Properties

Label 5550.2.a.w
Level $5550$
Weight $2$
Character orbit 5550.a
Self dual yes
Analytic conductor $44.317$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5550 = 2 \cdot 3 \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5550.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.3169731218\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - q^{6} - 4q^{7} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - q^{6} - 4q^{7} + q^{8} + q^{9} + 4q^{11} - q^{12} - 2q^{13} - 4q^{14} + q^{16} + 2q^{17} + q^{18} + 4q^{21} + 4q^{22} - q^{24} - 2q^{26} - q^{27} - 4q^{28} - 6q^{29} + q^{32} - 4q^{33} + 2q^{34} + q^{36} + q^{37} + 2q^{39} - 6q^{41} + 4q^{42} - 4q^{43} + 4q^{44} - 12q^{47} - q^{48} + 9q^{49} - 2q^{51} - 2q^{52} + 6q^{53} - q^{54} - 4q^{56} - 6q^{58} + 8q^{59} + 10q^{61} - 4q^{63} + q^{64} - 4q^{66} - 4q^{67} + 2q^{68} + q^{72} + 6q^{73} + q^{74} - 16q^{77} + 2q^{78} + q^{81} - 6q^{82} + 4q^{83} + 4q^{84} - 4q^{86} + 6q^{87} + 4q^{88} - 14q^{89} + 8q^{91} - 12q^{94} - q^{96} + 10q^{97} + 9q^{98} + 4q^{99} + O(q^{100}) \)

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
0
1.00000 −1.00000 1.00000 0 −1.00000 −4.00000 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(37\) \(-1\)

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 5550.2.a.w 1
5.b even 2 1 1110.2.a.g 1
15.d odd 2 1 3330.2.a.ba 1
20.d odd 2 1 8880.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.a.g 1 5.b even 2 1
3330.2.a.ba 1 15.d odd 2 1
5550.2.a.w 1 1.a even 1 1 trivial
8880.2.a.a 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5550))\):

\( T_{7} + 4 \)
\( T_{11} - 4 \)
\( T_{13} + 2 \)
\( T_{17} - 2 \)

Hecke characteristic polynomials

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