↑

Properties

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

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
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

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

Display 100 coefficients 10 coefficients

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(9))\) 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
9.12.a.a 9.a 1.a $1$ $6.915$ \(\Q\) None \(-78\) \(0\) \(5370\) \(-27760\) $-$ $\mathrm{SU}(2)$ \(q-78q^{2}+4036q^{4}+5370q^{5}-27760q^{7}+\cdots\)
9.12.a.b 9.a 1.a $1$ $6.915$ \(\Q\) None \(24\) \(0\) \(-4830\) \(-16744\) $-$ $\mathrm{SU}(2)$ \(q+24q^{2}-1472q^{4}-4830q^{5}-16744q^{7}+\cdots\)
9.12.a.c 9.a 1.a $2$ $6.915$ \(\Q(\sqrt{70}) \) None \(0\) \(0\) \(0\) \(116200\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+472q^{4}+224\beta q^{5}+58100q^{7}+\cdots\)

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

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