Properties

Label 38.2
Level 38
Weight 2
Dimension 14
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 180
Trace bound 2

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Defining parameters

Level: \( N \) = \( 38 = 2 \cdot 19 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(180\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(38))\).

Total New Old
Modular forms 63 14 49
Cusp forms 28 14 14
Eisenstein series 35 0 35

Trace form

\( 14q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} + O(q^{10}) \) \( 14q - q^{2} - 4q^{3} - q^{4} - 6q^{5} - 4q^{6} - 8q^{7} - q^{8} - 13q^{9} - 6q^{10} - 12q^{11} + 2q^{12} + 10q^{13} + 10q^{14} + 12q^{15} - q^{16} + 14q^{18} + 23q^{19} + 12q^{20} + 10q^{21} + 15q^{22} - 6q^{23} - 4q^{24} + 5q^{25} + 4q^{26} - 7q^{27} - 2q^{28} - 12q^{29} - 24q^{30} - 14q^{31} - q^{32} + 6q^{33} - 18q^{34} - 12q^{35} - 13q^{36} - 20q^{37} - 19q^{38} - 2q^{39} - 6q^{40} - 24q^{41} - 32q^{42} - 2q^{43} - 3q^{44} + 30q^{45} + 12q^{46} + 24q^{47} + 5q^{48} + 3q^{49} + 41q^{50} + 27q^{51} - 8q^{52} + 18q^{53} + 14q^{54} + 28q^{56} + 14q^{57} + 6q^{58} + 30q^{59} + 12q^{60} + 16q^{61} + 22q^{62} - 8q^{63} + 5q^{64} + 24q^{65} + 24q^{66} + 10q^{67} - 9q^{68} - 42q^{69} - 12q^{70} - 4q^{72} - 41q^{73} - 38q^{74} - 82q^{75} - 19q^{76} - 60q^{77} - 20q^{78} - 2q^{79} - 6q^{80} - 22q^{81} - 6q^{82} - 12q^{83} + 22q^{84} - 36q^{85} - 8q^{86} + 60q^{87} - 12q^{88} - 18q^{89} + 12q^{90} - 16q^{91} + 12q^{92} + 22q^{93} + 24q^{94} - 24q^{95} - 4q^{96} - 8q^{97} + 15q^{98} + 51q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(38))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
38.2.a \(\chi_{38}(1, \cdot)\) 38.2.a.a 1 1
38.2.a.b 1
38.2.c \(\chi_{38}(7, \cdot)\) 38.2.c.a 2 2
38.2.c.b 4
38.2.e \(\chi_{38}(5, \cdot)\) 38.2.e.a 6 6

