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Properties

Label 4080.2.a.bt
Level $4080$
Weight $2$
Character orbit 4080.a
Self dual yes
Analytic conductor $32.579$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4080,2,Mod(1,4080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4080.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4080.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,-4,0,-4,0,-4,0,4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(32.5789640247\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13768.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 2x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 255)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{1}\)\(=\) \( \nu^{3} - 6\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 2\nu + 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} + 3\beta_{2} - \beta _1 + 5 ) / 2 \) 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
−1.89761
−0.489088
2.53744
0.849256
0 −1.00000 0 −1.00000 0 −4.55250 0 1.00000 0
1.2 0 −1.00000 0 −1.00000 0 −2.81754 0 1.00000 0
1.3 0 −1.00000 0 −1.00000 0 −1.11299 0 1.00000 0
1.4 0 −1.00000 0 −1.00000 0 4.48302 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(5\) \( +1 \)
\(17\) \( +1 \)

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 4080.2.a.bt 4
4.b odd 2 1 255.2.a.d ✓ 4
12.b even 2 1 765.2.a.m 4
20.d odd 2 1 1275.2.a.t 4
20.e even 4 2 1275.2.b.k 8
60.h even 2 1 3825.2.a.bi 4
68.d odd 2 1 4335.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.a.d ✓ 4 4.b odd 2 1
765.2.a.m 4 12.b even 2 1
1275.2.a.t 4 20.d odd 2 1
1275.2.b.k 8 20.e even 4 2
3825.2.a.bi 4 60.h even 2 1
4080.2.a.bt 4 1.a even 1 1 trivial
4335.2.a.z 4 68.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4080))\):

\( T_{7}^{4} + 4T_{7}^{3} - 17T_{7}^{2} - 80T_{7} - 64 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 31T_{11}^{2} - 112T_{11} - 96 \) Copy content Toggle raw display
\( T_{13}^{4} + 2T_{13}^{3} - 48T_{13}^{2} - 120T_{13} + 208 \) Copy content Toggle raw display
\( T_{19}^{4} + 12T_{19}^{3} + 31T_{19}^{2} + 8T_{19} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( (T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots - 96 \) Copy content Toggle raw display
$13$ \( T^{4} + 2 T^{3} + \cdots + 208 \) Copy content Toggle raw display
$17$ \( (T + 1)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 12 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 2304 \) Copy content Toggle raw display
$29$ \( T^{4} - 4 T^{3} + \cdots + 12 \) Copy content Toggle raw display
$31$ \( T^{4} + 6 T^{3} + \cdots + 128 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} + \cdots + 1124 \) Copy content Toggle raw display
$41$ \( T^{4} - 109 T^{2} + \cdots + 1308 \) Copy content Toggle raw display
$43$ \( T^{4} + 4 T^{3} + \cdots + 512 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + \cdots - 96 \) Copy content Toggle raw display
$53$ \( T^{4} - 109 T^{2} + \cdots + 1308 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + \cdots - 192 \) Copy content Toggle raw display
$61$ \( T^{4} + 2 T^{3} + \cdots + 208 \) Copy content Toggle raw display
$67$ \( T^{4} + 22 T^{3} + \cdots - 13184 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + \cdots - 768 \) Copy content Toggle raw display
$73$ \( T^{4} + 10 T^{3} + \cdots - 108 \) Copy content Toggle raw display
$79$ \( T^{4} - 84 T^{2} + \cdots - 128 \) Copy content Toggle raw display
$83$ \( T^{4} - 20 T^{3} + \cdots - 768 \) Copy content Toggle raw display
$89$ \( T^{4} - 10 T^{3} + \cdots + 3888 \) Copy content Toggle raw display
$97$ \( T^{4} - 12 T^{3} + \cdots + 944 \) Copy content Toggle raw display
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