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Properties

Label 441.3.m
Level $441$
Weight $3$
Character orbit 441.m
Rep. character $\chi_{441}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $62$
Newform subspaces $13$
Sturm bound $168$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 441.m (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 13 \)
Sturm bound: \(168\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(441, [\chi])\).

Total New Old
Modular forms 256 70 186
Cusp forms 192 62 130
Eisenstein series 64 8 56

Trace form

\( 62 q - 3 q^{2} - 49 q^{4} - 6 q^{5} + 18 q^{8} + O(q^{10}) \) \( 62 q - 3 q^{2} - 49 q^{4} - 6 q^{5} + 18 q^{8} - 48 q^{10} - 28 q^{11} - 49 q^{16} + 48 q^{17} + 78 q^{19} - 100 q^{22} - 110 q^{23} + 153 q^{25} - 126 q^{26} + 36 q^{29} + 66 q^{31} - 49 q^{32} + 88 q^{37} + 174 q^{38} + 348 q^{40} - 152 q^{43} - 278 q^{44} - 94 q^{46} - 222 q^{47} - 290 q^{50} - 492 q^{52} + 54 q^{53} + 70 q^{58} + 84 q^{59} - 180 q^{61} + 554 q^{64} + 62 q^{65} - 64 q^{67} + 36 q^{68} + 416 q^{71} + 150 q^{73} - 304 q^{74} + 208 q^{79} + 48 q^{80} - 324 q^{82} - 224 q^{85} + 360 q^{86} + 414 q^{88} - 60 q^{89} + 412 q^{92} + 204 q^{94} + 46 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(441, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
441.3.m.a 441.m 7.d $2$ $12.016$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) \(-3\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q-3\zeta_{6}q^{2}+(-5+5\zeta_{6})q^{4}+3q^{8}+\cdots\)
441.3.m.b 441.m 7.d $2$ $12.016$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-2\zeta_{6}q^{2}+(-4+2\zeta_{6})q^{5}-8q^{8}+\cdots\)
441.3.m.c 441.m 7.d $2$ $12.016$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(4-4\zeta_{6})q^{4}+(7-14\zeta_{6})q^{13}-2^{4}\zeta_{6}q^{16}+\cdots\)
441.3.m.d 441.m 7.d $2$ $12.016$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{4}+(-8+4\zeta_{6})q^{5}+\cdots\)
441.3.m.e 441.m 7.d $2$ $12.016$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(-9\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{4}+(-6+3\zeta_{6})q^{5}+\cdots\)
441.3.m.f 441.m 7.d $2$ $12.016$ \(\Q(\sqrt{-3}) \) None \(1\) \(0\) \(12\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{6}q^{2}+(3-3\zeta_{6})q^{4}+(8-4\zeta_{6})q^{5}+\cdots\)
441.3.m.g 441.m 7.d $2$ $12.016$ \(\Q(\sqrt{-3}) \) None \(3\) \(0\) \(9\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+3\zeta_{6}q^{2}+(-5+5\zeta_{6})q^{4}+(6-3\zeta_{6})q^{5}+\cdots\)
441.3.m.h 441.m 7.d $4$ $12.016$ \(\Q(\sqrt{-3}, \sqrt{13})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{2}+(-9+9\beta _{1})q^{4}+(-\beta _{2}-2\beta _{3})q^{5}+\cdots\)
441.3.m.i 441.m 7.d $4$ $12.016$ \(\Q(\sqrt{-3}, \sqrt{7})\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\beta _{1}q^{2}+3\beta _{2}q^{4}-\beta _{3}q^{8}+(8\beta _{1}+8\beta _{3})q^{11}+\cdots\)
441.3.m.j 441.m 7.d $8$ $12.016$ 8.0.339738624.1 None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\beta _{3}+\beta _{5}-\beta _{7})q^{2}+(2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
441.3.m.k 441.m 7.d $8$ $12.016$ 8.0.339738624.1 None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\beta _{3}+\beta _{5}-\beta _{7})q^{2}+(2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
441.3.m.l 441.m 7.d $8$ $12.016$ 8.0.339738624.1 None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{4}-\beta _{5})q^{2}+(1-2\beta _{2}+\beta _{4}-2\beta _{5}+\cdots)q^{4}+\cdots\)
441.3.m.m 441.m 7.d $16$ $12.016$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{2}q^{2}+(1-\beta _{3}+\beta _{5})q^{4}+\beta _{15}q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(441, [\chi])\) into lower level spaces

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