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

\(f(q)\) \(=\) \( q - 10 q^{2} - 14 q^{3} + 68 q^{4} - 56 q^{5} + 140 q^{6} - 49 q^{7} - 360 q^{8} - 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 10 q^{2} - 14 q^{3} + 68 q^{4} - 56 q^{5} + 140 q^{6} - 49 q^{7} - 360 q^{8} - 47 q^{9} + 560 q^{10} + 232 q^{11} - 952 q^{12} - 140 q^{13} + 490 q^{14} + 784 q^{15} + 1424 q^{16} - 1722 q^{17} + 470 q^{18} - 98 q^{19} - 3808 q^{20} + 686 q^{21} - 2320 q^{22} + 1824 q^{23} + 5040 q^{24} + 11 q^{25} + 1400 q^{26} + 4060 q^{27} - 3332 q^{28} + 3418 q^{29} - 7840 q^{30} - 7644 q^{31} - 2720 q^{32} - 3248 q^{33} + 17220 q^{34} + 2744 q^{35} - 3196 q^{36} - 10398 q^{37} + 980 q^{38} + 1960 q^{39} + 20160 q^{40} - 17962 q^{41} - 6860 q^{42} + 10880 q^{43} + 15776 q^{44} + 2632 q^{45} - 18240 q^{46} + 9324 q^{47} - 19936 q^{48} + 2401 q^{49} - 110 q^{50} + 24108 q^{51} - 9520 q^{52} + 2262 q^{53} - 40600 q^{54} - 12992 q^{55} + 17640 q^{56} + 1372 q^{57} - 34180 q^{58} - 2730 q^{59} + 53312 q^{60} + 25648 q^{61} + 76440 q^{62} + 2303 q^{63} - 18368 q^{64} + 7840 q^{65} + 32480 q^{66} - 48404 q^{67} - 117096 q^{68} - 25536 q^{69} - 27440 q^{70} - 58560 q^{71} + 16920 q^{72} + 68082 q^{73} + 103980 q^{74} - 154 q^{75} - 6664 q^{76} - 11368 q^{77} - 19600 q^{78} + 31784 q^{79} - 79744 q^{80} - 45419 q^{81} + 179620 q^{82} - 20538 q^{83} + 46648 q^{84} + 96432 q^{85} - 108800 q^{86} - 47852 q^{87} - 83520 q^{88} - 50582 q^{89} - 26320 q^{90} + 6860 q^{91} + 124032 q^{92} + 107016 q^{93} - 93240 q^{94} + 5488 q^{95} + 38080 q^{96} - 58506 q^{97} - 24010 q^{98} - 10904 q^{99}+O(q^{100}) \) 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 −49.0000 −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 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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 10 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(7))\). 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
$7$ \( T + 49 \) 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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