Properties

Label 4608.2.a.y
Level $4608$
Weight $2$
Character orbit 4608.a
Self dual yes
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM no
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: \(\Q(\sqrt{2}, \sqrt{5})\)
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + \beta_{2} q^{7} - \beta_{3} q^{11} - \beta_{3} q^{13} + 2 \beta_{2} q^{17} + 2 \beta_1 q^{19} + 4 q^{23} - 3 q^{25} - 3 \beta_1 q^{29} - \beta_{2} q^{31} + \beta_{3} q^{35} - \beta_{3} q^{37} + 2 \beta_{2} q^{41} + 6 \beta_1 q^{43} + 12 q^{47} + 3 q^{49} + 5 \beta_1 q^{53} - 2 \beta_{2} q^{55} + 3 \beta_{3} q^{61} - 2 \beta_{2} q^{65} + 8 q^{71} - 4 q^{73} - 10 \beta_1 q^{77} + \beta_{2} q^{79} + \beta_{3} q^{83} + 2 \beta_{3} q^{85} - 10 \beta_1 q^{91} + 4 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 16 q^{23} - 12 q^{25} + 48 q^{47} + 12 q^{49} + 32 q^{71} - 16 q^{73} + 16 q^{95} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 8\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{2} + 4\beta_1 \) 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
−2.28825
0.874032
−0.874032
2.28825
0 0 0 −1.41421 0 −3.16228 0 0 0
1.2 0 0 0 −1.41421 0 3.16228 0 0 0
1.3 0 0 0 1.41421 0 −3.16228 0 0 0
1.4 0 0 0 1.41421 0 3.16228 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
8.b even 2 1 inner
12.b even 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.y yes 4
3.b odd 2 1 4608.2.a.x 4
4.b odd 2 1 4608.2.a.x 4
8.b even 2 1 inner 4608.2.a.y yes 4
8.d odd 2 1 4608.2.a.x 4
12.b even 2 1 inner 4608.2.a.y yes 4
16.e even 4 2 4608.2.d.i 4
16.f odd 4 2 4608.2.d.l 4
24.f even 2 1 inner 4608.2.a.y yes 4
24.h odd 2 1 4608.2.a.x 4
48.i odd 4 2 4608.2.d.l 4
48.k even 4 2 4608.2.d.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.a.x 4 3.b odd 2 1
4608.2.a.x 4 4.b odd 2 1
4608.2.a.x 4 8.d odd 2 1
4608.2.a.x 4 24.h odd 2 1
4608.2.a.y yes 4 1.a even 1 1 trivial
4608.2.a.y yes 4 8.b even 2 1 inner
4608.2.a.y yes 4 12.b even 2 1 inner
4608.2.a.y yes 4 24.f even 2 1 inner
4608.2.d.i 4 16.e even 4 2
4608.2.d.i 4 48.k even 4 2
4608.2.d.l 4 16.f odd 4 2
4608.2.d.l 4 48.i odd 4 2

Hecke kernels

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

\( T_{5}^{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{2} - 10 \) Copy content Toggle raw display
\( T_{11}^{2} - 20 \) Copy content Toggle raw display
\( T_{17}^{2} - 40 \) Copy content Toggle raw display
\( T_{23} - 4 \) 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
$5$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$23$ \( (T - 4)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 40)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 72)^{2} \) Copy content Toggle raw display
$47$ \( (T - 12)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 50)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 180)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T - 8)^{4} \) Copy content Toggle raw display
$73$ \( (T + 4)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 10)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 20)^{2} \) 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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