Properties

Label 7800.2.a.bk
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: 3.3.2700.1
Defining polynomial: \(x^{3} - 15 x - 20\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{1} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + \beta_{1} q^{7} + q^{9} -\beta_{1} q^{11} - q^{13} + ( -2 + \beta_{2} ) q^{17} + ( 1 - \beta_{2} ) q^{19} -\beta_{1} q^{21} + 2 \beta_{2} q^{23} - q^{27} + q^{29} -\beta_{1} q^{31} + \beta_{1} q^{33} + ( -3 - \beta_{2} ) q^{37} + q^{39} + ( 1 + \beta_{2} ) q^{41} + 2 q^{43} + ( -1 + \beta_{1} - \beta_{2} ) q^{47} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{49} + ( 2 - \beta_{2} ) q^{51} + ( -1 - 2 \beta_{2} ) q^{53} + ( -1 + \beta_{2} ) q^{57} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{59} + ( -2 - 3 \beta_{2} ) q^{61} + \beta_{1} q^{63} + ( 1 + \beta_{1} + \beta_{2} ) q^{67} -2 \beta_{2} q^{69} + ( 3 + \beta_{2} ) q^{71} + ( 4 + 2 \beta_{1} ) q^{73} + ( -10 - 2 \beta_{1} - \beta_{2} ) q^{77} + ( 5 + 2 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{83} - q^{87} + ( 2 + 2 \beta_{2} ) q^{89} -\beta_{1} q^{91} + \beta_{1} q^{93} + ( -4 - 2 \beta_{2} ) q^{97} -\beta_{1} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 15 x - 20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 10 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 10\)

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
−2.80560
−1.61323
4.41883
0 −1.00000 0 0 0 −2.80560 0 1.00000 0
1.2 0 −1.00000 0 0 0 −1.61323 0 1.00000 0
1.3 0 −1.00000 0 0 0 4.41883 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.bk 3
5.b even 2 1 7800.2.a.bq yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bk 3 1.a even 1 1 trivial
7800.2.a.bq yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{3} - 15 T_{7} - 20 \)
\( T_{11}^{3} - 15 T_{11} + 20 \)
\( T_{17}^{3} + 6 T_{17}^{2} - 3 T_{17} - 12 \)
\( T_{19}^{3} - 3 T_{19}^{2} - 12 T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( -20 - 15 T + T^{3} \)
$11$ \( 20 - 15 T + T^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( -12 - 3 T + 6 T^{2} + T^{3} \)
$19$ \( 4 - 12 T - 3 T^{2} + T^{3} \)
$23$ \( 80 - 60 T + T^{3} \)
$29$ \( ( -1 + T )^{3} \)
$31$ \( 20 - 15 T + T^{3} \)
$37$ \( -28 + 12 T + 9 T^{2} + T^{3} \)
$41$ \( 24 - 12 T - 3 T^{2} + T^{3} \)
$43$ \( ( -2 + T )^{3} \)
$47$ \( 31 - 27 T + 3 T^{2} + T^{3} \)
$53$ \( -139 - 57 T + 3 T^{2} + T^{3} \)
$59$ \( -58 - 63 T - 6 T^{2} + T^{3} \)
$61$ \( -532 - 123 T + 6 T^{2} + T^{3} \)
$67$ \( 49 - 27 T - 3 T^{2} + T^{3} \)
$71$ \( 28 + 12 T - 9 T^{2} + T^{3} \)
$73$ \( 16 - 12 T - 12 T^{2} + T^{3} \)
$79$ \( 100 - 15 T^{2} + T^{3} \)
$83$ \( 342 - 63 T - 6 T^{2} + T^{3} \)
$89$ \( 192 - 48 T - 6 T^{2} + T^{3} \)
$97$ \( -256 - 12 T + 12 T^{2} + T^{3} \)
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