Properties

Label 170.2.g
Level $170$
Weight $2$
Character orbit 170.g
Rep. character $\chi_{170}(89,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $16$
Newform subspaces $6$
Sturm bound $54$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 170 = 2 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 170.g (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 85 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 6 \)
Sturm bound: \(54\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(170, [\chi])\).

Total New Old
Modular forms 64 16 48
Cusp forms 48 16 32
Eisenstein series 16 0 16

Trace form

\( 16q + 16q^{4} + 4q^{5} + O(q^{10}) \) \( 16q + 16q^{4} + 4q^{5} - 4q^{10} - 16q^{11} + 16q^{16} + 4q^{20} - 32q^{21} + 8q^{29} - 8q^{34} - 40q^{35} - 24q^{39} - 4q^{40} - 16q^{44} + 12q^{45} - 24q^{46} - 8q^{50} - 24q^{51} - 24q^{54} - 8q^{55} + 8q^{61} + 16q^{64} + 32q^{65} + 80q^{69} + 48q^{71} + 16q^{74} - 56q^{75} + 40q^{79} + 4q^{80} + 48q^{81} - 32q^{84} + 36q^{85} - 32q^{86} + 8q^{89} + 28q^{90} + 40q^{91} + 48q^{95} + 40q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(170, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
170.2.g.a \(2\) \(1.357\) \(\Q(\sqrt{-1}) \) None \(-2\) \(2\) \(2\) \(-6\) \(q-q^{2}+(1-i)q^{3}+q^{4}+(1-2i)q^{5}+\cdots\)
170.2.g.b \(2\) \(1.357\) \(\Q(\sqrt{-1}) \) None \(-2\) \(2\) \(2\) \(2\) \(q-q^{2}+(1-i)q^{3}+q^{4}+(1+2i)q^{5}+\cdots\)
170.2.g.c \(2\) \(1.357\) \(\Q(\sqrt{-1}) \) None \(2\) \(-2\) \(-4\) \(6\) \(q+q^{2}+(-1+i)q^{3}+q^{4}+(-2+i)q^{5}+\cdots\)
170.2.g.d \(2\) \(1.357\) \(\Q(\sqrt{-1}) \) None \(2\) \(-2\) \(4\) \(-2\) \(q+q^{2}+(-1+i)q^{3}+q^{4}+(2+i)q^{5}+\cdots\)
170.2.g.e \(4\) \(1.357\) \(\Q(\zeta_{8})\) None \(-4\) \(-4\) \(0\) \(4\) \(q-q^{2}+(-1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+q^{4}+\cdots\)
170.2.g.f \(4\) \(1.357\) \(\Q(\zeta_{8})\) None \(4\) \(4\) \(0\) \(-4\) \(q+q^{2}+(1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+q^{4}+(\zeta_{8}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(170, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(170, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(85, [\chi])\)\(^{\oplus 2}\)