Properties

Label 861.2.a
Level 861
Weight 2
Character orbit a
Rep. character \(\chi_{861}(1,\cdot)\)
Character field \(\Q\)
Dimension 39
Newforms 13
Sturm bound 224
Trace bound 5

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Defining parameters

Level: \( N \) = \( 861 = 3 \cdot 7 \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 861.a (trivial)
Character field: \(\Q\)
Newforms: \( 13 \)
Sturm bound: \(224\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(861))\).

Total New Old
Modular forms 116 39 77
Cusp forms 109 39 70
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.

\(3\)\(7\)\(41\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(4\)
\(+\)\(+\)\(-\)\(-\)\(5\)
\(+\)\(-\)\(+\)\(-\)\(8\)
\(+\)\(-\)\(-\)\(+\)\(3\)
\(-\)\(+\)\(+\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(3\)
\(-\)\(-\)\(-\)\(-\)\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(25\)

Trace form

\(39q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 45q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 5q^{6} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(39q \) \(\mathstrut +\mathstrut q^{2} \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 45q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 5q^{6} \) \(\mathstrut +\mathstrut 3q^{7} \) \(\mathstrut -\mathstrut 3q^{8} \) \(\mathstrut +\mathstrut 39q^{9} \) \(\mathstrut -\mathstrut 10q^{10} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 7q^{12} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut +\mathstrut q^{14} \) \(\mathstrut +\mathstrut 2q^{15} \) \(\mathstrut +\mathstrut 45q^{16} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut q^{18} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 2q^{20} \) \(\mathstrut -\mathstrut q^{21} \) \(\mathstrut +\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 8q^{23} \) \(\mathstrut +\mathstrut 9q^{24} \) \(\mathstrut +\mathstrut 9q^{25} \) \(\mathstrut +\mathstrut 14q^{26} \) \(\mathstrut -\mathstrut q^{27} \) \(\mathstrut +\mathstrut 5q^{28} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 10q^{30} \) \(\mathstrut +\mathstrut 20q^{31} \) \(\mathstrut +\mathstrut 5q^{32} \) \(\mathstrut -\mathstrut 8q^{33} \) \(\mathstrut -\mathstrut 22q^{34} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut +\mathstrut 45q^{36} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 12q^{38} \) \(\mathstrut +\mathstrut 2q^{39} \) \(\mathstrut +\mathstrut 30q^{40} \) \(\mathstrut -\mathstrut q^{41} \) \(\mathstrut -\mathstrut 3q^{42} \) \(\mathstrut -\mathstrut 32q^{43} \) \(\mathstrut +\mathstrut 20q^{44} \) \(\mathstrut -\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 24q^{46} \) \(\mathstrut -\mathstrut 24q^{47} \) \(\mathstrut +\mathstrut q^{48} \) \(\mathstrut +\mathstrut 39q^{49} \) \(\mathstrut +\mathstrut 31q^{50} \) \(\mathstrut +\mathstrut 2q^{51} \) \(\mathstrut -\mathstrut 66q^{52} \) \(\mathstrut -\mathstrut 14q^{53} \) \(\mathstrut +\mathstrut 5q^{54} \) \(\mathstrut -\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 21q^{56} \) \(\mathstrut -\mathstrut 4q^{57} \) \(\mathstrut +\mathstrut 54q^{58} \) \(\mathstrut +\mathstrut 4q^{59} \) \(\mathstrut +\mathstrut 6q^{60} \) \(\mathstrut +\mathstrut 6q^{61} \) \(\mathstrut -\mathstrut 40q^{62} \) \(\mathstrut +\mathstrut 3q^{63} \) \(\mathstrut +\mathstrut 45q^{64} \) \(\mathstrut -\mathstrut 28q^{65} \) \(\mathstrut +\mathstrut 12q^{66} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut -\mathstrut 38q^{68} \) \(\mathstrut +\mathstrut 8q^{69} \) \(\mathstrut +\mathstrut 14q^{70} \) \(\mathstrut +\mathstrut 32q^{71} \) \(\mathstrut -\mathstrut 3q^{72} \) \(\mathstrut -\mathstrut 38q^{73} \) \(\mathstrut -\mathstrut 50q^{74} \) \(\mathstrut -\mathstrut 15q^{75} \) \(\mathstrut +\mathstrut 12q^{76} \) \(\mathstrut +\mathstrut 20q^{77} \) \(\mathstrut -\mathstrut 18q^{78} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut -\mathstrut 50q^{80} \) \(\mathstrut +\mathstrut 39q^{81} \) \(\mathstrut -\mathstrut 3q^{82} \) \(\mathstrut +\mathstrut 92q^{83} \) \(\mathstrut -\mathstrut 7q^{84} \) \(\mathstrut -\mathstrut 4q^{85} \) \(\mathstrut -\mathstrut 36q^{86} \) \(\mathstrut -\mathstrut 26q^{87} \) \(\mathstrut +\mathstrut 4q^{88} \) \(\mathstrut +\mathstrut 38q^{89} \) \(\mathstrut -\mathstrut 10q^{90} \) \(\mathstrut -\mathstrut 14q^{91} \) \(\mathstrut -\mathstrut 24q^{92} \) \(\mathstrut -\mathstrut 24q^{93} \) \(\mathstrut +\mathstrut 48q^{94} \) \(\mathstrut +\mathstrut 40q^{95} \) \(\mathstrut +\mathstrut 33q^{96} \) \(\mathstrut -\mathstrut 18q^{97} \) \(\mathstrut +\mathstrut q^{98} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(861))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 7 41
861.2.a.a \(1\) \(6.875\) \(\Q\) None \(-1\) \(-1\) \(2\) \(1\) \(+\) \(-\) \(+\) \(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+q^{7}+\cdots\)
861.2.a.b \(1\) \(6.875\) \(\Q\) None \(-1\) \(1\) \(-3\) \(1\) \(-\) \(-\) \(+\) \(q-q^{2}+q^{3}-q^{4}-3q^{5}-q^{6}+q^{7}+\cdots\)
861.2.a.c \(1\) \(6.875\) \(\Q\) None \(-1\) \(1\) \(3\) \(-1\) \(-\) \(+\) \(-\) \(q-q^{2}+q^{3}-q^{4}+3q^{5}-q^{6}-q^{7}+\cdots\)
861.2.a.d \(1\) \(6.875\) \(\Q\) None \(1\) \(1\) \(-1\) \(-1\) \(-\) \(+\) \(-\) \(q+q^{2}+q^{3}-q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
861.2.a.e \(2\) \(6.875\) \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(-2\) \(2\) \(-\) \(-\) \(+\) \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-q^{5}+\cdots\)
861.2.a.f \(2\) \(6.875\) \(\Q(\sqrt{17}) \) None \(-1\) \(-2\) \(4\) \(2\) \(+\) \(-\) \(+\) \(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+2q^{5}+\beta q^{6}+\cdots\)
861.2.a.g \(2\) \(6.875\) \(\Q(\sqrt{17}) \) None \(-1\) \(2\) \(-3\) \(-2\) \(-\) \(+\) \(-\) \(q-\beta q^{2}+q^{3}+(2+\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
861.2.a.h \(3\) \(6.875\) 3.3.148.1 None \(-1\) \(-3\) \(1\) \(3\) \(+\) \(-\) \(-\) \(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)
861.2.a.i \(4\) \(6.875\) 4.4.8468.1 None \(1\) \(-4\) \(-3\) \(-4\) \(+\) \(+\) \(+\) \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
861.2.a.j \(5\) \(6.875\) 5.5.981328.1 None \(-3\) \(-5\) \(1\) \(-5\) \(+\) \(+\) \(-\) \(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
861.2.a.k \(5\) \(6.875\) 5.5.1197392.1 None \(3\) \(-5\) \(-9\) \(5\) \(+\) \(-\) \(+\) \(q+(1-\beta _{1})q^{2}-q^{3}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
861.2.a.l \(5\) \(6.875\) 5.5.626512.1 None \(3\) \(5\) \(3\) \(-5\) \(-\) \(+\) \(+\) \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
861.2.a.m \(7\) \(6.875\) \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(4\) \(7\) \(1\) \(7\) \(-\) \(-\) \(-\) \(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(861))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(861)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(123))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(287))\)\(^{\oplus 2}\)