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

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

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(168))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 7
168.4.a.a \(1\) \(9.912\) \(\Q\) None \(0\) \(-3\) \(-10\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}-10q^{5}+7q^{7}+9q^{9}-12q^{11}+\cdots\)
168.4.a.b \(1\) \(9.912\) \(\Q\) None \(0\) \(-3\) \(-2\) \(7\) \(+\) \(+\) \(-\) \(q-3q^{3}-2q^{5}+7q^{7}+9q^{9}+12q^{11}+\cdots\)
168.4.a.c \(1\) \(9.912\) \(\Q\) None \(0\) \(-3\) \(4\) \(-7\) \(-\) \(+\) \(+\) \(q-3q^{3}+4q^{5}-7q^{7}+9q^{9}-26q^{11}+\cdots\)
168.4.a.d \(1\) \(9.912\) \(\Q\) None \(0\) \(3\) \(-16\) \(7\) \(-\) \(-\) \(-\) \(q+3q^{3}-2^{4}q^{5}+7q^{7}+9q^{9}-18q^{11}+\cdots\)
168.4.a.e \(1\) \(9.912\) \(\Q\) None \(0\) \(3\) \(-10\) \(-7\) \(+\) \(-\) \(+\) \(q+3q^{3}-10q^{5}-7q^{7}+9q^{9}-52q^{11}+\cdots\)
168.4.a.f \(1\) \(9.912\) \(\Q\) None \(0\) \(3\) \(-2\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}-2q^{5}-7q^{7}+9q^{9}+52q^{11}+\cdots\)
168.4.a.g \(2\) \(9.912\) \(\Q(\sqrt{337}) \) None \(0\) \(-6\) \(-6\) \(-14\) \(+\) \(+\) \(+\) \(q-3q^{3}+(-3-\beta )q^{5}-7q^{7}+9q^{9}+\cdots\)
168.4.a.h \(2\) \(9.912\) \(\Q(\sqrt{177}) \) None \(0\) \(6\) \(14\) \(14\) \(+\) \(-\) \(-\) \(q+3q^{3}+(7-\beta )q^{5}+7q^{7}+9q^{9}+(9+\cdots)q^{11}+\cdots\)

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

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