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Properties

Label 121.16.a.a
Level $121$
Weight $16$
Character orbit 121.a
Self dual yes
Analytic conductor $172.659$
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] = [121,16,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 121 = 11^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 121.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-216] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(172.659141171\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1)
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)
 
\(f(q)\) \(=\) \( q - 216 q^{2} - 3348 q^{3} + 13888 q^{4} + 52110 q^{5} + 723168 q^{6} - 2822456 q^{7} + 4078080 q^{8} - 3139803 q^{9} - 11255760 q^{10} - 46497024 q^{12} + 190073338 q^{13} + 609650496 q^{14} - 174464280 q^{15}+ \cdots - 695238414190488 q^{98}+O(q^{100}) \) 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
−216.000 −3348.00 13888.0 52110.0 723168. −2.82246e6 4.07808e6 −3.13980e6 −1.12558e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \( -1 \)

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 121.16.a.a 1
11.b odd 2 1 1.16.a.a ✓ 1
33.d even 2 1 9.16.a.a 1
44.c even 2 1 16.16.a.d 1
55.d odd 2 1 25.16.a.a 1
55.e even 4 2 25.16.b.a 2
77.b even 2 1 49.16.a.a 1
77.h odd 6 2 49.16.c.c 2
77.i even 6 2 49.16.c.b 2
88.b odd 2 1 64.16.a.i 1
88.g even 2 1 64.16.a.c 1
132.d odd 2 1 144.16.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.16.a.a ✓ 1 11.b odd 2 1
9.16.a.a 1 33.d even 2 1
16.16.a.d 1 44.c even 2 1
25.16.a.a 1 55.d odd 2 1
25.16.b.a 2 55.e even 4 2
49.16.a.a 1 77.b even 2 1
49.16.c.b 2 77.i even 6 2
49.16.c.c 2 77.h odd 6 2
64.16.a.c 1 88.g even 2 1
64.16.a.i 1 88.b odd 2 1
121.16.a.a 1 1.a even 1 1 trivial
144.16.a.f 1 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 216 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(121))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 216 \) Copy content Toggle raw display
$3$ \( T + 3348 \) Copy content Toggle raw display
$5$ \( T - 52110 \) Copy content Toggle raw display
$7$ \( T + 2822456 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 190073338 \) Copy content Toggle raw display
$17$ \( T + 1646527986 \) Copy content Toggle raw display
$19$ \( T + 1563257180 \) Copy content Toggle raw display
$23$ \( T - 9451116072 \) Copy content Toggle raw display
$29$ \( T - 36902568330 \) Copy content Toggle raw display
$31$ \( T - 71588483552 \) Copy content Toggle raw display
$37$ \( T + 1033652081554 \) Copy content Toggle raw display
$41$ \( T + 1641974018202 \) Copy content Toggle raw display
$43$ \( T - 492403109308 \) Copy content Toggle raw display
$47$ \( T + 3410684952624 \) Copy content Toggle raw display
$53$ \( T - 6797151655902 \) Copy content Toggle raw display
$59$ \( T - 9858856815540 \) Copy content Toggle raw display
$61$ \( T + 4931842626902 \) Copy content Toggle raw display
$67$ \( T + 28837826625364 \) Copy content Toggle raw display
$71$ \( T - 125050114914552 \) Copy content Toggle raw display
$73$ \( T - 82171455513478 \) Copy content Toggle raw display
$79$ \( T - 25413078694480 \) Copy content Toggle raw display
$83$ \( T - 281736730890468 \) Copy content Toggle raw display
$89$ \( T - 715618564776810 \) Copy content Toggle raw display
$97$ \( T - 612786136081826 \) Copy content Toggle raw display
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