Properties

Label 70.2.k
Level 70
Weight 2
Character orbit k
Rep. character \(\chi_{70}(3,\cdot)\)
Character field \(\Q(\zeta_{12})\)
Dimension 16
Newform subspaces 1
Sturm bound 24
Trace bound 0

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Defining parameters

Level: \( N \) = \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 70.k (of order \(12\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 35 \)
Character field: \(\Q(\zeta_{12})\)
Newform subspaces: \( 1 \)
Sturm bound: \(24\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(70, [\chi])\).

Total New Old
Modular forms 64 16 48
Cusp forms 32 16 16
Eisenstein series 32 0 32

Trace form

\( 16q - 12q^{5} + 8q^{7} + O(q^{10}) \) \( 16q - 12q^{5} + 8q^{7} - 12q^{10} - 12q^{11} + 16q^{15} + 8q^{16} - 36q^{17} - 8q^{18} - 28q^{21} - 8q^{22} - 4q^{23} + 12q^{25} + 12q^{26} + 4q^{28} + 20q^{30} + 24q^{31} + 48q^{33} + 8q^{35} - 8q^{36} + 4q^{37} + 24q^{38} + 36q^{42} - 8q^{43} - 12q^{45} - 8q^{46} + 12q^{47} - 32q^{50} - 16q^{51} - 28q^{53} - 4q^{56} + 8q^{57} - 32q^{58} + 8q^{60} - 12q^{61} - 36q^{63} - 8q^{65} + 32q^{67} - 36q^{68} - 12q^{70} + 16q^{71} - 8q^{72} - 12q^{73} - 48q^{75} + 16q^{77} + 16q^{78} - 12q^{80} - 48q^{82} + 24q^{85} + 12q^{86} - 24q^{87} - 4q^{88} - 16q^{91} + 8q^{92} + 28q^{93} + 20q^{95} + 12q^{96} + 40q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(70, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
70.2.k.a \(16\) \(0.559\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(-12\) \(8\) \(q+(-\beta _{7}-\beta _{15})q^{2}+(-\beta _{4}+\beta _{13})q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(70, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(70, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{4} + T^{8} )^{2} \)
$3$ \( 1 + 3 T^{4} - 24 T^{5} + 48 T^{7} + 5 T^{8} - 72 T^{9} + 288 T^{10} + 120 T^{11} - 570 T^{12} - 720 T^{13} + 2016 T^{14} - 2136 T^{15} - 2570 T^{16} - 6408 T^{17} + 18144 T^{18} - 19440 T^{19} - 46170 T^{20} + 29160 T^{21} + 209952 T^{22} - 157464 T^{23} + 32805 T^{24} + 944784 T^{25} - 4251528 T^{27} + 1594323 T^{28} + 43046721 T^{32} \)
$5$ \( 1 + 12 T + 66 T^{2} + 216 T^{3} + 450 T^{4} + 468 T^{5} - 864 T^{6} - 6252 T^{7} - 18241 T^{8} - 31260 T^{9} - 21600 T^{10} + 58500 T^{11} + 281250 T^{12} + 675000 T^{13} + 1031250 T^{14} + 937500 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 8 T + 32 T^{2} - 80 T^{3} + 73 T^{4} + 160 T^{5} - 416 T^{6} - 696 T^{7} + 4944 T^{8} - 4872 T^{9} - 20384 T^{10} + 54880 T^{11} + 175273 T^{12} - 1344560 T^{13} + 3764768 T^{14} - 6588344 T^{15} + 5764801 T^{16} \)
$11$ \( ( 1 + 6 T + 5 T^{2} - 18 T^{3} - 29 T^{4} - 252 T^{5} - 544 T^{6} - 516 T^{7} - 6650 T^{8} - 5676 T^{9} - 65824 T^{10} - 335412 T^{11} - 424589 T^{12} - 2898918 T^{13} + 8857805 T^{14} + 116923026 T^{15} + 214358881 T^{16} )^{2} \)
