Properties

Label 150.2.g
Level 150
Weight 2
Character orbit g
Rep. character \(\chi_{150}(31,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 16
Newform subspaces 3
Sturm bound 60
Trace bound 2

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Defining parameters

Level: \( N \) = \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 150.g (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 3 \)
Sturm bound: \(60\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(150, [\chi])\).

Total New Old
Modular forms 136 16 120
Cusp forms 104 16 88
Eisenstein series 32 0 32

Trace form

\( 16q + 2q^{2} - 4q^{4} + 6q^{5} + 2q^{6} + 12q^{7} + 2q^{8} - 4q^{9} + O(q^{10}) \) \( 16q + 2q^{2} - 4q^{4} + 6q^{5} + 2q^{6} + 12q^{7} + 2q^{8} - 4q^{9} + 4q^{10} - 12q^{11} + 12q^{13} + 6q^{15} - 4q^{16} - 8q^{17} - 8q^{18} - 8q^{19} - 4q^{20} + 4q^{21} - 8q^{22} - 36q^{23} - 8q^{24} + 4q^{25} + 12q^{26} + 2q^{28} - 12q^{29} - 16q^{30} - 6q^{31} - 8q^{32} - 8q^{33} - 10q^{34} + 16q^{35} - 4q^{36} + 26q^{37} + 16q^{38} + 4q^{40} - 24q^{41} - 6q^{42} - 32q^{43} + 8q^{44} + 6q^{45} - 4q^{46} + 8q^{47} - 4q^{49} + 26q^{50} - 16q^{51} + 12q^{52} - 30q^{53} + 2q^{54} - 40q^{55} + 24q^{57} + 24q^{58} - 4q^{60} + 8q^{61} - 20q^{62} - 8q^{63} - 4q^{64} + 62q^{65} + 8q^{66} + 32q^{67} + 12q^{68} + 16q^{69} - 6q^{70} + 8q^{71} + 2q^{72} - 8q^{73} + 28q^{74} + 24q^{75} - 8q^{76} + 16q^{77} + 16q^{78} - 4q^{79} + 6q^{80} - 4q^{81} - 44q^{82} - 8q^{83} + 4q^{84} - 14q^{85} + 24q^{86} + 16q^{87} + 2q^{88} + 6q^{89} + 4q^{90} + 12q^{91} + 24q^{92} - 16q^{93} + 32q^{94} + 16q^{95} + 2q^{96} + 26q^{97} - 14q^{98} + 8q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(150, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
150.2.g.a \(4\) \(1.198\) \(\Q(\zeta_{10})\) None \(-1\) \(1\) \(5\) \(6\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
150.2.g.b \(4\) \(1.198\) \(\Q(\zeta_{10})\) None \(1\) \(1\) \(5\) \(8\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\zeta_{10}^{3}q^{3}+\cdots\)
150.2.g.c \(8\) \(1.198\) 8.0.1064390625.3 None \(2\) \(-2\) \(-4\) \(-2\) \(q-\beta _{2}q^{2}+\beta _{5}q^{3}+\beta _{5}q^{4}+(-1+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(150, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(150, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))
$3$ (\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( 1 - T + T^{2} - T^{3} + T^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))
$5$ (\( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} \))(\( 1 - 5 T + 15 T^{2} - 25 T^{3} + 25 T^{4} \))(\( 1 + 4 T + T^{2} - 16 T^{3} - 39 T^{4} - 80 T^{5} + 25 T^{6} + 500 T^{7} + 625 T^{8} \))
$7$ (\( ( 1 - 3 T + 15 T^{2} - 21 T^{3} + 49 T^{4} )^{2} \))(\( ( 1 - 2 T + 7 T^{2} )^{4} \))(\( ( 1 + T + 9 T^{2} + 17 T^{3} + 104 T^{4} + 119 T^{5} + 441 T^{6} + 343 T^{7} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + 5 T - T^{2} - 55 T^{3} - 184 T^{4} - 605 T^{5} - 121 T^{6} + 6655 T^{7} + 14641 T^{8} \))(\( 1 + 2 T + 13 T^{2} + 34 T^{3} + 225 T^{4} + 374 T^{5} + 1573 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 + 5 T + 38 T^{2} + 210 T^{3} + 1048 T^{4} + 4455 T^{5} + 19301 T^{6} + 69050 T^{7} + 243280 T^{8} + 759550 T^{9} + 2335421 T^{10} + 5929605 T^{11} + 15343768 T^{12} + 33820710 T^{13} + 67319318 T^{14} + 97435855 T^{15} + 214358881 T^{16} \))
