Properties

Label 49.6.a.a
Level $49$
Weight $6$
Character orbit 49.a
Self dual yes
Analytic conductor $7.859$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.85880717084\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 10 q^{2} + 14 q^{3} + 68 q^{4} + 56 q^{5} - 140 q^{6} - 360 q^{8} - 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 10 q^{2} + 14 q^{3} + 68 q^{4} + 56 q^{5} - 140 q^{6} - 360 q^{8} - 47 q^{9} - 560 q^{10} + 232 q^{11} + 952 q^{12} + 140 q^{13} + 784 q^{15} + 1424 q^{16} + 1722 q^{17} + 470 q^{18} + 98 q^{19} + 3808 q^{20} - 2320 q^{22} + 1824 q^{23} - 5040 q^{24} + 11 q^{25} - 1400 q^{26} - 4060 q^{27} + 3418 q^{29} - 7840 q^{30} + 7644 q^{31} - 2720 q^{32} + 3248 q^{33} - 17220 q^{34} - 3196 q^{36} - 10398 q^{37} - 980 q^{38} + 1960 q^{39} - 20160 q^{40} + 17962 q^{41} + 10880 q^{43} + 15776 q^{44} - 2632 q^{45} - 18240 q^{46} - 9324 q^{47} + 19936 q^{48} - 110 q^{50} + 24108 q^{51} + 9520 q^{52} + 2262 q^{53} + 40600 q^{54} + 12992 q^{55} + 1372 q^{57} - 34180 q^{58} + 2730 q^{59} + 53312 q^{60} - 25648 q^{61} - 76440 q^{62} - 18368 q^{64} + 7840 q^{65} - 32480 q^{66} - 48404 q^{67} + 117096 q^{68} + 25536 q^{69} - 58560 q^{71} + 16920 q^{72} - 68082 q^{73} + 103980 q^{74} + 154 q^{75} + 6664 q^{76} - 19600 q^{78} + 31784 q^{79} + 79744 q^{80} - 45419 q^{81} - 179620 q^{82} + 20538 q^{83} + 96432 q^{85} - 108800 q^{86} + 47852 q^{87} - 83520 q^{88} + 50582 q^{89} + 26320 q^{90} + 124032 q^{92} + 107016 q^{93} + 93240 q^{94} + 5488 q^{95} - 38080 q^{96} + 58506 q^{97} - 10904 q^{99}+O(q^{100}) \) 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
0
−10.0000 14.0000 68.0000 56.0000 −140.000 0 −360.000 −47.0000 −560.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 49.6.a.a 1
3.b odd 2 1 441.6.a.k 1
4.b odd 2 1 784.6.a.c 1
7.b odd 2 1 7.6.a.a 1
7.c even 3 2 49.6.c.b 2
7.d odd 6 2 49.6.c.c 2
21.c even 2 1 63.6.a.e 1
28.d even 2 1 112.6.a.g 1
35.c odd 2 1 175.6.a.b 1
35.f even 4 2 175.6.b.a 2
56.e even 2 1 448.6.a.c 1
56.h odd 2 1 448.6.a.m 1
77.b even 2 1 847.6.a.b 1
84.h odd 2 1 1008.6.a.y 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 7.b odd 2 1
49.6.a.a 1 1.a even 1 1 trivial
49.6.c.b 2 7.c even 3 2
49.6.c.c 2 7.d odd 6 2
63.6.a.e 1 21.c even 2 1
112.6.a.g 1 28.d even 2 1
175.6.a.b 1 35.c odd 2 1
175.6.b.a 2 35.f even 4 2
441.6.a.k 1 3.b odd 2 1
448.6.a.c 1 56.e even 2 1
448.6.a.m 1 56.h odd 2 1
784.6.a.c 1 4.b odd 2 1
847.6.a.b 1 77.b even 2 1
1008.6.a.y 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2} + 10 \) Copy content Toggle raw display
\( T_{3} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 10 \) Copy content Toggle raw display
$3$ \( T - 14 \) Copy content Toggle raw display
$5$ \( T - 56 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 232 \) Copy content Toggle raw display
$13$ \( T - 140 \) Copy content Toggle raw display
$17$ \( T - 1722 \) Copy content Toggle raw display
$19$ \( T - 98 \) Copy content Toggle raw display
$23$ \( T - 1824 \) Copy content Toggle raw display
$29$ \( T - 3418 \) Copy content Toggle raw display
$31$ \( T - 7644 \) Copy content Toggle raw display
$37$ \( T + 10398 \) Copy content Toggle raw display
$41$ \( T - 17962 \) Copy content Toggle raw display
$43$ \( T - 10880 \) Copy content Toggle raw display
$47$ \( T + 9324 \) Copy content Toggle raw display
$53$ \( T - 2262 \) Copy content Toggle raw display
$59$ \( T - 2730 \) Copy content Toggle raw display
$61$ \( T + 25648 \) Copy content Toggle raw display
$67$ \( T + 48404 \) Copy content Toggle raw display
$71$ \( T + 58560 \) Copy content Toggle raw display
$73$ \( T + 68082 \) Copy content Toggle raw display
$79$ \( T - 31784 \) Copy content Toggle raw display
$83$ \( T - 20538 \) Copy content Toggle raw display
$89$ \( T - 50582 \) Copy content Toggle raw display
$97$ \( T - 58506 \) Copy content Toggle raw display
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