Properties

Label 144.2.x
Level 144
Weight 2
Character orbit x
Rep. character \(\chi_{144}(13,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 88
Newform subspaces 5
Sturm bound 48
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 144.x (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 144 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 5 \)
Sturm bound: \(48\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(144, [\chi])\).

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

Trace form

\( 88q - 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} - 10q^{6} - 8q^{8} + O(q^{10}) \) \( 88q - 2q^{2} - 4q^{3} - 2q^{4} - 2q^{5} - 10q^{6} - 8q^{8} - 8q^{10} - 2q^{11} + 8q^{12} - 2q^{13} - 10q^{14} - 8q^{15} - 2q^{16} - 16q^{17} - 18q^{18} - 8q^{19} + 12q^{20} - 10q^{21} - 2q^{22} - 40q^{24} - 40q^{26} + 8q^{27} - 24q^{28} - 2q^{29} + 6q^{30} - 4q^{31} - 22q^{32} - 8q^{33} - 6q^{34} - 28q^{35} + 14q^{36} - 8q^{37} + 26q^{38} - 2q^{40} - 16q^{42} - 2q^{43} + 36q^{44} - 14q^{45} + 8q^{46} - 44q^{47} + 2q^{48} + 16q^{49} - 36q^{50} - 36q^{51} - 2q^{52} - 8q^{53} + 74q^{54} + 52q^{56} - 20q^{58} + 10q^{59} + 30q^{60} - 2q^{61} + 100q^{62} - 36q^{63} - 44q^{64} - 4q^{65} + 8q^{66} - 2q^{67} + 16q^{68} - 10q^{69} + 12q^{70} + 114q^{72} + 26q^{74} + 56q^{75} + 10q^{76} - 30q^{77} + 88q^{78} - 4q^{79} + 144q^{80} - 8q^{81} - 52q^{82} - 22q^{83} - 22q^{84} - 12q^{85} + 70q^{86} - 26q^{88} + 72q^{90} - 36q^{91} - 56q^{92} - 22q^{93} + 6q^{94} + 60q^{95} + 118q^{96} - 4q^{97} + 40q^{98} + 10q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(144, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
144.2.x.a \(4\) \(1.150\) \(\Q(\zeta_{12})\) None \(-4\) \(0\) \(-4\) \(12\) \(q+(-1-\zeta_{12}^{3})q^{2}+(2\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
144.2.x.b \(4\) \(1.150\) \(\Q(\zeta_{12})\) None \(-2\) \(-6\) \(-8\) \(-6\) \(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-2+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
144.2.x.c \(4\) \(1.150\) \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(4\) \(6\) \(q+(\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+\cdots\)
144.2.x.d \(4\) \(1.150\) \(\Q(\zeta_{12})\) None \(2\) \(0\) \(2\) \(-12\) \(q+(1+\zeta_{12}-\zeta_{12}^{2})q^{2}+(-1+2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
144.2.x.e \(72\) \(1.150\) None \(4\) \(2\) \(4\) \(0\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 + 2 T + 2 T^{2} )^{2} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 + 2 T + 2 T^{2} + 4 T^{3} + 4 T^{4} \))(\( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \))
$3$ (\( ( 1 - 3 T^{2} )^{2} \))(\( ( 1 + 3 T + 3 T^{2} )^{2} \))(\( 1 + 3 T^{2} + 9 T^{4} \))(\( ( 1 + 3 T^{2} )^{2} \))
