Properties

Label 684.3.h
Level $684$
Weight $3$
Character orbit 684.h
Rep. character $\chi_{684}(37,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $5$
Sturm bound $360$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(360\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(684, [\chi])\).

Total New Old
Modular forms 252 16 236
Cusp forms 228 16 212
Eisenstein series 24 0 24

Trace form

\( 16q + q^{5} - 13q^{7} + O(q^{10}) \) \( 16q + q^{5} - 13q^{7} + q^{11} + 19q^{17} + 22q^{19} + 34q^{23} + 141q^{25} + 53q^{35} + 137q^{43} + q^{47} + 291q^{49} + 13q^{55} + 89q^{61} + 71q^{73} + 197q^{77} + 160q^{83} + 457q^{85} - 107q^{95} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
684.3.h.a \(2\) \(18.638\) \(\Q(\sqrt{57}) \) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-9\) \(5\) \(q+(-4-\beta )q^{5}+(4-3\beta )q^{7}+(4-5\beta )q^{11}+\cdots\)
684.3.h.b \(2\) \(18.638\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-4\) \(q-2q^{7}+\zeta_{6}q^{13}+(-13-\zeta_{6})q^{19}+\cdots\)
684.3.h.c \(2\) \(18.638\) \(\Q(\sqrt{-29}) \) None \(0\) \(0\) \(8\) \(-2\) \(q+4q^{5}-q^{7}-14q^{11}-\beta q^{13}-23q^{17}+\cdots\)
684.3.h.d \(4\) \(18.638\) \(\Q(\sqrt{3}, \sqrt{19})\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(0\) \(-10\) \(q+\beta _{1}q^{5}+(-3-\beta _{3})q^{7}-\beta _{2}q^{11}+\cdots\)
684.3.h.e \(6\) \(18.638\) 6.0.219615408.1 None \(0\) \(0\) \(2\) \(-2\) \(q+\beta _{3}q^{5}-\beta _{2}q^{7}+(4+\beta _{2})q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)