Properties

Label 84.2.o
Level 84
Weight 2
Character orbit o
Rep. character \(\chi_{84}(19,\cdot)\)
Character field \(\Q(\zeta_{6})\)
Dimension 16
Newform subspaces 2
Sturm bound 32
Trace bound 3

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Defining parameters

Level: \( N \) = \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 84.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 2 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(84, [\chi])\).

Total New Old
Modular forms 40 16 24
Cusp forms 24 16 8
Eisenstein series 16 0 16

Trace form

\( 16q + 2q^{2} - 2q^{4} + 8q^{8} - 8q^{9} + O(q^{10}) \) \( 16q + 2q^{2} - 2q^{4} + 8q^{8} - 8q^{9} - 18q^{10} - 22q^{14} - 10q^{16} + 2q^{18} - 8q^{21} - 12q^{22} + 18q^{24} + 4q^{25} + 30q^{26} + 6q^{28} - 32q^{29} + 8q^{30} + 12q^{32} - 12q^{33} + 4q^{36} + 12q^{37} + 18q^{38} - 30q^{40} + 16q^{42} - 4q^{44} + 12q^{46} + 8q^{49} + 4q^{50} + 36q^{52} - 8q^{53} + 40q^{56} + 24q^{57} + 14q^{58} - 22q^{60} + 24q^{61} + 4q^{64} + 8q^{65} - 36q^{66} - 36q^{68} + 38q^{70} - 4q^{72} - 36q^{73} - 38q^{74} + 16q^{77} - 12q^{78} - 72q^{80} - 8q^{81} - 24q^{82} - 40q^{84} - 64q^{85} - 6q^{86} - 26q^{88} - 56q^{92} + 12q^{93} + 30q^{96} - 72q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(84, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
84.2.o.a \(8\) \(0.671\) 8.0.562828176.1 None \(1\) \(-4\) \(0\) \(2\) \(q-\beta _{4}q^{2}+(-1+\beta _{3})q^{3}+(\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots\)
84.2.o.b \(8\) \(0.671\) 8.0.562828176.1 None \(1\) \(4\) \(0\) \(-2\) \(q+\beta _{6}q^{2}+\beta _{3}q^{3}+(\beta _{1}-\beta _{2})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(84, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(84, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + T^{2} - 2 T^{3} + 6 T^{4} - 4 T^{5} + 4 T^{6} - 8 T^{7} + 16 T^{8} \))(\( 1 - T + T^{2} - 2 T^{3} - 4 T^{5} + 4 T^{6} - 8 T^{7} + 16 T^{8} \))
$3$ (\( ( 1 + T + T^{2} )^{4} \))(\( ( 1 - T + T^{2} )^{4} \))
$5$ (\( 1 + 9 T^{2} + 45 T^{4} + 96 T^{5} + 66 T^{6} + 864 T^{7} - 394 T^{8} + 4320 T^{9} + 1650 T^{10} + 12000 T^{11} + 28125 T^{12} + 140625 T^{14} + 390625 T^{16} \))(\( 1 + 9 T^{2} + 45 T^{4} + 96 T^{5} + 66 T^{6} + 864 T^{7} - 394 T^{8} + 4320 T^{9} + 1650 T^{10} + 12000 T^{11} + 28125 T^{12} + 140625 T^{14} + 390625 T^{16} \))
$7$ (\( 1 - 2 T - 16 T^{3} + 65 T^{4} - 112 T^{5} - 686 T^{7} + 2401 T^{8} \))(\( 1 + 2 T + 16 T^{3} + 65 T^{4} + 112 T^{5} + 686 T^{7} + 2401 T^{8} \))
$11$ (\( ( 1 - 8 T + 34 T^{2} - 112 T^{3} + 339 T^{4} - 1232 T^{5} + 4114 T^{6} - 10648 T^{7} + 14641 T^{8} )( 1 + 2 T + 25 T^{2} + 58 T^{3} + 372 T^{4} + 638 T^{5} + 3025 T^{6} + 2662 T^{7} + 14641 T^{8} ) \))(\( ( 1 - 2 T + 25 T^{2} - 58 T^{3} + 372 T^{4} - 638 T^{5} + 3025 T^{6} - 2662 T^{7} + 14641 T^{8} )( 1 + 8 T + 34 T^{2} + 112 T^{3} + 339 T^{4} + 1232 T^{5} + 4114 T^{6} + 10648 T^{7} + 14641 T^{8} ) \))
