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Properties

Label 1116.2.g.b
Level $1116$
Weight $2$
Character orbit 1116.g
Analytic conductor $8.911$
Analytic rank $0$
Dimension $4$
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] = [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: \(4\)
Coefficient field: \(\Q(\sqrt{6}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 124)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_1 - 2) q^{4} + 2 q^{5} + ( - \beta_1 + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_1 - 2) q^{4} + 2 q^{5} + ( - \beta_1 + 2) q^{8} + 2 \beta_1 q^{10} + \beta_{3} q^{11} + (\beta_{3} + 2 \beta_{2}) q^{13} + (3 \beta_1 + 2) q^{16} + ( - 4 \beta_1 - 2) q^{19} + ( - 2 \beta_1 - 4) q^{20} + ( - \beta_{3} - \beta_{2}) q^{22} - 2 \beta_{3} q^{23} - q^{25} + (3 \beta_{3} - \beta_{2}) q^{26} + (\beta_{3} + 2 \beta_{2}) q^{29} + ( - 2 \beta_{3} + 2 \beta_1 + 1) q^{31} + ( - \beta_1 - 6) q^{32} + (\beta_{3} + 2 \beta_{2}) q^{37} + (2 \beta_1 + 8) q^{38} + ( - 2 \beta_1 + 4) q^{40} + 8 q^{41} - \beta_{3} q^{43} + ( - \beta_{3} + \beta_{2}) q^{44} + (2 \beta_{3} + 2 \beta_{2}) q^{46} + (8 \beta_1 + 4) q^{47} + 7 q^{49} - \beta_1 q^{50} + ( - 5 \beta_{3} - 3 \beta_{2}) q^{52} + (\beta_{3} + 2 \beta_{2}) q^{53} + 2 \beta_{3} q^{55} + (3 \beta_{3} - \beta_{2}) q^{58} + ( - 4 \beta_1 - 2) q^{59} + ( - \beta_{3} - 2 \beta_{2}) q^{61} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 4) q^{62} + ( - 5 \beta_1 + 2) q^{64} + (2 \beta_{3} + 4 \beta_{2}) q^{65} + ( - 4 \beta_1 - 2) q^{67} + (4 \beta_1 + 2) q^{71} + (2 \beta_{3} + 4 \beta_{2}) q^{73} + (3 \beta_{3} - \beta_{2}) q^{74} + (6 \beta_1 - 4) q^{76} + 4 \beta_{3} q^{79} + (6 \beta_1 + 4) q^{80} + 8 \beta_1 q^{82} + 5 \beta_{3} q^{83} + (\beta_{3} + \beta_{2}) q^{86} + (3 \beta_{3} + \beta_{2}) q^{88} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{89} + (2 \beta_{3} - 2 \beta_{2}) q^{92} + ( - 4 \beta_1 - 16) q^{94} + ( - 8 \beta_1 - 4) q^{95} + 4 q^{97} + 7 \beta_1 q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 6 q^{4} + 8 q^{5} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 6 q^{4} + 8 q^{5} + 10 q^{8} - 4 q^{10} + 2 q^{16} - 12 q^{20} - 4 q^{25} - 22 q^{32} + 28 q^{38} + 20 q^{40} + 32 q^{41} + 28 q^{49} + 2 q^{50} - 14 q^{62} + 18 q^{64} - 28 q^{76} + 4 q^{80} - 16 q^{82} - 56 q^{94} + 16 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 7x^{2} + 8x + 58 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} - 16 ) / 31 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 14\nu^{2} - 31\nu - 54 ) / 31 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{3} + 3\nu^{2} + 31\nu - 16 ) / 31 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{3} - \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{3} + 2\beta_{2} - \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} + 3\beta_{2} - 17\beta _1 - 2 \) 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
−1.94949 + 1.32288i
2.94949 + 1.32288i
−1.94949 − 1.32288i
2.94949 − 1.32288i
−0.500000 − 1.32288i 0 −1.50000 + 1.32288i 2.00000 0 0 2.50000 + 1.32288i 0 −1.00000 − 2.64575i
991.2 −0.500000 − 1.32288i 0 −1.50000 + 1.32288i 2.00000 0 0 2.50000 + 1.32288i 0 −1.00000 − 2.64575i
991.3 −0.500000 + 1.32288i 0 −1.50000 − 1.32288i 2.00000 0 0 2.50000 − 1.32288i 0 −1.00000 + 2.64575i
991.4 −0.500000 + 1.32288i 0 −1.50000 − 1.32288i 2.00000 0 0 2.50000 − 1.32288i 0 −1.00000 + 2.64575i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
31.b odd 2 1 inner
124.d even 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.b 4
3.b odd 2 1 124.2.d.b ✓ 4
4.b odd 2 1 inner 1116.2.g.b 4
12.b even 2 1 124.2.d.b ✓ 4
24.f even 2 1 1984.2.h.e 4
24.h odd 2 1 1984.2.h.e 4
31.b odd 2 1 inner 1116.2.g.b 4
93.c even 2 1 124.2.d.b ✓ 4
124.d even 2 1 inner 1116.2.g.b 4
372.b odd 2 1 124.2.d.b ✓ 4
744.m odd 2 1 1984.2.h.e 4
744.o even 2 1 1984.2.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.2.d.b ✓ 4 3.b odd 2 1
124.2.d.b ✓ 4 12.b even 2 1
124.2.d.b ✓ 4 93.c even 2 1
124.2.d.b ✓ 4 372.b odd 2 1
1116.2.g.b 4 1.a even 1 1 trivial
1116.2.g.b 4 4.b odd 2 1 inner
1116.2.g.b 4 31.b odd 2 1 inner
1116.2.g.b 4 124.d even 2 1 inner
1984.2.h.e 4 24.f even 2 1
1984.2.h.e 4 24.h odd 2 1
1984.2.h.e 4 744.m odd 2 1
1984.2.h.e 4 744.o even 2 1

Hecke kernels

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

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T - 2)^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 34T^{2} + 961 \) Copy content Toggle raw display
$37$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$41$ \( (T - 8)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 42)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 168)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 150)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 168)^{2} \) Copy content Toggle raw display
$97$ \( (T - 4)^{4} \) Copy content Toggle raw display
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