Properties

Label 8550.2.a.bn
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( -3 + \beta ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + ( -3 + \beta ) q^{7} - q^{8} -\beta q^{11} + ( -3 + 2 \beta ) q^{13} + ( 3 - \beta ) q^{14} + q^{16} + q^{17} - q^{19} + \beta q^{22} + ( 5 + 3 \beta ) q^{23} + ( 3 - 2 \beta ) q^{26} + ( -3 + \beta ) q^{28} + ( 3 + 2 \beta ) q^{29} + ( 2 - 3 \beta ) q^{31} - q^{32} - q^{34} -6 \beta q^{37} + q^{38} -3 \beta q^{41} + ( -6 - 3 \beta ) q^{43} -\beta q^{44} + ( -5 - 3 \beta ) q^{46} + ( 4 - 6 \beta ) q^{49} + ( -3 + 2 \beta ) q^{52} + ( -3 - 6 \beta ) q^{53} + ( 3 - \beta ) q^{56} + ( -3 - 2 \beta ) q^{58} + ( 3 + 7 \beta ) q^{59} + ( 10 - 3 \beta ) q^{61} + ( -2 + 3 \beta ) q^{62} + q^{64} + ( -9 - 3 \beta ) q^{67} + q^{68} + ( 12 - \beta ) q^{71} + ( -3 + 6 \beta ) q^{73} + 6 \beta q^{74} - q^{76} + ( -2 + 3 \beta ) q^{77} + ( 2 + 6 \beta ) q^{79} + 3 \beta q^{82} + ( 6 - 6 \beta ) q^{83} + ( 6 + 3 \beta ) q^{86} + \beta q^{88} + 5 \beta q^{89} + ( 13 - 9 \beta ) q^{91} + ( 5 + 3 \beta ) q^{92} + ( 6 + 4 \beta ) q^{97} + ( -4 + 6 \beta ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 6q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 6q^{7} - 2q^{8} - 6q^{13} + 6q^{14} + 2q^{16} + 2q^{17} - 2q^{19} + 10q^{23} + 6q^{26} - 6q^{28} + 6q^{29} + 4q^{31} - 2q^{32} - 2q^{34} + 2q^{38} - 12q^{43} - 10q^{46} + 8q^{49} - 6q^{52} - 6q^{53} + 6q^{56} - 6q^{58} + 6q^{59} + 20q^{61} - 4q^{62} + 2q^{64} - 18q^{67} + 2q^{68} + 24q^{71} - 6q^{73} - 2q^{76} - 4q^{77} + 4q^{79} + 12q^{83} + 12q^{86} + 26q^{91} + 10q^{92} + 12q^{97} - 8q^{98} + O(q^{100}) \)

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
−1.41421
1.41421
−1.00000 0 1.00000 0 0 −4.41421 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 −1.58579 −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.bn 2
3.b odd 2 1 950.2.a.g 2
5.b even 2 1 8550.2.a.cb 2
5.c odd 4 2 1710.2.d.c 4
12.b even 2 1 7600.2.a.bg 2
15.d odd 2 1 950.2.a.f 2
15.e even 4 2 190.2.b.a 4
60.h even 2 1 7600.2.a.v 2
60.l odd 4 2 1520.2.d.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.b.a 4 15.e even 4 2
950.2.a.f 2 15.d odd 2 1
950.2.a.g 2 3.b odd 2 1
1520.2.d.e 4 60.l odd 4 2
1710.2.d.c 4 5.c odd 4 2
7600.2.a.v 2 60.h even 2 1
7600.2.a.bg 2 12.b even 2 1
8550.2.a.bn 2 1.a even 1 1 trivial
8550.2.a.cb 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{2} + 6 T_{7} + 7 \)
\( T_{11}^{2} - 2 \)
\( T_{13}^{2} + 6 T_{13} + 1 \)
\( T_{17} - 1 \)
\( T_{23}^{2} - 10 T_{23} + 7 \)
\( T_{53}^{2} + 6 T_{53} - 63 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 7 + 6 T + T^{2} \)
$11$ \( -2 + T^{2} \)
$13$ \( 1 + 6 T + T^{2} \)
$17$ \( ( -1 + T )^{2} \)
$19$ \( ( 1 + T )^{2} \)
$23$ \( 7 - 10 T + T^{2} \)
$29$ \( 1 - 6 T + T^{2} \)
$31$ \( -14 - 4 T + T^{2} \)
$37$ \( -72 + T^{2} \)
$41$ \( -18 + T^{2} \)
$43$ \( 18 + 12 T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( -63 + 6 T + T^{2} \)
$59$ \( -89 - 6 T + T^{2} \)
$61$ \( 82 - 20 T + T^{2} \)
$67$ \( 63 + 18 T + T^{2} \)
$71$ \( 142 - 24 T + T^{2} \)
$73$ \( -63 + 6 T + T^{2} \)
$79$ \( -68 - 4 T + T^{2} \)
$83$ \( -36 - 12 T + T^{2} \)
$89$ \( -50 + T^{2} \)
$97$ \( 4 - 12 T + T^{2} \)
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