Properties

Label 999.2.c
Level $999$
Weight $2$
Character orbit 999.c
Rep. character $\chi_{999}(406,\cdot)$
Character field $\Q$
Dimension $50$
Newform subspaces $9$
Sturm bound $228$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 999 = 3^{3} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 999.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 37 \)
Character field: \(\Q\)
Newform subspaces: \( 9 \)
Sturm bound: \(228\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(999, [\chi])\).

Total New Old
Modular forms 120 50 70
Cusp forms 108 50 58
Eisenstein series 12 0 12

Trace form

\( 50 q - 48 q^{4} - 2 q^{7} + O(q^{10}) \) \( 50 q - 48 q^{4} - 2 q^{7} - 16 q^{10} + 76 q^{16} - 54 q^{25} - 12 q^{28} - 40 q^{34} - 25 q^{37} + 76 q^{40} - 52 q^{46} + 48 q^{49} - 40 q^{58} - 100 q^{64} + 58 q^{67} + 52 q^{70} + 22 q^{73} - 88 q^{85} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(999, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
999.2.c.a 999.c 37.b $2$ $7.977$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{2}-q^{4}-\zeta_{6}q^{5}+q^{7}-\zeta_{6}q^{8}+\cdots\)
999.2.c.b 999.c 37.b $2$ $7.977$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{2}-q^{4}-\zeta_{6}q^{5}+q^{7}-\zeta_{6}q^{8}+\cdots\)
999.2.c.c 999.c 37.b $2$ $7.977$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}-iq^{5}-3q^{7}+3iq^{8}+\cdots\)
999.2.c.d 999.c 37.b $2$ $7.977$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}-iq^{5}-3q^{7}+3iq^{8}+\cdots\)
999.2.c.e 999.c 37.b $2$ $7.977$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(2\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{4}+q^{7}+(-1+2\zeta_{6})q^{13}+4q^{16}+\cdots\)
999.2.c.f 999.c 37.b $8$ $7.977$ 8.0.\(\cdots\).1 \(\Q(\sqrt{-111}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{4}q^{2}+(-2+\beta _{7})q^{4}-\beta _{3}q^{5}+\beta _{7}q^{7}+\cdots\)
999.2.c.g 999.c 37.b $8$ $7.977$ 8.0.\(\cdots\).1 \(\Q(\sqrt{-111}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{4}q^{2}+(-2+\beta _{7})q^{4}+\beta _{2}q^{5}+(\beta _{3}+\cdots)q^{7}+\cdots\)
999.2.c.h 999.c 37.b $12$ $7.977$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}-\beta _{10}q^{5}+\cdots\)
999.2.c.i 999.c 37.b $12$ $7.977$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}-\beta _{10}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(999, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(999, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(37, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(111, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(333, [\chi])\)\(^{\oplus 2}\)