Properties

Label 77.2.a
Level 77
Weight 2
Character orbit a
Rep. character \(\chi_{77}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 4
Sturm bound 16
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 77 = 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 77.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(16\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(77))\).

Total New Old
Modular forms 10 5 5
Cusp forms 7 5 2
Eisenstein series 3 0 3

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

\(7\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(3\)
Plus space\(+\)\(1\)
Minus space\(-\)\(4\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(77))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 7 11
77.2.a.a \(1\) \(0.615\) \(\Q\) None \(0\) \(-3\) \(-1\) \(-1\) \(+\) \(+\) \(q-3q^{3}-2q^{4}-q^{5}-q^{7}+6q^{9}-q^{11}+\cdots\)
77.2.a.b \(1\) \(0.615\) \(\Q\) None \(0\) \(1\) \(3\) \(1\) \(-\) \(+\) \(q+q^{3}-2q^{4}+3q^{5}+q^{7}-2q^{9}-q^{11}+\cdots\)
77.2.a.c \(1\) \(0.615\) \(\Q\) None \(1\) \(2\) \(-2\) \(-1\) \(+\) \(-\) \(q+q^{2}+2q^{3}-q^{4}-2q^{5}+2q^{6}-q^{7}+\cdots\)
77.2.a.d \(2\) \(0.615\) \(\Q(\sqrt{5}) \) None \(0\) \(2\) \(-4\) \(2\) \(-\) \(+\) \(q-\beta q^{2}+(1+\beta )q^{3}+3q^{4}-2q^{5}+(-5+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(77))\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - T^{2} + 4 T^{4} \))
$3$ (\( 1 + 3 T + 3 T^{2} \))(\( 1 - T + 3 T^{2} \))(\( 1 - 2 T + 3 T^{2} \))(\( 1 - 2 T + 2 T^{2} - 6 T^{3} + 9 T^{4} \))
$5$ (\( 1 + T + 5 T^{2} \))(\( 1 - 3 T + 5 T^{2} \))(\( 1 + 2 T + 5 T^{2} \))(\( ( 1 + 2 T + 5 T^{2} )^{2} \))
$7$ (\( 1 + T \))(\( 1 - T \))(\( 1 + T \))(\( ( 1 - T )^{2} \))
$11$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( ( 1 + T )^{2} \))
$13$ (\( 1 + 4 T + 13 T^{2} \))(\( 1 + 4 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))(\( 1 - 2 T + 22 T^{2} - 26 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 + 6 T + 17 T^{2} \))(\( 1 - 4 T + 17 T^{2} \))(\( 1 + 2 T + 30 T^{2} + 34 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 - 2 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 5 T + 23 T^{2} \))(\( 1 - 3 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 + 4 T + 30 T^{2} + 92 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 10 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 + 6 T + 29 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4} \))
$31$ (\( 1 - T + 31 T^{2} \))(\( 1 - 5 T + 31 T^{2} \))(\( 1 - 10 T + 31 T^{2} \))(\( 1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 5 T + 37 T^{2} \))(\( 1 - 11 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 + 8 T + 70 T^{2} + 296 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 2 T + 41 T^{2} \))(\( 1 - 6 T + 41 T^{2} \))(\( 1 - 4 T + 41 T^{2} \))(\( 1 + 18 T + 158 T^{2} + 738 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 8 T + 43 T^{2} \))(\( 1 - 8 T + 43 T^{2} \))(\( 1 - 12 T + 43 T^{2} \))(\( ( 1 - 8 T + 43 T^{2} )^{2} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 + 47 T^{2} \))(\( 1 + 10 T + 47 T^{2} \))(\( 1 - 10 T + 114 T^{2} - 470 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 3 T + 59 T^{2} \))(\( 1 + 9 T + 59 T^{2} \))(\( 1 - 2 T + 59 T^{2} \))(\( 1 - 2 T + 114 T^{2} - 118 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( 1 + 10 T + 61 T^{2} \))(\( 1 + 61 T^{2} \))(\( 1 + 10 T + 142 T^{2} + 610 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 3 T + 67 T^{2} \))(\( 1 - 5 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 - 20 T + 214 T^{2} - 1340 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - T + 71 T^{2} \))(\( 1 - 9 T + 71 T^{2} \))(\( 1 + 12 T + 71 T^{2} \))(\( 1 + 12 T + 158 T^{2} + 852 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 73 T^{2} \))(\( 1 + 6 T + 150 T^{2} + 438 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 6 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 - 8 T + 79 T^{2} \))(\( 1 + 78 T^{2} + 6241 T^{4} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 12 T + 83 T^{2} \))(\( 1 + 83 T^{2} \))(\( 1 - 4 T - 10 T^{2} - 332 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 15 T + 89 T^{2} \))(\( 1 + 3 T + 89 T^{2} \))(\( 1 + 6 T + 89 T^{2} \))(\( ( 1 - 2 T + 89 T^{2} )^{2} \))
$97$ (\( 1 + 5 T + 97 T^{2} \))(\( 1 + T + 97 T^{2} \))(\( 1 + 10 T + 97 T^{2} \))(\( 1 - 8 T + 30 T^{2} - 776 T^{3} + 9409 T^{4} \))
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