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Properties

Label 162.6.a
Level $162$
Weight $6$
Character orbit 162.a
Rep. character $\chi_{162}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $10$
Sturm bound $162$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(162\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(162))\).

Total New Old
Modular forms 147 20 127
Cusp forms 123 20 103
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.

\(2\)\(3\)FrickeDim
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(5\)
\(-\)\(+\)$-$\(6\)
\(-\)\(-\)$+$\(4\)
Plus space\(+\)\(9\)
Minus space\(-\)\(11\)

Trace form

\( 20 q + 320 q^{4} - 116 q^{7} + O(q^{10}) \) \( 20 q + 320 q^{4} - 116 q^{7} - 24 q^{10} - 470 q^{13} + 5120 q^{16} - 770 q^{19} + 1896 q^{22} + 5414 q^{25} - 1856 q^{28} + 9952 q^{31} + 13392 q^{34} + 15358 q^{37} - 384 q^{40} + 56854 q^{43} - 18960 q^{46} + 4308 q^{49} - 7520 q^{52} - 64800 q^{55} + 36024 q^{58} + 90046 q^{61} + 81920 q^{64} - 8246 q^{67} + 126528 q^{70} - 154724 q^{73} - 12320 q^{76} - 139688 q^{79} + 13368 q^{82} - 196038 q^{85} + 30336 q^{88} + 19664 q^{91} + 206160 q^{94} - 29114 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(162))\) 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
162.6.a.a 162.a 1.a $1$ $25.982$ \(\Q\) None \(-4\) \(0\) \(-21\) \(74\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}-21q^{5}+74q^{7}-2^{6}q^{8}+\cdots\)
162.6.a.b 162.a 1.a $1$ $25.982$ \(\Q\) None \(4\) \(0\) \(21\) \(74\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+21q^{5}+74q^{7}+2^{6}q^{8}+\cdots\)
162.6.a.c 162.a 1.a $2$ $25.982$ \(\Q(\sqrt{3}) \) None \(-8\) \(0\) \(12\) \(-176\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(6+5\beta )q^{5}+(-88+\cdots)q^{7}+\cdots\)
162.6.a.d 162.a 1.a $2$ $25.982$ \(\Q(\sqrt{921}) \) None \(-8\) \(0\) \(12\) \(-14\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(6+\beta )q^{5}+(-7-\beta )q^{7}+\cdots\)
162.6.a.e 162.a 1.a $2$ $25.982$ \(\Q(\sqrt{6}) \) None \(-8\) \(0\) \(54\) \(-74\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(3^{3}+2\beta )q^{5}+(-37+\cdots)q^{7}+\cdots\)
162.6.a.f 162.a 1.a $2$ $25.982$ \(\Q(\sqrt{6}) \) None \(8\) \(0\) \(-54\) \(-74\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-3^{3}+2\beta )q^{5}+(-37+\cdots)q^{7}+\cdots\)
162.6.a.g 162.a 1.a $2$ $25.982$ \(\Q(\sqrt{3}) \) None \(8\) \(0\) \(-12\) \(-176\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-6+5\beta )q^{5}+(-88+\cdots)q^{7}+\cdots\)
162.6.a.h 162.a 1.a $2$ $25.982$ \(\Q(\sqrt{921}) \) None \(8\) \(0\) \(-12\) \(-14\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(-6-\beta )q^{5}+(-7+\cdots)q^{7}+\cdots\)
162.6.a.i 162.a 1.a $3$ $25.982$ 3.3.125628.1 None \(-12\) \(0\) \(-54\) \(132\) $+$ $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+2^{4}q^{4}+(-18+\beta _{1})q^{5}+(44+\cdots)q^{7}+\cdots\)
162.6.a.j 162.a 1.a $3$ $25.982$ 3.3.125628.1 None \(12\) \(0\) \(54\) \(132\) $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{2}+2^{4}q^{4}+(18-\beta _{1})q^{5}+(44+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(162))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(162)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)