Properties

Label 8550.2.a.cj
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: 3.3.257.1
Defining polynomial: \(x^{3} - x^{2} - 4 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 950)
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{7} - q^{8} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{11} + ( -2 \beta_{1} - \beta_{2} ) q^{13} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{14} + q^{16} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{17} - q^{19} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{22} + ( -5 + \beta_{1} - \beta_{2} ) q^{23} + ( 2 \beta_{1} + \beta_{2} ) q^{26} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{28} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} + 2 \beta_{2} ) q^{31} - q^{32} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{34} + ( -5 - 2 \beta_{1} + 2 \beta_{2} ) q^{37} + q^{38} + ( 4 - 4 \beta_{1} ) q^{41} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{43} + ( -2 \beta_{1} + 2 \beta_{2} ) q^{44} + ( 5 - \beta_{1} + \beta_{2} ) q^{46} + ( -3 - 4 \beta_{1} + 2 \beta_{2} ) q^{47} + ( 9 - 2 \beta_{1} - 3 \beta_{2} ) q^{49} + ( -2 \beta_{1} - \beta_{2} ) q^{52} + ( 3 + \beta_{1} - 2 \beta_{2} ) q^{53} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{56} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{58} + ( 1 + 3 \beta_{1} - \beta_{2} ) q^{59} + ( 8 - 2 \beta_{1} + 4 \beta_{2} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{2} ) q^{62} + q^{64} + ( -1 + 3 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{68} + 2 \beta_{1} q^{71} + ( -3 - 3 \beta_{1} + 3 \beta_{2} ) q^{73} + ( 5 + 2 \beta_{1} - 2 \beta_{2} ) q^{74} - q^{76} + ( 18 - 4 \beta_{1} - 6 \beta_{2} ) q^{77} + ( 8 + 4 \beta_{2} ) q^{79} + ( -4 + 4 \beta_{1} ) q^{82} + ( -2 - 6 \beta_{1} + 4 \beta_{2} ) q^{83} + ( -2 \beta_{1} - 2 \beta_{2} ) q^{86} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{88} + 10 q^{89} -7 \beta_{1} q^{91} + ( -5 + \beta_{1} - \beta_{2} ) q^{92} + ( 3 + 4 \beta_{1} - 2 \beta_{2} ) q^{94} + 6 \beta_{2} q^{97} + ( -9 + 2 \beta_{1} + 3 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{4} + 2q^{7} - 3q^{8} - 2q^{11} - 2q^{13} - 2q^{14} + 3q^{16} + 4q^{17} - 3q^{19} + 2q^{22} - 14q^{23} + 2q^{26} + 2q^{28} - 14q^{29} - 4q^{31} - 3q^{32} - 4q^{34} - 17q^{37} + 3q^{38} + 8q^{41} + 2q^{43} - 2q^{44} + 14q^{46} - 13q^{47} + 25q^{49} - 2q^{52} + 10q^{53} - 2q^{56} + 14q^{58} + 6q^{59} + 22q^{61} + 4q^{62} + 3q^{64} + 4q^{68} + 2q^{71} - 12q^{73} + 17q^{74} - 3q^{76} + 50q^{77} + 24q^{79} - 8q^{82} - 12q^{83} - 2q^{86} + 2q^{88} + 30q^{89} - 7q^{91} - 14q^{92} + 13q^{94} - 25q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 4 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
0.713538
2.19869
−1.91223
−1.00000 0 1.00000 0 0 −4.69527 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 2.46980 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 4.22547 −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.cj 3
3.b odd 2 1 950.2.a.m yes 3
5.b even 2 1 8550.2.a.co 3
12.b even 2 1 7600.2.a.bm 3
15.d odd 2 1 950.2.a.k 3
15.e even 4 2 950.2.b.g 6
60.h even 2 1 7600.2.a.cb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.a.k 3 15.d odd 2 1
950.2.a.m yes 3 3.b odd 2 1
950.2.b.g 6 15.e even 4 2
7600.2.a.bm 3 12.b even 2 1
7600.2.a.cb 3 60.h even 2 1
8550.2.a.cj 3 1.a even 1 1 trivial
8550.2.a.co 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{3} - 2 T_{7}^{2} - 21 T_{7} + 49 \)
\( T_{11}^{3} + 2 T_{11}^{2} - 32 T_{11} - 24 \)
\( T_{13}^{3} + 2 T_{13}^{2} - 23 T_{13} + 21 \)
\( T_{17}^{3} - 4 T_{17}^{2} - 15 T_{17} - 7 \)
\( T_{23}^{3} + 14 T_{23}^{2} + 57 T_{23} + 63 \)
\( T_{53}^{3} - 10 T_{53}^{2} + 11 T_{53} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( 49 - 21 T - 2 T^{2} + T^{3} \)
$11$ \( -24 - 32 T + 2 T^{2} + T^{3} \)
$13$ \( 21 - 23 T + 2 T^{2} + T^{3} \)
$17$ \( -7 - 15 T - 4 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( 63 + 57 T + 14 T^{2} + T^{3} \)
$29$ \( 25 + 41 T + 14 T^{2} + T^{3} \)
$31$ \( -152 - 36 T + 4 T^{2} + T^{3} \)
$37$ \( -9 + 63 T + 17 T^{2} + T^{3} \)
$41$ \( 64 - 48 T - 8 T^{2} + T^{3} \)
$43$ \( -72 - 40 T - 2 T^{2} + T^{3} \)
$47$ \( -525 - 25 T + 13 T^{2} + T^{3} \)
$53$ \( 3 + 11 T - 10 T^{2} + T^{3} \)
$59$ \( 175 - 29 T - 6 T^{2} + T^{3} \)
$61$ \( 536 + 72 T - 22 T^{2} + T^{3} \)
$67$ \( -469 - 131 T + T^{3} \)
$71$ \( 24 - 16 T - 2 T^{2} + T^{3} \)
$73$ \( -243 - 27 T + 12 T^{2} + T^{3} \)
$79$ \( 320 + 112 T - 24 T^{2} + T^{3} \)
$83$ \( -1544 - 164 T + 12 T^{2} + T^{3} \)
$89$ \( ( -10 + T )^{3} \)
$97$ \( 648 - 180 T + T^{3} \)
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