# 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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