Properties

Label 672.2.a
Level $672$
Weight $2$
Character orbit 672.a
Rep. character $\chi_{672}(1,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $10$
Sturm bound $256$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 672.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(256\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(672))\).

Total New Old
Modular forms 144 12 132
Cusp forms 113 12 101
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.

\(2\)\(3\)\(7\)FrickeDim
\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(1\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(1\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(8\)

Trace form

\( 12 q - 8 q^{5} + 12 q^{9} + O(q^{10}) \) \( 12 q - 8 q^{5} + 12 q^{9} - 8 q^{13} + 24 q^{17} + 36 q^{25} - 8 q^{29} - 8 q^{37} + 24 q^{41} - 8 q^{45} + 12 q^{49} - 8 q^{53} - 8 q^{61} + 48 q^{65} + 24 q^{73} + 12 q^{81} - 32 q^{85} - 8 q^{89} - 16 q^{93} - 8 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(672))\) 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 7
672.2.a.a 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(-1\) \(-4\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}-q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
672.2.a.b 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(-1\) \(-2\) \(1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
672.2.a.c 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(-1\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
672.2.a.d 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(-1\) \(2\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-q^{7}+q^{9}+2q^{13}-2q^{15}+\cdots\)
672.2.a.e 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(1\) \(-4\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}+q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
672.2.a.f 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(1\) \(-2\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}-q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
672.2.a.g 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(1\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
672.2.a.h 672.a 1.a $1$ $5.366$ \(\Q\) None \(0\) \(1\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{7}+q^{9}+2q^{13}+2q^{15}+\cdots\)
672.2.a.i 672.a 1.a $2$ $5.366$ \(\Q(\sqrt{3}) \) None \(0\) \(-2\) \(0\) \(2\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
672.2.a.j 672.a 1.a $2$ $5.366$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(0\) \(-2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}-q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(672))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(672)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(96))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)