# Properties

 Label 13.8.a Level 13 Weight 8 Character orbit a Rep. character $$\chi_{13}(1,\cdot)$$ Character field $$\Q$$ Dimension 7 Newform subspaces 3 Sturm bound 9 Trace bound 1

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$13$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 13.a (trivial) Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$9$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{8}(\Gamma_0(13))$$.

Total New Old
Modular forms 9 7 2
Cusp forms 7 7 0
Eisenstein series 2 0 2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$13$$Dim.
$$+$$$$4$$
$$-$$$$3$$

## Trace form

 $$7q + 6q^{2} + 52q^{3} + 318q^{4} - 390q^{5} - 84q^{6} + 1056q^{7} - 1320q^{8} + 4791q^{9} + O(q^{10})$$ $$7q + 6q^{2} + 52q^{3} + 318q^{4} - 390q^{5} - 84q^{6} + 1056q^{7} - 1320q^{8} + 4791q^{9} - 2238q^{10} - 7620q^{11} + 10086q^{12} - 2197q^{13} + 13014q^{14} - 21704q^{15} - 27694q^{16} - 17694q^{17} + 23462q^{18} + 46580q^{19} - 73740q^{20} - 70952q^{21} + 105472q^{22} + 128712q^{23} + 53196q^{24} + 154757q^{25} - 52728q^{26} + 33400q^{27} - 273952q^{28} + 34218q^{29} - 150142q^{30} - 316488q^{31} - 250704q^{32} + 169144q^{33} + 56716q^{34} + 255168q^{35} + 100504q^{36} + 684986q^{37} - 1099260q^{38} - 237276q^{39} + 1315346q^{40} + 843054q^{41} - 10158q^{42} + 5052q^{43} - 590604q^{44} - 2624614q^{45} - 496316q^{46} - 1610664q^{47} + 1839262q^{48} + 678811q^{49} + 5871426q^{50} - 3096904q^{51} - 413036q^{52} + 2453442q^{53} + 1043412q^{54} - 42296q^{55} + 760050q^{56} - 2563680q^{57} - 4502440q^{58} + 399996q^{59} - 11818512q^{60} + 5212914q^{61} + 1682232q^{62} + 10585472q^{63} - 9723598q^{64} - 1990482q^{65} + 19895328q^{66} - 6391324q^{67} - 2959782q^{68} + 4028200q^{69} - 13184060q^{70} - 4184352q^{71} - 4021200q^{72} - 17699298q^{73} + 18282570q^{74} + 16262716q^{75} + 27841480q^{76} - 14950008q^{77} - 7122674q^{78} + 18741312q^{79} + 4873068q^{80} - 10908753q^{81} + 2920092q^{82} - 13691388q^{83} - 21686852q^{84} - 6734372q^{85} - 31170660q^{86} + 30644360q^{87} + 21699000q^{88} + 33292110q^{89} - 50053200q^{90} - 5114616q^{91} + 35518716q^{92} - 9696336q^{93} + 5824414q^{94} + 1053576q^{95} - 36116084q^{96} - 25966114q^{97} - 12157482q^{98} - 30302100q^{99} + O(q^{100})$$

## Decomposition of $$S_{8}^{\mathrm{new}}(\Gamma_0(13))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 13
13.8.a.a $$1$$ $$4.061$$ $$\Q$$ None $$10$$ $$-73$$ $$-295$$ $$1373$$ $$-$$ $$q+10q^{2}-73q^{3}-28q^{4}-295q^{5}+\cdots$$
13.8.a.b $$2$$ $$4.061$$ $$\Q(\sqrt{337})$$ None $$-19$$ $$45$$ $$-353$$ $$-2009$$ $$-$$ $$q+(-9-\beta )q^{2}+(21+3\beta )q^{3}+(37+19\beta )q^{4}+\cdots$$
