Properties

Label 7800.2.a.x
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7800 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.2833135766\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{3} + 5q^{7} + q^{9} + O(q^{10}) \) \( q + q^{3} + 5q^{7} + q^{9} + q^{11} - q^{13} + 3q^{17} + 6q^{19} + 5q^{21} + 7q^{23} + q^{27} + 6q^{29} - 2q^{31} + q^{33} - q^{37} - q^{39} + 7q^{41} - 8q^{43} - 2q^{47} + 18q^{49} + 3q^{51} - 13q^{53} + 6q^{57} - 8q^{59} - 7q^{61} + 5q^{63} - 12q^{67} + 7q^{69} + q^{71} + 10q^{73} + 5q^{77} + 3q^{79} + q^{81} - 8q^{83} + 6q^{87} + 13q^{89} - 5q^{91} - 2q^{93} - 7q^{97} + q^{99} + O(q^{100}) \)

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
0
0 1.00000 0 0 0 5.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(13\) \(1\)

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 7800.2.a.x 1
5.b even 2 1 1560.2.a.c 1
15.d odd 2 1 4680.2.a.a 1
20.d odd 2 1 3120.2.a.ba 1
60.h even 2 1 9360.2.a.bb 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.a.c 1 5.b even 2 1
3120.2.a.ba 1 20.d odd 2 1
4680.2.a.a 1 15.d odd 2 1
7800.2.a.x 1 1.a even 1 1 trivial
9360.2.a.bb 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7} - 5 \)
\( T_{11} - 1 \)
\( T_{17} - 3 \)
\( T_{19} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( -1 + T \)
$5$ \( T \)
$7$ \( -5 + T \)
$11$ \( -1 + T \)
$13$ \( 1 + T \)
$17$ \( -3 + T \)
$19$ \( -6 + T \)
$23$ \( -7 + T \)
$29$ \( -6 + T \)
$31$ \( 2 + T \)
$37$ \( 1 + T \)
$41$ \( -7 + T \)
$43$ \( 8 + T \)
$47$ \( 2 + T \)
$53$ \( 13 + T \)
$59$ \( 8 + T \)
$61$ \( 7 + T \)
$67$ \( 12 + T \)
$71$ \( -1 + T \)
$73$ \( -10 + T \)
$79$ \( -3 + T \)
$83$ \( 8 + T \)
$89$ \( -13 + T \)
$97$ \( 7 + T \)
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