Properties

Label 9.8.a.a
Level 9
Weight 8
Character orbit 9.a
Self dual yes
Analytic conductor 2.811
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.81146522936\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 6q^{2} - 92q^{4} - 390q^{5} - 64q^{7} + 1320q^{8} + O(q^{10}) \) \( q - 6q^{2} - 92q^{4} - 390q^{5} - 64q^{7} + 1320q^{8} + 2340q^{10} + 948q^{11} - 5098q^{13} + 384q^{14} + 3856q^{16} - 28386q^{17} - 8620q^{19} + 35880q^{20} - 5688q^{22} + 15288q^{23} + 73975q^{25} + 30588q^{26} + 5888q^{28} - 36510q^{29} - 276808q^{31} - 192096q^{32} + 170316q^{34} + 24960q^{35} + 268526q^{37} + 51720q^{38} - 514800q^{40} + 629718q^{41} + 685772q^{43} - 87216q^{44} - 91728q^{46} - 583296q^{47} - 819447q^{49} - 443850q^{50} + 469016q^{52} + 428058q^{53} - 369720q^{55} - 84480q^{56} + 219060q^{58} - 1306380q^{59} + 300662q^{61} + 1660848q^{62} + 659008q^{64} + 1988220q^{65} - 507244q^{67} + 2611512q^{68} - 149760q^{70} - 5560632q^{71} + 1369082q^{73} - 1611156q^{74} + 793040q^{76} - 60672q^{77} - 6913720q^{79} - 1503840q^{80} - 3778308q^{82} + 4376748q^{83} + 11070540q^{85} - 4114632q^{86} + 1251360q^{88} + 8528310q^{89} + 326272q^{91} - 1406496q^{92} + 3499776q^{94} + 3361800q^{95} - 8826814q^{97} + 4916682q^{98} + O(q^{100}) \)

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 0 −92.0000 −390.000 0 −64.0000 1320.00 0 2340.00
\(n\): 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 9.8.a.a 1
3.b odd 2 1 3.8.a.a 1
4.b odd 2 1 144.8.a.b 1
5.b even 2 1 225.8.a.i 1
5.c odd 4 2 225.8.b.f 2
7.b odd 2 1 441.8.a.a 1
8.b even 2 1 576.8.a.w 1
8.d odd 2 1 576.8.a.x 1
9.c even 3 2 81.8.c.c 2
9.d odd 6 2 81.8.c.a 2
12.b even 2 1 48.8.a.g 1
15.d odd 2 1 75.8.a.a 1
15.e even 4 2 75.8.b.c 2
21.c even 2 1 147.8.a.b 1
21.g even 6 2 147.8.e.a 2
21.h odd 6 2 147.8.e.b 2
24.f even 2 1 192.8.a.a 1
24.h odd 2 1 192.8.a.i 1
33.d even 2 1 363.8.a.b 1
39.d odd 2 1 507.8.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 3.b odd 2 1
9.8.a.a 1 1.a even 1 1 trivial
48.8.a.g 1 12.b even 2 1
75.8.a.a 1 15.d odd 2 1
75.8.b.c 2 15.e even 4 2
81.8.c.a 2 9.d odd 6 2
81.8.c.c 2 9.c even 3 2
144.8.a.b 1 4.b odd 2 1
147.8.a.b 1 21.c even 2 1
147.8.e.a 2 21.g even 6 2
147.8.e.b 2 21.h odd 6 2
192.8.a.a 1 24.f even 2 1
192.8.a.i 1 24.h odd 2 1
225.8.a.i 1 5.b even 2 1
225.8.b.f 2 5.c odd 4 2
363.8.a.b 1 33.d even 2 1
441.8.a.a 1 7.b odd 2 1
507.8.a.a 1 39.d odd 2 1
576.8.a.w 1 8.b even 2 1
576.8.a.x 1 8.d odd 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 6 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 6 T + 128 T^{2} \)
$3$ 1
$5$ \( 1 + 390 T + 78125 T^{2} \)
$7$ \( 1 + 64 T + 823543 T^{2} \)
$11$ \( 1 - 948 T + 19487171 T^{2} \)
$13$ \( 1 + 5098 T + 62748517 T^{2} \)
$17$ \( 1 + 28386 T + 410338673 T^{2} \)
$19$ \( 1 + 8620 T + 893871739 T^{2} \)
$23$ \( 1 - 15288 T + 3404825447 T^{2} \)
$29$ \( 1 + 36510 T + 17249876309 T^{2} \)
$31$ \( 1 + 276808 T + 27512614111 T^{2} \)
$37$ \( 1 - 268526 T + 94931877133 T^{2} \)
$41$ \( 1 - 629718 T + 194754273881 T^{2} \)
$43$ \( 1 - 685772 T + 271818611107 T^{2} \)
$47$ \( 1 + 583296 T + 506623120463 T^{2} \)
$53$ \( 1 - 428058 T + 1174711139837 T^{2} \)
$59$ \( 1 + 1306380 T + 2488651484819 T^{2} \)
$61$ \( 1 - 300662 T + 3142742836021 T^{2} \)
$67$ \( 1 + 507244 T + 6060711605323 T^{2} \)
$71$ \( 1 + 5560632 T + 9095120158391 T^{2} \)
$73$ \( 1 - 1369082 T + 11047398519097 T^{2} \)
$79$ \( 1 + 6913720 T + 19203908986159 T^{2} \)
$83$ \( 1 - 4376748 T + 27136050989627 T^{2} \)
$89$ \( 1 - 8528310 T + 44231334895529 T^{2} \)
$97$ \( 1 + 8826814 T + 80798284478113 T^{2} \)
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