Properties

Label 159.2.a
Level 159
Weight 2
Character orbit a
Rep. character \(\chi_{159}(1,\cdot)\)
Character field \(\Q\)
Dimension 9
Newform subspaces 2
Sturm bound 36
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 159 = 3 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 159.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(159))\).

Total New Old
Modular forms 20 9 11
Cusp forms 17 9 8
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.

\(3\)\(53\)FrickeDim.
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(4\)
Plus space\(+\)\(0\)
Minus space\(-\)\(9\)

Trace form

\( 9q + 3q^{2} - q^{3} + 13q^{4} + 2q^{5} + 3q^{6} + 3q^{8} + 9q^{9} + O(q^{10}) \) \( 9q + 3q^{2} - q^{3} + 13q^{4} + 2q^{5} + 3q^{6} + 3q^{8} + 9q^{9} - 2q^{10} + 8q^{11} - 7q^{12} + 2q^{13} - 4q^{14} + 2q^{15} + 9q^{16} + 10q^{17} + 3q^{18} - 4q^{19} - 22q^{20} - 8q^{21} - 16q^{22} - 4q^{23} + 3q^{24} + 19q^{25} - 22q^{26} - q^{27} - 32q^{28} + 26q^{29} - 18q^{30} - 4q^{31} - 21q^{32} + 4q^{33} - 34q^{34} - 24q^{35} + 13q^{36} - 6q^{37} - 44q^{38} - 14q^{39} - 32q^{40} + 30q^{41} - 6q^{42} - 4q^{43} + 4q^{44} + 2q^{45} + 26q^{46} - 4q^{47} - 15q^{48} + 29q^{49} + 17q^{50} + 10q^{51} + 12q^{52} + q^{53} + 3q^{54} - 20q^{55} + 28q^{56} - 8q^{57} + 2q^{58} - 8q^{59} - 8q^{60} + 26q^{61} + q^{64} + 40q^{65} + 20q^{66} + 22q^{68} + 8q^{69} + 18q^{70} - 4q^{71} + 3q^{72} + 18q^{73} + 14q^{74} - 7q^{75} - 4q^{76} - 12q^{77} + 6q^{78} + 8q^{79} - 22q^{80} + 9q^{81} + 28q^{82} - 24q^{83} - 8q^{84} + 36q^{86} - 14q^{87} + 12q^{88} + 22q^{89} - 2q^{90} - 16q^{91} + 44q^{92} - 20q^{93} + 16q^{94} + 8q^{95} + 29q^{96} - 10q^{97} + 19q^{98} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(159))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 53
159.2.a.a \(4\) \(1.270\) 4.4.1957.1 None \(3\) \(4\) \(2\) \(-4\) \(-\) \(+\) \(q+(1+\beta _{2})q^{2}+q^{3}+(1-\beta _{1}-\beta _{3})q^{4}+\cdots\)
159.2.a.b \(5\) \(1.270\) 5.5.1054013.1 None \(0\) \(-5\) \(0\) \(4\) \(+\) \(-\) \(q+(-\beta _{1}-\beta _{3})q^{2}-q^{3}+(2-\beta _{4})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(159))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(159)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(53))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 3 T + 7 T^{2} - 11 T^{3} + 17 T^{4} - 22 T^{5} + 28 T^{6} - 24 T^{7} + 16 T^{8} \))(\( 1 + 2 T^{4} + 5 T^{5} + 4 T^{6} + 32 T^{10} \))
$3$ (\( ( 1 - T )^{4} \))(\( ( 1 + T )^{5} \))
$5$ (\( 1 - 2 T + 9 T^{2} + 2 T^{3} + 19 T^{4} + 10 T^{5} + 225 T^{6} - 250 T^{7} + 625 T^{8} \))(\( 1 + 6 T^{2} + 6 T^{3} + 32 T^{4} + 58 T^{5} + 160 T^{6} + 150 T^{7} + 750 T^{8} + 3125 T^{10} \))
$7$ (\( 1 + 4 T + 21 T^{2} + 40 T^{3} + 153 T^{4} + 280 T^{5} + 1029 T^{6} + 1372 T^{7} + 2401 T^{8} \))(\( 1 - 4 T + 12 T^{2} - 20 T^{3} + 124 T^{4} - 360 T^{5} + 868 T^{6} - 980 T^{7} + 4116 T^{8} - 9604 T^{9} + 16807 T^{10} \))
