Properties

Label 8550.2.a.cm
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: 3.3.788.1
Defining polynomial: \(x^{3} - x^{2} - 7 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - x^{2} - 7 x - 3\):

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

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
−1.87740
−0.476452
3.35386
1.00000 0 1.00000 0 0 −2.87740 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 −1.47645 1.00000 0 0
1.3 1.00000 0 1.00000 0 0 2.35386 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.cm yes 3
3.b odd 2 1 8550.2.a.cc 3
5.b even 2 1 8550.2.a.ch yes 3
15.d odd 2 1 8550.2.a.ct yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8550.2.a.cc 3 3.b odd 2 1
8550.2.a.ch yes 3 5.b even 2 1
8550.2.a.cm yes 3 1.a even 1 1 trivial
8550.2.a.ct 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{3} + 2 T_{7}^{2} - 6 T_{7} - 10 \)
\( T_{11}^{3} + 5 T_{11}^{2} - 5 T_{11} - 27 \)
\( T_{13}^{3} + 4 T_{13}^{2} - 8 T_{13} - 6 \)
\( T_{17}^{3} - 2 T_{17}^{2} - 6 T_{17} + 10 \)
\( T_{23}^{3} + T_{23}^{2} - 45 T_{23} - 81 \)
\( T_{53}^{3} - 5 T_{53}^{2} - 73 T_{53} - 117 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{3} \)
$3$ \( T^{3} \)
$5$ \( T^{3} \)
$7$ \( -10 - 6 T + 2 T^{2} + T^{3} \)
$11$ \( -27 - 5 T + 5 T^{2} + T^{3} \)
$13$ \( -6 - 8 T + 4 T^{2} + T^{3} \)
$17$ \( 10 - 6 T - 2 T^{2} + T^{3} \)
$19$ \( ( 1 + T )^{3} \)
$23$ \( -81 - 45 T + T^{2} + T^{3} \)
$29$ \( -197 - 43 T + 5 T^{2} + T^{3} \)
$31$ \( 1 - 21 T - 5 T^{2} + T^{3} \)
$37$ \( -144 - 48 T + 4 T^{2} + T^{3} \)
$41$ \( 4 + 24 T + 10 T^{2} + T^{3} \)
$43$ \( 162 - 100 T + 2 T^{2} + T^{3} \)
$47$ \( -204 - 88 T - 4 T^{2} + T^{3} \)
$53$ \( -117 - 73 T - 5 T^{2} + T^{3} \)
$59$ \( -50 + 28 T + 12 T^{2} + T^{3} \)
$61$ \( 275 - 81 T - T^{2} + T^{3} \)
$67$ \( -845 - 143 T + 9 T^{2} + T^{3} \)
$71$ \( -354 + 2 T + 16 T^{2} + T^{3} \)
$73$ \( -243 - 9 T + 15 T^{2} + T^{3} \)
$79$ \( -709 - 101 T + 9 T^{2} + T^{3} \)
$83$ \( -127 - 35 T + 3 T^{2} + T^{3} \)
$89$ \( -277 - 57 T + 9 T^{2} + T^{3} \)
$97$ \( 270 - 54 T - 6 T^{2} + T^{3} \)
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