Properties

Label 56.4.a
Level $56$
Weight $4$
Character orbit 56.a
Rep. character $\chi_{56}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $3$
Sturm bound $32$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 56.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(56))\).

Total New Old
Modular forms 28 4 24
Cusp forms 20 4 16
Eisenstein series 8 0 8

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

Trace form

\( 4q + 2q^{3} + 14q^{5} + 48q^{9} + O(q^{10}) \) \( 4q + 2q^{3} + 14q^{5} + 48q^{9} + 116q^{11} - 54q^{13} - 56q^{15} - 28q^{17} - 90q^{19} - 42q^{21} - 24q^{23} + 176q^{25} - 244q^{27} - 452q^{29} - 196q^{31} - 432q^{33} + 210q^{35} + 636q^{37} + 496q^{39} - 132q^{41} + 444q^{43} + 1350q^{45} - 924q^{47} + 196q^{49} + 28q^{51} - 536q^{53} + 1144q^{55} - 828q^{57} - 602q^{59} - 530q^{61} + 532q^{63} + 756q^{65} + 1256q^{67} - 1280q^{69} + 248q^{71} + 2480q^{73} - 3242q^{75} - 308q^{77} - 616q^{79} - 1224q^{81} - 162q^{83} - 2180q^{85} - 3436q^{87} + 488q^{89} + 966q^{91} + 1208q^{93} + 1512q^{95} - 1212q^{97} + 2436q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(56))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 7
56.4.a.a \(1\) \(3.304\) \(\Q\) None \(0\) \(-2\) \(-16\) \(-7\) \(-\) \(+\) \(q-2q^{3}-2^{4}q^{5}-7q^{7}-23q^{9}+24q^{11}+\cdots\)
56.4.a.b \(1\) \(3.304\) \(\Q\) None \(0\) \(6\) \(8\) \(-7\) \(+\) \(+\) \(q+6q^{3}+8q^{5}-7q^{7}+9q^{9}+56q^{11}+\cdots\)
56.4.a.c \(2\) \(3.304\) \(\Q(\sqrt{57}) \) None \(0\) \(-2\) \(22\) \(14\) \(-\) \(-\) \(q+(-1-\beta )q^{3}+(11+\beta )q^{5}+7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces

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