Properties

Label 10.4.a
Level 10
Weight 4
Character orbit a
Rep. character \(\chi_{10}(1,\cdot)\)
Character field \(\Q\)
Dimension 1
Newform subspaces 1
Sturm bound 6
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 10.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(6\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(10))\).

Total New Old
Modular forms 7 1 6
Cusp forms 3 1 2
Eisenstein series 4 0 4

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

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

Trace form

\( q + 2q^{2} - 8q^{3} + 4q^{4} + 5q^{5} - 16q^{6} - 4q^{7} + 8q^{8} + 37q^{9} + O(q^{10}) \) \( q + 2q^{2} - 8q^{3} + 4q^{4} + 5q^{5} - 16q^{6} - 4q^{7} + 8q^{8} + 37q^{9} + 10q^{10} + 12q^{11} - 32q^{12} - 58q^{13} - 8q^{14} - 40q^{15} + 16q^{16} + 66q^{17} + 74q^{18} - 100q^{19} + 20q^{20} + 32q^{21} + 24q^{22} + 132q^{23} - 64q^{24} + 25q^{25} - 116q^{26} - 80q^{27} - 16q^{28} - 90q^{29} - 80q^{30} + 152q^{31} + 32q^{32} - 96q^{33} + 132q^{34} - 20q^{35} + 148q^{36} - 34q^{37} - 200q^{38} + 464q^{39} + 40q^{40} - 438q^{41} + 64q^{42} + 32q^{43} + 48q^{44} + 185q^{45} + 264q^{46} - 204q^{47} - 128q^{48} - 327q^{49} + 50q^{50} - 528q^{51} - 232q^{52} + 222q^{53} - 160q^{54} + 60q^{55} - 32q^{56} + 800q^{57} - 180q^{58} + 420q^{59} - 160q^{60} + 902q^{61} + 304q^{62} - 148q^{63} + 64q^{64} - 290q^{65} - 192q^{66} - 1024q^{67} + 264q^{68} - 1056q^{69} - 40q^{70} + 432q^{71} + 296q^{72} + 362q^{73} - 68q^{74} - 200q^{75} - 400q^{76} - 48q^{77} + 928q^{78} - 160q^{79} + 80q^{80} - 359q^{81} - 876q^{82} + 72q^{83} + 128q^{84} + 330q^{85} + 64q^{86} + 720q^{87} + 96q^{88} + 810q^{89} + 370q^{90} + 232q^{91} + 528q^{92} - 1216q^{93} - 408q^{94} - 500q^{95} - 256q^{96} + 1106q^{97} - 654q^{98} + 444q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(10))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 5
10.4.a.a \(1\) \(0.590\) \(\Q\) None \(2\) \(-8\) \(5\) \(-4\) \(-\) \(-\) \(q+2q^{2}-8q^{3}+4q^{4}+5q^{5}-2^{4}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(10))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(10)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T \)
$3$ \( 1 + 8 T + 27 T^{2} \)
$5$ \( 1 - 5 T \)
$7$ \( 1 + 4 T + 343 T^{2} \)
$11$ \( 1 - 12 T + 1331 T^{2} \)
$13$ \( 1 + 58 T + 2197 T^{2} \)
$17$ \( 1 - 66 T + 4913 T^{2} \)
$19$ \( 1 + 100 T + 6859 T^{2} \)
$23$ \( 1 - 132 T + 12167 T^{2} \)
$29$ \( 1 + 90 T + 24389 T^{2} \)
$31$ \( 1 - 152 T + 29791 T^{2} \)
$37$ \( 1 + 34 T + 50653 T^{2} \)
$41$ \( 1 + 438 T + 68921 T^{2} \)
$43$ \( 1 - 32 T + 79507 T^{2} \)
$47$ \( 1 + 204 T + 103823 T^{2} \)
$53$ \( 1 - 222 T + 148877 T^{2} \)
$59$ \( 1 - 420 T + 205379 T^{2} \)
$61$ \( 1 - 902 T + 226981 T^{2} \)
$67$ \( 1 + 1024 T + 300763 T^{2} \)
$71$ \( 1 - 432 T + 357911 T^{2} \)
$73$ \( 1 - 362 T + 389017 T^{2} \)
$79$ \( 1 + 160 T + 493039 T^{2} \)
$83$ \( 1 - 72 T + 571787 T^{2} \)
$89$ \( 1 - 810 T + 704969 T^{2} \)
$97$ \( 1 - 1106 T + 912673 T^{2} \)
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