$13$ (\( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16} \))(\( 1 - 66 T^{2} + 2241 T^{4} - 49362 T^{6} + 759092 T^{8} - 8342178 T^{10} + 64005201 T^{12} - 318569394 T^{14} + 815730721 T^{16} \))
$17$ (\( 1 + 40 T^{2} + 786 T^{4} + 1104 T^{5} + 10208 T^{6} + 39072 T^{7} + 129731 T^{8} + 664224 T^{9} + 2950112 T^{10} + 5423952 T^{11} + 65647506 T^{12} + 965502760 T^{14} + 6975757441 T^{16} \))(\( 1 + 40 T^{2} + 786 T^{4} + 1104 T^{5} + 10208 T^{6} + 39072 T^{7} + 129731 T^{8} + 664224 T^{9} + 2950112 T^{10} + 5423952 T^{11} + 65647506 T^{12} + 965502760 T^{14} + 6975757441 T^{16} \))
$19$ (\( 1 + 6 T - 33 T^{2} - 150 T^{3} + 1165 T^{4} + 1968 T^{5} - 34182 T^{6} - 11868 T^{7} + 759066 T^{8} - 225492 T^{9} - 12339702 T^{10} + 13498512 T^{11} + 151823965 T^{12} - 371414850 T^{13} - 1552514073 T^{14} + 5363230434 T^{15} + 16983563041 T^{16} \))(\( 1 - 6 T - 33 T^{2} + 150 T^{3} + 1165 T^{4} - 1968 T^{5} - 34182 T^{6} + 11868 T^{7} + 759066 T^{8} + 225492 T^{9} - 12339702 T^{10} - 13498512 T^{11} + 151823965 T^{12} + 371414850 T^{13} - 1552514073 T^{14} - 5363230434 T^{15} + 16983563041 T^{16} \))
$23$ (\( 1 + 52 T^{2} + 1242 T^{4} + 2784 T^{5} + 24080 T^{6} + 140352 T^{7} + 497843 T^{8} + 3228096 T^{9} + 12738320 T^{10} + 33872928 T^{11} + 347562522 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} \))(\( 1 + 52 T^{2} + 1242 T^{4} - 2784 T^{5} + 24080 T^{6} - 140352 T^{7} + 497843 T^{8} - 3228096 T^{9} + 12738320 T^{10} - 33872928 T^{11} + 347562522 T^{12} + 7697866228 T^{14} + 78310985281 T^{16} \))
$29$ (\( ( 1 + 8 T + 71 T^{2} + 344 T^{3} + 1924 T^{4} + 9976 T^{5} + 59711 T^{6} + 195112 T^{7} + 707281 T^{8} )^{2} \))(\( ( 1 + 8 T + 71 T^{2} + 344 T^{3} + 1924 T^{4} + 9976 T^{5} + 59711 T^{6} + 195112 T^{7} + 707281 T^{8} )^{2} \))
$31$ (\( 1 - 6 T - 4 T^{2} + 336 T^{3} - 2729 T^{4} + 10764 T^{5} + 2216 T^{6} - 444234 T^{7} + 3877768 T^{8} - 13771254 T^{9} + 2129576 T^{10} + 320670324 T^{11} - 2520288809 T^{12} + 9619394736 T^{13} - 3550014724 T^{14} - 165075684666 T^{15} + 852891037441 T^{16} \))(\( 1 + 6 T - 4 T^{2} - 336 T^{3} - 2729 T^{4} - 10764 T^{5} + 2216 T^{6} + 444234 T^{7} + 3877768 T^{8} + 13771254 T^{9} + 2129576 T^{10} - 320670324 T^{11} - 2520288809 T^{12} - 9619394736 T^{13} - 3550014724 T^{14} + 165075684666 T^{15} + 852891037441 T^{16} \))
$37$ (\( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 5408808 T^{9} - 208890234 T^{10} + 495386340 T^{11} + 8907887233 T^{12} - 1248191226 T^{13} - 177035122221 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))(\( 1 - 6 T - 69 T^{2} - 18 T^{3} + 4753 T^{4} + 9780 T^{5} - 152586 T^{6} - 146184 T^{7} + 2893194 T^{8} - 5408808 T^{9} - 208890234 T^{10} + 495386340 T^{11} + 8907887233 T^{12} - 1248191226 T^{13} - 177035122221 T^{14} - 569591262798 T^{15} + 3512479453921 T^{16} \))
