Properties

Label 84.8.a
Level $84$
Weight $8$
Character orbit 84.a
Rep. character $\chi_{84}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $4$
Sturm bound $128$
Trace bound $3$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 84.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(128\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(84))\).

Total New Old
Modular forms 118 6 112
Cusp forms 106 6 100
Eisenstein series 12 0 12

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
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(4\)
Minus space\(-\)\(2\)

Trace form

\( 6 q + 220 q^{5} + 4374 q^{9} + O(q^{10}) \) \( 6 q + 220 q^{5} + 4374 q^{9} + 2636 q^{11} - 8172 q^{13} - 13716 q^{15} + 50852 q^{17} - 36840 q^{19} - 18522 q^{21} + 43148 q^{23} + 76722 q^{25} + 217132 q^{29} + 405096 q^{31} + 190296 q^{33} - 58996 q^{35} + 289548 q^{37} + 505224 q^{39} + 671988 q^{41} - 54984 q^{43} + 160380 q^{45} - 1190472 q^{47} + 705894 q^{49} + 91476 q^{51} + 3968604 q^{53} + 4476072 q^{55} - 1691280 q^{57} + 1268632 q^{59} - 3562380 q^{61} + 1812888 q^{65} + 2076000 q^{67} + 299160 q^{69} - 8216956 q^{71} + 3913404 q^{73} - 1881360 q^{75} - 3838856 q^{77} - 13392600 q^{79} + 3188646 q^{81} + 12297392 q^{83} - 16242432 q^{85} + 3946536 q^{87} - 18163500 q^{89} - 6873720 q^{91} - 8126136 q^{93} - 5207728 q^{95} + 21231516 q^{97} + 1921644 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(84))\) 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
84.8.a.a 84.a 1.a $1$ $26.240$ \(\Q\) None 84.8.a.a \(0\) \(-27\) \(100\) \(-343\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3^{3}q^{3}+10^{2}q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)
84.8.a.b 84.a 1.a $1$ $26.240$ \(\Q\) None 84.8.a.b \(0\) \(27\) \(-240\) \(343\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3^{3}q^{3}-240q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
84.8.a.c 84.a 1.a $2$ $26.240$ \(\Q(\sqrt{3649}) \) None 84.8.a.c \(0\) \(-54\) \(264\) \(686\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3^{3}q^{3}+(132-\beta )q^{5}+7^{3}q^{7}+3^{6}q^{9}+\cdots\)
84.8.a.d 84.a 1.a $2$ $26.240$ \(\Q(\sqrt{21961}) \) None 84.8.a.d \(0\) \(54\) \(96\) \(-686\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3^{3}q^{3}+(48-\beta )q^{5}-7^{3}q^{7}+3^{6}q^{9}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(84))\) into lower level spaces

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