Properties

Label 338.6.a
Level $338$
Weight $6$
Character orbit 338.a
Rep. character $\chi_{338}(1,\cdot)$
Character field $\Q$
Dimension $65$
Newform subspaces $17$
Sturm bound $273$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 338.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(273\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(338))\).

Total New Old
Modular forms 241 65 176
Cusp forms 213 65 148
Eisenstein series 28 0 28

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(13\)FrickeDim
\(+\)\(+\)\(+\)\(15\)
\(+\)\(-\)\(-\)\(17\)
\(-\)\(+\)\(-\)\(19\)
\(-\)\(-\)\(+\)\(14\)
Plus space\(+\)\(29\)
Minus space\(-\)\(36\)

Trace form

\( 65 q + 4 q^{2} - 18 q^{3} + 1040 q^{4} - 22 q^{5} - 312 q^{7} + 64 q^{8} + 4987 q^{9} + O(q^{10}) \) \( 65 q + 4 q^{2} - 18 q^{3} + 1040 q^{4} - 22 q^{5} - 312 q^{7} + 64 q^{8} + 4987 q^{9} - 496 q^{10} + 384 q^{11} - 288 q^{12} - 80 q^{14} + 2504 q^{15} + 16640 q^{16} + 150 q^{17} + 2900 q^{18} + 1332 q^{19} - 352 q^{20} + 3044 q^{21} + 1416 q^{22} + 2312 q^{23} + 50493 q^{25} - 18672 q^{27} - 4992 q^{28} - 2024 q^{29} - 752 q^{30} + 6368 q^{31} + 1024 q^{32} + 4340 q^{33} + 3080 q^{34} - 1212 q^{35} + 79792 q^{36} - 42102 q^{37} + 14552 q^{38} - 7936 q^{40} - 734 q^{41} + 26032 q^{42} + 2630 q^{43} + 6144 q^{44} + 38878 q^{45} + 7728 q^{46} + 64560 q^{47} - 4608 q^{48} + 99973 q^{49} - 29348 q^{50} + 48212 q^{51} - 10868 q^{53} + 52272 q^{54} + 26560 q^{55} - 1280 q^{56} + 77648 q^{57} + 9288 q^{58} + 32292 q^{59} + 40064 q^{60} - 37740 q^{61} - 46240 q^{62} - 147600 q^{63} + 266240 q^{64} - 119584 q^{66} - 41680 q^{67} + 2400 q^{68} + 76708 q^{69} - 77584 q^{70} + 14176 q^{71} + 46400 q^{72} - 50422 q^{73} + 35728 q^{74} - 26898 q^{75} + 21312 q^{76} - 34956 q^{77} - 85748 q^{79} - 5632 q^{80} + 287017 q^{81} - 102312 q^{82} + 202896 q^{83} + 48704 q^{84} - 136096 q^{85} + 54656 q^{86} + 253412 q^{87} + 22656 q^{88} - 28670 q^{89} - 13584 q^{90} + 36992 q^{92} + 263712 q^{93} - 78752 q^{94} + 152300 q^{95} - 136986 q^{97} + 82212 q^{98} - 70148 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(338))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 13
338.6.a.a 338.a 1.a $1$ $54.210$ \(\Q\) None 26.6.b.a \(-4\) \(-13\) \(51\) \(-105\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-13q^{3}+2^{4}q^{4}+51q^{5}+52q^{6}+\cdots\)
338.6.a.b 338.a 1.a $1$ $54.210$ \(\Q\) None 26.6.b.b \(-4\) \(4\) \(-68\) \(82\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+4q^{3}+2^{4}q^{4}-68q^{5}-2^{4}q^{6}+\cdots\)
338.6.a.c 338.a 1.a $1$ $54.210$ \(\Q\) None 26.6.b.a \(4\) \(-13\) \(-51\) \(105\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-13q^{3}+2^{4}q^{4}-51q^{5}-52q^{6}+\cdots\)
338.6.a.d 338.a 1.a $1$ $54.210$ \(\Q\) None 26.6.a.a \(4\) \(0\) \(14\) \(170\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+14q^{5}+170q^{7}+\cdots\)
338.6.a.e 338.a 1.a $1$ $54.210$ \(\Q\) None 26.6.b.b \(4\) \(4\) \(68\) \(-82\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+4q^{3}+2^{4}q^{4}+68q^{5}+2^{4}q^{6}+\cdots\)
338.6.a.f 338.a 1.a $2$ $54.210$ \(\Q(\sqrt{849}) \) None 26.6.a.c \(-8\) \(9\) \(-73\) \(-155\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(5-\beta )q^{3}+2^{4}q^{4}+(-35+\cdots)q^{5}+\cdots\)
338.6.a.g 338.a 1.a $2$ $54.210$ \(\Q(\sqrt{2785}) \) None 26.6.a.b \(8\) \(9\) \(37\) \(-327\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(5-\beta )q^{3}+2^{4}q^{4}+(19-\beta )q^{5}+\cdots\)
338.6.a.h 338.a 1.a $3$ $54.210$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 26.6.c.a \(-12\) \(0\) \(-1\) \(-202\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(-\beta _{1}+\beta _{2})q^{5}+\cdots\)
338.6.a.i 338.a 1.a $3$ $54.210$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 26.6.c.a \(12\) \(0\) \(1\) \(202\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(\beta _{1}-\beta _{2})q^{5}+\cdots\)
338.6.a.j 338.a 1.a $4$ $54.210$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 26.6.c.b \(-16\) \(0\) \(-12\) \(-126\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+(-3-\beta _{1}+\cdots)q^{5}+\cdots\)
338.6.a.k 338.a 1.a $4$ $54.210$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 26.6.c.b \(16\) \(0\) \(12\) \(126\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+(3+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
338.6.a.l 338.a 1.a $6$ $54.210$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 338.6.a.l \(-24\) \(-36\) \(124\) \(116\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+(-6-\beta _{2})q^{3}+2^{4}q^{4}+(23+\cdots)q^{5}+\cdots\)
338.6.a.m 338.a 1.a $6$ $54.210$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 26.6.e.a \(-24\) \(0\) \(92\) \(284\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+(15-\beta _{1}+\cdots)q^{5}+\cdots\)
338.6.a.n 338.a 1.a $6$ $54.210$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 338.6.a.l \(24\) \(-36\) \(-124\) \(-116\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+(-6-\beta _{2})q^{3}+2^{4}q^{4}+(-23+\cdots)q^{5}+\cdots\)
338.6.a.o 338.a 1.a $6$ $54.210$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None 26.6.e.a \(24\) \(0\) \(-92\) \(-284\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-\beta _{1}q^{3}+2^{4}q^{4}+(-15+\beta _{1}+\cdots)q^{5}+\cdots\)
338.6.a.p 338.a 1.a $9$ $54.210$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 338.6.a.p \(-36\) \(27\) \(-62\) \(-40\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+(3+\beta _{2})q^{3}+2^{4}q^{4}+(-7+\cdots)q^{5}+\cdots\)
338.6.a.q 338.a 1.a $9$ $54.210$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None 338.6.a.p \(36\) \(27\) \(62\) \(40\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+(3+\beta _{2})q^{3}+2^{4}q^{4}+(7+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(338)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)