Properties

Label 1728.3.h
Level $1728$
Weight $3$
Character orbit 1728.h
Rep. character $\chi_{1728}(161,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $10$
Sturm bound $864$
Trace bound $49$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 24 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(864\)
Trace bound: \(49\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1728, [\chi])\).

Total New Old
Modular forms 612 64 548
Cusp forms 540 64 476
Eisenstein series 72 0 72

Trace form

\( 64 q + O(q^{10}) \) \( 64 q + 320 q^{25} + 416 q^{49} - 160 q^{73} - 128 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1728, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1728.3.h.a 1728.h 24.h $4$ $47.085$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(-24\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-6q^{5}+\beta _{2}q^{7}+2\beta _{2}q^{11}+\beta _{3}q^{13}+\cdots\)
1728.3.h.b 1728.h 24.h $4$ $47.085$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}^{2}q^{5}+\zeta_{12}^{2}q^{7}-3q^{11}+\zeta_{12}^{3}q^{13}+\cdots\)
1728.3.h.c 1728.h 24.h $4$ $47.085$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{12}^{3}q^{7}-5\zeta_{12}^{2}q^{13}-11\zeta_{12}q^{19}+\cdots\)
1728.3.h.d 1728.h 24.h $4$ $47.085$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-5\zeta_{12}^{3}q^{7}-7\zeta_{12}^{2}q^{13}+37\zeta_{12}q^{19}+\cdots\)
1728.3.h.e 1728.h 24.h $4$ $47.085$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}^{2}q^{5}+\zeta_{12}^{2}q^{7}+3q^{11}+\zeta_{12}^{3}q^{13}+\cdots\)
1728.3.h.f 1728.h 24.h $4$ $47.085$ \(\Q(i, \sqrt{11})\) None \(0\) \(0\) \(24\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+6q^{5}+\beta _{2}q^{7}-2\beta _{2}q^{11}-\beta _{3}q^{13}+\cdots\)
1728.3.h.g 1728.h 24.h $8$ $47.085$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{24}^{4}q^{5}+\zeta_{24}^{3}q^{7}-\zeta_{24}^{7}q^{11}+\cdots\)
1728.3.h.h 1728.h 24.h $8$ $47.085$ 8.0.12960000.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{5}-5\beta _{2}q^{7}-\beta _{4}q^{11}+3\beta _{1}q^{13}+\cdots\)
1728.3.h.i 1728.h 24.h $12$ $47.085$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{8}q^{5}+(\beta _{3}-\beta _{8})q^{7}+(-1-\beta _{1}+\cdots)q^{11}+\cdots\)
1728.3.h.j 1728.h 24.h $12$ $47.085$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{8}q^{5}+(\beta _{3}-\beta _{8})q^{7}+(1+\beta _{1})q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1728, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)