Properties

Label 1122.2.a.e
Level $1122$
Weight $2$
Character orbit 1122.a
Self dual yes
Analytic conductor $8.959$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1122 = 2 \cdot 3 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1122.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.95921510679\)
Analytic rank: \(1\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{8} + q^{9} + O(q^{10}) \) \( q + q^{2} - q^{3} + q^{4} - 2q^{5} - q^{6} + q^{8} + q^{9} - 2q^{10} - q^{11} - q^{12} - 2q^{13} + 2q^{15} + q^{16} + q^{17} + q^{18} - 4q^{19} - 2q^{20} - q^{22} - q^{24} - q^{25} - 2q^{26} - q^{27} - 2q^{29} + 2q^{30} - 8q^{31} + q^{32} + q^{33} + q^{34} + q^{36} - 10q^{37} - 4q^{38} + 2q^{39} - 2q^{40} - 6q^{41} + 4q^{43} - q^{44} - 2q^{45} + 8q^{47} - q^{48} - 7q^{49} - q^{50} - q^{51} - 2q^{52} + 6q^{53} - q^{54} + 2q^{55} + 4q^{57} - 2q^{58} - 12q^{59} + 2q^{60} - 10q^{61} - 8q^{62} + q^{64} + 4q^{65} + q^{66} + 4q^{67} + q^{68} + 16q^{71} + q^{72} + 2q^{73} - 10q^{74} + q^{75} - 4q^{76} + 2q^{78} - 8q^{79} - 2q^{80} + q^{81} - 6q^{82} - 12q^{83} - 2q^{85} + 4q^{86} + 2q^{87} - q^{88} + 10q^{89} - 2q^{90} + 8q^{93} + 8q^{94} + 8q^{95} - q^{96} + 2q^{97} - 7q^{98} - q^{99} + O(q^{100}) \)

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 −2.00000 −1.00000 0 1.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(11\) \(1\)
\(17\) \(-1\)

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 1122.2.a.e 1
3.b odd 2 1 3366.2.a.k 1
4.b odd 2 1 8976.2.a.w 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1122.2.a.e 1 1.a even 1 1 trivial
3366.2.a.k 1 3.b odd 2 1
8976.2.a.w 1 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1122))\):

\( T_{5} + 2 \)
\( T_{7} \)
\( T_{13} + 2 \)
\( T_{19} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 1 + T \)
$5$ \( 2 + T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( 2 + T \)
$17$ \( -1 + T \)
$19$ \( 4 + T \)
$23$ \( T \)
$29$ \( 2 + T \)
$31$ \( 8 + T \)
$37$ \( 10 + T \)
$41$ \( 6 + T \)
$43$ \( -4 + T \)
$47$ \( -8 + T \)
$53$ \( -6 + T \)
$59$ \( 12 + T \)
$61$ \( 10 + T \)
$67$ \( -4 + T \)
$71$ \( -16 + T \)
$73$ \( -2 + T \)
$79$ \( 8 + T \)
$83$ \( 12 + T \)
$89$ \( -10 + T \)
$97$ \( -2 + T \)
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