Properties

Label 8550.2.a.cu
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \(x^{4} - 4 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{3} \)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{1} q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{1} q^{7} - q^{8} -\beta_{3} q^{11} -\beta_{1} q^{13} -\beta_{1} q^{14} + q^{16} + ( 4 - \beta_{2} ) q^{17} - q^{19} + \beta_{3} q^{22} + ( 2 + \beta_{2} ) q^{23} + \beta_{1} q^{26} + \beta_{1} q^{28} + ( 2 \beta_{1} + 3 \beta_{3} ) q^{29} + ( -4 - 2 \beta_{2} ) q^{31} - q^{32} + ( -4 + \beta_{2} ) q^{34} + ( -\beta_{1} - 2 \beta_{3} ) q^{37} + q^{38} + ( \beta_{1} + 2 \beta_{3} ) q^{41} + ( -2 \beta_{1} - \beta_{3} ) q^{43} -\beta_{3} q^{44} + ( -2 - \beta_{2} ) q^{46} + ( 6 - \beta_{2} ) q^{47} + ( 1 + 2 \beta_{2} ) q^{49} -\beta_{1} q^{52} + 6 q^{53} -\beta_{1} q^{56} + ( -2 \beta_{1} - 3 \beta_{3} ) q^{58} + ( \beta_{1} - 3 \beta_{3} ) q^{59} -2 q^{61} + ( 4 + 2 \beta_{2} ) q^{62} + q^{64} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{67} + ( 4 - \beta_{2} ) q^{68} -4 \beta_{3} q^{71} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{73} + ( \beta_{1} + 2 \beta_{3} ) q^{74} - q^{76} + 4 q^{77} -4 q^{79} + ( -\beta_{1} - 2 \beta_{3} ) q^{82} + ( 6 + 3 \beta_{2} ) q^{83} + ( 2 \beta_{1} + \beta_{3} ) q^{86} + \beta_{3} q^{88} -\beta_{1} q^{89} + ( -8 - 2 \beta_{2} ) q^{91} + ( 2 + \beta_{2} ) q^{92} + ( -6 + \beta_{2} ) q^{94} + ( 2 \beta_{1} + 5 \beta_{3} ) q^{97} + ( -1 - 2 \beta_{2} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + O(q^{10}) \) \( 4 q - 4 q^{2} + 4 q^{4} - 4 q^{8} + 4 q^{16} + 16 q^{17} - 4 q^{19} + 8 q^{23} - 16 q^{31} - 4 q^{32} - 16 q^{34} + 4 q^{38} - 8 q^{46} + 24 q^{47} + 4 q^{49} + 24 q^{53} - 8 q^{61} + 16 q^{62} + 4 q^{64} + 16 q^{68} - 4 q^{76} + 16 q^{77} - 16 q^{79} + 24 q^{83} - 32 q^{91} + 8 q^{92} - 24 q^{94} - 4 q^{98} + O(q^{100}) \)

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
−1.93185
−0.517638
0.517638
1.93185
−1.00000 0 1.00000 0 0 −3.86370 −1.00000 0 0
1.2 −1.00000 0 1.00000 0 0 −1.03528 −1.00000 0 0
1.3 −1.00000 0 1.00000 0 0 1.03528 −1.00000 0 0
1.4 −1.00000 0 1.00000 0 0 3.86370 −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

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8550.2.a.cu 4
3.b odd 2 1 8550.2.a.cv 4
5.b even 2 1 8550.2.a.cv 4
5.c odd 4 2 1710.2.d.g 8
15.d odd 2 1 inner 8550.2.a.cu 4
15.e even 4 2 1710.2.d.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1710.2.d.g 8 5.c odd 4 2
1710.2.d.g 8 15.e even 4 2
8550.2.a.cu 4 1.a even 1 1 trivial
8550.2.a.cu 4 15.d odd 2 1 inner
8550.2.a.cv 4 3.b odd 2 1
8550.2.a.cv 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8550))\):

\( T_{7}^{4} - 16 T_{7}^{2} + 16 \)
\( T_{11}^{4} - 16 T_{11}^{2} + 16 \)
\( T_{13}^{4} - 16 T_{13}^{2} + 16 \)
\( T_{17}^{2} - 8 T_{17} + 4 \)
\( T_{23}^{2} - 4 T_{23} - 8 \)
\( T_{53} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{4} \)
$3$ \( T^{4} \)
$5$ \( T^{4} \)
$7$ \( 16 - 16 T^{2} + T^{4} \)
$11$ \( 16 - 16 T^{2} + T^{4} \)
$13$ \( 16 - 16 T^{2} + T^{4} \)
$17$ \( ( 4 - 8 T + T^{2} )^{2} \)
$19$ \( ( 1 + T )^{4} \)
$23$ \( ( -8 - 4 T + T^{2} )^{2} \)
$29$ \( 1936 - 112 T^{2} + T^{4} \)
$31$ \( ( -32 + 8 T + T^{2} )^{2} \)
$37$ \( 144 - 48 T^{2} + T^{4} \)
$41$ \( 144 - 48 T^{2} + T^{4} \)
$43$ \( 144 - 48 T^{2} + T^{4} \)
$47$ \( ( 24 - 12 T + T^{2} )^{2} \)
$53$ \( ( -6 + T )^{4} \)
$59$ \( 7744 - 208 T^{2} + T^{4} \)
$61$ \( ( 2 + T )^{4} \)
$67$ \( 2304 - 192 T^{2} + T^{4} \)
$71$ \( 4096 - 256 T^{2} + T^{4} \)
$73$ \( 2304 - 192 T^{2} + T^{4} \)
$79$ \( ( 4 + T )^{4} \)
$83$ \( ( -72 - 12 T + T^{2} )^{2} \)
$89$ \( 16 - 16 T^{2} + T^{4} \)
$97$ \( 1936 - 304 T^{2} + T^{4} \)
show more
show less