Properties

Label 59.2.a
Level 59
Weight 2
Character orbit a
Rep. character \(\chi_{59}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 1
Sturm bound 10
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 59.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(59))\).

Total New Old
Modular forms 6 6 0
Cusp forms 5 5 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 and the Fricke involution.

\(59\)Dim.
\(-\)\(5\)

Trace form

\( 5q - 2q^{3} + 8q^{4} + 2q^{5} - 4q^{6} + 2q^{7} - 6q^{8} + 5q^{9} + O(q^{10}) \) \( 5q - 2q^{3} + 8q^{4} + 2q^{5} - 4q^{6} + 2q^{7} - 6q^{8} + 5q^{9} - 8q^{10} - 2q^{11} - 22q^{12} + 8q^{13} - 18q^{14} - 9q^{15} + 10q^{16} - q^{17} - 2q^{18} + 6q^{19} + 6q^{20} + 15q^{21} + 8q^{22} - 8q^{23} + 6q^{24} + 7q^{25} + 8q^{26} - 11q^{27} - 2q^{28} + 14q^{29} + 14q^{30} - 2q^{32} - 14q^{33} - 2q^{34} - 9q^{35} + 42q^{36} + 18q^{37} - 18q^{39} - 18q^{40} - 10q^{41} + 34q^{42} - 4q^{43} + 12q^{44} + q^{45} + 16q^{46} - 20q^{47} - 54q^{48} + q^{49} + 8q^{50} - 12q^{51} + 28q^{52} - 10q^{53} - 26q^{54} - 20q^{55} - 38q^{56} - 3q^{57} - 38q^{58} + 5q^{59} - 28q^{60} + 22q^{61} + 48q^{62} - 12q^{63} + 18q^{64} - 16q^{65} + 28q^{66} - 14q^{68} - 4q^{69} - 24q^{70} + 3q^{71} + 28q^{72} - 8q^{73} + 8q^{74} + 8q^{75} + 14q^{76} + 2q^{77} + 20q^{78} + 10q^{79} + 50q^{80} - 3q^{81} - 48q^{82} + 6q^{83} + 28q^{84} + 40q^{85} - 8q^{86} + 11q^{87} - 24q^{88} + 10q^{89} - 8q^{90} + 6q^{91} + 4q^{92} + 6q^{93} - 36q^{94} + 26q^{95} + 42q^{96} - 22q^{97} + 24q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(59))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 59
59.2.a.a \(5\) \(0.471\) 5.5.138136.1 None \(0\) \(-2\) \(2\) \(2\) \(-\) \(q+(\beta _{1}-\beta _{3})q^{2}+(-1-\beta _{2}+\beta _{4})q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} + 2 T^{3} + 2 T^{4} + 4 T^{6} + 8 T^{7} + 8 T^{8} + 32 T^{10} \)
$3$ \( 1 + 2 T + 7 T^{2} + 13 T^{3} + 31 T^{4} + 41 T^{5} + 93 T^{6} + 117 T^{7} + 189 T^{8} + 162 T^{9} + 243 T^{10} \)
$5$ \( 1 - 2 T + 11 T^{2} - 17 T^{3} + 59 T^{4} - 69 T^{5} + 295 T^{6} - 425 T^{7} + 1375 T^{8} - 1250 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 2 T + 19 T^{2} - 13 T^{3} + 167 T^{4} - 57 T^{5} + 1169 T^{6} - 637 T^{7} + 6517 T^{8} - 4802 T^{9} + 16807 T^{10} \)
$11$ \( 1 + 2 T + 31 T^{2} + 64 T^{3} + 546 T^{4} + 860 T^{5} + 6006 T^{6} + 7744 T^{7} + 41261 T^{8} + 29282 T^{9} + 161051 T^{10} \)
$13$ \( 1 - 8 T + 65 T^{2} - 328 T^{3} + 1642 T^{4} - 6048 T^{5} + 21346 T^{6} - 55432 T^{7} + 142805 T^{8} - 228488 T^{9} + 371293 T^{10} \)
$17$ \( 1 + T + 40 T^{2} - 13 T^{3} + 819 T^{4} - 608 T^{5} + 13923 T^{6} - 3757 T^{7} + 196520 T^{8} + 83521 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 6 T + 67 T^{2} - 239 T^{3} + 1847 T^{4} - 5219 T^{5} + 35093 T^{6} - 86279 T^{7} + 459553 T^{8} - 781926 T^{9} + 2476099 T^{10} \)
