Properties

Label 50.12.a
Level $50$
Weight $12$
Character orbit 50.a
Rep. character $\chi_{50}(1,\cdot)$
Character field $\Q$
Dimension $17$
Newform subspaces $10$
Sturm bound $90$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(90\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(50))\).

Total New Old
Modular forms 89 17 72
Cusp forms 77 17 60
Eisenstein series 12 0 12

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

\(2\)\(5\)FrickeDim
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(5\)
Plus space\(+\)\(10\)
Minus space\(-\)\(7\)

Trace form

\( 17 q - 32 q^{2} - 1012 q^{3} + 17408 q^{4} + 12608 q^{6} + 45224 q^{7} - 32768 q^{8} + 871259 q^{9} - 687626 q^{11} - 1036288 q^{12} + 2191418 q^{13} + 2201984 q^{14} + 17825792 q^{16} - 12748506 q^{17}+ \cdots - 368375403752 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(50))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
50.12.a.a 50.a 1.a $1$ $38.417$ \(\Q\) None 50.12.a.a \(-32\) \(-207\) \(0\) \(19514\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}-207q^{3}+2^{10}q^{4}+6624q^{6}+\cdots\)
50.12.a.b 50.a 1.a $1$ $38.417$ \(\Q\) None 10.12.a.c \(-32\) \(318\) \(0\) \(70714\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+318q^{3}+2^{10}q^{4}-10176q^{6}+\cdots\)
50.12.a.c 50.a 1.a $1$ $38.417$ \(\Q\) None 10.12.a.b \(32\) \(-738\) \(0\) \(-25574\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}-738q^{3}+2^{10}q^{4}-23616q^{6}+\cdots\)
50.12.a.d 50.a 1.a $1$ $38.417$ \(\Q\) None 10.12.a.a \(32\) \(12\) \(0\) \(14176\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+12q^{3}+2^{10}q^{4}+384q^{6}+\cdots\)
50.12.a.e 50.a 1.a $1$ $38.417$ \(\Q\) None 50.12.a.a \(32\) \(207\) \(0\) \(-19514\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+207q^{3}+2^{10}q^{4}+6624q^{6}+\cdots\)
50.12.a.f 50.a 1.a $2$ $38.417$ \(\Q(\sqrt{1969}) \) None 10.12.a.d \(-64\) \(-604\) \(0\) \(-14092\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+(-302-\beta )q^{3}+2^{10}q^{4}+\cdots\)
50.12.a.g 50.a 1.a $2$ $38.417$ \(\Q(\sqrt{10129}) \) None 50.12.a.g \(-64\) \(56\) \(0\) \(-54712\) $+$ $+$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+(28-\beta )q^{3}+2^{10}q^{4}+(-896+\cdots)q^{6}+\cdots\)
50.12.a.h 50.a 1.a $2$ $38.417$ \(\Q(\sqrt{10129}) \) None 50.12.a.g \(64\) \(-56\) \(0\) \(54712\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+(-28-\beta )q^{3}+2^{10}q^{4}+\cdots\)
50.12.a.i 50.a 1.a $3$ $38.417$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 10.12.b.a \(-96\) \(-266\) \(0\) \(-33218\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2^{5}q^{2}+(-89+\beta _{1})q^{3}+2^{10}q^{4}+\cdots\)
50.12.a.j 50.a 1.a $3$ $38.417$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None 10.12.b.a \(96\) \(266\) \(0\) \(33218\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2^{5}q^{2}+(89-\beta _{1})q^{3}+2^{10}q^{4}+(2848+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(50))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(50)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)