Properties

Label 51.2.a
Level 51
Weight 2
Character orbit a
Rep. character \(\chi_{51}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 12
Trace bound 1

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 51.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 \(M_{2}(\Gamma_0(51))\).

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.

\(3\)\(17\)FrickeDim.
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q - q^{2} - q^{3} + 3q^{4} + 6q^{5} + q^{6} - 4q^{7} - 9q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - q^{2} - q^{3} + 3q^{4} + 6q^{5} + q^{6} - 4q^{7} - 9q^{8} + 3q^{9} - 10q^{10} - 4q^{11} - 7q^{12} + 4q^{13} + 7q^{16} + q^{17} - q^{18} + 2q^{19} + 10q^{20} - 4q^{21} - 8q^{22} + 9q^{24} + 7q^{25} + 6q^{26} - q^{27} + 8q^{28} + 6q^{29} + 10q^{30} - 9q^{32} - 2q^{33} - q^{34} - 12q^{35} + 3q^{36} - 6q^{37} + 24q^{38} - 6q^{39} - 22q^{40} - 6q^{41} - 10q^{43} + 12q^{44} + 6q^{45} - 4q^{46} - 20q^{47} + q^{48} - 5q^{49} - 27q^{50} - 3q^{51} + 6q^{52} + 2q^{53} + q^{54} - 2q^{55} - 4q^{57} + 34q^{58} + 12q^{59} - 22q^{60} + 18q^{61} - 16q^{62} - 4q^{63} - q^{64} - 4q^{65} + 8q^{66} + 4q^{67} + 7q^{68} + 18q^{69} + 16q^{71} - 9q^{72} - 6q^{73} + 18q^{74} + q^{75} - 16q^{76} + 12q^{77} - 6q^{78} - 4q^{79} + 42q^{80} + 3q^{81} + 10q^{82} - 16q^{83} + 8q^{84} - 24q^{86} + 6q^{87} - 4q^{88} + 6q^{89} - 10q^{90} + 4q^{91} - 32q^{92} + 4q^{93} + 24q^{94} - 24q^{95} + 9q^{96} - 30q^{97} + 7q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(51))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 17
51.2.a.a \(1\) \(0.407\) \(\Q\) None \(0\) \(1\) \(3\) \(-4\) \(-\) \(+\) \(q+q^{3}-2q^{4}+3q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
51.2.a.b \(2\) \(0.407\) \(\Q(\sqrt{17}) \) None \(-1\) \(-2\) \(3\) \(0\) \(+\) \(-\) \(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(51))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(51)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 2 T^{2} \))(\( 1 + T + 2 T^{3} + 4 T^{4} \))
$3$ (\( 1 - T \))(\( ( 1 + T )^{2} \))
$5$ (\( 1 - 3 T + 5 T^{2} \))(\( 1 - 3 T + 8 T^{2} - 15 T^{3} + 25 T^{4} \))
$7$ (\( 1 + 4 T + 7 T^{2} \))(\( ( 1 + 7 T^{2} )^{2} \))
$11$ (\( 1 + 3 T + 11 T^{2} \))(\( 1 + T + 18 T^{2} + 11 T^{3} + 121 T^{4} \))
$13$ (\( 1 + T + 13 T^{2} \))(\( 1 - 5 T + 28 T^{2} - 65 T^{3} + 169 T^{4} \))
$17$ (\( 1 + T \))(\( ( 1 - T )^{2} \))
$19$ (\( 1 + T + 19 T^{2} \))(\( 1 - 3 T + 2 T^{2} - 57 T^{3} + 361 T^{4} \))
$23$ (\( 1 - 9 T + 23 T^{2} \))(\( 1 + 9 T + 62 T^{2} + 207 T^{3} + 529 T^{4} \))
$29$ (\( 1 - 6 T + 29 T^{2} \))(\( 1 - 10 T^{2} + 841 T^{4} \))
$31$ (\( 1 - 2 T + 31 T^{2} \))(\( 1 + 2 T + 46 T^{2} + 62 T^{3} + 961 T^{4} \))
$37$ (\( 1 + 4 T + 37 T^{2} \))(\( 1 + 2 T + 58 T^{2} + 74 T^{3} + 1369 T^{4} \))
$41$ (\( 1 + 3 T + 41 T^{2} \))(\( 1 + 3 T + 80 T^{2} + 123 T^{3} + 1681 T^{4} \))
$43$ (\( 1 + 7 T + 43 T^{2} \))(\( 1 + 3 T + 50 T^{2} + 129 T^{3} + 1849 T^{4} \))
$47$ (\( 1 + 6 T + 47 T^{2} \))(\( 1 + 14 T + 126 T^{2} + 658 T^{3} + 2209 T^{4} \))
$53$ (\( 1 + 6 T + 53 T^{2} \))(\( 1 - 8 T + 54 T^{2} - 424 T^{3} + 2809 T^{4} \))
$59$ (\( 1 - 6 T + 59 T^{2} \))(\( 1 - 6 T + 110 T^{2} - 354 T^{3} + 3481 T^{4} \))
$61$ (\( 1 - 8 T + 61 T^{2} \))(\( 1 - 10 T + 130 T^{2} - 610 T^{3} + 3721 T^{4} \))
$67$ (\( 1 + 4 T + 67 T^{2} \))(\( ( 1 - 4 T + 67 T^{2} )^{2} \))
$71$ (\( 1 - 12 T + 71 T^{2} \))(\( 1 - 4 T + 78 T^{2} - 284 T^{3} + 5041 T^{4} \))
$73$ (\( 1 - 2 T + 73 T^{2} \))(\( 1 + 8 T + 94 T^{2} + 584 T^{3} + 5329 T^{4} \))
$79$ (\( 1 + 10 T + 79 T^{2} \))(\( 1 - 6 T + 14 T^{2} - 474 T^{3} + 6241 T^{4} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 10 T + 174 T^{2} + 830 T^{3} + 6889 T^{4} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 6 T + 170 T^{2} - 534 T^{3} + 7921 T^{4} \))
$97$ (\( 1 + 16 T + 97 T^{2} \))(\( 1 + 14 T + 226 T^{2} + 1358 T^{3} + 9409 T^{4} \))
show more
show less