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Properties

Label 180.2.k
Level $180$
Weight $2$
Character orbit 180.k
Rep. character $\chi_{180}(127,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $26$
Newform subspaces $5$
Sturm bound $72$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 180.k (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 5 \)
Sturm bound: \(72\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(7\), \(13\), \(17\)

Dimensions

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

Total New Old
Modular forms 88 34 54
Cusp forms 56 26 30
Eisenstein series 32 8 24

Trace form

\( 26 q + 2 q^{2} + 4 q^{5} + 8 q^{8} + O(q^{10}) \) \( 26 q + 2 q^{2} + 4 q^{5} + 8 q^{8} - 10 q^{10} - 2 q^{13} - 16 q^{16} + 14 q^{17} - 16 q^{20} - 16 q^{22} + 6 q^{25} - 20 q^{26} - 32 q^{28} - 28 q^{32} - 6 q^{37} - 16 q^{38} + 20 q^{40} + 16 q^{46} + 30 q^{50} + 52 q^{52} - 22 q^{53} + 64 q^{56} + 24 q^{58} - 40 q^{61} + 56 q^{62} - 22 q^{65} + 28 q^{68} + 16 q^{70} - 38 q^{73} - 48 q^{76} - 48 q^{77} - 20 q^{80} - 32 q^{82} - 26 q^{85} - 64 q^{86} + 32 q^{88} - 56 q^{92} - 54 q^{97} - 38 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
180.2.k.a 180.k 20.e $2$ $1.437$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(-2\) \(0\) \(2\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1-i)q^{2}+2iq^{4}+(1-2i)q^{5}+\cdots\)
180.2.k.b 180.k 20.e $2$ $1.437$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(-2\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(1+i)q^{2}+2iq^{4}+(-1+2i)q^{5}+\cdots\)
180.2.k.c 180.k 20.e $2$ $1.437$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(1+i)q^{2}+2iq^{4}+(2-i)q^{5}+(-2+\cdots)q^{8}+\cdots\)
180.2.k.d 180.k 20.e $8$ $1.437$ 8.0.157351936.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{2}+\beta _{4})q^{2}+(\beta _{3}+\beta _{5})q^{4}+(2\beta _{4}+\cdots)q^{5}+\cdots\)
180.2.k.e 180.k 20.e $12$ $1.437$ 12.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-1-\beta _{1}+\beta _{7}+\cdots)q^{5}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(180, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)