Properties

Label 4608.2.a.z
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: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + \beta_{1} q^{7} +O(q^{10})\) \( q + \beta_{2} q^{5} + \beta_{1} q^{7} -\beta_{3} q^{11} + 2 \beta_{2} q^{13} + 4 q^{17} -2 \beta_{3} q^{19} + 4 \beta_{1} q^{23} + q^{25} + \beta_{2} q^{29} -\beta_{1} q^{31} + \beta_{3} q^{35} + 2 \beta_{2} q^{37} + 4 q^{41} + 2 \beta_{3} q^{43} -4 \beta_{1} q^{47} -5 q^{49} -3 \beta_{2} q^{53} -6 \beta_{1} q^{55} + 4 \beta_{3} q^{59} + 2 \beta_{2} q^{61} + 12 q^{65} + 8 \beta_{1} q^{71} -4 q^{73} -2 \beta_{2} q^{77} + 5 \beta_{1} q^{79} -3 \beta_{3} q^{83} + 4 \beta_{2} q^{85} + 16 q^{89} + 2 \beta_{3} q^{91} -12 \beta_{1} q^{95} + 6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + O(q^{10}) \) \( 4 q + 16 q^{17} + 4 q^{25} + 16 q^{41} - 20 q^{49} + 48 q^{65} - 16 q^{73} + 64 q^{89} + 24 q^{97} + O(q^{100}) \)

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
−1.93185
−0.517638
0.517638
1.93185
0 0 0 −2.44949 0 −1.41421 0 0 0
1.2 0 0 0 −2.44949 0 1.41421 0 0 0
1.3 0 0 0 2.44949 0 −1.41421 0 0 0
1.4 0 0 0 2.44949 0 1.41421 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.z yes 4
3.b odd 2 1 4608.2.a.v 4
4.b odd 2 1 inner 4608.2.a.z yes 4
8.b even 2 1 inner 4608.2.a.z yes 4
8.d odd 2 1 inner 4608.2.a.z yes 4
12.b even 2 1 4608.2.a.v 4
16.e even 4 2 4608.2.d.m 4
16.f odd 4 2 4608.2.d.m 4
24.f even 2 1 4608.2.a.v 4
24.h odd 2 1 4608.2.a.v 4
48.i odd 4 2 4608.2.d.e 4
48.k even 4 2 4608.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4608.2.a.v 4 3.b odd 2 1
4608.2.a.v 4 12.b even 2 1
4608.2.a.v 4 24.f even 2 1
4608.2.a.v 4 24.h odd 2 1
4608.2.a.z yes 4 1.a even 1 1 trivial
4608.2.a.z yes 4 4.b odd 2 1 inner
4608.2.a.z yes 4 8.b even 2 1 inner
4608.2.a.z yes 4 8.d odd 2 1 inner
4608.2.d.e 4 48.i odd 4 2
4608.2.d.e 4 48.k even 4 2
4608.2.d.m 4 16.e even 4 2
4608.2.d.m 4 16.f odd 4 2

Hecke kernels

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

\( T_{5}^{2} - 6 \)
\( T_{7}^{2} - 2 \)
\( T_{11}^{2} - 12 \)
\( T_{17} - 4 \)
\( T_{23}^{2} - 32 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( -6 + T^{2} )^{2} \)
$7$ \( ( -2 + T^{2} )^{2} \)
$11$ \( ( -12 + T^{2} )^{2} \)
$13$ \( ( -24 + T^{2} )^{2} \)
$17$ \( ( -4 + T )^{4} \)
$19$ \( ( -48 + T^{2} )^{2} \)
$23$ \( ( -32 + T^{2} )^{2} \)
$29$ \( ( -6 + T^{2} )^{2} \)
$31$ \( ( -2 + T^{2} )^{2} \)
$37$ \( ( -24 + T^{2} )^{2} \)
$41$ \( ( -4 + T )^{4} \)
$43$ \( ( -48 + T^{2} )^{2} \)
$47$ \( ( -32 + T^{2} )^{2} \)
$53$ \( ( -54 + T^{2} )^{2} \)
$59$ \( ( -192 + T^{2} )^{2} \)
$61$ \( ( -24 + T^{2} )^{2} \)
$67$ \( T^{4} \)
$71$ \( ( -128 + T^{2} )^{2} \)
$73$ \( ( 4 + T )^{4} \)
$79$ \( ( -50 + T^{2} )^{2} \)
$83$ \( ( -108 + T^{2} )^{2} \)
$89$ \( ( -16 + T )^{4} \)
$97$ \( ( -6 + T )^{4} \)
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