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Properties

Label 1001.2.a
Level $1001$
Weight $2$
Character orbit 1001.a
Rep. character $\chi_{1001}(1,\cdot)$
Character field $\Q$
Dimension $59$
Newform subspaces $14$
Sturm bound $224$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1001 = 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1001.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(224\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1001))\).

Total New Old
Modular forms 116 59 57
Cusp forms 109 59 50
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.

\(7\)\(11\)\(13\)FrickeDim
\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(-\)$-$\(8\)
\(+\)\(-\)\(+\)$-$\(12\)
\(+\)\(-\)\(-\)$+$\(5\)
\(-\)\(+\)\(+\)$-$\(8\)
\(-\)\(+\)\(-\)$+$\(5\)
\(-\)\(-\)\(+\)$+$\(5\)
\(-\)\(-\)\(-\)$-$\(11\)
Plus space\(+\)\(20\)
Minus space\(-\)\(39\)

Trace form

\( 59 q - 3 q^{2} + 69 q^{4} - 2 q^{5} + 4 q^{6} - q^{7} - 15 q^{8} + 51 q^{9} + O(q^{10}) \) \( 59 q - 3 q^{2} + 69 q^{4} - 2 q^{5} + 4 q^{6} - q^{7} - 15 q^{8} + 51 q^{9} - 2 q^{10} + 7 q^{11} - 12 q^{12} - q^{13} - 3 q^{14} + 12 q^{15} + 77 q^{16} - 10 q^{17} + 17 q^{18} + 20 q^{19} + 30 q^{20} - 4 q^{21} - 3 q^{22} - 8 q^{23} + 44 q^{24} + 45 q^{25} - 3 q^{26} - 12 q^{27} - 7 q^{28} - 18 q^{29} + 24 q^{30} + 12 q^{31} - 23 q^{32} + 26 q^{34} - 2 q^{35} + 89 q^{36} - 10 q^{37} + 12 q^{38} - 4 q^{39} + 6 q^{40} - 18 q^{41} + 28 q^{42} - 32 q^{43} + 13 q^{44} - 22 q^{45} + 24 q^{46} - 16 q^{47} - 28 q^{48} + 59 q^{49} - 53 q^{50} - 24 q^{51} - 7 q^{52} + 30 q^{53} - 56 q^{54} + 6 q^{55} - 15 q^{56} - 64 q^{57} - 82 q^{58} - 32 q^{59} + 32 q^{60} + 10 q^{61} - 40 q^{62} + 3 q^{63} + 117 q^{64} - 2 q^{65} - 12 q^{66} - 8 q^{67} - 70 q^{68} + 44 q^{69} - 10 q^{70} + 76 q^{71} - 43 q^{72} - 2 q^{73} - 2 q^{74} + 60 q^{75} + 36 q^{76} - q^{77} - 4 q^{78} + 36 q^{79} + 6 q^{80} + 19 q^{81} - 46 q^{82} + 28 q^{83} - 28 q^{84} + 12 q^{85} + 20 q^{86} + 136 q^{87} - 15 q^{88} - 38 q^{89} - 34 q^{90} + 7 q^{91} - 40 q^{92} + 12 q^{93} + 32 q^{94} + 44 q^{95} + 52 q^{96} - 6 q^{97} - 3 q^{98} + 31 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 11 13
1001.2.a.a 1001.a 1.a $1$ $7.993$ \(\Q\) None \(-2\) \(-3\) \(-3\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+2q^{4}-3q^{5}+6q^{6}+\cdots\)
1001.2.a.b 1001.a 1.a $1$ $7.993$ \(\Q\) None \(-1\) \(0\) \(-2\) \(-1\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-2q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
1001.2.a.c 1001.a 1.a $1$ $7.993$ \(\Q\) None \(0\) \(2\) \(-1\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{4}-q^{5}-q^{7}+q^{9}-q^{11}+\cdots\)
1001.2.a.d 1001.a 1.a $2$ $7.993$ \(\Q(\sqrt{5}) \) None \(0\) \(3\) \(3\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-2\beta )q^{2}+(2-\beta )q^{3}+3q^{4}+(1+\cdots)q^{5}+\cdots\)
1001.2.a.e 1001.a 1.a $2$ $7.993$ \(\Q(\sqrt{21}) \) None \(2\) \(1\) \(-1\) \(2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+(1-\beta )q^{3}-q^{4}-\beta q^{5}+(1-\beta )q^{6}+\cdots\)
1001.2.a.f 1001.a 1.a $3$ $7.993$ 3.3.1101.1 None \(3\) \(-1\) \(1\) \(-3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-\beta _{1}q^{3}-q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
1001.2.a.g 1001.a 1.a $4$ $7.993$ 4.4.23301.1 None \(-2\) \(-1\) \(-5\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{2}-\beta _{1}q^{3}+(2-\beta _{2})q^{4}+\cdots\)
1001.2.a.h 1001.a 1.a $4$ $7.993$ 4.4.1957.1 None \(0\) \(-1\) \(5\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{1}-\beta _{2}-\beta _{3})q^{2}+(\beta _{1}+\beta _{2})q^{3}+\cdots\)
1001.2.a.i 1001.a 1.a $5$ $7.993$ 5.5.106069.1 None \(-2\) \(-3\) \(0\) \(5\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1001.2.a.j 1001.a 1.a $5$ $7.993$ 5.5.216637.1 None \(-2\) \(0\) \(-4\) \(-5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1001.2.a.k 1001.a 1.a $5$ $7.993$ 5.5.81509.1 None \(0\) \(-4\) \(-6\) \(5\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{4}q^{2}+(-1+\beta _{2})q^{3}+(1-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1001.2.a.l 1001.a 1.a $7$ $7.993$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(6\) \(9\) \(-7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{5}q^{2}+(1-\beta _{1})q^{3}+(4+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
1001.2.a.m 1001.a 1.a $8$ $7.993$ 8.8.3212905625.1 None \(2\) \(-1\) \(-5\) \(-8\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{7}q^{2}-\beta _{2}q^{3}+(2-\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1001.2.a.n 1001.a 1.a $11$ $7.993$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-1\) \(2\) \(7\) \(11\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1001))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1001)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(143))\)\(^{\oplus 2}\)