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Properties

Label 270.3.b
Level $270$
Weight $3$
Character orbit 270.b
Rep. character $\chi_{270}(269,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $4$
Sturm bound $162$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 270.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(162\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(270, [\chi])\).

Total New Old
Modular forms 120 16 104
Cusp forms 96 16 80
Eisenstein series 24 0 24

Trace form

\( 16 q + 32 q^{4} + O(q^{10}) \) \( 16 q + 32 q^{4} + 8 q^{10} + 64 q^{16} + 48 q^{19} + 72 q^{25} - 8 q^{31} + 176 q^{34} + 16 q^{40} + 112 q^{46} - 552 q^{49} - 136 q^{55} - 400 q^{61} + 128 q^{64} + 104 q^{70} + 96 q^{76} + 288 q^{79} - 568 q^{85} + 72 q^{91} - 224 q^{94} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(270, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
270.3.b.a 270.b 15.d $4$ $7.357$ 4.0.31744.1 None \(0\) \(0\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+2q^{4}+(-3-\beta _{1}-\beta _{2})q^{5}+\cdots\)
270.3.b.b 270.b 15.d $4$ $7.357$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\zeta_{8}+\zeta_{8}^{3})q^{2}+2q^{4}+(3\zeta_{8}-4\zeta_{8}^{3})q^{5}+\cdots\)
270.3.b.c 270.b 15.d $4$ $7.357$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{8}-\zeta_{8}^{3})q^{2}+2q^{4}+5\zeta_{8}q^{5}+5\zeta_{8}^{2}q^{7}+\cdots\)
270.3.b.d 270.b 15.d $4$ $7.357$ 4.0.31744.1 None \(0\) \(0\) \(12\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+2q^{4}+(3-\beta _{1}+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(270, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(270, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)