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Properties

Label 2592.2.i.z
Level $2592$
Weight $2$
Character orbit 2592.i
Analytic conductor $20.697$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [2592,2,Mod(865,2592)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2592, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) 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("2592.865"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-2,0,0,0,0,0,0,0,-6] 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: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
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.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{5} + \beta_{2} q^{7} + \beta_{2} q^{11} + (3 \beta_1 - 3) q^{13} + 5 q^{17} + \beta_{3} q^{19} + (\beta_{3} - \beta_{2}) q^{23} + 4 \beta_1 q^{25} - 5 \beta_1 q^{29} - \beta_{3} q^{35}+ \cdots + 6 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} - 6 q^{13} + 20 q^{17} + 8 q^{25} - 10 q^{29} - 20 q^{37} + 4 q^{41} - 34 q^{49} + 8 q^{53} + 26 q^{61} - 6 q^{65} + 12 q^{73} - 48 q^{77} - 10 q^{85} + 52 q^{89} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

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

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

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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
−1.22474 + 0.707107i
1.22474 − 0.707107i
−1.22474 − 0.707107i
1.22474 + 0.707107i
0 0 0 −0.500000 − 0.866025i 0 −2.44949 + 4.24264i 0 0 0
865.2 0 0 0 −0.500000 − 0.866025i 0 2.44949 − 4.24264i 0 0 0
1729.1 0 0 0 −0.500000 + 0.866025i 0 −2.44949 − 4.24264i 0 0 0
1729.2 0 0 0 −0.500000 + 0.866025i 0 2.44949 + 4.24264i 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
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.i.z 4
3.b odd 2 1 2592.2.i.bd 4
4.b odd 2 1 inner 2592.2.i.z 4
9.c even 3 1 2592.2.a.r yes 2
9.c even 3 1 inner 2592.2.i.z 4
9.d odd 6 1 2592.2.a.m ✓ 2
9.d odd 6 1 2592.2.i.bd 4
12.b even 2 1 2592.2.i.bd 4
36.f odd 6 1 2592.2.a.r yes 2
36.f odd 6 1 inner 2592.2.i.z 4
36.h even 6 1 2592.2.a.m ✓ 2
36.h even 6 1 2592.2.i.bd 4
72.j odd 6 1 5184.2.a.bv 2
72.l even 6 1 5184.2.a.bv 2
72.n even 6 1 5184.2.a.bk 2
72.p odd 6 1 5184.2.a.bk 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2592.2.a.m ✓ 2 9.d odd 6 1
2592.2.a.m ✓ 2 36.h even 6 1
2592.2.a.r yes 2 9.c even 3 1
2592.2.a.r yes 2 36.f odd 6 1
2592.2.i.z 4 1.a even 1 1 trivial
2592.2.i.z 4 4.b odd 2 1 inner
2592.2.i.z 4 9.c even 3 1 inner
2592.2.i.z 4 36.f odd 6 1 inner
2592.2.i.bd 4 3.b odd 2 1
2592.2.i.bd 4 9.d odd 6 1
2592.2.i.bd 4 12.b even 2 1
2592.2.i.bd 4 36.h even 6 1
5184.2.a.bk 2 72.n even 6 1
5184.2.a.bk 2 72.p odd 6 1
5184.2.a.bv 2 72.j odd 6 1
5184.2.a.bv 2 72.l even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

\( T_{5}^{2} + T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 24T_{7}^{2} + 576 \) Copy content Toggle raw display
\( T_{11}^{4} + 24T_{11}^{2} + 576 \) 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
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T - 5)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( (T + 5)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$47$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$53$ \( (T - 2)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$61$ \( (T^{2} - 13 T + 169)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 24T^{2} + 576 \) Copy content Toggle raw display
$71$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$73$ \( (T - 3)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 216 T^{2} + 46656 \) Copy content Toggle raw display
$83$ \( T^{4} + 96T^{2} + 9216 \) Copy content Toggle raw display
$89$ \( (T - 13)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
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