Properties

Label 1728.3.e
Level $1728$
Weight $3$
Character orbit 1728.e
Rep. character $\chi_{1728}(1025,\cdot)$
Character field $\Q$
Dimension $64$
Newform subspaces $22$
Sturm bound $864$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 22 \)
Sturm bound: \(864\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(7\)

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

\( 64 q + O(q^{10}) \) \( 64 q - 16 q^{13} - 320 q^{25} + 80 q^{37} + 448 q^{49} - 80 q^{61} - 160 q^{85} + 32 q^{97} + O(q^{100}) \)

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.e.a 1728.e 3.b $1$ $47.085$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-13\) $\mathrm{U}(1)[D_{2}]$ \(q-13q^{7}+q^{13}-11q^{19}+5^{2}q^{25}+\cdots\)
1728.3.e.b 1728.e 3.b $1$ $47.085$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-11\) $\mathrm{U}(1)[D_{2}]$ \(q-11q^{7}-23q^{13}-37q^{19}+5^{2}q^{25}+\cdots\)
1728.3.e.c 1728.e 3.b $1$ $47.085$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(11\) $\mathrm{U}(1)[D_{2}]$ \(q+11q^{7}-23q^{13}+37q^{19}+5^{2}q^{25}+\cdots\)
1728.3.e.d 1728.e 3.b $1$ $47.085$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(13\) $\mathrm{U}(1)[D_{2}]$ \(q+13q^{7}+q^{13}+11q^{19}+5^{2}q^{25}+\cdots\)
1728.3.e.e 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-14\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{5}-7q^{7}+iq^{11}-14q^{13}-2iq^{17}+\cdots\)
1728.3.e.f 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{5}-5q^{7}+\beta q^{11}+q^{13}+3\beta q^{17}+\cdots\)
1728.3.e.g 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{5}-5q^{7}-5iq^{11}+10q^{13}+\cdots\)
1728.3.e.h 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{5}-3q^{7}+7\beta q^{11}-7q^{13}-5\beta q^{17}+\cdots\)
1728.3.e.i 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{5}-3q^{7}-\beta q^{11}+17q^{13}-5\beta q^{17}+\cdots\)
1728.3.e.j 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{5}+3q^{7}-7\beta q^{11}-7q^{13}-5\beta q^{17}+\cdots\)
1728.3.e.k 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{5}+3q^{7}+\beta q^{11}+17q^{13}-5\beta q^{17}+\cdots\)
1728.3.e.l 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{5}+5q^{7}-\beta q^{11}+q^{13}+3\beta q^{17}+\cdots\)
1728.3.e.m 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{5}+5q^{7}+5iq^{11}+10q^{13}+\cdots\)
1728.3.e.n 1728.e 3.b $2$ $47.085$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(14\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{5}+7q^{7}-iq^{11}-14q^{13}-2iq^{17}+\cdots\)
1728.3.e.o 1728.e 3.b $4$ $47.085$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}+\zeta_{8}^{2})q^{5}+(-3-\zeta_{8}^{3})q^{7}+(-\zeta_{8}+\cdots)q^{11}+\cdots\)
1728.3.e.p 1728.e 3.b $4$ $47.085$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\zeta_{8}+\zeta_{8}^{2})q^{5}+(-3+\zeta_{8}^{3})q^{7}+\cdots\)
1728.3.e.q 1728.e 3.b $4$ $47.085$ \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{5}+(-3+\beta _{1})q^{7}+(-\beta _{2}-3\beta _{3})q^{11}+\cdots\)
1728.3.e.r 1728.e 3.b $4$ $47.085$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}+\zeta_{8}^{2})q^{5}+(3+\zeta_{8}^{3})q^{7}+(\zeta_{8}+\cdots)q^{11}+\cdots\)
1728.3.e.s 1728.e 3.b $4$ $47.085$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\zeta_{8}+\zeta_{8}^{2})q^{5}+(3-\zeta_{8}^{3})q^{7}+(-3\zeta_{8}+\cdots)q^{11}+\cdots\)
1728.3.e.t 1728.e 3.b $4$ $47.085$ \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{5}+(3+\beta _{1})q^{7}+(\beta _{2}-3\beta _{3})q^{11}+\cdots\)
1728.3.e.u 1728.e 3.b $8$ $47.085$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{24}^{2}q^{5}+\zeta_{24}^{4}q^{7}+\zeta_{24}^{6}q^{11}+\cdots\)
1728.3.e.v 1728.e 3.b $8$ $47.085$ 8.0.2441150464.4 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{6}q^{5}-\beta _{4}q^{7}+(-\beta _{1}-\beta _{2})q^{11}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1728, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(288, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(576, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(864, [\chi])\)\(^{\oplus 2}\)