Properties

Label 312.2.a
Level $312$
Weight $2$
Character orbit 312.a
Rep. character $\chi_{312}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $6$
Sturm bound $112$
Trace bound $5$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 312 = 2^{3} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 312.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(112\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(312))\).

Total New Old
Modular forms 64 6 58
Cusp forms 49 6 43
Eisenstein series 15 0 15

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

\(2\)\(3\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(2\)
Minus space\(-\)\(4\)

Trace form

\( 6 q - 4 q^{7} + 6 q^{9} + O(q^{10}) \) \( 6 q - 4 q^{7} + 6 q^{9} - 2 q^{13} - 4 q^{15} - 4 q^{17} + 12 q^{19} - 4 q^{21} + 24 q^{23} + 10 q^{25} - 4 q^{29} - 4 q^{31} + 8 q^{33} + 8 q^{35} - 20 q^{37} - 8 q^{41} - 16 q^{43} - 16 q^{47} + 6 q^{49} + 4 q^{53} - 12 q^{57} - 24 q^{59} + 4 q^{61} - 4 q^{63} + 4 q^{67} - 8 q^{69} - 8 q^{71} - 12 q^{73} + 16 q^{77} - 24 q^{79} + 6 q^{81} + 24 q^{83} + 32 q^{85} - 8 q^{87} - 32 q^{89} + 12 q^{91} + 12 q^{93} - 8 q^{95} - 12 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(312))\) 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 13
312.2.a.a 312.a 1.a $1$ $2.491$ \(\Q\) None \(0\) \(-1\) \(-2\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+4q^{7}+q^{9}+q^{13}+2q^{15}+\cdots\)
312.2.a.b 312.a 1.a $1$ $2.491$ \(\Q\) None \(0\) \(-1\) \(0\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.c 312.a 1.a $1$ $2.491$ \(\Q\) None \(0\) \(-1\) \(4\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{5}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.d 312.a 1.a $1$ $2.491$ \(\Q\) None \(0\) \(1\) \(-4\) \(-4\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-4q^{7}+q^{9}-2q^{11}+\cdots\)
312.2.a.e 312.a 1.a $1$ $2.491$ \(\Q\) None \(0\) \(1\) \(0\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+6q^{11}-q^{13}+2q^{17}+\cdots\)
312.2.a.f 312.a 1.a $1$ $2.491$ \(\Q\) None \(0\) \(1\) \(2\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{9}+q^{13}+2q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(312)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 2}\)