# Properties

 Label 15.10.a Level 15 Weight 10 Character orbit a Rep. character $$\chi_{15}(1,\cdot)$$ Character field $$\Q$$ Dimension 6 Newform subspaces 4 Sturm bound 20 Trace bound 2

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 15.a (trivial) Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$20$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{10}(\Gamma_0(15))$$.

Total New Old
Modular forms 20 6 14
Cusp forms 16 6 10
Eisenstein series 4 0 4

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

$$3$$$$5$$FrickeDim.
$$+$$$$+$$$$+$$$$1$$
$$+$$$$-$$$$-$$$$2$$
$$-$$$$+$$$$-$$$$2$$
$$-$$$$-$$$$+$$$$1$$
Plus space$$+$$$$2$$
Minus space$$-$$$$4$$

## Trace form

 $$6q + 68q^{2} + 1538q^{4} - 3078q^{6} - 11428q^{7} + 67932q^{8} + 39366q^{9} + O(q^{10})$$ $$6q + 68q^{2} + 1538q^{4} - 3078q^{6} - 11428q^{7} + 67932q^{8} + 39366q^{9} - 8750q^{10} - 87076q^{11} + 41472q^{12} + 55888q^{13} + 115452q^{14} - 101250q^{15} + 1085330q^{16} - 420520q^{17} + 446148q^{18} + 658912q^{19} - 905000q^{20} - 2241756q^{21} - 7156540q^{22} + 3122496q^{23} + 3193506q^{24} + 2343750q^{25} - 5892868q^{26} + 3780524q^{28} - 6996536q^{29} - 1620000q^{30} - 4504264q^{31} + 17859412q^{32} - 1195236q^{33} + 5709836q^{34} + 15182500q^{35} + 10090818q^{36} + 25085112q^{37} - 2385976q^{38} - 5547852q^{39} - 4751250q^{40} + 24274636q^{41} - 44096724q^{42} - 4089728q^{43} - 102869396q^{44} - 90064944q^{46} + 51128264q^{47} + 65192688q^{48} + 84594310q^{49} + 26562500q^{50} + 62822628q^{51} - 52677360q^{52} - 114062224q^{53} - 20194758q^{54} - 80472500q^{55} - 68883420q^{56} - 2667816q^{57} - 205991852q^{58} + 180942428q^{59} - 130916250q^{60} + 479495740q^{61} - 31830936q^{62} - 74979108q^{63} + 464210442q^{64} - 192047500q^{65} - 57426084q^{66} - 484354360q^{67} + 717440536q^{68} + 5244912q^{69} + 543322500q^{70} + 83143192q^{71} + 445701852q^{72} - 591042044q^{73} - 773558108q^{74} - 849378912q^{76} + 72064224q^{77} + 265485600q^{78} - 274527880q^{79} + 103030000q^{80} + 258280326q^{81} + 1526220184q^{82} + 804884184q^{83} - 1559325492q^{84} - 192897500q^{85} - 752029576q^{86} + 37145628q^{87} - 2168339028q^{88} - 329760348q^{89} - 57408750q^{90} - 871611944q^{91} - 2807106528q^{92} - 830087352q^{93} + 3271510568q^{94} + 363665000q^{95} + 2378839914q^{96} + 1387556460q^{97} + 5935831732q^{98} - 571305636q^{99} + O(q^{100})$$

## Decomposition of $$S_{10}^{\mathrm{new}}(\Gamma_0(15))$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces A-L signs $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$ 3 5
15.10.a.a $$1$$ $$7.726$$ $$\Q$$ None $$-4$$ $$81$$ $$625$$ $$-7680$$ $$-$$ $$-$$ $$q-4q^{2}+3^{4}q^{3}-496q^{4}+5^{4}q^{5}+\cdots$$
15.10.a.b $$1$$ $$7.726$$ $$\Q$$ None $$22$$ $$-81$$ $$-625$$ $$-5988$$ $$+$$ $$+$$ $$q+22q^{2}-3^{4}q^{3}-28q^{4}-5^{4}q^{5}+\cdots$$
15.10.a.c $$2$$ $$7.726$$ $$\Q(\sqrt{4729})$$ None $$19$$ $$162$$ $$-1250$$ $$-11872$$ $$-$$ $$+$$ $$q+(10-\beta )q^{2}+3^{4}q^{3}+(770-19\beta )q^{4}+\cdots$$
15.10.a.d $$2$$ $$7.726$$ $$\Q(\sqrt{241})$$ None $$31$$ $$-162$$ $$1250$$ $$14112$$ $$+$$ $$-$$ $$q+(15-\beta )q^{2}-3^{4}q^{3}+(255-31\beta )q^{4}+\cdots$$

