Properties

Label 450.2.q
Level 450
Weight 2
Character orbit q
Rep. character \(\chi_{450}(31,\cdot)\)
Character field \(\Q(\zeta_{15})\)
Dimension 240
Newform subspaces 3
Sturm bound 180
Trace bound 1

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Defining parameters

Level: \( N \) = \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 450.q (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 225 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 3 \)
Sturm bound: \(180\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(450, [\chi])\).

Total New Old
Modular forms 752 240 512
Cusp forms 688 240 448
Eisenstein series 64 0 64

Trace form

\( 240q + 4q^{3} + 30q^{4} + 8q^{5} + 4q^{9} + O(q^{10}) \) \( 240q + 4q^{3} + 30q^{4} + 8q^{5} + 4q^{9} + 4q^{11} - 6q^{12} + 8q^{14} + 36q^{15} + 30q^{16} + 24q^{17} + 8q^{18} - 2q^{20} - 24q^{21} + 24q^{25} + 96q^{26} - 14q^{27} + 12q^{29} - 2q^{30} - 12q^{31} - 26q^{33} + 8q^{35} - 8q^{36} + 24q^{37} - 12q^{38} - 52q^{39} + 16q^{41} - 20q^{42} - 8q^{44} + 16q^{45} + 26q^{47} + 12q^{48} - 120q^{49} - 12q^{50} + 32q^{51} - 72q^{53} - 24q^{54} + 8q^{56} - 60q^{57} + 12q^{58} + 18q^{59} - 14q^{60} - 36q^{62} - 86q^{63} - 60q^{64} - 104q^{65} + 16q^{66} + 18q^{67} + 48q^{68} - 68q^{69} - 24q^{70} + 4q^{71} - 16q^{72} - 80q^{74} - 122q^{75} + 32q^{77} - 64q^{78} + 12q^{79} + 4q^{80} + 76q^{81} - 48q^{82} - 104q^{83} - 18q^{84} + 12q^{85} + 20q^{86} - 22q^{87} - 188q^{89} - 74q^{90} + 8q^{95} + 12q^{97} + 48q^{98} + 12q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(450, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
450.2.q.a \(8\) \(3.593\) \(\Q(\zeta_{15})\) None \(1\) \(0\) \(-5\) \(-8\) \(q+(-\zeta_{15}-\zeta_{15}^{6})q^{2}+(\zeta_{15}^{2}-\zeta_{15}^{7})q^{3}+\cdots\)
450.2.q.b \(112\) \(3.593\) None \(14\) \(2\) \(9\) \(-8\)
450.2.q.c \(120\) \(3.593\) None \(-15\) \(2\) \(4\) \(16\)

