Properties

Label 684.5.h
Level $684$
Weight $5$
Character orbit 684.h
Rep. character $\chi_{684}(37,\cdot)$
Character field $\Q$
Dimension $34$
Newform subspaces $6$
Sturm bound $600$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 684.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(600\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(684, [\chi])\).

Total New Old
Modular forms 492 34 458
Cusp forms 468 34 434
Eisenstein series 24 0 24

Trace form

\( 34q - 9q^{5} - 15q^{7} + O(q^{10}) \) \( 34q - 9q^{5} - 15q^{7} - 297q^{11} + 225q^{17} - 36q^{19} - 186q^{23} + 3421q^{25} + 327q^{35} - 5977q^{43} - 3825q^{47} + 13875q^{49} + 9071q^{55} + 63q^{61} - 7195q^{73} - 13161q^{77} - 10500q^{83} - 3325q^{85} + 19107q^{95} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(684, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
684.5.h.a \(2\) \(70.705\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(188\) \(q+94q^{7}+11\zeta_{6}q^{13}+(23+13\zeta_{6})q^{19}+\cdots\)
684.5.h.b \(2\) \(70.705\) \(\Q(\sqrt{57}) \) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(31\) \(73\) \(q+(17+3\beta )q^{5}+(39+5\beta )q^{7}+(-119+\cdots)q^{11}+\cdots\)
684.5.h.c \(4\) \(70.705\) \(\mathbb{Q}[x]/(x^{4} + \cdots)\) None \(0\) \(0\) \(-22\) \(24\) \(q+(-6+\beta _{3})q^{5}+(7-2\beta _{3})q^{7}+(48+\cdots)q^{11}+\cdots\)
684.5.h.d \(4\) \(70.705\) \(\Q(\sqrt{3}, \sqrt{19})\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(0\) \(-146\) \(q+(-2\beta _{1}-\beta _{2})q^{5}+(-34+5\beta _{3})q^{7}+\cdots\)
684.5.h.e \(8\) \(70.705\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(-68\) \(q+\beta _{4}q^{5}+(-9+\beta _{1})q^{7}+(-2\beta _{4}+\beta _{5}+\cdots)q^{11}+\cdots\)
684.5.h.f \(14\) \(70.705\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(0\) \(-18\) \(-86\) \(q+(-1-\beta _{1})q^{5}+(-6+\beta _{2})q^{7}+(-18+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(684, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(684, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(228, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(342, [\chi])\)\(^{\oplus 2}\)