## Decomposition of $$S_{10}^{\mathrm{old}}(\Gamma_0(15))$$ into lower level spaces

$$S_{10}^{\mathrm{old}}(\Gamma_0(15)) \cong$$ $$S_{10}^{\mathrm{new}}(\Gamma_0(3))$$$$^{\oplus 2}$$$$\oplus$$$$S_{10}^{\mathrm{new}}(\Gamma_0(5))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ ($$1 + 4 T + 512 T^{2}$$)($$1 - 22 T + 512 T^{2}$$)($$1 - 19 T - 68 T^{2} - 9728 T^{3} + 262144 T^{4}$$)($$1 - 31 T + 722 T^{2} - 15872 T^{3} + 262144 T^{4}$$)
$3$ ($$1 - 81 T$$)($$1 + 81 T$$)($$( 1 - 81 T )^{2}$$)($$( 1 + 81 T )^{2}$$)
$5$ ($$1 - 625 T$$)($$1 + 625 T$$)($$( 1 + 625 T )^{2}$$)($$( 1 - 625 T )^{2}$$)
$7$ ($$1 + 7680 T + 40353607 T^{2}$$)($$1 + 5988 T + 40353607 T^{2}$$)($$1 + 11872 T + 112235774 T^{2} + 479078022304 T^{3} + 1628413597910449 T^{4}$$)($$1 - 14112 T + 103286414 T^{2} - 569470101984 T^{3} + 1628413597910449 T^{4}$$)
$11$ ($$1 + 86404 T + 2357947691 T^{2}$$)($$1 + 14648 T + 2357947691 T^{2}$$)($$1 - 35488 T + 526013014 T^{2} - 83678847658208 T^{3} + 5559917313492231481 T^{4}$$)($$1 + 21512 T + 1790961254 T^{2} + 50724170728792 T^{3} + 5559917313492231481 T^{4}$$)
$13$ ($$1 + 149978 T + 10604499373 T^{2}$$)($$1 - 37906 T + 10604499373 T^{2}$$)($$1 - 143676 T + 24105149134 T^{2} - 1523612051915148 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)($$1 - 24284 T + 5870669214 T^{2} - 257519662773932 T^{3} +$$$$11\!\cdots\!29$$$$T^{4}$$)
$17$ ($$1 + 207622 T + 118587876497 T^{2}$$)($$1 + 441098 T + 118587876497 T^{2}$$)($$1 - 385156 T + 268539949078 T^{2} - 45674832160078532 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)($$1 + 156956 T + 212670095078 T^{2} + 18613078743463132 T^{3} +$$$$14\!\cdots\!09$$$$T^{4}$$)
$19$ ($$1 - 716284 T + 322687697779 T^{2}$$)($$1 - 441820 T + 322687697779 T^{2}$$)($$1 + 403296 T + 684929514838 T^{2} + 130138657763479584 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)($$1 + 95896 T + 629883192438 T^{2} + 30944459466214984 T^{3} +$$$$10\!\cdots\!41$$$$T^{4}$$)
$23$ ($$1 - 1369920 T + 1801152661463 T^{2}$$)($$1 - 2264136 T + 1801152661463 T^{2}$$)($$1 - 223704 T - 1375273107794 T^{2} - 402925054979918952 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)($$1 + 735264 T + 3363187908526 T^{2} + 1324322710477931232 T^{3} +$$$$32\!\cdots\!69$$$$T^{4}$$)
$29$ ($$1 + 3194402 T + 14507145975869 T^{2}$$)($$1 + 1049350 T + 14507145975869 T^{2}$$)($$1 + 74572 T + 28833430018078 T^{2} + 1081826889712503068 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 + 2678212 T + 15397908029438 T^{2} + 38853212438324066228 T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)
$31$ ($$1 + 2349000 T + 26439622160671 T^{2}$$)($$1 + 7910568 T + 26439622160671 T^{2}$$)($$1 + 5027128 T + 52415931233342 T^{2} +$$$$13\!\cdots\!88$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)($$1 - 10782432 T + 69294691361342 T^{2} -$$$$28\!\cdots\!72$$$$T^{3} +$$$$69\!\cdots\!41$$$$T^{4}$$)
$37$ ($$1 - 18735710 T + 129961739795077 T^{2}$$)($$1 + 20992558 T + 129961739795077 T^{2}$$)($$1 - 5373628 T + 231061724951934 T^{2} -$$$$69\!\cdots\!56$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)($$1 - 21968332 T + 359210373327534 T^{2} -$$$$28\!\cdots\!64$$$$T^{3} +$$$$16\!\cdots\!29$$$$T^{4}$$)
$41$ ($$1 + 29282630 T + 327381934393961 T^{2}$$)($$1 - 13285562 T + 327381934393961 T^{2}$$)($$1 - 14211332 T + 443988635955862 T^{2} -$$$$46\!\cdots\!52$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)($$1 - 26060372 T + 693239183881142 T^{2} -$$$$85\!\cdots\!92$$$$T^{3} +$$$$10\!\cdots\!21$$$$T^{4}$$)
$43$ ($$1 + 1516724 T + 502592611936843 T^{2}$$)($$1 + 23130764 T + 502592611936843 T^{2}$$)($$1 - 27748920 T + 1170232974699430 T^{2} -$$$$13\!\cdots\!60$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)($$1 + 7191160 T + 750634586008230 T^{2} +$$$$36\!\cdots\!80$$$$T^{3} +$$$$25\!\cdots\!49$$$$T^{4}$$)
