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Properties

Label 27.9.b
Level $27$
Weight $9$
Character orbit 27.b
Rep. character $\chi_{27}(26,\cdot)$
Character field $\Q$
Dimension $11$
Newform subspaces $4$
Sturm bound $27$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(27\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(27, [\chi])\).

Total New Old
Modular forms 27 11 16
Cusp forms 21 11 10
Eisenstein series 6 0 6

Trace form

\( 11 q - 1774 q^{4} + 1117 q^{7} + O(q^{10}) \) \( 11 q - 1774 q^{4} + 1117 q^{7} - 11934 q^{10} - 54353 q^{13} + 395114 q^{16} + 92191 q^{19} + 947322 q^{22} - 1413355 q^{25} - 3150662 q^{28} + 624796 q^{31} + 6399324 q^{34} - 11812253 q^{37} + 9131238 q^{40} + 14397742 q^{43} - 33774192 q^{46} + 8794032 q^{49} + 36565696 q^{52} - 58811454 q^{55} + 66246444 q^{58} - 21311981 q^{61} - 88126594 q^{64} + 93618691 q^{67} + 122431770 q^{70} + 33184849 q^{73} - 194322908 q^{76} + 2014147 q^{79} - 47688696 q^{82} - 72370908 q^{85} - 309154914 q^{88} + 32956613 q^{91} + 40926492 q^{94} + 20086669 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(27, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
27.9.b.a 27.b 3.b $1$ $10.999$ \(\Q\) \(\Q(\sqrt{-3}) \) 27.9.b.a \(0\) \(0\) \(0\) \(239\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{8}q^{4}+239q^{7}+56447q^{13}+2^{16}q^{16}+\cdots\)
27.9.b.b 27.b 3.b $2$ $10.999$ \(\Q(\sqrt{-6}) \) None 27.9.b.b \(0\) \(0\) \(0\) \(3934\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-608q^{4}-28\beta q^{5}+1967q^{7}+\cdots\)
27.9.b.c 27.b 3.b $2$ $10.999$ \(\Q(\sqrt{-30}) \) None 27.9.b.c \(0\) \(0\) \(0\) \(-1358\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-14q^{4}+26\beta q^{5}-679q^{7}+\cdots\)
27.9.b.d 27.b 3.b $6$ $10.999$ 6.0.6171673600.1 None 27.9.b.d \(0\) \(0\) \(0\) \(-1698\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-131+\beta _{2})q^{4}+(20\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(27, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(27, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(3, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 2}\)