Properties

Label 7800.2.a.bx
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: \( x^{4} - x^{3} - 14x^{2} - 12x + 6 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_1 + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_1 + 1) q^{7} + q^{9} + ( - \beta_{2} + \beta_1 + 1) q^{11} + q^{13} + (\beta_{3} + \beta_1 + 2) q^{17} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{19} + ( - \beta_1 + 1) q^{21} + (\beta_{2} + 2) q^{23} + q^{27} + ( - 2 \beta_1 + 1) q^{29} + (\beta_{2} + \beta_1 + 1) q^{31} + ( - \beta_{2} + \beta_1 + 1) q^{33} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{37} + q^{39} + ( - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{41} + ( - \beta_{2} - 2 \beta_1 + 2) q^{43} + ( - \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{47} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{49} + (\beta_{3} + \beta_1 + 2) q^{51} + (2 \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{53} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{57} + ( - 2 \beta_{3} - \beta_1 - 3) q^{59} + ( - \beta_{3} + \beta_1 + 4) q^{61} + ( - \beta_1 + 1) q^{63} + (\beta_{3} - \beta_{2} - 2 \beta_1) q^{67} + (\beta_{2} + 2) q^{69} + (\beta_{3} - \beta_{2} + \beta_1 + 5) q^{71} + ( - 2 \beta_{3} - \beta_{2} - 2 \beta_1 - 2) q^{73} + ( - \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 4) q^{77} + ( - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{79} + q^{81} + (\beta_1 + 5) q^{83} + ( - 2 \beta_1 + 1) q^{87} + (2 \beta_{2} - 2 \beta_1 + 4) q^{89} + ( - \beta_1 + 1) q^{91} + (\beta_{2} + \beta_1 + 1) q^{93} + (2 \beta_1 - 2) q^{97} + ( - \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 3 q^{7} + 4 q^{9} + 5 q^{11} + 4 q^{13} + 7 q^{17} + 3 q^{19} + 3 q^{21} + 8 q^{23} + 4 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - q^{37} + 4 q^{39} + 7 q^{41} + 6 q^{43} + 3 q^{49} + 7 q^{51} + 6 q^{53} + 3 q^{57} - 9 q^{59} + 19 q^{61} + 3 q^{63} - 4 q^{67} + 8 q^{69} + 19 q^{71} - 6 q^{73} - 17 q^{77} - 9 q^{79} + 4 q^{81} + 21 q^{83} + 2 q^{87} + 14 q^{89} + 3 q^{91} + 5 q^{93} - 6 q^{97} + 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 3\nu^{2} - 8\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} - 11\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} + 3\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} - 2\beta_{2} + 17\beta _1 + 17 \) 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
4.57275
0.352593
−1.60110
−2.32424
0 1.00000 0 0 0 −3.57275 0 1.00000 0
1.2 0 1.00000 0 0 0 0.647407 0 1.00000 0
1.3 0 1.00000 0 0 0 2.60110 0 1.00000 0
1.4 0 1.00000 0 0 0 3.32424 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.bx yes 4
5.b even 2 1 7800.2.a.bu 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bu 4 5.b even 2 1
7800.2.a.bx yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

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