Properties

Label 9.6.a.a
Level 9
Weight 6
Character orbit 9.a
Self dual yes
Analytic conductor 1.443
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.44345437832\)
Analytic rank: \(0\)
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

\(f(q)\) \(=\) \( q + 6q^{2} + 4q^{4} - 6q^{5} - 40q^{7} - 168q^{8} + O(q^{10}) \) \( q + 6q^{2} + 4q^{4} - 6q^{5} - 40q^{7} - 168q^{8} - 36q^{10} + 564q^{11} + 638q^{13} - 240q^{14} - 1136q^{16} - 882q^{17} - 556q^{19} - 24q^{20} + 3384q^{22} + 840q^{23} - 3089q^{25} + 3828q^{26} - 160q^{28} - 4638q^{29} + 4400q^{31} - 1440q^{32} - 5292q^{34} + 240q^{35} - 2410q^{37} - 3336q^{38} + 1008q^{40} + 6870q^{41} + 9644q^{43} + 2256q^{44} + 5040q^{46} + 18672q^{47} - 15207q^{49} - 18534q^{50} + 2552q^{52} - 33750q^{53} - 3384q^{55} + 6720q^{56} - 27828q^{58} + 18084q^{59} + 39758q^{61} + 26400q^{62} + 27712q^{64} - 3828q^{65} - 23068q^{67} - 3528q^{68} + 1440q^{70} + 4248q^{71} - 41110q^{73} - 14460q^{74} - 2224q^{76} - 22560q^{77} + 21920q^{79} + 6816q^{80} + 41220q^{82} - 82452q^{83} + 5292q^{85} + 57864q^{86} - 94752q^{88} + 94086q^{89} - 25520q^{91} + 3360q^{92} + 112032q^{94} + 3336q^{95} + 49442q^{97} - 91242q^{98} + O(q^{100}) \)

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 4.00000 −6.00000 0 −40.0000 −168.000 0 −36.0000
\(n\): 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 9.6.a.a 1
3.b odd 2 1 3.6.a.a 1
4.b odd 2 1 144.6.a.f 1
5.b even 2 1 225.6.a.a 1
5.c odd 4 2 225.6.b.b 2
7.b odd 2 1 441.6.a.i 1
8.b even 2 1 576.6.a.s 1
8.d odd 2 1 576.6.a.t 1
9.c even 3 2 81.6.c.a 2
9.d odd 6 2 81.6.c.c 2
11.b odd 2 1 1089.6.a.b 1
12.b even 2 1 48.6.a.a 1
15.d odd 2 1 75.6.a.e 1
15.e even 4 2 75.6.b.b 2
21.c even 2 1 147.6.a.a 1
21.g even 6 2 147.6.e.k 2
21.h odd 6 2 147.6.e.h 2
24.f even 2 1 192.6.a.l 1
24.h odd 2 1 192.6.a.d 1
33.d even 2 1 363.6.a.d 1
39.d odd 2 1 507.6.a.b 1
48.i odd 4 2 768.6.d.k 2
48.k even 4 2 768.6.d.h 2
51.c odd 2 1 867.6.a.a 1
57.d even 2 1 1083.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.6.a.a 1 3.b odd 2 1
9.6.a.a 1 1.a even 1 1 trivial
48.6.a.a 1 12.b even 2 1
75.6.a.e 1 15.d odd 2 1
75.6.b.b 2 15.e even 4 2
81.6.c.a 2 9.c even 3 2
81.6.c.c 2 9.d odd 6 2
144.6.a.f 1 4.b odd 2 1
147.6.a.a 1 21.c even 2 1
147.6.e.h 2 21.h odd 6 2
147.6.e.k 2 21.g even 6 2
192.6.a.d 1 24.h odd 2 1
192.6.a.l 1 24.f even 2 1
225.6.a.a 1 5.b even 2 1
225.6.b.b 2 5.c odd 4 2
363.6.a.d 1 33.d even 2 1
441.6.a.i 1 7.b odd 2 1
507.6.a.b 1 39.d odd 2 1
576.6.a.s 1 8.b even 2 1
576.6.a.t 1 8.d odd 2 1
768.6.d.h 2 48.k even 4 2
768.6.d.k 2 48.i odd 4 2
867.6.a.a 1 51.c odd 2 1
1083.6.a.c 1 57.d even 2 1
1089.6.a.b 1 11.b odd 2 1

Atkin-Lehner signs

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

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 6 T + 32 T^{2} \)
$3$ 1
$5$ \( 1 + 6 T + 3125 T^{2} \)
$7$ \( 1 + 40 T + 16807 T^{2} \)
$11$ \( 1 - 564 T + 161051 T^{2} \)
$13$ \( 1 - 638 T + 371293 T^{2} \)
$17$ \( 1 + 882 T + 1419857 T^{2} \)
$19$ \( 1 + 556 T + 2476099 T^{2} \)
$23$ \( 1 - 840 T + 6436343 T^{2} \)
$29$ \( 1 + 4638 T + 20511149 T^{2} \)
$31$ \( 1 - 4400 T + 28629151 T^{2} \)
$37$ \( 1 + 2410 T + 69343957 T^{2} \)
$41$ \( 1 - 6870 T + 115856201 T^{2} \)
$43$ \( 1 - 9644 T + 147008443 T^{2} \)
$47$ \( 1 - 18672 T + 229345007 T^{2} \)
$53$ \( 1 + 33750 T + 418195493 T^{2} \)
$59$ \( 1 - 18084 T + 714924299 T^{2} \)
$61$ \( 1 - 39758 T + 844596301 T^{2} \)
$67$ \( 1 + 23068 T + 1350125107 T^{2} \)
$71$ \( 1 - 4248 T + 1804229351 T^{2} \)
$73$ \( 1 + 41110 T + 2073071593 T^{2} \)
$79$ \( 1 - 21920 T + 3077056399 T^{2} \)
$83$ \( 1 + 82452 T + 3939040643 T^{2} \)
$89$ \( 1 - 94086 T + 5584059449 T^{2} \)
$97$ \( 1 - 49442 T + 8587340257 T^{2} \)
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