Properties

Label 1003.2.a.c
Level 1003
Weight 2
Character orbit 1003.a
Self dual yes
Analytic conductor 8.009
Analytic rank 0
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: \(0\)
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

\(f(q)\) \(=\) \( q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + 3q^{7} - 3q^{8} - 2q^{9} + O(q^{10}) \) \( q + q^{2} + q^{3} - q^{4} + q^{5} + q^{6} + 3q^{7} - 3q^{8} - 2q^{9} + q^{10} + 4q^{11} - q^{12} + 3q^{14} + q^{15} - q^{16} + q^{17} - 2q^{18} + 7q^{19} - q^{20} + 3q^{21} + 4q^{22} + 2q^{23} - 3q^{24} - 4q^{25} - 5q^{27} - 3q^{28} + 3q^{29} + q^{30} + 4q^{31} + 5q^{32} + 4q^{33} + q^{34} + 3q^{35} + 2q^{36} + 7q^{38} - 3q^{40} - 3q^{41} + 3q^{42} + 4q^{43} - 4q^{44} - 2q^{45} + 2q^{46} + 6q^{47} - q^{48} + 2q^{49} - 4q^{50} + q^{51} - 9q^{53} - 5q^{54} + 4q^{55} - 9q^{56} + 7q^{57} + 3q^{58} - q^{59} - q^{60} - 10q^{61} + 4q^{62} - 6q^{63} + 7q^{64} + 4q^{66} - 4q^{67} - q^{68} + 2q^{69} + 3q^{70} - 4q^{71} + 6q^{72} - 2q^{73} - 4q^{75} - 7q^{76} + 12q^{77} + 5q^{79} - q^{80} + q^{81} - 3q^{82} + 16q^{83} - 3q^{84} + q^{85} + 4q^{86} + 3q^{87} - 12q^{88} - 2q^{90} - 2q^{92} + 4q^{93} + 6q^{94} + 7q^{95} + 5q^{96} - 8q^{97} + 2q^{98} - 8q^{99} + O(q^{100}) \)

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 1.00000 1.00000 3.00000 −3.00000 −2.00000 1.00000
\(n\): 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.c 1
3.b odd 2 1 9027.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1003.2.a.c 1 1.a even 1 1 trivial
9027.2.a.b 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\):

\( T_{2} - 1 \)
\( T_{3} - 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + 2 T^{2} \)
$3$ \( 1 - T + 3 T^{2} \)
$5$ \( 1 - T + 5 T^{2} \)
$7$ \( 1 - 3 T + 7 T^{2} \)
$11$ \( 1 - 4 T + 11 T^{2} \)
$13$ \( 1 + 13 T^{2} \)
$17$ \( 1 - T \)
$19$ \( 1 - 7 T + 19 T^{2} \)
$23$ \( 1 - 2 T + 23 T^{2} \)
$29$ \( 1 - 3 T + 29 T^{2} \)
$31$ \( 1 - 4 T + 31 T^{2} \)
$37$ \( 1 + 37 T^{2} \)
$41$ \( 1 + 3 T + 41 T^{2} \)
$43$ \( 1 - 4 T + 43 T^{2} \)
$47$ \( 1 - 6 T + 47 T^{2} \)
$53$ \( 1 + 9 T + 53 T^{2} \)
$59$ \( 1 + T \)
$61$ \( 1 + 10 T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 4 T + 71 T^{2} \)
$73$ \( 1 + 2 T + 73 T^{2} \)
$79$ \( 1 - 5 T + 79 T^{2} \)
$83$ \( 1 - 16 T + 83 T^{2} \)
$89$ \( 1 + 89 T^{2} \)
$97$ \( 1 + 8 T + 97 T^{2} \)
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