Properties

Label 77.2.b
Level 77
Weight 2
Character orbit b
Rep. character \(\chi_{77}(76,\cdot)\)
Character field \(\Q\)
Dimension 6
Newform subspaces 2
Sturm bound 16
Trace bound 1

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Defining parameters

Level: \( N \) = \( 77 = 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 77.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 77 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(77, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\( 6q - 10q^{4} - 2q^{9} + O(q^{10}) \) \( 6q - 10q^{4} - 2q^{9} - 8q^{11} - 2q^{14} + 20q^{15} + 6q^{16} + 22q^{22} - 28q^{23} + 10q^{25} - 30q^{36} - 16q^{37} + 20q^{42} - 20q^{44} - 22q^{49} + 20q^{53} + 66q^{56} + 48q^{58} - 58q^{64} + 36q^{67} - 20q^{70} + 4q^{71} + 2q^{77} - 80q^{78} - 26q^{81} - 4q^{86} - 26q^{88} - 40q^{91} + 80q^{92} + 60q^{93} + 36q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(77, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
77.2.b.a \(2\) \(0.615\) \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{2}-5q^{4}-\beta q^{7}+3\beta q^{8}+3q^{9}+\cdots\)
77.2.b.b \(4\) \(0.615\) \(\Q(\sqrt{-2}, \sqrt{-5})\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{2}q^{2}+\beta _{3}q^{3}-\beta _{3}q^{5}+(2\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - T + 2 T^{2} )( 1 + T + 2 T^{2} ) \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 - T^{2} + 9 T^{4} )^{2} \))
$5$ (\( ( 1 - 5 T^{2} )^{2} \))(\( ( 1 - 5 T^{2} + 25 T^{4} )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( 1 + 4 T^{2} + 49 T^{4} \))
$11$ (\( 1 - 4 T + 11 T^{2} \))(\( ( 1 + 6 T + 11 T^{2} )^{2} \))
$13$ (\( ( 1 + 13 T^{2} )^{2} \))(\( ( 1 - 14 T^{2} + 169 T^{4} )^{2} \))
$17$ (\( ( 1 + 17 T^{2} )^{2} \))(\( ( 1 + 17 T^{2} )^{4} \))
$19$ (\( ( 1 + 19 T^{2} )^{2} \))(\( ( 1 + 28 T^{2} + 361 T^{4} )^{2} \))
$23$ (\( ( 1 + 8 T + 23 T^{2} )^{2} \))(\( ( 1 + 3 T + 23 T^{2} )^{4} \))
$29$ (\( ( 1 - 2 T + 29 T^{2} )( 1 + 2 T + 29 T^{2} ) \))(\( ( 1 - 56 T^{2} + 841 T^{4} )^{2} \))
$31$ (\( ( 1 - 31 T^{2} )^{2} \))(\( ( 1 - 17 T^{2} + 961 T^{4} )^{2} \))
$37$ (\( ( 1 + 6 T + 37 T^{2} )^{2} \))(\( ( 1 + T + 37 T^{2} )^{4} \))
$41$ (\( ( 1 + 41 T^{2} )^{2} \))(\( ( 1 - 8 T^{2} + 1681 T^{4} )^{2} \))
$43$ (\( ( 1 - 12 T + 43 T^{2} )( 1 + 12 T + 43 T^{2} ) \))(\( ( 1 - 68 T^{2} + 1849 T^{4} )^{2} \))
$47$ (\( ( 1 - 47 T^{2} )^{2} \))(\( ( 1 - 74 T^{2} + 2209 T^{4} )^{2} \))
$53$ (\( ( 1 - 10 T + 53 T^{2} )^{2} \))(\( ( 1 + 53 T^{2} )^{4} \))
$59$ (\( ( 1 - 59 T^{2} )^{2} \))(\( ( 1 - 113 T^{2} + 3481 T^{4} )^{2} \))
$61$ (\( ( 1 + 61 T^{2} )^{2} \))(\( ( 1 + 112 T^{2} + 3721 T^{4} )^{2} \))
$67$ (\( ( 1 + 4 T + 67 T^{2} )^{2} \))(\( ( 1 - 11 T + 67 T^{2} )^{4} \))
$71$ (\( ( 1 + 16 T + 71 T^{2} )^{2} \))(\( ( 1 - 9 T + 71 T^{2} )^{4} \))
$73$ (\( ( 1 + 73 T^{2} )^{2} \))(\( ( 1 + 136 T^{2} + 5329 T^{4} )^{2} \))
$79$ (\( ( 1 - 8 T + 79 T^{2} )( 1 + 8 T + 79 T^{2} ) \))(\( ( 1 - 86 T^{2} + 6241 T^{4} )^{2} \))
$83$ (\( ( 1 + 83 T^{2} )^{2} \))(\( ( 1 + 76 T^{2} + 6889 T^{4} )^{2} \))
$89$ (\( ( 1 - 89 T^{2} )^{2} \))(\( ( 1 - 173 T^{2} + 7921 T^{4} )^{2} \))
$97$ (\( ( 1 - 97 T^{2} )^{2} \))(\( ( 1 - 149 T^{2} + 9409 T^{4} )^{2} \))
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