Properties

Label 7800.2.a.bt
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $1$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.30056.1
Defining polynomial: \(x^{4} - 11 x^{2} + 26\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1560)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( -2 - \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -2 - \beta_{1} ) q^{7} + q^{9} + ( \beta_{1} - \beta_{2} ) q^{11} + q^{13} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( 2 + \beta_{1} ) q^{21} + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{23} - q^{27} + ( 2 + \beta_{2} + \beta_{3} ) q^{29} + ( -\beta_{1} + \beta_{2} ) q^{33} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} - q^{39} + ( 2 - \beta_{1} - \beta_{3} ) q^{41} + ( -2 + \beta_{2} + \beta_{3} ) q^{43} + ( -2 + \beta_{2} ) q^{47} + ( 3 + 3 \beta_{1} + \beta_{2} + \beta_{3} ) q^{49} + ( \beta_{1} - \beta_{2} + \beta_{3} ) q^{51} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{53} + ( 2 + 2 \beta_{2} - \beta_{3} ) q^{59} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{61} + ( -2 - \beta_{1} ) q^{63} + ( -4 - 2 \beta_{2} ) q^{67} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{69} + ( 4 + 3 \beta_{1} + \beta_{3} ) q^{71} + ( 2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{73} + ( -2 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{77} + ( \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{79} + q^{81} + ( -6 - 4 \beta_{1} + \beta_{2} ) q^{83} + ( -2 - \beta_{2} - \beta_{3} ) q^{87} + ( 2 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{89} + ( -2 - \beta_{1} ) q^{91} + ( 5 \beta_{1} - 2 \beta_{2} ) q^{97} + ( \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{3} - 7q^{7} + 4q^{9} + O(q^{10}) \) \( 4q - 4q^{3} - 7q^{7} + 4q^{9} + q^{11} + 4q^{13} - 3q^{17} + 7q^{21} - 5q^{23} - 4q^{27} + 8q^{29} - q^{33} - 5q^{37} - 4q^{39} + 7q^{41} - 8q^{43} - 10q^{47} + 9q^{49} + 3q^{51} + 5q^{53} + 2q^{59} + 11q^{61} - 7q^{63} - 12q^{67} + 5q^{69} + 15q^{71} + 10q^{73} - 15q^{77} - q^{79} + 4q^{81} - 22q^{83} - 8q^{87} + 9q^{89} - 7q^{91} - q^{97} + q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 11 x^{2} + 26\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + \nu^{2} - 6 \nu - 6 \)\()/2\)
\(\beta_{2}\)\(=\)\( \nu^{2} + \nu - 6 \)
\(\beta_{3}\)\(=\)\( -\nu^{2} + \nu + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + \beta_{2} + 12\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(7 \beta_{3} + 5 \beta_{2} + 4 \beta_{1}\)\()/2\)

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
2.74983
−1.85431
−2.74983
1.85431
0 −1.00000 0 0 0 −4.92778 0 1.00000 0
1.2 0 −1.00000 0 0 0 −3.09417 0 1.00000 0
1.3 0 −1.00000 0 0 0 −0.633776 0 1.00000 0
1.4 0 −1.00000 0 0 0 1.65573 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.bt 4
5.b even 2 1 7800.2.a.by 4
5.c odd 4 2 1560.2.l.d 8
15.e even 4 2 4680.2.l.g 8
20.e even 4 2 3120.2.l.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.d 8 5.c odd 4 2
3120.2.l.n 8 20.e even 4 2
4680.2.l.g 8 15.e even 4 2
7800.2.a.bt 4 1.a even 1 1 trivial
7800.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{4} + 7 T_{7}^{3} + 6 T_{7}^{2} - 24 T_{7} - 16 \)
\( T_{11}^{4} - T_{11}^{3} - 20 T_{11}^{2} - 26 T_{11} - 4 \)
\( T_{17}^{4} + 3 T_{17}^{3} - 26 T_{17}^{2} - 36 T_{17} + 8 \)
\( T_{19} \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( -16 - 24 T + 6 T^{2} + 7 T^{3} + T^{4} \)
$11$ \( -4 - 26 T - 20 T^{2} - T^{3} + T^{4} \)
$13$ \( ( -1 + T )^{4} \)
$17$ \( 8 - 36 T - 26 T^{2} + 3 T^{3} + T^{4} \)
$19$ \( T^{4} \)
$23$ \( -32 - 72 T - 20 T^{2} + 5 T^{3} + T^{4} \)
$29$ \( 256 + 144 T - 20 T^{2} - 8 T^{3} + T^{4} \)
$31$ \( T^{4} \)
$37$ \( 376 - 140 T - 54 T^{2} + 5 T^{3} + T^{4} \)
$41$ \( -4 + 22 T - 8 T^{2} - 7 T^{3} + T^{4} \)
$43$ \( 256 - 144 T - 20 T^{2} + 8 T^{3} + T^{4} \)
$47$ \( -128 - 52 T + 18 T^{2} + 10 T^{3} + T^{4} \)
$53$ \( 376 + 140 T - 54 T^{2} - 5 T^{3} + T^{4} \)
$59$ \( 1072 + 36 T - 86 T^{2} - 2 T^{3} + T^{4} \)
$61$ \( 32 + 128 T - 22 T^{2} - 11 T^{3} + T^{4} \)
$67$ \( -256 - 224 T - 24 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 1256 + 530 T - 30 T^{2} - 15 T^{3} + T^{4} \)
$73$ \( 32 + 168 T - 12 T^{2} - 10 T^{3} + T^{4} \)
$79$ \( 7328 - 52 T - 176 T^{2} + T^{3} + T^{4} \)
$83$ \( -2704 - 1612 T + 10 T^{2} + 22 T^{3} + T^{4} \)
$89$ \( 268 + 378 T - 136 T^{2} - 9 T^{3} + T^{4} \)
$97$ \( 6464 + 308 T - 272 T^{2} + T^{3} + T^{4} \)
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