$13$ (\( 1 - 12 T + 51 T^{2} - 76 T^{3} + 9 T^{4} - 988 T^{5} + 8619 T^{6} - 26364 T^{7} + 28561 T^{8} \))(\( 1 + 6 T + 23 T^{2} + 120 T^{3} + 601 T^{4} + 1560 T^{5} + 3887 T^{6} + 13182 T^{7} + 28561 T^{8} \))(\( 1 - 6 T + 6 T^{2} + 26 T^{3} + 3 T^{4} - 1366 T^{5} + 5284 T^{6} - 2112 T^{7} - 15227 T^{8} - 27456 T^{9} + 892996 T^{10} - 3001102 T^{11} + 85683 T^{12} + 9653618 T^{13} + 28960854 T^{14} - 376491102 T^{15} + 815730721 T^{16} \))
$17$ (\( 1 - 6 T - T^{2} + 18 T^{3} + 169 T^{4} + 306 T^{5} - 289 T^{6} - 29478 T^{7} + 83521 T^{8} \))(\( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 3060 T^{5} + 10693 T^{6} + 58956 T^{7} + 83521 T^{8} \))(\( 1 + 2 T + 44 T^{2} + 12 T^{3} + 1273 T^{4} - 258 T^{5} + 28916 T^{6} - 13984 T^{7} + 544693 T^{8} - 237728 T^{9} + 8356724 T^{10} - 1267554 T^{11} + 106322233 T^{12} + 17038284 T^{13} + 1062053036 T^{14} + 820677346 T^{15} + 6975757441 T^{16} \))
$19$ (\( 1 + 16 T + 117 T^{2} + 578 T^{3} + 2525 T^{4} + 10982 T^{5} + 42237 T^{6} + 109744 T^{7} + 130321 T^{8} \))(\( 1 + 21 T^{2} - 10 T^{3} + 381 T^{4} - 190 T^{5} + 7581 T^{6} + 130321 T^{8} \))(\( ( 1 - 4 T - 3 T^{2} - 62 T^{3} + 605 T^{4} - 1178 T^{5} - 1083 T^{6} - 27436 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 4600 T^{5} + 19573 T^{6} + 121670 T^{7} + 279841 T^{8} \))(\( 1 + 6 T + 13 T^{2} - 60 T^{3} - 659 T^{4} - 1380 T^{5} + 6877 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( ( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 4600 T^{5} + 19573 T^{6} + 121670 T^{7} + 279841 T^{8} )^{2} \))
$29$ (\( 1 - 6 T + 47 T^{2} - 288 T^{3} + 2365 T^{4} - 8352 T^{5} + 39527 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 - 19 T^{2} + 120 T^{3} + 721 T^{4} + 3480 T^{5} - 15979 T^{6} + 707281 T^{8} \))(\( 1 + 18 T + 160 T^{2} + 1050 T^{3} + 5925 T^{4} + 26514 T^{5} + 92732 T^{6} + 300180 T^{7} + 1280525 T^{8} + 8705220 T^{9} + 77987612 T^{10} + 646649946 T^{11} + 4190639925 T^{12} + 21536706450 T^{13} + 95171731360 T^{14} + 310497773562 T^{15} + 500246412961 T^{16} \))
$31$ (\( ( 1 - T - 39 T^{2} - 31 T^{3} + 961 T^{4} )( 1 + 4 T + 46 T^{2} + 124 T^{3} + 961 T^{4} ) \))(\( 1 + 12 T + 113 T^{2} + 834 T^{3} + 5605 T^{4} + 25854 T^{5} + 108593 T^{6} + 357492 T^{7} + 923521 T^{8} \))(\( 1 - 9 T - 20 T^{2} + 300 T^{3} + 690 T^{4} - 8217 T^{5} - 2797 T^{6} + 139770 T^{7} - 752500 T^{8} + 4332870 T^{9} - 2687917 T^{10} - 244792647 T^{11} + 637229490 T^{12} + 8588745300 T^{13} - 17750073620 T^{14} - 247613526999 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 + 8 T + 27 T^{2} - 80 T^{3} - 1639 T^{4} - 2960 T^{5} + 36963 T^{6} + 405224 T^{7} + 1874161 T^{8} \))(\( 1 - 13 T + 27 T^{2} + 445 T^{3} - 4264 T^{4} + 16465 T^{5} + 36963 T^{6} - 658489 T^{7} + 1874161 T^{8} \))(\( 1 - 21 T + 258 T^{2} - 2462 T^{3} + 19746 T^{4} - 143741 T^{5} + 980785 T^{6} - 6308436 T^{7} + 38864404 T^{8} - 233412132 T^{9} + 1342694665 T^{10} - 7280912873 T^{11} + 37007183106 T^{12} - 170724822134 T^{13} + 661957413522 T^{14} - 1993569419793 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 + 14 T + 95 T^{2} + 786 T^{3} + 6569 T^{4} + 32226 T^{5} + 159695 T^{6} + 964894 T^{7} + 2825761 T^{8} \))(\( 1 + 12 T + 53 T^{2} + 444 T^{3} + 4405 T^{4} + 18204 T^{5} + 89093 T^{6} + 827052 T^{7} + 2825761 T^{8} \))(\( 1 - 2 T + 36 T^{2} - 168 T^{3} + 2201 T^{4} - 17286 T^{5} + 84044 T^{6} - 959864 T^{7} + 2420157 T^{8} - 39354424 T^{9} + 141277964 T^{10} - 1191368406 T^{11} + 6219499961 T^{12} - 19463841768 T^{13} + 171003752676 T^{14} - 389508547762 T^{15} + 7984925229121 T^{16} \))
