Properties

Label 210.2.a
Level 210
Weight 2
Character orbit a
Rep. character \(\chi_{210}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newforms 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: \(\Q\)
Newforms: \( 5 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(210))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 5 7
210.2.a.a \(1\) \(1.677\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
210.2.a.b \(1\) \(1.677\) \(\Q\) None \(-1\) \(1\) \(1\) \(1\) \(+\) \(-\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
210.2.a.c \(1\) \(1.677\) \(\Q\) None \(1\) \(-1\) \(1\) \(1\) \(-\) \(+\) \(-\) \(-\) \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
210.2.a.d \(1\) \(1.677\) \(\Q\) None \(1\) \(1\) \(-1\) \(1\) \(-\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
210.2.a.e \(1\) \(1.677\) \(\Q\) None \(1\) \(1\) \(1\) \(-1\) \(-\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)

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

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