Properties

Label 49.10.a
Level $49$
Weight $10$
Character orbit 49.a
Rep. character $\chi_{49}(1,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $7$
Sturm bound $46$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(46\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(49))\).

Total New Old
Modular forms 46 33 13
Cusp forms 38 28 10
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.

\(7\)TotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(22\)\(15\)\(7\)\(18\)\(13\)\(5\)\(4\)\(2\)\(2\)
\(-\)\(24\)\(18\)\(6\)\(20\)\(15\)\(5\)\(4\)\(3\)\(1\)

Trace form

\( 28 q - 32 q^{2} + 2 q^{3} + 6828 q^{4} + 684 q^{5} - 926 q^{6} - 33660 q^{8} + 192958 q^{9} + 54476 q^{10} - 14354 q^{11} + 54250 q^{12} + 46312 q^{13} + 91970 q^{15} + 1275700 q^{16} - 552774 q^{17} - 757708 q^{18}+ \cdots - 2046116492 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(49))\) 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
49.10.a.a 49.a 1.a $1$ $25.237$ \(\Q\) \(\Q(\sqrt{-7}) \) 49.10.a.a \(-5\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q-5q^{2}-487q^{4}+4995q^{8}-3^{9}q^{9}+\cdots\)
49.10.a.b 49.a 1.a $2$ $25.237$ \(\Q(\sqrt{193}) \) None 7.10.a.a \(-6\) \(86\) \(2238\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{2}+(43-11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
49.10.a.c 49.a 1.a $3$ $25.237$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 7.10.a.b \(21\) \(-84\) \(-1554\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(7-\beta _{2})q^{2}+(-28+\beta _{1}+\beta _{2})q^{3}+\cdots\)
49.10.a.d 49.a 1.a $4$ $25.237$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None 49.10.a.d \(-12\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{3})q^{2}-\beta _{1}q^{3}+(698-6\beta _{3})q^{4}+\cdots\)
49.10.a.e 49.a 1.a $5$ $25.237$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 7.10.c.a \(18\) \(-161\) \(-1533\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(4-\beta _{1})q^{2}+(-33+\beta _{1}+\beta _{2})q^{3}+\cdots\)
49.10.a.f 49.a 1.a $5$ $25.237$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 7.10.c.a \(18\) \(161\) \(1533\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(4-\beta _{1})q^{2}+(33-\beta _{1}-\beta _{2})q^{3}+(191+\cdots)q^{4}+\cdots\)
49.10.a.g 49.a 1.a $8$ $25.237$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 49.10.a.g \(-66\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(-8+\beta _{1})q^{2}+(-\beta _{3}-\beta _{5})q^{3}+(209+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(49))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(49)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)