Properties

Label 7800.2.a.br
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: 3.3.788.1
Defining polynomial: \(x^{3} - x^{2} - 7 x - 3\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 7 x - 3\):

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

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
−1.87740
−0.476452
3.35386
0 1.00000 0 0 0 −1.87740 0 1.00000 0
1.2 0 1.00000 0 0 0 −0.476452 0 1.00000 0
1.3 0 1.00000 0 0 0 3.35386 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.br yes 3
5.b even 2 1 7800.2.a.bg 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bg 3 5.b even 2 1
7800.2.a.br 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{3} - T_{7}^{2} - 7 T_{7} - 3 \)
\( T_{11}^{3} + 3 T_{11}^{2} - 17 T_{11} - 53 \)
\( T_{17} + 1 \)
\( T_{19}^{3} + 6 T_{19}^{2} - 20 T_{19} - 100 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \)
$3$ \( ( -1 + T )^{3} \)
$5$ \( T^{3} \)
$7$ \( -3 - 7 T - T^{2} + T^{3} \)
$11$ \( -53 - 17 T + 3 T^{2} + T^{3} \)
$13$ \( ( 1 + T )^{3} \)
$17$ \( ( 1 + T )^{3} \)
$19$ \( -100 - 20 T + 6 T^{2} + T^{3} \)
$23$ \( 44 - 32 T + T^{3} \)
$29$ \( -93 - 53 T + T^{2} + T^{3} \)
$31$ \( -425 - 65 T + 7 T^{2} + T^{3} \)
$37$ \( 60 + 56 T + 14 T^{2} + T^{3} \)
$41$ \( -52 - 84 T + T^{3} \)
$43$ \( 12 - 4 T - 4 T^{2} + T^{3} \)
$47$ \( 89 - 83 T - T^{2} + T^{3} \)
$53$ \( -781 - 125 T + 5 T^{2} + T^{3} \)
$59$ \( 9 - 35 T + 7 T^{2} + T^{3} \)
$61$ \( -9 - 45 T + 5 T^{2} + T^{3} \)
$67$ \( 15 - 17 T + 3 T^{2} + T^{3} \)
$71$ \( -180 - 12 T + 10 T^{2} + T^{3} \)
$73$ \( -52 - 60 T - 10 T^{2} + T^{3} \)
$79$ \( 100 + 76 T + 16 T^{2} + T^{3} \)
$83$ \( -255 - 11 T + 11 T^{2} + T^{3} \)
$89$ \( -48 - 32 T + 8 T^{2} + T^{3} \)
$97$ \( 440 + 196 T + 26 T^{2} + T^{3} \)
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