Properties

Label 144.2.k
Level 144
Weight 2
Character orbit k
Rep. character \(\chi_{144}(37,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 18
Newform subspaces 3
Sturm bound 48
Trace bound 4

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Defining parameters

Level: \( N \) = \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 144.k (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(48\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

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

Total New Old
Modular forms 56 22 34
Cusp forms 40 18 22
Eisenstein series 16 4 12

Trace form

\( 18q + 2q^{2} + 2q^{5} + 8q^{8} + O(q^{10}) \) \( 18q + 2q^{2} + 2q^{5} + 8q^{8} + 8q^{10} + 6q^{11} - 2q^{13} - 8q^{14} - 8q^{16} + 4q^{17} - 10q^{19} - 20q^{20} - 8q^{22} - 24q^{26} - 16q^{28} + 10q^{29} - 16q^{31} - 8q^{32} - 36q^{34} - 20q^{35} + 6q^{37} + 20q^{38} - 16q^{40} - 22q^{43} + 44q^{44} + 36q^{46} - 16q^{47} - 10q^{49} + 42q^{50} + 36q^{52} - 6q^{53} + 8q^{56} + 12q^{58} - 26q^{59} + 14q^{61} - 4q^{62} + 48q^{64} + 12q^{65} + 6q^{67} - 32q^{68} + 32q^{70} - 52q^{74} - 36q^{76} - 20q^{77} + 32q^{79} - 16q^{80} + 42q^{83} - 28q^{85} + 16q^{86} + 24q^{88} + 52q^{91} + 40q^{92} + 8q^{94} + 60q^{95} - 4q^{97} + 46q^{98} + O(q^{100}) \)

