Properties

Label 41.2.a
Level 41
Weight 2
Character orbit a
Rep. character \(\chi_{41}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 1
Sturm bound 7
Trace bound 0

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Defining parameters

Level: \( N \) = \( 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 41.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(7\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(41))\).

Total New Old
Modular forms 4 4 0
Cusp forms 3 3 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.

\(41\)Dim.
\(-\)\(3\)

Trace form

\( 3q - q^{2} + 5q^{4} - 2q^{5} - 6q^{6} + 6q^{7} - 9q^{8} - q^{9} + O(q^{10}) \) \( 3q - q^{2} + 5q^{4} - 2q^{5} - 6q^{6} + 6q^{7} - 9q^{8} - q^{9} - 10q^{10} + 2q^{11} - 4q^{12} - 2q^{13} + 4q^{14} + 6q^{15} + 13q^{16} - 6q^{17} + 11q^{18} + 4q^{19} + 6q^{20} - 8q^{21} - 4q^{22} + 4q^{23} - 2q^{24} - 3q^{25} + 10q^{26} - 6q^{27} + 14q^{28} - 6q^{29} + 10q^{30} + 16q^{31} - 29q^{32} - 12q^{33} + 2q^{34} - 10q^{35} - 11q^{36} - 6q^{37} + 2q^{38} + 4q^{39} - 14q^{40} + 3q^{41} - 20q^{42} - 4q^{43} + 34q^{44} - 10q^{45} - 4q^{46} + 24q^{48} - q^{49} + 13q^{50} - 30q^{52} + 6q^{53} + 14q^{54} + 2q^{55} - 16q^{56} + 12q^{57} + 14q^{58} - 8q^{59} + 18q^{60} + 2q^{61} + 16q^{62} + 4q^{63} + 13q^{64} - 8q^{65} - 16q^{66} - 2q^{67} - 10q^{68} + 20q^{69} - 30q^{70} + 20q^{71} + 23q^{72} - 2q^{73} - 30q^{74} - 16q^{75} - 24q^{76} + 16q^{77} + 8q^{78} + 32q^{79} + 34q^{80} - 13q^{81} - q^{82} - 12q^{84} + 4q^{85} - 8q^{86} - 16q^{87} - 40q^{88} - 6q^{89} + 2q^{90} - 8q^{91} - 28q^{92} - 12q^{93} - 46q^{94} - 6q^{95} + 2q^{96} + 6q^{97} + 35q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(41))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 41
41.2.a.a \(3\) \(0.327\) 3.3.148.1 None \(-1\) \(0\) \(-2\) \(6\) \(-\) \(q+(-\beta _{1}-\beta _{2})q^{2}+\beta _{2}q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} + 3 T^{3} + 2 T^{4} + 4 T^{5} + 8 T^{6} \)
$3$ \( 1 + 5 T^{2} + 2 T^{3} + 15 T^{4} + 27 T^{6} \)
$5$ \( 1 + 2 T + 11 T^{2} + 16 T^{3} + 55 T^{4} + 50 T^{5} + 125 T^{6} \)
$7$ \( 1 - 6 T + 29 T^{2} - 86 T^{3} + 203 T^{4} - 294 T^{5} + 343 T^{6} \)
$11$ \( 1 - 2 T + 13 T^{2} + 6 T^{3} + 143 T^{4} - 242 T^{5} + 1331 T^{6} \)
$13$ \( 1 + 2 T + 27 T^{2} + 44 T^{3} + 351 T^{4} + 338 T^{5} + 2197 T^{6} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{3} \)
$19$ \( 1 - 4 T + 41 T^{2} - 162 T^{3} + 779 T^{4} - 1444 T^{5} + 6859 T^{6} \)
$23$ \( 1 - 4 T + 37 T^{2} - 216 T^{3} + 851 T^{4} - 2116 T^{5} + 12167 T^{6} \)
$29$ \( 1 + 6 T + 83 T^{2} + 308 T^{3} + 2407 T^{4} + 5046 T^{5} + 24389 T^{6} \)
$31$ \( 1 - 16 T + 157 T^{2} - 1024 T^{3} + 4867 T^{4} - 15376 T^{5} + 29791 T^{6} \)
$37$ \( 1 + 6 T + 75 T^{2} + 336 T^{3} + 2775 T^{4} + 8214 T^{5} + 50653 T^{6} \)
$41$ \( ( 1 - T )^{3} \)
$43$ \( 1 + 4 T + 121 T^{2} + 328 T^{3} + 5203 T^{4} + 7396 T^{5} + 79507 T^{6} \)
$47$ \( 1 + 21 T^{2} - 502 T^{3} + 987 T^{4} + 103823 T^{6} \)
$53$ \( 1 - 6 T + 155 T^{2} - 628 T^{3} + 8215 T^{4} - 16854 T^{5} + 148877 T^{6} \)
$59$ \( 1 + 8 T + 161 T^{2} + 784 T^{3} + 9499 T^{4} + 27848 T^{5} + 205379 T^{6} \)
$61$ \( 1 - 2 T + 131 T^{2} - 60 T^{3} + 7991 T^{4} - 7442 T^{5} + 226981 T^{6} \)
$67$ \( 1 + 2 T + 181 T^{2} + 218 T^{3} + 12127 T^{4} + 8978 T^{5} + 300763 T^{6} \)
$71$ \( 1 - 20 T + 297 T^{2} - 2706 T^{3} + 21087 T^{4} - 100820 T^{5} + 357911 T^{6} \)
$73$ \( 1 + 2 T + 39 T^{2} + 536 T^{3} + 2847 T^{4} + 10658 T^{5} + 389017 T^{6} \)
$79$ \( 1 - 32 T + 565 T^{2} - 6146 T^{3} + 44635 T^{4} - 199712 T^{5} + 493039 T^{6} \)
$83$ \( 1 + 185 T^{2} - 128 T^{3} + 15355 T^{4} + 571787 T^{6} \)
$89$ \( 1 + 6 T + 119 T^{2} + 148 T^{3} + 10591 T^{4} + 47526 T^{5} + 704969 T^{6} \)
$97$ \( 1 - 6 T + 239 T^{2} - 916 T^{3} + 23183 T^{4} - 56454 T^{5} + 912673 T^{6} \)
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