Properties

Label 208.2.a
Level $208$
Weight $2$
Character orbit 208.a
Rep. character $\chi_{208}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $5$
Sturm bound $56$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 208.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(56\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(208))\).

Total New Old
Modular forms 34 6 28
Cusp forms 23 6 17
Eisenstein series 11 0 11

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

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

Trace form

\( 6 q - 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 6 q - 2 q^{7} + 2 q^{9} + 2 q^{11} + 8 q^{15} - 4 q^{17} - 2 q^{19} - 8 q^{21} + 8 q^{23} - 2 q^{25} + 12 q^{27} - 14 q^{31} - 8 q^{33} + 8 q^{37} - 4 q^{39} + 4 q^{41} - 12 q^{43} - 8 q^{45} - 10 q^{47} - 2 q^{49} - 28 q^{51} + 4 q^{53} + 4 q^{55} + 18 q^{59} + 12 q^{61} + 6 q^{63} - 26 q^{67} + 8 q^{69} - 6 q^{71} - 20 q^{73} + 12 q^{75} + 4 q^{77} - 28 q^{79} + 6 q^{81} + 22 q^{83} + 8 q^{87} + 4 q^{89} + 6 q^{91} - 16 q^{93} + 36 q^{95} - 4 q^{97} + 34 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(208))\) 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 13
208.2.a.a 208.a 1.a $1$ $1.661$ \(\Q\) None \(0\) \(-1\) \(-3\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-3q^{5}+q^{7}-2q^{9}-6q^{11}+\cdots\)
208.2.a.b 208.a 1.a $1$ $1.661$ \(\Q\) None \(0\) \(-1\) \(-1\) \(-5\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-5q^{7}-2q^{9}+2q^{11}+\cdots\)
208.2.a.c 208.a 1.a $1$ $1.661$ \(\Q\) None \(0\) \(0\) \(2\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{5}+2q^{7}-3q^{9}+2q^{11}-q^{13}+\cdots\)
208.2.a.d 208.a 1.a $1$ $1.661$ \(\Q\) None \(0\) \(3\) \(-1\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}-q^{5}-q^{7}+6q^{9}+2q^{11}+\cdots\)
208.2.a.e 208.a 1.a $2$ $1.661$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(3\) \(1\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(2-\beta )q^{5}+\beta q^{7}+(1+\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(208))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(208)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 2}\)