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.k.a \(2\) \(1.150\) \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(2\) \(0\) \(q+(1+i)q^{2}+2iq^{4}+(1+i)q^{5}-2iq^{7}+\cdots\)
144.2.k.b \(8\) \(1.150\) 8.0.18939904.2 None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{6}q^{2}+(-1-\beta _{1}-\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
144.2.k.c \(8\) \(1.150\) 8.0.629407744.1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{4}-\beta _{6})q^{5}+(-1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(144, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(144, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T + 2 T^{2} \))(\( 1 + 2 T^{2} - 4 T^{3} + 2 T^{4} - 8 T^{5} + 8 T^{6} + 16 T^{8} \))(\( 1 - 2 T^{2} + 2 T^{4} - 8 T^{6} + 16 T^{8} \))
$3$ (\( \))(\( \))(\( \))
$5$ (\( ( 1 - 4 T + 5 T^{2} )( 1 + 2 T + 5 T^{2} ) \))(\( 1 + 16 T^{3} - 12 T^{4} - 48 T^{5} + 128 T^{6} + 32 T^{7} - 506 T^{8} + 160 T^{9} + 3200 T^{10} - 6000 T^{11} - 7500 T^{12} + 50000 T^{13} + 390625 T^{16} \))(\( 1 - 12 T^{4} - 506 T^{8} - 7500 T^{12} + 390625 T^{16} \))
$7$ (\( 1 - 10 T^{2} + 49 T^{4} \))(\( 1 - 24 T^{2} + 292 T^{4} - 2440 T^{6} + 17222 T^{8} - 119560 T^{10} + 701092 T^{12} - 2823576 T^{14} + 5764801 T^{16} \))(\( ( 1 - 12 T^{2} + 106 T^{4} - 588 T^{6} + 2401 T^{8} )^{2} \))
$11$ (\( 1 + 2 T + 2 T^{2} + 22 T^{3} + 121 T^{4} \))(\( 1 - 8 T + 32 T^{2} - 88 T^{3} + 132 T^{4} - 344 T^{5} + 2400 T^{6} - 13000 T^{7} + 54374 T^{8} - 143000 T^{9} + 290400 T^{10} - 457864 T^{11} + 1932612 T^{12} - 14172488 T^{13} + 56689952 T^{14} - 155897368 T^{15} + 214358881 T^{16} \))(\( 1 - 60 T^{4} - 14618 T^{8} - 878460 T^{12} + 214358881 T^{16} \))
$13$ (\( ( 1 - 4 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \))(\( 1 - 64 T^{3} - 4 T^{4} + 704 T^{5} + 2048 T^{6} - 1408 T^{7} - 53466 T^{8} - 18304 T^{9} + 346112 T^{10} + 1546688 T^{11} - 114244 T^{12} - 23762752 T^{13} + 815730721 T^{16} \))(\( ( 1 - 194 T^{4} + 28561 T^{8} )^{2} \))
$17$ (\( ( 1 - 2 T + 17 T^{2} )^{2} \))(\( ( 1 + 36 T^{2} - 64 T^{3} + 662 T^{4} - 1088 T^{5} + 10404 T^{6} + 83521 T^{8} )^{2} \))(\( ( 1 + 28 T^{2} + 662 T^{4} + 8092 T^{6} + 83521 T^{8} )^{2} \))
$19$ (\( 1 - 6 T + 18 T^{2} - 114 T^{3} + 361 T^{4} \))(\( 1 + 8 T + 32 T^{2} + 120 T^{3} + 452 T^{4} + 2168 T^{5} + 10080 T^{6} + 37832 T^{7} + 138918 T^{8} + 718808 T^{9} + 3638880 T^{10} + 14870312 T^{11} + 58905092 T^{12} + 297131880 T^{13} + 1505468192 T^{14} + 7150973912 T^{15} + 16983563041 T^{16} \))(\( ( 1 + 4 T + 8 T^{2} + 28 T^{3} - 46 T^{4} + 532 T^{5} + 2888 T^{6} + 27436 T^{7} + 130321 T^{8} )^{2} \))
$23$ (\( 1 - 10 T^{2} + 529 T^{4} \))(\( ( 1 - 38 T^{2} + 529 T^{4} )^{4} \))(\( ( 1 - 12 T^{2} + 646 T^{4} - 6348 T^{6} + 279841 T^{8} )^{2} \))
$29$ (\( ( 1 - 4 T + 29 T^{2} )( 1 + 10 T + 29 T^{2} ) \))(\( 1 - 16 T + 128 T^{2} - 928 T^{3} + 6580 T^{4} - 38208 T^{5} + 199680 T^{6} - 1073680 T^{7} + 5802054 T^{8} - 31136720 T^{9} + 167930880 T^{10} - 931854912 T^{11} + 4653908980 T^{12} - 19034346272 T^{13} + 76137385088 T^{14} - 275998020944 T^{15} + 500246412961 T^{16} \))(\( 1 + 180 T^{4} - 772538 T^{8} + 127310580 T^{12} + 500246412961 T^{16} \))
$31$ (\( ( 1 + 8 T + 31 T^{2} )^{2} \))(\( ( 1 - 12 T + 164 T^{2} - 1140 T^{3} + 8218 T^{4} - 35340 T^{5} + 157604 T^{6} - 357492 T^{7} + 923521 T^{8} )^{2} \))(\( ( 1 + 6 T + 64 T^{2} + 186 T^{3} + 961 T^{4} )^{4} \))
$37$ (\( 1 - 6 T + 18 T^{2} - 222 T^{3} + 1369 T^{4} \))(\( 1 + 16 T + 128 T^{2} + 1008 T^{3} + 5948 T^{4} + 15248 T^{5} - 9344 T^{6} - 717840 T^{7} - 7530650 T^{8} - 26560080 T^{9} - 12791936 T^{10} + 772356944 T^{11} + 11147509628 T^{12} + 69898708656 T^{13} + 328412980352 T^{14} + 1518910034128 T^{15} + 3512479453921 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 248 T^{3} + 1886 T^{4} - 9176 T^{5} + 43808 T^{6} - 405224 T^{7} + 1874161 T^{8} )^{2} \))
$41$ (\( ( 1 - 41 T^{2} )^{2} \))(\( 1 - 200 T^{2} + 19452 T^{4} - 1244536 T^{6} + 58583750 T^{8} - 2092065016 T^{10} + 54966702972 T^{12} - 950020848200 T^{14} + 7984925229121 T^{16} \))(\( ( 1 - 60 T^{2} + 4150 T^{4} - 100860 T^{6} + 2825761 T^{8} )^{2} \))