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(38))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(38)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T \))(\( 1 - T \))(\( 1 - T + T^{2} \))(\( ( 1 + T + T^{2} )^{2} \))(\( 1 + T^{3} + T^{6} \))
$3$ (\( 1 - T + 3 T^{2} \))(\( 1 + T + 3 T^{2} \))(\( 1 + T - 2 T^{2} + 3 T^{3} + 9 T^{4} \))(\( 1 + T^{2} - 8 T^{4} + 9 T^{6} + 81 T^{8} \))(\( 1 + 3 T + 6 T^{2} + 8 T^{3} + 21 T^{4} + 51 T^{5} + 109 T^{6} + 153 T^{7} + 189 T^{8} + 216 T^{9} + 486 T^{10} + 729 T^{11} + 729 T^{12} \))
$5$ (\( 1 + 5 T^{2} \))(\( 1 + 4 T + 5 T^{2} \))(\( 1 - 5 T^{2} + 25 T^{4} \))(\( 1 + 2 T - 12 T^{3} - 29 T^{4} - 60 T^{5} + 250 T^{7} + 625 T^{8} \))(\( 1 - 22 T^{3} + 359 T^{6} - 2750 T^{9} + 15625 T^{12} \))
$7$ (\( 1 + T + 7 T^{2} \))(\( 1 - 3 T + 7 T^{2} \))(\( ( 1 + 4 T + 7 T^{2} )^{2} \))(\( ( 1 - 2 T + 8 T^{2} - 14 T^{3} + 49 T^{4} )^{2} \))(\( 1 + 6 T + 15 T^{2} + 6 T^{3} - 66 T^{4} - 210 T^{5} - 565 T^{6} - 1470 T^{7} - 3234 T^{8} + 2058 T^{9} + 36015 T^{10} + 100842 T^{11} + 117649 T^{12} \))
$11$ (\( 1 + 6 T + 11 T^{2} \))(\( 1 - 2 T + 11 T^{2} \))(\( ( 1 - 3 T + 11 T^{2} )^{2} \))(\( ( 1 + 4 T + 19 T^{2} + 44 T^{3} + 121 T^{4} )^{2} \))(\( 1 + 6 T - 10 T^{3} + 222 T^{4} + 42 T^{5} - 3181 T^{6} + 462 T^{7} + 26862 T^{8} - 13310 T^{9} + 966306 T^{11} + 1771561 T^{12} \))
$13$ (\( 1 - 5 T + 13 T^{2} \))(\( 1 + T + 13 T^{2} \))(\( ( 1 - 5 T + 13 T^{2} )( 1 + 7 T + 13 T^{2} ) \))(\( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \))(\( 1 - 12 T + 48 T^{2} + 14 T^{3} - 684 T^{4} + 1854 T^{5} - 2757 T^{6} + 24102 T^{7} - 115596 T^{8} + 30758 T^{9} + 1370928 T^{10} - 4455516 T^{11} + 4826809 T^{12} \))
$17$ (\( 1 - 3 T + 17 T^{2} \))(\( 1 - 3 T + 17 T^{2} \))(\( 1 - 6 T + 19 T^{2} - 102 T^{3} + 289 T^{4} \))(\( ( 1 - 17 T^{2} + 289 T^{4} )^{2} \))(\( 1 + 12 T + 108 T^{2} + 702 T^{3} + 3744 T^{4} + 17904 T^{5} + 75187 T^{6} + 304368 T^{7} + 1082016 T^{8} + 3448926 T^{9} + 9020268 T^{10} + 17038284 T^{11} + 24137569 T^{12} \))
$19$ (\( 1 - T \))(\( 1 + T \))(\( 1 + 7 T + 19 T^{2} \))(\( 1 - 12 T + 67 T^{2} - 228 T^{3} + 361 T^{4} \))(\( 1 - 18 T + 162 T^{2} - 883 T^{3} + 3078 T^{4} - 6498 T^{5} + 6859 T^{6} \))
$23$ (\( 1 - 3 T + 23 T^{2} \))(\( 1 + T + 23 T^{2} \))(\( 1 - 6 T + 13 T^{2} - 138 T^{3} + 529 T^{4} \))(\( 1 + 2 T - 36 T^{2} - 12 T^{3} + 979 T^{4} - 276 T^{5} - 19044 T^{6} + 24334 T^{7} + 279841 T^{8} \))(\( 1 + 12 T + 60 T^{2} + 152 T^{3} - 84 T^{4} - 4734 T^{5} - 34459 T^{6} - 108882 T^{7} - 44436 T^{8} + 1849384 T^{9} + 16790460 T^{10} + 77236116 T^{11} + 148035889 T^{12} \))
$29$ (\( 1 - 9 T + 29 T^{2} \))(\( 1 + 5 T + 29 T^{2} \))(\( 1 - 29 T^{2} + 841 T^{4} \))(\( 1 - 2 T - 48 T^{2} + 12 T^{3} + 1747 T^{4} + 348 T^{5} - 40368 T^{6} - 48778 T^{7} + 707281 T^{8} \))(\( 1 + 18 T + 144 T^{2} + 580 T^{3} + 1440 T^{4} + 7542 T^{5} + 56483 T^{6} + 218718 T^{7} + 1211040 T^{8} + 14145620 T^{9} + 101848464 T^{10} + 369200682 T^{11} + 594823321 T^{12} \))
