Properties

Label 37.2.f
Level 37
Weight 2
Character orbit f
Rep. character \(\chi_{37}(7,\cdot)\)
Character field \(\Q(\zeta_{9})\)
Dimension 12
Newform subspaces 2
Sturm bound 6
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 37.f (of order \(9\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 37 \)
Character field: \(\Q(\zeta_{9})\)
Newform subspaces: \( 2 \)
Sturm bound: \(6\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(37, [\chi])\).

Total New Old
Modular forms 24 24 0
Cusp forms 12 12 0
Eisenstein series 12 12 0

Trace form

\( 12q - 3q^{2} - 3q^{3} - 3q^{4} - 12q^{6} - 9q^{7} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( 12q - 3q^{2} - 3q^{3} - 3q^{4} - 12q^{6} - 9q^{7} + 3q^{8} - 3q^{9} + 3q^{10} + 9q^{11} + 9q^{12} - 21q^{13} - 9q^{14} + 9q^{15} + 21q^{16} + 6q^{17} + 12q^{18} - 6q^{19} - 27q^{20} + 18q^{21} + 15q^{22} + 15q^{23} + 15q^{24} - 12q^{25} + 6q^{26} - 3q^{27} + 15q^{28} - 6q^{29} - 6q^{30} - 12q^{31} - 30q^{32} - 24q^{33} + 12q^{34} + 12q^{35} - 42q^{36} - 18q^{37} - 18q^{38} - 12q^{39} - 3q^{40} + 15q^{41} + 3q^{42} - 24q^{43} - 6q^{44} + 24q^{45} - 18q^{46} + 24q^{47} + 15q^{48} + 9q^{49} + 33q^{50} - 33q^{51} + 39q^{52} + 45q^{53} + 39q^{54} + 24q^{55} - 21q^{56} + 33q^{57} + 30q^{58} + 30q^{60} - 36q^{61} - 12q^{62} - 15q^{63} - 9q^{64} - 3q^{65} - 15q^{66} + 6q^{67} + 18q^{68} - 30q^{69} - 66q^{70} + 21q^{71} - 3q^{72} + 12q^{73} - 54q^{74} - 66q^{75} + 15q^{76} - 27q^{77} - 9q^{78} + 30q^{79} - 36q^{80} + 3q^{81} - 27q^{82} + 24q^{83} - 18q^{84} - 15q^{85} + 12q^{86} - 9q^{87} + 6q^{88} - 12q^{89} + 27q^{90} + 45q^{91} + 18q^{93} + 6q^{94} - 3q^{95} + 33q^{96} - 9q^{97} + 39q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(37, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
37.2.f.a \(6\) \(0.295\) \(\Q(\zeta_{18})\) None \(-6\) \(3\) \(-6\) \(3\) \(q+(-1+\zeta_{18}^{5})q^{2}+(\zeta_{18}-\zeta_{18}^{2}+\zeta_{18}^{3}+\cdots)q^{3}+\cdots\)
37.2.f.b \(6\) \(0.295\) \(\Q(\zeta_{18})\) None \(3\) \(-6\) \(6\) \(-12\) \(q+(1-\zeta_{18}+\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{2}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 6 T + 15 T^{2} + 19 T^{3} + 6 T^{4} - 27 T^{5} - 61 T^{6} - 54 T^{7} + 24 T^{8} + 152 T^{9} + 240 T^{10} + 192 T^{11} + 64 T^{12} \))(\( 1 - 3 T + 9 T^{2} - 18 T^{3} + 36 T^{4} - 57 T^{5} + 91 T^{6} - 114 T^{7} + 144 T^{8} - 144 T^{9} + 144 T^{10} - 96 T^{11} + 64 T^{12} \))
$3$ (\( 1 - 3 T + 9 T^{2} - 15 T^{3} + 36 T^{4} - 54 T^{5} + 117 T^{6} - 162 T^{7} + 324 T^{8} - 405 T^{9} + 729 T^{10} - 729 T^{11} + 729 T^{12} \))(\( 1 + 6 T + 15 T^{2} + 19 T^{3} + 3 T^{4} - 51 T^{5} - 134 T^{6} - 153 T^{7} + 27 T^{8} + 513 T^{9} + 1215 T^{10} + 1458 T^{11} + 729 T^{12} \))
