Properties

Label 1002.2.a.b
Level $1002$
Weight $2$
Character orbit 1002.a
Self dual yes
Analytic conductor $8.001$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1002 = 2 \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.00101028253\)
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

\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} - q^{8} + q^{9} - 3 q^{10} + 6 q^{11} - q^{12} + 2 q^{13} - 3 q^{14} - 3 q^{15} + q^{16} - q^{18} + 3 q^{20} - 3 q^{21} - 6 q^{22} + 4 q^{23} + q^{24} + 4 q^{25} - 2 q^{26} - q^{27} + 3 q^{28} - 6 q^{29} + 3 q^{30} - 9 q^{31} - q^{32} - 6 q^{33} + 9 q^{35} + q^{36} - q^{37} - 2 q^{39} - 3 q^{40} + 2 q^{41} + 3 q^{42} - 8 q^{43} + 6 q^{44} + 3 q^{45} - 4 q^{46} - q^{47} - q^{48} + 2 q^{49} - 4 q^{50} + 2 q^{52} - 3 q^{53} + q^{54} + 18 q^{55} - 3 q^{56} + 6 q^{58} + 3 q^{59} - 3 q^{60} - 12 q^{61} + 9 q^{62} + 3 q^{63} + q^{64} + 6 q^{65} + 6 q^{66} + q^{67} - 4 q^{69} - 9 q^{70} + 8 q^{71} - q^{72} + 10 q^{73} + q^{74} - 4 q^{75} + 18 q^{77} + 2 q^{78} - 12 q^{79} + 3 q^{80} + q^{81} - 2 q^{82} + 17 q^{83} - 3 q^{84} + 8 q^{86} + 6 q^{87} - 6 q^{88} - 15 q^{89} - 3 q^{90} + 6 q^{91} + 4 q^{92} + 9 q^{93} + q^{94} + q^{96} - 13 q^{97} - 2 q^{98} + 6 q^{99} + O(q^{100}) \)

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 3.00000 1.00000 3.00000 −1.00000 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(-1\)

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 1002.2.a.b 1
3.b odd 2 1 3006.2.a.c 1
4.b odd 2 1 8016.2.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.b 1 1.a even 1 1 trivial
3006.2.a.c 1 3.b odd 2 1
8016.2.a.j 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 3 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -3 + T \)
$7$ \( -3 + T \)
$11$ \( -6 + T \)
$13$ \( -2 + T \)
$17$ \( T \)
$19$ \( T \)
$23$ \( -4 + T \)
$29$ \( 6 + T \)
$31$ \( 9 + T \)
$37$ \( 1 + T \)
$41$ \( -2 + T \)
$43$ \( 8 + T \)
$47$ \( 1 + T \)
$53$ \( 3 + T \)
$59$ \( -3 + T \)
$61$ \( 12 + T \)
$67$ \( -1 + T \)
$71$ \( -8 + T \)
$73$ \( -10 + T \)
$79$ \( 12 + T \)
$83$ \( -17 + T \)
$89$ \( 15 + T \)
$97$ \( 13 + T \)
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