Properties

Label 224.6.a
Level $224$
Weight $6$
Character orbit 224.a
Rep. character $\chi_{224}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $10$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 224 = 2^{5} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 224.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(224))\).

Total New Old
Modular forms 168 30 138
Cusp forms 152 30 122
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.
\(+\)\(+\)\(+\)\(7\)
\(+\)\(-\)\(-\)\(9\)
\(-\)\(+\)\(-\)\(8\)
\(-\)\(-\)\(+\)\(6\)
Plus space\(+\)\(13\)
Minus space\(-\)\(17\)

Trace form

\( 30 q - 76 q^{5} + 2342 q^{9} + O(q^{10}) \) \( 30 q - 76 q^{5} + 2342 q^{9} + 244 q^{13} - 3220 q^{17} + 28098 q^{25} + 8564 q^{29} - 7504 q^{33} + 18820 q^{37} + 42252 q^{41} - 102828 q^{45} + 72030 q^{49} + 117956 q^{53} + 198912 q^{57} - 91580 q^{61} - 118568 q^{65} + 285072 q^{69} + 122364 q^{73} + 115246 q^{81} + 131080 q^{85} + 426364 q^{89} + 364128 q^{93} - 470132 q^{97} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(224))\) 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
224.6.a.a 224.a 1.a $1$ $35.926$ \(\Q\) None \(0\) \(-14\) \(-64\) \(49\) $-$ $-$ $\mathrm{SU}(2)$ \(q-14q^{3}-2^{6}q^{5}+7^{2}q^{7}-47q^{9}+\cdots\)
224.6.a.b 224.a 1.a $1$ $35.926$ \(\Q\) None \(0\) \(14\) \(-64\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+14q^{3}-2^{6}q^{5}-7^{2}q^{7}-47q^{9}+\cdots\)
224.6.a.c 224.a 1.a $2$ $35.926$ \(\Q(\sqrt{61}) \) None \(0\) \(-14\) \(34\) \(-98\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-7-\beta )q^{3}+(17-7\beta )q^{5}-7^{2}q^{7}+\cdots\)
224.6.a.d 224.a 1.a $2$ $35.926$ \(\Q(\sqrt{61}) \) None \(0\) \(14\) \(34\) \(98\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(7-\beta )q^{3}+(17+7\beta )q^{5}+7^{2}q^{7}+\cdots\)
224.6.a.e 224.a 1.a $3$ $35.926$ 3.3.367637.1 None \(0\) \(-8\) \(-14\) \(147\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta _{1})q^{3}+(-5-\beta _{2})q^{5}+7^{2}q^{7}+\cdots\)
224.6.a.f 224.a 1.a $3$ $35.926$ 3.3.367637.1 None \(0\) \(8\) \(-14\) \(-147\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{3}+(-5-\beta _{2})q^{5}-7^{2}q^{7}+\cdots\)
224.6.a.g 224.a 1.a $4$ $35.926$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(-18\) \(-30\) \(-196\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta _{1})q^{3}+(-7-\beta _{1}-\beta _{2})q^{5}+\cdots\)
224.6.a.h 224.a 1.a $4$ $35.926$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(18\) \(-30\) \(196\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(4+\beta _{1})q^{3}+(-7-\beta _{1}-\beta _{2})q^{5}+\cdots\)
224.6.a.i 224.a 1.a $5$ $35.926$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-10\) \(36\) \(245\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{3}+(7-\beta _{3})q^{5}+7^{2}q^{7}+\cdots\)
224.6.a.j 224.a 1.a $5$ $35.926$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(10\) \(36\) \(-245\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{3}+(7-\beta _{3})q^{5}-7^{2}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(224))\) into lower level spaces

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