Properties

Label 9.12.a.a
Level 9
Weight 12
Character orbit 9.a
Self dual yes
Analytic conductor 6.915
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.91508862504\)
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

\(f(q)\) \(=\) \( q - 78q^{2} + 4036q^{4} + 5370q^{5} - 27760q^{7} - 155064q^{8} + O(q^{10}) \) \( q - 78q^{2} + 4036q^{4} + 5370q^{5} - 27760q^{7} - 155064q^{8} - 418860q^{10} - 637836q^{11} + 766214q^{13} + 2165280q^{14} + 3829264q^{16} - 3084354q^{17} - 19511404q^{19} + 21673320q^{20} + 49751208q^{22} - 15312360q^{23} - 19991225q^{25} - 59764692q^{26} - 112039360q^{28} - 10751262q^{29} - 50937400q^{31} + 18888480q^{32} + 240579612q^{34} - 149071200q^{35} + 664740830q^{37} + 1521889512q^{38} - 832693680q^{40} - 898833450q^{41} - 957947188q^{43} - 2574306096q^{44} + 1194364080q^{46} + 1555741344q^{47} - 1206709143q^{49} + 1559315550q^{50} + 3092439704q^{52} - 3792417030q^{53} - 3425179320q^{55} + 4304576640q^{56} + 838598436q^{58} - 555306924q^{59} + 4950420998q^{61} + 3973117200q^{62} - 9315634112q^{64} + 4114569180q^{65} + 5292399284q^{67} - 12448452744q^{68} + 11627553600q^{70} + 14831086248q^{71} + 13971005210q^{73} - 51849784740q^{74} - 78748026544q^{76} + 17706327360q^{77} + 3720542360q^{79} + 20563147680q^{80} + 70109009100q^{82} - 8768454036q^{83} - 16562980980q^{85} + 74719880664q^{86} + 98905401504q^{88} + 25472769174q^{89} - 21270100640q^{91} - 61800684960q^{92} - 121347824832q^{94} - 104776239480q^{95} - 39092494846q^{97} + 94123313154q^{98} + O(q^{100}) \)

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
−78.0000 0 4036.00 5370.00 0 −27760.0 −155064. 0 −418860.
\(n\): 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 9.12.a.a 1
3.b odd 2 1 3.12.a.a 1
4.b odd 2 1 144.12.a.l 1
5.b even 2 1 225.12.a.f 1
5.c odd 4 2 225.12.b.a 2
9.c even 3 2 81.12.c.e 2
9.d odd 6 2 81.12.c.a 2
12.b even 2 1 48.12.a.f 1
15.d odd 2 1 75.12.a.a 1
15.e even 4 2 75.12.b.a 2
21.c even 2 1 147.12.a.c 1
24.f even 2 1 192.12.a.g 1
24.h odd 2 1 192.12.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 3.b odd 2 1
9.12.a.a 1 1.a even 1 1 trivial
48.12.a.f 1 12.b even 2 1
75.12.a.a 1 15.d odd 2 1
75.12.b.a 2 15.e even 4 2
81.12.c.a 2 9.d odd 6 2
81.12.c.e 2 9.c even 3 2
144.12.a.l 1 4.b odd 2 1
147.12.a.c 1 21.c even 2 1
192.12.a.g 1 24.f even 2 1
192.12.a.q 1 24.h odd 2 1
225.12.a.f 1 5.b even 2 1
225.12.b.a 2 5.c odd 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 78 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 78 T + 2048 T^{2} \)
$3$ \( \)
$5$ \( 1 - 5370 T + 48828125 T^{2} \)
$7$ \( 1 + 27760 T + 1977326743 T^{2} \)
$11$ \( 1 + 637836 T + 285311670611 T^{2} \)
$13$ \( 1 - 766214 T + 1792160394037 T^{2} \)
$17$ \( 1 + 3084354 T + 34271896307633 T^{2} \)
$19$ \( 1 + 19511404 T + 116490258898219 T^{2} \)
$23$ \( 1 + 15312360 T + 952809757913927 T^{2} \)
$29$ \( 1 + 10751262 T + 12200509765705829 T^{2} \)
$31$ \( 1 + 50937400 T + 25408476896404831 T^{2} \)
$37$ \( 1 - 664740830 T + 177917621779460413 T^{2} \)
$41$ \( 1 + 898833450 T + 550329031716248441 T^{2} \)
$43$ \( 1 + 957947188 T + 929293739471222707 T^{2} \)
$47$ \( 1 - 1555741344 T + 2472159215084012303 T^{2} \)
$53$ \( 1 + 3792417030 T + 9269035929372191597 T^{2} \)
$59$ \( 1 + 555306924 T + 30155888444737842659 T^{2} \)
$61$ \( 1 - 4950420998 T + 43513917611435838661 T^{2} \)
$67$ \( 1 - 5292399284 T + \)\(12\!\cdots\!83\)\( T^{2} \)
$71$ \( 1 - 14831086248 T + \)\(23\!\cdots\!71\)\( T^{2} \)
$73$ \( 1 - 13971005210 T + \)\(31\!\cdots\!77\)\( T^{2} \)
$79$ \( 1 - 3720542360 T + \)\(74\!\cdots\!79\)\( T^{2} \)
$83$ \( 1 + 8768454036 T + \)\(12\!\cdots\!67\)\( T^{2} \)
$89$ \( 1 - 25472769174 T + \)\(27\!\cdots\!89\)\( T^{2} \)
$97$ \( 1 + 39092494846 T + \)\(71\!\cdots\!53\)\( T^{2} \)
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