Properties

Label 1680.2.f
Level $1680$
Weight $2$
Character orbit 1680.f
Rep. character $\chi_{1680}(881,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $12$
Sturm bound $768$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(768\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(17\), \(41\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1680, [\chi])\).

Total New Old
Modular forms 408 64 344
Cusp forms 360 64 296
Eisenstein series 48 0 48

Trace form

\( 64q - 4q^{7} + O(q^{10}) \) \( 64q - 4q^{7} + 4q^{21} + 64q^{25} + 12q^{39} - 56q^{43} + 8q^{49} + 28q^{51} + 20q^{63} + 40q^{67} - 24q^{79} + 8q^{81} - 32q^{93} + 84q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1680, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1680.2.f.a \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(2\) \(-4\) \(q+(-2+\zeta_{6})q^{3}+q^{5}+(-1-2\zeta_{6})q^{7}+\cdots\)
1680.2.f.b \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-2\) \(-4\) \(q+\zeta_{6}q^{3}-q^{5}+(-2-\zeta_{6})q^{7}-3q^{9}+\cdots\)
1680.2.f.c \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(-4\) \(q-\zeta_{6}q^{3}+q^{5}+(-2+\zeta_{6})q^{7}-3q^{9}+\cdots\)
1680.2.f.d \(2\) \(13.415\) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(-2\) \(-4\) \(q+(2-\zeta_{6})q^{3}-q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
1680.2.f.e \(4\) \(13.415\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(-4\) \(-6\) \(q+(\beta _{1}+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.f \(4\) \(13.415\) \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(-4\) \(6\) \(q+(-1-\beta _{1})q^{3}-q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.g \(4\) \(13.415\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(-1\) \(4\) \(8\) \(q-\beta _{1}q^{3}+q^{5}+(2+\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
1680.2.f.h \(4\) \(13.415\) \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(0\) \(1\) \(-4\) \(8\) \(q+\beta _{1}q^{3}-q^{5}+(2-\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
1680.2.f.i \(4\) \(13.415\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(4\) \(-6\) \(q+(-\beta _{1}-\beta _{2})q^{3}+q^{5}+(-1-2\beta _{2}+\cdots)q^{7}+\cdots\)
1680.2.f.j \(4\) \(13.415\) \(\Q(i, \sqrt{5})\) None \(0\) \(2\) \(4\) \(6\) \(q+(1+\beta _{1})q^{3}+q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots\)
1680.2.f.k \(16\) \(13.415\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-16\) \(-2\) \(q+\beta _{3}q^{3}-q^{5}-\beta _{9}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots\)
1680.2.f.l \(16\) \(13.415\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(16\) \(-2\) \(q-\beta _{3}q^{3}+q^{5}-\beta _{4}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1680, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1680, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(336, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(420, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(840, [\chi])\)\(^{\oplus 2}\)