Properties

Label 7800.2.a.bj
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + q^{9} +O(q^{10})\) \( q - q^{3} + q^{9} + ( -2 - \beta_{1} ) q^{11} - q^{13} + ( 1 - \beta_{1} - \beta_{2} ) q^{17} + ( -1 - \beta_{1} - \beta_{2} ) q^{19} + ( 3 - \beta_{1} - \beta_{2} ) q^{23} - q^{27} + ( -4 - 2 \beta_{2} ) q^{29} + ( 4 + 2 \beta_{1} ) q^{31} + ( 2 + \beta_{1} ) q^{33} + ( -2 - 2 \beta_{1} ) q^{37} + q^{39} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} ) q^{43} + ( 5 - 2 \beta_{1} - \beta_{2} ) q^{47} -7 q^{49} + ( -1 + \beta_{1} + \beta_{2} ) q^{51} + ( 4 + 2 \beta_{2} ) q^{53} + ( 1 + \beta_{1} + \beta_{2} ) q^{57} + ( -2 + 3 \beta_{1} ) q^{59} + ( 1 - \beta_{1} + \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{67} + ( -3 + \beta_{1} + \beta_{2} ) q^{69} + ( 3 + \beta_{2} ) q^{71} + ( 2 + 4 \beta_{1} ) q^{73} + ( 7 + \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{83} + ( 4 + 2 \beta_{2} ) q^{87} + ( -7 - 2 \beta_{1} + \beta_{2} ) q^{89} + ( -4 - 2 \beta_{1} ) q^{93} + 6 q^{97} + ( -2 - \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 3q^{9} - 6q^{11} - 3q^{13} + 4q^{17} - 2q^{19} + 10q^{23} - 3q^{27} - 10q^{29} + 12q^{31} + 6q^{33} - 6q^{37} + 3q^{39} - 10q^{41} - 4q^{43} + 16q^{47} - 21q^{49} - 4q^{51} + 10q^{53} + 2q^{57} - 6q^{59} + 2q^{61} + 4q^{67} - 10q^{69} + 8q^{71} + 6q^{73} + 20q^{79} + 3q^{81} + 4q^{83} + 10q^{87} - 22q^{89} - 12q^{93} + 18q^{97} - 6q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 6 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{2}\)\(=\)\( -\nu^{2} + 3 \nu + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + 3 \beta_{1} + 9\)\()/2\)

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
3.12489
−1.76156
−0.363328
0 −1.00000 0 0 0 0 0 1.00000 0
1.2 0 −1.00000 0 0 0 0 0 1.00000 0
1.3 0 −1.00000 0 0 0 0 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.bj 3
5.b even 2 1 7800.2.a.bp 3
5.c odd 4 2 1560.2.l.c 6
15.e even 4 2 4680.2.l.e 6
20.e even 4 2 3120.2.l.m 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.c 6 5.c odd 4 2
3120.2.l.m 6 20.e even 4 2
4680.2.l.e 6 15.e even 4 2
7800.2.a.bj 3 1.a even 1 1 trivial
7800.2.a.bp 3 5.b even 2 1

Hecke kernels

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

\( T_{7} \)
\( T_{11}^{3} + 6 T_{11}^{2} + 2 T_{11} - 20 \)
\( T_{17}^{3} - 4 T_{17}^{2} - 20 T_{17} + 64 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 24 T_{19} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( T^{3} \)
$11$ \( -20 + 2 T + 6 T^{2} + T^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( 64 - 20 T - 4 T^{2} + T^{3} \)
$19$ \( 16 - 24 T + 2 T^{2} + T^{3} \)
$23$ \( 80 + 8 T - 10 T^{2} + T^{3} \)
$29$ \( -472 - 44 T + 10 T^{2} + T^{3} \)
$31$ \( 160 + 8 T - 12 T^{2} + T^{3} \)
$37$ \( -136 - 28 T + 6 T^{2} + T^{3} \)
$41$ \( -332 - 34 T + 10 T^{2} + T^{3} \)
$43$ \( -64 - 20 T + 4 T^{2} + T^{3} \)
$47$ \( 256 + 34 T - 16 T^{2} + T^{3} \)
$53$ \( 472 - 44 T - 10 T^{2} + T^{3} \)
$59$ \( 44 - 78 T + 6 T^{2} + T^{3} \)
$61$ \( 32 - 32 T - 2 T^{2} + T^{3} \)
$67$ \( 256 - 128 T - 4 T^{2} + T^{3} \)
$71$ \( 64 + 2 T - 8 T^{2} + T^{3} \)
$73$ \( 824 - 148 T - 6 T^{2} + T^{3} \)
$79$ \( -160 + 108 T - 20 T^{2} + T^{3} \)
$83$ \( 232 - 62 T - 4 T^{2} + T^{3} \)
$89$ \( -244 + 94 T + 22 T^{2} + T^{3} \)
$97$ \( ( -6 + T )^{3} \)
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