Properties

Label 8550.2.a.bq
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 8550 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(68.2720937282\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{10}) \)
Defining polynomial: \(x^{2} - 10\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2850)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta q^{7} - q^{8} + ( -1 - \beta ) q^{11} + ( 2 + \beta ) q^{13} -\beta q^{14} + q^{16} + ( -4 + \beta ) q^{17} - q^{19} + ( 1 + \beta ) q^{22} + ( -1 + 2 \beta ) q^{23} + ( -2 - \beta ) q^{26} + \beta q^{28} + ( 7 - \beta ) q^{29} + ( -1 + 2 \beta ) q^{31} - q^{32} + ( 4 - \beta ) q^{34} + 10 q^{37} + q^{38} + 2 \beta q^{41} + ( -6 - \beta ) q^{43} + ( -1 - \beta ) q^{44} + ( 1 - 2 \beta ) q^{46} -6 q^{47} + 3 q^{49} + ( 2 + \beta ) q^{52} + ( -5 + \beta ) q^{53} -\beta q^{56} + ( -7 + \beta ) q^{58} + ( -2 + 3 \beta ) q^{59} + ( -1 - \beta ) q^{61} + ( 1 - 2 \beta ) q^{62} + q^{64} + ( 3 - 3 \beta ) q^{67} + ( -4 + \beta ) q^{68} + ( 2 - \beta ) q^{71} - q^{73} -10 q^{74} - q^{76} + ( -10 - \beta ) q^{77} + ( 1 - 2 \beta ) q^{79} -2 \beta q^{82} + ( 3 - 3 \beta ) q^{83} + ( 6 + \beta ) q^{86} + ( 1 + \beta ) q^{88} + ( 1 + 2 \beta ) q^{89} + ( 10 + 2 \beta ) q^{91} + ( -1 + 2 \beta ) q^{92} + 6 q^{94} + ( 10 - 3 \beta ) q^{97} -3 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{8} - 2q^{11} + 4q^{13} + 2q^{16} - 8q^{17} - 2q^{19} + 2q^{22} - 2q^{23} - 4q^{26} + 14q^{29} - 2q^{31} - 2q^{32} + 8q^{34} + 20q^{37} + 2q^{38} - 12q^{43} - 2q^{44} + 2q^{46} - 12q^{47} + 6q^{49} + 4q^{52} - 10q^{53} - 14q^{58} - 4q^{59} - 2q^{61} + 2q^{62} + 2q^{64} + 6q^{67} - 8q^{68} + 4q^{71} - 2q^{73} - 20q^{74} - 2q^{76} - 20q^{77} + 2q^{79} + 6q^{83} + 12q^{86} + 2q^{88} + 2q^{89} + 20q^{91} - 2q^{92} + 12q^{94} + 20q^{97} - 6q^{98} + O(q^{100}) \)

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
−3.16228
3.16228
−1.00000 0 1.00000 0 0 −3.16228 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 3.16228 −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(5\) \(1\)
\(19\) \(1\)

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 8550.2.a.bq 2
3.b odd 2 1 2850.2.a.bg yes 2
5.b even 2 1 8550.2.a.ca 2
15.d odd 2 1 2850.2.a.bf 2
15.e even 4 2 2850.2.d.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bf 2 15.d odd 2 1
2850.2.a.bg yes 2 3.b odd 2 1
2850.2.d.v 4 15.e even 4 2
8550.2.a.bq 2 1.a even 1 1 trivial
8550.2.a.ca 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{2} - 10 \)
\( T_{11}^{2} + 2 T_{11} - 9 \)
\( T_{13}^{2} - 4 T_{13} - 6 \)
\( T_{17}^{2} + 8 T_{17} + 6 \)
\( T_{23}^{2} + 2 T_{23} - 39 \)
\( T_{53}^{2} + 10 T_{53} + 15 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -10 + T^{2} \)
$11$ \( -9 + 2 T + T^{2} \)
$13$ \( -6 - 4 T + T^{2} \)
$17$ \( 6 + 8 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -39 + 2 T + T^{2} \)
$29$ \( 39 - 14 T + T^{2} \)
$31$ \( -39 + 2 T + T^{2} \)
$37$ \( ( -10 + T )^{2} \)
$41$ \( -40 + T^{2} \)
$43$ \( 26 + 12 T + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( 15 + 10 T + T^{2} \)
$59$ \( -86 + 4 T + T^{2} \)
$61$ \( -9 + 2 T + T^{2} \)
$67$ \( -81 - 6 T + T^{2} \)
$71$ \( -6 - 4 T + T^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( -39 - 2 T + T^{2} \)
$83$ \( -81 - 6 T + T^{2} \)
$89$ \( -39 - 2 T + T^{2} \)
$97$ \( 10 - 20 T + T^{2} \)
show more
show less