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

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

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

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