Properties

Label 8550.2.a.cv
Level $8550$
Weight $2$
Character orbit 8550.a
Self dual yes
Analytic conductor $68.272$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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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: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
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.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + \beta_1 q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + \beta_1 q^{7} + q^{8} + \beta_{3} q^{11} - \beta_1 q^{13} + \beta_1 q^{14} + q^{16} + (\beta_{2} - 4) q^{17} - q^{19} + \beta_{3} q^{22} + ( - \beta_{2} - 2) q^{23} - \beta_1 q^{26} + \beta_1 q^{28} + ( - 3 \beta_{3} - 2 \beta_1) q^{29} + ( - 2 \beta_{2} - 4) q^{31} + q^{32} + (\beta_{2} - 4) q^{34} + ( - 2 \beta_{3} - \beta_1) q^{37} - q^{38} + ( - 2 \beta_{3} - \beta_1) q^{41} + ( - \beta_{3} - 2 \beta_1) q^{43} + \beta_{3} q^{44} + ( - \beta_{2} - 2) q^{46} + (\beta_{2} - 6) q^{47} + (2 \beta_{2} + 1) q^{49} - \beta_1 q^{52} - 6 q^{53} + \beta_1 q^{56} + ( - 3 \beta_{3} - 2 \beta_1) q^{58} + (3 \beta_{3} - \beta_1) q^{59} - 2 q^{61} + ( - 2 \beta_{2} - 4) q^{62} + q^{64} + (2 \beta_{3} + 4 \beta_1) q^{67} + (\beta_{2} - 4) q^{68} + 4 \beta_{3} q^{71} + ( - 4 \beta_{3} - 2 \beta_1) q^{73} + ( - 2 \beta_{3} - \beta_1) q^{74} - q^{76} - 4 q^{77} - 4 q^{79} + ( - 2 \beta_{3} - \beta_1) q^{82} + ( - 3 \beta_{2} - 6) q^{83} + ( - \beta_{3} - 2 \beta_1) q^{86} + \beta_{3} q^{88} + \beta_1 q^{89} + ( - 2 \beta_{2} - 8) q^{91} + ( - \beta_{2} - 2) q^{92} + (\beta_{2} - 6) q^{94} + (5 \beta_{3} + 2 \beta_1) q^{97} + (2 \beta_{2} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} + 4 q^{16} - 16 q^{17} - 4 q^{19} - 8 q^{23} - 16 q^{31} + 4 q^{32} - 16 q^{34} - 4 q^{38} - 8 q^{46} - 24 q^{47} + 4 q^{49} - 24 q^{53} - 8 q^{61} - 16 q^{62} + 4 q^{64} - 16 q^{68} - 4 q^{76} - 16 q^{77} - 16 q^{79} - 24 q^{83} - 32 q^{91} - 8 q^{92} - 24 q^{94} + 4 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{24} + \zeta_{24}^{-1}\):

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} - 8\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} + 4\beta_1 ) / 2 \) 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
−1.93185
−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.cv 4
3.b odd 2 1 8550.2.a.cu 4
5.b even 2 1 8550.2.a.cu 4
5.c odd 4 2 1710.2.d.g 8
15.d odd 2 1 inner 8550.2.a.cv 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 3.b odd 2 1
8550.2.a.cu 4 5.b even 2 1
8550.2.a.cv 4 1.a even 1 1 trivial
8550.2.a.cv 4 15.d odd 2 1 inner

Hecke kernels

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

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

Hecke characteristic polynomials

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