Properties

Label 1408.2.a
Level $1408$
Weight $2$
Character orbit 1408.a
Rep. character $\chi_{1408}(1,\cdot)$
Character field $\Q$
Dimension $40$
Newform subspaces $20$
Sturm bound $384$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1408 = 2^{7} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1408.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 20 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1408))\).

Total New Old
Modular forms 208 40 168
Cusp forms 177 40 137
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.

\(2\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(11\)
\(-\)\(+\)\(-\)\(11\)
\(-\)\(-\)\(+\)\(9\)
Plus space\(+\)\(18\)
Minus space\(-\)\(22\)

Trace form

\( 40 q + 40 q^{9} + O(q^{10}) \) \( 40 q + 40 q^{9} + 16 q^{17} + 56 q^{25} + 16 q^{41} - 24 q^{49} - 64 q^{57} - 32 q^{65} - 112 q^{73} + 104 q^{81} + 16 q^{89} - 48 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1408))\) 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 11
1408.2.a.a 1408.a 1.a $1$ $11.243$ \(\Q\) None 1408.2.a.a \(0\) \(0\) \(-2\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-4q^{7}-3q^{9}-q^{11}+6q^{17}+\cdots\)
1408.2.a.b 1408.a 1.a $1$ $11.243$ \(\Q\) None 1408.2.a.a \(0\) \(0\) \(-2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{5}+4q^{7}-3q^{9}+q^{11}+6q^{17}+\cdots\)
1408.2.a.c 1408.a 1.a $1$ $11.243$ \(\Q\) None 1408.2.a.a \(0\) \(0\) \(2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-4q^{7}-3q^{9}+q^{11}+6q^{17}+\cdots\)
1408.2.a.d 1408.a 1.a $1$ $11.243$ \(\Q\) None 1408.2.a.a \(0\) \(0\) \(2\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}+4q^{7}-3q^{9}-q^{11}+6q^{17}+\cdots\)
1408.2.a.e 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{6}) \) None 1408.2.a.e \(0\) \(-2\) \(-6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-3q^{5}+\beta q^{7}+(4-2\beta )q^{9}+\cdots\)
1408.2.a.f 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.f \(0\) \(-2\) \(-2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}-\beta q^{7}+\cdots\)
1408.2.a.g 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.g \(0\) \(-2\) \(-2\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}+(2-\beta )q^{7}-2\beta q^{9}+\cdots\)
1408.2.a.h 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.g \(0\) \(-2\) \(2\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+q^{5}+(-2+\beta )q^{7}+\cdots\)
1408.2.a.i 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.f \(0\) \(-2\) \(2\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+(1-2\beta )q^{5}+\beta q^{7}+\cdots\)
1408.2.a.j 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{6}) \) None 1408.2.a.e \(0\) \(-2\) \(6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}+3q^{5}-\beta q^{7}+(4-2\beta )q^{9}+\cdots\)
1408.2.a.k 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{6}) \) None 1408.2.a.e \(0\) \(2\) \(-6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-3q^{5}+\beta q^{7}+(4+2\beta )q^{9}+\cdots\)
1408.2.a.l 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.g \(0\) \(2\) \(-2\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+(-2-\beta )q^{7}+2\beta q^{9}+\cdots\)
1408.2.a.m 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.f \(0\) \(2\) \(-2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-1-2\beta )q^{5}-\beta q^{7}+\cdots\)
1408.2.a.n 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.f \(0\) \(2\) \(2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(1+2\beta )q^{5}+\beta q^{7}+2\beta q^{9}+\cdots\)
1408.2.a.o 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{2}) \) None 1408.2.a.g \(0\) \(2\) \(2\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+(2+\beta )q^{7}+2\beta q^{9}+\cdots\)
1408.2.a.p 1408.a 1.a $2$ $11.243$ \(\Q(\sqrt{6}) \) None 1408.2.a.e \(0\) \(2\) \(6\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+3q^{5}-\beta q^{7}+(4+2\beta )q^{9}+\cdots\)
1408.2.a.q 1408.a 1.a $3$ $11.243$ 3.3.568.1 None 1408.2.a.q \(0\) \(-2\) \(-4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{2})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
1408.2.a.r 1408.a 1.a $3$ $11.243$ 3.3.568.1 None 1408.2.a.q \(0\) \(-2\) \(4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}+(1-\beta _{2})q^{5}+(\beta _{1}+\beta _{2})q^{7}+\cdots\)
1408.2.a.s 1408.a 1.a $3$ $11.243$ 3.3.568.1 None 1408.2.a.q \(0\) \(2\) \(-4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(-1+\beta _{2})q^{5}+(\beta _{1}+\beta _{2})q^{7}+\cdots\)
1408.2.a.t 1408.a 1.a $3$ $11.243$ 3.3.568.1 None 1408.2.a.q \(0\) \(2\) \(4\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{3}+(1-\beta _{2})q^{5}+(-\beta _{1}-\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1408))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1408)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 7}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(64))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(128))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(352))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(704))\)\(^{\oplus 2}\)