Properties

Label 4608.2.a.bc
Level $4608$
Weight $2$
Character orbit 4608.a
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 2) q^{5} + (\beta_{3} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 2) q^{5} + (\beta_{3} + \beta_1) q^{7} + \beta_{3} q^{11} + (4 \beta_{2} + 5) q^{25} + ( - 3 \beta_{2} + 2) q^{29} + ( - \beta_{3} - 5 \beta_1) q^{31} + (3 \beta_{3} + 8 \beta_1) q^{35} + (4 \beta_{2} + 7) q^{49} + (\beta_{2} + 10) q^{53} + (2 \beta_{3} + 6 \beta_1) q^{55} - 8 \beta_1 q^{59} - 4 \beta_{2} q^{73} + (2 \beta_{2} + 12) q^{77} + ( - 3 \beta_{3} + 5 \beta_1) q^{79} - 5 \beta_{3} q^{83} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{5} + 20 q^{25} + 8 q^{29} + 28 q^{49} + 40 q^{53} + 48 q^{77} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−0.517638
−1.93185
0.517638
1.93185
0 0 0 −0.449490 0 −2.04989 0 0 0
1.2 0 0 0 −0.449490 0 2.04989 0 0 0
1.3 0 0 0 4.44949 0 −4.87832 0 0 0
1.4 0 0 0 4.44949 0 4.87832 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
4.b odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.a.bc yes 4
3.b odd 2 1 4608.2.a.s 4
4.b odd 2 1 inner 4608.2.a.bc yes 4
8.b even 2 1 4608.2.a.s 4
8.d odd 2 1 4608.2.a.s 4
12.b even 2 1 4608.2.a.s 4
16.e even 4 2 4608.2.d.q 8
16.f odd 4 2 4608.2.d.q 8
24.f even 2 1 inner 4608.2.a.bc yes 4
24.h odd 2 1 CM 4608.2.a.bc yes 4
48.i odd 4 2 4608.2.d.q 8
48.k even 4 2 4608.2.d.q 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.a.s 4 3.b odd 2 1
4608.2.a.s 4 8.b even 2 1
4608.2.a.s 4 8.d odd 2 1
4608.2.a.s 4 12.b even 2 1
4608.2.a.bc yes 4 1.a even 1 1 trivial
4608.2.a.bc yes 4 4.b odd 2 1 inner
4608.2.a.bc yes 4 24.f even 2 1 inner
4608.2.a.bc yes 4 24.h odd 2 1 CM
4608.2.d.q 8 16.e even 4 2
4608.2.d.q 8 16.f odd 4 2
4608.2.d.q 8 48.i odd 4 2
4608.2.d.q 8 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4608))\):

\( T_{5}^{2} - 4T_{5} - 2 \) Copy content Toggle raw display
\( T_{7}^{4} - 28T_{7}^{2} + 100 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{23} \) 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
$5$ \( (T^{2} - 4 T - 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 28T^{2} + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 4 T - 50)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 124T^{2} + 1444 \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 20 T + 94)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 316T^{2} + 3364 \) Copy content Toggle raw display
$83$ \( (T^{2} - 300)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T + 2)^{4} \) Copy content Toggle raw display
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