13.8.a.c $$4$$ $$4.061$$ $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ None $$15$$ $$80$$ $$258$$ $$1692$$ $$+$$ $$q+(4-\beta _{1})q^{2}+(21-2\beta _{1}+\beta _{2})q^{3}+\cdots$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 - 10 T + 128 T^{2}$$)($$1 + 19 T + 262 T^{2} + 2432 T^{3} + 16384 T^{4}$$)($$1 - 15 T + 242 T^{2} - 2496 T^{3} + 42064 T^{4} - 319488 T^{5} + 3964928 T^{6} - 31457280 T^{7} + 268435456 T^{8}$$)
$3$ ($$1 + 73 T + 2187 T^{2}$$)($$1 - 45 T + 4122 T^{2} - 98415 T^{3} + 4782969 T^{4}$$)($$1 - 80 T + 5827 T^{2} - 295440 T^{3} + 17561808 T^{4} - 646127280 T^{5} + 27870360363 T^{6} - 836828256240 T^{7} + 22876792454961 T^{8}$$)
$5$ ($$1 + 295 T + 78125 T^{2}$$)($$1 + 353 T + 177208 T^{2} + 27578125 T^{3} + 6103515625 T^{4}$$)($$1 - 258 T + 79825 T^{2} + 26674950 T^{3} - 6698488500 T^{4} + 2083980468750 T^{5} + 487213134765625 T^{6} - 123023986816406250 T^{7} + 37252902984619140625 T^{8}$$)
$7$ ($$1 - 1373 T + 823543 T^{2}$$)($$1 + 2009 T + 2649282 T^{2} + 1654497887 T^{3} + 678223072849 T^{4}$$)($$1 - 1692 T + 3462207 T^{2} - 3922575624 T^{3} + 4347732494636 T^{4} - 3230409697115832 T^{5} + 2348148670379317743 T^{6} -$$$$94\!\cdots\!44$$$$T^{7} +$$$$45\!\cdots\!01$$$$T^{8}$$)
$11$ ($$1 + 7646 T + 19487171 T^{2}$$)($$1 + 1810 T + 7198390 T^{2} + 35271779510 T^{3} + 379749833583241 T^{4}$$)($$1 - 1836 T + 55670940 T^{2} - 77527625436 T^{3} + 1430012052531782 T^{4} - 1510794094095281556 T^{5} +$$$$21\!\cdots\!40$$$$T^{6} -$$$$13\!\cdots\!96$$$$T^{7} +$$$$14\!\cdots\!81$$$$T^{8}$$)
$13$ ($$1 - 2197 T$$)($$( 1 - 2197 T )^{2}$$)($$( 1 + 2197 T )^{4}$$)
$17$ ($$1 + 4147 T + 410338673 T^{2}$$)($$1 + 25361 T + 643460668 T^{2} + 10406599085953 T^{3} + 168377826559400929 T^{4}$$)($$1 - 11814 T + 685788809 T^{2} - 12346679658534 T^{3} + 400824270251066740 T^{4} -$$$$50\!\cdots\!82$$$$T^{5} +$$$$11\!\cdots\!61$$$$T^{6} -$$$$81\!\cdots\!38$$$$T^{7} +$$$$28\!\cdots\!41$$$$T^{8}$$)
$19$ ($$1 + 3186 T + 893871739 T^{2}$$)($$1 - 22106 T + 141106790 T^{2} - 19759928662334 T^{3} + 799006685782884121 T^{4}$$)($$1 - 27660 T + 1718678892 T^{2} - 32486632008348 T^{3} + 1911429179618741894 T^{4} -$$$$29\!\cdots\!72$$$$T^{5} +$$$$13\!\cdots\!32$$$$T^{6} -$$$$19\!\cdots\!40$$$$T^{7} +$$$$63\!\cdots\!41$$$$T^{8}$$)
$23$ ($$1 + 17784 T + 3404825447 T^{2}$$)($$1 + 26424 T + 3097648846 T^{2} + 89969107611528 T^{3} + 11592836324538749809 T^{4}$$)($$1 - 172920 T + 23964752172 T^{2} - 2015760149818584 T^{3} +$$$$14\!\cdots\!86$$$$T^{4} -$$$$68\!\cdots\!48$$$$T^{5} +$$$$27\!\cdots\!48$$$$T^{6} -$$$$68\!\cdots\!60$$$$T^{7} +$$$$13\!\cdots\!81$$$$T^{8}$$)
$29$ ($$1 + 93322 T + 17249876309 T^{2}$$)($$1 + 5804 T + 33449999614 T^{2} + 100118282097436 T^{3} +$$$$29\!\cdots\!81$$$$T^{4}$$)($$1 - 133344 T + 26660421180 T^{2} - 3188251722846816 T^{3} +$$$$29\!\cdots\!38$$$$T^{4} -$$$$54\!\cdots\!44$$$$T^{5} +$$$$79\!\cdots\!80$$$$T^{6} -$$$$68\!\cdots\!76$$$$T^{7} +$$$$88\!\cdots\!61$$$$T^{8}$$)
