Properties

Label 490.2.a.f
Level $490$
Weight $2$
Character orbit 490.a
Self dual yes
Analytic conductor $3.913$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.91266969904\)
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

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

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 −2.00000 1.00000 −1.00000 −2.00000 0 1.00000 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)

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 490.2.a.f 1
3.b odd 2 1 4410.2.a.s 1
4.b odd 2 1 3920.2.a.bg 1
5.b even 2 1 2450.2.a.n 1
5.c odd 4 2 2450.2.c.b 2
7.b odd 2 1 490.2.a.i yes 1
7.c even 3 2 490.2.e.e 2
7.d odd 6 2 490.2.e.b 2
21.c even 2 1 4410.2.a.i 1
28.d even 2 1 3920.2.a.j 1
35.c odd 2 1 2450.2.a.d 1
35.f even 4 2 2450.2.c.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.f 1 1.a even 1 1 trivial
490.2.a.i yes 1 7.b odd 2 1
490.2.e.b 2 7.d odd 6 2
490.2.e.e 2 7.c even 3 2
2450.2.a.d 1 35.c odd 2 1
2450.2.a.n 1 5.b even 2 1
2450.2.c.b 2 5.c odd 4 2
2450.2.c.n 2 35.f even 4 2
3920.2.a.j 1 28.d even 2 1
3920.2.a.bg 1 4.b odd 2 1
4410.2.a.i 1 21.c even 2 1
4410.2.a.s 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(490))\):

\( T_{3} + 2 \)
\( T_{11} + 4 \)
\( T_{13} + 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 2 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 4 + T \)
$13$ \( 2 + T \)
$17$ \( 8 + T \)
$19$ \( -6 + T \)
$23$ \( 4 + T \)
$29$ \( 6 + T \)
$31$ \( 4 + T \)
$37$ \( 10 + T \)
$41$ \( 4 + T \)
$43$ \( -4 + T \)
$47$ \( 4 + T \)
$53$ \( -10 + T \)
$59$ \( -14 + T \)
$61$ \( -10 + T \)
$67$ \( 4 + T \)
$71$ \( -12 + T \)
$73$ \( 4 + T \)
$79$ \( -4 + T \)
$83$ \( -2 + T \)
$89$ \( 8 + T \)
$97$ \( T \)
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