Properties

Label 420.2.l
Level $420$
Weight $2$
Character orbit 420.l
Rep. character $\chi_{420}(239,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $8$
Sturm bound $192$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 420.l (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 60 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(17\), \(43\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(420, [\chi])\).

Total New Old
Modular forms 104 72 32
Cusp forms 88 72 16
Eisenstein series 16 0 16

Trace form

\( 72 q + 4 q^{4} + O(q^{10}) \) \( 72 q + 4 q^{4} + 8 q^{10} - 12 q^{16} - 44 q^{24} - 24 q^{30} - 32 q^{34} + 4 q^{36} - 8 q^{40} - 24 q^{45} - 24 q^{46} + 72 q^{49} + 68 q^{54} + 52 q^{60} - 64 q^{61} + 52 q^{64} - 76 q^{66} - 48 q^{69} + 12 q^{70} - 24 q^{76} - 16 q^{81} + 24 q^{85} - 56 q^{90} - 152 q^{94} + 60 q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(420, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
420.2.l.a 420.l 60.h $4$ $3.354$ \(\Q(\zeta_{8})\) None \(0\) \(-4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}^{2}q^{2}+(-1+\zeta_{8}^{2})q^{3}-2q^{4}+\cdots\)
420.2.l.b 420.l 60.h $4$ $3.354$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}+(1-\zeta_{8}^{2})q^{3}-2q^{4}+(\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
420.2.l.c 420.l 60.h $8$ $3.354$ 8.0.386672896.3 None \(0\) \(-2\) \(-8\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.d 420.l 60.h $8$ $3.354$ 8.0.386672896.3 None \(0\) \(-2\) \(8\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}-\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.e 420.l 60.h $8$ $3.354$ 8.0.386672896.3 None \(0\) \(2\) \(-8\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+\beta _{6}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.f 420.l 60.h $8$ $3.354$ 8.0.386672896.3 None \(0\) \(2\) \(8\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}-\beta _{2}q^{3}+(1-\beta _{4}-\beta _{6})q^{4}+\cdots\)
420.2.l.g 420.l 60.h $16$ $3.354$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+\beta _{4}q^{3}+\beta _{11}q^{4}+\beta _{5}q^{5}+\cdots\)
420.2.l.h 420.l 60.h $16$ $3.354$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(16\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+\beta _{6}q^{3}+\beta _{11}q^{4}+(-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(420, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(420, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 2}\)