Properties

Label 936.2.g.b
Level $936$
Weight $2$
Character orbit 936.g
Analytic conductor $7.474$
Analytic rank $1$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [936,2,Mod(469,936)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(936, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("936.469");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 936.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: \(7.47399762919\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 104)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{4} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{7} + ( - 2 \beta_{2} - 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - \beta_1) q^{2} + (\beta_{3} + \beta_1) q^{4} + ( - 2 \beta_{3} + 2 \beta_1 - 2) q^{5} + ( - \beta_{3} + \beta_{2} - \beta_1 - 3) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + ( - 2 \beta_{2} - 2 \beta_1 + 4) q^{10} + (\beta_{3} + 3 \beta_{2} - \beta_1 + 1) q^{11} + \beta_{2} q^{13} + (\beta_{3} - 2 \beta_{2} + 3 \beta_1 + 2) q^{14} + ( - 2 \beta_{3} + 2 \beta_1) q^{16} + (2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{17} + (\beta_{3} - \beta_{2} - \beta_1 + 1) q^{19} + ( - 2 \beta_{3} + 8 \beta_{2} - 2 \beta_1) q^{20} + (3 \beta_{3} - 2 \beta_{2} + \beta_1 - 2) q^{22} - 4 q^{23} - 7 q^{25} + (\beta_{3} - \beta_{2}) q^{26} + ( - 2 \beta_{3} - 4 \beta_1 - 2) q^{28} - 2 \beta_{2} q^{29} + (\beta_{3} - \beta_{2} + \beta_1 - 5) q^{31} + ( - 4 \beta_1 + 4) q^{32} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots - 4) q^{34}+ \cdots + ( - 6 \beta_{3} - \beta_{2} + \cdots - 12) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 12 q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 12 q^{7} - 8 q^{8} + 12 q^{10} + 12 q^{14} + 8 q^{16} - 8 q^{17} - 12 q^{22} - 16 q^{23} - 28 q^{25} - 2 q^{26} - 12 q^{28} - 20 q^{31} + 8 q^{32} - 8 q^{34} - 4 q^{38} - 8 q^{41} - 12 q^{44} + 8 q^{46} - 20 q^{47} + 20 q^{49} + 14 q^{50} - 4 q^{52} + 24 q^{55} + 24 q^{56} + 4 q^{58} + 4 q^{62} + 24 q^{68} - 24 q^{70} + 12 q^{71} - 16 q^{73} + 28 q^{74} + 4 q^{76} + 8 q^{79} - 48 q^{80} - 20 q^{82} - 4 q^{86} + 24 q^{88} + 4 q^{94} + 24 q^{95} - 16 q^{97} - 46 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{2} + \zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( -\beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/936\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(469\) \(703\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
469.1
0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
−1.36603 0.366025i 0 1.73205 + 1.00000i 3.46410i 0 −4.73205 −2.00000 2.00000i 0 1.26795 4.73205i
469.2 −1.36603 + 0.366025i 0 1.73205 1.00000i 3.46410i 0 −4.73205 −2.00000 + 2.00000i 0 1.26795 + 4.73205i
469.3 0.366025 1.36603i 0 −1.73205 1.00000i 3.46410i 0 −1.26795 −2.00000 + 2.00000i 0 4.73205 + 1.26795i
469.4 0.366025 + 1.36603i 0 −1.73205 + 1.00000i 3.46410i 0 −1.26795 −2.00000 2.00000i 0 4.73205 1.26795i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 936.2.g.b 4
3.b odd 2 1 104.2.b.b 4
4.b odd 2 1 3744.2.g.b 4
8.b even 2 1 inner 936.2.g.b 4
8.d odd 2 1 3744.2.g.b 4
12.b even 2 1 416.2.b.b 4
24.f even 2 1 416.2.b.b 4
24.h odd 2 1 104.2.b.b 4
48.i odd 4 1 3328.2.a.n 2
48.i odd 4 1 3328.2.a.bd 2
48.k even 4 1 3328.2.a.m 2
48.k even 4 1 3328.2.a.bc 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.2.b.b 4 3.b odd 2 1
104.2.b.b 4 24.h odd 2 1
416.2.b.b 4 12.b even 2 1
416.2.b.b 4 24.f even 2 1
936.2.g.b 4 1.a even 1 1 trivial
936.2.g.b 4 8.b even 2 1 inner
3328.2.a.m 2 48.k even 4 1
3328.2.a.n 2 48.i odd 4 1
3328.2.a.bc 2 48.k even 4 1
3328.2.a.bd 2 48.i odd 4 1
3744.2.g.b 4 4.b odd 2 1
3744.2.g.b 4 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 12 \) acting on \(S_{2}^{\mathrm{new}}(936, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T - 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T + 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 10 T + 22)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
$41$ \( (T^{2} + 4 T - 44)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 56T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T^{2} + 10 T + 22)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$59$ \( T^{4} + 104T^{2} + 4 \) Copy content Toggle raw display
$61$ \( T^{4} + 128T^{2} + 1024 \) Copy content Toggle raw display
$67$ \( T^{4} + 8T^{2} + 4 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 18)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 8 T + 4)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$89$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 8 T - 92)^{2} \) Copy content Toggle raw display
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