Properties

Label 7800.2.a.bu
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.461844.1
Defining polynomial: \(x^{4} - x^{3} - 14 x^{2} - 12 x + 6\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 14 x^{2} - 12 x + 6\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - 3 \nu^{2} - 8 \nu + 4 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 11 \nu - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - \beta_{2} + 3 \beta_{1} + 7\)
\(\nu^{3}\)\(=\)\(3 \beta_{3} - 2 \beta_{2} + 17 \beta_{1} + 17\)

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
−2.32424
−1.60110
0.352593
4.57275
0 −1.00000 0 0 0 −3.32424 0 1.00000 0
1.2 0 −1.00000 0 0 0 −2.60110 0 1.00000 0
1.3 0 −1.00000 0 0 0 −0.647407 0 1.00000 0
1.4 0 −1.00000 0 0 0 3.57275 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.bu 4
5.b even 2 1 7800.2.a.bx yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bu 4 1.a even 1 1 trivial
7800.2.a.bx yes 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{4} + 3 T_{7}^{3} - 11 T_{7}^{2} - 39 T_{7} - 20 \)
\( T_{11}^{4} - 5 T_{11}^{3} - 29 T_{11}^{2} + 159 T_{11} - 72 \)
\( T_{17}^{4} + 7 T_{17}^{3} - 29 T_{17}^{2} - 171 T_{17} + 216 \)
\( T_{19}^{4} - 3 T_{19}^{3} - 42 T_{19}^{2} + 144 T_{19} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( -20 - 39 T - 11 T^{2} + 3 T^{3} + T^{4} \)
$11$ \( -72 + 159 T - 29 T^{2} - 5 T^{3} + T^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( 216 - 171 T - 29 T^{2} + 7 T^{3} + T^{4} \)
$19$ \( 36 + 144 T - 42 T^{2} - 3 T^{3} + T^{4} \)
$23$ \( -192 - 132 T - 8 T^{2} + 8 T^{3} + T^{4} \)
$29$ \( -57 + 210 T - 56 T^{2} - 2 T^{3} + T^{4} \)
$31$ \( -428 + 307 T - 45 T^{2} - 5 T^{3} + T^{4} \)
$37$ \( 636 - 12 T - 74 T^{2} - T^{3} + T^{4} \)
$41$ \( -2244 + 960 T - 92 T^{2} - 7 T^{3} + T^{4} \)
$43$ \( -8 - 60 T - 92 T^{2} + 6 T^{3} + T^{4} \)
$47$ \( 3949 - 18 T - 128 T^{2} + T^{4} \)
$53$ \( 10147 - 978 T - 224 T^{2} + 6 T^{3} + T^{4} \)
$59$ \( 1026 - 1017 T - 129 T^{2} + 9 T^{3} + T^{4} \)
$61$ \( -680 + 191 T + 75 T^{2} - 19 T^{3} + T^{4} \)
$67$ \( 2767 + 422 T - 126 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( -764 + 80 T + 90 T^{2} - 19 T^{3} + T^{4} \)
$73$ \( 20016 + 828 T - 276 T^{2} - 6 T^{3} + T^{4} \)
$79$ \( 388 - 240 T - 44 T^{2} + 9 T^{3} + T^{4} \)
$83$ \( 466 + 447 T + 151 T^{2} + 21 T^{3} + T^{4} \)
$89$ \( 1440 + 336 T - 80 T^{2} - 14 T^{3} + T^{4} \)
$97$ \( -320 + 312 T - 44 T^{2} - 6 T^{3} + T^{4} \)
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