$31$ ($$1 + 124484 T + 27512614111 T^{2}$$)($$1 - 39744 T + 54390419758 T^{2} - 1093461335227584 T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 + 231748 T + 63192780624 T^{2} + 4752659145995828 T^{3} +$$$$11\!\cdots\!78$$$$T^{4} +$$$$13\!\cdots\!08$$$$T^{5} +$$$$47\!\cdots\!04$$$$T^{6} +$$$$48\!\cdots\!88$$$$T^{7} +$$$$57\!\cdots\!41$$$$T^{8}$$)
$37$ ($$1 - 273661 T + 94931877133 T^{2}$$)($$1 - 163299 T + 181994892160 T^{2} - 15502280603941767 T^{3} +$$$$90\!\cdots\!89$$$$T^{4}$$)($$1 - 248026 T + 189788542425 T^{2} - 80359538860800338 T^{3} +$$$$18\!\cdots\!48$$$$T^{4} -$$$$76\!\cdots\!54$$$$T^{5} +$$$$17\!\cdots\!25$$$$T^{6} -$$$$21\!\cdots\!62$$$$T^{7} +$$$$81\!\cdots\!21$$$$T^{8}$$)
$41$ ($$1 - 585816 T + 194754273881 T^{2}$$)($$1 + 330870 T + 274020654610 T^{2} + 64438346599006470 T^{3} +$$$$37\!\cdots\!61$$$$T^{4}$$)($$1 - 588108 T + 893291225384 T^{2} - 351834276353141028 T^{3} +$$$$27\!\cdots\!22$$$$T^{4} -$$$$68\!\cdots\!68$$$$T^{5} +$$$$33\!\cdots\!24$$$$T^{6} -$$$$43\!\cdots\!28$$$$T^{7} +$$$$14\!\cdots\!21$$$$T^{8}$$)
$43$ ($$1 + 533559 T + 271818611107 T^{2}$$)($$1 - 229307 T + 555183291698 T^{2} - 62329910257112849 T^{3} +$$$$73\!\cdots\!49$$$$T^{4}$$)($$1 - 309304 T + 330901386963 T^{2} - 96124591764168032 T^{3} +$$$$43\!\cdots\!36$$$$T^{4} -$$$$26\!\cdots\!24$$$$T^{5} +$$$$24\!\cdots\!87$$$$T^{6} -$$$$62\!\cdots\!72$$$$T^{7} +$$$$54\!\cdots\!01$$$$T^{8}$$)
$47$ ($$1 + 530055 T + 506623120463 T^{2}$$)($$1 + 1638525 T + 1682936215978 T^{2} + 830114648456637075 T^{3} +$$$$25\!\cdots\!69$$$$T^{4}$$)($$1 - 557916 T + 1395712852759 T^{2} - 547681742284229736 T^{3} +$$$$88\!\cdots\!36$$$$T^{4} -$$$$27\!\cdots\!68$$$$T^{5} +$$$$35\!\cdots\!71$$$$T^{6} -$$$$72\!\cdots\!52$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8}$$)
$53$ ($$1 + 615288 T + 1174711139837 T^{2}$$)($$1 - 1046382 T + 1701182112730 T^{2} - 1229196591924919734 T^{3} +$$$$13\!\cdots\!69$$$$T^{4}$$)($$1 - 2022348 T + 5983020325800 T^{2} - 7336956290945884116 T^{3} +$$$$11\!\cdots\!98$$$$T^{4} -$$$$86\!\cdots\!92$$$$T^{5} +$$$$82\!\cdots\!00$$$$T^{6} -$$$$32\!\cdots\!44$$$$T^{7} +$$$$19\!\cdots\!61$$$$T^{8}$$)
$59$ ($$1 + 392514 T + 2488651484819 T^{2}$$)($$1 + 370158 T + 4837245961366 T^{2} + 921194256317631402 T^{3} +$$$$61\!\cdots\!61$$$$T^{4}$$)($$1 - 1162668 T + 6954372243836 T^{2} - 4790410082035974684 T^{3} +$$$$21\!\cdots\!26$$$$T^{4} -$$$$11\!\cdots\!96$$$$T^{5} +$$$$43\!\cdots\!96$$$$T^{6} -$$$$17\!\cdots\!12$$$$T^{7} +$$$$38\!\cdots\!21$$$$T^{8}$$)
