Properties

Label 7800.2.a.bh
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: 3.3.3732.1
Defining polynomial: \(x^{3} - x^{2} - 13 x + 19\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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.

\(f(q)\) \(=\) \( q - q^{3} -\beta_{2} q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} -\beta_{2} q^{7} + q^{9} + ( 2 - \beta_{1} ) q^{11} + q^{13} -3 q^{17} + ( -\beta_{1} - \beta_{2} ) q^{19} + \beta_{2} q^{21} + ( 2 - \beta_{1} - \beta_{2} ) q^{23} - q^{27} + ( -1 - 2 \beta_{1} ) q^{29} + ( 2 - \beta_{1} ) q^{31} + ( -2 + \beta_{1} ) q^{33} + ( 2 + \beta_{1} - \beta_{2} ) q^{37} - q^{39} + ( -\beta_{1} + \beta_{2} ) q^{41} + ( \beta_{1} - \beta_{2} ) q^{43} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{47} + ( 8 + 3 \beta_{1} - \beta_{2} ) q^{49} + 3 q^{51} + ( 1 - \beta_{1} - \beta_{2} ) q^{53} + ( \beta_{1} + \beta_{2} ) q^{57} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{59} + ( 3 + 2 \beta_{1} ) q^{61} -\beta_{2} q^{63} + ( -8 - \beta_{1} ) q^{67} + ( -2 + \beta_{1} + \beta_{2} ) q^{69} + ( 3 \beta_{1} - \beta_{2} ) q^{71} + ( -3 \beta_{1} + \beta_{2} ) q^{73} + ( -1 + 2 \beta_{1} ) q^{77} + ( 4 + \beta_{1} - \beta_{2} ) q^{79} + q^{81} + ( 6 - 2 \beta_{1} - \beta_{2} ) q^{83} + ( 1 + 2 \beta_{1} ) q^{87} + ( 6 - 2 \beta_{1} ) q^{89} -\beta_{2} q^{91} + ( -2 + \beta_{1} ) q^{93} + ( -4 - 2 \beta_{2} ) q^{97} + ( 2 - \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} - q^{7} + 3 q^{9} + 5 q^{11} + 3 q^{13} - 9 q^{17} - 2 q^{19} + q^{21} + 4 q^{23} - 3 q^{27} - 5 q^{29} + 5 q^{31} - 5 q^{33} + 6 q^{37} - 3 q^{39} - 13 q^{47} + 26 q^{49} + 9 q^{51} + q^{53} + 2 q^{57} + 13 q^{59} + 11 q^{61} - q^{63} - 25 q^{67} - 4 q^{69} + 2 q^{71} - 2 q^{73} - q^{77} + 12 q^{79} + 3 q^{81} + 15 q^{83} + 5 q^{87} + 16 q^{89} - q^{91} - 5 q^{93} - 14 q^{97} + 5 q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 13 x + 19\):

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

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
3.20633
−3.77576
1.56943
0 −1.00000 0 0 0 −4.48688 0 1.00000 0
1.2 0 −1.00000 0 0 0 −1.48059 0 1.00000 0
1.3 0 −1.00000 0 0 0 4.96747 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.bh 3
5.b even 2 1 7800.2.a.bs yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bh 3 1.a even 1 1 trivial
7800.2.a.bs 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{3} + T_{7}^{2} - 23 T_{7} - 33 \)
\( T_{11}^{3} - 5 T_{11}^{2} - 5 T_{11} + 3 \)
\( T_{17} + 3 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 36 T_{19} + 60 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( 1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( -33 - 23 T + T^{2} + T^{3} \)
$11$ \( 3 - 5 T - 5 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( ( 3 + T )^{3} \)
$19$ \( 60 - 36 T + 2 T^{2} + T^{3} \)
$23$ \( 132 - 32 T - 4 T^{2} + T^{3} \)
$29$ \( -201 - 45 T + 5 T^{2} + T^{3} \)
$31$ \( 3 - 5 T - 5 T^{2} + T^{3} \)
$37$ \( 20 - 24 T - 6 T^{2} + T^{3} \)
$41$ \( 44 - 36 T + T^{3} \)
$43$ \( -44 - 36 T + T^{3} \)
$47$ \( -361 - 19 T + 13 T^{2} + T^{3} \)
$53$ \( 97 - 37 T - T^{2} + T^{3} \)
$59$ \( 361 - 19 T - 13 T^{2} + T^{3} \)
$61$ \( 263 - 13 T - 11 T^{2} + T^{3} \)
$67$ \( 453 + 195 T + 25 T^{2} + T^{3} \)
$71$ \( 636 - 140 T - 2 T^{2} + T^{3} \)
$73$ \( -636 - 140 T + 2 T^{2} + T^{3} \)
$79$ \( 36 + 12 T - 12 T^{2} + T^{3} \)
$83$ \( 463 - 3 T - 15 T^{2} + T^{3} \)
$89$ \( 16 + 32 T - 16 T^{2} + T^{3} \)
$97$ \( -536 - 28 T + 14 T^{2} + T^{3} \)
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