$11$ (\( 1 - 6 T + 16 T^{2} + 34 T^{3} - 226 T^{4} + 374 T^{5} + 1936 T^{6} - 7986 T^{7} + 14641 T^{8} \))(\( 1 - 2 T + 27 T^{2} - 16 T^{3} + 302 T^{4} + 68 T^{5} + 3322 T^{6} - 1936 T^{7} + 35937 T^{8} - 29282 T^{9} + 161051 T^{10} \))
$13$ (\( 1 + 6 T + 43 T^{2} + 164 T^{3} + 781 T^{4} + 2132 T^{5} + 7267 T^{6} + 13182 T^{7} + 28561 T^{8} \))(\( 1 - 8 T + 52 T^{2} - 280 T^{3} + 1284 T^{4} - 4686 T^{5} + 16692 T^{6} - 47320 T^{7} + 114244 T^{8} - 228488 T^{9} + 371293 T^{10} \))
$17$ (\( 1 - 10 T + 56 T^{2} - 230 T^{3} + 894 T^{4} - 3910 T^{5} + 16184 T^{6} - 49130 T^{7} + 83521 T^{8} \))(\( 1 + 45 T^{2} + 1202 T^{4} - 160 T^{5} + 20434 T^{6} + 221085 T^{8} + 1419857 T^{10} \))
$19$ (\( 1 + 6 T + 40 T^{2} + 62 T^{3} + 430 T^{4} + 1178 T^{5} + 14440 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( 1 - 2 T + 67 T^{2} - 80 T^{3} + 2030 T^{4} - 1660 T^{5} + 38570 T^{6} - 28880 T^{7} + 459553 T^{8} - 260642 T^{9} + 2476099 T^{10} \))
$23$ (\( 1 - 2 T + 57 T^{2} - 34 T^{3} + 1543 T^{4} - 782 T^{5} + 30153 T^{6} - 24334 T^{7} + 279841 T^{8} \))(\( 1 + 6 T + 76 T^{2} + 356 T^{3} + 2826 T^{4} + 10300 T^{5} + 64998 T^{6} + 188324 T^{7} + 924692 T^{8} + 1679046 T^{9} + 6436343 T^{10} \))
$29$ (\( 1 - 6 T + 88 T^{2} - 290 T^{3} + 3086 T^{4} - 8410 T^{5} + 74008 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 - 20 T + 241 T^{2} - 2128 T^{3} + 15034 T^{4} - 88280 T^{5} + 435986 T^{6} - 1789648 T^{7} + 5877749 T^{8} - 14145620 T^{9} + 20511149 T^{10} \))
$31$ (\( 1 + 12 T + 100 T^{2} + 548 T^{3} + 3334 T^{4} + 16988 T^{5} + 96100 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 - 8 T + 147 T^{2} - 824 T^{3} + 8530 T^{4} - 35584 T^{5} + 264430 T^{6} - 791864 T^{7} + 4379277 T^{8} - 7388168 T^{9} + 28629151 T^{10} \))
$37$ (\( 1 + 10 T + 151 T^{2} + 1016 T^{3} + 8489 T^{4} + 37592 T^{5} + 206719 T^{6} + 506530 T^{7} + 1874161 T^{8} \))(\( 1 - 4 T + 120 T^{2} - 344 T^{3} + 6260 T^{4} - 14470 T^{5} + 231620 T^{6} - 470936 T^{7} + 6078360 T^{8} - 7496644 T^{9} + 69343957 T^{10} \))
$41$ (\( 1 - 12 T + 79 T^{2} - 734 T^{3} + 6359 T^{4} - 30094 T^{5} + 132799 T^{6} - 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 18 T + 268 T^{2} - 2552 T^{3} + 22006 T^{4} - 145274 T^{5} + 902246 T^{6} - 4289912 T^{7} + 18470828 T^{8} - 50863698 T^{9} + 115856201 T^{10} \))
$43$ (\( 1 + 16 T + 205 T^{2} + 1708 T^{3} + 12829 T^{4} + 73444 T^{5} + 379045 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( 1 - 12 T + 176 T^{2} - 1672 T^{3} + 13788 T^{4} - 102420 T^{5} + 592884 T^{6} - 3091528 T^{7} + 13993232 T^{8} - 41025612 T^{9} + 147008443 T^{10} \))
