Properties

Label 9.12.a.c
Level 9
Weight 12
Character orbit 9.a
Self dual yes
Analytic conductor 6.915
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{70}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 6\sqrt{70}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + 472 q^{4} + 224 \beta q^{5} + 58100 q^{7} -1576 \beta q^{8} +O(q^{10})\) \( q + \beta q^{2} + 472 q^{4} + 224 \beta q^{5} + 58100 q^{7} -1576 \beta q^{8} + 564480 q^{10} -3200 \beta q^{11} + 762650 q^{13} + 58100 \beta q^{14} -4938176 q^{16} -178752 \beta q^{17} -10301704 q^{19} + 105728 \beta q^{20} -8064000 q^{22} + 207104 \beta q^{23} + 77615395 q^{25} + 762650 \beta q^{26} + 27423200 q^{28} -2808800 \beta q^{29} + 106159508 q^{31} -1710528 \beta q^{32} -450455040 q^{34} + 13014400 \beta q^{35} -9574450 q^{37} -10301704 \beta q^{38} -889620480 q^{40} -2161600 \beta q^{41} + 1590697400 q^{43} -1510400 \beta q^{44} + 521902080 q^{46} -28695296 \beta q^{47} + 1398283257 q^{49} + 77615395 \beta q^{50} + 359970800 q^{52} + 20943648 \beta q^{53} -1806336000 q^{55} -91565600 \beta q^{56} -7078176000 q^{58} -115091200 \beta q^{59} -3092621098 q^{61} + 106159508 \beta q^{62} + 5802853888 q^{64} + 170833600 \beta q^{65} -9113820400 q^{67} -84370944 \beta q^{68} + 32796288000 q^{70} + 66739200 \beta q^{71} + 620142950 q^{73} -9574450 \beta q^{74} -4862404288 q^{76} -185920000 \beta q^{77} + 10618486484 q^{79} -1106151424 \beta q^{80} -5447232000 q^{82} + 1195288192 \beta q^{83} -100901928960 q^{85} + 1590697400 \beta q^{86} + 12708864000 q^{88} -1223376000 \beta q^{89} + 44309965000 q^{91} + 97753088 \beta q^{92} -72312145920 q^{94} -2307581696 \beta q^{95} + 131872902350 q^{97} + 1398283257 \beta q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 944q^{4} + 116200q^{7} + O(q^{10}) \) \( 2q + 944q^{4} + 116200q^{7} + 1128960q^{10} + 1525300q^{13} - 9876352q^{16} - 20603408q^{19} - 16128000q^{22} + 155230790q^{25} + 54846400q^{28} + 212319016q^{31} - 900910080q^{34} - 19148900q^{37} - 1779240960q^{40} + 3181394800q^{43} + 1043804160q^{46} + 2796566514q^{49} + 719941600q^{52} - 3612672000q^{55} - 14156352000q^{58} - 6185242196q^{61} + 11605707776q^{64} - 18227640800q^{67} + 65592576000q^{70} + 1240285900q^{73} - 9724808576q^{76} + 21236972968q^{79} - 10894464000q^{82} - 201803857920q^{85} + 25417728000q^{88} + 88619930000q^{91} - 144624291840q^{94} + 263745804700q^{97} + O(q^{100}) \)

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
−8.36660
8.36660
−50.1996 0 472.000 −11244.7 0 58100.0 79114.6 0 564480.
1.2 50.1996 0 472.000 11244.7 0 58100.0 −79114.6 0 564480.
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 9.12.a.c 2
3.b odd 2 1 inner 9.12.a.c 2
4.b odd 2 1 144.12.a.r 2
5.b even 2 1 225.12.a.j 2
5.c odd 4 2 225.12.b.g 4
9.c even 3 2 81.12.c.g 4
9.d odd 6 2 81.12.c.g 4
12.b even 2 1 144.12.a.r 2
15.d odd 2 1 225.12.a.j 2
15.e even 4 2 225.12.b.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.12.a.c 2 1.a even 1 1 trivial
9.12.a.c 2 3.b odd 2 1 inner
81.12.c.g 4 9.c even 3 2
81.12.c.g 4 9.d odd 6 2
144.12.a.r 2 4.b odd 2 1
144.12.a.r 2 12.b even 2 1
225.12.a.j 2 5.b even 2 1
225.12.a.j 2 15.d odd 2 1
225.12.b.g 4 5.c odd 4 2
225.12.b.g 4 15.e even 4 2

Atkin-Lehner signs

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 2520 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 1576 T^{2} + 4194304 T^{4} \)
$3$ \( \)
$5$ \( 1 - 28787270 T^{2} + 2384185791015625 T^{4} \)
$7$ \( ( 1 - 58100 T + 1977326743 T^{2} )^{2} \)
$11$ \( 1 + 544818541222 T^{2} + \)\(81\!\cdots\!21\)\( T^{4} \)
$13$ \( ( 1 - 762650 T + 1792160394037 T^{2} )^{2} \)
$17$ \( 1 - 11975946694814 T^{2} + \)\(11\!\cdots\!89\)\( T^{4} \)
$19$ \( ( 1 + 10301704 T + 116490258898219 T^{2} )^{2} \)
$23$ \( 1 + 1797531507451534 T^{2} + \)\(90\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 + 4519838782611658 T^{2} + \)\(14\!\cdots\!41\)\( T^{4} \)
$31$ \( ( 1 - 106159508 T + 25408476896404831 T^{2} )^{2} \)
$37$ \( ( 1 + 9574450 T + 177917621779460413 T^{2} )^{2} \)
$41$ \( 1 + 1088883326741296882 T^{2} + \)\(30\!\cdots\!81\)\( T^{4} \)
$43$ \( ( 1 - 1590697400 T + 929293739471222707 T^{2} )^{2} \)
$47$ \( 1 + 2869299998598432286 T^{2} + \)\(61\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 + 17432708152043665114 T^{2} + \)\(85\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 + 26931896409526885318 T^{2} + \)\(90\!\cdots\!81\)\( T^{4} \)
$61$ \( ( 1 + 3092621098 T + 43513917611435838661 T^{2} )^{2} \)
$67$ \( ( 1 + 9113820400 T + \)\(12\!\cdots\!83\)\( T^{2} )^{2} \)
$71$ \( 1 + \)\(45\!\cdots\!42\)\( T^{2} + \)\(53\!\cdots\!41\)\( T^{4} \)
$73$ \( ( 1 - 620142950 T + \)\(31\!\cdots\!77\)\( T^{2} )^{2} \)
$79$ \( ( 1 - 10618486484 T + \)\(74\!\cdots\!79\)\( T^{2} )^{2} \)
$83$ \( 1 - \)\(10\!\cdots\!46\)\( T^{2} + \)\(16\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 + \)\(17\!\cdots\!78\)\( T^{2} + \)\(77\!\cdots\!21\)\( T^{4} \)
$97$ \( ( 1 - 131872902350 T + \)\(71\!\cdots\!53\)\( T^{2} )^{2} \)
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