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Properties

Label 1288.2.a
Level $1288$
Weight $2$
Character orbit 1288.a
Rep. character $\chi_{1288}(1,\cdot)$
Character field $\Q$
Dimension $32$
Newform subspaces $16$
Sturm bound $384$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1288.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(384\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1288))\).

Total New Old
Modular forms 200 32 168
Cusp forms 185 32 153
Eisenstein series 15 0 15

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

\(2\)\(7\)\(23\)FrickeDim
\(+\)\(+\)\(+\)$+$\(3\)
\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(14\)
Minus space\(-\)\(18\)

Trace form

\( 32 q - 4 q^{3} + 4 q^{5} + 40 q^{9} + O(q^{10}) \) \( 32 q - 4 q^{3} + 4 q^{5} + 40 q^{9} - 4 q^{11} - 4 q^{13} + 16 q^{17} - 12 q^{19} + 24 q^{25} - 16 q^{27} - 16 q^{29} + 12 q^{35} + 28 q^{37} + 24 q^{39} + 8 q^{41} - 12 q^{43} - 4 q^{45} - 8 q^{47} + 32 q^{49} - 8 q^{51} + 4 q^{53} - 8 q^{55} + 24 q^{57} - 36 q^{59} + 20 q^{61} - 16 q^{65} - 20 q^{67} + 8 q^{71} + 32 q^{73} - 28 q^{75} - 8 q^{77} - 24 q^{79} + 48 q^{81} - 12 q^{83} - 8 q^{85} - 16 q^{87} - 8 q^{89} - 48 q^{93} - 16 q^{95} - 8 q^{97} - 20 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1288))\) 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 7 23
1288.2.a.a 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(-3\) \(-4\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-4q^{5}-q^{7}+6q^{9}+2q^{11}+\cdots\)
1288.2.a.b 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(-3\) \(0\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{7}+6q^{9}-6q^{11}+q^{13}+\cdots\)
1288.2.a.c 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(-2\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{7}+q^{9}-6q^{17}+6q^{19}+\cdots\)
1288.2.a.d 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-q^{7}-3q^{9}-4q^{13}-4q^{17}+\cdots\)
1288.2.a.e 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}-2q^{9}-6q^{11}-3q^{13}+\cdots\)
1288.2.a.f 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(2\) \(0\) \(-1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1288.2.a.g 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(2\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{5}+q^{7}+q^{9}+2q^{11}+\cdots\)
1288.2.a.h 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(2\) \(4\) \(1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{5}+q^{7}+q^{9}+4q^{11}+\cdots\)
1288.2.a.i 1288.a 1.a $1$ $10.285$ \(\Q\) None \(0\) \(3\) \(2\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+2q^{5}-q^{7}+6q^{9}+2q^{11}+\cdots\)
1288.2.a.j 1288.a 1.a $2$ $10.285$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-4\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-2q^{5}-q^{7}+(1+\beta )q^{9}+2\beta q^{11}+\cdots\)
1288.2.a.k 1288.a 1.a $2$ $10.285$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-4\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(-2-\beta )q^{5}-q^{7}-q^{9}+2q^{11}+\cdots\)
1288.2.a.l 1288.a 1.a $3$ $10.285$ 3.3.148.1 None \(0\) \(0\) \(4\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(2-2\beta _{1}-\beta _{2})q^{5}-q^{7}+\cdots\)
1288.2.a.m 1288.a 1.a $3$ $10.285$ 3.3.568.1 None \(0\) \(1\) \(0\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{5}+q^{7}+(1+\cdots)q^{9}+\cdots\)
1288.2.a.n 1288.a 1.a $4$ $10.285$ 4.4.8468.1 None \(0\) \(-3\) \(0\) \(4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-\beta _{2}+\beta _{3})q^{5}+q^{7}+\cdots\)
1288.2.a.o 1288.a 1.a $4$ $10.285$ 4.4.34196.1 None \(0\) \(0\) \(0\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{1}q^{5}-q^{7}+(1+\beta _{2})q^{9}+\cdots\)
1288.2.a.p 1288.a 1.a $5$ $10.285$ 5.5.3385684.1 None \(0\) \(-3\) \(2\) \(5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{3}+\beta _{1}q^{5}+q^{7}+(3-\beta _{4})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1288))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1288)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(322))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(644))\)\(^{\oplus 2}\)