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: \( 2\cdot 3 \)
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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Atkin-Lehner signs

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

Hecke kernels

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