Properties

Label 42.3.h
Level $42$
Weight $3$
Character orbit 42.h
Rep. character $\chi_{42}(11,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $12$
Newform subspaces $2$
Sturm bound $24$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 42.h (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(24\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(42, [\chi])\).

Total New Old
Modular forms 40 12 28
Cusp forms 24 12 12
Eisenstein series 16 0 16

Trace form

\( 12q + 12q^{4} + 8q^{6} - 14q^{7} + 4q^{9} + O(q^{10}) \) \( 12q + 12q^{4} + 8q^{6} - 14q^{7} + 4q^{9} - 8q^{10} - 68q^{13} - 68q^{15} - 24q^{16} - 24q^{18} + 66q^{19} - 2q^{21} - 16q^{22} + 8q^{24} + 70q^{25} + 252q^{27} + 16q^{28} + 76q^{30} + 34q^{31} - 106q^{33} - 16q^{34} + 16q^{36} + 6q^{37} - 74q^{39} + 16q^{40} - 144q^{42} + 84q^{43} - 154q^{45} + 80q^{46} - 138q^{49} - 34q^{51} - 68q^{52} + 8q^{54} - 472q^{55} - 144q^{57} - 128q^{58} - 68q^{60} + 112q^{61} + 276q^{63} - 96q^{64} + 224q^{66} + 26q^{67} + 260q^{69} + 392q^{70} + 48q^{72} + 326q^{73} + 70q^{75} + 264q^{76} + 224q^{78} + 154q^{79} + 64q^{81} - 48q^{82} - 152q^{84} + 440q^{85} - 164q^{87} - 16q^{88} - 592q^{90} - 338q^{91} - 224q^{93} - 480q^{94} - 16q^{96} - 88q^{97} - 140q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(42, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
42.3.h.a \(4\) \(1.144\) \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(2\) \(0\) \(-14\) \(q+\beta _{1}q^{2}+(2\beta _{1}+\beta _{2}-2\beta _{3})q^{3}+2\beta _{2}q^{4}+\cdots\)
42.3.h.b \(8\) \(1.144\) 8.0.4857532416.2 None \(0\) \(-2\) \(0\) \(0\) \(q+(\beta _{2}-\beta _{6})q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(42, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(42, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 2 T^{2} + 4 T^{4} \))(\( ( 1 - 2 T^{2} + 4 T^{4} )^{2} \))
$3$ (\( 1 - 2 T - 5 T^{2} - 18 T^{3} + 81 T^{4} \))(\( 1 + 2 T + 7 T^{2} - 42 T^{3} - 108 T^{4} - 378 T^{5} + 567 T^{6} + 1458 T^{7} + 6561 T^{8} \))
$5$ (\( 1 - 22 T^{2} - 141 T^{4} - 13750 T^{6} + 390625 T^{8} \))(\( 1 + 62 T^{2} + 1985 T^{4} + 37758 T^{6} + 662756 T^{8} + 23598750 T^{10} + 775390625 T^{12} + 15136718750 T^{14} + 152587890625 T^{16} \))
$7$ (\( ( 1 + 7 T + 49 T^{2} )^{2} \))(\( ( 1 + 10 T^{2} + 2401 T^{4} )^{2} \))
$11$ (\( ( 1 - 11 T + 121 T^{2} )^{2}( 1 + 11 T + 121 T^{2} )^{2} \))(\( 1 - 70 T^{2} - 23407 T^{4} + 68250 T^{6} + 515186468 T^{8} + 999248250 T^{10} - 5017498327567 T^{12} - 219689986370470 T^{14} + 45949729863572161 T^{16} \))
$13$ (\( ( 1 + T + 169 T^{2} )^{4} \))(\( ( 1 + 16 T + 314 T^{2} + 2704 T^{3} + 28561 T^{4} )^{4} \))
$17$ (\( 1 + 506 T^{2} + 172515 T^{4} + 42261626 T^{6} + 6975757441 T^{8} \))(\( 1 + 1118 T^{2} + 770753 T^{4} + 348960222 T^{6} + 118234187396 T^{8} + 29145506701662 T^{10} + 5376585974923073 T^{12} + 651371661222872798 T^{14} + 48661191875666868481 T^{16} \))
$19$ (\( ( 1 - 31 T + 600 T^{2} - 11191 T^{3} + 130321 T^{4} )^{2} \))(\( ( 1 - 2 T - 521 T^{2} + 394 T^{3} + 143860 T^{4} + 142234 T^{5} - 67897241 T^{6} - 94091762 T^{7} + 16983563041 T^{8} )^{2} \))
$23$ (\( 1 + 986 T^{2} + 692355 T^{4} + 275923226 T^{6} + 78310985281 T^{8} \))(\( 1 + 1370 T^{2} + 955793 T^{4} + 495152250 T^{6} + 244893547268 T^{8} + 138563900792250 T^{10} + 74849091554682833 T^{12} + 30023035471867839770 T^{14} + \)\(61\!\cdots\!61\)\( T^{16} \))
