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Properties

Label 1350.2.e
Level $1350$
Weight $2$
Character orbit 1350.e
Rep. character $\chi_{1350}(451,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $38$
Newform subspaces $14$
Sturm bound $540$
Trace bound $17$

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Defining parameters

Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 14 \)
Sturm bound: \(540\)
Trace bound: \(17\)
Distinguishing \(T_p\): \(7\), \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1350, [\chi])\).

Total New Old
Modular forms 612 38 574
Cusp forms 468 38 430
Eisenstein series 144 0 144

Trace form

\( 38 q + q^{2} - 19 q^{4} - 2 q^{7} - 2 q^{8} + O(q^{10}) \) \( 38 q + q^{2} - 19 q^{4} - 2 q^{7} - 2 q^{8} + q^{11} - 2 q^{13} - 2 q^{14} - 19 q^{16} - 18 q^{17} - 2 q^{19} + 3 q^{22} - 6 q^{23} + 20 q^{26} + 4 q^{28} - 18 q^{29} - 8 q^{31} + q^{32} + 3 q^{34} + 16 q^{37} + 11 q^{38} + 13 q^{41} + q^{43} - 2 q^{44} + 12 q^{46} + 18 q^{47} - 27 q^{49} - 2 q^{52} + 48 q^{53} - 2 q^{56} + 6 q^{58} - 43 q^{59} - 20 q^{61} - 16 q^{62} + 38 q^{64} - 17 q^{67} + 9 q^{68} + 24 q^{71} + 22 q^{73} - 8 q^{74} + q^{76} + 42 q^{77} - 8 q^{79} - 30 q^{82} - 24 q^{83} - 35 q^{86} + 3 q^{88} - 92 q^{89} + 56 q^{91} - 6 q^{92} - 6 q^{94} + 7 q^{97} - 42 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1350.2.e.a 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-1+\zeta_{6})q^{7}+\cdots\)
1350.2.e.b 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-1+\zeta_{6})q^{7}+\cdots\)
1350.2.e.c 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(2-2\zeta_{6})q^{7}+\cdots\)
1350.2.e.d 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(2-2\zeta_{6})q^{7}+\cdots\)
1350.2.e.e 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+(4-4\zeta_{6})q^{7}+\cdots\)
1350.2.e.f 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-4+4\zeta_{6})q^{7}+\cdots\)
1350.2.e.g 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-4+4\zeta_{6})q^{7}+\cdots\)
1350.2.e.h 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(-2+2\zeta_{6})q^{7}+\cdots\)
1350.2.e.i 1350.e 9.c $2$ $10.780$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+(1-\zeta_{6})q^{7}+\cdots\)
1350.2.e.j 1350.e 9.c $4$ $10.780$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(-2\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1})q^{2}-\beta _{1}q^{4}+(-2+2\beta _{1}+\cdots)q^{7}+\cdots\)
1350.2.e.k 1350.e 9.c $4$ $10.780$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(-2\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1})q^{2}-\beta _{1}q^{4}+(-2+2\beta _{1}+\cdots)q^{7}+\cdots\)
1350.2.e.l 1350.e 9.c $4$ $10.780$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(0\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{1})q^{4}+\beta _{3}q^{7}-q^{8}+\cdots\)
1350.2.e.m 1350.e 9.c $4$ $10.780$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(2\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1350.2.e.n 1350.e 9.c $4$ $10.780$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(2\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1})q^{2}-\beta _{1}q^{4}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1350, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1350, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 3}\)