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Properties

Label 270.4.a.k
Level $270$
Weight $4$
Character orbit 270.a
Self dual yes
Analytic conductor $15.931$
Analytic rank $1$
Dimension $1$
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] = [270,4,Mod(1,270)] 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("270.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(270, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,2,0,4,5,0,-34] 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: \(15.9305157015\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{2} + 4 q^{4} + 5 q^{5} - 34 q^{7} + 8 q^{8} + 10 q^{10} - 48 q^{11} - 70 q^{13} - 68 q^{14} + 16 q^{16} - 27 q^{17} + 119 q^{19} + 20 q^{20} - 96 q^{22} - 51 q^{23} + 25 q^{25} - 140 q^{26} - 136 q^{28}+ \cdots + 1626 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
0
2.00000 0 4.00000 5.00000 0 −34.0000 8.00000 0 10.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 270.4.a.k yes 1
3.b odd 2 1 270.4.a.a ✓ 1
4.b odd 2 1 2160.4.a.t 1
5.b even 2 1 1350.4.a.n 1
5.c odd 4 2 1350.4.c.c 2
9.c even 3 2 810.4.e.d 2
9.d odd 6 2 810.4.e.x 2
12.b even 2 1 2160.4.a.j 1
15.d odd 2 1 1350.4.a.bb 1
15.e even 4 2 1350.4.c.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
270.4.a.a ✓ 1 3.b odd 2 1
270.4.a.k yes 1 1.a even 1 1 trivial
810.4.e.d 2 9.c even 3 2
810.4.e.x 2 9.d odd 6 2
1350.4.a.n 1 5.b even 2 1
1350.4.a.bb 1 15.d odd 2 1
1350.4.c.c 2 5.c odd 4 2
1350.4.c.r 2 15.e even 4 2
2160.4.a.j 1 12.b even 2 1
2160.4.a.t 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(270))\):

\( T_{7} + 34 \) Copy content Toggle raw display
\( T_{11} + 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5 \) Copy content Toggle raw display
$7$ \( T + 34 \) Copy content Toggle raw display
$11$ \( T + 48 \) Copy content Toggle raw display
$13$ \( T + 70 \) Copy content Toggle raw display
$17$ \( T + 27 \) Copy content Toggle raw display
$19$ \( T - 119 \) Copy content Toggle raw display
$23$ \( T + 51 \) Copy content Toggle raw display
$29$ \( T + 30 \) Copy content Toggle raw display
$31$ \( T + 133 \) Copy content Toggle raw display
$37$ \( T - 218 \) Copy content Toggle raw display
$41$ \( T - 156 \) Copy content Toggle raw display
$43$ \( T + 88 \) Copy content Toggle raw display
$47$ \( T + 516 \) Copy content Toggle raw display
$53$ \( T + 639 \) Copy content Toggle raw display
$59$ \( T + 654 \) Copy content Toggle raw display
$61$ \( T - 461 \) Copy content Toggle raw display
$67$ \( T - 182 \) Copy content Toggle raw display
$71$ \( T - 900 \) Copy content Toggle raw display
$73$ \( T - 704 \) Copy content Toggle raw display
$79$ \( T + 1375 \) Copy content Toggle raw display
$83$ \( T - 915 \) Copy content Toggle raw display
$89$ \( T + 1116 \) Copy content Toggle raw display
$97$ \( T + 16 \) Copy content Toggle raw display
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