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Properties

Label 1368.4.a
Level $1368$
Weight $4$
Character orbit 1368.a
Rep. character $\chi_{1368}(1,\cdot)$
Character field $\Q$
Dimension $67$
Newform subspaces $16$
Sturm bound $960$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1368.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(960\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1368))\).

Total New Old
Modular forms 736 67 669
Cusp forms 704 67 637
Eisenstein series 32 0 32

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

\(2\)\(3\)\(19\)FrickeDim
\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(10\)
\(+\)\(-\)\(-\)$+$\(10\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(8\)
\(-\)\(-\)\(+\)$+$\(12\)
\(-\)\(-\)\(-\)$-$\(9\)
Plus space\(+\)\(35\)
Minus space\(-\)\(32\)

Trace form

\( 67 q + 24 q^{5} + 12 q^{7} + O(q^{10}) \) \( 67 q + 24 q^{5} + 12 q^{7} - 66 q^{11} - 66 q^{13} + 154 q^{17} + 57 q^{19} + 42 q^{23} + 1379 q^{25} - 194 q^{29} - 60 q^{31} - 606 q^{35} + 206 q^{37} - 474 q^{41} + 570 q^{43} + 1602 q^{47} + 3303 q^{49} - 1694 q^{53} - 1866 q^{55} - 280 q^{59} + 148 q^{61} - 616 q^{65} + 620 q^{67} + 2312 q^{71} + 1514 q^{73} - 574 q^{77} - 1548 q^{79} - 2156 q^{83} - 1462 q^{85} - 618 q^{89} - 1424 q^{91} - 1014 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1368))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 19
1368.4.a.a 1368.a 1.a $2$ $80.715$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(5\) \(-6\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta )q^{5}+(-5+4\beta )q^{7}+(11-7\beta )q^{11}+\cdots\)
1368.4.a.b 1368.a 1.a $2$ $80.715$ \(\Q(\sqrt{105}) \) None \(0\) \(0\) \(9\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(5-\beta )q^{5}+(1+\beta )q^{7}+(-11+5\beta )q^{11}+\cdots\)
1368.4.a.c 1368.a 1.a $2$ $80.715$ \(\Q(\sqrt{97}) \) None \(0\) \(0\) \(13\) \(3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(7-\beta )q^{5}+(3-3\beta )q^{7}+(19+3\beta )q^{11}+\cdots\)
1368.4.a.d 1368.a 1.a $3$ $80.715$ 3.3.7057.1 None \(0\) \(0\) \(-7\) \(7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-2\beta _{1}-\beta _{2})q^{5}+(2+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1368.4.a.e 1368.a 1.a $3$ $80.715$ 3.3.3221.1 None \(0\) \(0\) \(-2\) \(-35\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}-2\beta _{2})q^{5}+(-11+2\beta _{1}+\cdots)q^{7}+\cdots\)
1368.4.a.f 1368.a 1.a $3$ $80.715$ 3.3.24665.1 None \(0\) \(0\) \(1\) \(7\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{5}+(3+2\beta _{1}-3\beta _{2})q^{7}+(-5+\cdots)q^{11}+\cdots\)
1368.4.a.g 1368.a 1.a $4$ $80.715$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(-9\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1368.4.a.h 1368.a 1.a $4$ $80.715$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(-4\) \(-14\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{5}+(-3+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1368.4.a.i 1368.a 1.a $4$ $80.715$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(8\) \(10\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{5}+(3+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1368.4.a.j 1368.a 1.a $4$ $80.715$ 4.4.4914253.1 None \(0\) \(0\) \(16\) \(-14\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(4-\beta _{2}-\beta _{3})q^{5}+(-4-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
1368.4.a.k 1368.a 1.a $5$ $80.715$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-6\) \(10\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{5}+(2+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1368.4.a.l 1368.a 1.a $5$ $80.715$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-6\) \(10\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{5}+(2-\beta _{2})q^{7}+(7-\beta _{1}+\cdots)q^{11}+\cdots\)
1368.4.a.m 1368.a 1.a $5$ $80.715$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(0\) \(22\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2})q^{5}+(4+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1368.4.a.n 1368.a 1.a $5$ $80.715$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(6\) \(10\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{5}+(2+\beta _{2}-\beta _{3})q^{7}+(5+\cdots)q^{11}+\cdots\)
1368.4.a.o 1368.a 1.a $8$ $80.715$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(-16\) \(-4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(-2+\cdots)q^{11}+\cdots\)
1368.4.a.p 1368.a 1.a $8$ $80.715$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(16\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(2+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1368))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1368)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(456))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(684))\)\(^{\oplus 2}\)