$5$ (\( 1 + 6 T + 24 T^{2} + 64 T^{3} + 132 T^{4} + 252 T^{5} + 479 T^{6} + 1260 T^{7} + 3300 T^{8} + 8000 T^{9} + 15000 T^{10} + 18750 T^{11} + 15625 T^{12} \))(\( 1 - 6 T + 18 T^{2} - 45 T^{3} + 81 T^{4} - 87 T^{5} + 109 T^{6} - 435 T^{7} + 2025 T^{8} - 5625 T^{9} + 11250 T^{10} - 18750 T^{11} + 15625 T^{12} \))
$7$ (\( 1 - 3 T + 12 T^{2} - 4 T^{3} + 81 T^{4} - 135 T^{5} + 1023 T^{6} - 945 T^{7} + 3969 T^{8} - 1372 T^{9} + 28812 T^{10} - 50421 T^{11} + 117649 T^{12} \))(\( 1 + 12 T + 60 T^{2} + 152 T^{3} + 108 T^{4} - 702 T^{5} - 3051 T^{6} - 4914 T^{7} + 5292 T^{8} + 52136 T^{9} + 144060 T^{10} + 201684 T^{11} + 117649 T^{12} \))
$11$ (\( 1 - 30 T^{2} - 2 T^{3} + 570 T^{4} + 30 T^{5} - 7237 T^{6} + 330 T^{7} + 68970 T^{8} - 2662 T^{9} - 439230 T^{10} + 1771561 T^{12} \))(\( 1 - 9 T + 30 T^{2} - 45 T^{3} + 177 T^{4} - 1548 T^{5} + 7099 T^{6} - 17028 T^{7} + 21417 T^{8} - 59895 T^{9} + 439230 T^{10} - 1449459 T^{11} + 1771561 T^{12} \))
$13$ (\( 1 + 9 T + 27 T^{2} - 11 T^{3} - 198 T^{4} + 360 T^{5} + 4161 T^{6} + 4680 T^{7} - 33462 T^{8} - 24167 T^{9} + 771147 T^{10} + 3341637 T^{11} + 4826809 T^{12} \))(\( 1 + 12 T + 78 T^{2} + 359 T^{3} + 1485 T^{4} + 6129 T^{5} + 23841 T^{6} + 79677 T^{7} + 250965 T^{8} + 788723 T^{9} + 2227758 T^{10} + 4455516 T^{11} + 4826809 T^{12} \))
$17$ (\( 1 - 3 T + 36 T^{2} + 90 T^{4} + 3489 T^{5} - 8513 T^{6} + 59313 T^{7} + 26010 T^{8} + 3006756 T^{10} - 4259571 T^{11} + 24137569 T^{12} \))(\( 1 - 3 T - 72 T^{3} - 117 T^{4} + 1275 T^{5} + 1369 T^{6} + 21675 T^{7} - 33813 T^{8} - 353736 T^{9} - 4259571 T^{11} + 24137569 T^{12} \))
$19$ (\( 1 - 3 T - 36 T^{2} + 150 T^{3} + 45 T^{4} - 1767 T^{5} + 10583 T^{6} - 33573 T^{7} + 16245 T^{8} + 1028850 T^{9} - 4691556 T^{10} - 7428297 T^{11} + 47045881 T^{12} \))(\( 1 + 9 T + 18 T^{2} - 142 T^{3} - 837 T^{4} + 513 T^{5} + 15087 T^{6} + 9747 T^{7} - 302157 T^{8} - 973978 T^{9} + 2345778 T^{10} + 22284891 T^{11} + 47045881 T^{12} \))
$23$ (\( 1 + 3 T - 60 T^{2} - 67 T^{3} + 2763 T^{4} + 1518 T^{5} - 71161 T^{6} + 34914 T^{7} + 1461627 T^{8} - 815189 T^{9} - 16790460 T^{10} + 19309029 T^{11} + 148035889 T^{12} \))(\( ( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} )^{3} \))
$29$ (\( 1 + 15 T + 75 T^{2} + 396 T^{3} + 4245 T^{4} + 21165 T^{5} + 66382 T^{6} + 613785 T^{7} + 3570045 T^{8} + 9658044 T^{9} + 53046075 T^{10} + 307667235 T^{11} + 594823321 T^{12} \))(\( 1 - 9 T - 6 T^{2} + 207 T^{3} + 591 T^{4} - 5220 T^{5} + 6181 T^{6} - 151380 T^{7} + 497031 T^{8} + 5048523 T^{9} - 4243686 T^{10} - 184600341 T^{11} + 594823321 T^{12} \))
