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Properties

Label 360.2.q
Level $360$
Weight $2$
Character orbit 360.q
Rep. character $\chi_{360}(121,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $5$
Sturm bound $144$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 360.q (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(144\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(360, [\chi])\).

Total New Old
Modular forms 160 24 136
Cusp forms 128 24 104
Eisenstein series 32 0 32

Trace form

\( 24 q - 2 q^{3} - 2 q^{5} + 4 q^{9} + O(q^{10}) \) \( 24 q - 2 q^{3} - 2 q^{5} + 4 q^{9} + 2 q^{11} + 28 q^{17} - 12 q^{19} + 20 q^{21} - 12 q^{25} + 4 q^{27} - 6 q^{29} + 12 q^{31} - 14 q^{33} - 24 q^{35} - 36 q^{39} - 4 q^{41} + 18 q^{43} - 12 q^{45} + 12 q^{47} - 6 q^{49} + 14 q^{51} + 8 q^{53} + 18 q^{57} + 2 q^{59} + 6 q^{61} + 8 q^{63} - 8 q^{65} + 6 q^{67} - 58 q^{69} - 8 q^{71} + 36 q^{73} - 2 q^{75} - 40 q^{77} + 4 q^{81} + 32 q^{83} + 8 q^{87} - 36 q^{89} - 24 q^{91} + 8 q^{93} - 18 q^{97} - 80 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(360, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
360.2.q.a 360.q 9.c $2$ $2.875$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+3\zeta_{6}q^{9}+\cdots\)
360.2.q.b 360.q 9.c $4$ $2.875$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{3}+\zeta_{12}q^{5}+(-2+2\zeta_{12}+\cdots)q^{7}+\cdots\)
360.2.q.c 360.q 9.c $4$ $2.875$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(2\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{2}-\beta _{3})q^{3}+\beta _{2}q^{5}+(-1+\cdots)q^{7}+\cdots\)
360.2.q.d 360.q 9.c $6$ $2.875$ 6.0.954288.1 None \(0\) \(-1\) \(-3\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{3}+(-1-\beta _{3})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
360.2.q.e 360.q 9.c $8$ $2.875$ 8.0.856615824.2 None \(0\) \(0\) \(-4\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{4}q^{3}-\beta _{1}q^{5}+(-\beta _{2}+\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(360, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(360, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 2}\)