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

\(f(q)\) \(=\) \( q - 78 q^{2} + 4036 q^{4} + 5370 q^{5} - 27760 q^{7} - 155064 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 78 q^{2} + 4036 q^{4} + 5370 q^{5} - 27760 q^{7} - 155064 q^{8} - 418860 q^{10} - 637836 q^{11} + 766214 q^{13} + 2165280 q^{14} + 3829264 q^{16} - 3084354 q^{17} - 19511404 q^{19} + 21673320 q^{20} + 49751208 q^{22} - 15312360 q^{23} - 19991225 q^{25} - 59764692 q^{26} - 112039360 q^{28} - 10751262 q^{29} - 50937400 q^{31} + 18888480 q^{32} + 240579612 q^{34} - 149071200 q^{35} + 664740830 q^{37} + 1521889512 q^{38} - 832693680 q^{40} - 898833450 q^{41} - 957947188 q^{43} - 2574306096 q^{44} + 1194364080 q^{46} + 1555741344 q^{47} - 1206709143 q^{49} + 1559315550 q^{50} + 3092439704 q^{52} - 3792417030 q^{53} - 3425179320 q^{55} + 4304576640 q^{56} + 838598436 q^{58} - 555306924 q^{59} + 4950420998 q^{61} + 3973117200 q^{62} - 9315634112 q^{64} + 4114569180 q^{65} + 5292399284 q^{67} - 12448452744 q^{68} + 11627553600 q^{70} + 14831086248 q^{71} + 13971005210 q^{73} - 51849784740 q^{74} - 78748026544 q^{76} + 17706327360 q^{77} + 3720542360 q^{79} + 20563147680 q^{80} + 70109009100 q^{82} - 8768454036 q^{83} - 16562980980 q^{85} + 74719880664 q^{86} + 98905401504 q^{88} + 25472769174 q^{89} - 21270100640 q^{91} - 61800684960 q^{92} - 121347824832 q^{94} - 104776239480 q^{95} - 39092494846 q^{97} + 94123313154 q^{98}+O(q^{100}) \) 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
−78.0000 0 4036.00 5370.00 0 −27760.0 −155064. 0 −418860.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 78 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 5370 \) Copy content Toggle raw display
$7$ \( T + 27760 \) Copy content Toggle raw display
$11$ \( T + 637836 \) Copy content Toggle raw display
$13$ \( T - 766214 \) Copy content Toggle raw display
$17$ \( T + 3084354 \) Copy content Toggle raw display
$19$ \( T + 19511404 \) Copy content Toggle raw display
$23$ \( T + 15312360 \) Copy content Toggle raw display
$29$ \( T + 10751262 \) Copy content Toggle raw display
$31$ \( T + 50937400 \) Copy content Toggle raw display
$37$ \( T - 664740830 \) Copy content Toggle raw display
$41$ \( T + 898833450 \) Copy content Toggle raw display
$43$ \( T + 957947188 \) Copy content Toggle raw display
$47$ \( T - 1555741344 \) Copy content Toggle raw display
$53$ \( T + 3792417030 \) Copy content Toggle raw display
$59$ \( T + 555306924 \) Copy content Toggle raw display
$61$ \( T - 4950420998 \) Copy content Toggle raw display
$67$ \( T - 5292399284 \) Copy content Toggle raw display
$71$ \( T - 14831086248 \) Copy content Toggle raw display
$73$ \( T - 13971005210 \) Copy content Toggle raw display
$79$ \( T - 3720542360 \) Copy content Toggle raw display
$83$ \( T + 8768454036 \) Copy content Toggle raw display
$89$ \( T - 25472769174 \) Copy content Toggle raw display
$97$ \( T + 39092494846 \) Copy content Toggle raw display
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