Properties

Label 7.8.a.a
Level 7
Weight 8
Character orbit 7.a
Self dual yes
Analytic conductor 2.187
Analytic rank 1
Dimension 1
CM no
Inner twists 1

Related objects

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Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 7.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.18669517839\)
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

\(f(q)\) \(=\) \( q - 6q^{2} - 42q^{3} - 92q^{4} - 84q^{5} + 252q^{6} + 343q^{7} + 1320q^{8} - 423q^{9} + O(q^{10}) \) \( q - 6q^{2} - 42q^{3} - 92q^{4} - 84q^{5} + 252q^{6} + 343q^{7} + 1320q^{8} - 423q^{9} + 504q^{10} - 5568q^{11} + 3864q^{12} - 5152q^{13} - 2058q^{14} + 3528q^{15} + 3856q^{16} - 13986q^{17} + 2538q^{18} + 55370q^{19} + 7728q^{20} - 14406q^{21} + 33408q^{22} - 91272q^{23} - 55440q^{24} - 71069q^{25} + 30912q^{26} + 109620q^{27} - 31556q^{28} + 41610q^{29} - 21168q^{30} + 150332q^{31} - 192096q^{32} + 233856q^{33} + 83916q^{34} - 28812q^{35} + 38916q^{36} - 136366q^{37} - 332220q^{38} + 216384q^{39} - 110880q^{40} - 510258q^{41} + 86436q^{42} - 172072q^{43} + 512256q^{44} + 35532q^{45} + 547632q^{46} - 519036q^{47} - 161952q^{48} + 117649q^{49} + 426414q^{50} + 587412q^{51} + 473984q^{52} - 59202q^{53} - 657720q^{54} + 467712q^{55} + 452760q^{56} - 2325540q^{57} - 249660q^{58} + 1979250q^{59} - 324576q^{60} - 2988748q^{61} - 901992q^{62} - 145089q^{63} + 659008q^{64} + 432768q^{65} - 1403136q^{66} + 2409404q^{67} + 1286712q^{68} + 3833424q^{69} + 172872q^{70} + 1504512q^{71} - 558360q^{72} - 1821022q^{73} + 818196q^{74} + 2984898q^{75} - 5094040q^{76} - 1909824q^{77} - 1298304q^{78} - 1669240q^{79} - 323904q^{80} - 3678939q^{81} + 3061548q^{82} + 696738q^{83} + 1325352q^{84} + 1174824q^{85} + 1032432q^{86} - 1747620q^{87} - 7349760q^{88} + 5558490q^{89} - 213192q^{90} - 1767136q^{91} + 8397024q^{92} - 6313944q^{93} + 3114216q^{94} - 4651080q^{95} + 8068032q^{96} + 9876734q^{97} - 705894q^{98} + 2355264q^{99} + O(q^{100}) \)

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
−6.00000 −42.0000 −92.0000 −84.0000 252.000 343.000 1320.00 −423.000 504.000
\(n\): 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 7.8.a.a 1
3.b odd 2 1 63.8.a.b 1
4.b odd 2 1 112.8.a.c 1
5.b even 2 1 175.8.a.a 1
5.c odd 4 2 175.8.b.a 2
7.b odd 2 1 49.8.a.b 1
7.c even 3 2 49.8.c.b 2
7.d odd 6 2 49.8.c.a 2
8.b even 2 1 448.8.a.g 1
8.d odd 2 1 448.8.a.d 1
21.c even 2 1 441.8.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.8.a.a 1 1.a even 1 1 trivial
49.8.a.b 1 7.b odd 2 1
49.8.c.a 2 7.d odd 6 2
49.8.c.b 2 7.c even 3 2
63.8.a.b 1 3.b odd 2 1
112.8.a.c 1 4.b odd 2 1
175.8.a.a 1 5.b even 2 1
175.8.b.a 2 5.c odd 4 2
441.8.a.e 1 21.c even 2 1
448.8.a.d 1 8.d odd 2 1
448.8.a.g 1 8.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 \( T_{2} + 6 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 128 T^{2} \)
$3$ \( 1 + 42 T + 2187 T^{2} \)
$5$ \( 1 + 84 T + 78125 T^{2} \)
$7$ \( 1 - 343 T \)
$11$ \( 1 + 5568 T + 19487171 T^{2} \)
$13$ \( 1 + 5152 T + 62748517 T^{2} \)
$17$ \( 1 + 13986 T + 410338673 T^{2} \)
$19$ \( 1 - 55370 T + 893871739 T^{2} \)
$23$ \( 1 + 91272 T + 3404825447 T^{2} \)
$29$ \( 1 - 41610 T + 17249876309 T^{2} \)
$31$ \( 1 - 150332 T + 27512614111 T^{2} \)
$37$ \( 1 + 136366 T + 94931877133 T^{2} \)
$41$ \( 1 + 510258 T + 194754273881 T^{2} \)
$43$ \( 1 + 172072 T + 271818611107 T^{2} \)
$47$ \( 1 + 519036 T + 506623120463 T^{2} \)
$53$ \( 1 + 59202 T + 1174711139837 T^{2} \)
$59$ \( 1 - 1979250 T + 2488651484819 T^{2} \)
$61$ \( 1 + 2988748 T + 3142742836021 T^{2} \)
$67$ \( 1 - 2409404 T + 6060711605323 T^{2} \)
$71$ \( 1 - 1504512 T + 9095120158391 T^{2} \)
$73$ \( 1 + 1821022 T + 11047398519097 T^{2} \)
$79$ \( 1 + 1669240 T + 19203908986159 T^{2} \)
$83$ \( 1 - 696738 T + 27136050989627 T^{2} \)
$89$ \( 1 - 5558490 T + 44231334895529 T^{2} \)
$97$ \( 1 - 9876734 T + 80798284478113 T^{2} \)
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