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Properties

Label 1116.2.g.g
Level $1116$
Weight $2$
Character orbit 1116.g
Analytic conductor $8.911$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1116,2,Mod(991,1116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1116, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1116.991");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1116 = 2^{2} \cdot 3^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1116.g (of order \(2\), degree \(1\), 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: \(8.91130486557\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{5}) q^{2} + ( - \beta_{3} + 1) q^{4} + (2 \beta_{7} + \beta_{5}) q^{5} + (\beta_{7} + 3 \beta_{5}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{5}) q^{2} + ( - \beta_{3} + 1) q^{4} + (2 \beta_{7} + \beta_{5}) q^{5} + (\beta_{7} + 3 \beta_{5}) q^{8} + ( - \beta_{3} + 3) q^{10} + ( - \beta_{4} - 2 \beta_{2}) q^{11} + ( - \beta_{6} - 2 \beta_1) q^{13} + ( - 3 \beta_{3} - 1) q^{16} + 3 \beta_{4} q^{17} + (2 \beta_{3} + 1) q^{19} + (3 \beta_{7} + 5 \beta_{5}) q^{20} + ( - 3 \beta_{6} + \beta_1) q^{22} + (2 \beta_{4} + 4 \beta_{2}) q^{23} + 2 q^{25} + (3 \beta_{4} - \beta_{2}) q^{26} - 4 \beta_{4} q^{29} + (\beta_{6} - 4 \beta_{3} - 2) q^{31} + ( - \beta_{7} + 5 \beta_{5}) q^{32} + ( - 3 \beta_{6} - 3 \beta_1) q^{34} + (2 \beta_{6} + 4 \beta_1) q^{37} + (\beta_{7} - 3 \beta_{5}) q^{38} + ( - 5 \beta_{3} + 1) q^{40} + ( - 4 \beta_{7} - 2 \beta_{5}) q^{41} + 2 \beta_{6} q^{43} + ( - 5 \beta_{4} - 3 \beta_{2}) q^{44} + (6 \beta_{6} - 2 \beta_1) q^{46} - 11 \beta_{5} q^{47} + 7 q^{49} + (2 \beta_{7} + 2 \beta_{5}) q^{50} + ( - 5 \beta_{6} - 3 \beta_1) q^{52} - 4 \beta_{4} q^{53} - 7 \beta_{6} q^{55} + (4 \beta_{6} + 4 \beta_1) q^{58} - 8 \beta_{5} q^{59} + (\beta_{6} + 2 \beta_1) q^{61} + ( - 2 \beta_{7} + 6 \beta_{5} + \cdots + \beta_{2}) q^{62}+ \cdots + (7 \beta_{7} + 7 \beta_{5}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{4} + 28 q^{10} + 4 q^{16} + 16 q^{25} + 28 q^{40} + 56 q^{49} - 36 q^{64} + 28 q^{76} - 56 q^{82} + 44 q^{94} - 88 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 15\nu^{5} + 25\nu^{3} + 72\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 5\nu^{4} - 5\nu^{2} + 2 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 7 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{6} + 5\nu^{4} + 15\nu^{2} + 26 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} - 16\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - \nu^{5} + \nu^{3} + 8\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} + 15\nu^{5} + 25\nu^{3} + 48\nu ) / 40 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{3} - \beta_{2} - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{5} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{4} + 3\beta_{3} + 3\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 5\beta_{6} - 5\beta_{5} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -5\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} + 3\beta_{6} - 3\beta_{5} - 7\beta_1 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1116\mathbb{Z}\right)^\times\).

\(n\) \(497\) \(559\) \(685\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
991.1
0.228425 + 1.39564i
1.09445 − 0.895644i
0.228425 − 1.39564i
1.09445 + 0.895644i
−0.228425 + 1.39564i
−1.09445 − 0.895644i
−0.228425 − 1.39564i
−1.09445 + 0.895644i
−1.32288 − 0.500000i 0 1.50000 + 1.32288i −2.64575 0 0 −1.32288 − 2.50000i 0 3.50000 + 1.32288i
991.2 −1.32288 − 0.500000i 0 1.50000 + 1.32288i −2.64575 0 0 −1.32288 − 2.50000i 0 3.50000 + 1.32288i
991.3 −1.32288 + 0.500000i 0 1.50000 − 1.32288i −2.64575 0 0 −1.32288 + 2.50000i 0 3.50000 − 1.32288i
991.4 −1.32288 + 0.500000i 0 1.50000 − 1.32288i −2.64575 0 0 −1.32288 + 2.50000i 0 3.50000 − 1.32288i
991.5 1.32288 − 0.500000i 0 1.50000 − 1.32288i 2.64575 0 0 1.32288 − 2.50000i 0 3.50000 − 1.32288i
991.6 1.32288 − 0.500000i 0 1.50000 − 1.32288i 2.64575 0 0 1.32288 − 2.50000i 0 3.50000 − 1.32288i
991.7 1.32288 + 0.500000i 0 1.50000 + 1.32288i 2.64575 0 0 1.32288 + 2.50000i 0 3.50000 + 1.32288i
991.8 1.32288 + 0.500000i 0 1.50000 + 1.32288i 2.64575 0 0 1.32288 + 2.50000i 0 3.50000 + 1.32288i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 991.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
31.b odd 2 1 inner
93.c even 2 1 inner
124.d even 2 1 inner
372.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1116.2.g.g ✓ 8
3.b odd 2 1 inner 1116.2.g.g ✓ 8
4.b odd 2 1 inner 1116.2.g.g ✓ 8
12.b even 2 1 inner 1116.2.g.g ✓ 8
31.b odd 2 1 inner 1116.2.g.g ✓ 8
93.c even 2 1 inner 1116.2.g.g ✓ 8
124.d even 2 1 inner 1116.2.g.g ✓ 8
372.b odd 2 1 inner 1116.2.g.g ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1116.2.g.g ✓ 8 1.a even 1 1 trivial
1116.2.g.g ✓ 8 3.b odd 2 1 inner
1116.2.g.g ✓ 8 4.b odd 2 1 inner
1116.2.g.g ✓ 8 12.b even 2 1 inner
1116.2.g.g ✓ 8 31.b odd 2 1 inner
1116.2.g.g ✓ 8 93.c even 2 1 inner
1116.2.g.g ✓ 8 124.d even 2 1 inner
1116.2.g.g ✓ 8 372.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1116, [\chi])\):

\( T_{5}^{2} - 7 \) Copy content Toggle raw display
\( T_{11}^{2} - 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 3 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 7)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} - 21)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 27)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 84)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 50 T^{2} + 961)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 84)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 28)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 121)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 21)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 175)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 49)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 84)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 75)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 21)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$97$ \( (T + 11)^{8} \) Copy content Toggle raw display
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