Properties

Label 4800.2.a.bx
Level $4800$
Weight $2$
Character orbit 4800.a
Self dual yes
Analytic conductor $38.328$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4800 = 2^{6} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.3281929702\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2400)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - q^{7} + q^{9} + q^{13} - 3 q^{19} - q^{21} - 4 q^{23} + q^{27} - 4 q^{29} - 7 q^{31} - 6 q^{37} + q^{39} + 6 q^{41} + 9 q^{43} - 6 q^{47} - 6 q^{49} + 2 q^{53} - 3 q^{57} - 10 q^{59} + q^{61} - q^{63} - 3 q^{67} - 4 q^{69} - 14 q^{71} - 10 q^{73} + 8 q^{79} + q^{81} + 18 q^{83} - 4 q^{87} - q^{91} - 7 q^{93} + 3 q^{97}+O(q^{100}) \) 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
0 1.00000 0 0 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4800.2.a.bx 1
4.b odd 2 1 4800.2.a.x 1
5.b even 2 1 4800.2.a.w 1
5.c odd 4 2 4800.2.f.t 2
8.b even 2 1 2400.2.a.f 1
8.d odd 2 1 2400.2.a.bb yes 1
20.d odd 2 1 4800.2.a.bw 1
20.e even 4 2 4800.2.f.q 2
24.f even 2 1 7200.2.a.bi 1
24.h odd 2 1 7200.2.a.r 1
40.e odd 2 1 2400.2.a.g yes 1
40.f even 2 1 2400.2.a.bc yes 1
40.i odd 4 2 2400.2.f.h 2
40.k even 4 2 2400.2.f.k 2
120.i odd 2 1 7200.2.a.bj 1
120.m even 2 1 7200.2.a.s 1
120.q odd 4 2 7200.2.f.r 2
120.w even 4 2 7200.2.f.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2400.2.a.f 1 8.b even 2 1
2400.2.a.g yes 1 40.e odd 2 1
2400.2.a.bb yes 1 8.d odd 2 1
2400.2.a.bc yes 1 40.f even 2 1
2400.2.f.h 2 40.i odd 4 2
2400.2.f.k 2 40.k even 4 2
4800.2.a.w 1 5.b even 2 1
4800.2.a.x 1 4.b odd 2 1
4800.2.a.bw 1 20.d odd 2 1
4800.2.a.bx 1 1.a even 1 1 trivial
4800.2.f.q 2 20.e even 4 2
4800.2.f.t 2 5.c odd 4 2
7200.2.a.r 1 24.h odd 2 1
7200.2.a.s 1 120.m even 2 1
7200.2.a.bi 1 24.f even 2 1
7200.2.a.bj 1 120.i odd 2 1
7200.2.f.l 2 120.w even 4 2
7200.2.f.r 2 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4800))\):

\( T_{7} + 1 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} - 1 \) Copy content Toggle raw display
\( T_{19} + 3 \) Copy content Toggle raw display
\( T_{23} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 3 \) Copy content Toggle raw display
$23$ \( T + 4 \) Copy content Toggle raw display
$29$ \( T + 4 \) Copy content Toggle raw display
$31$ \( T + 7 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T - 9 \) Copy content Toggle raw display
$47$ \( T + 6 \) Copy content Toggle raw display
$53$ \( T - 2 \) Copy content Toggle raw display
$59$ \( T + 10 \) Copy content Toggle raw display
$61$ \( T - 1 \) Copy content Toggle raw display
$67$ \( T + 3 \) Copy content Toggle raw display
$71$ \( T + 14 \) Copy content Toggle raw display
$73$ \( T + 10 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 18 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 3 \) Copy content Toggle raw display
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