Decomposition of \(S_{2}^{\mathrm{old}}(450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(450, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(225, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \))
$3$ (\( 1 - 3 T^{2} + 9 T^{4} - 27 T^{6} + 81 T^{8} \))
$5$ (\( 1 + 5 T + 10 T^{2} + 25 T^{3} + 75 T^{4} + 125 T^{5} + 250 T^{6} + 625 T^{7} + 625 T^{8} \))
$7$ (\( ( 1 + 4 T + 3 T^{2} - 4 T^{3} + 8 T^{4} - 28 T^{5} + 147 T^{6} + 1372 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 - 5 T + 26 T^{2} - 105 T^{3} + 460 T^{4} - 1290 T^{5} + 5624 T^{6} - 14960 T^{7} + 52369 T^{8} - 164560 T^{9} + 680504 T^{10} - 1716990 T^{11} + 6734860 T^{12} - 16910355 T^{13} + 46060586 T^{14} - 97435855 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 - 6 T + 33 T^{2} - 178 T^{3} + 888 T^{4} - 3508 T^{5} + 14851 T^{6} - 54384 T^{7} + 204103 T^{8} - 706992 T^{9} + 2509819 T^{10} - 7707076 T^{11} + 25362168 T^{12} - 66090154 T^{13} + 159284697 T^{14} - 376491102 T^{15} + 815730721 T^{16} \))
$17$ (\( ( 1 - T + 14 T^{2} - 37 T^{3} + 359 T^{4} - 629 T^{5} + 4046 T^{6} - 4913 T^{7} + 83521 T^{8} )^{2} \))
$19$ (\( ( 1 - 7 T + 73 T^{3} - 151 T^{4} + 1387 T^{5} - 48013 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 5 T + 33 T^{2} + 90 T^{3} - 730 T^{4} + 6315 T^{5} - 5362 T^{6} - 65600 T^{7} + 727839 T^{8} - 1508800 T^{9} - 2836498 T^{10} + 76834605 T^{11} - 204283930 T^{12} + 579270870 T^{13} + 4885184337 T^{14} - 17024127235 T^{15} + 78310985281 T^{16} \))
$29$ (\( 1 + 6 T + 49 T^{2} + 306 T^{3} + 1896 T^{4} + 3828 T^{5} + 40931 T^{6} + 46368 T^{7} + 168167 T^{8} + 1344672 T^{9} + 34422971 T^{10} + 93361092 T^{11} + 1341004776 T^{12} + 6276411594 T^{13} + 29146342729 T^{14} + 103499257854 T^{15} + 500246412961 T^{16} \))
$31$ (\( 1 - 15 T + 121 T^{2} - 210 T^{3} - 3690 T^{4} + 40305 T^{5} - 113986 T^{6} - 550800 T^{7} + 7157039 T^{8} - 17074800 T^{9} - 109540546 T^{10} + 1200726255 T^{11} - 3407792490 T^{12} - 6012121710 T^{13} + 107387945401 T^{14} - 412689211665 T^{15} + 852891037441 T^{16} \))
$37$ (\( ( 1 + 12 T + 27 T^{2} - 430 T^{3} - 4539 T^{4} - 15910 T^{5} + 36963 T^{6} + 607836 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( 1 - 4 T + 51 T^{2} - 108 T^{3} + 572 T^{4} + 9936 T^{5} - 41743 T^{6} + 458012 T^{7} - 1989729 T^{8} + 18778492 T^{9} - 70169983 T^{10} + 684799056 T^{11} + 1616335292 T^{12} - 12512469708 T^{13} + 242255316291 T^{14} - 779017095524 T^{15} + 7984925229121 T^{16} \))
$43$ (\( ( 1 - 2 T - 63 T^{2} + 38 T^{3} + 2468 T^{4} + 1634 T^{5} - 116487 T^{6} - 159014 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 + 14 T + 147 T^{2} + 330 T^{3} - 2530 T^{4} - 51204 T^{5} + 11531 T^{6} + 2583170 T^{7} + 34974699 T^{8} + 121408990 T^{9} + 25471979 T^{10} - 5316152892 T^{11} - 12345592930 T^{12} + 75683852310 T^{13} + 1584544653363 T^{14} + 7092723686482 T^{15} + 23811286661761 T^{16} \))
$53$ (\( ( 1 - 7 T + 71 T^{2} - 601 T^{3} + 6944 T^{4} - 31853 T^{5} + 199439 T^{6} - 1042139 T^{7} + 7890481 T^{8} )^{2} \))
$59$ (\( 1 + 14 T + 179 T^{2} + 1134 T^{3} + 9436 T^{4} + 46452 T^{5} + 737981 T^{6} + 5890472 T^{7} + 63809287 T^{8} + 347537848 T^{9} + 2568911861 T^{10} + 9540265308 T^{11} + 114339418396 T^{12} + 810724155066 T^{13} + 7550315521739 T^{14} + 34841120787466 T^{15} + 146830437604321 T^{16} \))
$61$ (\( ( 1 - 2 T - 57 T^{2} + 236 T^{3} + 3005 T^{4} + 14396 T^{5} - 212097 T^{6} - 453962 T^{7} + 13845841 T^{8} )( 1 + 31 T + 480 T^{2} + 5069 T^{3} + 42899 T^{4} + 309209 T^{5} + 1786080 T^{6} + 7036411 T^{7} + 13845841 T^{8} ) \))
$67$ (\( 1 - 7 T + 82 T^{2} + 541 T^{3} - 5812 T^{4} + 76154 T^{5} - 153016 T^{6} - 2655272 T^{7} + 36651313 T^{8} - 177903224 T^{9} - 686888824 T^{10} + 22904305502 T^{11} - 117118315252 T^{12} + 730417682887 T^{13} + 7417587337858 T^{14} - 42424981237261 T^{15} + 406067677556641 T^{16} \))
$71$ (\( ( 1 + 6 T + 65 T^{2} + 294 T^{3} + 889 T^{4} + 20874 T^{5} + 327665 T^{6} + 2147466 T^{7} + 25411681 T^{8} )^{2} \))
$73$ (\( ( 1 - 6 T - 37 T^{2} + 660 T^{3} - 1259 T^{4} + 48180 T^{5} - 197173 T^{6} - 2334102 T^{7} + 28398241 T^{8} )^{2} \))
$79$ (\( 1 + 18 T + 79 T^{2} - 1386 T^{3} - 24138 T^{4} - 213012 T^{5} - 504217 T^{6} + 16949034 T^{7} + 266385251 T^{8} + 1338973686 T^{9} - 3146818297 T^{10} - 105023223468 T^{11} - 940177055178 T^{12} - 4264800169014 T^{13} + 19203908986159 T^{14} + 345670361750862 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 - 4 T - 67 T^{2} + 768 T^{3} - 6122 T^{4} - 4812 T^{5} + 366161 T^{6} - 2942446 T^{7} + 15150763 T^{8} - 244223018 T^{9} + 2522483129 T^{10} - 2751439044 T^{11} - 290539841162 T^{12} + 3025183213824 T^{13} - 21905005015723 T^{14} - 108544203958508 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( ( 1 + 15 T + 46 T^{2} + 675 T^{3} + 13951 T^{4} + 60075 T^{5} + 364366 T^{6} + 10574535 T^{7} + 62742241 T^{8} )^{2} \))
$97$ (\( 1 + 8 T + 97 T^{2} - 1424 T^{3} - 17872 T^{4} - 192016 T^{5} - 262081 T^{6} + 16836748 T^{7} + 190506703 T^{8} + 1633164556 T^{9} - 2465920129 T^{10} - 175247818768 T^{11} - 1582195310032 T^{12} - 12228372525968 T^{13} + 80798284478113 T^{14} + 646386275824904 T^{15} + 7837433594376961 T^{16} \))
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