Properties

Label 490.6.a.a
Level 490
Weight 6
Character orbit 490.a
Self dual yes
Analytic conductor 78.588
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: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(78.5880717084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} - 24q^{3} + 16q^{4} - 25q^{5} + 96q^{6} - 64q^{8} + 333q^{9} + O(q^{10}) \) \( q - 4q^{2} - 24q^{3} + 16q^{4} - 25q^{5} + 96q^{6} - 64q^{8} + 333q^{9} + 100q^{10} + 132q^{11} - 384q^{12} + 946q^{13} + 600q^{15} + 256q^{16} + 222q^{17} - 1332q^{18} - 500q^{19} - 400q^{20} - 528q^{22} + 3564q^{23} + 1536q^{24} + 625q^{25} - 3784q^{26} - 2160q^{27} + 2190q^{29} - 2400q^{30} - 2312q^{31} - 1024q^{32} - 3168q^{33} - 888q^{34} + 5328q^{36} - 11242q^{37} + 2000q^{38} - 22704q^{39} + 1600q^{40} - 1242q^{41} + 20624q^{43} + 2112q^{44} - 8325q^{45} - 14256q^{46} - 6588q^{47} - 6144q^{48} - 2500q^{50} - 5328q^{51} + 15136q^{52} - 21066q^{53} + 8640q^{54} - 3300q^{55} + 12000q^{57} - 8760q^{58} - 7980q^{59} + 9600q^{60} - 16622q^{61} + 9248q^{62} + 4096q^{64} - 23650q^{65} + 12672q^{66} + 1808q^{67} + 3552q^{68} - 85536q^{69} - 24528q^{71} - 21312q^{72} - 20474q^{73} + 44968q^{74} - 15000q^{75} - 8000q^{76} + 90816q^{78} - 46240q^{79} - 6400q^{80} - 29079q^{81} + 4968q^{82} + 51576q^{83} - 5550q^{85} - 82496q^{86} - 52560q^{87} - 8448q^{88} + 110310q^{89} + 33300q^{90} + 57024q^{92} + 55488q^{93} + 26352q^{94} + 12500q^{95} + 24576q^{96} + 78382q^{97} + 43956q^{99} + O(q^{100}) \)

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
−4.00000 −24.0000 16.0000 −25.0000 96.0000 0 −64.0000 333.000 100.000
\(n\): 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 490.6.a.a 1
7.b odd 2 1 10.6.a.b 1
21.c even 2 1 90.6.a.d 1
28.d even 2 1 80.6.a.a 1
35.c odd 2 1 50.6.a.d 1
35.f even 4 2 50.6.b.a 2
56.e even 2 1 320.6.a.o 1
56.h odd 2 1 320.6.a.b 1
84.h odd 2 1 720.6.a.j 1
105.g even 2 1 450.6.a.l 1
105.k odd 4 2 450.6.c.h 2
140.c even 2 1 400.6.a.n 1
140.j odd 4 2 400.6.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 7.b odd 2 1
50.6.a.d 1 35.c odd 2 1
50.6.b.a 2 35.f even 4 2
80.6.a.a 1 28.d even 2 1
90.6.a.d 1 21.c even 2 1
320.6.a.b 1 56.h odd 2 1
320.6.a.o 1 56.e even 2 1
400.6.a.n 1 140.c even 2 1
400.6.c.b 2 140.j odd 4 2
450.6.a.l 1 105.g even 2 1
450.6.c.h 2 105.k odd 4 2
490.6.a.a 1 1.a even 1 1 trivial
720.6.a.j 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 24 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T \)
$3$ \( 1 + 24 T + 243 T^{2} \)
$5$ \( 1 + 25 T \)
$7$ 1
$11$ \( 1 - 132 T + 161051 T^{2} \)
$13$ \( 1 - 946 T + 371293 T^{2} \)
$17$ \( 1 - 222 T + 1419857 T^{2} \)
$19$ \( 1 + 500 T + 2476099 T^{2} \)
$23$ \( 1 - 3564 T + 6436343 T^{2} \)
$29$ \( 1 - 2190 T + 20511149 T^{2} \)
$31$ \( 1 + 2312 T + 28629151 T^{2} \)
$37$ \( 1 + 11242 T + 69343957 T^{2} \)
$41$ \( 1 + 1242 T + 115856201 T^{2} \)
$43$ \( 1 - 20624 T + 147008443 T^{2} \)
$47$ \( 1 + 6588 T + 229345007 T^{2} \)
$53$ \( 1 + 21066 T + 418195493 T^{2} \)
$59$ \( 1 + 7980 T + 714924299 T^{2} \)
$61$ \( 1 + 16622 T + 844596301 T^{2} \)
$67$ \( 1 - 1808 T + 1350125107 T^{2} \)
$71$ \( 1 + 24528 T + 1804229351 T^{2} \)
$73$ \( 1 + 20474 T + 2073071593 T^{2} \)
$79$ \( 1 + 46240 T + 3077056399 T^{2} \)
$83$ \( 1 - 51576 T + 3939040643 T^{2} \)
$89$ \( 1 - 110310 T + 5584059449 T^{2} \)
$97$ \( 1 - 78382 T + 8587340257 T^{2} \)
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