$41$ (\( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} \))(\( 1 - 120 T^{2} + 9948 T^{4} - 607176 T^{6} + 27583238 T^{8} - 1020662856 T^{10} + 28110670428 T^{12} - 570012508920 T^{14} + 7984925229121 T^{16} \))
$43$ (\( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16} \))(\( 1 - 210 T^{2} + 23793 T^{4} - 1729746 T^{6} + 88400276 T^{8} - 3198300354 T^{10} + 81343532193 T^{12} - 1327486240290 T^{14} + 11688200277601 T^{16} \))
$47$ (\( 1 - 4 T - 144 T^{2} + 456 T^{3} + 12722 T^{4} - 27948 T^{5} - 805600 T^{6} + 556940 T^{7} + 41968563 T^{8} + 26176180 T^{9} - 1779570400 T^{10} - 2901645204 T^{11} + 62079301682 T^{12} + 104581323192 T^{13} - 1552207007376 T^{14} - 2026492481852 T^{15} + 23811286661761 T^{16} \))(\( 1 + 4 T - 144 T^{2} - 456 T^{3} + 12722 T^{4} + 27948 T^{5} - 805600 T^{6} - 556940 T^{7} + 41968563 T^{8} - 26176180 T^{9} - 1779570400 T^{10} + 2901645204 T^{11} + 62079301682 T^{12} - 104581323192 T^{13} - 1552207007376 T^{14} + 2026492481852 T^{15} + 23811286661761 T^{16} \))
$53$ (\( 1 + 4 T - 135 T^{2} - 900 T^{3} + 9413 T^{4} + 70368 T^{5} - 312982 T^{6} - 1938608 T^{7} + 10598262 T^{8} - 102746224 T^{9} - 879166438 T^{10} + 10476176736 T^{11} + 74273097653 T^{12} - 376375943700 T^{13} - 2992188752415 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} \))(\( 1 + 4 T - 135 T^{2} - 900 T^{3} + 9413 T^{4} + 70368 T^{5} - 312982 T^{6} - 1938608 T^{7} + 10598262 T^{8} - 102746224 T^{9} - 879166438 T^{10} + 10476176736 T^{11} + 74273097653 T^{12} - 376375943700 T^{13} - 2992188752415 T^{14} + 4698844559348 T^{15} + 62259690411361 T^{16} \))
$59$ (\( 1 + 14 T - 13 T^{2} - 1110 T^{3} - 3463 T^{4} + 11848 T^{5} - 87914 T^{6} + 1416852 T^{7} + 32978194 T^{8} + 83594268 T^{9} - 306028634 T^{10} + 2433330392 T^{11} - 41962421143 T^{12} - 793565971890 T^{13} - 548346937333 T^{14} + 34841120787466 T^{15} + 146830437604321 T^{16} \))(\( 1 - 14 T - 13 T^{2} + 1110 T^{3} - 3463 T^{4} - 11848 T^{5} - 87914 T^{6} - 1416852 T^{7} + 32978194 T^{8} - 83594268 T^{9} - 306028634 T^{10} - 2433330392 T^{11} - 41962421143 T^{12} + 793565971890 T^{13} - 548346937333 T^{14} - 34841120787466 T^{15} + 146830437604321 T^{16} \))
$61$ (\( 1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 218182116 T^{9} + 37329072 T^{10} - 10636329660 T^{11} + 172048420266 T^{12} - 1337840540784 T^{13} + 9273667384980 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} \))(\( 1 - 12 T + 180 T^{2} - 1584 T^{3} + 12426 T^{4} - 46860 T^{5} + 10032 T^{6} + 3576756 T^{7} - 33274477 T^{8} + 218182116 T^{9} + 37329072 T^{10} - 10636329660 T^{11} + 172048420266 T^{12} - 1337840540784 T^{13} + 9273667384980 T^{14} - 37712914032252 T^{15} + 191707312997281 T^{16} \))
