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

\(f(q)\) \(=\) \( q + q^{2} - q^{4} + q^{5} - 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - q^{4} + q^{5} - 3 q^{8} - 3 q^{9} + q^{10} - q^{11} + 2 q^{13} - q^{16} - 6 q^{17} - 3 q^{18} + q^{19} - q^{20} - q^{22} - 8 q^{23} + q^{25} + 2 q^{26} - 6 q^{29} + 4 q^{31} + 5 q^{32} - 6 q^{34} + 3 q^{36} - 2 q^{37} + q^{38} - 3 q^{40} - 10 q^{41} + 4 q^{43} + q^{44} - 3 q^{45} - 8 q^{46} - 7 q^{49} + q^{50} - 2 q^{52} - 2 q^{53} - q^{55} - 6 q^{58} - 8 q^{59} + 14 q^{61} + 4 q^{62} + 7 q^{64} + 2 q^{65} + 8 q^{67} + 6 q^{68} - 4 q^{71} + 9 q^{72} + 2 q^{73} - 2 q^{74} - q^{76} - 16 q^{79} - q^{80} + 9 q^{81} - 10 q^{82} - 4 q^{83} - 6 q^{85} + 4 q^{86} + 3 q^{88} + 10 q^{89} - 3 q^{90} + 8 q^{92} + q^{95} + 10 q^{97} - 7 q^{98} + 3 q^{99}+O(q^{100}) \) 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 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\). Copy content Toggle raw display

Hecke characteristic polynomials

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