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Properties

Label 117.2.ba.a
Level $117$
Weight $2$
Character orbit 117.ba
Analytic conductor $0.934$
Analytic rank $0$
Dimension $8$
CM no
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(71,117)]
 
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([6, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 117.ba (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: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{4}) q^{2} - 2 \beta_1 q^{4} + (\beta_{6} + \beta_{4}) q^{5} + (\beta_{3} - \beta_{2}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{4}) q^{2} - 2 \beta_1 q^{4} + (\beta_{6} + \beta_{4}) q^{5} + (\beta_{3} - \beta_{2}) q^{7} + (2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 4) q^{10} + ( - 3 \beta_{7} - 3 \beta_{6} + \cdots + 2 \beta_{4}) q^{11}+ \cdots + (4 \beta_{7} - 6 \beta_{6} + \cdots - 6 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{7} - 24 q^{10} + 8 q^{13} - 16 q^{16} + 20 q^{19} - 8 q^{22} + 8 q^{28} + 44 q^{31} + 48 q^{34} - 28 q^{37} - 12 q^{43} - 24 q^{46} - 12 q^{49} - 24 q^{52} - 32 q^{55} - 8 q^{58} - 32 q^{61} - 40 q^{67} + 16 q^{70} + 44 q^{73} + 40 q^{76} + 16 q^{79} + 48 q^{82} + 72 q^{85} - 28 q^{91} + 40 q^{94} - 52 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

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

\(n\) \(28\) \(92\)
\(\chi(n)\) \(\beta_{1}\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.258819 + 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 − 0.965926i
−0.258819 + 0.965926i
0.965926 − 0.258819i
−0.965926 + 0.258819i
−1.93185 + 0.517638i 0 1.73205 − 1.00000i 0.517638 + 0.517638i 0 −0.500000 + 1.86603i 0 0 −1.26795 − 0.732051i
71.2 1.93185 − 0.517638i 0 1.73205 − 1.00000i −0.517638 − 0.517638i 0 −0.500000 + 1.86603i 0 0 −1.26795 − 0.732051i
80.1 −0.517638 + 1.93185i 0 −1.73205 − 1.00000i 1.93185 + 1.93185i 0 −0.500000 + 0.133975i 0 0 −4.73205 + 2.73205i
80.2 0.517638 − 1.93185i 0 −1.73205 − 1.00000i −1.93185 − 1.93185i 0 −0.500000 + 0.133975i 0 0 −4.73205 + 2.73205i
89.1 −1.93185 − 0.517638i 0 1.73205 + 1.00000i 0.517638 − 0.517638i 0 −0.500000 − 1.86603i 0 0 −1.26795 + 0.732051i
89.2 1.93185 + 0.517638i 0 1.73205 + 1.00000i −0.517638 + 0.517638i 0 −0.500000 − 1.86603i 0 0 −1.26795 + 0.732051i
98.1 −0.517638 − 1.93185i 0 −1.73205 + 1.00000i 1.93185 − 1.93185i 0 −0.500000 − 0.133975i 0 0 −4.73205 − 2.73205i
98.2 0.517638 + 1.93185i 0 −1.73205 + 1.00000i −1.93185 + 1.93185i 0 −0.500000 − 0.133975i 0 0 −4.73205 − 2.73205i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
13.f odd 12 1 inner
39.k 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.ba.a ✓ 8
3.b odd 2 1 inner 117.2.ba.a ✓ 8
13.c even 3 1 1521.2.i.d 8
13.e even 6 1 1521.2.i.e 8
13.f odd 12 1 inner 117.2.ba.a ✓ 8
13.f odd 12 1 1521.2.i.d 8
13.f odd 12 1 1521.2.i.e 8
39.h odd 6 1 1521.2.i.e 8
39.i odd 6 1 1521.2.i.d 8
39.k even 12 1 inner 117.2.ba.a ✓ 8
39.k even 12 1 1521.2.i.d 8
39.k even 12 1 1521.2.i.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
117.2.ba.a ✓ 8 1.a even 1 1 trivial
117.2.ba.a ✓ 8 3.b odd 2 1 inner
117.2.ba.a ✓ 8 13.f odd 12 1 inner
117.2.ba.a ✓ 8 39.k even 12 1 inner
1521.2.i.d 8 13.c even 3 1
1521.2.i.d 8 13.f odd 12 1
1521.2.i.d 8 39.i odd 6 1
1521.2.i.d 8 39.k even 12 1
1521.2.i.e 8 13.e even 6 1
1521.2.i.e 8 13.f odd 12 1
1521.2.i.e 8 39.h odd 6 1
1521.2.i.e 8 39.k even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 16T_{2}^{4} + 256 \) acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 16T^{4} + 256 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 56T^{4} + 16 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} + 5 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 36 T^{6} + \cdots + 456976 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 48 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$19$ \( (T^{4} - 10 T^{3} + \cdots + 484)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 22 T^{3} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 14 T^{3} + \cdots + 676)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 48 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$43$ \( (T^{4} + 6 T^{3} + \cdots + 1089)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 2168 T^{4} + 234256 \) Copy content Toggle raw display
$53$ \( (T^{4} + 112 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 180 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$61$ \( (T^{4} + 16 T^{3} + \cdots + 1369)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 20 T^{3} + \cdots + 5329)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 420 T^{6} + \cdots + 456976 \) Copy content Toggle raw display
$73$ \( (T^{4} - 22 T^{3} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 23)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 3104T^{4} + 256 \) Copy content Toggle raw display
$89$ \( T^{8} + 192 T^{6} + \cdots + 7311616 \) Copy content Toggle raw display
$97$ \( (T^{4} + 26 T^{3} + \cdots + 3721)^{2} \) Copy content Toggle raw display
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