$43$ (\( ( 1 + 2 T + 42 T^{2} + 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 - 2 T + 82 T^{2} - 86 T^{3} + 1849 T^{4} )^{2} \))(\( ( 1 + 16 T + 168 T^{2} + 1280 T^{3} + 9326 T^{4} + 55040 T^{5} + 310632 T^{6} + 1272112 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( 1 - 20 T + 113 T^{2} + 270 T^{3} - 5851 T^{4} + 12690 T^{5} + 249617 T^{6} - 2076460 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T + 17 T^{2} - 130 T^{3} + 761 T^{4} - 6110 T^{5} + 37553 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 10 T + 66 T^{2} + 210 T^{3} + 1787 T^{4} - 6510 T^{5} - 94072 T^{6} - 974600 T^{7} - 2764935 T^{8} - 45806200 T^{9} - 207805048 T^{10} - 675887730 T^{11} + 8719989947 T^{12} + 48162451470 T^{13} + 711428211714 T^{14} + 5066231204630 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 16 T + 53 T^{2} - 200 T^{3} - 1759 T^{4} - 10600 T^{5} + 148877 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 21 T + 253 T^{2} + 2535 T^{3} + 21196 T^{4} + 134355 T^{5} + 710677 T^{6} + 3126417 T^{7} + 7890481 T^{8} \))(\( 1 - 7 T + 62 T^{2} - 816 T^{3} + 3586 T^{4} + 23943 T^{5} - 77365 T^{6} + 2388872 T^{7} - 30975296 T^{8} + 126610216 T^{9} - 217318285 T^{10} + 3564562011 T^{11} + 28295264866 T^{12} - 341247522288 T^{13} + 1374190389998 T^{14} - 8222977978859 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 - 5 T - 49 T^{2} + 295 T^{3} + 1736 T^{4} + 17405 T^{5} - 170569 T^{6} - 1026895 T^{7} + 12117361 T^{8} \))(\( 1 - 20 T + 181 T^{2} - 1600 T^{3} + 14601 T^{4} - 94400 T^{5} + 630061 T^{6} - 4107580 T^{7} + 12117361 T^{8} \))(\( 1 + 25 T + 212 T^{2} + 150 T^{3} - 12692 T^{4} - 128475 T^{5} - 252121 T^{6} + 7858000 T^{7} + 98067400 T^{8} + 463622000 T^{9} - 877633201 T^{10} - 26386067025 T^{11} - 153793545812 T^{12} + 107238644850 T^{13} + 8942273131892 T^{14} + 62216287120475 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 21 T^{2} - 410 T^{3} + 2901 T^{4} - 25010 T^{5} - 78141 T^{6} + 13845841 T^{8} \))(\( 1 + 2 T + 3 T^{2} + 424 T^{3} + 4265 T^{4} + 25864 T^{5} + 11163 T^{6} + 453962 T^{7} + 13845841 T^{8} \))(\( 1 - 10 T - 62 T^{2} + 700 T^{3} + 5543 T^{4} - 26410 T^{5} - 544444 T^{6} + 851800 T^{7} + 32679625 T^{8} + 51959800 T^{9} - 2025876124 T^{10} - 5994568210 T^{11} + 76747496663 T^{12} + 591217410700 T^{13} - 3194263210382 T^{14} - 31427428360210 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 - 16 T + 29 T^{2} + 868 T^{3} - 9191 T^{4} + 58156 T^{5} + 130181 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))(\( 1 - 18 T + 77 T^{2} - 30 T^{3} + 961 T^{4} - 2010 T^{5} + 345653 T^{6} - 5413734 T^{7} + 20151121 T^{8} \))(\( 1 + 2 T + 34 T^{2} + 232 T^{3} - 637 T^{4} - 10798 T^{5} + 66956 T^{6} + 2488096 T^{7} - 7257587 T^{8} + 166702432 T^{9} + 300565484 T^{10} - 3247638874 T^{11} - 12836264077 T^{12} + 313229024824 