Properties

Label 338.6.b
Level $338$
Weight $6$
Character orbit 338.b
Rep. character $\chi_{338}(337,\cdot)$
Character field $\Q$
Dimension $66$
Newform subspaces $8$
Sturm bound $273$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(273\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(338, [\chi])\).

Total New Old
Modular forms 242 66 176
Cusp forms 214 66 148
Eisenstein series 28 0 28

Trace form

\( 66 q + 18 q^{3} - 1056 q^{4} + 6272 q^{9} + O(q^{10}) \) \( 66 q + 18 q^{3} - 1056 q^{4} + 6272 q^{9} + 136 q^{10} - 288 q^{12} - 272 q^{14} + 16896 q^{16} - 1816 q^{17} + 3352 q^{22} - 4464 q^{23} - 41676 q^{25} - 7008 q^{27} + 15686 q^{29} + 5776 q^{30} + 35492 q^{35} - 100352 q^{36} + 856 q^{38} - 2176 q^{40} + 26768 q^{42} + 5686 q^{43} + 4608 q^{48} - 177822 q^{49} - 48284 q^{51} - 72150 q^{53} + 69080 q^{55} + 4352 q^{56} + 152658 q^{61} - 90048 q^{62} - 270336 q^{64} - 3680 q^{66} + 29056 q^{68} - 19452 q^{69} + 10904 q^{74} + 111678 q^{75} + 162372 q^{77} - 8724 q^{79} + 456554 q^{81} + 142096 q^{82} - 45956 q^{87} - 53632 q^{88} - 137400 q^{90} + 71424 q^{92} - 202144 q^{94} - 325780 q^{95} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(338, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
338.6.b.a 338.b 13.b $2$ $54.210$ \(\Q(\sqrt{-1}) \) None 26.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}-2^{4}q^{4}-7iq^{5}+85iq^{7}+\cdots\)
338.6.b.b 338.b 13.b $4$ $54.210$ \(\Q(i, \sqrt{849})\) None 26.6.a.c \(0\) \(18\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{2}q^{2}+(5-\beta _{3})q^{3}-2^{4}q^{4}+(3\beta _{1}+\cdots)q^{5}+\cdots\)
338.6.b.c 338.b 13.b $4$ $54.210$ \(\Q(i, \sqrt{2785})\) None 26.6.a.b \(0\) \(18\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{2}q^{2}+(4+\beta _{3})q^{3}-2^{4}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
338.6.b.d 338.b 13.b $6$ $54.210$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 26.6.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{2}q^{2}+\beta _{3}q^{3}-2^{4}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
338.6.b.e 338.b 13.b $8$ $54.210$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 26.6.c.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{2}q^{2}-\beta _{3}q^{3}-2^{4}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
338.6.b.f 338.b 13.b $12$ $54.210$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 338.6.a.l \(0\) \(-72\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4\beta _{6}q^{2}+(-6+\beta _{4})q^{3}-2^{4}q^{4}+\cdots\)
338.6.b.g 338.b 13.b $12$ $54.210$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 26.6.e.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{1}q^{3}-2^{4}q^{4}+(\beta _{2}+4\beta _{3}+\cdots)q^{5}+\cdots\)
338.6.b.h 338.b 13.b $18$ $54.210$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None 338.6.a.p \(0\) \(54\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{10}q^{2}+(3+\beta _{2})q^{3}-2^{4}q^{4}+(-\beta _{9}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(338, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(338, [\chi]) \simeq \) \(S_{6}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)