Properties

Label 490.2.a.m
Level $490$
Weight $2$
Character orbit 490.a
Self dual yes
Analytic conductor $3.913$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

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

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
−1.41421
1.41421
−1.00000 0.585786 1.00000 1.00000 −0.585786 0 −1.00000 −2.65685 −1.00000
1.2 −1.00000 3.41421 1.00000 1.00000 −3.41421 0 −1.00000 8.65685 −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.m yes 2
3.b odd 2 1 4410.2.a.bt 2
4.b odd 2 1 3920.2.a.bm 2
5.b even 2 1 2450.2.a.bn 2
5.c odd 4 2 2450.2.c.t 4
7.b odd 2 1 490.2.a.l 2
7.c even 3 2 490.2.e.i 4
7.d odd 6 2 490.2.e.j 4
21.c even 2 1 4410.2.a.by 2
28.d even 2 1 3920.2.a.ca 2
35.c odd 2 1 2450.2.a.bs 2
35.f even 4 2 2450.2.c.w 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
490.2.a.l 2 7.b odd 2 1
490.2.a.m yes 2 1.a even 1 1 trivial
490.2.e.i 4 7.c even 3 2
490.2.e.j 4 7.d odd 6 2
2450.2.a.bn 2 5.b even 2 1
2450.2.a.bs 2 35.c odd 2 1
2450.2.c.t 4 5.c odd 4 2
2450.2.c.w 4 35.f even 4 2
3920.2.a.bm 2 4.b odd 2 1
3920.2.a.ca 2 28.d even 2 1
4410.2.a.bt 2 3.b odd 2 1
4410.2.a.by 2 21.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(490))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{2} \)
$3$ \( 2 - 4 T + T^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( T^{2} \)
$11$ \( -4 - 4 T + T^{2} \)
$13$ \( -4 + 4 T + T^{2} \)
$17$ \( 14 - 8 T + T^{2} \)
$19$ \( 2 - 4 T + T^{2} \)
$23$ \( 8 + 8 T + T^{2} \)
$29$ \( -4 + 4 T + T^{2} \)
$31$ \( -8 + T^{2} \)
$37$ \( -28 + 4 T + T^{2} \)
$41$ \( -34 - 8 T + T^{2} \)
$43$ \( 28 + 12 T + T^{2} \)
$47$ \( 56 - 16 T + T^{2} \)
$53$ \( -68 + 4 T + T^{2} \)
$59$ \( 98 - 20 T + T^{2} \)
$61$ \( -124 + 4 T + T^{2} \)
$67$ \( -16 + 8 T + T^{2} \)
$71$ \( -56 - 8 T + T^{2} \)
$73$ \( 62 + 16 T + T^{2} \)
$79$ \( 8 + 8 T + T^{2} \)
$83$ \( -14 - 4 T + T^{2} \)
$89$ \( -162 + T^{2} \)
$97$ \( 126 + 24 T + T^{2} \)
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