$67$ (\( 1 - 42 T + 1023 T^{2} - 18270 T^{3} + 261141 T^{4} - 3133152 T^{5} + 32837970 T^{6} - 307895844 T^{7} + 2631022010 T^{8} - 20629021548 T^{9} + 147409647330 T^{10} - 942336194976 T^{11} + 5262283889061 T^{12} - 24666785704890 T^{13} + 92538924958887 T^{14} - 254549887423566 T^{15} + 406067677556641 T^{16} \))(\( 1 + 42 T + 1023 T^{2} + 18270 T^{3} + 261141 T^{4} + 3133152 T^{5} + 32837970 T^{6} + 307895844 T^{7} + 2631022010 T^{8} + 20629021548 T^{9} + 147409647330 T^{10} + 942336194976 T^{11} + 5262283889061 T^{12} + 24666785704890 T^{13} + 92538924958887 T^{14} + 254549887423566 T^{15} + 406067677556641 T^{16} \))
$71$ (\( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16} \))(\( 1 - 288 T^{2} + 41724 T^{4} - 4336608 T^{6} + 350671046 T^{8} - 21860840928 T^{10} + 1060276978044 T^{12} - 36892881769248 T^{14} + 645753531245761 T^{16} \))
$73$ (\( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 2552644728 T^{9} + 23365309582 T^{10} + 180169333380 T^{11} + 1436979392841 T^{12} + 8918353993086 T^{13} + 52512976522283 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16} \))(\( 1 + 18 T + 347 T^{2} + 4302 T^{3} + 50601 T^{4} + 463140 T^{5} + 4384558 T^{6} + 34967736 T^{7} + 313616978 T^{8} + 2552644728 T^{9} + 23365309582 T^{10} + 180169333380 T^{11} + 1436979392841 T^{12} + 8918353993086 T^{13} + 52512976522283 T^{14} + 198853173343746 T^{15} + 806460091894081 T^{16} \))
$79$ (\( 1 + 6 T + 132 T^{2} + 720 T^{3} + 9999 T^{4} + 52644 T^{5} - 314880 T^{6} + 1496442 T^{7} - 48671848 T^{8} + 118218918 T^{9} - 1965166080 T^{10} + 25955545116 T^{11} + 389461859919 T^{12} + 2215480607280 T^{13} + 32087544128772 T^{14} + 115223453916954 T^{15} + 1517108809906561 T^{16} \))(\( 1 - 6 T + 132 T^{2} - 720 T^{3} + 9999 T^{4} - 52644 T^{5} - 314880 T^{6} - 1496442 T^{7} - 48671848 T^{8} - 118218918 T^{9} - 1965166080 T^{10} - 25955545116 T^{11} + 389461859919 T^{12} - 2215480607280 T^{13} + 32087544128772 T^{14} - 115223453916954 T^{15} + 1517108809906561 T^{16} \))
$83$ (\( ( 1 - 2 T + 229 T^{2} - 802 T^{3} + 24432 T^{4} - 66566 T^{5} + 1577581 T^{6} - 1143574 T^{7} + 47458321 T^{8} )^{2} \))(\( ( 1 + 2 T + 229 T^{2} + 802 T^{3} + 24432 T^{4} + 66566 T^{5} + 1577581 T^{6} + 1143574 T^{7} + 47458321 T^{8} )^{2} \))
$89$ (\( 1 + 264 T^{2} + 38610 T^{4} + 50400 T^{5} + 4052064 T^{6} + 8952768 T^{7} + 353995811 T^{8} + 796796352 T^{9} + 32096398944 T^{10} + 35530437600 T^{11} + 2422477925010 T^{12} + 131203060813704 T^{14} + 3936588805702081 T^{16} \))(\( 1 + 264 T^{2} + 38610 T^{4} + 50400 T^{5} + 4052064 T^{6} + 8952768 T^{7} + 353995811 T^{8} + 796796352 T^{9} + 32096398944 T^{10} + 35530437600 T^{11} + 2422477925010 T^{12} + 131203060813704 T^{14} + 3936588805702081 T^{16} \))
$97$ (\( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16} \))(\( 1 - 594 T^{2} + 165777 T^{4} - 28537554 T^{6} + 3324136868 T^{8} - 268509845586 T^{10} + 14676118616337 T^{12} - 494785370927826 T^{14} + 7837433594376961 T^{16} \))
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