Properties

Label 117.2.z.a
Level $117$
Weight $2$
Character orbit 117.z
Analytic conductor $0.934$
Analytic rank $1$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,2,Mod(5,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([10, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.5");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.z (of order \(12\), degree \(4\), minimal)

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: no
Analytic conductor: \(0.934249703649\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots - 1) q^{2}+ \cdots + ( - 3 \zeta_{12}^{2} + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots - 1) q^{2}+ \cdots + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + \cdots - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{2} - 6 q^{3} - 6 q^{5} + 12 q^{6} - 4 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{2} - 6 q^{3} - 6 q^{5} + 12 q^{6} - 4 q^{7} + 6 q^{9} - 12 q^{11} - 6 q^{13} + 24 q^{14} + 6 q^{15} + 8 q^{16} - 18 q^{18} - 16 q^{19} + 24 q^{20} + 24 q^{22} - 12 q^{24} - 32 q^{28} - 12 q^{29} - 2 q^{31} + 24 q^{33} + 6 q^{34} + 4 q^{37} + 18 q^{39} - 24 q^{40} - 18 q^{41} - 24 q^{42} + 12 q^{46} + 6 q^{47} + 6 q^{50} + 16 q^{52} + 18 q^{54} + 48 q^{55} + 24 q^{57} + 12 q^{58} - 18 q^{59} - 48 q^{60} - 6 q^{61} + 12 q^{63} + 30 q^{65} - 72 q^{66} - 16 q^{67} - 24 q^{68} - 24 q^{70} + 36 q^{72} - 28 q^{73} - 12 q^{74} + 32 q^{76} - 30 q^{78} - 6 q^{79} - 18 q^{81} + 6 q^{83} + 48 q^{84} - 6 q^{85} + 54 q^{86} + 24 q^{87} + 8 q^{91} - 24 q^{92} + 6 q^{93} - 12 q^{94} - 20 q^{97} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\zeta_{12}^{3}\) \(\zeta_{12}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−0.633975 + 2.36603i −1.50000 0.866025i −3.46410 2.00000i −2.36603 + 0.633975i 3.00000 3.00000i 0.732051 2.73205i 3.46410 3.46410i 1.50000 + 2.59808i 6.00000i
47.1 −0.633975 2.36603i −1.50000 + 0.866025i −3.46410 + 2.00000i −2.36603 0.633975i 3.00000 + 3.00000i 0.732051 + 2.73205i 3.46410 + 3.46410i 1.50000 2.59808i 6.00000i
83.1 −2.36603 + 0.633975i −1.50000 + 0.866025i 3.46410 2.00000i −0.633975 + 2.36603i 3.00000 3.00000i −2.73205 + 0.732051i −3.46410 + 3.46410i 1.50000 2.59808i 6.00000i
86.1 −2.36603 0.633975i −1.50000 0.866025i 3.46410 + 2.00000i −0.633975 2.36603i 3.00000 + 3.00000i −2.73205 0.732051i −3.46410 3.46410i 1.50000 + 2.59808i 6.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
13.d odd 4 1 inner
117.z even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 117.2.z.a 4
3.b odd 2 1 351.2.bc.a 4
9.c even 3 1 351.2.bc.a 4
9.d odd 6 1 inner 117.2.z.a 4
13.d odd 4 1 inner 117.2.z.a 4
39.f even 4 1 351.2.bc.a 4
117.y odd 12 1 351.2.bc.a 4
117.z even 12 1 inner 117.2.z.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.z.a 4 1.a even 1 1 trivial
117.2.z.a 4 9.d odd 6 1 inner
117.2.z.a 4 13.d odd 4 1 inner
117.2.z.a 4 117.z even 12 1 inner
351.2.bc.a 4 3.b odd 2 1
351.2.bc.a 4 9.c even 3 1
351.2.bc.a 4 39.f even 4 1
351.2.bc.a 4 117.y odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6T_{2}^{3} + 18T_{2}^{2} + 36T_{2} + 36 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{4} + 12 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + \cdots + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8 T + 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$29$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$37$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 18 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$43$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$47$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$53$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 18 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$61$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 16 T^{3} + \cdots + 16384 \) Copy content Toggle raw display
$71$ \( T^{4} + 576 \) Copy content Toggle raw display
$73$ \( (T^{2} + 14 T + 98)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$89$ \( T^{4} + 46656 \) Copy content Toggle raw display
$97$ \( T^{4} + 20 T^{3} + \cdots + 40000 \) Copy content Toggle raw display
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