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

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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

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