Properties

Label 8550.2.a.bw
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1710)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

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

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 −2.00000 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 −2.00000 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.bw 2
3.b odd 2 1 8550.2.a.bo 2
5.b even 2 1 1710.2.a.u 2
15.d odd 2 1 1710.2.a.y yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.a.u 2 5.b even 2 1
1710.2.a.y yes 2 15.d odd 2 1
8550.2.a.bo 2 3.b odd 2 1
8550.2.a.bw 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7} + 2 \)
\( T_{11}^{2} - 12 \)
\( T_{13}^{2} + 4 T_{13} - 8 \)
\( T_{17}^{2} - 12 \)
\( T_{23}^{2} - 48 \)
\( T_{53}^{2} + 12 T_{53} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( ( 2 + T )^{2} \)
$11$ \( -12 + T^{2} \)
$13$ \( -8 + 4 T + T^{2} \)
$17$ \( -12 + T^{2} \)
$19$ \( ( -1 + T )^{2} \)
$23$ \( -48 + T^{2} \)
$29$ \( -12 + T^{2} \)
$31$ \( -8 - 4 T + T^{2} \)
$37$ \( -8 + 4 T + T^{2} \)
$41$ \( T^{2} \)
$43$ \( -8 + 4 T + T^{2} \)
$47$ \( -48 + T^{2} \)
$53$ \( -12 + 12 T + T^{2} \)
$59$ \( -12 + T^{2} \)
$61$ \( ( -2 + T )^{2} \)
$67$ \( 16 + 16 T + T^{2} \)
$71$ \( -48 + T^{2} \)
$73$ \( -44 + 4 T + T^{2} \)
$79$ \( -8 - 4 T + T^{2} \)
$83$ \( 24 + 12 T + T^{2} \)
$89$ \( -48 + T^{2} \)
$97$ \( -44 + 16 T + T^{2} \)
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