Properties

Label 1248.2.a
Level $1248$
Weight $2$
Character orbit 1248.a
Rep. character $\chi_{1248}(1,\cdot)$
Character field $\Q$
Dimension $24$
Newform subspaces $16$
Sturm bound $448$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1248 = 2^{5} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1248.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(448\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1248))\).

Total New Old
Modular forms 240 24 216
Cusp forms 209 24 185
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.

\(2\)\(3\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(4\)
Plus space\(+\)\(10\)
Minus space\(-\)\(14\)

Trace form

\( 24 q + 24 q^{9} + O(q^{10}) \) \( 24 q + 24 q^{9} + 8 q^{25} - 16 q^{33} + 8 q^{49} - 16 q^{53} - 16 q^{61} - 16 q^{73} + 80 q^{77} + 24 q^{81} + 64 q^{85} - 32 q^{93} + 48 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1248))\) 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 13
1248.2.a.a 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(-1\) \(-2\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}-2q^{7}+q^{9}+6q^{11}+\cdots\)
1248.2.a.b 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}+4q^{11}+q^{13}+\cdots\)
1248.2.a.c 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}+q^{13}+2q^{17}+\cdots\)
1248.2.a.d 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(-1\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}+4q^{11}+q^{13}+\cdots\)
1248.2.a.e 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(-1\) \(2\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-2q^{7}+q^{9}+2q^{11}+\cdots\)
1248.2.a.f 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(1\) \(-2\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+q^{9}-4q^{11}+q^{13}+\cdots\)
1248.2.a.g 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(1\) \(-2\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}+2q^{7}+q^{9}-6q^{11}+\cdots\)
1248.2.a.h 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(1\) \(0\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}+q^{9}-4q^{11}+q^{13}+\cdots\)
1248.2.a.i 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(1\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}+q^{13}+2q^{17}+\cdots\)
1248.2.a.j 1248.a 1.a $1$ $9.965$ \(\Q\) None \(0\) \(1\) \(2\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
1248.2.a.k 1248.a 1.a $2$ $9.965$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-2\) \(6\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(-1-\beta )q^{5}+(3-\beta )q^{7}+q^{9}+\cdots\)
1248.2.a.l 1248.a 1.a $2$ $9.965$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(2\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+(1+\beta )q^{5}+(-1-\beta )q^{7}+q^{9}+\cdots\)
1248.2.a.m 1248.a 1.a $2$ $9.965$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-2\) \(-6\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(-1-\beta )q^{5}+(-3+\beta )q^{7}+\cdots\)
1248.2.a.n 1248.a 1.a $2$ $9.965$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(2\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta )q^{5}+(1+\beta )q^{7}+q^{9}+\cdots\)
1248.2.a.o 1248.a 1.a $3$ $9.965$ 3.3.148.1 None \(0\) \(-3\) \(2\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+(1+\beta _{1})q^{5}+\beta _{2}q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1248.2.a.p 1248.a 1.a $3$ $9.965$ 3.3.148.1 None \(0\) \(3\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta _{1})q^{5}-\beta _{2}q^{7}+q^{9}+(1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1248))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1248)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(208))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(416))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(624))\)\(^{\oplus 2}\)