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Properties

Label 675.2.l
Level $675$
Weight $2$
Character orbit 675.l
Rep. character $\chi_{675}(76,\cdot)$
Character field $\Q(\zeta_{9})$
Dimension $324$
Newform subspaces $8$
Sturm bound $180$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.l (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 27 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 8 \)
Sturm bound: \(180\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(675, [\chi])\).

Total New Old
Modular forms 576 360 216
Cusp forms 504 324 180
Eisenstein series 72 36 36

Trace form

\( 324 q + 6 q^{2} + 6 q^{3} + 6 q^{4} - 24 q^{6} + 6 q^{7} + 12 q^{8} + O(q^{10}) \) \( 324 q + 6 q^{2} + 6 q^{3} + 6 q^{4} - 24 q^{6} + 6 q^{7} + 12 q^{8} - 21 q^{11} + 24 q^{12} + 6 q^{13} + 15 q^{14} - 24 q^{16} + 15 q^{17} - 21 q^{18} + 3 q^{19} + 12 q^{21} + 15 q^{22} - 24 q^{23} + 12 q^{24} + 18 q^{26} - 21 q^{27} + 12 q^{28} - 30 q^{29} - 27 q^{31} - 72 q^{32} + 6 q^{33} + 33 q^{34} + 18 q^{36} + 3 q^{37} + 6 q^{38} - 33 q^{39} + 3 q^{41} - 78 q^{42} + 15 q^{43} - 3 q^{44} - 9 q^{46} - 57 q^{47} - 99 q^{48} - 12 q^{49} - 108 q^{51} + 45 q^{52} - 54 q^{53} + 90 q^{54} + 69 q^{56} - 27 q^{57} + 33 q^{58} - 42 q^{59} - 36 q^{61} - 6 q^{62} - 21 q^{63} - 96 q^{64} - 15 q^{66} + 33 q^{67} - 75 q^{68} + 93 q^{69} - 63 q^{71} + 126 q^{72} + 12 q^{73} + 111 q^{74} - 36 q^{76} + 75 q^{77} - 36 q^{78} + 42 q^{79} - 108 q^{81} + 12 q^{82} - 9 q^{83} - 12 q^{84} - 111 q^{86} + 57 q^{87} - 78 q^{88} - 87 q^{89} - 9 q^{92} + 21 q^{93} + 69 q^{94} - 264 q^{96} + 51 q^{97} + 75 q^{98} - 99 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(675, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
675.2.l.a 675.l 27.e $6$ $5.390$ \(\Q(\zeta_{18})\) None \(-6\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-1-\zeta_{18}+\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)
675.2.l.b 675.l 27.e $6$ $5.390$ \(\Q(\zeta_{18})\) None \(6\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1+\zeta_{18}-\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+(-\zeta_{18}^{2}+\cdots)q^{3}+\cdots\)
675.2.l.c 675.l 27.e $12$ $5.390$ 12.0.\(\cdots\).1 None \(6\) \(6\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1+\beta _{3}-\beta _{8})q^{2}+(1+\beta _{2}-\beta _{6}+\beta _{8}+\cdots)q^{3}+\cdots\)
675.2.l.d 675.l 27.e $30$ $5.390$ None \(0\) \(-3\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$
675.2.l.e 675.l 27.e $42$ $5.390$ None \(0\) \(3\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$
675.2.l.f 675.l 27.e $66$ $5.390$ None \(-6\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{9}]$
675.2.l.g 675.l 27.e $66$ $5.390$ None \(6\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{9}]$
675.2.l.h 675.l 27.e $96$ $5.390$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$

Decomposition of \(S_{2}^{\mathrm{old}}(675, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(675, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)