Properties

Label 338.4.e
Level $338$
Weight $4$
Character orbit 338.e
Rep. character $\chi_{338}(23,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $76$
Newform subspaces $9$
Sturm bound $182$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.e (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 9 \)
Sturm bound: \(182\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(338, [\chi])\).

Total New Old
Modular forms 300 76 224
Cusp forms 244 76 168
Eisenstein series 56 0 56

Trace form

\( 76 q + 6 q^{3} + 152 q^{4} - 18 q^{7} - 356 q^{9} + O(q^{10}) \) \( 76 q + 6 q^{3} + 152 q^{4} - 18 q^{7} - 356 q^{9} + 8 q^{10} + 18 q^{11} + 48 q^{12} - 40 q^{14} + 192 q^{15} - 608 q^{16} - 110 q^{17} - 594 q^{19} - 72 q^{20} + 100 q^{22} + 318 q^{23} - 2048 q^{25} - 192 q^{27} - 72 q^{28} + 22 q^{29} - 244 q^{30} + 42 q^{33} + 190 q^{35} + 1424 q^{36} + 768 q^{37} + 824 q^{38} + 64 q^{40} + 564 q^{41} + 820 q^{42} - 38 q^{43} - 630 q^{45} - 576 q^{46} + 96 q^{48} + 2362 q^{49} - 1968 q^{50} - 1384 q^{51} - 384 q^{53} - 648 q^{54} - 1488 q^{55} - 80 q^{56} + 1848 q^{58} + 1110 q^{59} - 1046 q^{61} - 1824 q^{62} + 1980 q^{63} - 4864 q^{64} + 3392 q^{66} - 510 q^{67} + 440 q^{68} + 2730 q^{69} + 1470 q^{71} - 576 q^{72} + 376 q^{74} + 1320 q^{75} - 2376 q^{76} - 3708 q^{77} - 2024 q^{79} - 288 q^{80} - 6302 q^{81} - 440 q^{82} + 2136 q^{84} + 2898 q^{85} - 2410 q^{87} - 400 q^{88} + 4434 q^{89} + 4536 q^{90} + 2544 q^{92} - 3108 q^{93} + 2908 q^{94} + 580 q^{95} - 1854 q^{97} - 4272 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(338, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
338.4.e.a 338.e 13.e $4$ $19.943$ \(\Q(\zeta_{12})\) None 26.4.a.c \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(-4+4\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots\)
338.4.e.b 338.e 13.e $4$ $19.943$ \(\Q(\zeta_{12})\) None 26.4.a.a \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+2\zeta_{12}q^{2}+(-3+3\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots\)
338.4.e.c 338.e 13.e $4$ $19.943$ \(\Q(\zeta_{12})\) None 26.4.a.b \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+2\zeta_{12}q^{2}+(1-\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots\)
338.4.e.d 338.e 13.e $4$ $19.943$ \(\Q(\zeta_{12})\) None 26.4.c.a \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+2\zeta_{12}q^{2}+(3-3\zeta_{12}^{2})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots\)
338.4.e.e 338.e 13.e $8$ $19.943$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 26.4.e.a \(0\) \(-6\) \(0\) \(-18\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(-2+2\beta _{1}+\beta _{4}-\beta _{7})q^{3}+\cdots\)
338.4.e.f 338.e 13.e $8$ $19.943$ 8.0.\(\cdots\).49 None 26.4.c.b \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-2\beta _{6}q^{2}+(2-\beta _{2}+\beta _{4}+\beta _{5})q^{3}+\cdots\)
338.4.e.g 338.e 13.e $8$ $19.943$ 8.0.\(\cdots\).49 None 26.4.b.a \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{7}q^{2}+(1-2\beta _{2}-\beta _{4}-\beta _{6})q^{3}+\cdots\)
338.4.e.h 338.e 13.e $12$ $19.943$ 12.0.\(\cdots\).1 None 338.4.a.j \(0\) \(24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+2\beta _{6}q^{2}+(2-3\beta _{4}-2\beta _{7}-3\beta _{9}+\cdots)q^{3}+\cdots\)
338.4.e.i 338.e 13.e $24$ $19.943$ None 338.4.a.n \(0\) \(-18\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{4}^{\mathrm{old}}(338, [\chi])\) into lower level spaces

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