Properties

Label 15.12.a
Level 15
Weight 12
Character orbit a
Rep. character \(\chi_{15}(1,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 4
Sturm bound 24
Trace bound 2

Related objects

Downloads

Learn more about

Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 15.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(15))\).

Total New Old
Modular forms 24 8 16
Cusp forms 20 8 12
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.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(5\)
Minus space\(-\)\(3\)

Trace form

\( 8q - 92q^{2} + 486q^{3} + 2098q^{4} + 11178q^{6} + 10296q^{7} + 36156q^{8} + 472392q^{9} + O(q^{10}) \) \( 8q - 92q^{2} + 486q^{3} + 2098q^{4} + 11178q^{6} + 10296q^{7} + 36156q^{8} + 472392q^{9} - 68750q^{10} + 362368q^{11} + 1492992q^{12} - 1732192q^{13} + 3055740q^{14} + 1518750q^{15} - 2528078q^{16} + 15690832q^{17} - 5432508q^{18} + 6097088q^{19} + 29975000q^{20} - 14881320q^{21} + 80817764q^{22} - 54036408q^{23} - 34419006q^{24} + 78125000q^{25} - 42340964q^{26} + 28697814q^{27} - 206985012q^{28} - 344119264q^{29} + 48600000q^{30} - 244730640q^{31} + 392770196q^{32} - 1145016q^{33} - 1039380484q^{34} + 51425000q^{35} + 123884802q^{36} + 411385216q^{37} + 925664872q^{38} - 207359676q^{39} + 10518750q^{40} + 50518160q^{41} - 39765492q^{42} + 3382732760q^{43} + 993772268q^{44} + 1792516800q^{46} - 6579674120q^{47} + 486330480q^{48} + 7476115336q^{49} - 898437500q^{50} - 235693476q^{51} - 11457178448q^{52} + 2398000672q^{53} + 660049722q^{54} + 1546300000q^{55} + 6876152580q^{56} - 421367832q^{57} - 15688092236q^{58} + 11641748272q^{59} + 906693750q^{60} - 1339937296q^{61} - 14092986648q^{62} + 607968504q^{63} - 22506773446q^{64} + 3813400000q^{65} + 3967032348q^{66} + 17395245672q^{67} - 16371169096q^{68} - 18746409960q^{69} + 14610862500q^{70} + 17460704816q^{71} + 2134975644q^{72} + 13234384592q^{73} - 73427688700q^{74} + 4746093750q^{75} + 91790636368q^{76} + 39494634432q^{77} - 6583379328q^{78} - 29704597920q^{79} + 6801350000q^{80} + 27894275208q^{81} - 7271225288q^{82} + 29154140856q^{83} - 38819245140q^{84} + 44187850000q^{85} + 230761968568q^{86} + 16697096676q^{87} - 159023820564q^{88} - 22145809872q^{89} - 4059618750q^{90} - 121971765360q^{91} - 185727975744q^{92} + 104018918256q^{93} + 205545366584q^{94} + 6142450000q^{95} + 21391744410q^{96} - 385635691568q^{97} - 216115979308q^{98} + 21397468032q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(15))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
15.12.a.a \(1\) \(11.525\) \(\Q\) None \(-56\) \(-243\) \(3125\) \(27984\) \(+\) \(-\) \(q-56q^{2}-3^{5}q^{3}+1088q^{4}+5^{5}q^{5}+\cdots\)
15.12.a.b \(2\) \(11.525\) \(\Q(\sqrt{1609}) \) None \(-22\) \(486\) \(-6250\) \(-10864\) \(-\) \(+\) \(q+(-11-\beta )q^{2}+3^{5}q^{3}+(-318+22\beta )q^{4}+\cdots\)
15.12.a.c \(2\) \(11.525\) \(\Q(\sqrt{1801}) \) None \(-13\) \(-486\) \(-6250\) \(7784\) \(+\) \(+\) \(q+(-6-\beta )q^{2}-3^{5}q^{3}+(-1562+13\beta )q^{4}+\cdots\)
15.12.a.d \(3\) \(11.525\) \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-1\) \(729\) \(9375\) \(-14608\) \(-\) \(-\) \(q-\beta _{1}q^{2}+3^{5}q^{3}+(1585+3\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(15))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(15)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 56 T + 2048 T^{2} \))(\( 1 + 22 T + 2608 T^{2} + 45056 T^{3} + 4194304 T^{4} \))(\( 1 + 13 T + 3688 T^{2} + 26624 T^{3} + 4194304 T^{4} \))(\( 1 + T + 694 T^{2} + 11344 T^{3} + 1421312 T^{4} + 4194304 T^{5} + 8589934592 T^{6} \))
$3$ (\( 1 + 243 T \))(\( ( 1 - 243 T )^{2} \))(\( ( 1 + 243 T )^{2} \))(\( ( 1 - 243 T )^{3} \))
$5$ (\( 1 - 3125 T \))(\( ( 1 + 3125 T )^{2} \))(\( ( 1 + 3125 T )^{2} \))(\( ( 1 - 3125 T )^{3} \))
