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Properties

Label 210.2.a
Level $210$
Weight $2$
Character orbit 210.a
Rep. character $\chi_{210}(1,\cdot)$
Character field $\Q$
Dimension $5$
Newform subspaces $5$
Sturm bound $96$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 210 = 2 \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 210.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(96\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(11\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(210))\).

Total New Old
Modular forms 56 5 51
Cusp forms 41 5 36
Eisenstein series 15 0 15

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

\(2\)\(3\)\(5\)\(7\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(1\)
\(-\)\(-\)\(-\)\(+\)$-$\(1\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

\( 5 q + q^{2} + q^{3} + 5 q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + 5 q^{9} + O(q^{10}) \) \( 5 q + q^{2} + q^{3} + 5 q^{4} + q^{5} + q^{6} + q^{7} + q^{8} + 5 q^{9} + q^{10} - 4 q^{11} + q^{12} - 2 q^{13} + q^{14} + q^{15} + 5 q^{16} - 14 q^{17} + q^{18} + 4 q^{19} + q^{20} + q^{21} + 4 q^{22} - 24 q^{23} + q^{24} + 5 q^{25} - 2 q^{26} + q^{27} + q^{28} + 14 q^{29} - 3 q^{30} - 24 q^{31} + q^{32} - 4 q^{33} + 10 q^{34} + q^{35} + 5 q^{36} - 2 q^{37} - 12 q^{38} + 6 q^{39} + q^{40} - 6 q^{41} - 3 q^{42} - 4 q^{43} - 4 q^{44} + q^{45} - 8 q^{46} - 16 q^{47} + q^{48} + 5 q^{49} + q^{50} - 6 q^{51} - 2 q^{52} + 6 q^{53} + q^{54} + 4 q^{55} + q^{56} + 12 q^{57} - 18 q^{58} - 4 q^{59} + q^{60} - 2 q^{61} + q^{63} + 5 q^{64} - 2 q^{65} - 12 q^{66} + 4 q^{67} - 14 q^{68} + 8 q^{69} - 3 q^{70} + q^{72} - 6 q^{73} + 14 q^{74} + q^{75} + 4 q^{76} + 12 q^{77} - 2 q^{78} + q^{80} + 5 q^{81} + 10 q^{82} + 20 q^{83} + q^{84} + 10 q^{85} - 12 q^{86} - 18 q^{87} + 4 q^{88} + 26 q^{89} + q^{90} + 6 q^{91} - 24 q^{92} + 8 q^{93} - 24 q^{94} + 12 q^{95} + q^{96} + 18 q^{97} + q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(210))\) 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 5 7
210.2.a.a 210.a 1.a $1$ $1.677$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
210.2.a.b 210.a 1.a $1$ $1.677$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
210.2.a.c 210.a 1.a $1$ $1.677$ \(\Q\) None \(1\) \(-1\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
210.2.a.d 210.a 1.a $1$ $1.677$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
210.2.a.e 210.a 1.a $1$ $1.677$ \(\Q\) None \(1\) \(1\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(210))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(210)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)