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Properties

Label 840.2.a
Level $840$
Weight $2$
Character orbit 840.a
Rep. character $\chi_{840}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $11$
Sturm bound $384$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 840.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(384\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(840))\).

Total New Old
Modular forms 208 12 196
Cusp forms 177 12 165
Eisenstein series 31 0 31

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

\(2\)\(3\)\(5\)\(7\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(+\)\(+\)\(11\)\(1\)\(10\)\(10\)\(1\)\(9\)\(1\)\(0\)\(1\)
\(+\)\(+\)\(+\)\(-\)\(-\)\(14\)\(1\)\(13\)\(12\)\(1\)\(11\)\(2\)\(0\)\(2\)
\(+\)\(+\)\(-\)\(+\)\(-\)\(12\)\(1\)\(11\)\(10\)\(1\)\(9\)\(2\)\(0\)\(2\)
\(+\)\(+\)\(-\)\(-\)\(+\)\(13\)\(0\)\(13\)\(11\)\(0\)\(11\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(+\)\(+\)\(-\)\(13\)\(1\)\(12\)\(11\)\(1\)\(10\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(+\)\(14\)\(0\)\(14\)\(12\)\(0\)\(12\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(-\)\(+\)\(+\)\(12\)\(0\)\(12\)\(10\)\(0\)\(10\)\(2\)\(0\)\(2\)
\(+\)\(-\)\(-\)\(-\)\(-\)\(15\)\(2\)\(13\)\(13\)\(2\)\(11\)\(2\)\(0\)\(2\)
\(-\)\(+\)\(+\)\(+\)\(-\)\(15\)\(0\)\(15\)\(13\)\(0\)\(13\)\(2\)\(0\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(+\)\(12\)\(1\)\(11\)\(10\)\(1\)\(9\)\(2\)\(0\)\(2\)
\(-\)\(+\)\(-\)\(+\)\(+\)\(14\)\(1\)\(13\)\(12\)\(1\)\(11\)\(2\)\(0\)\(2\)
\(-\)\(+\)\(-\)\(-\)\(-\)\(13\)\(1\)\(12\)\(11\)\(1\)\(10\)\(2\)\(0\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(+\)\(13\)\(1\)\(12\)\(11\)\(1\)\(10\)\(2\)\(0\)\(2\)
\(-\)\(-\)\(+\)\(-\)\(-\)\(12\)\(1\)\(11\)\(10\)\(1\)\(9\)\(2\)\(0\)\(2\)
\(-\)\(-\)\(-\)\(+\)\(-\)\(14\)\(1\)\(13\)\(12\)\(1\)\(11\)\(2\)\(0\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(+\)\(11\)\(0\)\(11\)\(9\)\(0\)\(9\)\(2\)\(0\)\(2\)
Plus space\(+\)\(100\)\(4\)\(96\)\(85\)\(4\)\(81\)\(15\)\(0\)\(15\)
Minus space\(-\)\(108\)\(8\)\(100\)\(92\)\(8\)\(84\)\(16\)\(0\)\(16\)

Trace form

\( 12 q + 12 q^{9} + 16 q^{23} + 12 q^{25} + 24 q^{31} + 8 q^{37} + 8 q^{39} + 16 q^{41} - 16 q^{43} - 32 q^{47} + 12 q^{49} + 8 q^{51} + 16 q^{53} + 8 q^{55} + 8 q^{57} - 32 q^{59} + 16 q^{61} - 24 q^{71}+ \cdots - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(840))\) 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 3 5 7
840.2.a.a 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.a \(0\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
840.2.a.b 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.b \(0\) \(-1\) \(-1\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
840.2.a.c 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.c \(0\) \(-1\) \(-1\) \(1\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}+q^{7}+q^{9}-2q^{13}+q^{15}+\cdots\)
840.2.a.d 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.d \(0\) \(-1\) \(1\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-q^{7}+q^{9}-4q^{11}-2q^{13}+\cdots\)
840.2.a.e 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.e \(0\) \(-1\) \(1\) \(-1\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
840.2.a.f 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.f \(0\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
840.2.a.g 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.g \(0\) \(1\) \(-1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
840.2.a.h 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.h \(0\) \(1\) \(-1\) \(-1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-q^{7}+q^{9}+6q^{13}-q^{15}+\cdots\)
840.2.a.i 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.i \(0\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}+q^{7}+q^{9}+2q^{13}-q^{15}+\cdots\)
840.2.a.j 840.a 1.a $1$ $6.707$ \(\Q\) None 840.2.a.j \(0\) \(1\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}-q^{7}+q^{9}+4q^{11}-2q^{13}+\cdots\)
840.2.a.k 840.a 1.a $2$ $6.707$ \(\Q(\sqrt{2}) \) None 840.2.a.k \(0\) \(2\) \(2\) \(2\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{5}+q^{7}+q^{9}+\beta q^{11}+2q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(840))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(840)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(280))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 2}\)