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Properties

Label 5472.2.d.e
Level $5472$
Weight $2$
Character orbit 5472.d
Analytic conductor $43.694$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5472,2,Mod(2015,5472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 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("5472.2015"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(23)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(43.6941399860\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - 2 \beta_{4}) q^{5} + ( - 2 \beta_{2} - \beta_1) q^{7} + ( - \beta_{6} + 2 \beta_{5}) q^{11} + (2 \beta_{3} + 2) q^{13} + (4 \beta_{7} + \beta_{4}) q^{17} - \beta_1 q^{19} + (\beta_{6} + 3 \beta_{5}) q^{23}+ \cdots + (4 \beta_{3} + 4) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{13} - 8 q^{25} + 48 q^{37} - 48 q^{49} - 16 q^{61} + 8 q^{73} + 24 q^{85} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} - \zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \zeta_{24}^{7} + \zeta_{24}^{5} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{6} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{5} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(2053\) \(3745\) \(4447\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2015.1
0.258819 − 0.965926i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
−0.965926 − 0.258819i
0.965926 − 0.258819i
−0.258819 + 0.965926i
0.258819 + 0.965926i
0 0 0 − 3.34607i 0 − 2.46410i 0 0 0
2015.2 0 0 0 − 3.34607i 0 2.46410i 0 0 0
2015.3 0 0 0 − 0.896575i 0 − 4.46410i 0 0 0
2015.4 0 0 0 − 0.896575i 0 4.46410i 0 0 0
2015.5 0 0 0 0.896575i 0 − 4.46410i 0 0 0
2015.6 0 0 0 0.896575i 0 4.46410i 0 0 0
2015.7 0 0 0 3.34607i 0 − 2.46410i 0 0 0
2015.8 0 0 0 3.34607i 0 2.46410i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2015.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5472.2.d.e ✓ 8
3.b odd 2 1 inner 5472.2.d.e ✓ 8
4.b odd 2 1 inner 5472.2.d.e ✓ 8
12.b even 2 1 inner 5472.2.d.e ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5472.2.d.e ✓ 8 1.a even 1 1 trivial
5472.2.d.e ✓ 8 3.b odd 2 1 inner
5472.2.d.e ✓ 8 4.b odd 2 1 inner
5472.2.d.e ✓ 8 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(5472, [\chi])\):

\( T_{5}^{4} + 12T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{11}^{4} - 28T_{11}^{2} + 169 \) Copy content Toggle raw display
\( T_{23}^{4} - 28T_{23}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + 12 T^{2} + 9)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 26 T^{2} + 121)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 28 T^{2} + 169)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 4 T - 8)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 52 T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 28 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 12 T + 24)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 8)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 38 T^{2} + 169)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 84 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 208 T^{2} + 8464)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 144 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 23)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 104 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 112 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 2 T - 47)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 152 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 76 T^{2} + 676)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 112 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T - 32)^{4} \) Copy content Toggle raw display
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