$7$ (\( 1 - 27984 T + 1977326743 T^{2} \))(\( 1 + 10864 T + 1364424926 T^{2} + 21481677735952 T^{3} + 3909821048582988049 T^{4} \))(\( 1 - 7784 T - 1813685314 T^{2} - 15391511367512 T^{3} + 3909821048582988049 T^{4} \))(\( 1 + 14608 T + 3230740485 T^{2} + 80112730650848 T^{3} + 6388229560683290355 T^{4} + \)\(57\!\cdots\!92\)\( T^{5} + \)\(77\!\cdots\!07\)\( T^{6} \))
$11$ (\( 1 + 112028 T + 285311670611 T^{2} \))(\( 1 + 361792 T + 429852327334 T^{2} + 103223479933694912 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))(\( 1 - 295568 T + 160442914534 T^{2} - 84328999859152048 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))(\( 1 - 540620 T + 871024121641 T^{2} - 293660175406171784 T^{3} + \)\(24\!\cdots\!51\)\( T^{4} - \)\(44\!\cdots\!20\)\( T^{5} + \)\(23\!\cdots\!31\)\( T^{6} \))
$13$ (\( 1 + 1096922 T + 1792160394037 T^{2} \))(\( 1 + 2133732 T + 4716617120926 T^{2} + 3823989981889356084 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 657492 T + 1362446380366 T^{2} - 1178331121796175204 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))(\( 1 - 840970 T + 1597595357099 T^{2} - 4490565644417681884 T^{3} + \)\(28\!\cdots\!63\)\( T^{4} - \)\(27\!\cdots\!30\)\( T^{5} + \)\(57\!\cdots\!53\)\( T^{6} \))
$17$ (\( 1 + 249566 T + 34271896307633 T^{2} \))(\( 1 + 7804588 T + 76431993653302 T^{2} + \)\(26\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( 1 - 8579948 T + 84202726534342 T^{2} - \)\(29\!\cdots\!84\)\( T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))(\( 1 - 15165038 T + 88864608966847 T^{2} - \)\(38\!\cdots\!44\)\( T^{3} + \)\(30\!\cdots\!51\)\( T^{4} - \)\(17\!\cdots\!82\)\( T^{5} + \)\(40\!\cdots\!37\)\( T^{6} \))
$19$ (\( 1 + 13712420 T + 116490258898219 T^{2} \))(\( 1 + 15562224 T + 223670591350918 T^{2} + \)\(18\!\cdots\!56\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))(\( 1 - 17627976 T + 302871116598118 T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))(\( 1 - 17743756 T + 229657463589617 T^{2} - \)\(22\!\cdots\!28\)\( T^{3} + \)\(26\!\cdots\!23\)\( T^{4} - \)\(24\!\cdots\!16\)\( T^{5} + \)\(15\!\cdots\!59\)\( T^{6} \))
$23$ (\( 1 - 41395728 T + 952809757913927 T^{2} \))(\( 1 + 37450248 T + 1485695255413774 T^{2} + \)\(35\!\cdots\!96\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))(\( 1 + 29841072 T + 1016542384821454 T^{2} + \)\(28\!\cdots\!44\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))(\( 1 + 28140816 T + 2322805026860565 T^{2} + \)\(38\!\cdots\!24\)\( T^{3} + \)\(22\!\cdots\!55\)\( T^{4} + \)\(25\!\cdots\!64\)\( T^{5} + \)\(86\!\cdots\!83\)\( T^{6} \))
$29$ (\( 1 + 4533850 T + 12200509765705829 T^{2} \))(\( 1 + 70320668 T + 10276443921751438 T^{2} + \)\(85\!\cdots\!72\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))(\( 1 + 201881948 T + 26564448133170958 T^{2} + \)\(24\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))(\( 1 + 67382798 T - 2080427172177893 T^{2} - \)\(21\!\cdots\!16\)\( T^{3} - \)\(25\!\cdots\!97\)\( T^{4} + \)\(10\!\cdots\!18\)\( T^{5} + \)\(18\!\cdots\!89\)\( T^{6} \))
$31$ (\( 1 + 265339008 T + 25408476896404831 T^{2} \))(\( 1 - 298584872 T + 71113481278713662 T^{2} - \)\(75\!\cdots\!32\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))(\( 1 + 71057008 T + 33404626522449662 T^{2} + \)\(18\!\cdots\!48\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))(\( 1 + 206919496 T + 29249324330157597 T^{2} + \)\(26\!\cdots\!52\)\( T^{3} + \)\(74\!\cdots\!07\)\( T^{4} + \)\(13\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!91\)\( T^{6} \))
$37$ (\( 1 + 212136946 T + 177917621779460413 T^{2} \))(\( 1 - 236000956 T + 238646163644168046 T^{2} - \)\(41\!\cdots\!28\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))(\( 1 - 705858484 T + 406267859630678046 T^{2} - \)\(12\!\cdots\!92\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))(\( 1 + 318337278 T + 488167879326286755 T^{2} + \)\(97\!\cdots\!08\)\( T^{3} + \)\(86\!\cdots\!15\)\( T^{4} + \)\(10\!\cdots\!82\)\( T^{5} + \)\(56\!\cdots\!97\)\( T^{6} \))