$47$ (\( 1 + 124 T^{2} - 200 T^{3} + 7190 T^{4} - 9400 T^{5} + 273916 T^{6} + 4879681 T^{8} \))(\( 1 + 4 T + 131 T^{2} + 296 T^{3} + 7698 T^{4} + 12200 T^{5} + 361806 T^{6} + 653864 T^{7} + 13600813 T^{8} + 19518724 T^{9} + 229345007 T^{10} \))
$53$ (\( ( 1 + T )^{4} \))(\( ( 1 - T )^{5} \))
$59$ (\( 1 - 10 T + 120 T^{2} - 866 T^{3} + 7246 T^{4} - 51094 T^{5} + 417720 T^{6} - 2053790 T^{7} + 12117361 T^{8} \))(\( 1 + 18 T + 315 T^{2} + 3024 T^{3} + 30910 T^{4} + 219548 T^{5} + 1823690 T^{6} + 10526544 T^{7} + 64694385 T^{8} + 218112498 T^{9} + 714924299 T^{10} \))
$61$ (\( 1 - 14 T + 240 T^{2} - 2106 T^{3} + 21086 T^{4} - 128466 T^{5} + 893040 T^{6} - 3177734 T^{7} + 13845841 T^{8} \))(\( 1 - 12 T + 113 T^{2} - 1712 T^{3} + 15706 T^{4} - 101288 T^{5} + 958066 T^{6} - 6370352 T^{7} + 25648853 T^{8} - 166150092 T^{9} + 844596301 T^{10} \))
$67$ (\( 1 + 6 T + 184 T^{2} + 926 T^{3} + 16174 T^{4} + 62042 T^{5} + 825976 T^{6} + 1804578 T^{7} + 20151121 T^{8} \))(\( 1 - 6 T + 147 T^{2} - 640 T^{3} + 14574 T^{4} - 66132 T^{5} + 976458 T^{6} - 2872960 T^{7} + 44212161 T^{8} - 120906726 T^{9} + 1350125107 T^{10} \))
$71$ (\( 1 + 8 T + 259 T^{2} + 1550 T^{3} + 27083 T^{4} + 110050 T^{5} + 1305619 T^{6} + 2863288 T^{7} + 25411681 T^{8} \))(\( 1 - 4 T + 62 T^{2} - 314 T^{3} + 604 T^{4} - 29252 T^{5} + 42884 T^{6} - 1582874 T^{7} + 22190482 T^{8} - 101646724 T^{9} + 1804229351 T^{10} \))
$73$ (\( 1 - 10 T + 156 T^{2} - 1326 T^{3} + 17190 T^{4} - 96798 T^{5} + 831324 T^{6} - 3890170 T^{7} + 28398241 T^{8} \))(\( 1 - 8 T + 65 T^{2} - 912 T^{3} + 10422 T^{4} - 56688 T^{5} + 760806 T^{6} - 4860048 T^{7} + 25286105 T^{8} - 227185928 T^{9} + 2073071593 T^{10} \))
$79$ (\( 1 - 10 T + 320 T^{2} - 2218 T^{3} + 37774 T^{4} - 175222 T^{5} + 1997120 T^{6} - 4930390 T^{7} + 38950081 T^{8} \))(\( 1 + 2 T + 279 T^{2} + 544 T^{3} + 36214 T^{4} + 59580 T^{5} + 2860906 T^{6} + 3395104 T^{7} + 137557881 T^{8} + 77900162 T^{9} + 3077056399 T^{10} \))
$83$ (\( 1 - 12 T + 311 T^{2} - 2986 T^{3} + 37851 T^{4} - 247838 T^{5} + 2142479 T^{6} - 6861444 T^{7} + 47458321 T^{8} \))(\( 1 + 36 T + 742 T^{2} + 10334 T^{3} + 114580 T^{4} + 1091016 T^{5} + 9510140 T^{6} + 71190926 T^{7} + 424265954 T^{8} + 1708499556 T^{9} + 3939040643 T^{10} \))
$89$ (\( 1 + 4 T + 156 T^{2} + 692 T^{3} + 12934 T^{4} + 61588 T^{5} + 1235676 T^{6} + 2819876 T^{7} + 62742241 T^{8} \))(\( 1 - 26 T + 533 T^{2} - 6736 T^{3} + 80530 T^{4} - 745612 T^{5} + 7167170 T^{6} - 53355856 T^{7} + 375748477 T^{8} - 1631298266 T^{9} + 5584059449 T^{10} \))
$97$ (\( 1 - 6 T + 315 T^{2} - 1244 T^{3} + 41861 T^{4} - 120668 T^{5} + 2963835 T^{6} - 5476038 T^{7} + 88529281 T^{8} \))(\( 1 + 16 T + 240 T^{2} + 2976 T^{3} + 38592 T^{4} + 424542 T^{5} + 3743424 T^{6} + 28001184 T^{7} + 219041520 T^{8} + 1416468496 T^{9} + 8587340257 T^{10} \))
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