$31$ (\( ( 1 - 3 T + 84 T^{2} - 167 T^{3} + 2604 T^{4} - 2883 T^{5} + 29791 T^{6} )^{2} \))(\( ( 1 + 9 T + 99 T^{2} + 505 T^{3} + 3069 T^{4} + 8649 T^{5} + 29791 T^{6} )^{2} \))
$37$ (\( 1 + 6 T + 96 T^{2} + 371 T^{3} + 3552 T^{4} + 8214 T^{5} + 50653 T^{6} \))(\( 1 + 12 T - 30 T^{2} - 803 T^{3} - 1110 T^{4} + 16428 T^{5} + 50653 T^{6} \))
$41$ (\( 1 + 9 T^{2} - 9 T^{3} + 369 T^{4} + 11889 T^{5} - 26774 T^{6} + 487449 T^{7} + 620289 T^{8} - 620289 T^{9} + 25431849 T^{10} + 4750104241 T^{12} \))(\( 1 - 15 T + 72 T^{2} + 162 T^{3} - 2889 T^{4} + 6123 T^{5} + 21493 T^{6} + 251043 T^{7} - 4856409 T^{8} + 11165202 T^{9} + 203454792 T^{10} - 1737843015 T^{11} + 4750104241 T^{12} \))
$43$ (\( ( 1 + 3 T + 69 T^{2} + 187 T^{3} + 2967 T^{4} + 5547 T^{5} + 79507 T^{6} )^{2} \))(\( ( 1 + 9 T + 153 T^{2} + 793 T^{3} + 6579 T^{4} + 16641 T^{5} + 79507 T^{6} )^{2} \))
$47$ (\( 1 - 15 T + 21 T^{2} - 126 T^{3} + 10023 T^{4} - 44763 T^{5} - 93242 T^{6} - 2103861 T^{7} + 22140807 T^{8} - 13081698 T^{9} + 102473301 T^{10} - 3440175105 T^{11} + 10779215329 T^{12} \))(\( 1 - 9 T - 24 T^{2} + 765 T^{3} - 1605 T^{4} - 20826 T^{5} + 241567 T^{6} - 978822 T^{7} - 3545445 T^{8} + 79424595 T^{9} - 117112344 T^{10} - 2064105063 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 - 24 T + 228 T^{2} - 866 T^{3} - 3012 T^{4} + 75276 T^{5} - 701113 T^{6} + 3989628 T^{7} - 8460708 T^{8} - 128927482 T^{9} + 1799029668 T^{10} - 10036691832 T^{11} + 22164361129 T^{12} \))(\( 1 - 21 T + 180 T^{2} - 756 T^{3} + 4158 T^{4} - 67395 T^{5} + 688951 T^{6} - 3571935 T^{7} + 11679822 T^{8} - 112551012 T^{9} + 1420286580 T^{10} - 8782105353 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 - 6 T - 12 T^{2} + 503 T^{3} - 3207 T^{4} - 32067 T^{5} + 445277 T^{6} - 1891953 T^{7} - 11163567 T^{8} + 103305637 T^{9} - 145408332 T^{10} - 4289545794 T^{11} + 42180533641 T^{12} \))(\( 1 + 6 T + 18 T^{2} + 180 T^{3} - 972 T^{4} - 38640 T^{5} - 291923 T^{6} - 2279760 T^{7} - 3383532 T^{8} + 36968220 T^{9} + 218112498 T^{10} + 4289545794 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 + 27 T + 324 T^{2} + 1924 T^{3} - 2538 T^{4} - 194049 T^{5} - 2164749 T^{6} - 11836989 T^{7} - 9443898 T^{8} + 436711444 T^{9} + 4486052484 T^{10} + 22804100127 T^{11} + 51520374361 T^{12} \))(\( 1 + 9 T + 45 T^{2} + 335 T^{3} + 2538 T^{4} + 11988 T^{5} - 65103 T^{6} + 731268 T^{7} + 9443898 T^{8} + 76038635 T^{9} + 623062845 T^{10} + 7601366709 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 + 18 T + 144 T^{2} + 821 T^{3} + 2133 T^{4} - 39825 T^{5} - 605283 T^{6} - 2668275 T^{7} + 9575037 T^{8} + 246926423 T^{9} + 2901761424 T^{10} + 24302251926 T^{11} + 90458382169 T^{12} \))(\( 1 - 24 T + 258 T^{2} - 1711 T^{3} + 603 T^{4} + 133227 T^{5} - 1538559 T^{6} + 8926209 T^{7} + 2706867 T^{8} - 514605493 T^{9} + 5198989218 T^{10} - 32403002568 T^{11} + 90458382169 T^{12} \))