$31$ (\( 1 + 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( ( 1 - 2 T + 31 T^{2} )^{2} \))(\( ( 1 + 6 T + 64 T^{2} + 186 T^{3} + 961 T^{4} )^{2} \))(\( 1 - 6 T - 33 T^{2} + 346 T^{3} + 342 T^{4} - 6318 T^{5} + 21795 T^{6} - 195858 T^{7} + 328662 T^{8} + 10307686 T^{9} - 30476193 T^{10} - 171774906 T^{11} + 887503681 T^{12} \))
$37$ (\( 1 - 2 T + 37 T^{2} \))(\( 1 + 2 T + 37 T^{2} \))(\( ( 1 + 10 T + 37 T^{2} )^{2} \))(\( ( 1 - 6 T + 76 T^{2} - 222 T^{3} + 1369 T^{4} )^{2} \))(\( ( 1 + 6 T + 87 T^{2} + 308 T^{3} + 3219 T^{4} + 8214 T^{5} + 50653 T^{6} )^{2} \))
$41$ (\( 1 + 41 T^{2} \))(\( 1 + 8 T + 41 T^{2} \))(\( 1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4} \))(\( 1 + 10 T + 21 T^{2} - 30 T^{3} + 460 T^{4} - 1230 T^{5} + 35301 T^{6} + 689210 T^{7} + 2825761 T^{8} \))(\( 1 - 3 T + 6 T^{2} - 8 T^{3} - 111 T^{4} + 9711 T^{5} - 83311 T^{6} + 398151 T^{7} - 186591 T^{8} - 551368 T^{9} + 16954566 T^{10} - 347568603 T^{11} + 4750104241 T^{12} \))
$43$ (\( 1 - 8 T + 43 T^{2} \))(\( 1 - 4 T + 43 T^{2} \))(\( 1 - 4 T - 27 T^{2} - 172 T^{3} + 1849 T^{4} \))(\( 1 + 12 T + 50 T^{2} + 96 T^{3} + 795 T^{4} + 4128 T^{5} + 92450 T^{6} + 954084 T^{7} + 3418801 T^{8} \))(\( 1 + 6 T + 42 T^{2} - 116 T^{3} + 1044 T^{4} - 6768 T^{5} + 34173 T^{6} - 291024 T^{7} + 1930356 T^{8} - 9222812 T^{9} + 143589642 T^{10} + 882050658 T^{11} + 6321363049 T^{12} \))
$47$ (\( 1 + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 - 47 T^{2} + 2209 T^{4} \))(\( 1 + 14 T + 60 T^{2} + 588 T^{3} + 7075 T^{4} + 27636 T^{5} + 132540 T^{6} + 1453522 T^{7} + 4879681 T^{8} \))(\( 1 - 30 T + 372 T^{2} - 2086 T^{3} - 2544 T^{4} + 153360 T^{5} - 1460047 T^{6} + 7207920 T^{7} - 5619696 T^{8} - 216574778 T^{9} + 1815241332 T^{10} - 6880350210 T^{11} + 10779215329 T^{12} \))
$53$ (\( 1 + 3 T + 53 T^{2} \))(\( 1 + T + 53 T^{2} \))(\( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \))(\( 1 - 4 T + 18 T^{2} + 432 T^{3} - 3653 T^{4} + 22896 T^{5} + 50562 T^{6} - 595508 T^{7} + 7890481 T^{8} \))(\( 1 - 24 T + 276 T^{2} - 2152 T^{3} + 18132 T^{4} - 179262 T^{5} + 1523855 T^{6} - 9500886 T^{7} + 50932788 T^{8} - 320383304 T^{9} + 2177772756 T^{10} - 10036691832 T^{11} + 22164361129 T^{12} \))
$59$ (\( 1 - 9 T + 59 T^{2} \))(\( 1 - 15 T + 59 T^{2} \))(\( 1 - 9 T + 22 T^{2} - 531 T^{3} + 3481 T^{4} \))(\( 1 - 55 T^{2} - 456 T^{4} - 191455 T^{6} + 12117361 T^{8} \))(\( 1 + 3 T + 54 T^{2} + 378 T^{3} + 5823 T^{4} + 40179 T^{5} + 259255 T^{6} + 2370561 T^{7} + 20269863 T^{8} + 77633262 T^{9} + 654337494 T^{10} + 2144772897 T^{11} + 42180533641 T^{12} \))
$61$ (\( 1 + 10 T + 61 T^{2} \))(\( 1 - 2 T + 61 T^{2} \))(\( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \))(\( 1 - 14 T + 88 T^{2} + 196 T^{3} - 4013 T^{4} + 11956 T^{5} + 327448 T^{6} - 3177734 T^{7} + 13845841 T^{8} \))(\( 1 - 6 T - 12 T^{2} - 586 T^{3} - 1008 T^{4} + 15732 T^{5} + 419787 T^{6} + 959652 T^{7} - 3750768 T^{8} - 133010866 T^{9} - 166150092 T^{10} - 5067577806 T^{11} + 51520374361 T^{12} \))
