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.

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

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 \(T_{2}^{2} \) \(\mathstrut -\mathstrut 2520 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\).