Properties

Label 8550.2.a.br
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: \( 1 \)
Twist minimal: no (minimal twist has level 190)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.56155
2.56155
−1.00000 0 1.00000 0 0 −1.56155 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 2.56155 −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.br 2
3.b odd 2 1 950.2.a.h 2
5.b even 2 1 1710.2.a.w 2
12.b even 2 1 7600.2.a.y 2
15.d odd 2 1 190.2.a.d 2
15.e even 4 2 950.2.b.f 4
60.h even 2 1 1520.2.a.n 2
105.g even 2 1 9310.2.a.bc 2
120.i odd 2 1 6080.2.a.bh 2
120.m even 2 1 6080.2.a.bb 2
285.b even 2 1 3610.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.a.d 2 15.d odd 2 1
950.2.a.h 2 3.b odd 2 1
950.2.b.f 4 15.e even 4 2
1520.2.a.n 2 60.h even 2 1
1710.2.a.w 2 5.b even 2 1
3610.2.a.t 2 285.b even 2 1
6080.2.a.bb 2 120.m even 2 1
6080.2.a.bh 2 120.i odd 2 1
7600.2.a.y 2 12.b even 2 1
8550.2.a.br 2 1.a even 1 1 trivial
9310.2.a.bc 2 105.g even 2 1

Hecke kernels

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

\( T_{7}^{2} - T_{7} - 4 \)
\( T_{11} + 4 \)
\( T_{13}^{2} - T_{13} - 38 \)
\( T_{17}^{2} - 11 T_{17} + 26 \)
\( T_{23}^{2} - 3 T_{23} - 36 \)
\( T_{53}^{2} + 5 T_{53} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( -4 - T + T^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( -38 - T + T^{2} \)
$17$ \( 26 - 11 T + T^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( -36 - 3 T + T^{2} \)
$29$ \( -38 + T + T^{2} \)
$31$ \( -16 + 2 T + T^{2} \)
$37$ \( ( -6 + T )^{2} \)
$41$ \( -52 + 8 T + T^{2} \)
$43$ \( 32 - 14 T + T^{2} \)
$47$ \( -64 + 4 T + T^{2} \)
$53$ \( 2 + 5 T + T^{2} \)
$59$ \( -4 + T + T^{2} \)
$61$ \( 32 - 14 T + T^{2} \)
$67$ \( -4 - T + T^{2} \)
$71$ \( -64 + 4 T + T^{2} \)
$73$ \( -18 + 9 T + T^{2} \)
$79$ \( -16 + 2 T + T^{2} \)
$83$ \( 32 - 14 T + T^{2} \)
$89$ \( ( 2 + T )^{2} \)
$97$ \( ( 6 + T )^{2} \)
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