Properties

Label 7800.2.a.bw
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: 4.4.10304.1
Defining polynomial: \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{3} q^{7} + q^{9} + ( -2 - \beta_{1} ) q^{11} - q^{13} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{19} + \beta_{3} q^{21} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{23} + q^{27} + ( 2 + \beta_{3} ) q^{29} + ( -4 - 2 \beta_{3} ) q^{31} + ( -2 - \beta_{1} ) q^{33} + ( -2 + 2 \beta_{1} ) q^{37} - q^{39} + ( -1 + \beta_{2} ) q^{41} + ( 1 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{43} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{47} + q^{49} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + 2 \beta_{2} q^{53} + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{57} + ( -2 - \beta_{1} ) q^{59} + ( 3 + \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{61} + \beta_{3} q^{63} + ( -6 - 2 \beta_{2} ) q^{67} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{69} + ( -3 - 2 \beta_{1} + \beta_{2} ) q^{71} + ( -6 + 2 \beta_{1} + \beta_{3} ) q^{73} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{77} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{79} + q^{81} + ( -3 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 2 + \beta_{3} ) q^{87} + ( -1 - 3 \beta_{2} - 2 \beta_{3} ) q^{89} -\beta_{3} q^{91} + ( -4 - 2 \beta_{3} ) q^{93} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{97} + ( -2 - \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{3} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{3} + 4q^{9} - 8q^{11} - 4q^{13} + 4q^{17} - 12q^{19} + 4q^{23} + 4q^{27} + 8q^{29} - 16q^{31} - 8q^{33} - 8q^{37} - 4q^{39} - 4q^{41} + 4q^{43} - 12q^{47} + 4q^{49} + 4q^{51} - 12q^{57} - 8q^{59} + 12q^{61} - 24q^{67} + 4q^{69} - 12q^{71} - 24q^{73} - 8q^{77} - 12q^{79} + 4q^{81} - 12q^{83} + 8q^{87} - 4q^{89} - 16q^{93} - 8q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} - \nu^{2} - 4 \nu \)\()/2\)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + \nu^{2} + 8 \nu - 2 \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(2 \beta_{3} + \beta_{2} + \beta_{1} + 9\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{3} + 5 \beta_{2} + 9 \beta_{1} + 13\)\()/2\)

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
−0.692297
1.69230
3.16053
−2.16053
0 1.00000 0 0 0 −2.82843 0 1.00000 0
1.2 0 1.00000 0 0 0 −2.82843 0 1.00000 0
1.3 0 1.00000 0 0 0 2.82843 0 1.00000 0
1.4 0 1.00000 0 0 0 2.82843 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.bw 4
5.b even 2 1 7800.2.a.bv 4
5.c odd 4 2 1560.2.l.e 8
15.e even 4 2 4680.2.l.f 8
20.e even 4 2 3120.2.l.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1560.2.l.e 8 5.c odd 4 2
3120.2.l.o 8 20.e even 4 2
4680.2.l.f 8 15.e even 4 2
7800.2.a.bv 4 5.b even 2 1
7800.2.a.bw 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{2} - 8 \)
\( T_{11}^{4} + 8 T_{11}^{3} + 6 T_{11}^{2} - 24 T_{11} + 8 \)
\( T_{17}^{4} - 4 T_{17}^{3} - 44 T_{17}^{2} + 224 T_{17} - 224 \)
\( T_{19}^{4} + 12 T_{19}^{3} + 12 T_{19}^{2} - 224 T_{19} - 448 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( ( -1 + T )^{4} \)
$5$ \( T^{4} \)
$7$ \( ( -8 + T^{2} )^{2} \)
$11$ \( 8 - 24 T + 6 T^{2} + 8 T^{3} + T^{4} \)
$13$ \( ( 1 + T )^{4} \)
$17$ \( -224 + 224 T - 44 T^{2} - 4 T^{3} + T^{4} \)
$19$ \( -448 - 224 T + 12 T^{2} + 12 T^{3} + T^{4} \)
$23$ \( -224 + 224 T - 44 T^{2} - 4 T^{3} + T^{4} \)
$29$ \( ( -4 - 4 T + T^{2} )^{2} \)
$31$ \( ( -16 + 8 T + T^{2} )^{2} \)
$37$ \( -16 - 384 T - 48 T^{2} + 8 T^{3} + T^{4} \)
$41$ \( -8 - 56 T - 14 T^{2} + 4 T^{3} + T^{4} \)
$43$ \( -512 + 384 T - 68 T^{2} - 4 T^{3} + T^{4} \)
$47$ \( -1024 - 512 T - 30 T^{2} + 12 T^{3} + T^{4} \)
$53$ \( 496 - 160 T - 80 T^{2} + T^{4} \)
$59$ \( 8 - 24 T + 6 T^{2} + 8 T^{3} + T^{4} \)
$61$ \( -1472 + 736 T - 44 T^{2} - 12 T^{3} + T^{4} \)
$67$ \( -128 + 64 T + 136 T^{2} + 24 T^{3} + T^{4} \)
$71$ \( 512 - 576 T - 46 T^{2} + 12 T^{3} + T^{4} \)
$73$ \( -5008 - 768 T + 112 T^{2} + 24 T^{3} + T^{4} \)
$79$ \( 6272 - 896 T - 132 T^{2} + 12 T^{3} + T^{4} \)
$83$ \( -4384 - 2448 T - 206 T^{2} + 12 T^{3} + T^{4} \)
$89$ \( 184 - 1288 T - 190 T^{2} + 4 T^{3} + T^{4} \)
$97$ \( -16 - 384 T - 48 T^{2} + 8 T^{3} + T^{4} \)
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