$13$ \( 1 + 714 T^{4} + 266977 T^{8} + 68653746 T^{12} + 13237862628 T^{16} + 1960819639506 T^{20} + 217781340700417 T^{24} + 16634832777451434 T^{28} + 665416609183179841 T^{32} \)
$17$ \( 1 + 36 T + 648 T^{2} + 7776 T^{3} + 70080 T^{4} + 507420 T^{5} + 3088368 T^{6} + 16357284 T^{7} + 77768530 T^{8} + 343436904 T^{9} + 1468660248 T^{10} + 6351048828 T^{11} + 28493424864 T^{12} + 131692528476 T^{13} + 608521516920 T^{14} + 2723900163624 T^{15} + 11569567164915 T^{16} + 46306302781608 T^{17} + 175862718389880 T^{18} + 647005392402588 T^{19} + 2379799338066144 T^{20} + 9017581135777596 T^{21} + 35449888073657112 T^{22} + 140925443446588392 T^{23} + 542494401823131730 T^{24} + 1939775574818354148 T^{25} + 6226131050341877232 T^{26} + 17390245624419136860 T^{27} + 40830166385061650880 T^{28} + 77017998783876566112 T^{29} + \)\(10\!\cdots\!92\)\( T^{30} + \)\(10\!\cdots\!48\)\( T^{31} + 48661191875666868481 T^{32} \)
$19$ \( 1 - 90 T^{2} + 4011 T^{4} - 123414 T^{6} + 3034057 T^{8} - 61998180 T^{10} + 1094493006 T^{12} - 18780034728 T^{14} + 344074548210 T^{16} - 6779592536808 T^{18} + 142635423034926 T^{20} - 2916758998496580 T^{22} + 51529098329487337 T^{24} - 756659411140252614 T^{26} + 8877606140374371771 T^{28} - 71910601720459570890 T^{30} + \)\(28\!\cdots\!81\)\( T^{32} \)
$23$ \( 1 + 4 T + 8 T^{2} - 128 T^{3} - 198 T^{4} - 292 T^{5} + 8608 T^{6} + 3564 T^{7} - 17335 T^{8} + 1017836 T^{9} + 5500848 T^{10} - 425132 T^{11} + 20115594 T^{12} + 929862264 T^{13} + 4560471720 T^{14} - 14572270428 T^{15} - 102219015884 T^{16} - 335162219844 T^{17} + 2412489539880 T^{18} + 11313634166088 T^{19} + 5629167940554 T^{20} - 2736295372276 T^{21} + 814322923933872 T^{22} + 3465553913672692 T^{23} - 1357520929846135 T^{24} + 6419308085454132 T^{25} + 356599408527090592 T^{26} - 278220449310866684 T^{27} - 4339095637540023558 T^{28} - 64516654327867825024 T^{29} + 92742690596309998472 T^{30} + \)\(10\!\cdots\!28\)\( T^{31} + \)\(61\!\cdots\!61\)\( T^{32} \)
$29$ \( ( 1 - 70 T^{2} + 2685 T^{4} - 80906 T^{6} + 2505752 T^{8} - 68041946 T^{10} + 1899049485 T^{12} - 41637632470 T^{14} + 500246412961 T^{16} )^{2} \)
$31$ \( ( 1 - 12 T + 156 T^{2} - 1296 T^{3} + 10686 T^{4} - 72996 T^{5} + 489600 T^{6} - 2919012 T^{7} + 17046947 T^{8} - 90489372 T^{9} + 470505600 T^{10} - 2174623836 T^{11} + 9868745406 T^{12} - 37103379696 T^{13} + 138450574236 T^{14} - 330151369332 T^{15} + 852891037441 T^{16} )^{2} \)
$37$ \( 1 - 4 T + 8 T^{2} + 240 T^{3} - 3049 T^{4} + 13544 T^{5} - 984 T^{6} - 514972 T^{7} + 3800209 T^{8} - 12256568 T^{9} - 16248928 T^{10} + 295990824 T^{11} + 119383314 T^{12} - 6433305488 T^{13} + 16607500784 T^{14} + 478842888760 T^{15} - 3828375485890 T^{16} + 17717186884120 T^{17} + 22735668573296 T^{18} - 325866222883664 T^{19} + 223743551149554 T^{20} + 20525174971850568 T^{21} - 41690303687539552 T^{22} - 1163539007448259544 T^{23} + 13348156033105669489 T^{24} - 66926657065750392844 T^{25} - 4731647022459163416 T^{26} + \)\(24\!\cdots\!72\)\( T^{27} - \)\(20\!\cdots\!69\)\( T^{28} + \)\(58\!\cdots\!80\)\( T^{29} + \)\(72\!\cdots\!12\)\( T^{30} - \)\(13\!\cdots\!72\)\( T^{31} + \)\(12\!\cdots\!41\)\( T^{32} \)
