Properties

Label 2001.2.a.a
Level $2001$
Weight $2$
Character orbit 2001.a
Self dual yes
Analytic conductor $15.978$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2001 = 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2001.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(15.9780654445\)
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

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

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 0 3.00000 1.00000 −3.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

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 2001.2.a.a 1
3.b odd 2 1 6003.2.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.a 1 1.a even 1 1 trivial
6003.2.a.c 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):

\( T_{2} + 1 \)
\( T_{5} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( 1 + T \)
$5$ \( -3 + T \)
$7$ \( T \)
$11$ \( -3 + T \)
$13$ \( -3 + T \)
$17$ \( T \)
$19$ \( -8 + T \)
$23$ \( 1 + T \)
$29$ \( -1 + T \)
$31$ \( -7 + T \)
$37$ \( 5 + T \)
$41$ \( -3 + T \)
$43$ \( -2 + T \)
$47$ \( 6 + T \)
$53$ \( 2 + T \)
$59$ \( -1 + T \)
$61$ \( -7 + T \)
$67$ \( 9 + T \)
$71$ \( -3 + T \)
$73$ \( 12 + T \)
$79$ \( 14 + T \)
$83$ \( -14 + T \)
$89$ \( 10 + T \)
$97$ \( -14 + T \)
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