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Properties

Label 224.4.a
Level $224$
Weight $4$
Character orbit 224.a
Rep. character $\chi_{224}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $8$
Sturm bound $128$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 224.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(128\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(224))\).

Total New Old
Modular forms 104 18 86
Cusp forms 88 18 70
Eisenstein series 16 0 16

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

\(2\)\(7\)FrickeDim
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(3\)
\(-\)\(+\)$-$\(4\)
\(-\)\(-\)$+$\(6\)
Plus space\(+\)\(11\)
Minus space\(-\)\(7\)

Trace form

\( 18 q - 4 q^{5} + 122 q^{9} + O(q^{10}) \) \( 18 q - 4 q^{5} + 122 q^{9} + 92 q^{13} + 308 q^{17} + 318 q^{25} + 284 q^{29} + 1232 q^{33} - 660 q^{37} - 588 q^{41} + 1788 q^{45} + 882 q^{49} + 1388 q^{53} - 768 q^{57} + 780 q^{61} - 1304 q^{65} - 1488 q^{69} - 1212 q^{73} - 2078 q^{81} + 1880 q^{85} + 388 q^{89} - 5856 q^{93} - 3628 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(224))\) 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
224.4.a.a 224.a 1.a $1$ $13.216$ \(\Q\) None \(0\) \(-2\) \(0\) \(7\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+7q^{7}-23q^{9}+20q^{11}-20q^{13}+\cdots\)
224.4.a.b 224.a 1.a $1$ $13.216$ \(\Q\) None \(0\) \(2\) \(0\) \(-7\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-7q^{7}-23q^{9}-20q^{11}-20q^{13}+\cdots\)
224.4.a.c 224.a 1.a $2$ $13.216$ \(\Q(\sqrt{37}) \) None \(0\) \(-6\) \(-6\) \(14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{3}+(-3+\beta )q^{5}+7q^{7}+\cdots\)
224.4.a.d 224.a 1.a $2$ $13.216$ \(\Q(\sqrt{37}) \) None \(0\) \(6\) \(-6\) \(-14\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{3}+(-3+\beta )q^{5}-7q^{7}+(19+\cdots)q^{9}+\cdots\)
224.4.a.e 224.a 1.a $3$ $13.216$ 3.3.2981.1 None \(0\) \(-8\) \(10\) \(-21\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{3}+(3+\beta _{2})q^{5}-7q^{7}+\cdots\)
224.4.a.f 224.a 1.a $3$ $13.216$ 3.3.621.1 None \(0\) \(0\) \(-6\) \(-21\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}-7q^{7}+\cdots\)
224.4.a.g 224.a 1.a $3$ $13.216$ 3.3.621.1 None \(0\) \(0\) \(-6\) \(21\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(-2-2\beta _{1}-\beta _{2})q^{5}+7q^{7}+\cdots\)
224.4.a.h 224.a 1.a $3$ $13.216$ 3.3.2981.1 None \(0\) \(8\) \(10\) \(21\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+(3+\beta _{2})q^{5}+7q^{7}+(11+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(224))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(224)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)