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Properties

Label 1224.4.a
Level $1224$
Weight $4$
Character orbit 1224.a
Rep. character $\chi_{1224}(1,\cdot)$
Character field $\Q$
Dimension $60$
Newform subspaces $17$
Sturm bound $864$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1224 = 2^{3} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1224.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 17 \)
Sturm bound: \(864\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1224))\).

Total New Old
Modular forms 664 60 604
Cusp forms 632 60 572
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.

\(2\)\(3\)\(17\)FrickeDim
\(+\)\(+\)\(+\)$+$\(7\)
\(+\)\(+\)\(-\)$-$\(5\)
\(+\)\(-\)\(+\)$-$\(8\)
\(+\)\(-\)\(-\)$+$\(10\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(7\)
\(-\)\(-\)\(+\)$+$\(9\)
\(-\)\(-\)\(-\)$-$\(9\)
Plus space\(+\)\(33\)
Minus space\(-\)\(27\)

Trace form

\( 60 q + 10 q^{5} + 48 q^{7} + O(q^{10}) \) \( 60 q + 10 q^{5} + 48 q^{7} - 94 q^{11} - 36 q^{13} + 34 q^{17} + 248 q^{19} - 180 q^{23} + 1248 q^{25} + 258 q^{29} - 136 q^{31} + 136 q^{35} + 178 q^{37} + 260 q^{41} - 196 q^{43} + 1032 q^{47} + 3132 q^{49} - 524 q^{53} - 1592 q^{55} + 492 q^{59} + 570 q^{61} - 2132 q^{65} + 596 q^{67} + 1292 q^{71} + 1648 q^{73} - 2208 q^{77} - 996 q^{79} + 828 q^{83} - 170 q^{85} - 1392 q^{89} - 840 q^{91} + 872 q^{95} + 188 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1224))\) 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
1224.4.a.a 1224.a 1.a $1$ $72.218$ \(\Q\) None \(0\) \(0\) \(-6\) \(-24\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-6q^{5}-24q^{7}-44q^{11}+6q^{13}+\cdots\)
1224.4.a.b 1224.a 1.a $1$ $72.218$ \(\Q\) None \(0\) \(0\) \(7\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+7q^{5}+4q^{7}+21q^{11}-5^{2}q^{13}+\cdots\)
1224.4.a.c 1224.a 1.a $2$ $72.218$ \(\Q(\sqrt{241}) \) None \(0\) \(0\) \(-7\) \(18\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{5}+(8+2\beta )q^{7}+(-9-3\beta )q^{11}+\cdots\)
1224.4.a.d 1224.a 1.a $2$ $72.218$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(12\) \(-36\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(6+2\beta )q^{5}+(-18-3\beta )q^{7}+(10+\cdots)q^{11}+\cdots\)
1224.4.a.e 1224.a 1.a $3$ $72.218$ 3.3.23321.1 None \(0\) \(0\) \(-5\) \(20\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{2})q^{5}+(7-\beta _{1})q^{7}+(-9+\cdots)q^{11}+\cdots\)
1224.4.a.f 1224.a 1.a $3$ $72.218$ 3.3.1556.1 None \(0\) \(0\) \(-2\) \(12\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{5}+(4-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1224.4.a.g 1224.a 1.a $3$ $72.218$ 3.3.12821.1 None \(0\) \(0\) \(4\) \(28\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1}-2\beta _{2})q^{5}+(9+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1224.4.a.h 1224.a 1.a $3$ $72.218$ 3.3.4481.1 None \(0\) \(0\) \(5\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1}-\beta _{2})q^{5}+(-2-2\beta _{1})q^{7}+\cdots\)
1224.4.a.i 1224.a 1.a $3$ $72.218$ 3.3.8396.1 None \(0\) \(0\) \(8\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1}+2\beta _{2})q^{5}+(-2-3\beta _{1}-\beta _{2})q^{7}+\cdots\)
1224.4.a.j 1224.a 1.a $3$ $72.218$ 3.3.17717.1 None \(0\) \(0\) \(10\) \(-22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{5}+(-8+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1224.4.a.k 1224.a 1.a $4$ $72.218$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(-10\) \(-4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{5}+(-1-\beta _{2}-\beta _{3})q^{7}+\cdots\)
1224.4.a.l 1224.a 1.a $4$ $72.218$ 4.4.550476.1 None \(0\) \(0\) \(-8\) \(-22\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}+\beta _{2}+2\beta _{3})q^{5}+(-4+\cdots)q^{7}+\cdots\)
1224.4.a.m 1224.a 1.a $4$ $72.218$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(2\) \(32\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{5}+(9+\beta _{1}-\beta _{2})q^{7}+(-15+\cdots)q^{11}+\cdots\)
1224.4.a.n 1224.a 1.a $5$ $72.218$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(-13\) \(-2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{3})q^{5}+\beta _{1}q^{7}+(1+\beta _{2}-2\beta _{3}+\cdots)q^{11}+\cdots\)
1224.4.a.o 1224.a 1.a $5$ $72.218$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(0\) \(13\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{3})q^{5}+\beta _{1}q^{7}+(-1-\beta _{2}+2\beta _{3}+\cdots)q^{11}+\cdots\)
1224.4.a.p 1224.a 1.a $7$ $72.218$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(-3\) \(26\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{4}q^{5}+(4+\beta _{3})q^{7}+(-9-\beta _{2}+\beta _{4}+\cdots)q^{11}+\cdots\)
1224.4.a.q 1224.a 1.a $7$ $72.218$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(0\) \(3\) \(26\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{5}+(4+\beta _{3})q^{7}+(9+\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1224))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1224)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(306))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(612))\)\(^{\oplus 2}\)