Properties

Label 112.2.a
Level $112$
Weight $2$
Character orbit 112.a
Rep. character $\chi_{112}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $32$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 112.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 \(M_{2}(\Gamma_0(112))\).

Total New Old
Modular forms 22 3 19
Cusp forms 11 3 8
Eisenstein series 11 0 11

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

Trace form

\( 3 q - 2 q^{5} - q^{7} - q^{9} + O(q^{10}) \) \( 3 q - 2 q^{5} - q^{7} - q^{9} + 4 q^{11} - 2 q^{13} + 8 q^{15} - 2 q^{17} - 8 q^{19} - 8 q^{23} + 5 q^{25} + 2 q^{29} - 8 q^{31} + 6 q^{35} - 6 q^{37} - 8 q^{39} + 6 q^{41} - 12 q^{43} - 10 q^{45} + 24 q^{47} + 3 q^{49} + 16 q^{51} + 2 q^{53} + 8 q^{55} - 8 q^{57} + 6 q^{61} - 5 q^{63} + 4 q^{65} + 20 q^{67} + 16 q^{69} + 8 q^{71} - 2 q^{73} - 32 q^{75} + 4 q^{77} - 16 q^{79} - 13 q^{81} - 8 q^{83} - 4 q^{85} - 16 q^{87} - 2 q^{89} + 6 q^{91} + 16 q^{93} - 24 q^{95} - 18 q^{97} - 12 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(112))\) 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 7
112.2.a.a 112.a 1.a $1$ $0.894$ \(\Q\) None \(0\) \(-2\) \(-4\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-4q^{5}-q^{7}+q^{9}+8q^{15}+\cdots\)
112.2.a.b 112.a 1.a $1$ $0.894$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+q^{7}-3q^{9}+4q^{11}+2q^{13}+\cdots\)
112.2.a.c 112.a 1.a $1$ $0.894$ \(\Q\) None \(0\) \(2\) \(0\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{7}+q^{9}-4q^{13}+6q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(112))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(112)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 2}\)