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Properties

Label 4640.2.a.u
Level $4640$
Weight $2$
Character orbit 4640.a
Self dual yes
Analytic conductor $37.051$
Analytic rank $0$
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] = [4640,2,Mod(1,4640)] 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("4640.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4640, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 4640 = 2^{5} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4640.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,1,0,-6,0,3,0,1,0,-4,0,9,0,-1,0,-11,0,8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(21)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(37.0505865379\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.26118032.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 12x^{2} - 4x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - q^{5} + ( - \beta_{4} - \beta_{3} + \cdots - \beta_1) q^{7} + (\beta_{5} - \beta_{3}) q^{9} + ( - \beta_{3} - \beta_1 - 1) q^{11} + ( - \beta_{3} + 1) q^{13} - \beta_1 q^{15} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \cdots - 3) q^{17}+ \cdots + (\beta_{3} + 2 \beta_{2} - 3 \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} - 6 q^{5} + 3 q^{7} + q^{9} - 4 q^{11} + 9 q^{13} - q^{15} - 11 q^{17} + 8 q^{19} + 4 q^{21} + q^{23} + 6 q^{25} + 4 q^{27} - 6 q^{29} + 3 q^{31} - 12 q^{33} - 3 q^{35} + 14 q^{37} + 9 q^{39}+ \cdots + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 9x^{4} + 6x^{3} + 12x^{2} - 4x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - \nu^{4} - 9\nu^{3} + 6\nu^{2} + 10\nu - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - \nu^{4} - 8\nu^{3} + 5\nu^{2} + 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 3\nu^{4} + 7\nu^{3} - 22\nu^{2} - 2\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - \nu^{4} - 8\nu^{3} + 6\nu^{2} + 4\nu - 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{5} + \beta_{4} - 8\beta_{3} - \beta_{2} + 2\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{5} + \beta_{4} - 2\beta_{3} - 17\beta_{2} + 46\beta _1 + 5 \) 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
−2.60580
−0.918183
−0.537069
0.824941
1.27382
2.96229
0 −2.60580 0 −1.00000 0 0.722669 0 3.79019 0
1.2 0 −0.918183 0 −1.00000 0 0.478673 0 −2.15694 0
1.3 0 −0.537069 0 −1.00000 0 0.349411 0 −2.71156 0
1.4 0 0.824941 0 −1.00000 0 −4.71237 0 −2.31947 0
1.5 0 1.27382 0 −1.00000 0 4.65201 0 −1.37738 0
1.6 0 2.96229 0 −1.00000 0 1.50961 0 5.77515 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
\(2\) \( -1 \)
\(5\) \( +1 \)
\(29\) \( +1 \)

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 4640.2.a.u yes 6
4.b odd 2 1 4640.2.a.s ✓ 6
8.b even 2 1 9280.2.a.cn 6
8.d odd 2 1 9280.2.a.cp 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4640.2.a.s ✓ 6 4.b odd 2 1
4640.2.a.u yes 6 1.a even 1 1 trivial
9280.2.a.cn 6 8.b even 2 1
9280.2.a.cp 6 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4640))\):

\( T_{3}^{6} - T_{3}^{5} - 9T_{3}^{4} + 6T_{3}^{3} + 12T_{3}^{2} - 4T_{3} - 4 \) Copy content Toggle raw display
\( T_{7}^{6} - 3T_{7}^{5} - 19T_{7}^{4} + 66T_{7}^{3} - 68T_{7}^{2} + 28T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{6} + 4T_{11}^{5} - 16T_{11}^{4} - 64T_{11}^{3} + 20T_{11}^{2} + 120T_{11} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - T^{5} - 9 T^{4} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( (T + 1)^{6} \) Copy content Toggle raw display
$7$ \( T^{6} - 3 T^{5} + \cdots - 4 \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + \cdots + 32 \) Copy content Toggle raw display
$17$ \( T^{6} + 11 T^{5} + \cdots + 6436 \) Copy content Toggle raw display
$19$ \( T^{6} - 8 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$23$ \( T^{6} - T^{5} + \cdots - 1772 \) Copy content Toggle raw display
$29$ \( (T + 1)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 3 T^{5} + \cdots - 20492 \) Copy content Toggle raw display
$37$ \( T^{6} - 14 T^{5} + \cdots + 5872 \) Copy content Toggle raw display
$41$ \( T^{6} + 2 T^{5} + \cdots - 143744 \) Copy content Toggle raw display
$43$ \( T^{6} - 23 T^{5} + \cdots - 75548 \) Copy content Toggle raw display
$47$ \( T^{6} + 2 T^{5} + \cdots - 1088 \) Copy content Toggle raw display
$53$ \( T^{6} - 15 T^{5} + \cdots + 276896 \) Copy content Toggle raw display
$59$ \( T^{6} - T^{5} + \cdots - 20048 \) Copy content Toggle raw display
$61$ \( T^{6} - 5 T^{5} + \cdots + 175648 \) Copy content Toggle raw display
$67$ \( T^{6} - 26 T^{5} + \cdots - 431552 \) Copy content Toggle raw display
$71$ \( T^{6} + 12 T^{5} + \cdots - 12032 \) Copy content Toggle raw display
$73$ \( T^{6} + 19 T^{5} + \cdots + 39668 \) Copy content Toggle raw display
$79$ \( T^{6} - 33 T^{5} + \cdots + 691276 \) Copy content Toggle raw display
$83$ \( T^{6} + 4 T^{5} + \cdots + 75952 \) Copy content Toggle raw display
$89$ \( T^{6} + 18 T^{5} + \cdots + 428608 \) Copy content Toggle raw display
$97$ \( T^{6} + 21 T^{5} + \cdots + 58852 \) Copy content Toggle raw display
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