Properties

Label 1323.2.h
Level $1323$
Weight $2$
Character orbit 1323.h
Rep. character $\chi_{1323}(226,\cdot)$
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.h (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 63 \)
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 88 296
Cusp forms 288 72 216
Eisenstein series 96 16 80

Trace form

\( 72 q - 2 q^{2} + 66 q^{4} + 5 q^{5} + 12 q^{8} + O(q^{10}) \) \( 72 q - 2 q^{2} + 66 q^{4} + 5 q^{5} + 12 q^{8} + 6 q^{10} - 3 q^{11} + 3 q^{13} + 54 q^{16} + 9 q^{17} + 4 q^{20} - 4 q^{23} - 21 q^{25} + 16 q^{26} + 18 q^{29} - 6 q^{31} + 82 q^{32} + 3 q^{37} + 19 q^{38} + 6 q^{40} + 10 q^{41} + 11 q^{44} - 12 q^{46} - 54 q^{47} + 45 q^{50} + 15 q^{52} - 16 q^{53} + 6 q^{55} + 9 q^{58} - 60 q^{59} - 12 q^{62} + 12 q^{64} + 60 q^{65} - 12 q^{67} + 30 q^{68} - 6 q^{71} - 12 q^{73} - 41 q^{74} - 6 q^{76} - 36 q^{79} + 19 q^{80} + 18 q^{83} + 3 q^{85} - 25 q^{86} - 9 q^{88} + 41 q^{89} + 52 q^{92} - 6 q^{94} - 34 q^{95} + 3 q^{97} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1323.2.h.a 1323.h 63.h $2$ $10.564$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-q^{2}-q^{4}+(1-\zeta_{6})q^{5}+3q^{8}+(-1+\cdots)q^{10}+\cdots\)
1323.2.h.b 1323.h 63.h $6$ $10.564$ \(\Q(\zeta_{18})\) None \(-6\) \(0\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{18}^{3}+\zeta_{18}^{4})q^{2}+(1-\zeta_{18}^{3}+\cdots)q^{4}+\cdots\)
1323.2.h.c 1323.h 63.h $6$ $10.564$ \(\Q(\zeta_{18})\) None \(-6\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{18}^{3}+\zeta_{18}^{4})q^{2}+(1-\zeta_{18}^{3}+\cdots)q^{4}+\cdots\)
1323.2.h.d 1323.h 63.h $6$ $10.564$ 6.0.309123.1 None \(2\) \(0\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(1-\beta _{1}+\beta _{3})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
1323.2.h.e 1323.h 63.h $6$ $10.564$ 6.0.309123.1 None \(2\) \(0\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(1-\beta _{1}+\beta _{3})q^{4}+(\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
1323.2.h.f 1323.h 63.h $10$ $10.564$ 10.0.\(\cdots\).1 None \(4\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{5})q^{2}+(1+\beta _{3})q^{4}+(-\beta _{6}+\cdots)q^{5}+\cdots\)
1323.2.h.g 1323.h 63.h $12$ $10.564$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{5})q^{4}+\beta _{2}q^{5}+\cdots\)
1323.2.h.h 1323.h 63.h $24$ $10.564$ None \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$

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

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