$67$ (\( 1 - 5 T + 67 T^{2} \))(\( 1 - 3 T + 67 T^{2} \))(\( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} \))(\( 1 - 4 T - 115 T^{2} + 12 T^{3} + 11600 T^{4} + 804 T^{5} - 516235 T^{6} - 1203052 T^{7} + 20151121 T^{8} \))(\( 1 + 9 T + 162 T^{2} + 1854 T^{3} + 22437 T^{4} + 180369 T^{5} + 1838609 T^{6} + 12084723 T^{7} + 100719693 T^{8} + 557614602 T^{9} + 3264481602 T^{10} + 12151125963 T^{11} + 90458382169 T^{12} \))
$71$ (\( 1 + 6 T + 71 T^{2} \))(\( 1 - 2 T + 71 T^{2} \))(\( 1 - 6 T - 35 T^{2} - 426 T^{3} + 5041 T^{4} \))(\( 1 - 16 T + 78 T^{2} - 576 T^{3} + 8467 T^{4} - 40896 T^{5} + 393198 T^{6} - 5726576 T^{7} + 25411681 T^{8} \))(\( 1 + 18 T + 216 T^{2} + 2516 T^{3} + 26244 T^{4} + 221256 T^{5} + 1869317 T^{6} + 15709176 T^{7} + 132296004 T^{8} + 900504076 T^{9} + 5488923096 T^{10} + 32476128318 T^{11} + 128100283921 T^{12} \))
$73$ (\( 1 + 7 T + 73 T^{2} \))(\( 1 - 9 T + 73 T^{2} \))(\( 1 - T - 72 T^{2} - 73 T^{3} + 5329 T^{4} \))(\( 1 + 14 T + 29 T^{2} + 294 T^{3} + 8252 T^{4} + 21462 T^{5} + 154541 T^{6} + 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 30 T + 279 T^{2} - 231 T^{3} - 13599 T^{4} + 110865 T^{5} + 2572046 T^{6} + 8093145 T^{7} - 72469071 T^{8} - 89862927 T^{9} + 7923109239 T^{10} + 62192147790 T^{11} + 151334226289 T^{12} \))
$79$ (\( 1 + 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( ( 1 - 17 T + 79 T^{2} )( 1 + 13 T + 79 T^{2} ) \))(\( ( 1 - 17 T + 79 T^{2} )^{2}( 1 + 13 T + 79 T^{2} )^{2} \))(\( 1 - 6 T - 48 T^{2} + 1430 T^{3} - 8820 T^{4} - 56934 T^{5} + 1359837 T^{6} - 4497786 T^{7} - 55045620 T^{8} + 705045770 T^{9} - 1869603888 T^{10} - 18462338394 T^{11} + 243087455521 T^{12} \))
$83$ (\( 1 + 6 T + 83 T^{2} \))(\( 1 + 6 T + 83 T^{2} \))(\( ( 1 - 3 T + 83 T^{2} )^{2} \))(\( ( 1 + 103 T^{2} + 6889 T^{4} )^{2} \))(\( 1 + 6 T - 186 T^{2} - 558 T^{3} + 25188 T^{4} + 30012 T^{5} - 2301977 T^{6} + 2490996 T^{7} + 173520132 T^{8} - 319057146 T^{9} - 8827247706 T^{10} + 23634243858 T^{11} + 326940373369 T^{12} \))
$89$ (\( 1 + 12 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 6 T - 53 T^{2} + 534 T^{3} + 7921 T^{4} \))(\( ( 1 - 89 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 36 T^{2} - 486 T^{3} - 1548 T^{4} - 34956 T^{5} + 586603 T^{6} - 3111084 T^{7} - 12261708 T^{8} - 342614934 T^{9} + 2258720676 T^{10} + 496981290961 T^{12} \))
$97$ (\( 1 + 10 T + 97 T^{2} \))(\( 1 + 2 T + 97 T^{2} \))(\( 1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4} \))(\( 1 - 18 T + 77 T^{2} - 954 T^{3} + 20172 T^{4} - 92538 T^{5} + 724493 T^{6} - 16428114 T^{7} + 88529281 T^{8} \))(\( 1 - 3 T + 6 T^{2} - 8 T^{3} + 585 T^{4} - 29097 T^{5} - 744183 T^{6} - 2822409 T^{7} + 5504265 T^{8} - 7301384 T^{9} + 531175686 T^{10} - 25762020771 T^{11} + 832972004929 T^{12} \))
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