Properties

Label 1045.2.a.b
Level $1045$
Weight $2$
Character orbit 1045.a
Self dual yes
Analytic conductor $8.344$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1045 = 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1045.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.34436701122\)
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

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + q^{5} - 3q^{8} - 3q^{9} + O(q^{10}) \) \( q + q^{2} - q^{4} + q^{5} - 3q^{8} - 3q^{9} + q^{10} - q^{11} + 2q^{13} - q^{16} - 6q^{17} - 3q^{18} + q^{19} - q^{20} - q^{22} - 8q^{23} + q^{25} + 2q^{26} - 6q^{29} + 4q^{31} + 5q^{32} - 6q^{34} + 3q^{36} - 2q^{37} + q^{38} - 3q^{40} - 10q^{41} + 4q^{43} + q^{44} - 3q^{45} - 8q^{46} - 7q^{49} + q^{50} - 2q^{52} - 2q^{53} - q^{55} - 6q^{58} - 8q^{59} + 14q^{61} + 4q^{62} + 7q^{64} + 2q^{65} + 8q^{67} + 6q^{68} - 4q^{71} + 9q^{72} + 2q^{73} - 2q^{74} - q^{76} - 16q^{79} - q^{80} + 9q^{81} - 10q^{82} - 4q^{83} - 6q^{85} + 4q^{86} + 3q^{88} + 10q^{89} - 3q^{90} + 8q^{92} + q^{95} + 10q^{97} - 7q^{98} + 3q^{99} + O(q^{100}) \)

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 0 −1.00000 1.00000 0 0 −3.00000 −3.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(11\) \(1\)
\(19\) \(-1\)

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 1045.2.a.b 1
3.b odd 2 1 9405.2.a.d 1
5.b even 2 1 5225.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.b 1 1.a even 1 1 trivial
5225.2.a.a 1 5.b even 2 1
9405.2.a.d 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( T \)
$5$ \( -1 + T \)
$7$ \( T \)
$11$ \( 1 + T \)
$13$ \( -2 + T \)
$17$ \( 6 + T \)
$19$ \( -1 + T \)
$23$ \( 8 + T \)
$29$ \( 6 + T \)
$31$ \( -4 + T \)
$37$ \( 2 + T \)
$41$ \( 10 + T \)
$43$ \( -4 + T \)
$47$ \( T \)
$53$ \( 2 + T \)
$59$ \( 8 + T \)
$61$ \( -14 + T \)
$67$ \( -8 + T \)
$71$ \( 4 + T \)
$73$ \( -2 + T \)
$79$ \( 16 + T \)
$83$ \( 4 + T \)
$89$ \( -10 + T \)
$97$ \( -10 + T \)
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