$41$ \( ( 1 - 188 T^{2} + 17802 T^{4} - 1146544 T^{6} + 54447827 T^{8} - 1927340464 T^{10} + 50304197322 T^{12} - 893019597308 T^{14} + 7984925229121 T^{16} )^{2} \)
$43$ \( ( 1 + 4 T + 8 T^{2} + 32 T^{3} + 3425 T^{4} + 20696 T^{5} + 55896 T^{6} + 651228 T^{7} + 6527152 T^{8} + 28002804 T^{9} + 103351704 T^{10} + 1645476872 T^{11} + 11709393425 T^{12} + 4704270176 T^{13} + 50570904392 T^{14} + 1087274444428 T^{15} + 11688200277601 T^{16} )^{2} \)
$47$ \( 1 - 12 T + 72 T^{2} - 288 T^{3} - 449 T^{4} + 2880 T^{5} + 39240 T^{6} - 500820 T^{7} + 44673 T^{8} + 37644384 T^{9} - 288946944 T^{10} + 1723474464 T^{11} + 9984376994 T^{12} - 179746348152 T^{13} + 1044056404368 T^{14} - 4096130163936 T^{15} + 8350203627806 T^{16} - 192518117704992 T^{17} + 2306320597248912 T^{18} - 18661805104185096 T^{19} + 48720574714458914 T^{20} + 395270263010401248 T^{21} - 3114621328032504576 T^{22} + 19071515289987429792 T^{23} + 1063721609040849153 T^{24} - \)\(56\!\cdots\!40\)\( T^{25} + \)\(20\!\cdots\!60\)\( T^{26} + \)\(71\!\cdots\!40\)\( T^{27} - \)\(52\!\cdots\!09\)\( T^{28} - \)\(15\!\cdots\!76\)\( T^{29} + \)\(18\!\cdots\!68\)\( T^{30} - \)\(14\!\cdots\!16\)\( T^{31} + \)\(56\!\cdots\!21\)\( T^{32} \)
$53$ \( 1 + 28 T + 392 T^{2} + 1808 T^{3} - 25321 T^{4} - 567704 T^{5} - 4335448 T^{6} + 1435172 T^{7} + 392677969 T^{8} + 4105212392 T^{9} + 14977011104 T^{10} - 124625519288 T^{11} - 2189877391278 T^{12} - 14281594759536 T^{13} - 14623477702416 T^{14} + 648224117593688 T^{15} + 7103959219709822 T^{16} + 34355878232465464 T^{17} - 41077348866086544 T^{18} - 2126200983015441072 T^{19} - 17279185948208624718 T^{20} - 52117830479026168984 T^{21} + \)\(33\!\cdots\!16\)\( T^{22} + \)\(48\!\cdots\!04\)\( T^{23} + \)\(24\!\cdots\!09\)\( T^{24} + \)\(47\!\cdots\!76\)\( T^{25} - \)\(75\!\cdots\!52\)\( T^{26} - \)\(52\!\cdots\!88\)\( T^{27} - \)\(12\!\cdots\!61\)\( T^{28} + \)\(47\!\cdots\!84\)\( T^{29} + \)\(54\!\cdots\!48\)\( T^{30} + \)\(20\!\cdots\!96\)\( T^{31} + \)\(38\!\cdots\!21\)\( T^{32} \)
$59$ \( 1 - 320 T^{2} + 54244 T^{4} - 6209920 T^{6} + 532238186 T^{8} - 36082424000 T^{10} + 2045329177232 T^{12} - 105925693186240 T^{14} + 5808829694261683 T^{16} - 368727337981301440 T^{18} + 24783992004353124752 T^{20} - \)\(15\!\cdots\!00\)\( T^{22} + \)\(78\!\cdots\!06\)\( T^{24} - \)\(31\!\cdots\!20\)\( T^{26} + \)\(96\!\cdots\!64\)\( T^{28} - \)\(19\!\cdots\!20\)\( T^{30} + \)\(21\!\cdots\!41\)\( T^{32} \)
$61$ \( ( 1 + 6 T + 207 T^{2} + 1170 T^{3} + 23079 T^{4} + 119796 T^{5} + 1969284 T^{6} + 9244740 T^{7} + 135493478 T^{8} + 563929140 T^{9} + 7327705764 T^{10} + 27191415876 T^{11} + 319548164439 T^{12} + 988177672170 T^{13} + 10664717492727 T^{14} + 18856457016126 T^{15} + 191707312997281 T^{16} )^{2} \)
