Properties

Label 35.4.b
Level $35$
Weight $4$
Character orbit 35.b
Rep. character $\chi_{35}(29,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $1$
Sturm bound $16$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 35.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(16\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(35, [\chi])\).

Total New Old
Modular forms 14 10 4
Cusp forms 10 10 0
Eisenstein series 4 0 4

Trace form

\( 10q - 36q^{4} + 6q^{5} + 12q^{6} - 46q^{9} + O(q^{10}) \) \( 10q - 36q^{4} + 6q^{5} + 12q^{6} - 46q^{9} - 16q^{10} + 84q^{11} - 56q^{14} + 8q^{15} + 148q^{16} + 72q^{19} - 68q^{20} + 140q^{21} + 72q^{24} - 362q^{25} - 620q^{26} + 88q^{29} + 52q^{30} + 120q^{31} + 964q^{34} - 28q^{35} - 420q^{36} + 212q^{39} + 1396q^{40} - 852q^{41} - 1424q^{44} - 510q^{45} + 176q^{46} - 490q^{49} + 1644q^{50} + 1276q^{51} - 996q^{54} - 1136q^{55} + 588q^{56} + 864q^{59} - 1692q^{60} - 884q^{61} - 2724q^{64} + 520q^{65} + 1148q^{66} + 4808q^{69} + 756q^{70} + 880q^{71} + 3024q^{74} + 720q^{75} - 2672q^{76} - 3580q^{79} - 2316q^{80} - 302q^{81} - 1904q^{84} + 1572q^{85} + 2432q^{86} - 1492q^{89} - 7748q^{90} + 476q^{91} + 1652q^{94} - 1628q^{95} + 4080q^{96} - 5304q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(35, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
35.4.b.a \(10\) \(2.065\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(0\) \(6\) \(0\) \(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(-4+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 22 T^{2} + 277 T^{4} - 2288 T^{6} + 16804 T^{8} - 119360 T^{10} + 1075456 T^{12} - 9371648 T^{14} + 72613888 T^{16} - 369098752 T^{18} + 1073741824 T^{20} \)
$3$ \( 1 - 112 T^{2} + 6658 T^{4} - 276902 T^{6} + 9205601 T^{8} - 263150732 T^{10} + 6710883129 T^{12} - 147157075782 T^{14} + 2579445615762 T^{16} - 31632108085872 T^{18} + 205891132094649 T^{20} \)
$5$ \( 1 - 6 T + 199 T^{2} + 1836 T^{3} + 20 T^{4} + 520100 T^{5} + 2500 T^{6} + 28687500 T^{7} + 388671875 T^{8} - 1464843750 T^{9} + 30517578125 T^{10} \)
$7$ \( ( 1 + 49 T^{2} )^{5} \)
$11$ \( ( 1 - 42 T + 4384 T^{2} - 71864 T^{3} + 6091387 T^{4} - 31351772 T^{5} + 8107636097 T^{6} - 127311459704 T^{7} + 10337242677344 T^{8} - 131813991822282 T^{9} + 4177248169415651 T^{10} )^{2} \)
$13$ \( 1 - 5148 T^{2} + 15695890 T^{4} - 26456649834 T^{6} + 12726832447985 T^{8} + 6726945022008644 T^{10} + 61429989401426029865 T^{12} - \)\(61\!\cdots\!54\)\( T^{14} + \)\(17\!\cdots\!10\)\( T^{16} - \)\(27\!\cdots\!28\)\( T^{18} + \)\(26\!\cdots\!49\)\( T^{20} \)
