Properties

Label 82.2.a
Level 82
Weight 2
Character orbit a
Rep. character \(\chi_{82}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 21
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 82 = 2 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 82.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(21\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(82))\).

Total New Old
Modular forms 12 3 9
Cusp forms 9 3 6
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.

\(2\)\(41\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(2\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(82))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 41
82.2.a.a \(1\) \(0.655\) \(\Q\) None \(-1\) \(-2\) \(-2\) \(-4\) \(+\) \(+\) \(q-q^{2}-2q^{3}+q^{4}-2q^{5}+2q^{6}-4q^{7}+\cdots\)
82.2.a.b \(2\) \(0.655\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(0\) \(-4\) \(-\) \(+\) \(q+q^{2}+\beta q^{3}+q^{4}-2\beta q^{5}+\beta q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(82))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(82)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(41))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( ( 1 - T )^{2} \))
$3$ (\( 1 + 2 T + 3 T^{2} \))(\( 1 + 4 T^{2} + 9 T^{4} \))
$5$ (\( 1 + 2 T + 5 T^{2} \))(\( 1 + 2 T^{2} + 25 T^{4} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( 1 + 4 T + 16 T^{2} + 28 T^{3} + 49 T^{4} \))
$11$ (\( 1 + 2 T + 11 T^{2} \))(\( 1 + 4 T^{2} + 121 T^{4} \))
$13$ (\( 1 - 4 T + 13 T^{2} \))(\( ( 1 + 13 T^{2} )^{2} \))
$17$ (\( 1 + 2 T + 17 T^{2} \))(\( 1 - 4 T + 6 T^{2} - 68 T^{3} + 289 T^{4} \))
$19$ (\( 1 - 6 T + 19 T^{2} \))(\( 1 + 8 T + 52 T^{2} + 152 T^{3} + 361 T^{4} \))
$23$ (\( 1 + 8 T + 23 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 184 T^{3} + 529 T^{4} \))
$29$ (\( 1 + 29 T^{2} \))(\( 1 - 8 T + 42 T^{2} - 232 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( 1 + 8 T + 70 T^{2} + 248 T^{3} + 961 T^{4} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T^{2} + 1369 T^{4} \))
$41$ (\( 1 + T \))(\( ( 1 + T )^{2} \))
$43$ (\( 1 + 12 T + 43 T^{2} \))(\( 1 - 8 T + 70 T^{2} - 344 T^{3} + 1849 T^{4} \))
$47$ (\( 1 - 4 T + 47 T^{2} \))(\( 1 + 4 T + 48 T^{2} + 188 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 4 T + 53 T^{2} \))(\( ( 1 - 12 T + 53 T^{2} )^{2} \))
$59$ (\( 1 - 8 T + 59 T^{2} \))(\( 1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 14 T + 61 T^{2} \))(\( ( 1 - 6 T + 61 T^{2} )^{2} \))
$67$ (\( 1 + 2 T + 67 T^{2} \))(\( 1 + 8 T + 132 T^{2} + 536 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 4 T + 144 T^{2} + 284 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 10 T + 73 T^{2} \))(\( 1 + 16 T + 178 T^{2} + 1168 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 4 T + 79 T^{2} \))(\( 1 + 12 T + 176 T^{2} + 948 T^{3} + 6241 T^{4} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 - 24 T + 278 T^{2} - 1992 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 14 T + 89 T^{2} \))(\( 1 + 12 T + 182 T^{2} + 1068 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 6 T + 97 T^{2} \))(\( 1 + 4 T + 166 T^{2} + 388 T^{3} + 9409 T^{4} \))
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