$5$ (\( 1 + 4 T + 5 T^{2} - 8 T^{3} - 44 T^{4} - 40 T^{5} + 125 T^{6} + 500 T^{7} + 625 T^{8} \))(\( 1 + 8 T + 20 T^{2} - 4 T^{3} - 89 T^{4} - 20 T^{5} + 500 T^{6} + 1000 T^{7} + 625 T^{8} \))(\( 1 - 4 T + 20 T^{2} - 52 T^{3} + 151 T^{4} - 260 T^{5} + 500 T^{6} - 500 T^{7} + 625 T^{8} \))(\( 1 - 2 T + 5 T^{2} - 14 T^{3} + 16 T^{4} - 70 T^{5} + 125 T^{6} - 250 T^{7} + 625 T^{8} \))
$7$ (\( 1 - 12 T + 73 T^{2} - 300 T^{3} + 912 T^{4} - 2100 T^{5} + 3577 T^{6} - 4116 T^{7} + 2401 T^{8} \))(\( 1 + 6 T + 28 T^{2} + 96 T^{3} + 291 T^{4} + 672 T^{5} + 1372 T^{6} + 2058 T^{7} + 2401 T^{8} \))(\( 1 - 6 T + 28 T^{2} - 96 T^{3} + 291 T^{4} - 672 T^{5} + 1372 T^{6} - 2058 T^{7} + 2401 T^{8} \))(\( 1 + 12 T + 73 T^{2} + 300 T^{3} + 912 T^{4} + 2100 T^{5} + 3577 T^{6} + 4116 T^{7} + 2401 T^{8} \))
$11$ (\( 1 + 2 T + 5 T^{2} + 26 T^{3} - 32 T^{4} + 286 T^{5} + 605 T^{6} + 2662 T^{7} + 14641 T^{8} \))(\( 1 + 10 T + 41 T^{2} + 82 T^{3} + 136 T^{4} + 902 T^{5} + 4961 T^{6} + 13310 T^{7} + 14641 T^{8} \))(\( 1 - 8 T + 41 T^{2} - 152 T^{3} + 532 T^{4} - 1672 T^{5} + 4961 T^{6} - 10648 T^{7} + 14641 T^{8} \))(\( 1 - 4 T + 5 T^{2} + 20 T^{3} - 164 T^{4} + 220 T^{5} + 605 T^{6} - 5324 T^{7} + 14641 T^{8} \))
$13$ (\( ( 1 - 6 T + 13 T^{2} )^{2}( 1 + 4 T + 3 T^{2} + 52 T^{3} + 169 T^{4} ) \))(\( ( 1 + 5 T + 13 T^{2} )^{2}( 1 + 23 T^{2} + 169 T^{4} ) \))(\( ( 1 - 7 T + 13 T^{2} )^{2}( 1 - T^{2} + 169 T^{4} ) \))(\( ( 1 - 4 T + 13 T^{2} )^{2}( 1 + 6 T + 23 T^{2} + 78 T^{3} + 169 T^{4} ) \))
$17$ (\( ( 1 - 4 T + 17 T^{2} )^{4} \))(\( ( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )^{2} \))(\( ( 1 - 4 T + 17 T^{2} )^{4} \))
$19$ (\( ( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))(\( 1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 2508 T^{5} + 6498 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 132 T^{3} + 959 T^{4} + 2508 T^{5} + 6498 T^{6} + 41154 T^{7} + 130321 T^{8} \))(\( ( 1 + 6 T + 18 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \))
$23$ (\( 1 + 12 T + 97 T^{2} + 588 T^{3} + 2976 T^{4} + 13524 T^{5} + 51313 T^{6} + 146004 T^{7} + 279841 T^{8} \))(\( 1 - 6 T + 52 T^{2} - 240 T^{3} + 1347 T^{4} - 5520 T^{5} + 27508 T^{6} - 73002 T^{7} + 279841 T^{8} \))(\( 1 + 6 T + 52 T^{2} + 240 T^{3} + 1347 T^{4} + 5520 T^{5} + 27508 T^{6} + 73002 T^{7} + 279841 T^{8} \))(\( 1 - 12 T + 97 T^{2} - 588 T^{3} + 2976 T^{4} - 13524 T^{5} + 51313 T^{6} - 146004 T^{7} + 279841 T^{8} \))
$29$ (\( 1 + 9 T^{2} - 156 T^{3} - 484 T^{4} - 4524 T^{5} + 7569 T^{6} + 707281 T^{8} \))(\( 1 + 6 T + 18 T^{2} + 36 T^{3} - 457 T^{4} + 1044 T^{5} + 15138 T^{6} + 146334 T^{7} + 707281 T^{8} \))(\( 1 - 6 T + 18 T^{2} - 36 T^{3} - 457 T^{4} - 1044 T^{5} + 15138 T^{6} - 146334 T^{7} + 707281 T^{8} \))(\( 1 + 6 T + 9 T^{2} - 174 T^{3} - 1528 T^{4} - 5046 T^{5} + 7569 T^{6} + 146334 T^{7} + 707281 T^{8} \))
