Properties

Label 4800.2.a.p
Level $4800$
Weight $2$
Character orbit 4800.a
Self dual yes
Analytic conductor $38.328$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
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 \( q - q^{3} - q^{7} + q^{9} + 6 q^{11} + 5 q^{13} + 6 q^{17} + 5 q^{19} + q^{21} - 6 q^{23} - q^{27} + 6 q^{29} + q^{31} - 6 q^{33} + 2 q^{37} - 5 q^{39} + q^{43} + 6 q^{47} - 6 q^{49} - 6 q^{51} - 12 q^{53} - 5 q^{57} - 6 q^{59} + 13 q^{61} - q^{63} - 11 q^{67} + 6 q^{69} - 2 q^{73} - 6 q^{77} - 8 q^{79} + q^{81} + 6 q^{83} - 6 q^{87} - 5 q^{91} - q^{93} + 7 q^{97} + 6 q^{99}+O(q^{100}) \) 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.p 1
4.b odd 2 1 4800.2.a.ce 1
5.b even 2 1 4800.2.a.cf 1
5.c odd 4 2 4800.2.f.bi 2
8.b even 2 1 1200.2.a.n 1
8.d odd 2 1 300.2.a.b 1
20.d odd 2 1 4800.2.a.o 1
20.e even 4 2 4800.2.f.b 2
24.f even 2 1 900.2.a.e 1
24.h odd 2 1 3600.2.a.s 1
40.e odd 2 1 300.2.a.c yes 1
40.f even 2 1 1200.2.a.f 1
40.i odd 4 2 1200.2.f.a 2
40.k even 4 2 300.2.d.a 2
120.i odd 2 1 3600.2.a.z 1
120.m even 2 1 900.2.a.c 1
120.q odd 4 2 900.2.d.a 2
120.w even 4 2 3600.2.f.v 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 8.d odd 2 1
300.2.a.c yes 1 40.e odd 2 1
300.2.d.a 2 40.k even 4 2
900.2.a.c 1 120.m even 2 1
900.2.a.e 1 24.f even 2 1
900.2.d.a 2 120.q odd 4 2
1200.2.a.f 1 40.f even 2 1
1200.2.a.n 1 8.b even 2 1
1200.2.f.a 2 40.i odd 4 2
3600.2.a.s 1 24.h odd 2 1
3600.2.a.z 1 120.i odd 2 1
3600.2.f.v 2 120.w even 4 2
4800.2.a.o 1 20.d odd 2 1
4800.2.a.p 1 1.a even 1 1 trivial
4800.2.a.ce 1 4.b odd 2 1
4800.2.a.cf 1 5.b even 2 1
4800.2.f.b 2 20.e even 4 2
4800.2.f.bi 2 5.c 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
\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display
\( T_{19} - 5 \) Copy content Toggle raw display
\( T_{23} + 6 \) 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
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T - 5 \) Copy content Toggle raw display
$17$ \( T - 6 \) Copy content Toggle raw display
$19$ \( T - 5 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 1 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 1 \) Copy content Toggle raw display
$47$ \( T - 6 \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T - 13 \) Copy content Toggle raw display
$67$ \( T + 11 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 2 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 7 \) Copy content Toggle raw display
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