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Properties

Label 9.10.a
Level $9$
Weight $10$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
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: \(\Q\)
Newform subspaces: \( 3 \)
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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(9))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3
9.10.a.a 9.a 1.a $1$ $4.635$ \(\Q\) None \(-18\) \(0\) \(1530\) \(9128\) $-$ $\mathrm{SU}(2)$ \(q-18q^{2}-188q^{4}+1530q^{5}+9128q^{7}+\cdots\)
9.10.a.b 9.a 1.a $1$ $4.635$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-12580\) $+$ $N(\mathrm{U}(1))$ \(q-2^{9}q^{4}-12580q^{7}+118370q^{13}+\cdots\)
9.10.a.c 9.a 1.a $1$ $4.635$ \(\Q\) None \(36\) \(0\) \(1314\) \(-4480\) $-$ $\mathrm{SU}(2)$ \(q+6^{2}q^{2}+28^{2}q^{4}+1314q^{5}-4480q^{7}+\cdots\)

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

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