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Properties

Label 9.24.a
Level $9$
Weight $24$
Character orbit 9.a
Rep. character $\chi_{9}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $4$
Sturm bound $24$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{24}(\Gamma_0(9))\).

Total New Old
Modular forms 25 10 15
Cusp forms 21 9 12
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
\(+\)\(4\)
\(-\)\(5\)

Trace form

\( 9 q - 966 q^{2} + 30861684 q^{4} + 22603530 q^{5} + 5266601496 q^{7} - 34541441880 q^{8} + O(q^{10}) \) \( 9 q - 966 q^{2} + 30861684 q^{4} + 22603530 q^{5} + 5266601496 q^{7} - 34541441880 q^{8} + 886716676260 q^{10} - 897511154628 q^{11} - 6887770456698 q^{13} + 8701800356352 q^{14} + 32349630866832 q^{16} - 37150925746266 q^{17} + 599997541119516 q^{19} - 6044593864920 q^{20} - 1899819158647608 q^{22} - 1987650893074632 q^{23} + 17754308269926975 q^{25} - 51662088915741060 q^{26} + 118532787336260928 q^{28} - 154391379960124254 q^{29} - 26743811323318272 q^{31} - 731194931198899296 q^{32} + 2132768596489114572 q^{34} - 2836356781972825200 q^{35} + 3918238867356436926 q^{37} - 8392658241594553080 q^{38} + 7595951182698635280 q^{40} - 8475315709619111586 q^{41} + 3024919086364021572 q^{43} - 16392836424453052080 q^{44} + 10975641067376596656 q^{46} - 18935640364611754416 q^{47} + 18796677438068369361 q^{49} + 99970864565759301750 q^{50} - 212401640943159438024 q^{52} + 225927387246338543898 q^{53} - 302068073481538569480 q^{55} + 579797236436170905600 q^{56} - 210906947093979795660 q^{58} + 350638550938944305292 q^{59} + 27466186178457161190 q^{61} + 232111973881349959728 q^{62} - 1187510039637911690688 q^{64} - 303270547779421626180 q^{65} + 1695483738691178949036 q^{67} - 3065914189380453976008 q^{68} + 6979463291552275814400 q^{70} - 6222552463425754983960 q^{71} - 1964203672838180405958 q^{73} - 10470901978395122078868 q^{74} + 20050844858285319735120 q^{76} - 4866613890407151650592 q^{77} - 2168731152633150004464 q^{79} + 558358014282973180320 q^{80} - 46494258897518705001348 q^{82} + 9111037715401318540548 q^{83} + 37960374120295317730380 q^{85} + 56654880463430679932472 q^{86} - 43825966383408668252640 q^{88} + 60511924806049292190318 q^{89} - 146152946814070058820144 q^{91} + 139327552337522937721824 q^{92} - 53798791959046118347776 q^{94} + 12073320699431071705080 q^{95} + 107073868128218643648546 q^{97} + 31217017057081829372682 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{24}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.24.a.a 9.a 1.a $1$ $30.168$ \(\Q\) None \(-1128\) \(0\) \(48863730\) \(-1723688680\) $-$ $\mathrm{SU}(2)$ \(q-1128q^{2}-7116224q^{4}+48863730q^{5}+\cdots\)
9.24.a.b 9.a 1.a $2$ $30.168$ \(\Q(\sqrt{144169}) \) None \(-1080\) \(0\) \(-73069020\) \(-1359184400\) $-$ $\mathrm{SU}(2)$ \(q+(-540-\beta )q^{2}+(12663328+1080\beta )q^{4}+\cdots\)
9.24.a.c 9.a 1.a $2$ $30.168$ \(\Q(\sqrt{530401}) \) None \(1242\) \(0\) \(46808820\) \(-211963904\) $-$ $\mathrm{SU}(2)$ \(q+(621-\beta )q^{2}+(-3229358-1242\beta )q^{4}+\cdots\)
9.24.a.d 9.a 1.a $4$ $30.168$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(0\) \(8561438480\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(4777492+\beta _{3})q^{4}+(17070\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{24}^{\mathrm{old}}(\Gamma_0(9))\) into lower level spaces

\( S_{24}^{\mathrm{old}}(\Gamma_0(9)) \cong \) \(S_{24}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 3}\)\(\oplus\)\(S_{24}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)