Properties

Label 7800.2.a.bo
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

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

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
2.17009
−1.48119
0.311108
0 1.00000 0 0 0 −3.24846 0 1.00000 0
1.2 0 1.00000 0 0 0 −1.86907 0 1.00000 0
1.3 0 1.00000 0 0 0 4.11753 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.bo yes 3
5.b even 2 1 7800.2.a.bl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bl 3 5.b even 2 1
7800.2.a.bo yes 3 1.a even 1 1 trivial

Hecke kernels

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

\( T_{7}^{3} + T_{7}^{2} - 15 T_{7} - 25 \)
\( T_{11}^{3} + 3 T_{11}^{2} - T_{11} - 1 \)
\( T_{17}^{3} + 5 T_{17}^{2} - 29 T_{17} - 137 \)
\( T_{19}^{3} + 2 T_{19}^{2} - 4 T_{19} - 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( -25 - 15 T + T^{2} + T^{3} \)
$11$ \( -1 - T + 3 T^{2} + T^{3} \)
$13$ \( ( -1 + T )^{3} \)
$17$ \( -137 - 29 T + 5 T^{2} + T^{3} \)
$19$ \( -4 - 4 T + 2 T^{2} + T^{3} \)
$23$ \( -268 - 72 T + 4 T^{2} + T^{3} \)
$29$ \( -5 - 13 T + T^{2} + T^{3} \)
$31$ \( 19 + 27 T + 11 T^{2} + T^{3} \)
$37$ \( -556 - 56 T + 10 T^{2} + T^{3} \)
$41$ \( -100 + 52 T + 16 T^{2} + T^{3} \)
$43$ \( -268 - 84 T + T^{3} \)
$47$ \( -13 + 25 T - 11 T^{2} + T^{3} \)
$53$ \( 221 + 115 T + 19 T^{2} + T^{3} \)
$59$ \( -675 - 207 T + 3 T^{2} + T^{3} \)
$61$ \( 215 + 131 T + 21 T^{2} + T^{3} \)
$67$ \( 37 - 37 T + T^{2} + T^{3} \)
$71$ \( 796 - 148 T - 2 T^{2} + T^{3} \)
$73$ \( -860 - 156 T + 2 T^{2} + T^{3} \)
$79$ \( -604 - 52 T + 12 T^{2} + T^{3} \)
$83$ \( 859 - 119 T - 7 T^{2} + T^{3} \)
$89$ \( -2096 - 160 T + 16 T^{2} + T^{3} \)
$97$ \( 8 - 28 T - 10 T^{2} + T^{3} \)
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