$17$ \( 1 - 27808 T^{2} + 398887406 T^{4} - 3863758657518 T^{6} + 27870658319090257 T^{8} - \)\(15\!\cdots\!08\)\( T^{10} + \)\(67\!\cdots\!33\)\( T^{12} - \)\(22\!\cdots\!98\)\( T^{14} + \)\(56\!\cdots\!54\)\( T^{16} - \)\(94\!\cdots\!68\)\( T^{18} + \)\(81\!\cdots\!49\)\( T^{20} \)
$19$ \( ( 1 - 36 T + 16945 T^{2} - 428680 T^{3} + 139308340 T^{4} - 2473195688 T^{5} + 955515904060 T^{6} - 20167628267080 T^{7} + 5467943038865155 T^{8} - 79679337086381796 T^{9} + 15181127029874798299 T^{10} )^{2} \)
$23$ \( 1 - 45414 T^{2} + 1274694781 T^{4} - 26147824589064 T^{6} + 429674577520286642 T^{8} - \)\(57\!\cdots\!04\)\( T^{10} + \)\(63\!\cdots\!38\)\( T^{12} - \)\(57\!\cdots\!44\)\( T^{14} + \)\(41\!\cdots\!89\)\( T^{16} - \)\(21\!\cdots\!74\)\( T^{18} + \)\(71\!\cdots\!49\)\( T^{20} \)
$29$ \( ( 1 - 44 T + 5930 T^{2} + 1541470 T^{3} + 551498365 T^{4} + 1453032508 T^{5} + 13450493623985 T^{6} + 916902304621870 T^{7} + 86027375636903170 T^{8} - 15567850461040637804 T^{9} + \)\(86\!\cdots\!49\)\( T^{10} )^{2} \)
$31$ \( ( 1 - 60 T + 93703 T^{2} - 4540464 T^{3} + 4639455974 T^{4} - 198463065128 T^{5} + 138214032921434 T^{6} - 4029678513447984 T^{7} + 2477471915321354713 T^{8} - 47259767027312985660 T^{9} + \)\(23\!\cdots\!51\)\( T^{10} )^{2} \)
$37$ \( 1 - 341874 T^{2} + 58168168629 T^{4} - 6417971253293144 T^{6} + \)\(50\!\cdots\!78\)\( T^{8} - \)\(29\!\cdots\!04\)\( T^{10} + \)\(12\!\cdots\!02\)\( T^{12} - \)\(42\!\cdots\!64\)\( T^{14} + \)\(98\!\cdots\!41\)\( T^{16} - \)\(14\!\cdots\!14\)\( T^{18} + \)\(11\!\cdots\!49\)\( T^{20} \)
$41$ \( ( 1 + 426 T + 295309 T^{2} + 96639912 T^{3} + 38701723370 T^{4} + 9383799279644 T^{5} + 2667361476383770 T^{6} + 459049655841066792 T^{7} + 96678831663946228949 T^{8} + \)\(96\!\cdots\!06\)\( T^{9} + \)\(15\!\cdots\!01\)\( T^{10} )^{2} \)
$43$ \( 1 - 571726 T^{2} + 155217037109 T^{4} - 26607197126991720 T^{6} + \)\(32\!\cdots\!58\)\( T^{8} - \)\(29\!\cdots\!88\)\( T^{10} + \)\(20\!\cdots\!42\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{14} + \)\(39\!\cdots\!41\)\( T^{16} - \)\(91\!\cdots\!26\)\( T^{18} + \)\(10\!\cdots\!49\)\( T^{20} \)
$47$ \( 1 - 790788 T^{2} + 293544900286 T^{4} - 68085993582616986 T^{6} + \)\(11\!\cdots\!37\)\( T^{8} - \)\(13\!\cdots\!72\)\( T^{10} + \)\(11\!\cdots\!73\)\( T^{12} - \)\(79\!\cdots\!26\)\( T^{14} + \)\(36\!\cdots\!54\)\( T^{16} - \)\(10\!\cdots\!28\)\( T^{18} + \)\(14\!\cdots\!49\)\( T^{20} \)
$53$ \( 1 - 972802 T^{2} + 426448129813 T^{4} - 113796393195924760 T^{6} + \)\(21\!\cdots\!46\)\( T^{8} - \)\(34\!\cdots\!56\)\( T^{10} + \)\(48\!\cdots\!34\)\( T^{12} - \)\(55\!\cdots\!60\)\( T^{14} + \)\(46\!\cdots\!57\)\( T^{16} - \)\(23\!\cdots\!62\)\( T^{18} + \)\(53\!\cdots\!49\)\( T^{20} \)