$23$ \( 1 + 8 T + 115 T^{2} + 648 T^{3} + 5178 T^{4} + 21312 T^{5} + 119094 T^{6} + 342792 T^{7} + 1399205 T^{8} + 2238728 T^{9} + 6436343 T^{10} \)
$29$ \( 1 - 14 T + 155 T^{2} - 1235 T^{3} + 8795 T^{4} - 49839 T^{5} + 255055 T^{6} - 1038635 T^{7} + 3780295 T^{8} - 9901934 T^{9} + 20511149 T^{10} \)
$31$ \( 1 + 39 T^{2} + 56 T^{3} + 102 T^{4} + 3728 T^{5} + 3162 T^{6} + 53816 T^{7} + 1161849 T^{8} + 28629151 T^{10} \)
$37$ \( 1 - 18 T + 265 T^{2} - 2600 T^{3} + 21978 T^{4} - 143084 T^{5} + 813186 T^{6} - 3559400 T^{7} + 13423045 T^{8} - 33734898 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 10 T + 135 T^{2} + 947 T^{3} + 8107 T^{4} + 44251 T^{5} + 332387 T^{6} + 1591907 T^{7} + 9304335 T^{8} + 28257610 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 4 T + 187 T^{2} + 632 T^{3} + 15134 T^{4} + 39432 T^{5} + 650762 T^{6} + 1168568 T^{7} + 14867809 T^{8} + 13675204 T^{9} + 147008443 T^{10} \)
$47$ \( 1 + 20 T + 359 T^{2} + 3952 T^{3} + 39254 T^{4} + 282872 T^{5} + 1844938 T^{6} + 8729968 T^{7} + 37272457 T^{8} + 97593620 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 10 T + 243 T^{2} + 2043 T^{3} + 24683 T^{4} + 160451 T^{5} + 1308199 T^{6} + 5738787 T^{7} + 36177111 T^{8} + 78904810 T^{9} + 418195493 T^{10} \)
$59$ \( ( 1 - T )^{5} \)
$61$ \( 1 - 22 T + 361 T^{2} - 4000 T^{3} + 39010 T^{4} - 313204 T^{5} + 2379610 T^{6} - 14884000 T^{7} + 81940141 T^{8} - 304608502 T^{9} + 844596301 T^{10} \)
$67$ \( 1 + 147 T^{2} - 200 T^{3} + 12574 T^{4} - 35696 T^{5} + 842458 T^{6} - 897800 T^{7} + 44212161 T^{8} + 1350125107 T^{10} \)
$71$ \( 1 - 3 T + 278 T^{2} - 837 T^{3} + 35705 T^{4} - 85184 T^{5} + 2535055 T^{6} - 4219317 T^{7} + 99499258 T^{8} - 76235043 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 8 T + 245 T^{2} + 1208 T^{3} + 26674 T^{4} + 101056 T^{5} + 1947202 T^{6} + 6437432 T^{7} + 95309165 T^{8} + 227185928 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 - 10 T + 335 T^{2} - 2155 T^{3} + 44559 T^{4} - 211747 T^{5} + 3520161 T^{6} - 13449355 T^{7} + 165168065 T^{8} - 389500810 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 6 T + 155 T^{2} - 1144 T^{3} + 19446 T^{4} - 78084 T^{5} + 1614018 T^{6} - 7881016 T^{7} + 88626985 T^{8} - 284749926 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 - 10 T + 365 T^{2} - 2960 T^{3} + 59834 T^{4} - 370444 T^{5} + 5325226 T^{6} - 23446160 T^{7} + 257313685 T^{8} - 627422410 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 22 T + 545 T^{2} + 7960 T^{3} + 111198 T^{4} + 1132900 T^{5} + 10786206 T^{6} + 74895640 T^{7} + 497406785 T^{8} + 1947644182 T^{9} + 8587340257 T^{10} \)
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