Properties

Label 784.2.a
Level $784$
Weight $2$
Character orbit 784.a
Rep. character $\chi_{784}(1,\cdot)$
Character field $\Q$
Dimension $18$
Newform subspaces $14$
Sturm bound $224$
Trace bound $11$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(784))\).

Total New Old
Modular forms 136 23 113
Cusp forms 89 18 71
Eisenstein series 47 5 42

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.
\(+\)\(+\)\(+\)\(4\)
\(+\)\(-\)\(-\)\(6\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(11\)

Trace form

\( 18 q + 2 q^{5} + 16 q^{9} + O(q^{10}) \) \( 18 q + 2 q^{5} + 16 q^{9} - 6 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 8 q^{19} + 6 q^{23} + 4 q^{25} - 4 q^{29} + 8 q^{31} + 10 q^{37} + 20 q^{39} - 6 q^{41} + 16 q^{43} + 10 q^{45} - 24 q^{47} + 14 q^{51} + 2 q^{53} - 8 q^{55} - 6 q^{57} - 6 q^{61} - 12 q^{65} - 30 q^{67} - 16 q^{69} + 8 q^{71} + 2 q^{73} + 32 q^{75} + 6 q^{79} + 2 q^{81} + 8 q^{83} - 10 q^{85} + 16 q^{87} + 2 q^{89} - 42 q^{93} + 66 q^{95} + 18 q^{97} + 20 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(784))\) 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
784.2.a.a 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(-3\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}+6q^{9}+q^{11}+2q^{13}+\cdots\)
784.2.a.b 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+4q^{13}-6q^{17}+2q^{19}+\cdots\)
784.2.a.c 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
784.2.a.d 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(-1\) \(3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
784.2.a.e 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(0\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
784.2.a.f 784.a 1.a $1$ $6.260$ \(\Q\) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q-3q^{9}-4q^{11}-8q^{23}-5q^{25}+2q^{29}+\cdots\)
784.2.a.g 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(1\) \(-3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
784.2.a.h 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(1\) \(-1\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
784.2.a.i 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(2\) \(4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{5}+q^{9}+8q^{15}+2q^{17}+\cdots\)
784.2.a.j 784.a 1.a $1$ $6.260$ \(\Q\) None \(0\) \(3\) \(1\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+6q^{9}+q^{11}-2q^{13}+\cdots\)
784.2.a.k 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}-q^{9}-6q^{11}+4\beta q^{13}+\cdots\)
784.2.a.l 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}-q^{9}+2q^{11}+4q^{15}+\cdots\)
784.2.a.m 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
784.2.a.n 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{5}+5q^{9}+4q^{11}+\beta q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(784))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(784)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)