Properties

Label 4608.2.a.k
Level $4608$
Weight $2$
Character orbit 4608.a
Self dual yes
Analytic conductor $36.795$
Analytic rank $1$
Dimension $2$
CM discriminant -8
Inner twists $2$

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q +O(q^{10})\) \( q + ( 2 - 3 \beta ) q^{11} + 4 \beta q^{17} + ( -6 + \beta ) q^{19} -5 q^{25} + 6 q^{41} + ( -6 + 5 \beta ) q^{43} -7 q^{49} + ( -10 + 3 \beta ) q^{59} + ( -6 - 7 \beta ) q^{67} -12 \beta q^{73} + ( 2 + 9 \beta ) q^{83} -4 \beta q^{89} + 12 \beta q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + O(q^{10}) \) \( 2 q + 4 q^{11} - 12 q^{19} - 10 q^{25} + 12 q^{41} - 12 q^{43} - 14 q^{49} - 20 q^{59} - 12 q^{67} + 4 q^{83} + O(q^{100}) \)

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
1.41421
−1.41421
0 0 0 0 0 0 0 0 0
1.2 0 0 0 0 0 0 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.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4608.2.a.k 2
3.b odd 2 1 512.2.a.a 2
4.b odd 2 1 4608.2.a.i 2
8.b even 2 1 4608.2.a.i 2
8.d odd 2 1 CM 4608.2.a.k 2
12.b even 2 1 512.2.a.f yes 2
16.e even 4 2 4608.2.d.k 4
16.f odd 4 2 4608.2.d.k 4
24.f even 2 1 512.2.a.a 2
24.h odd 2 1 512.2.a.f yes 2
48.i odd 4 2 512.2.b.c 4
48.k even 4 2 512.2.b.c 4
96.o even 8 2 1024.2.e.g 4
96.o even 8 2 1024.2.e.o 4
96.p odd 8 2 1024.2.e.g 4
96.p odd 8 2 1024.2.e.o 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
512.2.a.a 2 3.b odd 2 1
512.2.a.a 2 24.f even 2 1
512.2.a.f yes 2 12.b even 2 1
512.2.a.f yes 2 24.h odd 2 1
512.2.b.c 4 48.i odd 4 2
512.2.b.c 4 48.k even 4 2
1024.2.e.g 4 96.o even 8 2
1024.2.e.g 4 96.p odd 8 2
1024.2.e.o 4 96.o even 8 2
1024.2.e.o 4 96.p odd 8 2
4608.2.a.i 2 4.b odd 2 1
4608.2.a.i 2 8.b even 2 1
4608.2.a.k 2 1.a even 1 1 trivial
4608.2.a.k 2 8.d odd 2 1 CM
4608.2.d.k 4 16.e even 4 2
4608.2.d.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4608))\):

\( T_{5} \)
\( T_{7} \)
\( T_{11}^{2} - 4 T_{11} - 14 \)
\( T_{17}^{2} - 32 \)
\( T_{23} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( -14 - 4 T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( -32 + T^{2} \)
$19$ \( 34 + 12 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( T^{2} \)
$41$ \( ( -6 + T )^{2} \)
$43$ \( -14 + 12 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( 82 + 20 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( -62 + 12 T + T^{2} \)
$71$ \( T^{2} \)
$73$ \( -288 + T^{2} \)
$79$ \( T^{2} \)
$83$ \( -158 - 4 T + T^{2} \)
$89$ \( -32 + T^{2} \)
$97$ \( -288 + T^{2} \)
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