Properties

Label 7800.2.a.bv
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: 4.4.10304.1
Defining polynomial: \( x^{4} - 2x^{3} - 7x^{2} + 8x + 8 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{3} - \beta_{3} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \beta_{3} q^{7} + q^{9} + ( - \beta_1 - 2) q^{11} + q^{13} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_{3} - \beta_{2} + \beta_1 - 3) q^{19} + \beta_{3} q^{21} + (\beta_{3} + \beta_{2} + \beta_1 - 1) q^{23} - q^{27} + (\beta_{3} + 2) q^{29} + ( - 2 \beta_{3} - 4) q^{31} + (\beta_1 + 2) q^{33} + ( - 2 \beta_1 + 2) q^{37} - q^{39} + (\beta_{2} - 1) q^{41} + ( - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{43} + ( - \beta_{2} - 2 \beta_1 + 3) q^{47} + q^{49} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{51} - 2 \beta_{2} q^{53} + (\beta_{3} + \beta_{2} - \beta_1 + 3) q^{57} + ( - \beta_1 - 2) q^{59} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{61} - \beta_{3} q^{63} + (2 \beta_{2} + 6) q^{67} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{69} + (\beta_{2} - 2 \beta_1 - 3) q^{71} + ( - \beta_{3} - 2 \beta_1 + 6) q^{73} + (2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{77} + ( - 3 \beta_{2} - \beta_1 - 3) q^{79} + q^{81} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 3) q^{83} + ( - \beta_{3} - 2) q^{87} + ( - 2 \beta_{3} - 3 \beta_{2} - 1) q^{89} - \beta_{3} q^{91} + (2 \beta_{3} + 4) q^{93} + ( - \beta_{3} + 2 \beta_1 + 2) q^{97} + ( - \beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{9} - 8 q^{11} + 4 q^{13} - 4 q^{17} - 12 q^{19} - 4 q^{23} - 4 q^{27} + 8 q^{29} - 16 q^{31} + 8 q^{33} + 8 q^{37} - 4 q^{39} - 4 q^{41} - 4 q^{43} + 12 q^{47} + 4 q^{49} + 4 q^{51} + 12 q^{57} - 8 q^{59} + 12 q^{61} + 24 q^{67} + 4 q^{69} - 12 q^{71} + 24 q^{73} + 8 q^{77} - 12 q^{79} + 4 q^{81} + 12 q^{83} - 8 q^{87} - 4 q^{89} + 16 q^{93} + 8 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu^{2} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + \nu^{2} + 8\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{3} + \beta_{2} + \beta _1 + 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{3} + 5\beta_{2} + 9\beta _1 + 13 ) / 2 \) Copy content Toggle raw display

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
3.16053
−2.16053
−0.692297
1.69230
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.bv 4
5.b even 2 1 7800.2.a.bw 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 1.a even 1 1 trivial
7800.2.a.bw 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

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

Hecke characteristic polynomials

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