Properties

Label 336.4.a
Level $336$
Weight $4$
Character orbit 336.a
Rep. character $\chi_{336}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $15$
Sturm bound $256$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 336.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 15 \)
Sturm bound: \(256\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(336))\).

Total New Old
Modular forms 204 18 186
Cusp forms 180 18 162
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.

\(2\)\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(11\)
Minus space\(-\)\(7\)

Trace form

\( 18q - 4q^{5} - 14q^{7} + 162q^{9} + O(q^{10}) \) \( 18q - 4q^{5} - 14q^{7} + 162q^{9} + 20q^{11} + 92q^{13} - 60q^{15} + 52q^{17} - 24q^{19} + 164q^{23} + 406q^{25} + 484q^{29} + 264q^{31} - 124q^{37} - 312q^{39} - 236q^{41} + 488q^{43} - 36q^{45} + 744q^{47} + 882q^{49} + 1356q^{51} + 196q^{53} + 1080q^{55} - 120q^{59} + 1068q^{61} - 126q^{63} + 696q^{65} + 672q^{67} + 528q^{69} + 2060q^{71} + 740q^{73} + 1104q^{75} - 952q^{77} - 1784q^{79} + 1458q^{81} - 2224q^{83} + 512q^{85} - 1368q^{87} - 1580q^{89} + 1092q^{91} - 456q^{93} + 656q^{95} - 1884q^{97} + 180q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(336))\) 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
336.4.a.a \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(-16\) \(-7\) \(+\) \(+\) \(+\) \(q-3q^{3}-2^{4}q^{5}-7q^{7}+9q^{9}+18q^{11}+\cdots\)
336.4.a.b \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(-10\) \(7\) \(+\) \(+\) \(-\) \(q-3q^{3}-10q^{5}+7q^{7}+9q^{9}+52q^{11}+\cdots\)
336.4.a.c \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(-2\) \(7\) \(+\) \(+\) \(-\) \(q-3q^{3}-2q^{5}+7q^{7}+9q^{9}-52q^{11}+\cdots\)
336.4.a.d \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(2\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}+2q^{5}+7q^{7}+9q^{9}+8q^{11}+\cdots\)
336.4.a.e \(1\) \(19.825\) \(\Q\) None \(0\) \(-3\) \(14\) \(7\) \(-\) \(+\) \(-\) \(q-3q^{3}+14q^{5}+7q^{7}+9q^{9}-4q^{11}+\cdots\)
336.4.a.f \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-18\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}-18q^{5}-7q^{7}+9q^{9}+6^{2}q^{11}+\cdots\)
336.4.a.g \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-10\) \(-7\) \(+\) \(-\) \(+\) \(q+3q^{3}-10q^{5}-7q^{7}+9q^{9}+12q^{11}+\cdots\)
336.4.a.h \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-4\) \(7\) \(-\) \(-\) \(-\) \(q+3q^{3}-4q^{5}+7q^{7}+9q^{9}-62q^{11}+\cdots\)
336.4.a.i \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(-2\) \(-7\) \(+\) \(-\) \(+\) \(q+3q^{3}-2q^{5}-7q^{7}+9q^{9}-12q^{11}+\cdots\)
336.4.a.j \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(4\) \(7\) \(+\) \(-\) \(-\) \(q+3q^{3}+4q^{5}+7q^{7}+9q^{9}+26q^{11}+\cdots\)
336.4.a.k \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(6\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}+6q^{5}-7q^{7}+9q^{9}-6^{2}q^{11}+\cdots\)
336.4.a.l \(1\) \(19.825\) \(\Q\) None \(0\) \(3\) \(18\) \(-7\) \(-\) \(-\) \(+\) \(q+3q^{3}+18q^{5}-7q^{7}+9q^{9}+72q^{11}+\cdots\)
336.4.a.m \(2\) \(19.825\) \(\Q(\sqrt{57}) \) None \(0\) \(-6\) \(6\) \(-14\) \(-\) \(+\) \(+\) \(q-3q^{3}+(3+\beta )q^{5}-7q^{7}+9q^{9}+(3+\cdots)q^{11}+\cdots\)
336.4.a.n \(2\) \(19.825\) \(\Q(\sqrt{177}) \) None \(0\) \(-6\) \(14\) \(-14\) \(+\) \(+\) \(+\) \(q-3q^{3}+(7-\beta )q^{5}-7q^{7}+9q^{9}+(-9+\cdots)q^{11}+\cdots\)
336.4.a.o \(2\) \(19.825\) \(\Q(\sqrt{337}) \) None \(0\) \(6\) \(-6\) \(14\) \(+\) \(-\) \(-\) \(q+3q^{3}+(-3-\beta )q^{5}+7q^{7}+9q^{9}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(336))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(336)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)