Properties

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

Related objects

Downloads

Learn more about

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

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