Properties

Label 105.2.a
Level 105
Weight 2
Character orbit a
Rep. character \(\chi_{105}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 32
Trace bound 1

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Defining parameters

Level: \( N \) = \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 105.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(105))\).

Total New Old
Modular forms 20 3 17
Cusp forms 13 3 10
Eisenstein series 7 0 7

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

\(3\)\(5\)\(7\)FrickeDim.
\(+\)\(+\)\(-\)\(-\)\(2\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + q^{2} - q^{3} + 5q^{4} - q^{5} + q^{6} + 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} - q^{3} + 5q^{4} - q^{5} + q^{6} + 3q^{7} - 3q^{8} + 3q^{9} + q^{10} + 4q^{11} - 7q^{12} - 6q^{13} + q^{14} + 3q^{15} - 3q^{16} - 2q^{17} + q^{18} - 4q^{19} - 7q^{20} - q^{21} - 20q^{22} + 16q^{23} - 3q^{24} + 3q^{25} - 26q^{26} - q^{27} + 5q^{28} - 6q^{29} + q^{30} + 16q^{31} + 5q^{32} - 4q^{33} + 2q^{34} - q^{35} + 5q^{36} + 2q^{37} + 12q^{38} - 6q^{39} - 3q^{40} - 10q^{41} + q^{42} + 4q^{43} + 12q^{44} - q^{45} + 8q^{46} + 16q^{47} + q^{48} + 3q^{49} + q^{50} + 6q^{51} + 6q^{52} - 6q^{53} + q^{54} - 4q^{55} - 3q^{56} - 12q^{57} - 2q^{58} + 4q^{59} + 5q^{60} - 6q^{61} + 24q^{62} + 3q^{63} - 19q^{64} - 6q^{65} + 20q^{66} - 4q^{67} - 14q^{68} + q^{70} + 8q^{71} - 3q^{72} - 18q^{73} + 38q^{74} - q^{75} + 20q^{76} + 4q^{77} + 14q^{78} + 16q^{79} + q^{80} + 3q^{81} - 6q^{82} - 20q^{83} - 7q^{84} + 6q^{85} - 36q^{86} + 2q^{87} - 20q^{88} - 10q^{89} + q^{90} - 6q^{91} + 16q^{92} - 8q^{93} - 32q^{94} - 12q^{95} + 5q^{96} - 10q^{97} + q^{98} + 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(105))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5 7
105.2.a.a \(1\) \(0.838\) \(\Q\) None \(1\) \(1\) \(1\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}+q^{7}+\cdots\)
105.2.a.b \(2\) \(0.838\) \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-2\) \(2\) \(+\) \(+\) \(-\) \(q-\beta q^{2}-q^{3}+3q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(105))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(105)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + 2 T^{2} \))(\( 1 - T^{2} + 4 T^{4} \))
$3$ (\( 1 - T \))(\( ( 1 + T )^{2} \))
$5$ (\( 1 - T \))(\( ( 1 + T )^{2} \))
$7$ (\( 1 - T \))(\( ( 1 - T )^{2} \))
$11$ (\( 1 + 11 T^{2} \))(\( 1 - 4 T + 6 T^{2} - 44 T^{3} + 121 T^{4} \))
$13$ (\( 1 + 6 T + 13 T^{2} \))(\( 1 + 6 T^{2} + 169 T^{4} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( ( 1 + 2 T + 17 T^{2} )^{2} \))
$19$ (\( 1 + 8 T + 19 T^{2} \))(\( 1 - 4 T + 22 T^{2} - 76 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 8 T + 23 T^{2} \))(\( ( 1 - 4 T + 23 T^{2} )^{2} \))
$29$ (\( 1 + 2 T + 29 T^{2} \))(\( ( 1 + 2 T + 29 T^{2} )^{2} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 - 12 T + 78 T^{2} - 372 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 2 T + 37 T^{2} \))(\( 1 - 4 T - 2 T^{2} - 148 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 6 T + 41 T^{2} \))(\( ( 1 + 2 T + 41 T^{2} )^{2} \))
$43$ (\( 1 - 4 T + 43 T^{2} \))(\( 1 + 6 T^{2} + 1849 T^{4} \))
$47$ (\( 1 - 8 T + 47 T^{2} \))(\( 1 - 8 T + 30 T^{2} - 376 T^{3} + 2209 T^{4} \))
$53$ (\( 1 - 10 T + 53 T^{2} \))(\( 1 + 16 T + 150 T^{2} + 848 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 + 38 T^{2} + 3481 T^{4} \))
$61$ (\( 1 + 2 T + 61 T^{2} \))(\( ( 1 + 2 T + 61 T^{2} )^{2} \))
$67$ (\( 1 - 4 T + 67 T^{2} \))(\( ( 1 + 4 T + 67 T^{2} )^{2} \))
$71$ (\( 1 + 12 T + 71 T^{2} \))(\( 1 - 20 T + 222 T^{2} - 1420 T^{3} + 5041 T^{4} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 16 T + 190 T^{2} + 1168 T^{3} + 5329 T^{4} \))
$79$ (\( 1 - 8 T + 79 T^{2} \))(\( 1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4} \))
$83$ (\( 1 + 4 T + 83 T^{2} \))(\( 1 + 16 T + 150 T^{2} + 1328 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 6 T + 89 T^{2} \))(\( ( 1 + 2 T + 89 T^{2} )^{2} \))
$97$ (\( 1 + 18 T + 97 T^{2} \))(\( 1 - 8 T + 190 T^{2} - 776 T^{3} + 9409 T^{4} \))
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