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Properties

Label 168.4.a
Level $168$
Weight $4$
Character orbit 168.a
Rep. character $\chi_{168}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $8$
Sturm bound $128$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 168.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(128\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(168))\).

Total New Old
Modular forms 104 10 94
Cusp forms 88 10 78
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.

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(1\)
Plus space\(+\)\(6\)
Minus space\(-\)\(4\)

Trace form

\( 10 q - 28 q^{5} + 90 q^{9} + O(q^{10}) \) \( 10 q - 28 q^{5} + 90 q^{9} + 132 q^{13} - 172 q^{17} + 42 q^{21} + 152 q^{23} + 374 q^{25} + 556 q^{29} + 176 q^{31} - 372 q^{37} + 24 q^{39} + 324 q^{41} - 360 q^{43} - 252 q^{45} - 480 q^{47} + 490 q^{49} - 552 q^{51} + 844 q^{53} - 992 q^{55} - 120 q^{57} + 560 q^{59} - 236 q^{61} + 40 q^{65} - 760 q^{67} - 2248 q^{71} - 1084 q^{73} + 1568 q^{79} + 810 q^{81} - 592 q^{83} + 1144 q^{85} + 1776 q^{87} - 2476 q^{89} - 1512 q^{91} - 720 q^{93} - 3552 q^{95} + 612 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(168))\) 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 7
168.4.a.a 168.a 1.a $1$ $9.912$ \(\Q\) None \(0\) \(-3\) \(-10\) \(7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-10q^{5}+7q^{7}+9q^{9}-12q^{11}+\cdots\)
168.4.a.b 168.a 1.a $1$ $9.912$ \(\Q\) None \(0\) \(-3\) \(-2\) \(7\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{5}+7q^{7}+9q^{9}+12q^{11}+\cdots\)
168.4.a.c 168.a 1.a $1$ $9.912$ \(\Q\) None \(0\) \(-3\) \(4\) \(-7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+4q^{5}-7q^{7}+9q^{9}-26q^{11}+\cdots\)
168.4.a.d 168.a 1.a $1$ $9.912$ \(\Q\) None \(0\) \(3\) \(-16\) \(7\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-2^{4}q^{5}+7q^{7}+9q^{9}-18q^{11}+\cdots\)
168.4.a.e 168.a 1.a $1$ $9.912$ \(\Q\) None \(0\) \(3\) \(-10\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-10q^{5}-7q^{7}+9q^{9}-52q^{11}+\cdots\)
168.4.a.f 168.a 1.a $1$ $9.912$ \(\Q\) None \(0\) \(3\) \(-2\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-2q^{5}-7q^{7}+9q^{9}+52q^{11}+\cdots\)
168.4.a.g 168.a 1.a $2$ $9.912$ \(\Q(\sqrt{337}) \) None \(0\) \(-6\) \(-6\) \(-14\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+(-3-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
168.4.a.h 168.a 1.a $2$ $9.912$ \(\Q(\sqrt{177}) \) None \(0\) \(6\) \(14\) \(14\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+(7-\beta )q^{5}+7q^{7}+9q^{9}+(9+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(168))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(168)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 2}\)