Properties

Label 2006.2.a
Level $2006$
Weight $2$
Character orbit 2006.a
Rep. character $\chi_{2006}(1,\cdot)$
Character field $\Q$
Dimension $79$
Newform subspaces $23$
Sturm bound $540$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 2006 = 2 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2006.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(540\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\), \(11\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2006))\).

Total New Old
Modular forms 274 79 195
Cusp forms 267 79 188
Eisenstein series 7 0 7

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

\(2\)\(17\)\(59\)FrickeDim
\(+\)\(+\)\(+\)$+$\(7\)
\(+\)\(+\)\(-\)$-$\(12\)
\(+\)\(-\)\(+\)$-$\(13\)
\(+\)\(-\)\(-\)$+$\(8\)
\(-\)\(+\)\(+\)$-$\(13\)
\(-\)\(+\)\(-\)$+$\(6\)
\(-\)\(-\)\(+\)$+$\(7\)
\(-\)\(-\)\(-\)$-$\(13\)
Plus space\(+\)\(28\)
Minus space\(-\)\(51\)

Trace form

\( 79 q - q^{2} + 79 q^{4} - 2 q^{5} + 4 q^{6} + 8 q^{7} - q^{8} + 83 q^{9} + O(q^{10}) \) \( 79 q - q^{2} + 79 q^{4} - 2 q^{5} + 4 q^{6} + 8 q^{7} - q^{8} + 83 q^{9} - 6 q^{10} - 12 q^{11} + 10 q^{13} - 16 q^{15} + 79 q^{16} + 3 q^{17} - 5 q^{18} + 20 q^{19} - 2 q^{20} + 16 q^{21} - 8 q^{22} - 16 q^{23} + 4 q^{24} + 73 q^{25} + 2 q^{26} + 8 q^{28} + 14 q^{29} + 8 q^{30} + 8 q^{31} - q^{32} + 16 q^{33} - q^{34} - 24 q^{35} + 83 q^{36} + 18 q^{37} + 12 q^{38} + 8 q^{39} - 6 q^{40} + 14 q^{41} + 8 q^{42} + 4 q^{43} - 12 q^{44} - 2 q^{45} + 8 q^{46} + 32 q^{47} + 95 q^{49} + 17 q^{50} + 10 q^{52} - 38 q^{53} + 16 q^{54} - 16 q^{55} + 8 q^{57} - 6 q^{58} - q^{59} - 16 q^{60} + 42 q^{61} - 16 q^{63} + 79 q^{64} - 36 q^{65} + 40 q^{66} - 4 q^{67} + 3 q^{68} - 40 q^{69} - 24 q^{70} - 8 q^{71} - 5 q^{72} - 18 q^{73} - 18 q^{74} + 8 q^{75} + 20 q^{76} + 32 q^{77} + 48 q^{78} - 16 q^{79} - 2 q^{80} + 111 q^{81} + 6 q^{82} + 4 q^{83} + 16 q^{84} + 6 q^{85} - 20 q^{86} + 16 q^{87} - 8 q^{88} - 66 q^{89} - 6 q^{90} + 32 q^{91} - 16 q^{92} + 56 q^{93} + 48 q^{95} + 4 q^{96} - 10 q^{97} - 17 q^{98} - 28 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2006))\) 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 17 59
2006.2.a.a 2006.a 1.a $1$ $16.018$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-5q^{7}+\cdots\)
2006.2.a.b 2006.a 1.a $1$ $16.018$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
2006.2.a.c 2006.a 1.a $1$ $16.018$ \(\Q\) None \(-1\) \(-1\) \(2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+4q^{7}+\cdots\)
2006.2.a.d 2006.a 1.a $1$ $16.018$ \(\Q\) None \(-1\) \(0\) \(0\) \(2\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{9}+4q^{11}+\cdots\)
2006.2.a.e 2006.a 1.a $1$ $16.018$ \(\Q\) None \(-1\) \(1\) \(-3\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}-q^{7}+\cdots\)
2006.2.a.f 2006.a 1.a $1$ $16.018$ \(\Q\) None \(-1\) \(3\) \(2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}+q^{4}+2q^{5}-3q^{6}+4q^{7}+\cdots\)
2006.2.a.g 2006.a 1.a $1$ $16.018$ \(\Q\) None \(1\) \(-3\) \(-3\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-3q^{3}+q^{4}-3q^{5}-3q^{6}-q^{7}+\cdots\)
2006.2.a.h 2006.a 1.a $1$ $16.018$ \(\Q\) None \(1\) \(-1\) \(-3\) \(1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-3q^{5}-q^{6}+q^{7}+\cdots\)
2006.2.a.i 2006.a 1.a $1$ $16.018$ \(\Q\) None \(1\) \(-1\) \(1\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2006.2.a.j 2006.a 1.a $1$ $16.018$ \(\Q\) None \(1\) \(0\) \(0\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{7}+q^{8}-3q^{9}-6q^{11}+\cdots\)
2006.2.a.k 2006.a 1.a $1$ $16.018$ \(\Q\) None \(1\) \(2\) \(-2\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+2q^{7}+\cdots\)
2006.2.a.l 2006.a 1.a $2$ $16.018$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(0\) \(-6\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta q^{3}+q^{4}-\beta q^{5}+\beta q^{6}-3q^{7}+\cdots\)
2006.2.a.m 2006.a 1.a $2$ $16.018$ \(\Q(\sqrt{5}) \) None \(2\) \(1\) \(-5\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+\beta q^{3}+q^{4}+(-2-\beta )q^{5}+\beta q^{6}+\cdots\)
2006.2.a.n 2006.a 1.a $3$ $16.018$ 3.3.229.1 None \(-3\) \(2\) \(-2\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
2006.2.a.o 2006.a 1.a $3$ $16.018$ 3.3.568.1 None \(3\) \(-1\) \(5\) \(7\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}+q^{4}+(2-\beta _{1})q^{5}-\beta _{1}q^{6}+\cdots\)
2006.2.a.p 2006.a 1.a $4$ $16.018$ 4.4.2225.1 None \(-4\) \(-1\) \(-5\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-1-\beta _{1}+\beta _{3})q^{5}+\cdots\)
2006.2.a.q 2006.a 1.a $4$ $16.018$ \(\Q(\zeta_{20})^+\) None \(-4\) \(0\) \(2\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+(-\beta _{2}+\beta _{3})q^{3}+q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
2006.2.a.r 2006.a 1.a $4$ $16.018$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(4\) \(-4\) \(4\) \(-4\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{2})q^{3}+q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
2006.2.a.s 2006.a 1.a $4$ $16.018$ 4.4.13676.1 None \(4\) \(-3\) \(3\) \(-12\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(-1+\beta _{3})q^{3}+q^{4}+(1-\beta _{3})q^{5}+\cdots\)
2006.2.a.t 2006.a 1.a $8$ $16.018$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(8\) \(5\) \(-5\) \(0\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{2})q^{3}+q^{4}+(-1+\beta _{7})q^{5}+\cdots\)
2006.2.a.u 2006.a 1.a $9$ $16.018$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-9\) \(-1\) \(7\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+(1-\beta _{6})q^{5}+\beta _{1}q^{6}+\cdots\)
2006.2.a.v 2006.a 1.a $12$ $16.018$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-12\) \(-3\) \(-1\) \(4\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{8}q^{5}+\beta _{1}q^{6}+\cdots\)
2006.2.a.w 2006.a 1.a $13$ $16.018$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(13\) \(7\) \(1\) \(8\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+\beta _{8}q^{5}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2006))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2006)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(59))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(118))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1003))\)\(^{\oplus 2}\)