Properties

Label 8550.2.a.cc
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: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + (\beta_1 - 1) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + (\beta_1 - 1) q^{7} - q^{8} + (\beta_{2} + \beta_1 + 1) q^{11} + (\beta_{2} + \beta_1 - 2) q^{13} + ( - \beta_1 + 1) q^{14} + q^{16} + (\beta_1 - 1) q^{17} - q^{19} + ( - \beta_{2} - \beta_1 - 1) q^{22} + ( - \beta_{2} + 2 \beta_1) q^{23} + ( - \beta_{2} - \beta_1 + 2) q^{26} + (\beta_1 - 1) q^{28} + ( - 2 \beta_{2} + \beta_1 + 2) q^{29} + (2 \beta_1 + 1) q^{31} - q^{32} + ( - \beta_1 + 1) q^{34} + ( - 2 \beta_{2} - 2 \beta_1) q^{37} + q^{38} + (\beta_{2} + 3) q^{41} + ( - 3 \beta_{2} + \beta_1) q^{43} + (\beta_{2} + \beta_1 + 1) q^{44} + (\beta_{2} - 2 \beta_1) q^{46} + (3 \beta_{2} + 2 \beta_1 - 3) q^{47} + (\beta_{2} - 2) q^{49} + (\beta_{2} + \beta_1 - 2) q^{52} + (3 \beta_{2} + \beta_1 - 3) q^{53} + ( - \beta_1 + 1) q^{56} + (2 \beta_{2} - \beta_1 - 2) q^{58} + ( - \beta_{2} + \beta_1 + 4) q^{59} + (3 \beta_{2} + \beta_1 - 1) q^{61} + ( - 2 \beta_1 - 1) q^{62} + q^{64} + (4 \beta_{2} - \beta_1 - 4) q^{67} + (\beta_1 - 1) q^{68} + ( - 2 \beta_{2} - 3 \beta_1 + 7) q^{71} + (3 \beta_{2} - 6) q^{73} + (2 \beta_{2} + 2 \beta_1) q^{74} - q^{76} + ( - \beta_{2} + 3 \beta_1 + 2) q^{77} + (2 \beta_{2} + 4 \beta_1 - 5) q^{79} + ( - \beta_{2} - 3) q^{82} + (2 \beta_{2} + \beta_1) q^{83} + (3 \beta_{2} - \beta_1) q^{86} + ( - \beta_{2} - \beta_1 - 1) q^{88} + (3 \beta_{2} + 2) q^{89} + ( - \beta_{2} + 5) q^{91} + ( - \beta_{2} + 2 \beta_1) q^{92} + ( - 3 \beta_{2} - 2 \beta_1 + 3) q^{94} + ( - 3 \beta_1 + 3) q^{97} + ( - \beta_{2} + 2) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{2} + 3 q^{4} - 2 q^{7} - 3 q^{8} + 5 q^{11} - 4 q^{13} + 2 q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{19} - 5 q^{22} + q^{23} + 4 q^{26} - 2 q^{28} + 5 q^{29} + 5 q^{31} - 3 q^{32} + 2 q^{34} - 4 q^{37} + 3 q^{38} + 10 q^{41} - 2 q^{43} + 5 q^{44} - q^{46} - 4 q^{47} - 5 q^{49} - 4 q^{52} - 5 q^{53} + 2 q^{56} - 5 q^{58} + 12 q^{59} + q^{61} - 5 q^{62} + 3 q^{64} - 9 q^{67} - 2 q^{68} + 16 q^{71} - 15 q^{73} + 4 q^{74} - 3 q^{76} + 8 q^{77} - 9 q^{79} - 10 q^{82} + 3 q^{83} + 2 q^{86} - 5 q^{88} + 9 q^{89} + 14 q^{91} + q^{92} + 4 q^{94} + 6 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 7x - 3 \) : 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
−1.87740
−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.cc 3
3.b odd 2 1 8550.2.a.cm yes 3
5.b even 2 1 8550.2.a.ct yes 3
15.d odd 2 1 8550.2.a.ch yes 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8550.2.a.cc 3 1.a even 1 1 trivial
8550.2.a.ch yes 3 15.d odd 2 1
8550.2.a.cm yes 3 3.b odd 2 1
8550.2.a.ct yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

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