$71$ (\( 1 - 33 T + 459 T^{2} - 3015 T^{3} + 3528 T^{4} + 104358 T^{5} - 1167947 T^{6} + 7409418 T^{7} + 17784648 T^{8} - 1079101665 T^{9} + 11663961579 T^{10} - 59539568583 T^{11} + 128100283921 T^{12} \))(\( 1 + 12 T + 126 T^{2} + 1692 T^{3} + 16038 T^{4} + 158646 T^{5} + 1508509 T^{6} + 11263866 T^{7} + 80847558 T^{8} + 605585412 T^{9} + 3201871806 T^{10} + 21650752212 T^{11} + 128100283921 T^{12} \))
$73$ (\( ( 1 - 6 T + 168 T^{2} - 767 T^{3} + 12264 T^{4} - 31974 T^{5} + 389017 T^{6} )^{2} \))(\( ( 1 + 180 T^{2} - 89 T^{3} + 13140 T^{4} + 389017 T^{6} )^{2} \))
$79$ (\( 1 - 33 T + 516 T^{2} - 4424 T^{3} + 9234 T^{4} + 298269 T^{5} - 4310931 T^{6} + 23563251 T^{7} + 57629394 T^{8} - 2181204536 T^{9} + 20098241796 T^{10} - 101542861167 T^{11} + 243087455521 T^{12} \))(\( 1 + 3 T + 114 T^{2} + 656 T^{3} + 8793 T^{4} + 56367 T^{5} + 695565 T^{6} + 4452993 T^{7} + 54877113 T^{8} + 323433584 T^{9} + 4440309234 T^{10} + 9231169197 T^{11} + 243087455521 T^{12} \))
$83$ (\( 1 - 21 T + 144 T^{2} + 936 T^{3} - 15894 T^{4} - 5853 T^{5} + 1245277 T^{6} - 485799 T^{7} - 109493766 T^{8} + 535192632 T^{9} + 6833998224 T^{10} - 82719853503 T^{11} + 326940373369 T^{12} \))(\( 1 - 3 T - 117 T^{2} + 639 T^{3} - 1080 T^{4} - 30306 T^{5} + 685621 T^{6} - 2515398 T^{7} - 7440120 T^{8} + 365371893 T^{9} - 5552623557 T^{10} - 11817121929 T^{11} + 326940373369 T^{12} \))
$89$ (\( 1 - 12 T + 159 T^{2} - 2483 T^{3} + 27897 T^{4} - 256761 T^{5} + 2868014 T^{6} - 22851729 T^{7} + 220972137 T^{8} - 1750438027 T^{9} + 9976016319 T^{10} - 67008713388 T^{11} + 496981290961 T^{12} \))(\( 1 + 24 T + 207 T^{2} + 225 T^{3} - 6615 T^{4} + 1455 T^{5} + 482158 T^{6} + 129495 T^{7} - 52397415 T^{8} + 158618025 T^{9} + 12987643887 T^{10} + 134017426776 T^{11} + 496981290961 T^{12} \))
$97$ (\( 1 - 18 T - 36 T^{2} + 646 T^{3} + 30546 T^{4} - 194922 T^{5} - 1099401 T^{6} - 18907434 T^{7} + 287407314 T^{8} + 589586758 T^{9} - 3187054116 T^{10} - 154572124626 T^{11} + 832972004929 T^{12} \))(\( 1 + 27 T + 234 T^{2} + 2171 T^{3} + 45891 T^{4} + 431748 T^{5} + 2525169 T^{6} + 41879556 T^{7} + 431788419 T^{8} + 1981413083 T^{9} + 20715851754 T^{10} + 231858186939 T^{11} + 832972004929 T^{12} \))
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