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Properties

Label 441.2.e
Level $441$
Weight $2$
Character orbit 441.e
Rep. character $\chi_{441}(226,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $30$
Newform subspaces $10$
Sturm bound $112$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(112\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(441, [\chi])\).

Total New Old
Modular forms 144 38 106
Cusp forms 80 30 50
Eisenstein series 64 8 56

Trace form

\( 30 q - 3 q^{2} - 15 q^{4} + 2 q^{5} + 18 q^{8} + O(q^{10}) \) \( 30 q - 3 q^{2} - 15 q^{4} + 2 q^{5} + 18 q^{8} - 4 q^{10} + 2 q^{11} + 12 q^{13} - 11 q^{16} - 6 q^{19} - 8 q^{20} - 8 q^{22} - 9 q^{25} - 2 q^{26} + 28 q^{29} + 2 q^{31} - q^{32} - 12 q^{37} - 2 q^{38} - 20 q^{41} - 4 q^{43} - 24 q^{44} + 12 q^{46} + 6 q^{47} - 46 q^{50} + 16 q^{52} + 6 q^{53} - 8 q^{55} + 42 q^{58} + 12 q^{59} + 24 q^{61} + 36 q^{62} + 14 q^{64} + 30 q^{65} - 10 q^{67} - 4 q^{71} - 10 q^{73} + 4 q^{74} - 32 q^{76} - 42 q^{79} - 8 q^{80} - 20 q^{82} + 12 q^{83} - 64 q^{85} + 38 q^{86} + 72 q^{88} - 16 q^{89} - 96 q^{92} - 12 q^{94} - 34 q^{95} - 16 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(441, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
441.2.e.a 441.e 7.c $2$ $3.521$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-2\zeta_{6}q^{5}-3q^{8}+\cdots\)
441.2.e.b 441.e 7.c $2$ $3.521$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+2\zeta_{6}q^{5}-3q^{8}+\cdots\)
441.2.e.c 441.e 7.c $2$ $3.521$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+(2-2\zeta_{6})q^{4}+7q^{13}-4\zeta_{6}q^{16}+\cdots\)
441.2.e.d 441.e 7.c $2$ $3.521$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+3q^{8}+(4-4\zeta_{6})q^{11}+\cdots\)
441.2.e.e 441.e 7.c $2$ $3.521$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+2\zeta_{6}q^{5}+\cdots\)
441.2.e.f 441.e 7.c $4$ $3.521$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
441.2.e.g 441.e 7.c $4$ $3.521$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(-2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
441.2.e.h 441.e 7.c $4$ $3.521$ \(\Q(\sqrt{-3}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+\beta _{1}q^{2}+5\beta _{2}q^{4}+3\beta _{3}q^{8}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
441.2.e.i 441.e 7.c $4$ $3.521$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}-2\zeta_{12}^{2}q^{5}+\cdots\)
441.2.e.j 441.e 7.c $4$ $3.521$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+2\zeta_{12}^{2}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(441, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(441, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(147, [\chi])\)\(^{\oplus 2}\)