Properties

Label 1568.2.a
Level $1568$
Weight $2$
Character orbit 1568.a
Rep. character $\chi_{1568}(1,\cdot)$
Character field $\Q$
Dimension $41$
Newform subspaces $24$
Sturm bound $448$
Trace bound $25$

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Defining parameters

Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 24 \)
Sturm bound: \(448\)
Trace bound: \(25\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1568))\).

Total New Old
Modular forms 256 41 215
Cusp forms 193 41 152
Eisenstein series 63 0 63

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

\(2\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(8\)
\(+\)\(-\)\(-\)\(12\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(9\)
Plus space\(+\)\(17\)
Minus space\(-\)\(24\)

Trace form

\( 41 q - 2 q^{5} + 37 q^{9} + O(q^{10}) \) \( 41 q - 2 q^{5} + 37 q^{9} - 10 q^{13} + 2 q^{17} + 55 q^{25} + 6 q^{29} + 16 q^{33} - 18 q^{37} + 26 q^{41} + 22 q^{45} + 30 q^{53} + 16 q^{57} - 10 q^{61} + 36 q^{65} + 48 q^{69} - 6 q^{73} + 33 q^{81} - 20 q^{85} - 22 q^{89} + 48 q^{93} - 46 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1568))\) 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 7
1568.2.a.a 1568.a 1.a $1$ $12.521$ \(\Q\) None \(0\) \(-2\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots\)
1568.2.a.b 1568.a 1.a $1$ $12.521$ \(\Q\) None \(0\) \(-2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+4q^{11}+4q^{13}+2q^{17}+\cdots\)
1568.2.a.c 1568.a 1.a $1$ $12.521$ \(\Q\) None \(0\) \(-2\) \(2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots\)
1568.2.a.d 1568.a 1.a $1$ $12.521$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-4q^{5}-3q^{9}+4q^{13}-8q^{17}+11q^{25}+\cdots\)
1568.2.a.e 1568.a 1.a $1$ $12.521$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(2\) \(0\) $-$ $-$ $N(\mathrm{U}(1))$ \(q+2q^{5}-3q^{9}-6q^{13}-2q^{17}-q^{25}+\cdots\)
1568.2.a.f 1568.a 1.a $1$ $12.521$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) $+$ $-$ $N(\mathrm{U}(1))$ \(q+4q^{5}-3q^{9}-4q^{13}+8q^{17}+11q^{25}+\cdots\)
1568.2.a.g 1568.a 1.a $1$ $12.521$ \(\Q\) None \(0\) \(2\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{5}+q^{9}-4q^{11}-6q^{13}+\cdots\)
1568.2.a.h 1568.a 1.a $1$ $12.521$ \(\Q\) None \(0\) \(2\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{9}-4q^{11}+4q^{13}+2q^{17}+\cdots\)
1568.2.a.i 1568.a 1.a $1$ $12.521$ \(\Q\) None \(0\) \(2\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots\)
1568.2.a.j 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}-2\beta q^{9}+\cdots\)
1568.2.a.k 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
1568.2.a.l 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) None \(0\) \(-2\) \(2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}-2\beta q^{9}+\cdots\)
1568.2.a.m 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(-6\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-3q^{5}+4q^{9}-\beta q^{11}-4q^{13}+\cdots\)
1568.2.a.n 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{5}-3\beta q^{11}-\beta q^{15}-5q^{17}+\cdots\)
1568.2.a.o 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $+$ $+$ $N(\mathrm{U}(1))$ \(q+\beta q^{5}-3q^{9}-\beta q^{13}-5\beta q^{17}-3q^{25}+\cdots\)
1568.2.a.p 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $-$ $+$ $N(\mathrm{U}(1))$ \(q+3\beta q^{5}-3q^{9}+5\beta q^{13}-3\beta q^{17}+\cdots\)
1568.2.a.q 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{9}-2q^{11}-2\beta q^{13}-3\beta q^{17}+\cdots\)
1568.2.a.r 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-q^{9}+2q^{11}+2\beta q^{13}+3\beta q^{17}+\cdots\)
1568.2.a.s 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}+3\beta q^{11}+\beta q^{15}+5q^{17}+\cdots\)
1568.2.a.t 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+3q^{5}+4q^{9}+\beta q^{11}+4q^{13}+\cdots\)
1568.2.a.u 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+2\beta q^{9}+\cdots\)
1568.2.a.v 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots\)
1568.2.a.w 1568.a 1.a $2$ $12.521$ \(\Q(\sqrt{2}) \) None \(0\) \(2\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+2\beta q^{9}+(-1+\cdots)q^{11}+\cdots\)
1568.2.a.x 1568.a 1.a $4$ $12.521$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}-2\beta _{1}q^{5}+7q^{9}-\beta _{3}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1568))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1568)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(112))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(224))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(784))\)\(^{\oplus 2}\)