Properties

Label 1003.2.a.d
Level 1003
Weight 2
Character orbit 1003.a
Self dual yes
Analytic conductor 8.009
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1003 = 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1003.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.00899532273\)
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

\(f(q)\) \(=\) \( q + 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 3q^{9} + O(q^{10}) \) \( q + 2q^{2} + 2q^{4} - 2q^{5} - 2q^{7} - 3q^{9} - 4q^{10} - 3q^{11} + 4q^{13} - 4q^{14} - 4q^{16} + q^{17} - 6q^{18} - q^{19} - 4q^{20} - 6q^{22} + q^{23} - q^{25} + 8q^{26} - 4q^{28} - 6q^{29} - 8q^{31} - 8q^{32} + 2q^{34} + 4q^{35} - 6q^{36} - 2q^{37} - 2q^{38} + 4q^{43} - 6q^{44} + 6q^{45} + 2q^{46} + 12q^{47} - 3q^{49} - 2q^{50} + 8q^{52} - 3q^{53} + 6q^{55} - 12q^{58} + q^{59} + 5q^{61} - 16q^{62} + 6q^{63} - 8q^{64} - 8q^{65} + 10q^{67} + 2q^{68} + 8q^{70} - 4q^{71} + 15q^{73} - 4q^{74} - 2q^{76} + 6q^{77} + 6q^{79} + 8q^{80} + 9q^{81} + 2q^{83} - 2q^{85} + 8q^{86} - 14q^{89} + 12q^{90} - 8q^{91} + 2q^{92} + 24q^{94} + 2q^{95} - 19q^{97} - 6q^{98} + 9q^{99} + O(q^{100}) \)

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
2.00000 0 2.00000 −2.00000 0 −2.00000 0 −3.00000 −4.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 1003.2.a.d 1
3.b odd 2 1 9027.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.d 1 1.a even 1 1 trivial
9027.2.a.a 1 3.b odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(17\) \(-1\)
\(59\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\( T_{2} - 2 \)
\( T_{3} \)

Hecke Characteristic Polynomials

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