Properties

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

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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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Defining polynomial: \(x^{2} - 3\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( 1 + \beta ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( 1 + \beta ) q^{7} - q^{8} + ( -2 + \beta ) q^{11} + ( 1 - \beta ) q^{13} + ( -1 - \beta ) q^{14} + q^{16} + ( -1 + 3 \beta ) q^{17} - q^{19} + ( 2 - \beta ) q^{22} + ( -1 - 4 \beta ) q^{23} + ( -1 + \beta ) q^{26} + ( 1 + \beta ) q^{28} -\beta q^{29} + ( 1 + 2 \beta ) q^{31} - q^{32} + ( 1 - 3 \beta ) q^{34} + 2 q^{37} + q^{38} + ( -4 - 4 \beta ) q^{41} + ( 3 - 3 \beta ) q^{43} + ( -2 + \beta ) q^{44} + ( 1 + 4 \beta ) q^{46} -2 \beta q^{47} + ( -3 + 2 \beta ) q^{49} + ( 1 - \beta ) q^{52} + \beta q^{53} + ( -1 - \beta ) q^{56} + \beta q^{58} + ( 3 - 3 \beta ) q^{59} + ( 2 - 5 \beta ) q^{61} + ( -1 - 2 \beta ) q^{62} + q^{64} + ( 2 + \beta ) q^{67} + ( -1 + 3 \beta ) q^{68} + ( -7 + 3 \beta ) q^{71} + ( -1 - 2 \beta ) q^{73} -2 q^{74} - q^{76} + ( 1 - \beta ) q^{77} + ( -9 - 2 \beta ) q^{79} + ( 4 + 4 \beta ) q^{82} + ( 2 - \beta ) q^{83} + ( -3 + 3 \beta ) q^{86} + ( 2 - \beta ) q^{88} + ( 3 + 8 \beta ) q^{89} -2 q^{91} + ( -1 - 4 \beta ) q^{92} + 2 \beta q^{94} + ( 3 - 7 \beta ) q^{97} + ( 3 - 2 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 2q^{7} - 2q^{8} - 4q^{11} + 2q^{13} - 2q^{14} + 2q^{16} - 2q^{17} - 2q^{19} + 4q^{22} - 2q^{23} - 2q^{26} + 2q^{28} + 2q^{31} - 2q^{32} + 2q^{34} + 4q^{37} + 2q^{38} - 8q^{41} + 6q^{43} - 4q^{44} + 2q^{46} - 6q^{49} + 2q^{52} - 2q^{56} + 6q^{59} + 4q^{61} - 2q^{62} + 2q^{64} + 4q^{67} - 2q^{68} - 14q^{71} - 2q^{73} - 4q^{74} - 2q^{76} + 2q^{77} - 18q^{79} + 8q^{82} + 4q^{83} - 6q^{86} + 4q^{88} + 6q^{89} - 4q^{91} - 2q^{92} + 6q^{97} + 6q^{98} + O(q^{100}) \)

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
−1.73205
1.73205
−1.00000 0 1.00000 0 0 −0.732051 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 2.73205 −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.bs 2
3.b odd 2 1 2850.2.a.bi yes 2
5.b even 2 1 8550.2.a.by 2
15.d odd 2 1 2850.2.a.bd 2
15.e even 4 2 2850.2.d.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2850.2.a.bd 2 15.d odd 2 1
2850.2.a.bi yes 2 3.b odd 2 1
2850.2.d.x 4 15.e even 4 2
8550.2.a.bs 2 1.a even 1 1 trivial
8550.2.a.by 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{2} - 2 T_{7} - 2 \)
\( T_{11}^{2} + 4 T_{11} + 1 \)
\( T_{13}^{2} - 2 T_{13} - 2 \)
\( T_{17}^{2} + 2 T_{17} - 26 \)
\( T_{23}^{2} + 2 T_{23} - 47 \)
\( T_{53}^{2} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -2 - 2 T + T^{2} \)
$11$ \( 1 + 4 T + T^{2} \)
$13$ \( -2 - 2 T + T^{2} \)
$17$ \( -26 + 2 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -47 + 2 T + T^{2} \)
$29$ \( -3 + T^{2} \)
$31$ \( -11 - 2 T + T^{2} \)
$37$ \( ( -2 + T )^{2} \)
$41$ \( -32 + 8 T + T^{2} \)
$43$ \( -18 - 6 T + T^{2} \)
$47$ \( -12 + T^{2} \)
$53$ \( -3 + T^{2} \)
$59$ \( -18 - 6 T + T^{2} \)
$61$ \( -71 - 4 T + T^{2} \)
$67$ \( 1 - 4 T + T^{2} \)
$71$ \( 22 + 14 T + T^{2} \)
$73$ \( -11 + 2 T + T^{2} \)
$79$ \( 69 + 18 T + T^{2} \)
$83$ \( 1 - 4 T + T^{2} \)
$89$ \( -183 - 6 T + T^{2} \)
$97$ \( -138 - 6 T + T^{2} \)
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