Properties

Label 270.4.e
Level $270$
Weight $4$
Character orbit 270.e
Rep. character $\chi_{270}(91,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $5$
Sturm bound $216$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 270.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(216\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(270, [\chi])\).

Total New Old
Modular forms 348 24 324
Cusp forms 300 24 276
Eisenstein series 48 0 48

Trace form

\( 24 q + 4 q^{2} - 48 q^{4} + 10 q^{5} - 24 q^{7} - 32 q^{8} + O(q^{10}) \) \( 24 q + 4 q^{2} - 48 q^{4} + 10 q^{5} - 24 q^{7} - 32 q^{8} + 42 q^{11} + 48 q^{13} + 44 q^{14} - 192 q^{16} - 348 q^{17} - 420 q^{19} + 40 q^{20} + 36 q^{22} - 168 q^{23} - 300 q^{25} - 160 q^{26} + 192 q^{28} - 642 q^{29} - 60 q^{31} + 64 q^{32} - 180 q^{34} - 280 q^{35} - 672 q^{37} + 668 q^{38} - 1020 q^{41} - 258 q^{43} - 336 q^{44} - 504 q^{46} - 948 q^{47} + 90 q^{49} + 100 q^{50} + 192 q^{52} - 1320 q^{53} + 176 q^{56} - 1398 q^{59} - 1698 q^{61} - 1360 q^{62} + 1536 q^{64} + 1160 q^{65} - 1950 q^{67} + 696 q^{68} - 180 q^{70} + 1176 q^{71} + 516 q^{73} + 536 q^{74} + 840 q^{76} + 2520 q^{77} + 1776 q^{79} - 320 q^{80} + 2088 q^{82} + 552 q^{83} + 720 q^{85} - 1636 q^{86} + 144 q^{88} + 3972 q^{89} - 4008 q^{91} - 672 q^{92} - 612 q^{94} + 1040 q^{95} + 1038 q^{97} + 2232 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(270, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
270.4.e.a 270.e 9.c $2$ $15.931$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-5\) \(16\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
270.4.e.b 270.e 9.c $4$ $15.931$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(-4\) \(0\) \(-10\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-5+5\beta _{1}+\cdots)q^{5}+\cdots\)
270.4.e.c 270.e 9.c $4$ $15.931$ \(\Q(\sqrt{-3}, \sqrt{-5})\) None \(4\) \(0\) \(-10\) \(-16\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(-5+5\beta _{1}+\cdots)q^{5}+\cdots\)
270.4.e.d 270.e 9.c $6$ $15.931$ 6.0.41783472.1 None \(-6\) \(0\) \(15\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{1})q^{2}+4\beta _{1}q^{4}-5\beta _{1}q^{5}+\cdots\)
270.4.e.e 270.e 9.c $8$ $15.931$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(0\) \(20\) \(-23\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{1})q^{2}-4\beta _{1}q^{4}+5\beta _{1}q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(270, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(270, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)