Properties

Label 784.4.f
Level $784$
Weight $4$
Character orbit 784.f
Rep. character $\chi_{784}(783,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $10$
Sturm bound $448$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(448\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(784, [\chi])\).

Total New Old
Modular forms 360 60 300
Cusp forms 312 60 252
Eisenstein series 48 0 48

Trace form

\( 60q + 540q^{9} + O(q^{10}) \) \( 60q + 540q^{9} - 2004q^{25} - 168q^{29} + 504q^{37} + 1176q^{53} + 2376q^{57} + 552q^{65} + 3924q^{81} - 2712q^{85} - 4248q^{93} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(784, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
784.4.f.a \(2\) \(46.257\) \(\Q(\sqrt{-3}) \) None \(0\) \(-14\) \(0\) \(0\) \(q-7q^{3}+9\zeta_{6}q^{5}+22q^{9}-7\zeta_{6}q^{11}+\cdots\)
784.4.f.b \(2\) \(46.257\) \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-q^{3}+3\zeta_{6}q^{5}-26q^{9}-15\zeta_{6}q^{11}+\cdots\)
784.4.f.c \(2\) \(46.257\) \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(0\) \(0\) \(q+q^{3}+3\zeta_{6}q^{5}-26q^{9}+15\zeta_{6}q^{11}+\cdots\)
784.4.f.d \(2\) \(46.257\) \(\Q(\sqrt{-3}) \) None \(0\) \(14\) \(0\) \(0\) \(q+7q^{3}-9\zeta_{6}q^{5}+22q^{9}-7\zeta_{6}q^{11}+\cdots\)
784.4.f.e \(4\) \(46.257\) 4.0.2048.2 \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}-5\beta _{2})q^{5}-3^{3}q^{9}+(4\beta _{1}+21\beta _{2}+\cdots)q^{13}+\cdots\)
784.4.f.f \(4\) \(46.257\) \(\Q(\sqrt{-3}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-5\beta _{2}q^{5}-20q^{9}-15\beta _{3}q^{11}+\cdots\)
784.4.f.g \(6\) \(46.257\) 6.0.\(\cdots\).1 None \(0\) \(-14\) \(0\) \(0\) \(q+(-2+\beta _{2})q^{3}-\beta _{3}q^{5}+(26-\beta _{1}+\cdots)q^{9}+\cdots\)
784.4.f.h \(6\) \(46.257\) 6.0.\(\cdots\).1 None \(0\) \(14\) \(0\) \(0\) \(q+(2-\beta _{2})q^{3}+\beta _{3}q^{5}+(26-\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
784.4.f.i \(8\) \(46.257\) 8.0.\(\cdots\).32 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{3}+(2\beta _{2}+\beta _{4})q^{5}+(43+3\beta _{5}+\cdots)q^{9}+\cdots\)
784.4.f.j \(24\) \(46.257\) None \(0\) \(0\) \(0\) \(0\)

Decomposition of \(S_{4}^{\mathrm{old}}(784, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(784, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(196, [\chi])\)\(^{\oplus 3}\)