Properties

Label 1764.2.t
Level $1764$
Weight $2$
Character orbit 1764.t
Rep. character $\chi_{1764}(521,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $28$
Newform subspaces $3$
Sturm bound $672$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.t (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 3 \)
Sturm bound: \(672\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1764, [\chi])\).

Total New Old
Modular forms 768 28 740
Cusp forms 576 28 548
Eisenstein series 192 0 192

Trace form

\( 28 q + O(q^{10}) \) \( 28 q - 18 q^{19} - 6 q^{25} + 6 q^{31} + 10 q^{37} + 4 q^{43} + 12 q^{61} - 50 q^{67} - 54 q^{73} - 6 q^{79} + 48 q^{85} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1764, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1764.2.t.a 1764.t 21.g $4$ $14.086$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{5}+(\beta _{2}+\beta _{3})q^{11}+(1-2\beta _{1}+\cdots)q^{13}+\cdots\)
1764.2.t.b 1764.t 21.g $8$ $14.086$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\zeta_{24}^{2}q^{5}+(-\zeta_{24}^{3}-\zeta_{24}^{7})q^{11}+\cdots\)
1764.2.t.c 1764.t 21.g $16$ $14.086$ \(\Q(\zeta_{48})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{48}^{4}-\zeta_{48}^{15})q^{5}+(\zeta_{48}-\zeta_{48}^{5}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1764, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1764, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(294, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(441, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(588, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(882, [\chi])\)\(^{\oplus 2}\)