$43$ (\( 1 - 10 T + 50 T^{2} - 430 T^{3} + 1849 T^{4} \))(\( 1 + 8 T + 32 T^{2} + 56 T^{3} + 260 T^{4} + 504 T^{5} - 2720 T^{6} - 625528 T^{7} - 7635866 T^{8} - 26897704 T^{9} - 5029280 T^{10} + 40071528 T^{11} + 888888260 T^{12} + 8232472808 T^{13} + 202283617568 T^{14} + 2174548888856 T^{15} + 11688200277601 T^{16} \))(\( ( 1 + 12 T + 72 T^{2} + 564 T^{3} + 4402 T^{4} + 24252 T^{5} + 133128 T^{6} + 954084 T^{7} + 3418801 T^{8} )^{2} \))
$47$ (\( ( 1 + 8 T + 47 T^{2} )^{2} \))(\( ( 1 + 86 T^{2} + 2209 T^{4} )^{4} \))(\( ( 1 - 20 T^{2} + 4070 T^{4} - 44180 T^{6} + 4879681 T^{8} )^{2} \))
$53$ (\( ( 1 - 14 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} ) \))(\( 1 + 16 T + 128 T^{2} + 928 T^{3} + 8564 T^{4} + 82496 T^{5} + 654336 T^{6} + 5021328 T^{7} + 38116486 T^{8} + 266130384 T^{9} + 1838029824 T^{10} + 12281756992 T^{11} + 67574079284 T^{12} + 388085417504 T^{13} + 2837038224512 T^{14} + 18795378237392 T^{15} + 62259690411361 T^{16} \))(\( 1 - 5772 T^{4} + 23807110 T^{8} - 45543856332 T^{12} + 62259690411361 T^{16} \))
$59$ (\( 1 - 6 T + 18 T^{2} - 354 T^{3} + 3481 T^{4} \))(\( ( 1 + 8 T + 32 T^{2} + 472 T^{3} + 3481 T^{4} )^{4} \))(\( 1 - 7452 T^{4} + 27767206 T^{8} - 90298574172 T^{12} + 146830437604321 T^{16} \))
$61$ (\( 1 + 18 T + 162 T^{2} + 1098 T^{3} + 3721 T^{4} \))(\( 1 - 16 T + 128 T^{2} - 1392 T^{3} + 14204 T^{4} - 79760 T^{5} + 426880 T^{6} - 2945904 T^{7} + 19569574 T^{8} - 179700144 T^{9} + 1588420480 T^{10} - 18104004560 T^{11} + 196666325564 T^{12} - 1175678050992 T^{13} + 6594607918208 T^{14} - 50283885376336 T^{15} + 191707312997281 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 440 T^{3} + 6014 T^{4} - 26840 T^{5} + 119072 T^{6} - 1815848 T^{7} + 13845841 T^{8} )^{2} \))
$67$ (\( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} \))(\( 1 + 16 T + 128 T^{2} + 304 T^{3} + 4388 T^{4} + 107696 T^{5} + 1207680 T^{6} + 4800272 T^{7} + 13154790 T^{8} + 321618224 T^{9} + 5421275520 T^{10} + 32390972048 T^{11} + 88423118948 T^{12} + 410438032528 T^{13} + 11578672917632 T^{14} + 96971385685168 T^{15} + 406067677556641 T^{16} \))(\( ( 1 - 8 T + 32 T^{2} - 536 T^{3} + 4489 T^{4} )^{4} \))
$71$ (\( 1 - 42 T^{2} + 5041 T^{4} \))(\( 1 - 440 T^{2} + 90844 T^{4} - 11522952 T^{6} + 984512390 T^{8} - 58087201032 T^{10} + 2308498748764 T^{12} - 56364124925240 T^{14} + 645753531245761 T^{16} \))(\( ( 1 - 92 T^{2} + 5030 T^{4} - 463772 T^{6} + 25411681 T^{8} )^{2} \))
$73$ (\( 1 - 130 T^{2} + 5329 T^{4} \))(\( 1 - 328 T^{2} + 45404 T^{4} - 3734648 T^{6} + 259745542 T^{8} - 19901939192 T^{10} + 1289393734364 T^{12} - 49637626222792 T^{14} + 806460091894081 T^{16} \))(\( ( 1 - 228 T^{2} + 23206 T^{4} - 1215012 T^{6} + 28398241 T^{8} )^{2} \))
$79$ (\( ( 1 + 79 T^{2} )^{2} \))(\( ( 1 + 12 T + 148 T^{2} - 44 T^{3} + 794 T^{4} - 3476 T^{5} + 923668 T^{6} + 5916468 T^{7} + 38950081 T^{8} )^{2} \))(\( ( 1 - 14 T + 200 T^{2} - 1106 T^{3} + 6241 T^{4} )^{4} \))
$83$ (\( 1 - 2 T + 2 T^{2} - 166 T^{3} + 6889 T^{4} \))(\( 1 - 40 T + 800 T^{2} - 11000 T^{3} + 122436 T^{4} - 1297720 T^{5} + 14460000 T^{6} - 161033000 T^{7} + 1597489574 T^{8} - 13365739000 T^{9} + 99614940000 T^{10} - 742019425640 T^{11} + 5810606989956 T^{12} - 43329447073000 T^{13} + 261552298695200 T^{14} - 1085442039585080 T^{15} + 2252292232139041 T^{16} \))(\( 1 + 20100 T^{4} + 185294374 T^{8} + 953912252100 T^{12} + 2252292232139041 T^{16} \))
$89$ (\( 1 - 162 T^{2} + 7921 T^{4} \))(\( 1 - 248 T^{2} + 36316 T^{4} - 4626504 T^{6} + 476004998 T^{8} - 36646538184 T^{10} + 2278547224156 T^{12} - 123251360158328 T^{14} + 3936588805702081 T^{16} \))(\( ( 1 - 260 T^{2} + 30950 T^{4} - 2059460 T^{6} + 62742241 T^{8} )^{2} \))
$97$ (\( ( 1 + 2 T + 97 T^{2} )^{2} \))(\( ( 1 + 164 T^{2} + 768 T^{3} + 13510 T^{4} + 74496 T^{5} + 1543076 T^{6} + 88529281 T^{8} )^{2} \))(\( ( 1 + 82 T^{2} + 9409 T^{4} )^{4} \))
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