$47$ ($$1 - 615752 T + 1119130473102767 T^{2}$$)($$1 + 13873688 T + 1119130473102767 T^{2}$$)($$1 - 95966440 T + 4320659216802910 T^{2} -$$$$10\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)($$1 + 31580240 T + 382042606129310 T^{2} +$$$$35\!\cdots\!80$$$$T^{3} +$$$$12\!\cdots\!89$$$$T^{4}$$)
$53$ ($$1 - 4747430 T + 3299763591802133 T^{2}$$)($$1 + 57635174 T + 3299763591802133 T^{2}$$)($$1 + 64305596 T + 6611083028543086 T^{2} +$$$$21\!\cdots\!68$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)($$1 - 3131116 T + 6584849973489806 T^{2} -$$$$10\!\cdots\!28$$$$T^{3} +$$$$10\!\cdots\!89$$$$T^{4}$$)
$59$ ($$1 - 60616076 T + 8662995818654939 T^{2}$$)($$1 + 32042120 T + 8662995818654939 T^{2}$$)($$1 - 187863136 T + 23071633420288438 T^{2} -$$$$16\!\cdots\!04$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)($$1 + 35494664 T + 7388006896329158 T^{2} +$$$$30\!\cdots\!96$$$$T^{3} +$$$$75\!\cdots\!21$$$$T^{4}$$)
$61$ ($$1 + 126745682 T + 11694146092834141 T^{2}$$)($$1 - 110664022 T + 11694146092834141 T^{2}$$)($$1 - 154080060 T + 23302683905802238 T^{2} -$$$$18\!\cdots\!60$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)($$1 - 341497340 T + 52053805546777278 T^{2} -$$$$39\!\cdots\!40$$$$T^{3} +$$$$13\!\cdots\!81$$$$T^{4}$$)
$67$ ($$1 + 111182652 T + 27206534396294947 T^{2}$$)($$1 + 118568268 T + 27206534396294947 T^{2}$$)($$1 - 33592376 T - 10819815556424362 T^{2} -$$$$91\!\cdots\!72$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)($$1 + 288195816 T + 74605839041196758 T^{2} +$$$$78\!\cdots\!52$$$$T^{3} +$$$$74\!\cdots\!09$$$$T^{4}$$)
$71$ ($$1 + 175551608 T + 45848500718449031 T^{2}$$)($$1 - 276679712 T + 45848500718449031 T^{2}$$)($$1 + 228270976 T + 45777616900481806 T^{2} +$$$$10\!\cdots\!56$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)($$1 - 210286064 T + 91549406631588686 T^{2} -$$$$96\!\cdots\!84$$$$T^{3} +$$$$21\!\cdots\!61$$$$T^{4}$$)
$73$ ($$1 + 61233350 T + 58871586708267913 T^{2}$$)($$1 + 264023294 T + 58871586708267913 T^{2}$$)($$1 + 33122316 T + 68371107952007926 T^{2} +$$$$19\!\cdots\!08$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)($$1 + 232663084 T + 130536207391012086 T^{2} +$$$$13\!\cdots\!92$$$$T^{3} +$$$$34\!\cdots\!69$$$$T^{4}$$)
$79$ ($$1 - 234431160 T + 119851595982618319 T^{2}$$)($$1 - 448202760 T + 119851595982618319 T^{2}$$)($$1 + 932406760 T + 453226630902929438 T^{2} +$$$$11\!\cdots\!40$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)($$1 + 24755040 T + 43090694479668638 T^{2} +$$$$29\!\cdots\!60$$$$T^{3} +$$$$14\!\cdots\!61$$$$T^{4}$$)
$83$ ($$1 - 118910388 T + 186940255267540403 T^{2}$$)($$1 - 851015796 T + 186940255267540403 T^{2}$$)($$1 - 207040152 T + 372783310330485238 T^{2} -$$$$38\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)($$1 + 372082152 T + 379908828789982198 T^{2} +$$$$69\!\cdots\!56$$$$T^{3} +$$$$34\!\cdots\!09$$$$T^{4}$$)
$89$ ($$1 + 316534326 T + 350356403707485209 T^{2}$$)($$1 - 189894930 T + 350356403707485209 T^{2}$$)($$1 - 224518164 T + 610925899926766678 T^{2} -$$$$78\!\cdots\!76$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)($$1 + 427639116 T + 702028302670151638 T^{2} +$$$$14\!\cdots\!44$$$$T^{3} +$$$$12\!\cdots\!81$$$$T^{4}$$)
$97$ ($$1 - 242912258 T + 760231058654565217 T^{2}$$)($$1 + 1014149278 T + 760231058654565217 T^{2}$$)($$1 - 387134596 T - 734969029248610362 T^{2} -$$$$29\!\cdots\!32$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)($$1 - 1771658884 T + 2207048700436243398 T^{2} -$$$$13\!\cdots\!28$$$$T^{3} +$$$$57\!\cdots\!89$$$$T^{4}$$)