$31$ (\( 1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 2728 T^{5} + 12493 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 992 T^{5} - 1922 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T - 2 T^{2} + 32 T^{3} + 1411 T^{4} + 992 T^{5} - 1922 T^{6} + 238328 T^{7} + 923521 T^{8} \))(\( 1 + 8 T + 13 T^{2} - 88 T^{3} - 344 T^{4} - 2728 T^{5} + 12493 T^{6} + 238328 T^{7} + 923521 T^{8} \))
$37$ (\( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 91464 T^{5} + 394272 T^{6} + 1215672 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 21756 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 - 12 T + 72 T^{2} - 588 T^{3} + 4658 T^{4} - 21756 T^{5} + 98568 T^{6} - 607836 T^{7} + 1874161 T^{8} \))(\( 1 + 24 T + 288 T^{2} + 2472 T^{3} + 16862 T^{4} + 91464 T^{5} + 394272 T^{6} + 1215672 T^{7} + 1874161 T^{8} \))
$41$ (\( 1 + 18 T + 181 T^{2} + 1314 T^{3} + 8076 T^{4} + 53874 T^{5} + 304261 T^{6} + 1240578 T^{7} + 2825761 T^{8} \))(\( 1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8} \))(\( 1 + 73 T^{2} + 3648 T^{4} + 122713 T^{6} + 2825761 T^{8} \))(\( 1 - 18 T + 181 T^{2} - 1314 T^{3} + 8076 T^{4} - 53874 T^{5} + 304261 T^{6} - 1240578 T^{7} + 2825761 T^{8} \))
$43$ (\( 1 - 14 T + 65 T^{2} + 474 T^{3} - 6280 T^{4} + 20382 T^{5} + 120185 T^{6} - 1113098 T^{7} + 3418801 T^{8} \))(\( 1 + 16 T + 65 T^{2} - 624 T^{3} - 8092 T^{4} - 26832 T^{5} + 120185 T^{6} + 1272112 T^{7} + 3418801 T^{8} \))(\( 1 - 2 T + 65 T^{2} - 426 T^{3} + 2744 T^{4} - 18318 T^{5} + 120185 T^{6} - 159014 T^{7} + 3418801 T^{8} \))(\( 1 - 8 T + 65 T^{2} + 120 T^{3} - 604 T^{4} + 5160 T^{5} + 120185 T^{6} - 636056 T^{7} + 3418801 T^{8} \))
$47$ (\( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 4136 T^{5} - 41971 T^{6} - 830584 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 6956 T^{5} - 35344 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 + 2 T - 16 T^{2} - 148 T^{3} - 1997 T^{4} - 6956 T^{5} - 35344 T^{6} + 207646 T^{7} + 4879681 T^{8} \))(\( 1 - 8 T - 19 T^{2} + 88 T^{3} + 1672 T^{4} + 4136 T^{5} - 41971 T^{6} - 830584 T^{7} + 4879681 T^{8} \))
$53$ (\( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 66992 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 51728 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 + 16 T + 128 T^{2} + 976 T^{3} + 7378 T^{4} + 51728 T^{5} + 359552 T^{6} + 2382032 T^{7} + 7890481 T^{8} \))(\( 1 - 16 T + 128 T^{2} - 1264 T^{3} + 11806 T^{4} - 66992 T^{5} + 359552 T^{6} - 2382032 T^{7} + 7890481 T^{8} \))
$59$ (\( 1 + 6 T + 45 T^{2} + 462 T^{3} + 848 T^{4} + 27258 T^{5} + 156645 T^{6} + 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 6 T + 45 T^{2} + 594 T^{3} - 3376 T^{4} + 35046 T^{5} + 156645 T^{6} - 1232274 T^{7} + 12117361 T^{8} \))(\( 1 - 12 T + 45 T^{2} + 828 T^{3} - 9748 T^{4} + 48852 T^{5} + 156645 T^{6} - 2464548 T^{7} + 12117361 T^{8} \))(\( 1 - 12 T + 45 T^{2} + 300 T^{3} - 5524 T^{4} + 17700 T^{5} + 156645 T^{6} - 2464548 T^{7} + 12117361 T^{8} \))
