Properties

Label 8550.2.a.ce
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 570)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{2} q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{2} q^{7} - q^{8} + ( -3 - \beta_{1} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} -\beta_{2} q^{14} + q^{16} + ( -1 - \beta_{1} - \beta_{2} ) q^{17} + q^{19} + ( 3 + \beta_{1} ) q^{22} + ( 3 + \beta_{1} ) q^{23} + ( -1 + \beta_{1} + \beta_{2} ) q^{26} + \beta_{2} q^{28} + ( -3 + \beta_{1} ) q^{29} + \beta_{2} q^{31} - q^{32} + ( 1 + \beta_{1} + \beta_{2} ) q^{34} + ( 1 - \beta_{1} - \beta_{2} ) q^{37} - q^{38} + ( -2 - 2 \beta_{1} + \beta_{2} ) q^{41} + ( -1 + 3 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -3 - \beta_{1} ) q^{44} + ( -3 - \beta_{1} ) q^{46} + ( -3 - \beta_{1} ) q^{47} + ( 3 - 2 \beta_{1} - 2 \beta_{2} ) q^{49} + ( 1 - \beta_{1} - \beta_{2} ) q^{52} + 6 q^{53} -\beta_{2} q^{56} + ( 3 - \beta_{1} ) q^{58} + ( -3 + \beta_{1} - \beta_{2} ) q^{59} + ( 4 - 2 \beta_{1} ) q^{61} -\beta_{2} q^{62} + q^{64} + ( 6 - 2 \beta_{1} - 2 \beta_{2} ) q^{67} + ( -1 - \beta_{1} - \beta_{2} ) q^{68} + ( -6 + 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{73} + ( -1 + \beta_{1} + \beta_{2} ) q^{74} + q^{76} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{77} -\beta_{2} q^{79} + ( 2 + 2 \beta_{1} - \beta_{2} ) q^{82} + ( 1 - \beta_{1} + 3 \beta_{2} ) q^{83} + ( 1 - 3 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 3 + \beta_{1} ) q^{88} + ( -8 + \beta_{2} ) q^{89} + ( -8 + 4 \beta_{2} ) q^{91} + ( 3 + \beta_{1} ) q^{92} + ( 3 + \beta_{1} ) q^{94} + ( 7 - \beta_{1} ) q^{97} + ( -3 + 2 \beta_{1} + 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} - 8 q^{11} + 4 q^{13} + 3 q^{16} - 2 q^{17} + 3 q^{19} + 8 q^{22} + 8 q^{23} - 4 q^{26} - 10 q^{29} - 3 q^{32} + 2 q^{34} + 4 q^{37} - 3 q^{38} - 4 q^{41} - 6 q^{43} - 8 q^{44} - 8 q^{46} - 8 q^{47} + 11 q^{49} + 4 q^{52} + 18 q^{53} + 10 q^{58} - 10 q^{59} + 14 q^{61} + 3 q^{64} + 20 q^{67} - 2 q^{68} - 20 q^{71} + 8 q^{73} - 4 q^{74} + 3 q^{76} + 8 q^{77} + 4 q^{82} + 4 q^{83} + 6 q^{86} + 8 q^{88} - 24 q^{89} - 24 q^{91} + 8 q^{92} + 8 q^{94} + 22 q^{97} - 11 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 3 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu - 1 \)
\(\beta_{2}\)\(=\)\( 2 \nu^{2} - 2 \nu - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{1} + 1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{2} + \beta_{1} + 5\)\()/2\)

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
0.311108
2.17009
−1.48119
−1.00000 0 1.00000 0 0 −4.42864 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 1.07838 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 3.35026 −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.ce 3
3.b odd 2 1 2850.2.a.bm 3
5.b even 2 1 8550.2.a.cq 3
5.c odd 4 2 1710.2.d.f 6
15.d odd 2 1 2850.2.a.bl 3
15.e even 4 2 570.2.d.c 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
570.2.d.c 6 15.e even 4 2
1710.2.d.f 6 5.c odd 4 2
2850.2.a.bl 3 15.d odd 2 1
2850.2.a.bm 3 3.b odd 2 1
8550.2.a.ce 3 1.a even 1 1 trivial
8550.2.a.cq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{3} - 16 T_{7} + 16 \)
\( T_{11}^{3} + 8 T_{11}^{2} + 8 T_{11} - 16 \)
\( T_{13}^{3} - 4 T_{13}^{2} - 16 T_{13} + 32 \)
\( T_{17}^{3} + 2 T_{17}^{2} - 20 T_{17} - 8 \)
\( T_{23}^{3} - 8 T_{23}^{2} + 8 T_{23} + 16 \)
\( T_{53} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( 16 - 16 T + T^{3} \)
$11$ \( -16 + 8 T + 8 T^{2} + T^{3} \)
$13$ \( 32 - 16 T - 4 T^{2} + T^{3} \)
$17$ \( -8 - 20 T + 2 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( 16 + 8 T - 8 T^{2} + T^{3} \)
$29$ \( -8 + 20 T + 10 T^{2} + T^{3} \)
$31$ \( 16 - 16 T + T^{3} \)
$37$ \( 32 - 16 T - 4 T^{2} + T^{3} \)
$41$ \( -400 - 80 T + 4 T^{2} + T^{3} \)
$43$ \( -760 - 124 T + 6 T^{2} + T^{3} \)
$47$ \( -16 + 8 T + 8 T^{2} + T^{3} \)
$53$ \( ( -6 + T )^{3} \)
$59$ \( -8 - 4 T + 10 T^{2} + T^{3} \)
$61$ \( 152 + 12 T - 14 T^{2} + T^{3} \)
$67$ \( 320 + 48 T - 20 T^{2} + T^{3} \)
$71$ \( -320 + 48 T + 20 T^{2} + T^{3} \)
$73$ \( 256 - 64 T - 8 T^{2} + T^{3} \)
$79$ \( -16 - 16 T + T^{3} \)
$83$ \( 160 - 176 T - 4 T^{2} + T^{3} \)
$89$ \( 400 + 176 T + 24 T^{2} + T^{3} \)
$97$ \( -296 + 148 T - 22 T^{2} + T^{3} \)
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