Properties

Label 1001.2.a
Level 1001
Weight 2
Character orbit a
Rep. character \(\chi_{1001}(1,\cdot)\)
Character field \(\Q\)
Dimension 59
Newforms 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: \(\Q\)
Newforms: \( 14 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1001))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7 11 13
1001.2.a.a \(1\) \(7.993\) \(\Q\) None \(-2\) \(-3\) \(-3\) \(-1\) \(+\) \(-\) \(+\) \(q-2q^{2}-3q^{3}+2q^{4}-3q^{5}+6q^{6}+\cdots\)
1001.2.a.b \(1\) \(7.993\) \(\Q\) None \(-1\) \(0\) \(-2\) \(-1\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{4}-2q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
1001.2.a.c \(1\) \(7.993\) \(\Q\) None \(0\) \(2\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(q+2q^{3}-2q^{4}-q^{5}-q^{7}+q^{9}-q^{11}+\cdots\)
1001.2.a.d \(2\) \(7.993\) \(\Q(\sqrt{5}) \) None \(0\) \(3\) \(3\) \(2\) \(-\) \(+\) \(+\) \(q+(1-2\beta )q^{2}+(2-\beta )q^{3}+3q^{4}+(1+\cdots)q^{5}+\cdots\)
1001.2.a.e \(2\) \(7.993\) \(\Q(\sqrt{21}) \) None \(2\) \(1\) \(-1\) \(2\) \(-\) \(+\) \(+\) \(q+q^{2}+(1-\beta )q^{3}-q^{4}-\beta q^{5}+(1-\beta )q^{6}+\cdots\)
1001.2.a.f \(3\) \(7.993\) 3.3.1101.1 None \(3\) \(-1\) \(1\) \(-3\) \(+\) \(-\) \(+\) \(q+q^{2}-\beta _{1}q^{3}-q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
1001.2.a.g \(4\) \(7.993\) 4.4.23301.1 None \(-2\) \(-1\) \(-5\) \(4\) \(-\) \(+\) \(+\) \(q+(-1+\beta _{3})q^{2}-\beta _{1}q^{3}+(2-\beta _{2})q^{4}+\cdots\)
1001.2.a.h \(4\) \(7.993\) 4.4.1957.1 None \(0\) \(-1\) \(5\) \(-4\) \(+\) \(+\) \(+\) \(q+(-\beta _{1}-\beta _{2}-\beta _{3})q^{2}+(\beta _{1}+\beta _{2})q^{3}+\cdots\)
1001.2.a.i \(5\) \(7.993\) 5.5.106069.1 None \(-2\) \(-3\) \(0\) \(5\) \(-\) \(+\) \(-\) \(q-\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1001.2.a.j \(5\) \(7.993\) 5.5.216637.1 None \(-2\) \(0\) \(-4\) \(-5\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1001.2.a.k \(5\) \(7.993\) 5.5.81509.1 None \(0\) \(-4\) \(-6\) \(5\) \(-\) \(-\) \(+\) \(q+\beta _{4}q^{2}+(-1+\beta _{2})q^{3}+(1-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
1001.2.a.l \(7\) \(7.993\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(6\) \(9\) \(-7\) \(+\) \(-\) \(+\) \(q+\beta _{5}q^{2}+(1-\beta _{1})q^{3}+(4+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
1001.2.a.m \(8\) \(7.993\) 8.8.3212905625.1 None \(2\) \(-1\) \(-5\) \(-8\) \(+\) \(+\) \(-\) \(q+\beta _{7}q^{2}-\beta _{2}q^{3}+(2-\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1001.2.a.n \(11\) \(7.993\) \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-1\) \(2\) \(7\) \(11\) \(-\) \(-\) \(-\) \(q-\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{4}+\cdots)q^{5}+\cdots\)

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

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