Properties

Label 8550.2.a.bx
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \(x^{2} - x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1710)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
1.00000 0 1.00000 0 0 −5.12311 1.00000 0 0
1.2 1.00000 0 1.00000 0 0 3.12311 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.bx 2
3.b odd 2 1 8550.2.a.bp 2
5.b even 2 1 1710.2.a.v 2
15.d odd 2 1 1710.2.a.x yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.a.v 2 5.b even 2 1
1710.2.a.x yes 2 15.d odd 2 1
8550.2.a.bp 2 3.b odd 2 1
8550.2.a.bx 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{2} + 2 T_{7} - 16 \)
\( T_{11} + 2 \)
\( T_{13} + 4 \)
\( T_{17}^{2} + 2 T_{17} - 16 \)
\( T_{23} \)
\( T_{53}^{2} - 8 T_{53} - 52 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -16 + 2 T + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( ( 4 + T )^{2} \)
$17$ \( -16 + 2 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( T^{2} \)
$29$ \( ( -2 + T )^{2} \)
$31$ \( 8 - 10 T + T^{2} \)
$37$ \( T^{2} \)
$41$ \( -16 - 2 T + T^{2} \)
$43$ \( -64 + 4 T + T^{2} \)
$47$ \( -64 - 4 T + T^{2} \)
$53$ \( -52 - 8 T + T^{2} \)
$59$ \( -16 + 2 T + T^{2} \)
$61$ \( -52 - 8 T + T^{2} \)
$67$ \( -64 - 4 T + T^{2} \)
$71$ \( -64 - 4 T + T^{2} \)
$73$ \( ( 6 + T )^{2} \)
$79$ \( 8 - 10 T + T^{2} \)
$83$ \( 104 - 22 T + T^{2} \)
$89$ \( -152 + 2 T + T^{2} \)
$97$ \( ( -6 + T )^{2} \)
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