Properties

Label 1005.2.a
Level $1005$
Weight $2$
Character orbit 1005.a
Rep. character $\chi_{1005}(1,\cdot)$
Character field $\Q$
Dimension $43$
Newform subspaces $10$
Sturm bound $272$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 1005 = 3 \cdot 5 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1005.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(272\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1005))\).

Total New Old
Modular forms 140 43 97
Cusp forms 133 43 90
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.

\(3\)\(5\)\(67\)FrickeDim
\(+\)\(+\)\(+\)$+$\(4\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(5\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(4\)
\(-\)\(-\)\(-\)$-$\(7\)
Plus space\(+\)\(18\)
Minus space\(-\)\(25\)

Trace form

\( 43 q - 3 q^{2} - q^{3} + 41 q^{4} - q^{5} + q^{6} - 8 q^{7} - 15 q^{8} + 43 q^{9} + O(q^{10}) \) \( 43 q - 3 q^{2} - q^{3} + 41 q^{4} - q^{5} + q^{6} - 8 q^{7} - 15 q^{8} + 43 q^{9} + 5 q^{10} - 4 q^{11} - 7 q^{12} - 14 q^{13} + 3 q^{15} + 41 q^{16} - 14 q^{17} - 3 q^{18} + 9 q^{20} - 8 q^{21} - 28 q^{22} - 4 q^{23} - 3 q^{24} + 43 q^{25} + 38 q^{26} - q^{27} - 16 q^{28} - 26 q^{29} - 3 q^{30} + 17 q^{32} + 4 q^{33} + 26 q^{34} + 41 q^{36} - 2 q^{37} + 52 q^{38} - 14 q^{39} + 9 q^{40} + 6 q^{41} - 8 q^{42} - 28 q^{43} - 4 q^{44} - q^{45} + 32 q^{46} - 20 q^{47} + q^{48} + 27 q^{49} - 3 q^{50} - 2 q^{51} - 26 q^{52} - 30 q^{53} + q^{54} + 4 q^{55} + 48 q^{56} - 20 q^{57} + 30 q^{58} - 16 q^{59} + 5 q^{60} - 30 q^{61} + 16 q^{62} - 8 q^{63} + q^{64} + 10 q^{65} + 20 q^{66} + 7 q^{67} - 14 q^{68} + 8 q^{69} + 16 q^{70} + 24 q^{71} - 15 q^{72} - 6 q^{73} - 34 q^{74} - q^{75} + 76 q^{76} - 16 q^{77} - 2 q^{78} - 48 q^{79} + q^{80} + 43 q^{81} + 66 q^{82} + 20 q^{83} + 16 q^{84} - 2 q^{85} + 4 q^{86} - 22 q^{87} + 4 q^{88} + 42 q^{89} + 5 q^{90} - 8 q^{91} - 104 q^{92} - 24 q^{93} - 40 q^{94} - 20 q^{95} + 5 q^{96} + 14 q^{97} - 11 q^{98} - 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1005))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 67
1005.2.a.a 1005.a 1.a $1$ $8.025$ \(\Q\) None \(0\) \(1\) \(-1\) \(2\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}+2q^{7}+q^{9}-6q^{11}+\cdots\)
1005.2.a.b 1005.a 1.a $1$ $8.025$ \(\Q\) None \(1\) \(-1\) \(1\) \(0\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}+q^{5}-q^{6}-3q^{8}+\cdots\)
1005.2.a.c 1005.a 1.a $4$ $8.025$ 4.4.1957.1 None \(-4\) \(4\) \(4\) \(-9\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1005.2.a.d 1005.a 1.a $4$ $8.025$ 4.4.2525.1 None \(-2\) \(4\) \(-4\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}-q^{5}+\cdots\)
1005.2.a.e 1005.a 1.a $4$ $8.025$ 4.4.1957.1 None \(0\) \(-4\) \(-4\) \(1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
1005.2.a.f 1005.a 1.a $4$ $8.025$ 4.4.9301.1 None \(2\) \(-4\) \(4\) \(11\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}-q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1005.2.a.g 1005.a 1.a $5$ $8.025$ 5.5.273397.1 None \(-2\) \(-5\) \(5\) \(-9\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
1005.2.a.h 1005.a 1.a $5$ $8.025$ 5.5.772525.1 None \(1\) \(5\) \(-5\) \(-3\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
1005.2.a.i 1005.a 1.a $7$ $8.025$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(4\) \(7\) \(7\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
1005.2.a.j 1005.a 1.a $8$ $8.025$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-3\) \(-8\) \(-8\) \(-3\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1005))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1005)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(201))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(67))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(335))\)\(^{\oplus 2}\)