Properties

Label 9.12.a
Level 9
Weight 12
Character orbit a
Rep. character \(\chi_{9}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newforms 3
Sturm bound 12
Trace bound 2

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 9.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(12\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(9))\).

Total New Old
Modular forms 13 5 8
Cusp forms 9 4 5
Eisenstein series 4 1 3

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)Dim.
\(+\)\(2\)
\(-\)\(2\)

Trace form

\( 4q - 54q^{2} + 3508q^{4} + 540q^{5} + 71696q^{7} - 239544q^{8} + O(q^{10}) \) \( 4q - 54q^{2} + 3508q^{4} + 540q^{5} + 71696q^{7} - 239544q^{8} + 594180q^{10} - 1172448q^{11} + 1713776q^{13} + 1763424q^{14} - 5059952q^{16} + 3821580q^{17} - 29453392q^{19} + 28783080q^{20} + 20792520q^{22} - 33955632q^{23} + 109740340q^{25} - 73630404q^{26} - 32545792q^{28} - 139157892q^{29} + 108538448q^{31} + 215594784q^{32} - 494588052q^{34} - 68197680q^{35} + 463378616q^{37} + 1777763592q^{38} - 2203896240q^{40} - 1206953892q^{41} + 2206321904q^{43} - 1787357232q^{44} + 1790729712q^{46} - 1131607152q^{47} - 107107836q^{49} + 947334150q^{50} + 4662811640q^{52} - 2196361332q^{53} - 4455675360q^{55} + 5719109760q^{56} - 16399512684q^{58} + 4633896816q^{59} + 5721657464q^{61} + 2704881168q^{62} + 4989370432q^{64} + 6905043720q^{65} - 28417068400q^{67} - 22613987592q^{68} + 79161094080q^{70} + 5039600976q^{71} + 16675082432q^{73} - 56222904276q^{74} - 104166445360q^{76} + 26657870688q^{77} + 63074361008q^{79} + 15795280800q^{80} + 51819654492q^{82} + 20566645632q^{83} - 251722500120q^{85} + 74308863672q^{86} + 169487151264q^{88} + 50465686284q^{89} + 77023474432q^{91} - 34357788576q^{92} - 330468480576q^{94} - 156270898080q^{95} + 299666878400q^{97} + 53396148186q^{98} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
9.12.a.a \(1\) \(6.915\) \(\Q\) None \(-78\) \(0\) \(5370\) \(-27760\) \(-\) \(q-78q^{2}+4036q^{4}+5370q^{5}-27760q^{7}+\cdots\)
9.12.a.b \(1\) \(6.915\) \(\Q\) None \(24\) \(0\) \(-4830\) \(-16744\) \(-\) \(q+24q^{2}-1472q^{4}-4830q^{5}-16744q^{7}+\cdots\)
9.12.a.c \(2\) \(6.915\) \(\Q(\sqrt{70}) \) None \(0\) \(0\) \(0\) \(116200\) \(+\) \(q+\beta q^{2}+472q^{4}+224\beta q^{5}+58100q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)