Properties

Label 8550.2.a.cn
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: 3.3.564.1
Defining polynomial: \(x^{3} - x^{2} - 5 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -\beta_{1} + \beta_{2} ) q^{7} + q^{8} + ( -2 + \beta_{1} ) q^{11} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{13} + ( -\beta_{1} + \beta_{2} ) q^{14} + q^{16} + ( -2 + \beta_{1} + \beta_{2} ) q^{17} + q^{19} + ( -2 + \beta_{1} ) q^{22} -\beta_{2} q^{23} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{26} + ( -\beta_{1} + \beta_{2} ) q^{28} + ( -5 + \beta_{1} + \beta_{2} ) q^{29} + ( 1 - 4 \beta_{1} ) q^{31} + q^{32} + ( -2 + \beta_{1} + \beta_{2} ) q^{34} + ( 2 - 2 \beta_{1} ) q^{37} + q^{38} + ( -3 - 3 \beta_{2} ) q^{41} + ( -3 - \beta_{1} ) q^{43} + ( -2 + \beta_{1} ) q^{44} -\beta_{2} q^{46} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{47} -3 \beta_{2} q^{49} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{52} + ( 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -\beta_{1} + \beta_{2} ) q^{56} + ( -5 + \beta_{1} + \beta_{2} ) q^{58} + ( -1 - 7 \beta_{1} + 2 \beta_{2} ) q^{59} + ( -2 + 5 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 1 - 4 \beta_{1} ) q^{62} + q^{64} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{67} + ( -2 + \beta_{1} + \beta_{2} ) q^{68} + ( -8 + \beta_{1} + \beta_{2} ) q^{71} + ( -4 + 4 \beta_{1} + \beta_{2} ) q^{73} + ( 2 - 2 \beta_{1} ) q^{74} + q^{76} + ( -3 + 3 \beta_{1} - 2 \beta_{2} ) q^{77} + ( 1 + 2 \beta_{1} ) q^{79} + ( -3 - 3 \beta_{2} ) q^{82} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{83} + ( -3 - \beta_{1} ) q^{86} + ( -2 + \beta_{1} ) q^{88} + ( -8 + 4 \beta_{1} + 3 \beta_{2} ) q^{89} + ( -11 + 5 \beta_{2} ) q^{91} -\beta_{2} q^{92} + ( 1 - 2 \beta_{1} + \beta_{2} ) q^{94} + ( -6 + 5 \beta_{1} + \beta_{2} ) q^{97} -3 \beta_{2} q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} + O(q^{10}) \) \( 3 q + 3 q^{2} + 3 q^{4} - 2 q^{7} + 3 q^{8} - 5 q^{11} - 2 q^{14} + 3 q^{16} - 6 q^{17} + 3 q^{19} - 5 q^{22} + q^{23} - 2 q^{28} - 15 q^{29} - q^{31} + 3 q^{32} - 6 q^{34} + 4 q^{37} + 3 q^{38} - 6 q^{41} - 10 q^{43} - 5 q^{44} + q^{46} + 3 q^{49} + 5 q^{53} - 2 q^{56} - 15 q^{58} - 12 q^{59} + q^{61} - q^{62} + 3 q^{64} - q^{67} - 6 q^{68} - 24 q^{71} - 9 q^{73} + 4 q^{74} + 3 q^{76} - 4 q^{77} + 5 q^{79} - 6 q^{82} + 11 q^{83} - 10 q^{86} - 5 q^{88} - 23 q^{89} - 38 q^{91} + q^{92} - 14 q^{97} + 3 q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 5 x + 3\):

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

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
0.571993
2.51414
−2.08613
1.00000 0 1.00000 0 0 −4.24482 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 −0.193252 1.00000 0 0
1.3 1.00000 0 1.00000 0 0 2.43807 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.cn yes 3
3.b odd 2 1 8550.2.a.cd 3
5.b even 2 1 8550.2.a.cg yes 3
15.d odd 2 1 8550.2.a.cs yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8550.2.a.cd 3 3.b odd 2 1
8550.2.a.cg yes 3 5.b even 2 1
8550.2.a.cn yes 3 1.a even 1 1 trivial
8550.2.a.cs yes 3 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{3} + 2 T_{7}^{2} - 10 T_{7} - 2 \)
\( T_{11}^{3} + 5 T_{11}^{2} + 3 T_{11} - 3 \)
\( T_{13}^{3} - 36 T_{13} - 82 \)
\( T_{17}^{3} + 6 T_{17}^{2} - 6 T_{17} - 54 \)
\( T_{23}^{3} - T_{23}^{2} - 9 T_{23} - 3 \)
\( T_{53}^{3} - 5 T_{53}^{2} - 57 T_{53} + 183 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( -2 - 10 T + 2 T^{2} + T^{3} \)
$11$ \( -3 + 3 T + 5 T^{2} + T^{3} \)
$13$ \( -82 - 36 T + T^{3} \)
$17$ \( -54 - 6 T + 6 T^{2} + T^{3} \)
$19$ \( ( -1 + T )^{3} \)
$23$ \( -3 - 9 T - T^{2} + T^{3} \)
$29$ \( 9 + 57 T + 15 T^{2} + T^{3} \)
$31$ \( -109 - 85 T + T^{2} + T^{3} \)
$37$ \( 16 - 16 T - 4 T^{2} + T^{3} \)
$41$ \( -324 - 72 T + 6 T^{2} + T^{3} \)
$43$ \( 18 + 28 T + 10 T^{2} + T^{3} \)
$47$ \( -36 - 24 T + T^{3} \)
$53$ \( 183 - 57 T - 5 T^{2} + T^{3} \)
$59$ \( -2466 - 204 T + 12 T^{2} + T^{3} \)
$61$ \( 639 - 137 T - T^{2} + T^{3} \)
$67$ \( -141 - 47 T + T^{2} + T^{3} \)
$71$ \( 342 + 174 T + 24 T^{2} + T^{3} \)
$73$ \( -541 - 81 T + 9 T^{2} + T^{3} \)
$79$ \( 41 - 13 T - 5 T^{2} + T^{3} \)
$83$ \( 39 - 27 T - 11 T^{2} + T^{3} \)
$89$ \( -2307 - 33 T + 23 T^{2} + T^{3} \)
$97$ \( -974 - 94 T + 14 T^{2} + T^{3} \)
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