T^{13} + 3075584993746 T^{14} + 12121423210646 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 + 20 T + 89 T^{2} - 1420 T^{3} - 22639 T^{4} - 100820 T^{5} + 448649 T^{6} + 7158220 T^{7} + 25411681 T^{8} \))(\( 1 - 28 T + 313 T^{2} - 2126 T^{3} + 14805 T^{4} - 150946 T^{5} + 1577833 T^{6} - 10021508 T^{7} + 25411681 T^{8} \))(\( 1 - 182 T^{2} - 750 T^{3} + 7983 T^{4} + 134400 T^{5} + 952256 T^{6} - 5830800 T^{7} - 134438695 T^{8} - 413986800 T^{9} + 4800322496 T^{10} + 48103238400 T^{11} + 202861449423 T^{12} - 1353172013250 T^{13} - 23314251673622 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 - 2 T + 51 T^{2} - 376 T^{3} + 6389 T^{4} - 27448 T^{5} + 271779 T^{6} - 778034 T^{7} + 28398241 T^{8} \))(\( 1 - 14 T + 3 T^{2} + 980 T^{3} - 9259 T^{4} + 71540 T^{5} + 15987 T^{6} - 5446238 T^{7} + 28398241 T^{8} \))(\( 1 + 24 T + 226 T^{2} + 1116 T^{3} + 5643 T^{4} + 16584 T^{5} - 622036 T^{6} - 12920832 T^{7} - 137738827 T^{8} - 943220736 T^{9} - 3314829844 T^{10} + 6451457928 T^{11} + 160251273963 T^{12} + 2313547897788 T^{13} + 34201535141314 T^{14} + 265137564458328 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 - 2 T - 55 T^{2} + 578 T^{3} + 3679 T^{4} + 45662 T^{5} - 343255 T^{6} - 986078 T^{7} + 38950081 T^{8} \))(\( 1 - 79 T^{2} + 6241 T^{4} - 493039 T^{6} + 38950081 T^{8} \))(\( 1 + 6 T + 34 T^{2} + 336 T^{3} + 4761 T^{4} - 34122 T^{5} + 60506 T^{6} - 838512 T^{7} + 1917227 T^{8} - 66242448 T^{9} + 377617946 T^{10} - 16823476758 T^{11} + 185441335641 T^{12} + 1033890950064 T^{13} + 8264973487714 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( 1 + 13 T - 19 T^{2} - 951 T^{3} - 6196 T^{4} - 78933 T^{5} - 130891 T^{6} + 7433231 T^{7} + 47458321 T^{8} \))(\( 1 + 6 T - 47 T^{2} - 780 T^{3} - 779 T^{4} - 64740 T^{5} - 323783 T^{6} + 3430722 T^{7} + 47458321 T^{8} \))(\( 1 - 11 T - 4 T^{2} - 264 T^{3} + 5188 T^{4} + 21759 T^{5} + 513329 T^{6} - 7685372 T^{7} + 27478708 T^{8} - 637885876 T^{9} + 3536323481 T^{10} + 12441513333 T^{11} + 246213769348 T^{12} - 1039906729752 T^{13} - 1307761493476 T^{14} - 298496560885897 T^{15} + 2252292232139041 T^{16} \))
$89$ (\( 1 - 2 T + 35 T^{2} - 632 T^{3} + 8789 T^{4} - 56248 T^{5} + 277235 T^{6} - 1409938 T^{7} + 62742241 T^{8} \))(\( 1 + 5 T - 79 T^{2} - 445 T^{3} + 5276 T^{4} - 39605 T^{5} - 625759 T^{6} + 3524845 T^{7} + 62742241 T^{8} \))(\( 1 - 9 T - 166 T^{2} + 1356 T^{3} + 12786 T^{4} - 119727 T^{5} + 426031 T^{6} + 2824668 T^{7} - 98110168 T^{8} + 251395452 T^{9} + 3374591551 T^{10} - 84403823463 T^{11} + 802222293426 T^{12} + 7571984612844 T^{13} - 82498894299526 T^{14} - 398082014059761 T^{15} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 7 T - 73 T^{2} + 835 T^{3} + 1816 T^{4} + 80995 T^{5} - 686857 T^{6} - 6388711 T^{7} + 88529281 T^{8} \))(\( 1 - 18 T + 47 T^{2} + 1740 T^{3} - 27059 T^{4} + 168780 T^{5} + 442223 T^{6} - 16428114 T^{7} + 88529281 T^{8} \))(\( 1 - T + 28 T^{2} - 632 T^{3} + 5996 T^{4} - 78691 T^{5} + 707075 T^{6} - 3586376 T^{7} + 72919024 T^{8} - 347878472 T^{9} + 6652868675 T^{10} - 71819151043 T^{11} + 530821568876 T^{12} - 5427199042424 T^{13} + 23323216138012 T^{14} - 80798284478113 T^{15} + 7837433594376961 T^{16} \))
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