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: \( x^{3} - 15x - 20 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_1 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_1 q^{7} + q^{9} - \beta_1 q^{11} - q^{13} + (\beta_{2} - 2) q^{17} + ( - \beta_{2} + 1) q^{19} - \beta_1 q^{21} + 2 \beta_{2} q^{23} - q^{27} + q^{29} - \beta_1 q^{31} + \beta_1 q^{33} + ( - \beta_{2} - 3) q^{37} + q^{39} + (\beta_{2} + 1) q^{41} + 2 q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{47} + (\beta_{2} + 2 \beta_1 + 3) q^{49} + ( - \beta_{2} + 2) q^{51} + ( - 2 \beta_{2} - 1) q^{53} + (\beta_{2} - 1) q^{57} + (2 \beta_{2} - \beta_1 + 2) q^{59} + ( - 3 \beta_{2} - 2) q^{61} + \beta_1 q^{63} + (\beta_{2} + \beta_1 + 1) q^{67} - 2 \beta_{2} q^{69} + (\beta_{2} + 3) q^{71} + (2 \beta_1 + 4) q^{73} + ( - \beta_{2} - 2 \beta_1 - 10) q^{77} + (\beta_{2} + 2 \beta_1 + 5) q^{79} + q^{81} + ( - 2 \beta_{2} + \beta_1 + 2) q^{83} - q^{87} + (2 \beta_{2} + 2) q^{89} - \beta_1 q^{91} + \beta_1 q^{93} + ( - 2 \beta_{2} - 4) q^{97} - \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 3 q^{9} - 3 q^{13} - 6 q^{17} + 3 q^{19} - 3 q^{27} + 3 q^{29} - 9 q^{37} + 3 q^{39} + 3 q^{41} + 6 q^{43} - 3 q^{47} + 9 q^{49} + 6 q^{51} - 3 q^{53} - 3 q^{57} + 6 q^{59} - 6 q^{61} + 3 q^{67} + 9 q^{71} + 12 q^{73} - 30 q^{77} + 15 q^{79} + 3 q^{81} + 6 q^{83} - 3 q^{87} + 6 q^{89} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 10 \) 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
−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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

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

Hecke characteristic polynomials

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