Properties

Label 162.4.a
Level $162$
Weight $4$
Character orbit 162.a
Rep. character $\chi_{162}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $8$
Sturm bound $108$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 162.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(108\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(162))\).

Total New Old
Modular forms 93 12 81
Cusp forms 69 12 57
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.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(7\)
Minus space\(-\)\(5\)

Trace form

\( 12 q + 48 q^{4} + 24 q^{7} + O(q^{10}) \) \( 12 q + 48 q^{4} + 24 q^{7} - 36 q^{10} - 30 q^{13} + 192 q^{16} - 30 q^{19} + 36 q^{22} + 822 q^{25} + 96 q^{28} + 132 q^{31} + 216 q^{34} - 678 q^{37} - 144 q^{40} - 1434 q^{43} + 504 q^{46} - 72 q^{49} - 120 q^{52} - 324 q^{55} + 1332 q^{58} - 1002 q^{61} + 768 q^{64} - 2838 q^{67} - 288 q^{70} + 2184 q^{73} - 120 q^{76} + 780 q^{79} - 1980 q^{82} + 3294 q^{85} + 144 q^{88} - 1140 q^{91} - 3096 q^{94} - 6078 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(162))\) 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
162.4.a.a 162.a 1.a $1$ $9.558$ \(\Q\) None \(-2\) \(0\) \(9\) \(-31\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+9q^{5}-31q^{7}-8q^{8}+\cdots\)
162.4.a.b 162.a 1.a $1$ $9.558$ \(\Q\) None \(-2\) \(0\) \(21\) \(8\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+21q^{5}+8q^{7}-8q^{8}+\cdots\)
162.4.a.c 162.a 1.a $1$ $9.558$ \(\Q\) None \(2\) \(0\) \(-21\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-21q^{5}+8q^{7}+8q^{8}+\cdots\)
162.4.a.d 162.a 1.a $1$ $9.558$ \(\Q\) None \(2\) \(0\) \(-9\) \(-31\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-9q^{5}-31q^{7}+8q^{8}+\cdots\)
162.4.a.e 162.a 1.a $2$ $9.558$ \(\Q(\sqrt{3}) \) None \(-4\) \(0\) \(-12\) \(16\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-6+\beta )q^{5}+(8-2\beta )q^{7}+\cdots\)
162.4.a.f 162.a 1.a $2$ $9.558$ \(\Q(\sqrt{105}) \) None \(-4\) \(0\) \(-9\) \(19\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-5-\beta )q^{5}+(9-\beta )q^{7}+\cdots\)
162.4.a.g 162.a 1.a $2$ $9.558$ \(\Q(\sqrt{105}) \) None \(4\) \(0\) \(9\) \(19\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(4-\beta )q^{5}+(10+\beta )q^{7}+\cdots\)
162.4.a.h 162.a 1.a $2$ $9.558$ \(\Q(\sqrt{3}) \) None \(4\) \(0\) \(12\) \(16\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(6+\beta )q^{5}+(8+2\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(162))\) into lower level spaces

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