 6 q^{23} + 4 q^{25} - 26 q^{27} - 10 q^{29} - 12 q^{31} - 26 q^{35} - 12 q^{37} + 2 q^{39} + 17 q^{41} + 11 q^{43}+ \cdots - 21 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 10x^{4} + 7x^{3} + 24x^{2} - 15x - 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 8\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{4} + 2\nu^{3} + 7\nu^{2} - 8\nu - 7 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 8\nu^{2} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 7\nu^{3} - 10\nu^{2} - 9\nu + 6 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 2\beta_{4} + 7\beta_{3} + 9\beta_{2} + 2\beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} + 11\beta_{4} + 11\beta_{3} + 15\beta_{2} + 30\beta _1 + 19 \) 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
−2.18328
−2.05906
−0.245893
0.935683
1.68692
2.86564
0 −3.18328 0 −0.0919008 0 3.89322 0 7.13330 0
1.2 0 −3.05906 0 3.00593 0 −1.56490 0 6.35786 0
1.3 0 −1.24589 0 −3.75627 0 3.38413 0 −1.44775 0
1.4 0 −0.0643165 0 2.31301 0 −1.63066 0 −2.99586 0
1.5 0 0.686920 0 0.750338 0 −3.05529 0 −2.52814 0
1.6 0 1.86564 0 −2.22111 0 0.973509 0 0.480595 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(11\) \( -1 \)

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 3872.2.a.bo 6
4.b odd 2 1 3872.2.a.bp 6
8.b even 2 1 7744.2.a.dw 6
8.d odd 2 1 7744.2.a.dt 6
11.b odd 2 1 3872.2.a.bn 6
11.d odd 10 2 352.2.m.f yes 12
44.c even 2 1 3872.2.a.bq 6
44.g even 10 2 352.2.m.e ✓ 12
88.b odd 2 1 7744.2.a.dv 6
88.g even 2 1 7744.2.a.du 6
88.k even 10 2 704.2.m.m 12
88.p odd 10 2 704.2.m.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
352.2.m.e ✓ 12 44.g even 10 2
352.2.m.f yes 12 11.d odd 10 2
704.2.m.m 12 88.k even 10 2
704.2.m.n 12 88.p odd 10 2
3872.2.a.bn 6 11.b odd 2 1
3872.2.a.bo 6 1.a even 1 1 trivial
3872.2.a.bp 6 4.b odd 2 1
3872.2.a.bq 6 44.c even 2 1
7744.2.a.dt 6 8.d odd 2 1
7744.2.a.du 6 88.g even 2 1
7744.2.a.dv 6 88.b odd 2 1
7744.2.a.dw 6 8.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3872))\):

\( T_{3}^{6} + 5T_{3}^{5} - 23T_{3}^{3} - 10T_{3}^{2} + 15T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{6} - 17T_{5}^{4} + 8T_{5}^{3} + 61T_{5}^{2} - 38T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{6} - 2T_{7}^{5} - 19T_{7}^{4} + 20T_{7}^{3} + 105T_{7}^{2} - 100 \) Copy content Toggle raw display
\( T_{13}^{6} + 4T_{13}^{5} - 49T_{13}^{4} - 192T_{13}^{3} + 469T_{13}^{2} + 1334T_{13} - 1796 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 5 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} - 17 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots - 100 \) Copy content Toggle raw display
$11$ \( T^{6} \) Copy content Toggle raw display
$13$ \( T^{6} + 4 T^{5} + \cdots - 1796 \) Copy content Toggle raw display
$17$ \( T^{6} - 9 T^{5} + \cdots - 29 \) Copy content Toggle raw display
$19$ \( T^{6} - 5 T^{5} + \cdots + 2759 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + \cdots + 320 \) Copy content Toggle raw display
$29$ \( T^{6} + 10 T^{5} + \cdots + 2644 \) Copy content Toggle raw display
$31$ \( T^{6} + 12 T^{5} + \cdots - 1396 \) Copy content Toggle raw display
$37$ \( T^{6} + 12 T^{5} + \cdots + 1796 \) Copy content Toggle raw display
$41$ \( T^{6} - 17 T^{5} + \cdots + 28979 \) Copy content Toggle raw display
$43$ \( T^{6} - 11 T^{5} + \cdots - 81776 \) Copy content Toggle raw display
$47$ \( T^{6} + 22 T^{5} + \cdots + 4304 \) Copy content Toggle raw display
$53$ \( T^{6} - 18 T^{5} + \cdots + 191824 \) Copy content Toggle raw display
$59$ \( T^{6} + 7 T^{5} + \cdots - 1681 \) Copy content Toggle raw display
$61$ \( T^{6} + 18 T^{5} + \cdots - 15920 \) Copy content Toggle raw display
$67$ \( T^{6} + 31 T^{5} + \cdots - 36304 \) Copy content Toggle raw display
$71$ \( T^{6} + 22 T^{5} + \cdots + 32020 \) Copy content Toggle raw display
$73$ \( T^{6} + T^{5} + \cdots + 5975 \) Copy content Toggle raw display
$79$ \( T^{6} + 28 T^{5} + \cdots - 7396 \) Copy content Toggle raw display
$83$ \( T^{6} + 13 T^{5} + \cdots + 200201 \) Copy content Toggle raw display
$89$ \( T^{6} + T^{5} + \cdots + 7600 \) Copy content Toggle raw display
$97$ \( T^{6} + 21 T^{5} + \cdots + 11 \) Copy content Toggle raw display
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