Properties

Label 1323.2.f
Level $1323$
Weight $2$
Character orbit 1323.f
Rep. character $\chi_{1323}(442,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $72$
Newform subspaces $8$
Sturm bound $336$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1323.f (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 8 \)
Sturm bound: \(336\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1323, [\chi])\).

Total New Old
Modular forms 384 92 292
Cusp forms 288 72 216
Eisenstein series 96 20 76

Trace form

\( 72 q - 2 q^{2} - 30 q^{4} + 2 q^{5} + 24 q^{8} + O(q^{10}) \) \( 72 q - 2 q^{2} - 30 q^{4} + 2 q^{5} + 24 q^{8} - 6 q^{11} - 18 q^{16} - 12 q^{17} + 12 q^{19} + 22 q^{20} - 12 q^{22} - 16 q^{23} - 18 q^{25} - 8 q^{26} - 6 q^{31} - 26 q^{32} + 6 q^{34} + 12 q^{37} - 14 q^{38} + 12 q^{40} + 22 q^{41} - 12 q^{43} + 44 q^{44} + 6 q^{47} + 18 q^{50} - 18 q^{52} - 16 q^{53} + 12 q^{55} - 18 q^{58} + 12 q^{59} - 96 q^{62} - 24 q^{64} - 24 q^{65} - 12 q^{67} - 12 q^{68} + 60 q^{71} + 36 q^{73} + 22 q^{74} - 6 q^{76} - 8 q^{80} + 30 q^{83} + 18 q^{85} - 58 q^{86} - 6 q^{88} + 20 q^{89} + 4 q^{92} + 6 q^{94} + 26 q^{95} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1323, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1323.2.f.a 1323.f 9.c $2$ $10.564$ \(\Q(\sqrt{-3}) \) None 63.2.g.a \(1\) \(0\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}-\zeta_{6}q^{5}+3q^{8}+\cdots\)
1323.2.f.b 1323.f 9.c $2$ $10.564$ \(\Q(\sqrt{-3}) \) None 63.2.g.a \(1\) \(0\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+\zeta_{6}q^{5}+3q^{8}+\cdots\)
1323.2.f.c 1323.f 9.c $6$ $10.564$ 6.0.309123.1 None 63.2.f.b \(-1\) \(0\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{5})q^{2}+(-1+\beta _{2}+\beta _{4}+\beta _{5})q^{4}+\cdots\)
1323.2.f.d 1323.f 9.c $6$ $10.564$ \(\Q(\zeta_{18})\) None 63.2.f.a \(3\) \(0\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\zeta_{18}-\zeta_{18}^{3}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)
1323.2.f.e 1323.f 9.c $10$ $10.564$ 10.0.\(\cdots\).1 None 63.2.g.b \(-2\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(\beta _{6}+\cdots)q^{5}+\cdots\)
1323.2.f.f 1323.f 9.c $10$ $10.564$ 10.0.\(\cdots\).1 None 63.2.g.b \(-2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(-\beta _{6}+\cdots)q^{5}+\cdots\)
1323.2.f.g 1323.f 9.c $12$ $10.564$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 441.2.f.g \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{6}q^{2}+(-\beta _{1}+\beta _{4}-\beta _{5}+\beta _{6}-\beta _{7}+\cdots)q^{4}+\cdots\)
1323.2.f.h 1323.f 9.c $24$ $10.564$ None 441.2.f.h \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$

Decomposition of \(S_{2}^{\mathrm{old}}(1323, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1323, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 2}\)