Properties

Label 1368.2.a
Level $1368$
Weight $2$
Character orbit 1368.a
Rep. character $\chi_{1368}(1,\cdot)$
Character field $\Q$
Dimension $23$
Newform subspaces $15$
Sturm bound $480$
Trace bound $11$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1368.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(480\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1368))\).

Total New Old
Modular forms 256 23 233
Cusp forms 225 23 202
Eisenstein series 31 0 31

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.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(13\)

Trace form

\( 23 q - 4 q^{5} - 4 q^{7} + O(q^{10}) \) \( 23 q - 4 q^{5} - 4 q^{7} + 6 q^{11} - 2 q^{13} - 6 q^{17} - 3 q^{19} - 6 q^{23} + 35 q^{25} + 14 q^{29} - 12 q^{31} - 6 q^{35} - 2 q^{37} - 2 q^{41} - 14 q^{43} - 6 q^{47} + 51 q^{49} + 10 q^{53} - 10 q^{55} - 8 q^{59} + 16 q^{61} + 16 q^{65} - 20 q^{67} + 8 q^{71} + 10 q^{73} + 30 q^{77} - 12 q^{79} + 4 q^{83} + 14 q^{85} + 30 q^{89} - 16 q^{91} + 26 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\) 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
1368.2.a.a 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(-4\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{5}+6q^{11}+2q^{13}-4q^{17}-q^{19}+\cdots\)
1368.2.a.b 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(-4\) \(4\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{5}+4q^{7}+4q^{11}-4q^{13}-6q^{17}+\cdots\)
1368.2.a.c 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}+2q^{13}-2q^{17}-q^{19}-q^{25}+\cdots\)
1368.2.a.d 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(-1\) \(-3\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{7}+5q^{11}-2q^{13}+q^{17}+\cdots\)
1368.2.a.e 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{11}-2q^{13}-4q^{17}+q^{19}-2q^{23}+\cdots\)
1368.2.a.f 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{11}-2q^{13}+4q^{17}+q^{19}+2q^{23}+\cdots\)
1368.2.a.g 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{7}-2q^{11}+q^{13}+5q^{17}+q^{19}+\cdots\)
1368.2.a.h 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(1\) \(-3\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{7}+3q^{11}-4q^{13}-5q^{17}+\cdots\)
1368.2.a.i 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(3\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{5}-3q^{7}+q^{11}-2q^{13}+5q^{17}+\cdots\)
1368.2.a.j 1368.a 1.a $1$ $10.924$ \(\Q\) None \(0\) \(0\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+4q^{5}-6q^{11}+2q^{13}+4q^{17}-q^{19}+\cdots\)
1368.2.a.k 1368.a 1.a $2$ $10.924$ \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(-1\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{5}+\beta q^{7}+(-4+\beta )q^{11}-2\beta q^{13}+\cdots\)
1368.2.a.l 1368.a 1.a $2$ $10.924$ \(\Q(\sqrt{41}) \) None \(0\) \(0\) \(1\) \(-3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+(-2+\beta )q^{7}+(-2+\beta )q^{11}+\cdots\)
1368.2.a.m 1368.a 1.a $3$ $10.924$ 3.3.892.1 None \(0\) \(0\) \(-2\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1368.2.a.n 1368.a 1.a $3$ $10.924$ 3.3.961.1 None \(0\) \(0\) \(-1\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{5}+(1-\beta _{2})q^{7}+(2-\beta _{1})q^{11}+\cdots\)
1368.2.a.o 1368.a 1.a $3$ $10.924$ 3.3.892.1 None \(0\) \(0\) \(2\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1368))\) into lower level spaces

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