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Properties

Label 504.2.s
Level $504$
Weight $2$
Character orbit 504.s
Rep. character $\chi_{504}(289,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $20$
Newform subspaces $9$
Sturm bound $192$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.s (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 9 \)
Sturm bound: \(192\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(504, [\chi])\).

Total New Old
Modular forms 224 20 204
Cusp forms 160 20 140
Eisenstein series 64 0 64

Trace form

\( 20 q - 2 q^{5} + O(q^{10}) \) \( 20 q - 2 q^{5} - 2 q^{11} - 6 q^{17} - 10 q^{19} + 2 q^{23} - 20 q^{25} + 16 q^{29} + 6 q^{31} + 30 q^{35} - 2 q^{37} + 8 q^{43} + 6 q^{47} + 36 q^{49} + 18 q^{53} + 4 q^{55} + 18 q^{59} + 10 q^{61} + 20 q^{65} + 2 q^{67} - 48 q^{71} + 22 q^{73} - 50 q^{77} - 14 q^{79} - 56 q^{83} - 60 q^{85} - 34 q^{89} - 44 q^{91} - 38 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(504, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
504.2.s.a 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-4\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-3q^{13}+\cdots\)
504.2.s.b 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{5}+(-2-\zeta_{6})q^{7}+5\zeta_{6}q^{11}+\cdots\)
504.2.s.c 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots\)
504.2.s.d 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+3\zeta_{6}q^{11}+\cdots\)
504.2.s.e 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}-\zeta_{6}q^{11}+\cdots\)
504.2.s.f 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{5}+(-2-\zeta_{6})q^{7}-5\zeta_{6}q^{11}+\cdots\)
504.2.s.g 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(2\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-6\zeta_{6}q^{11}+\cdots\)
504.2.s.h 504.s 7.c $2$ $4.024$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(4\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}-3q^{13}+\cdots\)
504.2.s.i 504.s 7.c $4$ $4.024$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(0\) \(-1\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+2\beta _{1}-\beta _{3})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(504, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(504, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(84, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(168, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)