Properties

Label 55.2.a
Level 55
Weight 2
Character orbit a
Rep. character \(\chi_{55}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) = \( 55 = 5 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 55.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(55))\).

Total New Old
Modular forms 8 3 5
Cusp forms 5 3 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.

\(5\)\(11\)FrickeDim.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + 3q^{2} + q^{4} - q^{5} - 8q^{6} - 4q^{7} + 3q^{8} + 7q^{9} + O(q^{10}) \) \( 3q + 3q^{2} + q^{4} - q^{5} - 8q^{6} - 4q^{7} + 3q^{8} + 7q^{9} - q^{10} + q^{11} - 16q^{12} - 6q^{13} - 4q^{14} + 5q^{16} + 14q^{17} + 7q^{18} - 4q^{19} - 3q^{20} + q^{22} + 4q^{23} - 8q^{24} + 3q^{25} + 2q^{26} - 4q^{28} + 10q^{29} + 8q^{30} - 8q^{31} - q^{32} + 22q^{34} + 4q^{35} + 13q^{36} - 6q^{37} - 4q^{38} - 16q^{39} - 9q^{40} + 14q^{41} + 16q^{42} - 8q^{43} + 3q^{44} - 13q^{45} - 4q^{46} - 12q^{47} - 13q^{49} + 3q^{50} - 16q^{51} + 6q^{52} + 10q^{53} - 16q^{54} - 3q^{55} - 12q^{56} - 6q^{58} - 4q^{59} + 16q^{60} - 6q^{61} - 8q^{62} - 20q^{63} - 7q^{64} + 10q^{65} - 8q^{66} - 8q^{67} + 18q^{68} + 16q^{69} + 4q^{70} + 8q^{71} + 39q^{72} + 6q^{73} - 22q^{74} + 4q^{76} - 4q^{77} + 16q^{78} + 16q^{79} - 7q^{80} + 11q^{81} + 14q^{82} - 16q^{83} + 32q^{84} - 2q^{85} - 8q^{86} + 32q^{87} + 9q^{88} + 6q^{89} - 13q^{90} + 16q^{91} - 20q^{92} - 4q^{94} - 4q^{95} - 8q^{96} + 6q^{97} - 13q^{98} + 13q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(55))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 5 11
55.2.a.a \(1\) \(0.439\) \(\Q\) None \(1\) \(0\) \(1\) \(0\) \(-\) \(+\) \(q+q^{2}-q^{4}+q^{5}-3q^{8}-3q^{9}+q^{10}+\cdots\)
55.2.a.b \(2\) \(0.439\) \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(-2\) \(-4\) \(+\) \(-\) \(q+(1+\beta )q^{2}-2\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(55))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(55)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 3 T^{2} - 4 T^{3} + 4 T^{4} \))
$3$ (\( 1 + 3 T^{2} \))(\( 1 - 2 T^{2} + 9 T^{4} \))
$5$ (\( 1 - T \))(\( ( 1 + T )^{2} \))
$7$ (\( 1 + 7 T^{2} \))(\( ( 1 + 2 T + 7 T^{2} )^{2} \))
$11$ (\( 1 + T \))(\( ( 1 - T )^{2} \))
$13$ (\( 1 - 2 T + 13 T^{2} \))(\( 1 + 8 T + 34 T^{2} + 104 T^{3} + 169 T^{4} \))
$17$ (\( 1 - 6 T + 17 T^{2} \))(\( 1 - 8 T + 42 T^{2} - 136 T^{3} + 289 T^{4} \))
$19$ (\( 1 + 4 T + 19 T^{2} \))(\( ( 1 + 19 T^{2} )^{2} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 + 38 T^{2} + 529 T^{4} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 4 T + 30 T^{2} - 116 T^{3} + 841 T^{4} \))
$31$ (\( 1 + 8 T + 31 T^{2} \))(\( ( 1 + 31 T^{2} )^{2} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 + 4 T + 46 T^{2} + 148 T^{3} + 1369 T^{4} \))
$41$ (\( 1 - 2 T + 41 T^{2} \))(\( ( 1 - 6 T + 41 T^{2} )^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( ( 1 + 6 T + 43 T^{2} )^{2} \))
$47$ (\( 1 + 12 T + 47 T^{2} \))(\( 1 + 86 T^{2} + 2209 T^{4} \))
$53$ (\( 1 + 2 T + 53 T^{2} \))(\( 1 - 12 T + 110 T^{2} - 636 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 8 T + 102 T^{2} + 472 T^{3} + 3481 T^{4} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( 1 - 4 T - 2 T^{2} - 244 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 16 T + 67 T^{2} \))(\( 1 - 8 T + 78 T^{2} - 536 T^{3} + 4489 T^{4} \))
$71$ (\( 1 - 8 T + 71 T^{2} \))(\( 1 + 14 T^{2} + 5041 T^{4} \))
$73$ (\( 1 - 14 T + 73 T^{2} \))(\( 1 + 8 T + 154 T^{2} + 584 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( ( 1 - 4 T + 79 T^{2} )^{2} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( ( 1 + 6 T + 83 T^{2} )^{2} \))
$89$ (\( 1 - 10 T + 89 T^{2} \))(\( 1 + 4 T + 54 T^{2} + 356 T^{3} + 7921 T^{4} \))
$97$ (\( 1 - 10 T + 97 T^{2} \))(\( 1 + 4 T + 166 T^{2} + 388 T^{3} + 9409 T^{4} \))
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