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Properties

Label 3276.2.x.h
Level $3276$
Weight $2$
Character orbit 3276.x
Analytic conductor $26.159$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3276,2,Mod(2557,3276)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3276.2557"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3276, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3276 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3276.x (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-6,0,-8,0,0,0,4,0,0] 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: \(26.1589917022\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + (3 \beta_1 - 3) q^{5} + ( - 2 \beta_1 - 1) q^{7} + (\beta_{3} + 1) q^{11} - \beta_{3} q^{13} + (\beta_{3} + \beta_{2} + 2 \beta_1 - 2) q^{17} + 2 q^{19} + ( - \beta_{2} - 2 \beta_1) q^{23} - 4 \beta_1 q^{25}+ \cdots + ( - 3 \beta_{2} + 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{5} - 8 q^{7} + 4 q^{11} - 4 q^{17} + 8 q^{19} - 4 q^{23} - 8 q^{25} - 2 q^{29} + 12 q^{31} + 30 q^{35} - 6 q^{37} + 6 q^{41} + 4 q^{43} + 2 q^{47} + 4 q^{49} - 16 q^{53} - 6 q^{55} - 18 q^{59}+ \cdots + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu + 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - 4\nu^{2} + 28\nu - 9 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 5 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} + 7\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3276\mathbb{Z}\right)^\times\).

\(n\) \(1639\) \(2017\) \(2341\) \(2549\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(-1 + \beta_{1}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2557.1
1.15139 + 1.99426i
−0.651388 − 1.12824i
1.15139 − 1.99426i
−0.651388 + 1.12824i
0 0 0 −1.50000 + 2.59808i 0 −2.00000 − 1.73205i 0 0 0
2557.2 0 0 0 −1.50000 + 2.59808i 0 −2.00000 − 1.73205i 0 0 0
2629.1 0 0 0 −1.50000 − 2.59808i 0 −2.00000 + 1.73205i 0 0 0
2629.2 0 0 0 −1.50000 − 2.59808i 0 −2.00000 + 1.73205i 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
91.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3276.2.x.h yes 4
3.b odd 2 1 3276.2.x.i yes 4
7.c even 3 1 3276.2.u.h ✓ 4
13.c even 3 1 3276.2.u.h ✓ 4
21.h odd 6 1 3276.2.u.i yes 4
39.i odd 6 1 3276.2.u.i yes 4
91.g even 3 1 inner 3276.2.x.h yes 4
273.bm odd 6 1 3276.2.x.i yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3276.2.u.h ✓ 4 7.c even 3 1
3276.2.u.h ✓ 4 13.c even 3 1
3276.2.u.i yes 4 21.h odd 6 1
3276.2.u.i yes 4 39.i odd 6 1
3276.2.x.h yes 4 1.a even 1 1 trivial
3276.2.x.h yes 4 91.g even 3 1 inner
3276.2.x.i yes 4 3.b odd 2 1
3276.2.x.i yes 4 273.bm odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3276, [\chi])\):

\( T_{5}^{2} + 3T_{5} + 9 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 12 \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display
\( T_{31}^{4} - 12T_{31}^{3} + 121T_{31}^{2} - 276T_{31} + 529 \) 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
$5$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 12)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T - 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$29$ \( T^{4} + 2 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$31$ \( T^{4} - 12 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$37$ \( T^{4} + 6 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$41$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$47$ \( T^{4} - 2 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$53$ \( T^{4} + 16 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 6 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2 T - 116)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 6 T^{3} + \cdots + 1849 \) Copy content Toggle raw display
$79$ \( T^{4} - 12 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$83$ \( (T^{2} - 14 T + 36)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 4 T^{3} + \cdots + 81 \) Copy content Toggle raw display
$97$ \( T^{4} - 8 T^{3} + \cdots + 10201 \) Copy content Toggle raw display
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