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Properties

Label 1008.2.a
Level $1008$
Weight $2$
Character orbit 1008.a
Rep. character $\chi_{1008}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $14$
Sturm bound $384$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(384\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(5\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1008))\).

Total New Old
Modular forms 216 15 201
Cusp forms 169 15 154
Eisenstein series 47 0 47

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
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(2\)
\(+\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(9\)

Trace form

\( 15 q - 2 q^{5} - q^{7} + O(q^{10}) \) \( 15 q - 2 q^{5} - q^{7} - 8 q^{11} - 2 q^{13} + 6 q^{17} + 8 q^{19} + 4 q^{23} + 17 q^{25} + 2 q^{29} + 8 q^{31} - 6 q^{35} + 2 q^{37} - 2 q^{41} + 20 q^{43} + 15 q^{49} + 2 q^{53} - 8 q^{55} + 24 q^{59} - 18 q^{61} + 20 q^{65} - 20 q^{67} + 12 q^{71} - 2 q^{73} + 4 q^{77} + 8 q^{79} - 8 q^{83} - 12 q^{85} + 6 q^{89} + 6 q^{91} + 8 q^{95} - 18 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1008))\) 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
1008.2.a.a 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(-4\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-4q^{5}+q^{7}+2q^{11}-6q^{13}+4q^{17}+\cdots\)
1008.2.a.b 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-q^{7}+6q^{13}+2q^{17}-4q^{19}+\cdots\)
1008.2.a.c 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(-2\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-q^{7}+6q^{11}-6q^{13}+2q^{17}+\cdots\)
1008.2.a.d 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+q^{7}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
1008.2.a.e 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+q^{7}-2q^{13}-6q^{17}+4q^{19}+\cdots\)
1008.2.a.f 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(-2\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+q^{7}+2q^{11}+2q^{13}-6q^{17}+\cdots\)
1008.2.a.g 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-6q^{11}+2q^{13}+4q^{19}-6q^{23}+\cdots\)
1008.2.a.h 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{7}-4q^{13}-6q^{17}-2q^{19}-5q^{25}+\cdots\)
1008.2.a.i 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(2\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}-q^{7}-6q^{11}-6q^{13}-2q^{17}+\cdots\)
1008.2.a.j 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+q^{7}-4q^{11}+6q^{13}-2q^{17}+\cdots\)
1008.2.a.k 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+q^{7}-2q^{11}+2q^{13}+6q^{17}+\cdots\)
1008.2.a.l 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(2\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}+q^{7}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
1008.2.a.m 1008.a 1.a $1$ $8.049$ \(\Q\) None \(0\) \(0\) \(4\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+4q^{5}-q^{7}+2q^{17}+2q^{19}+8q^{23}+\cdots\)
1008.2.a.n 1008.a 1.a $2$ $8.049$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{5}-q^{7}+\beta q^{11}+2q^{13}-\beta q^{17}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1008))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1008)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(168))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(336))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(504))\)\(^{\oplus 2}\)