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: \( x^{2} - 2 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(490))\):

\( T_{3}^{2} - 4T_{3} + 2 \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} - 4 \) Copy content Toggle raw display
\( T_{13}^{2} + 4T_{13} - 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

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