Properties

Label 7800.2.a.bm
Level $7800$
Weight $2$
Character orbit 7800.a
Self dual yes
Analytic conductor $62.283$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(x^{3} - x^{2} - 6 x - 2\)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{7} + q^{9} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{11} + q^{13} + ( 1 + \beta_{2} ) q^{17} + ( 2 \beta_{1} - \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{21} + ( 2 - 2 \beta_{1} ) q^{23} - q^{27} + ( 1 + 2 \beta_{2} ) q^{29} + ( 5 + \beta_{1} + 2 \beta_{2} ) q^{31} + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{33} + ( 2 \beta_{1} + \beta_{2} ) q^{37} - q^{39} + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{41} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 - \beta_{1} + \beta_{2} ) q^{47} + ( -2 + \beta_{2} ) q^{49} + ( -1 - \beta_{2} ) q^{51} + ( -5 + 2 \beta_{1} + 2 \beta_{2} ) q^{53} + ( -2 \beta_{1} + \beta_{2} ) q^{57} + ( 1 + 3 \beta_{1} - 2 \beta_{2} ) q^{59} + ( 7 - 4 \beta_{1} - \beta_{2} ) q^{61} + ( 1 - \beta_{1} ) q^{63} + ( -1 + 5 \beta_{1} - \beta_{2} ) q^{67} + ( -2 + 2 \beta_{1} ) q^{69} + ( 8 - 2 \beta_{1} - \beta_{2} ) q^{71} + ( -2 + 2 \beta_{2} ) q^{73} + ( 1 - 2 \beta_{1} + 3 \beta_{2} ) q^{77} + ( 4 - 4 \beta_{1} + \beta_{2} ) q^{79} + q^{81} + ( -3 + 3 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -1 - 2 \beta_{2} ) q^{87} + ( -4 - 4 \beta_{1} - 2 \beta_{2} ) q^{89} + ( 1 - \beta_{1} ) q^{91} + ( -5 - \beta_{1} - 2 \beta_{2} ) q^{93} + ( -2 - 6 \beta_{2} ) q^{97} + ( 1 + \beta_{1} + 2 \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{3} + 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{3} + 2q^{7} + 3q^{9} + 2q^{11} + 3q^{13} + 2q^{17} + 3q^{19} - 2q^{21} + 4q^{23} - 3q^{27} + q^{29} + 14q^{31} - 2q^{33} + q^{37} - 3q^{39} + 5q^{41} + 6q^{43} - 5q^{47} - 7q^{49} - 2q^{51} - 15q^{53} - 3q^{57} + 8q^{59} + 18q^{61} + 2q^{63} + 3q^{67} - 4q^{69} + 23q^{71} - 8q^{73} - 2q^{77} + 7q^{79} + 3q^{81} - 8q^{83} - q^{87} - 14q^{89} + 2q^{91} - 14q^{93} + 2q^{99} + O(q^{100}) \)

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 4\)

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
3.12489
−0.363328
−1.76156
0 −1.00000 0 0 0 −2.12489 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 2.76156 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.bm 3
5.b even 2 1 7800.2.a.bn yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7800.2.a.bm 3 1.a even 1 1 trivial
7800.2.a.bn yes 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7800))\):

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

Hecke characteristic polynomials

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