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Properties

Label 336.3.r
Level $336$
Weight $3$
Character orbit 336.r
Rep. character $\chi_{336}(13,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $128$
Newform subspaces $2$
Sturm bound $192$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 336.r (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 112 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(192\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(336, [\chi])\).

Total New Old
Modular forms 264 128 136
Cusp forms 248 128 120
Eisenstein series 16 0 16

Trace form

\( 128 q + 12 q^{4} + 24 q^{8} + O(q^{10}) \) \( 128 q + 12 q^{4} + 24 q^{8} - 32 q^{11} + 16 q^{14} + 4 q^{16} - 12 q^{18} + 60 q^{22} - 32 q^{29} + 96 q^{35} + 96 q^{37} + 60 q^{42} - 96 q^{43} + 300 q^{44} + 360 q^{46} + 452 q^{50} - 160 q^{53} - 28 q^{56} - 468 q^{58} - 72 q^{60} - 660 q^{64} - 544 q^{67} + 72 q^{70} - 60 q^{72} + 748 q^{74} - 576 q^{78} - 1152 q^{81} + 72 q^{84} - 604 q^{86} - 724 q^{88} + 480 q^{91} + 1160 q^{92} + 1536 q^{95} - 896 q^{98} + 96 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(336, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
336.3.r.a 336.r 112.l $4$ $9.155$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-2\beta _{2}q^{2}-\beta _{3}q^{3}-4q^{4}+4\beta _{1}q^{5}+\cdots\)
336.3.r.b 336.r 112.l $124$ $9.155$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$

Decomposition of \(S_{3}^{\mathrm{old}}(336, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(336, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 2}\)