Properties

Label 9.10.a
Level 9
Weight 10
Character orbit a
Rep. character \(\chi_{9}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 3
Sturm bound 10
Trace bound 2

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Defining parameters

Level: \( N \) = \( 9 = 3^{2} \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 9.a (trivial)
Character field: \(\Q\)
Newforms: \( 3 \)
Sturm bound: \(10\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(9))\).

Total New Old
Modular forms 11 4 7
Cusp forms 7 3 4
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.
\(+\)\(1\)
\(-\)\(2\)

Trace form

\(3q \) \(\mathstrut +\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 84q^{4} \) \(\mathstrut +\mathstrut 2844q^{5} \) \(\mathstrut -\mathstrut 7932q^{7} \) \(\mathstrut +\mathstrut 22392q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 18q^{2} \) \(\mathstrut +\mathstrut 84q^{4} \) \(\mathstrut +\mathstrut 2844q^{5} \) \(\mathstrut -\mathstrut 7932q^{7} \) \(\mathstrut +\mathstrut 22392q^{8} \) \(\mathstrut +\mathstrut 19764q^{10} \) \(\mathstrut -\mathstrut 22608q^{11} \) \(\mathstrut -\mathstrut 1938q^{13} \) \(\mathstrut -\mathstrut 325584q^{14} \) \(\mathstrut +\mathstrut 82704q^{16} \) \(\mathstrut +\mathstrut 171180q^{17} \) \(\mathstrut -\mathstrut 239592q^{19} \) \(\mathstrut +\mathstrut 742536q^{20} \) \(\mathstrut +\mathstrut 327240q^{22} \) \(\mathstrut +\mathstrut 2732976q^{23} \) \(\mathstrut -\mathstrut 1791879q^{25} \) \(\mathstrut -\mathstrut 6016644q^{26} \) \(\mathstrut +\mathstrut 1212576q^{28} \) \(\mathstrut +\mathstrut 1950012q^{29} \) \(\mathstrut +\mathstrut 8392740q^{31} \) \(\mathstrut -\mathstrut 10875168q^{32} \) \(\mathstrut -\mathstrut 8921988q^{34} \) \(\mathstrut +\mathstrut 8079120q^{35} \) \(\mathstrut -\mathstrut 13509822q^{37} \) \(\mathstrut +\mathstrut 18758664q^{38} \) \(\mathstrut +\mathstrut 32144688q^{40} \) \(\mathstrut +\mathstrut 11576988q^{41} \) \(\mathstrut -\mathstrut 44083272q^{43} \) \(\mathstrut +\mathstrut 2815632q^{44} \) \(\mathstrut +\mathstrut 3158352q^{46} \) \(\mathstrut -\mathstrut 51281136q^{47} \) \(\mathstrut +\mathstrut 140586363q^{49} \) \(\mathstrut -\mathstrut 15134994q^{50} \) \(\mathstrut -\mathstrut 185266920q^{52} \) \(\mathstrut -\mathstrut 94588884q^{53} \) \(\mathstrut -\mathstrut 34271424q^{55} \) \(\mathstrut +\mathstrut 71144640q^{56} \) \(\mathstrut +\mathstrut 323595972q^{58} \) \(\mathstrut +\mathstrut 106386624q^{59} \) \(\mathstrut -\mathstrut 265029078q^{61} \) \(\mathstrut +\mathstrut 261185184q^{62} \) \(\mathstrut -\mathstrut 212374464q^{64} \) \(\mathstrut -\mathstrut 151342488q^{65} \) \(\mathstrut +\mathstrut 486602160q^{67} \) \(\mathstrut -\mathstrut 137315304q^{68} \) \(\mathstrut -\mathstrut 463307040q^{70} \) \(\mathstrut +\mathstrut 80513136q^{71} \) \(\mathstrut +\mathstrut 25718154q^{73} \) \(\mathstrut -\mathstrut 437254164q^{74} \) \(\mathstrut +\mathstrut 937970448q^{76} \) \(\mathstrut -\mathstrut 186280416q^{77} \) \(\mathstrut -\mathstrut 826612284q^{79} \) \(\mathstrut -\mathstrut 263981664q^{80} \) \(\mathstrut +\mathstrut 26522316q^{82} \) \(\mathstrut +\mathstrut 1020687984q^{83} \) \(\mathstrut +\mathstrut 285268392q^{85} \) \(\mathstrut +\mathstrut 259740792q^{86} \) \(\mathstrut -\mathstrut 280716192q^{88} \) \(\mathstrut -\mathstrut 90873684q^{89} \) \(\mathstrut -\mathstrut 525354648q^{91} \) \(\mathstrut +\mathstrut 428535072q^{92} \) \(\mathstrut -\mathstrut 605638944q^{94} \) \(\mathstrut +\mathstrut 999662976q^{95} \) \(\mathstrut +\mathstrut 1628401650q^{97} \) \(\mathstrut -\mathstrut 1503597438q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(9))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3
9.10.a.a \(1\) \(4.635\) \(\Q\) None \(-18\) \(0\) \(1530\) \(9128\) \(-\) \(q-18q^{2}-188q^{4}+1530q^{5}+9128q^{7}+\cdots\)
9.10.a.b \(1\) \(4.635\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-12580\) \(+\) \(q-2^{9}q^{4}-12580q^{7}+118370q^{13}+\cdots\)
9.10.a.c \(1\) \(4.635\) \(\Q\) None \(36\) \(0\) \(1314\) \(-4480\) \(-\) \(q+6^{2}q^{2}+28^{2}q^{4}+1314q^{5}-4480q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)