Properties

Label 338.8.b
Level $338$
Weight $8$
Character orbit 338.b
Rep. character $\chi_{338}(337,\cdot)$
Character field $\Q$
Dimension $88$
Newform subspaces $11$
Sturm bound $364$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 338.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(364\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(338, [\chi])\).

Total New Old
Modular forms 332 88 244
Cusp forms 304 88 216
Eisenstein series 28 0 28

Trace form

\( 88 q - 54 q^{3} - 5632 q^{4} + 57482 q^{9} + O(q^{10}) \) \( 88 q - 54 q^{3} - 5632 q^{4} + 57482 q^{9} + 1136 q^{10} + 3456 q^{12} - 5600 q^{14} + 360448 q^{16} + 17252 q^{17} - 13264 q^{22} - 130592 q^{23} - 1165546 q^{25} + 79248 q^{27} + 90702 q^{29} + 585568 q^{30} - 1181932 q^{35} - 3678848 q^{36} - 779920 q^{38} - 72704 q^{40} - 1832224 q^{42} - 2981042 q^{43} - 221184 q^{48} - 9548636 q^{49} - 1932476 q^{51} + 4648578 q^{53} - 5048840 q^{55} + 358400 q^{56} - 14202614 q^{61} + 1860416 q^{62} - 23068672 q^{64} + 7062592 q^{66} - 1104128 q^{68} + 344652 q^{69} - 20903216 q^{74} + 22539702 q^{75} + 5373020 q^{77} + 32332684 q^{79} - 5403568 q^{81} - 17895712 q^{82} - 62233460 q^{87} + 848896 q^{88} + 22099440 q^{90} + 8357888 q^{92} - 13879040 q^{94} - 4853268 q^{95} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(338, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
338.8.b.a 338.b 13.b $2$ $105.586$ \(\Q(\sqrt{-1}) \) None 26.8.a.b \(0\) \(-174\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8iq^{2}-87q^{3}-2^{6}q^{4}-321iq^{5}+\cdots\)
338.8.b.b 338.b 13.b $2$ $105.586$ \(\Q(\sqrt{-1}) \) None 26.8.a.a \(0\) \(-78\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8iq^{2}-39q^{3}-2^{6}q^{4}+385iq^{5}+\cdots\)
338.8.b.c 338.b 13.b $2$ $105.586$ \(\Q(\sqrt{-1}) \) None 26.8.a.c \(0\) \(-54\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8iq^{2}-3^{3}q^{3}-2^{6}q^{4}-245iq^{5}+\cdots\)
338.8.b.d 338.b 13.b $2$ $105.586$ \(\Q(\sqrt{-1}) \) None 2.8.a.a \(0\) \(24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-4iq^{2}+12q^{3}-2^{6}q^{4}-105iq^{5}+\cdots\)
338.8.b.e 338.b 13.b $4$ $105.586$ \(\Q(i, \sqrt{105})\) None 26.8.a.d \(0\) \(-24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{1}q^{2}+(-6+7\beta _{3})q^{3}-2^{6}q^{4}+\cdots\)
338.8.b.f 338.b 13.b $4$ $105.586$ \(\Q(i, \sqrt{2305})\) None 26.8.a.e \(0\) \(174\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4\beta _{2}q^{2}+(43+\beta _{3})q^{3}-2^{6}q^{4}+(5\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.b.g 338.b 13.b $6$ $105.586$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None 26.8.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{2}q^{2}+\beta _{3}q^{3}-2^{6}q^{4}+(2\beta _{1}-110\beta _{2}+\cdots)q^{5}+\cdots\)
338.8.b.h 338.b 13.b $8$ $105.586$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 26.8.c.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-8\beta _{2}q^{2}+\beta _{3}q^{3}-2^{6}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.b.i 338.b 13.b $16$ $105.586$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 26.8.e.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{10}q^{2}+\beta _{2}q^{3}-2^{6}q^{4}+(-\beta _{9}+\cdots)q^{5}+\cdots\)
338.8.b.j 338.b 13.b $18$ $105.586$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None 338.8.a.o \(0\) \(-138\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+8\beta _{3}q^{2}+(-7-\beta _{1}-\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
338.8.b.k 338.b 13.b $24$ $105.586$ None 338.8.a.q \(0\) \(216\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{8}^{\mathrm{old}}(338, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(338, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)