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: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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.

Atkin-Lehner signs

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

Hecke kernels

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

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