Properties

Label 8550.2.a.ck
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: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_1 + 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_1 + 1) q^{7} - q^{8} + (\beta_{2} + \beta_1) q^{11} + (\beta_{2} + 3) q^{13} + ( - \beta_1 - 1) q^{14} + q^{16} + (\beta_{2} + 1) q^{17} + q^{19} + ( - \beta_{2} - \beta_1) q^{22} + (2 \beta_{2} - \beta_1 + 1) q^{23} + ( - \beta_{2} - 3) q^{26} + (\beta_1 + 1) q^{28} + (3 \beta_{2} + 2 \beta_1 + 3) q^{29} + (3 \beta_{2} + \beta_1 + 2) q^{31} - q^{32} + ( - \beta_{2} - 1) q^{34} + (2 \beta_1 + 4) q^{37} - q^{38} + ( - \beta_{2} - 3 \beta_1) q^{41} + (\beta_{2} + \beta_1 + 6) q^{43} + (\beta_{2} + \beta_1) q^{44} + ( - 2 \beta_{2} + \beta_1 - 1) q^{46} + ( - 2 \beta_{2} + 4) q^{47} + (\beta_{2} + 4 \beta_1 - 2) q^{49} + (\beta_{2} + 3) q^{52} + (\beta_{2} - 5) q^{53} + ( - \beta_1 - 1) q^{56} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{58} + ( - 2 \beta_{2} - \beta_1 - 1) q^{59} + ( - \beta_{2} - \beta_1 - 10) q^{61} + ( - 3 \beta_{2} - \beta_1 - 2) q^{62} + q^{64} + ( - 2 \beta_{2} - 3 \beta_1 + 1) q^{67} + (\beta_{2} + 1) q^{68} + ( - \beta_{2} - 5 \beta_1 + 4) q^{71} + (3 \beta_{2} - 2 \beta_1 + 5) q^{73} + ( - 2 \beta_1 - 4) q^{74} + q^{76} + (\beta_{2} + 3 \beta_1 + 2) q^{77} + (6 \beta_1 - 2) q^{79} + (\beta_{2} + 3 \beta_1) q^{82} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{83} + ( - \beta_{2} - \beta_1 - 6) q^{86} + ( - \beta_{2} - \beta_1) q^{88} + ( - \beta_{2} + \beta_1 + 4) q^{89} + (3 \beta_1 + 1) q^{91} + (2 \beta_{2} - \beta_1 + 1) q^{92} + (2 \beta_{2} - 4) q^{94} + ( - 4 \beta_{2} + 2) q^{97} + ( - \beta_{2} - 4 \beta_1 + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} + 4 q^{7} - 3 q^{8} + 8 q^{13} - 4 q^{14} + 3 q^{16} + 2 q^{17} + 3 q^{19} - 8 q^{26} + 4 q^{28} + 8 q^{29} + 4 q^{31} - 3 q^{32} - 2 q^{34} + 14 q^{37} - 3 q^{38} - 2 q^{41} + 18 q^{43} + 14 q^{47} - 3 q^{49} + 8 q^{52} - 16 q^{53} - 4 q^{56} - 8 q^{58} - 2 q^{59} - 30 q^{61} - 4 q^{62} + 3 q^{64} + 2 q^{67} + 2 q^{68} + 8 q^{71} + 10 q^{73} - 14 q^{74} + 3 q^{76} + 8 q^{77} + 2 q^{82} + 6 q^{83} - 18 q^{86} + 14 q^{89} + 6 q^{91} - 14 q^{94} + 10 q^{97} + 3 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2\nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2\beta _1 + 4 \) Copy content Toggle raw display

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
−1.76156
−0.363328
3.12489
−1.00000 0 1.00000 0 0 −0.761557 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 0.636672 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 4.12489 −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.ck 3
3.b odd 2 1 950.2.a.n 3
5.b even 2 1 8550.2.a.cl 3
5.c odd 4 2 1710.2.d.d 6
12.b even 2 1 7600.2.a.bi 3
15.d odd 2 1 950.2.a.i 3
15.e even 4 2 190.2.b.b 6
60.h even 2 1 7600.2.a.cd 3
60.l odd 4 2 1520.2.d.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.b 6 15.e even 4 2
950.2.a.i 3 15.d odd 2 1
950.2.a.n 3 3.b odd 2 1
1520.2.d.j 6 60.l odd 4 2
1710.2.d.d 6 5.c odd 4 2
7600.2.a.bi 3 12.b even 2 1
7600.2.a.cd 3 60.h even 2 1
8550.2.a.ck 3 1.a even 1 1 trivial
8550.2.a.cl 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{3} - 4T_{7}^{2} - T_{7} + 2 \) Copy content Toggle raw display
\( T_{11}^{3} - 10T_{11} + 8 \) Copy content Toggle raw display
\( T_{13}^{3} - 8T_{13}^{2} + 13T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{3} - 2T_{17}^{2} - 7T_{17} + 4 \) Copy content Toggle raw display
\( T_{23}^{3} - 49T_{23} - 122 \) Copy content Toggle raw display
\( T_{53}^{3} + 16T_{53}^{2} + 77T_{53} + 106 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} - 4T^{2} - T + 2 \) Copy content Toggle raw display
$11$ \( T^{3} - 10T + 8 \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + 13 T + 2 \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} - 7 T + 4 \) Copy content Toggle raw display
$19$ \( (T - 1)^{3} \) Copy content Toggle raw display
$23$ \( T^{3} - 49T - 122 \) Copy content Toggle raw display
$29$ \( T^{3} - 8 T^{2} - 51 T + 410 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 62 T + 232 \) Copy content Toggle raw display
$37$ \( T^{3} - 14 T^{2} + 40 T - 16 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 50 T + 100 \) Copy content Toggle raw display
$43$ \( T^{3} - 18 T^{2} + 98 T - 148 \) Copy content Toggle raw display
$47$ \( T^{3} - 14 T^{2} + 32 T + 64 \) Copy content Toggle raw display
$53$ \( T^{3} + 16 T^{2} + 77 T + 106 \) Copy content Toggle raw display
$59$ \( T^{3} + 2 T^{2} - 29 T - 80 \) Copy content Toggle raw display
$61$ \( T^{3} + 30 T^{2} + 290 T + 892 \) Copy content Toggle raw display
$67$ \( T^{3} - 2 T^{2} - 61 T + 64 \) Copy content Toggle raw display
$71$ \( T^{3} - 8 T^{2} - 122 T + 1016 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} - 95 T - 164 \) Copy content Toggle raw display
$79$ \( T^{3} - 228T - 880 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 28 T + 8 \) Copy content Toggle raw display
$89$ \( T^{3} - 14 T^{2} + 46 T + 20 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} - 100 T + 488 \) Copy content Toggle raw display
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