Properties

Label 405.3.l
Level $405$
Weight $3$
Character orbit 405.l
Rep. character $\chi_{405}(28,\cdot)$
Character field $\Q(\zeta_{12})$
Dimension $184$
Newform subspaces $15$
Sturm bound $162$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 405.l (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 45 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 15 \)
Sturm bound: \(162\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 480 200 280
Cusp forms 384 184 200
Eisenstein series 96 16 80

Trace form

\( 184 q + 4 q^{7} + O(q^{10}) \) \( 184 q + 4 q^{7} - 8 q^{10} + 4 q^{13} + 328 q^{16} + 20 q^{22} + 16 q^{25} + 152 q^{28} + 8 q^{31} + 160 q^{37} - 60 q^{40} + 4 q^{43} + 728 q^{46} - 44 q^{52} - 472 q^{55} - 252 q^{58} - 184 q^{61} + 160 q^{67} - 96 q^{70} - 8 q^{73} + 660 q^{76} - 88 q^{82} + 52 q^{85} + 300 q^{88} - 352 q^{91} - 296 q^{97} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
405.3.l.a 405.l 45.k $4$ $11.035$ \(\Q(\zeta_{12})\) None 405.3.g.a \(-4\) \(0\) \(20\) \(22\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12})q^{2}+(1+2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
405.3.l.b 405.l 45.k $4$ $11.035$ \(\Q(\zeta_{12})\) None 405.3.g.a \(-2\) \(0\) \(10\) \(-2\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}+\cdots)q^{4}+\cdots\)
405.3.l.c 405.l 45.k $4$ $11.035$ \(\Q(\zeta_{12})\) None 405.3.g.a \(2\) \(0\) \(-10\) \(-2\) $\mathrm{SU}(2)[C_{12}]$ \(q+(1-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}+\cdots)q^{4}+\cdots\)
405.3.l.d 405.l 45.k $4$ $11.035$ \(\Q(\zeta_{12})\) None 405.3.g.a \(4\) \(0\) \(-20\) \(22\) $\mathrm{SU}(2)[C_{12}]$ \(q+(1-\zeta_{12})q^{2}+(1+2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
405.3.l.e 405.l 45.k $8$ $11.035$ 8.0.49787136.1 None 405.3.g.d \(-6\) \(0\) \(-6\) \(26\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1-\beta _{4}+\beta _{5})q^{2}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
405.3.l.f 405.l 45.k $8$ $11.035$ \(\Q(\zeta_{24})\) None 15.3.f.a \(-4\) \(0\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta_{7}+\beta_{4}+\cdots-\beta_1)q^{2}+\cdots+(2\beta_{6}+2\beta_{5}+\beta_1)q^{4}+\cdots\)
405.3.l.g 405.l 45.k $8$ $11.035$ 8.0.3317760000.2 None 45.3.g.a \(0\) \(0\) \(0\) \(20\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{7}q^{2}-\beta _{2}q^{4}+(2\beta _{3}+\beta _{5}-2\beta _{7})q^{5}+\cdots\)
405.3.l.h 405.l 45.k $8$ $11.035$ \(\Q(\zeta_{24})\) None 15.3.f.a \(4\) \(0\) \(4\) \(-4\) $\mathrm{SU}(2)[C_{12}]$ \(q+(\beta_{7}-\beta_{4}+\beta_{2}+\beta_1)q^{2}+(2\beta_{6}+2\beta_{5}+\beta_1)q^{4}+\cdots\)
405.3.l.i 405.l 45.k $8$ $11.035$ 8.0.49787136.1 None 405.3.g.d \(6\) \(0\) \(6\) \(26\) $\mathrm{SU}(2)[C_{12}]$ \(q+(1+\beta _{4}-\beta _{5})q^{2}+(-\beta _{1}-\beta _{2}-\beta _{5}+\cdots)q^{4}+\cdots\)
405.3.l.j 405.l 45.k $16$ $11.035$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 405.3.g.f \(-6\) \(0\) \(12\) \(-20\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{2}q^{2}+(-\beta _{1}+\beta _{4}-\beta _{7}-2\beta _{8}+\cdots)q^{4}+\cdots\)
405.3.l.k 405.l 45.k $16$ $11.035$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 405.3.g.c \(-2\) \(0\) \(-2\) \(-26\) $\mathrm{SU}(2)[C_{12}]$ \(q-\beta _{1}q^{2}+(\beta _{3}+3\beta _{5}-\beta _{6}+\beta _{12})q^{4}+\cdots\)
405.3.l.l 405.l 45.k $16$ $11.035$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 405.3.g.c \(2\) \(0\) \(2\) \(-26\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{1}q^{2}+(\beta _{3}+3\beta _{5}-\beta _{6}+\beta _{12})q^{4}+\cdots\)
405.3.l.m 405.l 45.k $16$ $11.035$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 405.3.g.f \(6\) \(0\) \(-12\) \(-20\) $\mathrm{SU}(2)[C_{12}]$ \(q+(\beta _{1}-\beta _{2})q^{2}+(\beta _{1}-\beta _{3}-\beta _{5}+\beta _{7}+\cdots)q^{4}+\cdots\)
405.3.l.n 405.l 45.k $32$ $11.035$ None 135.3.g.b \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{12}]$
405.3.l.o 405.l 45.k $32$ $11.035$ None 135.3.g.a \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{12}]$

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

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