Properties

Label 49.4.a
Level $49$
Weight $4$
Character orbit 49.a
Rep. character $\chi_{49}(1,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $5$
Sturm bound $18$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(18\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(49))\).

Total New Old
Modular forms 18 13 5
Cusp forms 10 8 2
Eisenstein series 8 5 3

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

\(7\)Dim
\(+\)\(5\)
\(-\)\(3\)

Trace form

\( 8 q + 2 q^{3} + 36 q^{4} - 16 q^{5} - 2 q^{6} - 12 q^{8} + 34 q^{9} + 16 q^{10} + 14 q^{11} - 14 q^{12} - 28 q^{13} + 2 q^{15} - 76 q^{16} - 54 q^{17} - 124 q^{18} + 110 q^{19} + 112 q^{20} - 12 q^{22}+ \cdots + 5140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(49))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7
49.4.a.a 49.a 1.a $1$ $2.891$ \(\Q\) \(\Q(\sqrt{-7}) \) 49.4.a.a \(-5\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q-5q^{2}+17q^{4}-45q^{8}-3^{3}q^{9}-68q^{11}+\cdots\)
49.4.a.b 49.a 1.a $1$ $2.891$ \(\Q\) None 7.4.a.a \(-1\) \(2\) \(-16\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-7q^{4}-2^{4}q^{5}-2q^{6}+\cdots\)
49.4.a.c 49.a 1.a $1$ $2.891$ \(\Q\) None 7.4.c.a \(2\) \(-7\) \(-7\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-7q^{3}-4q^{4}-7q^{5}-14q^{6}+\cdots\)
49.4.a.d 49.a 1.a $1$ $2.891$ \(\Q\) None 7.4.c.a \(2\) \(7\) \(7\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+7q^{3}-4q^{4}+7q^{5}+14q^{6}+\cdots\)
49.4.a.e 49.a 1.a $4$ $2.891$ \(\Q(\sqrt{2}, \sqrt{65})\) None 49.4.a.e \(2\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+\beta _{2}q^{3}+(9+\beta _{1})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)