Properties

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

Related objects

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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: \(1\)
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

\(f(q)\) \(=\) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + 2q^{11} + q^{12} - 6q^{13} + q^{14} - q^{15} + q^{16} - 4q^{17} - q^{18} - 4q^{19} - q^{20} - q^{21} - 2q^{22} + 8q^{23} - q^{24} - 4q^{25} + 6q^{26} + q^{27} - q^{28} - 2q^{29} + q^{30} - 5q^{31} - q^{32} + 2q^{33} + 4q^{34} + q^{35} + q^{36} + 3q^{37} + 4q^{38} - 6q^{39} + q^{40} - 6q^{41} + q^{42} - 8q^{43} + 2q^{44} - q^{45} - 8q^{46} + 3q^{47} + q^{48} - 6q^{49} + 4q^{50} - 4q^{51} - 6q^{52} - 7q^{53} - q^{54} - 2q^{55} + q^{56} - 4q^{57} + 2q^{58} + 11q^{59} - q^{60} + 4q^{61} + 5q^{62} - q^{63} + q^{64} + 6q^{65} - 2q^{66} - 7q^{67} - 4q^{68} + 8q^{69} - q^{70} - 4q^{71} - q^{72} + 6q^{73} - 3q^{74} - 4q^{75} - 4q^{76} - 2q^{77} + 6q^{78} + 4q^{79} - q^{80} + q^{81} + 6q^{82} - 15q^{83} - q^{84} + 4q^{85} + 8q^{86} - 2q^{87} - 2q^{88} + 9q^{89} + q^{90} + 6q^{91} + 8q^{92} - 5q^{93} - 3q^{94} + 4q^{95} - q^{96} - 13q^{97} + 6q^{98} + 2q^{99} + O(q^{100}) \)

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 −1.00000 −1.00000 1.00000 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 1002.2.a.c 1
3.b odd 2 1 3006.2.a.f 1
4.b odd 2 1 8016.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.c 1 1.a even 1 1 trivial
3006.2.a.f 1 3.b odd 2 1
8016.2.a.b 1 4.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1002))\).

Hecke Characteristic Polynomials

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