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Properties

Label 240.9.c.b
Level $240$
Weight $9$
Character orbit 240.c
Self dual yes
Analytic conductor $97.771$
Analytic rank $0$
Dimension $1$
CM discriminant -15
Inner twists $2$

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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] = [240,9,Mod(209,240)] 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("240.209"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(240, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 240.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,81,0,-625,0,0,0,6561] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(97.7708664147\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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)
 
\(f(q)\) \(=\) \( q + 81 q^{3} - 625 q^{5} + 6561 q^{9} - 50625 q^{15} + 21118 q^{17} + 203998 q^{19} - 550078 q^{23} + 390625 q^{25} + 531441 q^{27} - 1831682 q^{31} - 4100625 q^{45} + 8065922 q^{47} + 5764801 q^{49} + 1710558 q^{51}+ \cdots - 127498750 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
0
0 81.0000 0 −625.000 0 0 0 6561.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.9.c.b 1
3.b odd 2 1 240.9.c.a 1
4.b odd 2 1 15.9.d.a ✓ 1
5.b even 2 1 240.9.c.a 1
12.b even 2 1 15.9.d.b yes 1
15.d odd 2 1 CM 240.9.c.b 1
20.d odd 2 1 15.9.d.b yes 1
20.e even 4 2 75.9.c.d 2
60.h even 2 1 15.9.d.a ✓ 1
60.l odd 4 2 75.9.c.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.d.a ✓ 1 4.b odd 2 1
15.9.d.a ✓ 1 60.h even 2 1
15.9.d.b yes 1 12.b even 2 1
15.9.d.b yes 1 20.d odd 2 1
75.9.c.d 2 20.e even 4 2
75.9.c.d 2 60.l odd 4 2
240.9.c.a 1 3.b odd 2 1
240.9.c.a 1 5.b even 2 1
240.9.c.b 1 1.a even 1 1 trivial
240.9.c.b 1 15.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{9}^{\mathrm{new}}(240, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{17} - 21118 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 81 \) Copy content Toggle raw display
$5$ \( T + 625 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T - 21118 \) Copy content Toggle raw display
$19$ \( T - 203998 \) Copy content Toggle raw display
$23$ \( T + 550078 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T + 1831682 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T - 8065922 \) Copy content Toggle raw display
$53$ \( T - 12619678 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 14324642 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T - 69617278 \) Copy content Toggle raw display
$83$ \( T - 3847202 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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