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Properties

Label 816.2.a
Level $816$
Weight $2$
Character orbit 816.a
Rep. character $\chi_{816}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $13$
Sturm bound $288$
Trace bound $19$

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Defining parameters

Level: \( N \) \(=\) \( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 816.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(288\)
Trace bound: \(19\)
Distinguishing \(T_p\): \(5\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(816))\).

Total New Old
Modular forms 156 16 140
Cusp forms 133 16 117
Eisenstein series 23 0 23

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

\(2\)\(3\)\(17\)FrickeDim
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(5\)
Minus space\(-\)\(11\)

Trace form

\( 16 q - 2 q^{3} + 16 q^{9} + O(q^{10}) \) \( 16 q - 2 q^{3} + 16 q^{9} + 8 q^{11} - 4 q^{19} + 8 q^{23} + 24 q^{25} - 2 q^{27} - 16 q^{29} + 8 q^{31} + 8 q^{33} + 24 q^{35} - 16 q^{37} + 4 q^{39} + 16 q^{41} + 20 q^{43} + 24 q^{47} + 32 q^{49} - 6 q^{51} - 16 q^{53} + 4 q^{55} - 8 q^{57} + 16 q^{65} + 16 q^{67} + 8 q^{71} + 2 q^{75} - 16 q^{77} + 16 q^{81} + 40 q^{83} - 12 q^{87} + 32 q^{89} - 56 q^{91} + 16 q^{95} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(816))\) 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 3 17
816.2.a.a 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(-1\) \(-3\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}+4q^{7}+q^{9}-q^{11}-5q^{13}+\cdots\)
816.2.a.b 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)
816.2.a.c 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}+2q^{13}-q^{17}+\cdots\)
816.2.a.d 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(-1\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}+2q^{13}-q^{17}+\cdots\)
816.2.a.e 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{9}-5q^{11}-5q^{13}+\cdots\)
816.2.a.f 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(-1\) \(2\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
816.2.a.g 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(-1\) \(3\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}+4q^{7}+q^{9}+3q^{11}+\cdots\)
816.2.a.h 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(1\) \(-4\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}+2q^{7}+q^{9}-6q^{13}+\cdots\)
816.2.a.i 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(1\) \(-1\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-4q^{7}+q^{9}-3q^{11}+3q^{13}+\cdots\)
816.2.a.j 816.a 1.a $1$ $6.516$ \(\Q\) None \(0\) \(1\) \(3\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+3q^{5}+q^{9}+q^{11}+3q^{13}+\cdots\)
816.2.a.k 816.a 1.a $2$ $6.516$ \(\Q(\sqrt{57}) \) None \(0\) \(-2\) \(-1\) \(-8\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-\beta q^{5}-4q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
816.2.a.l 816.a 1.a $2$ $6.516$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(-1\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-\beta q^{5}+(2-2\beta )q^{7}+q^{9}+(4+\cdots)q^{11}+\cdots\)
816.2.a.m 816.a 1.a $2$ $6.516$ \(\Q(\sqrt{17}) \) None \(0\) \(2\) \(3\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+(1+\beta )q^{5}+q^{9}+(1-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(816))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(816)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)