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Properties

Label 2010.2.a
Level $2010$
Weight $2$
Character orbit 2010.a
Rep. character $\chi_{2010}(1,\cdot)$
Character field $\Q$
Dimension $45$
Newform subspaces $21$
Sturm bound $816$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 2010 = 2 \cdot 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2010.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 21 \)
Sturm bound: \(816\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2010))\).

Total New Old
Modular forms 416 45 371
Cusp forms 401 45 356
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.

\(2\)\(3\)\(5\)\(67\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(+\)\(-\)\(+\)$-$\(4\)
\(+\)\(+\)\(-\)\(-\)$+$\(2\)
\(+\)\(-\)\(+\)\(+\)$-$\(3\)
\(+\)\(-\)\(+\)\(-\)$+$\(2\)
\(+\)\(-\)\(-\)\(+\)$+$\(1\)
\(+\)\(-\)\(-\)\(-\)$-$\(5\)
\(-\)\(+\)\(+\)\(+\)$-$\(3\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(4\)
\(-\)\(-\)\(+\)\(+\)$+$\(2\)
\(-\)\(-\)\(+\)\(-\)$-$\(4\)
\(-\)\(-\)\(-\)\(+\)$-$\(6\)
Plus space\(+\)\(15\)
Minus space\(-\)\(30\)

Trace form

\( 45 q + q^{2} + q^{3} + 45 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 45 q^{9} + O(q^{10}) \) \( 45 q + q^{2} + q^{3} + 45 q^{4} + q^{5} + q^{6} + 8 q^{7} + q^{8} + 45 q^{9} - 3 q^{10} + 4 q^{11} + q^{12} + 14 q^{13} + q^{15} + 45 q^{16} + 10 q^{17} + q^{18} + 4 q^{19} + q^{20} + 8 q^{21} + 4 q^{22} + q^{24} + 45 q^{25} - 10 q^{26} + q^{27} + 8 q^{28} + 22 q^{29} + q^{30} + q^{32} - 12 q^{33} + 2 q^{34} + 45 q^{36} - 10 q^{37} - 12 q^{38} + 14 q^{39} - 3 q^{40} + 2 q^{41} - 8 q^{42} + 12 q^{43} + 4 q^{44} + q^{45} + 8 q^{47} + q^{48} + 77 q^{49} + q^{50} + 2 q^{51} + 14 q^{52} + 22 q^{53} + q^{54} + 12 q^{55} + 20 q^{57} - 2 q^{58} + 44 q^{59} + q^{60} + 30 q^{61} + 16 q^{62} + 8 q^{63} + 45 q^{64} + 6 q^{65} + 12 q^{66} - 3 q^{67} + 10 q^{68} + 8 q^{69} + 8 q^{70} - 16 q^{71} + q^{72} - 30 q^{73} - 2 q^{74} + q^{75} + 4 q^{76} - 32 q^{77} + 14 q^{78} + 32 q^{79} + q^{80} + 45 q^{81} - 22 q^{82} - 68 q^{83} + 8 q^{84} + 18 q^{85} - 20 q^{86} + 14 q^{87} + 4 q^{88} - 22 q^{89} - 3 q^{90} + 16 q^{91} + 24 q^{93} + 24 q^{94} + 20 q^{95} + q^{96} - 14 q^{97} - 7 q^{98} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2010))\) 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 67
2010.2.a.a 2010.a 1.a $1$ $16.050$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(0\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{8}+\cdots\)
2010.2.a.b 2010.a 1.a $1$ $16.050$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-3q^{7}+\cdots\)
2010.2.a.c 2010.a 1.a $1$ $16.050$ \(\Q\) None \(-1\) \(-1\) \(1\) \(2\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+2q^{7}+\cdots\)
2010.2.a.d 2010.a 1.a $1$ $16.050$ \(\Q\) None \(-1\) \(1\) \(-1\) \(4\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
2010.2.a.e 2010.a 1.a $1$ $16.050$ \(\Q\) None \(-1\) \(1\) \(1\) \(-2\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
2010.2.a.f 2010.a 1.a $1$ $16.050$ \(\Q\) None \(1\) \(-1\) \(1\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2010.2.a.g 2010.a 1.a $1$ $16.050$ \(\Q\) None \(1\) \(-1\) \(1\) \(0\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
2010.2.a.h 2010.a 1.a $1$ $16.050$ \(\Q\) None \(1\) \(1\) \(-1\) \(-4\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-4q^{7}+\cdots\)
2010.2.a.i 2010.a 1.a $1$ $16.050$ \(\Q\) None \(1\) \(1\) \(-1\) \(-2\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
2010.2.a.j 2010.a 1.a $1$ $16.050$ \(\Q\) None \(1\) \(1\) \(-1\) \(2\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
2010.2.a.k 2010.a 1.a $1$ $16.050$ \(\Q\) None \(1\) \(1\) \(1\) \(4\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
2010.2.a.l 2010.a 1.a $2$ $16.050$ \(\Q(\sqrt{3}) \) None \(-2\) \(2\) \(-2\) \(-2\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
2010.2.a.m 2010.a 1.a $2$ $16.050$ \(\Q(\sqrt{17}) \) None \(-2\) \(2\) \(-2\) \(1\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}+\beta q^{7}+\cdots\)
2010.2.a.n 2010.a 1.a $3$ $16.050$ 3.3.568.1 None \(3\) \(-3\) \(-3\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-\beta _{1}q^{7}+\cdots\)
2010.2.a.o 2010.a 1.a $3$ $16.050$ 3.3.316.1 None \(3\) \(-3\) \(-3\) \(2\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
2010.2.a.p 2010.a 1.a $3$ $16.050$ 3.3.316.1 None \(3\) \(-3\) \(3\) \(2\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
2010.2.a.q 2010.a 1.a $3$ $16.050$ 3.3.3132.1 None \(3\) \(3\) \(-3\) \(3\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
2010.2.a.r 2010.a 1.a $4$ $16.050$ 4.4.70292.1 None \(-4\) \(-4\) \(-4\) \(1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+\beta _{1}q^{7}+\cdots\)
2010.2.a.s 2010.a 1.a $4$ $16.050$ 4.4.11324.1 None \(-4\) \(-4\) \(4\) \(-2\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
2010.2.a.t 2010.a 1.a $5$ $16.050$ 5.5.31460256.1 None \(-5\) \(5\) \(5\) \(5\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}+(1-\beta _{2}+\cdots)q^{7}+\cdots\)
2010.2.a.u 2010.a 1.a $5$ $16.050$ 5.5.6517908.1 None \(5\) \(5\) \(5\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+\beta _{2}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2010))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2010)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(134))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(402))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(670))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\)\(^{\oplus 2}\)