Properties

Label 252.8.a
Level $252$
Weight $8$
Character orbit 252.a
Rep. character $\chi_{252}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $8$
Sturm bound $384$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(252))\).

Total New Old
Modular forms 348 18 330
Cusp forms 324 18 306
Eisenstein series 24 0 24

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

\(2\)\(3\)\(7\)FrickeDim
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(5\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(9\)
Minus space\(-\)\(9\)

Trace form

\( 18 q + 32 q^{5} + O(q^{10}) \) \( 18 q + 32 q^{5} - 6572 q^{11} + 11568 q^{13} - 37412 q^{17} - 4920 q^{19} - 66428 q^{23} + 402702 q^{25} - 243820 q^{29} + 31464 q^{31} - 56252 q^{35} - 444612 q^{37} - 233508 q^{41} + 804216 q^{43} + 538632 q^{47} + 2117682 q^{49} - 2161524 q^{53} - 1570584 q^{55} - 241480 q^{59} + 1899048 q^{61} - 9090672 q^{65} - 140160 q^{67} + 2557372 q^{71} + 11339844 q^{73} + 93296 q^{77} - 15423384 q^{79} + 16091248 q^{83} - 9278040 q^{85} + 7392516 q^{89} + 1885128 q^{91} + 30310048 q^{95} - 33749844 q^{97} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(252))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 7
252.8.a.a 252.a 1.a $1$ $78.721$ \(\Q\) None 84.8.a.a \(0\) \(0\) \(-100\) \(-343\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-10^{2}q^{5}-7^{3}q^{7}-2774q^{11}-3294q^{13}+\cdots\)
252.8.a.b 252.a 1.a $1$ $78.721$ \(\Q\) None 84.8.a.b \(0\) \(0\) \(240\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+240q^{5}+7^{3}q^{7}-702q^{11}-3958q^{13}+\cdots\)
252.8.a.c 252.a 1.a $2$ $78.721$ \(\Q(\sqrt{3649}) \) None 84.8.a.c \(0\) \(0\) \(-264\) \(686\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-132-\beta )q^{5}+7^{3}q^{7}+(2490-7\beta )q^{11}+\cdots\)
252.8.a.d 252.a 1.a $2$ $78.721$ \(\Q(\sqrt{21961}) \) None 84.8.a.d \(0\) \(0\) \(-96\) \(-686\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-48-\beta )q^{5}-7^{3}q^{7}+(-2070+\cdots)q^{11}+\cdots\)
252.8.a.e 252.a 1.a $2$ $78.721$ \(\Q(\sqrt{3529}) \) None 28.8.a.a \(0\) \(0\) \(-42\) \(686\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-21-\beta )q^{5}+7^{3}q^{7}+(-3714+\cdots)q^{11}+\cdots\)
252.8.a.f 252.a 1.a $2$ $78.721$ \(\Q(\sqrt{1009}) \) None 28.8.a.b \(0\) \(0\) \(294\) \(-686\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(147-11\beta )q^{5}-7^{3}q^{7}+(1746+182\beta )q^{11}+\cdots\)
252.8.a.g 252.a 1.a $4$ $78.721$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 252.8.a.g \(0\) \(0\) \(0\) \(-1372\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}-7^{3}q^{7}+(-5\beta _{1}+\beta _{3})q^{11}+\cdots\)
252.8.a.h 252.a 1.a $4$ $78.721$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 252.8.a.h \(0\) \(0\) \(0\) \(1372\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{5}+7^{3}q^{7}+(3\beta _{1}+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(252))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(252)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 9}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 2}\)