Properties

Label 7800.2.a.bn
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) 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 \(1,\beta_1,\beta_2\) 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 \( q + q^{3} + (\beta_1 - 1) q^{7} + q^{9} + (2 \beta_{2} + \beta_1 + 1) q^{11} - q^{13} + ( - \beta_{2} - 1) q^{17} + ( - \beta_{2} + 2 \beta_1) q^{19} + (\beta_1 - 1) q^{21} + (2 \beta_1 - 2) q^{23} + q^{27} + (2 \beta_{2} + 1) q^{29} + (2 \beta_{2} + \beta_1 + 5) q^{31} + (2 \beta_{2} + \beta_1 + 1) q^{33} + ( - \beta_{2} - 2 \beta_1) q^{37} - q^{39} + ( - \beta_{2} - 2 \beta_1 + 2) q^{41} + (2 \beta_{2} + 2 \beta_1 - 2) q^{43} + ( - \beta_{2} + \beta_1 + 1) q^{47} + (\beta_{2} - 2) q^{49} + ( - \beta_{2} - 1) q^{51} + ( - 2 \beta_{2} - 2 \beta_1 + 5) q^{53} + ( - \beta_{2} + 2 \beta_1) q^{57} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{59} + ( - \beta_{2} - 4 \beta_1 + 7) q^{61} + (\beta_1 - 1) q^{63} + (\beta_{2} - 5 \beta_1 + 1) q^{67} + (2 \beta_1 - 2) q^{69} + ( - \beta_{2} - 2 \beta_1 + 8) q^{71} + ( - 2 \beta_{2} + 2) q^{73} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{77} + (\beta_{2} - 4 \beta_1 + 4) q^{79} + q^{81} + ( - 2 \beta_{2} - 3 \beta_1 + 3) q^{83} + (2 \beta_{2} + 1) q^{87} + ( - 2 \beta_{2} - 4 \beta_1 - 4) q^{89} + ( - \beta_1 + 1) q^{91} + (2 \beta_{2} + \beta_1 + 5) q^{93} + (6 \beta_{2} + 2) q^{97} + (2 \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} - 3 q^{13} - 2 q^{17} + 3 q^{19} - 2 q^{21} - 4 q^{23} + 3 q^{27} + q^{29} + 14 q^{31} + 2 q^{33} - q^{37} - 3 q^{39} + 5 q^{41} - 6 q^{43} + 5 q^{47} - 7 q^{49} - 2 q^{51} + 15 q^{53} + 3 q^{57} + 8 q^{59} + 18 q^{61} - 2 q^{63} - 3 q^{67} - 4 q^{69} + 23 q^{71} + 8 q^{73} + 2 q^{77} + 7 q^{79} + 3 q^{81} + 8 q^{83} + q^{87} - 14 q^{89} + 2 q^{91} + 14 q^{93} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) 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
−1.76156
−0.363328
3.12489
0 1.00000 0 0 0 −2.76156 0 1.00000 0
1.2 0 1.00000 0 0 0 −1.36333 0 1.00000 0
1.3 0 1.00000 0 0 0 2.12489 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.bn yes 3
5.b even 2 1 7800.2.a.bm 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bm 3 5.b even 2 1
7800.2.a.bn yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

\( T_{7}^{3} + 2T_{7}^{2} - 5T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{3} - 2T_{11}^{2} - 29T_{11} + 80 \) Copy content Toggle raw display
\( T_{17}^{3} + 2T_{17}^{2} - 7T_{17} - 4 \) Copy content Toggle raw display
\( T_{19}^{3} - 3T_{19}^{2} - 40T_{19} + 100 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 2 T^{2} - 5 T - 8 \) Copy content Toggle raw display
$11$ \( T^{3} - 2 T^{2} - 29 T + 80 \) Copy content Toggle raw display
$13$ \( (T + 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} - 7 T - 4 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} - 40 T + 100 \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} - 20 T - 64 \) Copy content Toggle raw display
$29$ \( T^{3} - T^{2} - 33T + 1 \) Copy content Toggle raw display
$31$ \( T^{3} - 14 T^{2} + 35 T + 100 \) Copy content Toggle raw display
$37$ \( T^{3} + T^{2} - 24 T + 20 \) Copy content Toggle raw display
$41$ \( T^{3} - 5 T^{2} - 16 T + 64 \) Copy content Toggle raw display
$43$ \( T^{3} + 6 T^{2} - 28 T - 8 \) Copy content Toggle raw display
$47$ \( T^{3} - 5 T^{2} - 11 T + 59 \) Copy content Toggle raw display
$53$ \( T^{3} - 15 T^{2} + 35 T + 11 \) Copy content Toggle raw display
$59$ \( T^{3} - 8 T^{2} - 97 T + 670 \) Copy content Toggle raw display
$61$ \( T^{3} - 18 T^{2} + 17 T + 664 \) Copy content Toggle raw display
$67$ \( T^{3} + 3 T^{2} - 187 T - 61 \) Copy content Toggle raw display
$71$ \( T^{3} - 23 T^{2} + 152 T - 236 \) Copy content Toggle raw display
$73$ \( T^{3} - 8 T^{2} - 12 T + 80 \) Copy content Toggle raw display
$79$ \( T^{3} - 7 T^{2} - 112 T + 284 \) Copy content Toggle raw display
$83$ \( T^{3} - 8 T^{2} - 41 T + 170 \) Copy content Toggle raw display
$89$ \( T^{3} + 14 T^{2} - 32 T - 128 \) Copy content Toggle raw display
$97$ \( T^{3} - 300T - 272 \) Copy content Toggle raw display
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