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Properties

Label 2268.2.j.p
Level $2268$
Weight $2$
Character orbit 2268.j
Analytic conductor $18.110$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2268,2,Mod(757,2268)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2268, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2268.757"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.j (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,0,0,-2,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.1100711784\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{5} + (\beta_1 - 1) q^{7} + ( - \beta_{3} + \beta_{2}) q^{11} + 4 \beta_1 q^{13} + 2 \beta_{3} q^{17} + 2 q^{19} - 2 \beta_{2} q^{23} + (7 \beta_1 - 7) q^{25} + (4 \beta_{3} - 4 \beta_{2}) q^{29}+ \cdots + (2 \beta_1 - 2) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{7} + 8 q^{13} + 8 q^{19} - 14 q^{25} - 4 q^{31} + 44 q^{37} + 2 q^{43} - 2 q^{49} + 24 q^{55} - 16 q^{61} - 22 q^{67} - 64 q^{73} + 26 q^{79} - 24 q^{85} - 16 q^{91} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Display \(\zeta_{12}^j\) in terms of \(\beta_i\)

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

Display \(\beta_i\) in terms of \(\zeta_{12}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{1}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
757.1
0.866025 − 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 − 0.500000i
0 0 0 −1.73205 + 3.00000i 0 −0.500000 − 0.866025i 0 0 0
757.2 0 0 0 1.73205 − 3.00000i 0 −0.500000 − 0.866025i 0 0 0
1513.1 0 0 0 −1.73205 − 3.00000i 0 −0.500000 + 0.866025i 0 0 0
1513.2 0 0 0 1.73205 + 3.00000i 0 −0.500000 + 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.j.p 4
3.b odd 2 1 inner 2268.2.j.p 4
9.c even 3 1 2268.2.a.e ✓ 2
9.c even 3 1 inner 2268.2.j.p 4
9.d odd 6 1 2268.2.a.e ✓ 2
9.d odd 6 1 inner 2268.2.j.p 4
36.f odd 6 1 9072.2.a.bg 2
36.h even 6 1 9072.2.a.bg 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2268.2.a.e ✓ 2 9.c even 3 1
2268.2.a.e ✓ 2 9.d odd 6 1
2268.2.j.p 4 1.a even 1 1 trivial
2268.2.j.p 4 3.b odd 2 1 inner
2268.2.j.p 4 9.c even 3 1 inner
2268.2.j.p 4 9.d odd 6 1 inner
9072.2.a.bg 2 36.f odd 6 1
9072.2.a.bg 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5}^{4} + 12T_{5}^{2} + 144 \) Copy content Toggle raw display
\( T_{11}^{4} + 3T_{11}^{2} + 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$19$ \( (T - 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$29$ \( T^{4} + 48T^{2} + 2304 \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$37$ \( (T - 11)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 12T^{2} + 144 \) Copy content Toggle raw display
$43$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 108 T^{2} + 11664 \) Copy content Toggle raw display
$61$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 11 T + 121)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$73$ \( (T + 16)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 300 T^{2} + 90000 \) Copy content Toggle raw display
$89$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
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