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Properties

Label 180.2.i
Level $180$
Weight $2$
Character orbit 180.i
Rep. character $\chi_{180}(61,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $8$
Newform subspaces $2$
Sturm bound $72$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(72\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(180, [\chi])\).

Total New Old
Modular forms 84 8 76
Cusp forms 60 8 52
Eisenstein series 24 0 24

Trace form

\( 8 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 8 q^{9} + O(q^{10}) \) \( 8 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 8 q^{9} - 2 q^{13} - 2 q^{15} - 12 q^{17} + 16 q^{19} - 20 q^{21} + 6 q^{23} - 4 q^{25} + 2 q^{27} + 4 q^{31} + 12 q^{33} - 8 q^{35} + 4 q^{37} - 8 q^{39} - 12 q^{41} - 2 q^{43} - 8 q^{45} - 24 q^{47} - 12 q^{49} + 12 q^{51} - 24 q^{53} - 26 q^{57} - 20 q^{61} - 32 q^{63} + 10 q^{65} - 20 q^{67} + 24 q^{69} + 72 q^{71} + 40 q^{73} + 2 q^{75} - 6 q^{77} + 4 q^{79} + 20 q^{81} + 30 q^{83} + 6 q^{85} + 42 q^{87} + 8 q^{91} + 46 q^{93} + 4 q^{95} - 8 q^{97} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(180, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
180.2.i.a 180.i 9.c $2$ $1.437$ \(\Q(\sqrt{-3}) \) None 180.2.i.a \(0\) \(3\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots\)
180.2.i.b 180.i 9.c $6$ $1.437$ 6.0.954288.1 None 180.2.i.b \(0\) \(-1\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{3}+(1+\beta _{2})q^{5}+(\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(180, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(180, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)