Properties

Label 7.8.a
Level 7
Weight 8
Character orbit a
Rep. character \(\chi_{7}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 2
Sturm bound 5
Trace bound 1

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Defining parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 8 \)
Character orbit: \([\chi]\) = 7.a (trivial)
Character field: \(\Q\)
Newforms: \( 2 \)
Sturm bound: \(5\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(7))\).

Total New Old
Modular forms 5 3 2
Cusp forms 3 3 0
Eisenstein series 2 0 2

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

\(7\)Dim.
\(+\)\(2\)
\(-\)\(1\)

Trace form

\(3q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 52q^{3} \) \(\mathstrut +\mathstrut 89q^{4} \) \(\mathstrut +\mathstrut 246q^{5} \) \(\mathstrut -\mathstrut 754q^{6} \) \(\mathstrut -\mathstrut 343q^{7} \) \(\mathstrut +\mathstrut 135q^{8} \) \(\mathstrut +\mathstrut 1351q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut -\mathstrut 9q^{2} \) \(\mathstrut +\mathstrut 52q^{3} \) \(\mathstrut +\mathstrut 89q^{4} \) \(\mathstrut +\mathstrut 246q^{5} \) \(\mathstrut -\mathstrut 754q^{6} \) \(\mathstrut -\mathstrut 343q^{7} \) \(\mathstrut +\mathstrut 135q^{8} \) \(\mathstrut +\mathstrut 1351q^{9} \) \(\mathstrut -\mathstrut 4316q^{10} \) \(\mathstrut -\mathstrut 2724q^{11} \) \(\mathstrut +\mathstrut 14966q^{12} \) \(\mathstrut -\mathstrut 2618q^{13} \) \(\mathstrut -\mathstrut 1029q^{14} \) \(\mathstrut +\mathstrut 27688q^{15} \) \(\mathstrut -\mathstrut 31807q^{16} \) \(\mathstrut -\mathstrut 15474q^{17} \) \(\mathstrut -\mathstrut 81433q^{18} \) \(\mathstrut +\mathstrut 88180q^{19} \) \(\mathstrut +\mathstrut 50568q^{20} \) \(\mathstrut -\mathstrut 46648q^{21} \) \(\mathstrut +\mathstrut 79312q^{22} \) \(\mathstrut -\mathstrut 97848q^{23} \) \(\mathstrut -\mathstrut 82590q^{24} \) \(\mathstrut -\mathstrut 129619q^{25} \) \(\mathstrut +\mathstrut 408576q^{26} \) \(\mathstrut +\mathstrut 150040q^{27} \) \(\mathstrut -\mathstrut 93639q^{28} \) \(\mathstrut +\mathstrut 62250q^{29} \) \(\mathstrut -\mathstrut 403408q^{30} \) \(\mathstrut -\mathstrut 241504q^{31} \) \(\mathstrut -\mathstrut 55689q^{32} \) \(\mathstrut +\mathstrut 267184q^{33} \) \(\mathstrut -\mathstrut 53982q^{34} \) \(\mathstrut -\mathstrut 142002q^{35} \) \(\mathstrut +\mathstrut 443393q^{36} \) \(\mathstrut +\mathstrut 231026q^{37} \) \(\mathstrut -\mathstrut 10350q^{38} \) \(\mathstrut -\mathstrut 427448q^{39} \) \(\mathstrut -\mathstrut 163680q^{40} \) \(\mathstrut +\mathstrut 224406q^{41} \) \(\mathstrut +\mathstrut 431494q^{42} \) \(\mathstrut -\mathstrut 652548q^{43} \) \(\mathstrut +\mathstrut 619128q^{44} \) \(\mathstrut +\mathstrut 1141342q^{45} \) \(\mathstrut +\mathstrut 536736q^{46} \) \(\mathstrut -\mathstrut 1608144q^{47} \) \(\mathstrut -\mathstrut 1700578q^{48} \) \(\mathstrut +\mathstrut 352947q^{49} \) \(\mathstrut -\mathstrut 913011q^{50} \) \(\mathstrut +\mathstrut 797736q^{51} \) \(\mathstrut -\mathstrut 441084q^{52} \) \(\mathstrut +\mathstrut 2799642q^{53} \) \(\mathstrut -\mathstrut 3415420q^{54} \) \(\mathstrut +\mathstrut 435272q^{55} \) \(\mathstrut +\mathstrut 859215q^{56} \) \(\mathstrut -\mathstrut 1525640q^{57} \) \(\mathstrut +\mathstrut 3776230q^{58} \) \(\mathstrut +\mathstrut 2139420q^{59} \) \(\mathstrut +\mathstrut 2899904q^{60} \) \(\mathstrut -\mathstrut 3853394q^{61} \) \(\mathstrut -\mathstrut 1398948q^{62} \) \(\mathstrut -\mathstrut 753571q^{63} \) \(\mathstrut -\mathstrut 762831q^{64} \) \(\mathstrut -\mathstrut 2963772q^{65} \) \(\mathstrut -\mathstrut 325168q^{66} \) \(\mathstrut +\mathstrut 2080756q^{67} \) \(\mathstrut +\mathstrut 1572438q^{68} \) \(\mathstrut +\mathstrut 3565872q^{69} \) \(\mathstrut +\mathstrut 1826132q^{70} \) \(\mathstrut -\mathstrut 5995704q^{71} \) \(\mathstrut +\mathstrut 1073775q^{72} \) \(\mathstrut +\mathstrut 2480222q^{73} \) \(\mathstrut -\mathstrut 11488242q^{74} \) \(\mathstrut +\mathstrut 3087548q^{75} \) \(\mathstrut -\mathstrut 3237990q^{76} \) \(\mathstrut -\mathstrut 2885316q^{77} \) \(\mathstrut +\mathstrut 16500344q^{78} \) \(\mathstrut -\mathstrut 8077680q^{79} \) \(\mathstrut -\mathstrut 5520624q^{80} \) \(\mathstrut -\mathstrut 264797q^{81} \) \(\mathstrut +\mathstrut 12216722q^{82} \) \(\mathstrut +\mathstrut 12355812q^{83} \) \(\mathstrut -\mathstrut 2482634q^{84} \) \(\mathstrut +\mathstrut 2330604q^{85} \) \(\mathstrut +\mathstrut 4187256q^{86} \) \(\mathstrut -\mathstrut 8891240q^{87} \) \(\mathstrut -\mathstrut 10690440q^{88} \) \(\mathstrut +\mathstrut 15330750q^{89} \) \(\mathstrut -\mathstrut 19124332q^{90} \) \(\mathstrut -\mathstrut 2636298q^{91} \) \(\mathstrut +\mathstrut 7864176q^{92} \) \(\mathstrut -\mathstrut 22560816q^{93} \) \(\mathstrut -\mathstrut 11211372q^{94} \) \(\mathstrut -\mathstrut 2948280q^{95} \) \(\mathstrut +\mathstrut 26043346q^{96} \) \(\mathstrut +\mathstrut 20639486q^{97} \) \(\mathstrut -\mathstrut 1058841q^{98} \) \(\mathstrut -\mathstrut 4554068q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(7))\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7
7.8.a.a \(1\) \(2.187\) \(\Q\) None \(-6\) \(-42\) \(-84\) \(343\) \(-\) \(q-6q^{2}-42q^{3}-92q^{4}-84q^{5}+252q^{6}+\cdots\)
7.8.a.b \(2\) \(2.187\) \(\Q(\sqrt{865}) \) None \(-3\) \(94\) \(330\) \(-686\) \(+\) \(q+(-1-\beta )q^{2}+(46+2\beta )q^{3}+(89+3\beta )q^{4}+\cdots\)