Properties

Label 117.2.a.a
Level $117$
Weight $2$
Character orbit 117.a
Self dual yes
Analytic conductor $0.934$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(1,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.934249703649\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{4} - 2 q^{5} - 4 q^{7} + 3 q^{8} + 2 q^{10} - 4 q^{11} + q^{13} + 4 q^{14} - q^{16} - 2 q^{17} + 2 q^{20} + 4 q^{22} - q^{25} - q^{26} + 4 q^{28} + 10 q^{29} + 4 q^{31} - 5 q^{32} + 2 q^{34} + 8 q^{35} - 2 q^{37} - 6 q^{40} - 6 q^{41} - 12 q^{43} + 4 q^{44} + 9 q^{49} + q^{50} - q^{52} - 6 q^{53} + 8 q^{55} - 12 q^{56} - 10 q^{58} - 12 q^{59} - 2 q^{61} - 4 q^{62} + 7 q^{64} - 2 q^{65} - 8 q^{67} + 2 q^{68} - 8 q^{70} + 2 q^{73} + 2 q^{74} + 16 q^{77} + 8 q^{79} + 2 q^{80} + 6 q^{82} - 4 q^{83} + 4 q^{85} + 12 q^{86} - 12 q^{88} + 2 q^{89} - 4 q^{91} + 10 q^{97} - 9 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
−1.00000 0 −1.00000 −2.00000 0 −4.00000 3.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(13\) \(-1\)

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 117.2.a.a 1
3.b odd 2 1 39.2.a.a 1
4.b odd 2 1 1872.2.a.h 1
5.b even 2 1 2925.2.a.p 1
5.c odd 4 2 2925.2.c.e 2
7.b odd 2 1 5733.2.a.e 1
8.b even 2 1 7488.2.a.bl 1
8.d odd 2 1 7488.2.a.by 1
9.c even 3 2 1053.2.e.d 2
9.d odd 6 2 1053.2.e.b 2
12.b even 2 1 624.2.a.i 1
13.b even 2 1 1521.2.a.e 1
13.d odd 4 2 1521.2.b.b 2
15.d odd 2 1 975.2.a.f 1
15.e even 4 2 975.2.c.f 2
21.c even 2 1 1911.2.a.f 1
24.f even 2 1 2496.2.a.e 1
24.h odd 2 1 2496.2.a.q 1
33.d even 2 1 4719.2.a.c 1
39.d odd 2 1 507.2.a.a 1
39.f even 4 2 507.2.b.a 2
39.h odd 6 2 507.2.e.b 2
39.i odd 6 2 507.2.e.a 2
39.k even 12 4 507.2.j.e 4
156.h even 2 1 8112.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a 1 3.b odd 2 1
117.2.a.a 1 1.a even 1 1 trivial
507.2.a.a 1 39.d odd 2 1
507.2.b.a 2 39.f even 4 2
507.2.e.a 2 39.i odd 6 2
507.2.e.b 2 39.h odd 6 2
507.2.j.e 4 39.k even 12 4
624.2.a.i 1 12.b even 2 1
975.2.a.f 1 15.d odd 2 1
975.2.c.f 2 15.e even 4 2
1053.2.e.b 2 9.d odd 6 2
1053.2.e.d 2 9.c even 3 2
1521.2.a.e 1 13.b even 2 1
1521.2.b.b 2 13.d odd 4 2
1872.2.a.h 1 4.b odd 2 1
1911.2.a.f 1 21.c even 2 1
2496.2.a.e 1 24.f even 2 1
2496.2.a.q 1 24.h odd 2 1
2925.2.a.p 1 5.b even 2 1
2925.2.c.e 2 5.c odd 4 2
4719.2.a.c 1 33.d even 2 1
5733.2.a.e 1 7.b odd 2 1
7488.2.a.bl 1 8.b even 2 1
7488.2.a.by 1 8.d odd 2 1
8112.2.a.s 1 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(117))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 2 \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T + 4 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 10 \) Copy content Toggle raw display
$31$ \( T - 4 \) Copy content Toggle raw display
$37$ \( T + 2 \) Copy content Toggle raw display
$41$ \( T + 6 \) Copy content Toggle raw display
$43$ \( T + 12 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T + 12 \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T + 8 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T + 4 \) Copy content Toggle raw display
$89$ \( T - 2 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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