Properties

Label 7.6.a.a
Level 7
Weight 6
Character orbit 7.a
Self dual yes
Analytic conductor 1.123
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) \(=\) \( 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.12268673869\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 10q^{2} - 14q^{3} + 68q^{4} - 56q^{5} + 140q^{6} - 49q^{7} - 360q^{8} - 47q^{9} + O(q^{10}) \) \( q - 10q^{2} - 14q^{3} + 68q^{4} - 56q^{5} + 140q^{6} - 49q^{7} - 360q^{8} - 47q^{9} + 560q^{10} + 232q^{11} - 952q^{12} - 140q^{13} + 490q^{14} + 784q^{15} + 1424q^{16} - 1722q^{17} + 470q^{18} - 98q^{19} - 3808q^{20} + 686q^{21} - 2320q^{22} + 1824q^{23} + 5040q^{24} + 11q^{25} + 1400q^{26} + 4060q^{27} - 3332q^{28} + 3418q^{29} - 7840q^{30} - 7644q^{31} - 2720q^{32} - 3248q^{33} + 17220q^{34} + 2744q^{35} - 3196q^{36} - 10398q^{37} + 980q^{38} + 1960q^{39} + 20160q^{40} - 17962q^{41} - 6860q^{42} + 10880q^{43} + 15776q^{44} + 2632q^{45} - 18240q^{46} + 9324q^{47} - 19936q^{48} + 2401q^{49} - 110q^{50} + 24108q^{51} - 9520q^{52} + 2262q^{53} - 40600q^{54} - 12992q^{55} + 17640q^{56} + 1372q^{57} - 34180q^{58} - 2730q^{59} + 53312q^{60} + 25648q^{61} + 76440q^{62} + 2303q^{63} - 18368q^{64} + 7840q^{65} + 32480q^{66} - 48404q^{67} - 117096q^{68} - 25536q^{69} - 27440q^{70} - 58560q^{71} + 16920q^{72} + 68082q^{73} + 103980q^{74} - 154q^{75} - 6664q^{76} - 11368q^{77} - 19600q^{78} + 31784q^{79} - 79744q^{80} - 45419q^{81} + 179620q^{82} - 20538q^{83} + 46648q^{84} + 96432q^{85} - 108800q^{86} - 47852q^{87} - 83520q^{88} - 50582q^{89} - 26320q^{90} + 6860q^{91} + 124032q^{92} + 107016q^{93} - 93240q^{94} + 5488q^{95} + 38080q^{96} - 58506q^{97} - 24010q^{98} - 10904q^{99} + O(q^{100}) \)

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
−10.0000 −14.0000 68.0000 −56.0000 140.000 −49.0000 −360.000 −47.0000 560.000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

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 7.6.a.a 1
3.b odd 2 1 63.6.a.e 1
4.b odd 2 1 112.6.a.g 1
5.b even 2 1 175.6.a.b 1
5.c odd 4 2 175.6.b.a 2
7.b odd 2 1 49.6.a.a 1
7.c even 3 2 49.6.c.c 2
7.d odd 6 2 49.6.c.b 2
8.b even 2 1 448.6.a.m 1
8.d odd 2 1 448.6.a.c 1
11.b odd 2 1 847.6.a.b 1
12.b even 2 1 1008.6.a.y 1
21.c even 2 1 441.6.a.k 1
28.d even 2 1 784.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.6.a.a 1 1.a even 1 1 trivial
49.6.a.a 1 7.b odd 2 1
49.6.c.b 2 7.d odd 6 2
49.6.c.c 2 7.c even 3 2
63.6.a.e 1 3.b odd 2 1
112.6.a.g 1 4.b odd 2 1
175.6.a.b 1 5.b even 2 1
175.6.b.a 2 5.c odd 4 2
441.6.a.k 1 21.c even 2 1
448.6.a.c 1 8.d odd 2 1
448.6.a.m 1 8.b even 2 1
784.6.a.c 1 28.d even 2 1
847.6.a.b 1 11.b odd 2 1
1008.6.a.y 1 12.b even 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 10 T + 32 T^{2} \)
$3$ \( 1 + 14 T + 243 T^{2} \)
$5$ \( 1 + 56 T + 3125 T^{2} \)
$7$ \( 1 + 49 T \)
$11$ \( 1 - 232 T + 161051 T^{2} \)
$13$ \( 1 + 140 T + 371293 T^{2} \)
$17$ \( 1 + 1722 T + 1419857 T^{2} \)
$19$ \( 1 + 98 T + 2476099 T^{2} \)
$23$ \( 1 - 1824 T + 6436343 T^{2} \)
$29$ \( 1 - 3418 T + 20511149 T^{2} \)
$31$ \( 1 + 7644 T + 28629151 T^{2} \)
$37$ \( 1 + 10398 T + 69343957 T^{2} \)
$41$ \( 1 + 17962 T + 115856201 T^{2} \)
$43$ \( 1 - 10880 T + 147008443 T^{2} \)
$47$ \( 1 - 9324 T + 229345007 T^{2} \)
$53$ \( 1 - 2262 T + 418195493 T^{2} \)
$59$ \( 1 + 2730 T + 714924299 T^{2} \)
$61$ \( 1 - 25648 T + 844596301 T^{2} \)
$67$ \( 1 + 48404 T + 1350125107 T^{2} \)
$71$ \( 1 + 58560 T + 1804229351 T^{2} \)
$73$ \( 1 - 68082 T + 2073071593 T^{2} \)
$79$ \( 1 - 31784 T + 3077056399 T^{2} \)
$83$ \( 1 + 20538 T + 3939040643 T^{2} \)
$89$ \( 1 + 50582 T + 5584059449 T^{2} \)
$97$ \( 1 + 58506 T + 8587340257 T^{2} \)
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