Properties

Label 405.3.h
Level $405$
Weight $3$
Character orbit 405.h
Rep. character $\chi_{405}(134,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $92$
Newform subspaces $11$
Sturm bound $162$
Trace bound $5$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.h (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 45 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 11 \)
Sturm bound: \(162\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(405, [\chi])\).

Total New Old
Modular forms 240 100 140
Cusp forms 192 92 100
Eisenstein series 48 8 40

Trace form

\( 92 q - 84 q^{4} + O(q^{10}) \) \( 92 q - 84 q^{4} - 20 q^{10} - 140 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} - 10 q^{34} + 92 q^{40} - 508 q^{46} + 290 q^{49} - 156 q^{55} + 100 q^{61} - 316 q^{64} + 102 q^{70} + 474 q^{76} + 616 q^{79} + 76 q^{85} + 552 q^{91} - 748 q^{94} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(405, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
405.3.h.a 405.h 45.h $2$ $11.035$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-15}) \) \(-1\) \(0\) \(-5\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(-1+\zeta_{6})q^{2}+3\zeta_{6}q^{4}-5\zeta_{6}q^{5}+\cdots\)
405.3.h.b 405.h 45.h $2$ $11.035$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-15}) \) \(1\) \(0\) \(5\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(1-\zeta_{6})q^{2}+3\zeta_{6}q^{4}+5\zeta_{6}q^{5}+7q^{8}+\cdots\)
405.3.h.c 405.h 45.h $4$ $11.035$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(-2\) \(0\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{2}q^{2}+(3+3\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
405.3.h.d 405.h 45.h $4$ $11.035$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\zeta_{12}^{2})q^{2}+3\zeta_{12}^{2}q^{4}+(\zeta_{12}+\cdots)q^{5}+\cdots\)
405.3.h.e 405.h 45.h $4$ $11.035$ \(\Q(\sqrt{-3}, \sqrt{5})\) \(\Q(\sqrt{-15}) \) \(-1\) \(0\) \(10\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q-\beta _{2}q^{2}+(-1+7\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
405.3.h.f 405.h 45.h $4$ $11.035$ \(\Q(\sqrt{-3}, \sqrt{5})\) \(\Q(\sqrt{-15}) \) \(1\) \(0\) \(-10\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\beta _{2}q^{2}+(-1+7\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
405.3.h.g 405.h 45.h $4$ $11.035$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\zeta_{12}^{2})q^{2}+3\zeta_{12}^{2}q^{4}+(\zeta_{12}+\cdots)q^{5}+\cdots\)
405.3.h.h 405.h 45.h $4$ $11.035$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None \(2\) \(0\) \(1\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\beta _{2})q^{2}-3\beta _{2}q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
405.3.h.i 405.h 45.h $8$ $11.035$ 8.0.3317760000.2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{7}q^{2}-6\beta _{2}q^{4}+(-\beta _{4}+\beta _{6})q^{5}+\cdots\)
405.3.h.j 405.h 45.h $8$ $11.035$ 8.0.\(\cdots\).5 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{2}+(-3+3\beta _{2})q^{4}+(-\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
405.3.h.k 405.h 45.h $48$ $11.035$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(405, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(405, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(135, [\chi])\)\(^{\oplus 2}\)