Properties

Label 550.2.a.a
Level $550$
Weight $2$
Character orbit 550.a
Self dual yes
Analytic conductor $4.392$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 550 = 2 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.39177211117\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - 4 q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - 2 q^{3} + q^{4} + 2 q^{6} - 4 q^{7} - q^{8} + q^{9} - q^{11} - 2 q^{12} + 5 q^{13} + 4 q^{14} + q^{16} - q^{18} - 7 q^{19} + 8 q^{21} + q^{22} + 3 q^{23} + 2 q^{24} - 5 q^{26} + 4 q^{27} - 4 q^{28} + 3 q^{29} + 5 q^{31} - q^{32} + 2 q^{33} + q^{36} - 4 q^{37} + 7 q^{38} - 10 q^{39} + 12 q^{41} - 8 q^{42} + 5 q^{43} - q^{44} - 3 q^{46} - 2 q^{48} + 9 q^{49} + 5 q^{52} + 6 q^{53} - 4 q^{54} + 4 q^{56} + 14 q^{57} - 3 q^{58} + 12 q^{59} - 10 q^{61} - 5 q^{62} - 4 q^{63} + q^{64} - 2 q^{66} + 14 q^{67} - 6 q^{69} + 3 q^{71} - q^{72} + 8 q^{73} + 4 q^{74} - 7 q^{76} + 4 q^{77} + 10 q^{78} - 4 q^{79} - 11 q^{81} - 12 q^{82} - 15 q^{83} + 8 q^{84} - 5 q^{86} - 6 q^{87} + q^{88} + 3 q^{89} - 20 q^{91} + 3 q^{92} - 10 q^{93} + 2 q^{96} - 13 q^{97} - 9 q^{98} - q^{99}+O(q^{100}) \) 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
0
−1.00000 −2.00000 1.00000 0 2.00000 −4.00000 −1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(11\) \(1\)

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 550.2.a.a 1
3.b odd 2 1 4950.2.a.y 1
4.b odd 2 1 4400.2.a.bc 1
5.b even 2 1 550.2.a.m yes 1
5.c odd 4 2 550.2.b.d 2
11.b odd 2 1 6050.2.a.bb 1
15.d odd 2 1 4950.2.a.u 1
15.e even 4 2 4950.2.c.ba 2
20.d odd 2 1 4400.2.a.d 1
20.e even 4 2 4400.2.b.e 2
55.d odd 2 1 6050.2.a.n 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
550.2.a.a 1 1.a even 1 1 trivial
550.2.a.m yes 1 5.b even 2 1
550.2.b.d 2 5.c odd 4 2
4400.2.a.d 1 20.d odd 2 1
4400.2.a.bc 1 4.b odd 2 1
4400.2.b.e 2 20.e even 4 2
4950.2.a.u 1 15.d odd 2 1
4950.2.a.y 1 3.b odd 2 1
4950.2.c.ba 2 15.e even 4 2
6050.2.a.n 1 55.d odd 2 1
6050.2.a.bb 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(550))\):

\( T_{3} + 2 \) Copy content Toggle raw display
\( T_{7} + 4 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 2 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T + 4 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 5 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T + 7 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T - 3 \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T + 4 \) Copy content Toggle raw display
$41$ \( T - 12 \) Copy content Toggle raw display
$43$ \( T - 5 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T + 10 \) Copy content Toggle raw display
$67$ \( T - 14 \) Copy content Toggle raw display
$71$ \( T - 3 \) Copy content Toggle raw display
$73$ \( T - 8 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T + 15 \) Copy content Toggle raw display
$89$ \( T - 3 \) Copy content Toggle raw display
$97$ \( T + 13 \) Copy content Toggle raw display
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