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Properties

Label 1980.4.a.j
Level $1980$
Weight $4$
Character orbit 1980.a
Self dual yes
Analytic conductor $116.824$
Analytic rank $1$
Dimension $3$
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] = [1980,4,Mod(1,1980)] 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("1980.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1980.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,0,0,-15,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(116.823781811\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.953556.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 229x - 161 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 660)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 5 q^{5} + ( - \beta_1 - 3) q^{7} + 11 q^{11} + (\beta_{2} - \beta_1 + 28) q^{13} + (2 \beta_{2} + 3 \beta_1 - 7) q^{17} + ( - \beta_{2} + 2 \beta_1 - 31) q^{19} + ( - 4 \beta_{2} + 2 \beta_1 - 30) q^{23}+ \cdots + (10 \beta_{2} - 14 \beta_1 + 938) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 8 q^{7} + 33 q^{11} + 86 q^{13} - 22 q^{17} - 96 q^{19} - 96 q^{23} + 75 q^{25} - 86 q^{29} - 152 q^{31} + 40 q^{35} + 26 q^{37} - 274 q^{41} + 4 q^{43} + 8 q^{47} + 827 q^{49} - 162 q^{53}+ \cdots + 2838 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 229x - 161 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{2} - 8\nu - 301 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7\beta_{2} + 4\beta _1 + 305 ) / 2 \) 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
15.9703
−0.706780
−14.2636
0 0 0 −5.00000 0 −33.9407 0 0 0
1.2 0 0 0 −5.00000 0 −0.586440 0 0 0
1.3 0 0 0 −5.00000 0 26.5271 0 0 0
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( -1 \)

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 1980.4.a.j 3
3.b odd 2 1 660.4.a.e ✓ 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
660.4.a.e ✓ 3 3.b odd 2 1
1980.4.a.j 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1980))\):

\( T_{7}^{3} + 8T_{7}^{2} - 896T_{7} - 528 \) Copy content Toggle raw display
\( T_{17}^{3} + 22T_{17}^{2} - 11076T_{17} - 350952 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( (T + 5)^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 8 T^{2} + \cdots - 528 \) Copy content Toggle raw display
$11$ \( (T - 11)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 86 T^{2} + \cdots + 8984 \) Copy content Toggle raw display
$17$ \( T^{3} + 22 T^{2} + \cdots - 350952 \) Copy content Toggle raw display
$19$ \( T^{3} + 96 T^{2} + \cdots + 14560 \) Copy content Toggle raw display
$23$ \( T^{3} + 96 T^{2} + \cdots - 418176 \) Copy content Toggle raw display
$29$ \( T^{3} + 86 T^{2} + \cdots - 103800 \) Copy content Toggle raw display
$31$ \( T^{3} + 152 T^{2} + \cdots - 3292800 \) Copy content Toggle raw display
$37$ \( T^{3} - 26 T^{2} + \cdots - 752248 \) Copy content Toggle raw display
$41$ \( T^{3} + 274 T^{2} + \cdots - 1761000 \) Copy content Toggle raw display
$43$ \( T^{3} - 4 T^{2} + \cdots - 3351120 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} + \cdots + 27834240 \) Copy content Toggle raw display
$53$ \( T^{3} + 162 T^{2} + \cdots - 28879848 \) Copy content Toggle raw display
$59$ \( T^{3} - 132 T^{2} + \cdots + 732096 \) Copy content Toggle raw display
$61$ \( T^{3} - 1450 T^{2} + \cdots + 142213320 \) Copy content Toggle raw display
$67$ \( T^{3} - 380 T^{2} + \cdots + 17269952 \) Copy content Toggle raw display
$71$ \( T^{3} + 864 T^{2} + \cdots - 132036480 \) Copy content Toggle raw display
$73$ \( T^{3} + 158 T^{2} + \cdots + 178891560 \) Copy content Toggle raw display
$79$ \( T^{3} - 1428 T^{2} + \cdots + 274789408 \) Copy content Toggle raw display
$83$ \( T^{3} + 944 T^{2} + \cdots - 124496976 \) Copy content Toggle raw display
$89$ \( T^{3} + 2486 T^{2} + \cdots + 454201800 \) Copy content Toggle raw display
$97$ \( T^{3} - 2838 T^{2} + \cdots - 570278792 \) Copy content Toggle raw display
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