Properties

Label 1003.2.a.b
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

Related objects

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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

\(f(q)\) \(=\) \( q + 2q^{3} - 2q^{4} - 2q^{5} + 2q^{7} + q^{9} + O(q^{10}) \) \( q + 2q^{3} - 2q^{4} - 2q^{5} + 2q^{7} + q^{9} - 5q^{11} - 4q^{12} + 2q^{13} - 4q^{15} + 4q^{16} - q^{17} - 5q^{19} + 4q^{20} + 4q^{21} - 5q^{23} - q^{25} - 4q^{27} - 4q^{28} - 6q^{29} + 8q^{31} - 10q^{33} - 4q^{35} - 2q^{36} + 2q^{37} + 4q^{39} - 6q^{41} - 2q^{43} + 10q^{44} - 2q^{45} + 6q^{47} + 8q^{48} - 3q^{49} - 2q^{51} - 4q^{52} - 11q^{53} + 10q^{55} - 10q^{57} - q^{59} + 8q^{60} - q^{61} + 2q^{63} - 8q^{64} - 4q^{65} - 8q^{67} + 2q^{68} - 10q^{69} - 6q^{71} + 13q^{73} - 2q^{75} + 10q^{76} - 10q^{77} + 4q^{79} - 8q^{80} - 11q^{81} + 6q^{83} - 8q^{84} + 2q^{85} - 12q^{87} + 10q^{89} + 4q^{91} + 10q^{92} + 16q^{93} + 10q^{95} + 3q^{97} - 5q^{99} + O(q^{100}) \)

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
0 2.00000 −2.00000 −2.00000 0 2.00000 0 1.00000 0
\(n\): 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.b 1
3.b odd 2 1 9027.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.b 1 1.a even 1 1 trivial
9027.2.a.d 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

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

Hecke Characteristic Polynomials

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