Properties

Label 338.4.a
Level $338$
Weight $4$
Character orbit 338.a
Rep. character $\chi_{338}(1,\cdot)$
Character field $\Q$
Dimension $39$
Newform subspaces $15$
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.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(182\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(338))\).

Total New Old
Modular forms 151 39 112
Cusp forms 123 39 84
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
\(+\)\(+\)\(+\)\(11\)
\(+\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(12\)
Plus space\(+\)\(23\)
Minus space\(-\)\(16\)

Trace form

\( 39 q - 2 q^{2} - 6 q^{3} + 156 q^{4} - 10 q^{5} - 4 q^{7} - 8 q^{8} + 433 q^{9} + O(q^{10}) \) \( 39 q - 2 q^{2} - 6 q^{3} + 156 q^{4} - 10 q^{5} - 4 q^{7} - 8 q^{8} + 433 q^{9} + 24 q^{10} + 84 q^{11} - 24 q^{12} + 88 q^{14} + 56 q^{15} + 624 q^{16} + 6 q^{17} + 38 q^{18} - 168 q^{19} - 40 q^{20} - 172 q^{21} - 12 q^{22} + 44 q^{23} + 755 q^{25} + 372 q^{27} - 16 q^{28} - 116 q^{29} + 328 q^{30} - 212 q^{31} - 32 q^{32} + 308 q^{33} - 196 q^{34} + 684 q^{35} + 1732 q^{36} + 198 q^{37} - 100 q^{38} + 96 q^{40} + 358 q^{41} - 248 q^{42} - 642 q^{43} + 336 q^{44} + 442 q^{45} - 296 q^{46} + 372 q^{47} - 96 q^{48} + 1295 q^{49} - 734 q^{50} - 88 q^{51} - 968 q^{53} - 72 q^{54} - 240 q^{55} + 352 q^{56} - 112 q^{57} - 884 q^{58} - 48 q^{59} + 224 q^{60} + 316 q^{61} - 16 q^{62} - 348 q^{63} + 2496 q^{64} + 656 q^{66} + 1036 q^{67} + 24 q^{68} - 260 q^{69} + 2328 q^{70} + 1228 q^{71} + 152 q^{72} + 798 q^{73} - 1064 q^{74} - 162 q^{75} - 672 q^{76} + 2436 q^{77} - 1952 q^{79} - 160 q^{80} + 4015 q^{81} + 2228 q^{82} - 948 q^{83} - 688 q^{84} + 2072 q^{85} + 512 q^{86} + 1604 q^{87} - 48 q^{88} - 1562 q^{89} - 984 q^{90} + 176 q^{92} - 1056 q^{93} + 240 q^{94} - 3112 q^{95} + 2794 q^{97} - 1842 q^{98} - 1160 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(338))\) 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
338.4.a.a 338.a 1.a $1$ $19.943$ \(\Q\) None 26.4.c.a \(-2\) \(-3\) \(2\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}+2q^{5}+6q^{6}+\cdots\)
338.4.a.b 338.a 1.a $1$ $19.943$ \(\Q\) None 26.4.a.b \(-2\) \(-1\) \(-17\) \(35\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+4q^{4}-17q^{5}+2q^{6}+\cdots\)
338.4.a.c 338.a 1.a $1$ $19.943$ \(\Q\) None 26.4.a.c \(-2\) \(4\) \(18\) \(-20\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{3}+4q^{4}+18q^{5}-8q^{6}+\cdots\)
338.4.a.d 338.a 1.a $1$ $19.943$ \(\Q\) None 26.4.c.a \(2\) \(-3\) \(-2\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-2q^{5}-6q^{6}+\cdots\)
338.4.a.e 338.a 1.a $1$ $19.943$ \(\Q\) None 26.4.a.a \(2\) \(3\) \(-11\) \(-19\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}-11q^{5}+6q^{6}+\cdots\)
338.4.a.f 338.a 1.a $2$ $19.943$ \(\Q(\sqrt{217}) \) None 26.4.b.a \(-4\) \(-3\) \(-19\) \(-7\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1-\beta )q^{3}+4q^{4}+(-9+\cdots)q^{5}+\cdots\)
338.4.a.g 338.a 1.a $2$ $19.943$ \(\Q(\sqrt{217}) \) None 26.4.c.b \(-4\) \(-3\) \(7\) \(45\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-1-\beta )q^{3}+4q^{4}+(4-\beta )q^{5}+\cdots\)
338.4.a.h 338.a 1.a $2$ $19.943$ \(\Q(\sqrt{217}) \) None 26.4.c.b \(4\) \(-3\) \(-7\) \(-45\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1-\beta )q^{3}+4q^{4}+(-4+\cdots)q^{5}+\cdots\)
338.4.a.i 338.a 1.a $2$ $19.943$ \(\Q(\sqrt{217}) \) None 26.4.b.a \(4\) \(-3\) \(19\) \(7\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-1-\beta )q^{3}+4q^{4}+(9+\beta )q^{5}+\cdots\)
338.4.a.j 338.a 1.a $3$ $19.943$ \(\Q(\zeta_{14})^+\) None 338.4.a.j \(-6\) \(-12\) \(12\) \(-27\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(-5-3\beta _{2})q^{3}+4q^{4}+(1+\cdots)q^{5}+\cdots\)
338.4.a.k 338.a 1.a $3$ $19.943$ \(\Q(\zeta_{14})^+\) None 338.4.a.j \(6\) \(-12\) \(-12\) \(27\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+(-5-3\beta _{2})q^{3}+4q^{4}+(-1+\cdots)q^{5}+\cdots\)
338.4.a.l 338.a 1.a $4$ $19.943$ 4.4.1859472.2 None 26.4.e.a \(-8\) \(6\) \(4\) \(-20\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
338.4.a.m 338.a 1.a $4$ $19.943$ 4.4.1859472.2 None 26.4.e.a \(8\) \(6\) \(-4\) \(20\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
338.4.a.n 338.a 1.a $6$ $19.943$ 6.6.6681389953.1 None 338.4.a.n \(-12\) \(9\) \(-18\) \(-25\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(-3+\beta _{1}+\cdots)q^{5}+\cdots\)
338.4.a.o 338.a 1.a $6$ $19.943$ 6.6.6681389953.1 None 338.4.a.n \(12\) \(9\) \(18\) \(25\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(3-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(338))\) into lower level spaces

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