$67$ \( 1 - 32 T + 512 T^{2} - 3896 T^{3} - 10509 T^{4} + 663104 T^{5} - 8249312 T^{6} + 44834240 T^{7} + 165963557 T^{8} - 5505231928 T^{9} + 49681319072 T^{10} - 159136747544 T^{11} - 1412717145882 T^{12} + 21627811976512 T^{13} - 103082665870656 T^{14} - 335467607852040 T^{15} + 7552610469365206 T^{16} - 22476329726086680 T^{17} - 462738087093374784 T^{18} + 6504845613491678656 T^{19} - 28467834145442833722 T^{20} - \)\(21\!\cdots\!08\)\( T^{21} + \)\(44\!\cdots\!68\)\( T^{22} - \)\(33\!\cdots\!44\)\( T^{23} + \)\(67\!\cdots\!37\)\( T^{24} + \)\(12\!\cdots\!80\)\( T^{25} - \)\(15\!\cdots\!88\)\( T^{26} + \)\(80\!\cdots\!32\)\( T^{27} - \)\(85\!\cdots\!49\)\( T^{28} - \)\(21\!\cdots\!52\)\( T^{29} + \)\(18\!\cdots\!48\)\( T^{30} - \)\(78\!\cdots\!76\)\( T^{31} + \)\(16\!\cdots\!81\)\( T^{32} \)
$71$ \( ( 1 - 4 T + 94 T^{2} + 964 T^{3} - 1158 T^{4} + 68444 T^{5} + 473854 T^{6} - 1431644 T^{7} + 25411681 T^{8} )^{4} \)
$73$ \( 1 + 12 T + 72 T^{2} + 288 T^{3} + 12064 T^{4} + 83700 T^{5} + 177264 T^{6} - 3613524 T^{7} + 19255890 T^{8} - 96865032 T^{9} - 2225167272 T^{10} - 39441388620 T^{11} + 6056513120 T^{12} - 1050679100940 T^{13} - 2551121523144 T^{14} - 71385024146568 T^{15} + 865631164724531 T^{16} - 5211106762699464 T^{17} - 13594926596834376 T^{18} - 408732031810375980 T^{19} + 171994319201421920 T^{20} - 81764822336595471660 T^{21} - \)\(33\!\cdots\!08\)\( T^{22} - \)\(10\!\cdots\!04\)\( T^{23} + \)\(15\!\cdots\!90\)\( T^{24} - \)\(21\!\cdots\!12\)\( T^{25} + \)\(76\!\cdots\!36\)\( T^{26} + \)\(26\!\cdots\!00\)\( T^{27} + \)\(27\!\cdots\!44\)\( T^{28} + \)\(48\!\cdots\!04\)\( T^{29} + \)\(87\!\cdots\!48\)\( T^{30} + \)\(10\!\cdots\!84\)\( T^{31} + \)\(65\!\cdots\!61\)\( T^{32} \)
$79$ \( 1 + 344 T^{2} + 56836 T^{4} + 6614224 T^{6} + 630756554 T^{8} + 45125337896 T^{10} + 2052833339792 T^{12} + 54520165812136 T^{14} + 1660868031271123 T^{16} + 340260354833540776 T^{18} + 79958024864398923152 T^{20} + \)\(10\!\cdots\!16\)\( T^{22} + \)\(95\!\cdots\!94\)\( T^{24} + \)\(62\!\cdots\!24\)\( T^{26} + \)\(33\!\cdots\!76\)\( T^{28} + \)\(12\!\cdots\!64\)\( T^{30} + \)\(23\!\cdots\!21\)\( T^{32} \)
$83$ \( 1 - 25638 T^{4} + 318852241 T^{8} - 2957648820318 T^{12} + 22747905510477444 T^{16} - \)\(14\!\cdots\!78\)\( T^{20} + \)\(71\!\cdots\!81\)\( T^{24} - \)\(27\!\cdots\!18\)\( T^{28} + \)\(50\!\cdots\!81\)\( T^{32} \)
$89$ \( 1 - 522 T^{2} + 144915 T^{4} - 27754278 T^{6} + 4091335849 T^{8} - 494840689332 T^{10} + 51768593923950 T^{12} - 4926383904774360 T^{14} + 445675116984224850 T^{16} - 39021886909717705560 T^{18} + \)\(32\!\cdots\!50\)\( T^{20} - \)\(24\!\cdots\!52\)\( T^{22} + \)\(16\!\cdots\!69\)\( T^{24} - \)\(86\!\cdots\!78\)\( T^{26} + \)\(35\!\cdots\!15\)\( T^{28} - \)\(10\!\cdots\!02\)\( T^{30} + \)\(15\!\cdots\!61\)\( T^{32} \)
$97$ \( ( 1 + 3868 T^{4} + 90482118 T^{8} + 342431258908 T^{12} + 7837433594376961 T^{16} )^{2} \)
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