Properties

Label 338.4.c
Level $338$
Weight $4$
Character orbit 338.c
Rep. character $\chi_{338}(191,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $78$
Newform subspaces $16$
Sturm bound $182$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 338.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 16 \)
Sturm bound: \(182\)
Trace bound: \(5\)
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 302 78 224
Cusp forms 246 78 168
Eisenstein series 56 0 56

Trace form

\( 78 q + 2 q^{2} - 6 q^{3} - 156 q^{4} + 10 q^{5} - 50 q^{7} - 16 q^{8} - 337 q^{9} + O(q^{10}) \) \( 78 q + 2 q^{2} - 6 q^{3} - 156 q^{4} + 10 q^{5} - 50 q^{7} - 16 q^{8} - 337 q^{9} - 18 q^{10} + 18 q^{11} + 48 q^{12} + 200 q^{14} - 104 q^{15} - 624 q^{16} - 105 q^{17} - 308 q^{18} + 78 q^{19} - 20 q^{20} + 52 q^{21} + 12 q^{22} - 242 q^{23} + 2704 q^{25} + 672 q^{27} - 200 q^{28} - 157 q^{29} - 268 q^{30} - 352 q^{31} + 32 q^{32} - 806 q^{33} + 628 q^{34} + 318 q^{35} - 1348 q^{36} - 279 q^{37} + 40 q^{38} + 144 q^{40} + 437 q^{41} + 20 q^{42} + 258 q^{43} - 144 q^{44} + 83 q^{45} + 440 q^{46} - 1224 q^{47} - 96 q^{48} - 2135 q^{49} + 8 q^{50} - 1880 q^{51} - 514 q^{53} - 936 q^{54} + 528 q^{55} - 400 q^{56} + 676 q^{57} - 370 q^{58} + 102 q^{59} + 832 q^{60} + 683 q^{61} + 1528 q^{62} - 912 q^{63} + 4992 q^{64} + 3904 q^{66} + 710 q^{67} - 420 q^{68} - 142 q^{69} - 1104 q^{70} - 490 q^{71} + 616 q^{72} + 786 q^{73} - 2554 q^{74} + 840 q^{75} + 312 q^{76} + 204 q^{77} - 4288 q^{79} - 80 q^{80} - 4687 q^{81} - 170 q^{82} + 4992 q^{83} - 104 q^{84} - 359 q^{85} + 1168 q^{86} + 826 q^{87} + 48 q^{88} - 1912 q^{89} - 2772 q^{90} + 1936 q^{92} + 2496 q^{93} + 12 q^{94} + 484 q^{95} - 976 q^{97} + 1506 q^{98} - 4384 q^{99} + O(q^{100}) \)

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.c.a 338.c 13.c $2$ $19.943$ \(\Q(\sqrt{-3}) \) None 26.4.a.c \(-2\) \(-4\) \(-36\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}+(-4+4\zeta_{6})q^{3}+\cdots\)
338.4.c.b 338.c 13.c $2$ $19.943$ \(\Q(\sqrt{-3}) \) None 26.4.a.a \(-2\) \(-3\) \(-22\) \(19\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\cdots\)
338.4.c.c 338.c 13.c $2$ $19.943$ \(\Q(\sqrt{-3}) \) None 26.4.a.b \(-2\) \(1\) \(34\) \(35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
338.4.c.d 338.c 13.c $2$ $19.943$ \(\Q(\sqrt{-3}) \) None 26.4.c.a \(-2\) \(3\) \(-4\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}+(3-3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
338.4.c.e 338.c 13.c $2$ $19.943$ \(\Q(\sqrt{-3}) \) None 26.4.a.c \(2\) \(-4\) \(36\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+(-4+4\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
338.4.c.f 338.c 13.c $2$ $19.943$ \(\Q(\sqrt{-3}) \) None 26.4.a.a \(2\) \(-3\) \(22\) \(-19\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
338.4.c.g 338.c 13.c $2$ $19.943$ \(\Q(\sqrt{-3}) \) None 26.4.a.b \(2\) \(1\) \(-34\) \(-35\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-4\zeta_{6}q^{4}+\cdots\)
338.4.c.h 338.c 13.c $4$ $19.943$ \(\Q(\sqrt{-3}, \sqrt{217})\) None 26.4.b.a \(-4\) \(3\) \(38\) \(-7\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots\)
338.4.c.i 338.c 13.c $4$ $19.943$ \(\Q(\sqrt{-3}, \sqrt{217})\) None 26.4.b.a \(4\) \(3\) \(-38\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots\)
338.4.c.j 338.c 13.c $4$ $19.943$ \(\Q(\sqrt{-3}, \sqrt{217})\) None 26.4.c.b \(4\) \(3\) \(14\) \(-45\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{2}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots\)
338.4.c.k 338.c 13.c $6$ $19.943$ 6.0.64827.1 None 338.4.a.j \(-6\) \(12\) \(-24\) \(-27\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\beta _{5})q^{2}+(5-3\beta _{4}-5\beta _{5})q^{3}+\cdots\)
338.4.c.l 338.c 13.c $6$ $19.943$ 6.0.64827.1 None 338.4.a.j \(6\) \(12\) \(24\) \(27\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{5})q^{2}+(5-3\beta _{4}-5\beta _{5})q^{3}+\cdots\)
338.4.c.m 338.c 13.c $8$ $19.943$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 26.4.e.a \(-8\) \(-6\) \(-8\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{2})q^{2}+(-2-\beta _{2}+\beta _{3}+\cdots)q^{3}+\cdots\)
338.4.c.n 338.c 13.c $8$ $19.943$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None 26.4.e.a \(8\) \(-6\) \(8\) \(20\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2+2\beta _{2})q^{2}+(-2-\beta _{2}+\beta _{3}+\beta _{7})q^{3}+\cdots\)
338.4.c.o 338.c 13.c $12$ $19.943$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 338.4.a.n \(-12\) \(-9\) \(36\) \(-25\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\beta _{1})q^{2}+(-1+2\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
338.4.c.p 338.c 13.c $12$ $19.943$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 338.4.a.n \(12\) \(-9\) \(-36\) \(25\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{1})q^{2}+(-1+2\beta _{1}+\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots\)

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

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