$61$ (\( 1 + 12 T + 261 T^{2} + 2088 T^{3} + 24452 T^{4} + 127368 T^{5} + 971181 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 + 12 T + 180 T^{2} + 1596 T^{3} + 15143 T^{4} + 97356 T^{5} + 669780 T^{6} + 2723772 T^{7} + 13845841 T^{8} \))(\( 1 - 24 T + 180 T^{2} + 300 T^{3} - 11209 T^{4} + 18300 T^{5} + 669780 T^{6} - 5447544 T^{7} + 13845841 T^{8} \))(\( 1 - 30 T + 261 T^{2} + 702 T^{3} - 21664 T^{4} + 42822 T^{5} + 971181 T^{6} - 6809430 T^{7} + 13845841 T^{8} \))
$67$ (\( 1 + 2 T + 65 T^{2} + 762 T^{3} + 2600 T^{4} + 51054 T^{5} + 291785 T^{6} + 601526 T^{7} + 20151121 T^{8} \))(\( 1 - 16 T + 113 T^{2} - 384 T^{3} - 172 T^{4} - 25728 T^{5} + 507257 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))(\( 1 + 14 T + 113 T^{2} + 726 T^{3} + 3848 T^{4} + 48642 T^{5} + 507257 T^{6} + 4210682 T^{7} + 20151121 T^{8} \))(\( 1 - 16 T + 65 T^{2} + 960 T^{3} - 14284 T^{4} + 64320 T^{5} + 291785 T^{6} - 4812208 T^{7} + 20151121 T^{8} \))
$71$ (\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))(\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))(\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))(\( 1 - 156 T^{2} + 13094 T^{4} - 786396 T^{6} + 25411681 T^{8} \))
$73$ (\( 1 - 236 T^{2} + 23814 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))(\( 1 - 158 T^{2} + 16131 T^{4} - 841982 T^{6} + 28398241 T^{8} \))(\( 1 - 158 T^{2} + 16131 T^{4} - 841982 T^{6} + 28398241 T^{8} \))(\( 1 - 236 T^{2} + 23814 T^{4} - 1257644 T^{6} + 28398241 T^{8} \))
$79$ (\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))(\( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( ( 1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4} )^{2} \))(\( 1 - 155 T^{2} + 17784 T^{4} - 967355 T^{6} + 38950081 T^{8} \))
$83$ (\( 1 + 22 T + 185 T^{2} - 290 T^{3} - 14024 T^{4} - 24070 T^{5} + 1274465 T^{6} + 12579314 T^{7} + 47458321 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 2 T + 2 T^{2} - 328 T^{3} - 7217 T^{4} - 27224 T^{5} + 13778 T^{6} + 1143574 T^{7} + 47458321 T^{8} \))(\( 1 + 16 T + 185 T^{2} + 784 T^{3} + 4900 T^{4} + 65072 T^{5} + 1274465 T^{6} + 9148592 T^{7} + 47458321 T^{8} \))
$89$ (\( 1 + 36 T^{2} + 13094 T^{4} + 285156 T^{6} + 62742241 T^{8} \))(\( ( 1 - 174 T^{2} + 7921 T^{4} )^{2} \))(\( ( 1 - 174 T^{2} + 7921 T^{4} )^{2} \))(\( 1 + 36 T^{2} + 13094 T^{4} + 285156 T^{6} + 62742241 T^{8} \))
$97$ (\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))(\( 1 + 20 T + 109 T^{2} + 1940 T^{3} + 36472 T^{4} + 188180 T^{5} + 1025581 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))(\( 1 + 20 T + 109 T^{2} + 1940 T^{3} + 36472 T^{4} + 188180 T^{5} + 1025581 T^{6} + 18253460 T^{7} + 88529281 T^{8} \))(\( ( 1 + T - 96 T^{2} + 97 T^{3} + 9409 T^{4} )^{2} \))
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