$61$ ($$1 - 1878064 T + 3142742836021 T^{2}$$)($$1 - 4675422 T + 11232926947738 T^{2} - 14693648995874975862 T^{3} +$$$$98\!\cdots\!41$$$$T^{4}$$)($$1 + 1340572 T + 5788604730840 T^{2} + 8531245335271375364 T^{3} +$$$$28\!\cdots\!66$$$$T^{4} +$$$$26\!\cdots\!44$$$$T^{5} +$$$$57\!\cdots\!40$$$$T^{6} +$$$$41\!\cdots\!92$$$$T^{7} +$$$$97\!\cdots\!81$$$$T^{8}$$)
$67$ ($$1 + 3971438 T + 6060711605323 T^{2}$$)($$1 + 1821402 T + 6715905784390 T^{2} + 11038992239358522846 T^{3} +$$$$36\!\cdots\!29$$$$T^{4}$$)($$1 + 598484 T + 21069380047692 T^{2} + 12537255623338176004 T^{3} +$$$$18\!\cdots\!66$$$$T^{4} +$$$$75\!\cdots\!92$$$$T^{5} +$$$$77\!\cdots\!68$$$$T^{6} +$$$$13\!\cdots\!28$$$$T^{7} +$$$$13\!\cdots\!41$$$$T^{8}$$)
$71$ ($$1 + 3746601 T + 9095120158391 T^{2}$$)($$1 + 1135611 T - 6597281929502 T^{2} + 10328518498190561901 T^{3} +$$$$82\!\cdots\!81$$$$T^{4}$$)($$1 - 697860 T + 16434885332247 T^{2} - 30387847818551415048 T^{3} +$$$$15\!\cdots\!40$$$$T^{4} -$$$$27\!\cdots\!68$$$$T^{5} +$$$$13\!\cdots\!07$$$$T^{6} -$$$$52\!\cdots\!60$$$$T^{7} +$$$$68\!\cdots\!61$$$$T^{8}$$)
$73$ ($$1 - 2485802 T + 11047398519097 T^{2}$$)($$1 + 6459284 T + 32251656236358 T^{2} + 71358284496026946548 T^{3} +$$$$12\!\cdots\!09$$$$T^{4}$$)($$1 + 13725816 T + 106583767781868 T^{2} +$$$$56\!\cdots\!12$$$$T^{3} +$$$$21\!\cdots\!98$$$$T^{4} +$$$$62\!\cdots\!64$$$$T^{5} +$$$$13\!\cdots\!12$$$$T^{6} +$$$$18\!\cdots\!68$$$$T^{7} +$$$$14\!\cdots\!81$$$$T^{8}$$)
$79$ ($$1 + 1264456 T + 19203908986159 T^{2}$$)($$1 + 73808 T + 15935358386334 T^{2} + 1417402114450423472 T^{3} +$$$$36\!\cdots\!81$$$$T^{4}$$)($$1 - 20079576 T + 216425311961340 T^{2} -$$$$15\!\cdots\!56$$$$T^{3} +$$$$78\!\cdots\!58$$$$T^{4} -$$$$29\!\cdots\!04$$$$T^{5} +$$$$79\!\cdots\!40$$$$T^{6} -$$$$14\!\cdots\!04$$$$T^{7} +$$$$13\!\cdots\!61$$$$T^{8}$$)
$83$ ($$1 - 434308 T + 27136050989627 T^{2}$$)($$1 + 12100972 T + 88634017455958 T^{2} +$$$$32\!\cdots\!44$$$$T^{3} +$$$$73\!\cdots\!29$$$$T^{4}$$)($$1 + 2024724 T + 54338356180784 T^{2} + 99688027705706087700 T^{3} +$$$$21\!\cdots\!54$$$$T^{4} +$$$$27\!\cdots\!00$$$$T^{5} +$$$$40\!\cdots\!36$$$$T^{6} +$$$$40\!\cdots\!92$$$$T^{7} +$$$$54\!\cdots\!41$$$$T^{8}$$)
$89$ ($$1 - 5830810 T + 44231334895529 T^{2}$$)($$1 - 9815060 T + 78068771423926 T^{2} -$$$$43\!\cdots\!40$$$$T^{3} +$$$$19\!\cdots\!41$$$$T^{4}$$)($$1 - 17646240 T + 247581740134684 T^{2} -$$$$22\!\cdots\!00$$$$T^{3} +$$$$17\!\cdots\!90$$$$T^{4} -$$$$98\!\cdots\!00$$$$T^{5} +$$$$48\!\cdots\!44$$$$T^{6} -$$$$15\!\cdots\!60$$$$T^{7} +$$$$38\!\cdots\!81$$$$T^{8}$$)
$97$ ($$1 + 2045330 T + 80798284478113 T^{2}$$)($$1 + 17591688 T + 206995518168430 T^{2} +$$$$14\!\cdots\!44$$$$T^{3} +$$$$65\!\cdots\!69$$$$T^{4}$$)($$1 + 6329096 T + 139241493305964 T^{2} -$$$$30\!\cdots\!00$$$$T^{3} +$$$$50\!\cdots\!66$$$$T^{4} -$$$$24\!\cdots\!00$$$$T^{5} +$$$$90\!\cdots\!16$$$$T^{6} +$$$$33\!\cdots\!12$$$$T^{7} +$$$$42\!\cdots\!61$$$$T^{8}$$)