$59$ \( ( 1 - 432 T + 700701 T^{2} - 268108032 T^{3} + 242314293152 T^{4} - 78692992197632 T^{5} + 49766267213264608 T^{6} - 11308939863198304512 T^{7} + \)\(60\!\cdots\!39\)\( T^{8} - \)\(76\!\cdots\!92\)\( T^{9} + \)\(36\!\cdots\!99\)\( T^{10} )^{2} \)
$61$ \( ( 1 + 442 T + 798959 T^{2} + 288407548 T^{3} + 306199000900 T^{4} + 84731818304260 T^{5} + 69501355423282900 T^{6} + 14858864841498076828 T^{7} + \)\(93\!\cdots\!19\)\( T^{8} + \)\(11\!\cdots\!82\)\( T^{9} + \)\(60\!\cdots\!01\)\( T^{10} )^{2} \)
$67$ \( 1 - 1699398 T^{2} + 1483840354293 T^{4} - 859383478287164840 T^{6} + \)\(36\!\cdots\!46\)\( T^{8} - \)\(12\!\cdots\!24\)\( T^{10} + \)\(33\!\cdots\!74\)\( T^{12} - \)\(70\!\cdots\!40\)\( T^{14} + \)\(10\!\cdots\!37\)\( T^{16} - \)\(11\!\cdots\!58\)\( T^{18} + \)\(60\!\cdots\!49\)\( T^{20} \)
$71$ \( ( 1 - 440 T + 1467407 T^{2} - 492789840 T^{3} + 952423975862 T^{4} - 247789000795120 T^{5} + 340883017624744282 T^{6} - 63126518417384162640 T^{7} + \)\(67\!\cdots\!17\)\( T^{8} - \)\(72\!\cdots\!40\)\( T^{9} + \)\(58\!\cdots\!51\)\( T^{10} )^{2} \)
$73$ \( 1 - 2526314 T^{2} + 3202004346045 T^{4} - 2657464253457184440 T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(70\!\cdots\!52\)\( T^{10} + \)\(24\!\cdots\!90\)\( T^{12} - \)\(60\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!05\)\( T^{16} - \)\(13\!\cdots\!74\)\( T^{18} + \)\(79\!\cdots\!49\)\( T^{20} \)
$79$ \( ( 1 + 1790 T + 3366800 T^{2} + 3670804696 T^{3} + 3768250665575 T^{4} + 2749245121712708 T^{5} + 1857894539904432425 T^{6} + \)\(89\!\cdots\!16\)\( T^{7} + \)\(40\!\cdots\!00\)\( T^{8} + \)\(10\!\cdots\!90\)\( T^{9} + \)\(29\!\cdots\!99\)\( T^{10} )^{2} \)
$83$ \( 1 - 4190178 T^{2} + 8255172061705 T^{4} - 10209824219182076768 T^{6} + \)\(89\!\cdots\!30\)\( T^{8} - \)\(58\!\cdots\!68\)\( T^{10} + \)\(29\!\cdots\!70\)\( T^{12} - \)\(10\!\cdots\!48\)\( T^{14} + \)\(28\!\cdots\!45\)\( T^{16} - \)\(47\!\cdots\!38\)\( T^{18} + \)\(37\!\cdots\!49\)\( T^{20} \)
$89$ \( ( 1 + 746 T + 1733225 T^{2} - 126840160 T^{3} + 408516498430 T^{4} - 989188619900372 T^{5} + 287991467381698670 T^{6} - 63037186462499793760 T^{7} + \)\(60\!\cdots\!25\)\( T^{8} + \)\(18\!\cdots\!66\)\( T^{9} + \)\(17\!\cdots\!49\)\( T^{10} )^{2} \)
$97$ \( 1 - 7632832 T^{2} + 27232168586510 T^{4} - 59864233929174443022 T^{6} + \)\(89\!\cdots\!65\)\( T^{8} - \)\(96\!\cdots\!52\)\( T^{10} + \)\(74\!\cdots\!85\)\( T^{12} - \)\(41\!\cdots\!02\)\( T^{14} + \)\(15\!\cdots\!90\)\( T^{16} - \)\(36\!\cdots\!92\)\( T^{18} + \)\(40\!\cdots\!49\)\( T^{20} \)
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