Properties

Label 1224.2.a
Level $1224$
Weight $2$
Character orbit 1224.a
Rep. character $\chi_{1224}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $13$
Sturm bound $432$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1224.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(432\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1224))\).

Total New Old
Modular forms 232 20 212
Cusp forms 201 20 181
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.

\(2\)\(3\)\(17\)FrickeDim
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(3\)
\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(3\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(13\)

Trace form

\( 20 q - 2 q^{5} + O(q^{10}) \) \( 20 q - 2 q^{5} + 10 q^{11} + 4 q^{13} - 2 q^{17} + 8 q^{19} - 4 q^{23} + 32 q^{25} + 6 q^{29} - 8 q^{31} + 8 q^{35} + 6 q^{37} - 20 q^{41} - 4 q^{43} + 24 q^{47} + 36 q^{49} + 12 q^{53} + 24 q^{55} + 12 q^{59} + 14 q^{61} + 4 q^{65} + 20 q^{67} - 4 q^{71} + 16 q^{73} + 16 q^{77} + 28 q^{79} - 4 q^{83} + 2 q^{85} + 24 q^{91} - 24 q^{95} + 20 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1224))\) 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 17
1224.2.a.a 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(-3\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{5}+q^{11}+3q^{13}+q^{17}+q^{19}+\cdots\)
1224.2.a.b 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(-2\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-4q^{7}-4q^{11}+6q^{13}-q^{17}+\cdots\)
1224.2.a.c 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(-1\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}+3q^{11}-q^{13}-q^{17}+\cdots\)
1224.2.a.d 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{11}-6q^{13}+q^{17}+4q^{19}-4q^{23}+\cdots\)
1224.2.a.e 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(0\) \(2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{7}+2q^{13}+q^{17}+4q^{19}-2q^{23}+\cdots\)
1224.2.a.f 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(1\) \(-2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-3q^{11}-q^{13}+q^{17}+\cdots\)
1224.2.a.g 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(2\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-2q^{7}+6q^{11}+2q^{13}-q^{17}+\cdots\)
1224.2.a.h 1224.a 1.a $1$ $9.774$ \(\Q\) None \(0\) \(0\) \(3\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{5}-4q^{7}-q^{11}-5q^{13}-q^{17}+\cdots\)
1224.2.a.i 1224.a 1.a $2$ $9.774$ \(\Q(\sqrt{5}) \) None \(0\) \(0\) \(-4\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}+(1-\beta )q^{7}+(-1+\beta )q^{11}+\cdots\)
1224.2.a.j 1224.a 1.a $2$ $9.774$ \(\Q(\sqrt{17}) \) None \(0\) \(0\) \(1\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+(-2+2\beta )q^{7}+(4+\beta )q^{11}+\cdots\)
1224.2.a.k 1224.a 1.a $2$ $9.774$ \(\Q(\sqrt{57}) \) None \(0\) \(0\) \(1\) \(8\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+4q^{7}+(2-\beta )q^{11}+\beta q^{13}+\cdots\)
1224.2.a.l 1224.a 1.a $3$ $9.774$ 3.3.1304.1 None \(0\) \(0\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{2})q^{5}+(1+\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{11}+\cdots\)
1224.2.a.m 1224.a 1.a $3$ $9.774$ 3.3.1304.1 None \(0\) \(0\) \(1\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2})q^{5}+(1+\beta _{2})q^{7}+(-1+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1224))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1224)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(612))\)\(^{\oplus 2}\)