Properties

Label 61.2.a
Level 61
Weight 2
Character orbit a
Rep. character \(\chi_{61}(1,\cdot)\)
Character field \(\Q\)
Dimension 4
Newform subspaces 2
Sturm bound 10
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 61 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 61.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(61))\).

Total New Old
Modular forms 5 5 0
Cusp forms 4 4 0
Eisenstein series 1 1 0

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

\(61\)Dim.
\(+\)\(1\)
\(-\)\(3\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(61))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 61
61.2.a.a \(1\) \(0.487\) \(\Q\) None \(-1\) \(-2\) \(-3\) \(1\) \(+\) \(q-q^{2}-2q^{3}-q^{4}-3q^{5}+2q^{6}+q^{7}+\cdots\)
61.2.a.b \(3\) \(0.487\) 3.3.148.1 None \(1\) \(2\) \(-1\) \(-3\) \(-\) \(q+\beta _{1}q^{2}+(1-\beta _{1}-\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 - T + 3 T^{2} - 3 T^{3} + 6 T^{4} - 4 T^{5} + 8 T^{6} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 - 2 T + 5 T^{2} - 8 T^{3} + 15 T^{4} - 18 T^{5} + 27 T^{6} \))
$5$ (\( 1 + 3 T + 5 T^{2} \))(\( 1 + T + 6 T^{2} - 3 T^{3} + 30 T^{4} + 25 T^{5} + 125 T^{6} \))
$7$ (\( 1 - T + 7 T^{2} \))(\( 1 + 3 T + 20 T^{2} + 41 T^{3} + 140 T^{4} + 147 T^{5} + 343 T^{6} \))
$11$ (\( 1 + 5 T + 11 T^{2} \))(\( 1 - 13 T + 86 T^{2} - 353 T^{3} + 946 T^{4} - 1573 T^{5} + 1331 T^{6} \))
$13$ (\( 1 - T + 13 T^{2} \))(\( 1 + 9 T + 50 T^{2} + 197 T^{3} + 650 T^{4} + 1521 T^{5} + 2197 T^{6} \))
$17$ (\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 43 T^{2} + 72 T^{3} + 731 T^{4} + 578 T^{5} + 4913 T^{6} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( 1 + 9 T^{2} - 20 T^{3} + 171 T^{4} + 6859 T^{6} \))
$23$ (\( 1 + 9 T + 23 T^{2} \))(\( 1 - 5 T + 74 T^{2} - 229 T^{3} + 1702 T^{4} - 2645 T^{5} + 12167 T^{6} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 4 T + 83 T^{2} - 212 T^{3} + 2407 T^{4} - 3364 T^{5} + 24389 T^{6} \))
$31$ (\( 1 + 31 T^{2} \))(\( 1 + 2 T + 17 T^{2} + 240 T^{3} + 527 T^{4} + 1922 T^{5} + 29791 T^{6} \))
$37$ (\( 1 - 8 T + 37 T^{2} \))(\( 1 + 6 T + 75 T^{2} + 336 T^{3} + 2775 T^{4} + 8214 T^{5} + 50653 T^{6} \))
$41$ (\( 1 - 5 T + 41 T^{2} \))(\( 1 - 3 T + 62 T^{2} - 55 T^{3} + 2542 T^{4} - 5043 T^{5} + 68921 T^{6} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 + 14 T + 185 T^{2} + 1272 T^{3} + 7955 T^{4} + 25886 T^{5} + 79507 T^{6} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 53 T^{2} + 392 T^{3} + 2491 T^{4} + 8836 T^{5} + 103823 T^{6} \))
$53$ (\( 1 - 6 T + 53 T^{2} \))(\( 1 + 2 T + 147 T^{2} + 204 T^{3} + 7791 T^{4} + 5618 T^{5} + 148877 T^{6} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 - 29 T + 408 T^{2} - 3747 T^{3} + 24072 T^{4} - 100949 T^{5} + 205379 T^{6} \))
$61$ (\( 1 + T \))(\( ( 1 - T )^{3} \))
$67$ (\( 1 + 7 T + 67 T^{2} \))(\( 1 - 9 T + 116 T^{2} - 647 T^{3} + 7772 T^{4} - 40401 T^{5} + 300763 T^{6} \))
$71$ (\( 1 + 8 T + 71 T^{2} \))(\( 1 - 14 T + 201 T^{2} - 1896 T^{3} + 14271 T^{4} - 70574 T^{5} + 357911 T^{6} \))
$73$ (\( 1 + 11 T + 73 T^{2} \))(\( 1 + T + 174 T^{2} + 121 T^{3} + 12702 T^{4} + 5329 T^{5} + 389017 T^{6} \))
$79$ (\( 1 - 3 T + 79 T^{2} \))(\( 1 - 13 T + 186 T^{2} - 1429 T^{3} + 14694 T^{4} - 81133 T^{5} + 493039 T^{6} \))
$83$ (\( 1 - 4 T + 83 T^{2} \))(\( 1 + 8 T + 185 T^{2} + 1072 T^{3} + 15355 T^{4} + 55112 T^{5} + 571787 T^{6} \))
$89$ (\( 1 + 4 T + 89 T^{2} \))(\( 1 + 4 T + 211 T^{2} + 792 T^{3} + 18779 T^{4} + 31684 T^{5} + 704969 T^{6} \))
$97$ (\( 1 + 14 T + 97 T^{2} \))(\( 1 - 10 T + 175 T^{2} - 844 T^{3} + 16975 T^{4} - 94090 T^{5} + 912673 T^{6} \))
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