Properties

Label 392.4.a
Level $392$
Weight $4$
Character orbit 392.a
Rep. character $\chi_{392}(1,\cdot)$
Character field $\Q$
Dimension $31$
Newform subspaces $14$
Sturm bound $224$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 392.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(392))\).

Total New Old
Modular forms 184 31 153
Cusp forms 152 31 121
Eisenstein series 32 0 32

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(7\)
\(-\)\(+\)\(-\)\(7\)
\(-\)\(-\)\(+\)\(8\)
Plus space\(+\)\(17\)
Minus space\(-\)\(14\)

Trace form

\( 31q + 2q^{3} - 12q^{5} + 287q^{9} + O(q^{10}) \) \( 31q + 2q^{3} - 12q^{5} + 287q^{9} - 72q^{11} + 32q^{13} + 4q^{15} - 22q^{17} + 46q^{19} + 132q^{23} + 809q^{25} + 92q^{27} + 254q^{29} + 356q^{31} + 256q^{33} - 366q^{37} + 112q^{39} + 330q^{41} - 652q^{43} - 1372q^{45} + 396q^{47} - 44q^{51} + 1518q^{53} - 1232q^{55} + 1988q^{57} + 1270q^{59} - 20q^{61} - 344q^{65} - 1312q^{67} + 1056q^{69} - 2120q^{71} - 2634q^{73} + 2758q^{75} + 1100q^{79} + 3703q^{81} - 74q^{83} + 2756q^{85} + 4228q^{87} - 1202q^{89} - 1396q^{93} + 460q^{95} + 1690q^{97} - 1004q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(392))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
392.4.a.a \(1\) \(23.129\) \(\Q\) None \(0\) \(-6\) \(-8\) \(0\) \(+\) \(-\) \(q-6q^{3}-8q^{5}+9q^{9}+56q^{11}+28q^{13}+\cdots\)
392.4.a.b \(1\) \(23.129\) \(\Q\) None \(0\) \(-4\) \(12\) \(0\) \(+\) \(-\) \(q-4q^{3}+12q^{5}-11q^{9}+12q^{11}+\cdots\)
392.4.a.c \(1\) \(23.129\) \(\Q\) None \(0\) \(2\) \(16\) \(0\) \(-\) \(-\) \(q+2q^{3}+2^{4}q^{5}-23q^{9}+24q^{11}+\cdots\)
392.4.a.d \(1\) \(23.129\) \(\Q\) None \(0\) \(4\) \(-12\) \(0\) \(+\) \(-\) \(q+4q^{3}-12q^{5}-11q^{9}+12q^{11}+\cdots\)
392.4.a.e \(1\) \(23.129\) \(\Q\) None \(0\) \(4\) \(2\) \(0\) \(+\) \(-\) \(q+4q^{3}+2q^{5}-11q^{9}-44q^{11}-22q^{13}+\cdots\)
392.4.a.f \(2\) \(23.129\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+\beta q^{3}-10\beta q^{5}-5^{2}q^{9}+54q^{11}+\cdots\)
392.4.a.g \(2\) \(23.129\) \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) \(-\) \(-\) \(q+\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}-\beta q^{13}+\cdots\)
392.4.a.h \(2\) \(23.129\) \(\Q(\sqrt{57}) \) None \(0\) \(2\) \(-22\) \(0\) \(-\) \(-\) \(q+(1+\beta )q^{3}+(-11-\beta )q^{5}+(31+2\beta )q^{9}+\cdots\)
392.4.a.i \(3\) \(23.129\) 3.3.1929.1 None \(0\) \(-7\) \(-3\) \(0\) \(-\) \(+\) \(q+(-2+\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\cdots\)
392.4.a.j \(3\) \(23.129\) 3.3.1929.1 None \(0\) \(-1\) \(-13\) \(0\) \(+\) \(-\) \(q+\beta _{2}q^{3}+(-4-\beta _{1})q^{5}+(15+3\beta _{1}+\cdots)q^{9}+\cdots\)
392.4.a.k \(3\) \(23.129\) 3.3.1929.1 None \(0\) \(1\) \(13\) \(0\) \(+\) \(+\) \(q-\beta _{2}q^{3}+(4+\beta _{1})q^{5}+(15+3\beta _{1}-2\beta _{2})q^{9}+\cdots\)
392.4.a.l \(3\) \(23.129\) 3.3.1929.1 None \(0\) \(7\) \(3\) \(0\) \(-\) \(-\) \(q+(2-\beta _{1})q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+(4+\cdots)q^{9}+\cdots\)
392.4.a.m \(4\) \(23.129\) \(\Q(\sqrt{2}, \sqrt{65})\) None \(0\) \(0\) \(0\) \(0\) \(-\) \(+\) \(q+(2\beta _{1}+\beta _{3})q^{3}+(-3\beta _{1}-\beta _{3})q^{5}+\cdots\)
392.4.a.n \(4\) \(23.129\) \(\Q(\sqrt{2}, \sqrt{113})\) None \(0\) \(0\) \(0\) \(0\) \(+\) \(+\) \(q+(2\beta _{1}+\beta _{3})q^{3}+(-9\beta _{1}+\beta _{3})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(392))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(392)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)