$29$ (\( ( 1 - 1394 T^{2} + 707281 T^{4} )^{2} \))(\( ( 1 - 764 T^{2} + 680486 T^{4} - 540362684 T^{6} + 500246412961 T^{8} )^{2} \))
$31$ (\( ( 1 - 7 T - 912 T^{2} - 6727 T^{3} + 923521 T^{4} )^{2} \))(\( ( 1 - 10 T - 1297 T^{2} + 5250 T^{3} + 931988 T^{4} + 5045250 T^{5} - 1197806737 T^{6} - 8875036810 T^{7} + 852891037441 T^{8} )^{2} \))
$37$ (\( ( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{2} \))(\( ( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{4} \))
$41$ (\( ( 1 - 2210 T^{2} + 2825761 T^{4} )^{2} \))(\( ( 1 - 5788 T^{2} + 13912710 T^{4} - 16355504668 T^{6} + 7984925229121 T^{8} )^{2} \))
$43$ (\( ( 1 + 31 T + 1849 T^{2} )^{4} \))(\( ( 1 - 52 T + 3582 T^{2} - 96148 T^{3} + 3418801 T^{4} )^{4} \))
$47$ (\( 1 + 2618 T^{2} + 1974243 T^{4} + 12775004858 T^{6} + 23811286661761 T^{8} \))(\( 1 + 538 T^{2} - 7938479 T^{4} - 823914182 T^{6} + 42475035844804 T^{8} - 4020438379535942 T^{10} - \)\(18\!\cdots\!19\)\( T^{12} + \)\(62\!\cdots\!58\)\( T^{14} + \)\(56\!\cdots\!21\)\( T^{16} \))
$53$ (\( 1 + 4970 T^{2} + 16810419 T^{4} + 39215690570 T^{6} + 62259690411361 T^{8} \))(\( 1 + 5854 T^{2} + 10947457 T^{4} + 44144411038 T^{6} + 211248030023524 T^{8} + 348320636551529278 T^{10} + \)\(68\!\cdots\!77\)\( T^{12} + \)\(28\!\cdots\!14\)\( T^{14} + \)\(38\!\cdots\!21\)\( T^{16} \))
$59$ (\( 1 + 6890 T^{2} + 35354739 T^{4} + 83488617290 T^{6} + 146830437604321 T^{8} \))(\( 1 + 746 T^{2} - 10773535 T^{4} - 9626884566 T^{6} - 25203887215996 T^{8} - 116652435591550326 T^{10} - \)\(15\!\cdots\!35\)\( T^{12} + \)\(13\!\cdots\!26\)\( T^{14} + \)\(21\!\cdots\!41\)\( T^{16} \))
$61$ (\( ( 1 + 50 T - 1221 T^{2} + 186050 T^{3} + 13845841 T^{4} )^{2} \))(\( ( 1 - 106 T + 1337 T^{2} - 260442 T^{3} + 42335204 T^{4} - 969104682 T^{5} + 18511889417 T^{6} - 5461159682266 T^{7} + 191707312997281 T^{8} )^{2} \))
$67$ (\( ( 1 + 65 T - 264 T^{2} + 291785 T^{3} + 20151121 T^{4} )^{2} \))(\( ( 1 - 78 T - 697 T^{2} + 171366 T^{3} - 1480236 T^{4} + 769261974 T^{5} - 14045331337 T^{6} - 7055753809182 T^{7} + 406067677556641 T^{8} )^{2} \))
$71$ (\( ( 1 - 6554 T^{2} + 25411681 T^{4} )^{2} \))(\( ( 1 - 1316 T^{2} + 44745734 T^{4} - 33441772196 T^{6} + 645753531245761 T^{8} )^{2} \))
$73$ (\( ( 1 - 143 T + 5329 T^{2} )^{2}( 1 + 46 T + 5329 T^{2} )^{2} \))(\( ( 1 - 66 T - 7039 T^{2} - 48642 T^{3} + 78234660 T^{4} - 259213218 T^{5} - 199895218399 T^{6} - 9988058935074 T^{7} + 806460091894081 T^{8} )^{2} \))
$79$ (\( ( 1 - 103 T + 4368 T^{2} - 642823 T^{3} + 38950081 T^{4} )^{2} \))(\( ( 1 + 26 T - 5617 T^{2} - 160914 T^{3} - 3567148 T^{4} - 1004264274 T^{5} - 218782604977 T^{6} + 6320273843546 T^{7} + 1517108809906561 T^{8} )^{2} \))
$83$ (\( ( 1 - 11978 T^{2} + 47458321 T^{4} )^{2} \))(\( ( 1 - 18404 T^{2} + 168689894 T^{4} - 873422939684 T^{6} + 2252292232139041 T^{8} )^{2} \))
$89$ (\( 1 + 1730 T^{2} - 59749341 T^{4} + 108544076930 T^{6} + 3936588805702081 T^{8} \))(\( 1 + 2542 T^{2} + 3560113 T^{4} - 311605556402 T^{6} - 4333594781349404 T^{8} - 19550830916713376882 T^{10} + \)\(14\!\cdots\!53\)\( T^{12} + \)\(62\!\cdots\!82\)\( T^{14} + \)\(15\!\cdots\!61\)\( T^{16} \))
$97$ (\( ( 1 + 166 T + 9409 T^{2} )^{4} \))(\( ( 1 - 144 T + 21802 T^{2} - 1354896 T^{3} + 88529281 T^{4} )^{4} \))
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