Properties

Label 490.2.a.k
Level $490$
Weight $2$
Character orbit 490.a
Self dual yes
Analytic conductor $3.913$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 3.00000 1.00000 −1.00000 3.00000 0 1.00000 6.00000 −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.k 1
3.b odd 2 1 4410.2.a.r 1
4.b odd 2 1 3920.2.a.b 1
5.b even 2 1 2450.2.a.b 1
5.c odd 4 2 2450.2.c.s 2
7.b odd 2 1 490.2.a.e 1
7.c even 3 2 70.2.e.a 2
7.d odd 6 2 490.2.e.f 2
21.c even 2 1 4410.2.a.h 1
21.h odd 6 2 630.2.k.f 2
28.d even 2 1 3920.2.a.bk 1
28.g odd 6 2 560.2.q.i 2
35.c odd 2 1 2450.2.a.q 1
35.f even 4 2 2450.2.c.a 2
35.j even 6 2 350.2.e.l 2
35.l odd 12 4 350.2.j.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 7.c even 3 2
350.2.e.l 2 35.j even 6 2
350.2.j.f 4 35.l odd 12 4
490.2.a.e 1 7.b odd 2 1
490.2.a.k 1 1.a even 1 1 trivial
490.2.e.f 2 7.d odd 6 2
560.2.q.i 2 28.g odd 6 2
630.2.k.f 2 21.h odd 6 2
2450.2.a.b 1 5.b even 2 1
2450.2.a.q 1 35.c odd 2 1
2450.2.c.a 2 35.f even 4 2
2450.2.c.s 2 5.c odd 4 2
3920.2.a.b 1 4.b odd 2 1
3920.2.a.bk 1 28.d even 2 1
4410.2.a.h 1 21.c even 2 1
4410.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(490))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( -3 + T \)
$5$ \( 1 + T \)
$7$ \( T \)
$11$ \( 2 + T \)
$13$ \( T \)
$17$ \( 4 + T \)
$19$ \( 6 + T \)
$23$ \( -3 + T \)
$29$ \( -9 + T \)
$31$ \( 4 + T \)
$37$ \( 4 + T \)
$41$ \( 7 + T \)
$43$ \( 5 + T \)
$47$ \( -8 + T \)
$53$ \( 2 + T \)
$59$ \( -10 + T \)
$61$ \( -1 + T \)
$67$ \( 9 + T \)
$71$ \( -2 + T \)
$73$ \( 4 + T \)
$79$ \( -10 + T \)
$83$ \( 7 + T \)
$89$ \( -1 + T \)
$97$ \( -14 + T \)
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