Properties

Label 8550.2.a.cl
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
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.

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

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

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

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
3.12489
−0.363328
−1.76156
1.00000 0 1.00000 0 0 −4.12489 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 0.761557 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.cl 3
3.b odd 2 1 950.2.a.i 3
5.b even 2 1 8550.2.a.ck 3
5.c odd 4 2 1710.2.d.d 6
12.b even 2 1 7600.2.a.cd 3
15.d odd 2 1 950.2.a.n 3
15.e even 4 2 190.2.b.b 6
60.h even 2 1 7600.2.a.bi 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 3.b odd 2 1
950.2.a.n 3 15.d 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 60.h even 2 1
7600.2.a.cd 3 12.b even 2 1
8550.2.a.ck 3 5.b even 2 1
8550.2.a.cl 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

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

Hecke characteristic polynomials

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