$41$ (\( 1 + 1266969958 T + 550329031716248441 T^{2} \))(\( 1 + 464942588 T + 352557098230423222 T^{2} + \)\(25\!\cdots\!08\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))(\( 1 + 327655148 T + 975285625454018902 T^{2} + \)\(18\!\cdots\!68\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))(\( 1 - 2110085854 T + 3087038640326735767 T^{2} - \)\(26\!\cdots\!08\)\( T^{3} + \)\(16\!\cdots\!47\)\( T^{4} - \)\(63\!\cdots\!74\)\( T^{5} + \)\(16\!\cdots\!21\)\( T^{6} \))
$43$ (\( 1 - 14129548 T + 929293739471222707 T^{2} \))(\( 1 + 242208600 T - 840276877730003210 T^{2} + \)\(22\!\cdots\!00\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))(\( 1 - 3192552120 T + 4401027789107914870 T^{2} - \)\(29\!\cdots\!40\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))(\( 1 - 418259692 T + 1797645863194484297 T^{2} - \)\(99\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!79\)\( T^{4} - \)\(36\!\cdots\!08\)\( T^{5} + \)\(80\!\cdots\!43\)\( T^{6} \))
$47$ (\( 1 + 2657273336 T + 2472159215084012303 T^{2} \))(\( 1 + 4375796920 T + 9100022224681989790 T^{2} + \)\(10\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))(\( 1 - 2053064720 T + 5969854595602269790 T^{2} - \)\(50\!\cdots\!60\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))(\( 1 + 1599668584 T + 5091663441833436973 T^{2} + \)\(52\!\cdots\!80\)\( T^{3} + \)\(12\!\cdots\!19\)\( T^{4} + \)\(97\!\cdots\!56\)\( T^{5} + \)\(15\!\cdots\!27\)\( T^{6} \))
$53$ (\( 1 - 2402699278 T + 9269035929372191597 T^{2} \))(\( 1 + 2189541388 T + 18323810419577155774 T^{2} + \)\(20\!\cdots\!36\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))(\( 1 + 2304299452 T + 14223212019195836254 T^{2} + \)\(21\!\cdots\!44\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))(\( 1 - 4489142234 T + 28713145956507182035 T^{2} - \)\(73\!\cdots\!56\)\( T^{3} + \)\(26\!\cdots\!95\)\( T^{4} - \)\(38\!\cdots\!06\)\( T^{5} + \)\(79\!\cdots\!73\)\( T^{6} \))
$59$ (\( 1 - 7498737220 T + 30155888444737842659 T^{2} \))(\( 1 + 5480385856 T + 33382416716089423558 T^{2} + \)\(16\!\cdots\!04\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))(\( 1 + 1478770576 T + 34879178115908872198 T^{2} + \)\(44\!\cdots\!84\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))(\( 1 - 11102167484 T + 61748067044497465177 T^{2} - \)\(31\!\cdots\!12\)\( T^{3} + \)\(18\!\cdots\!43\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{5} + \)\(27\!\cdots\!79\)\( T^{6} \))
$61$ (\( 1 + 4064828858 T + 43513917611435838661 T^{2} \))(\( 1 - 14557903980 T + \)\(13\!\cdots\!58\)\( T^{2} - \)\(63\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))(\( 1 + 8264891460 T + 97961878141180941838 T^{2} + \)\(35\!\cdots\!60\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))(\( 1 + 3568120958 T + 27391638978954256379 T^{2} + \)\(52\!\cdots\!44\)\( T^{3} + \)\(11\!\cdots\!19\)\( T^{4} + \)\(67\!\cdots\!18\)\( T^{5} + \)\(82\!\cdots\!81\)\( T^{6} \))
$67$ (\( 1 - 6871514244 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 + 15918388888 T + \)\(30\!\cdots\!02\)\( T^{2} + \)\(19\!\cdots\!04\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))(\( 1 - 24212177528 T + \)\(37\!\cdots\!62\)\( T^{2} - \)\(29\!\cdots\!24\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))(\( 1 - 2229942788 T + \)\(29\!\cdots\!97\)\( T^{2} - \)\(25\!\cdots\!44\)\( T^{3} + \)\(36\!\cdots\!51\)\( T^{4} - \)\(33\!\cdots\!32\)\( T^{5} + \)\(18\!\cdots\!87\)\( T^{6} \))
$71$ (\( 1 + 13283734648 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 - 1120561024 T + \)\(32\!\cdots\!86\)\( T^{2} - \)\(25\!\cdots\!04\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))(\( 1 + 20218888256 T + \)\(25\!\cdots\!26\)\( T^{2} + \)\(46\!\cdots\!76\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))(\( 1 - 49842766696 T + \)\(14\!\cdots\!85\)\( T^{2} - \)\(26\!\cdots\!00\)\( T^{3} + \)\(33\!\cdots\!35\)\( T^{4} - \)\(26\!\cdots\!36\)\( T^{5} + \)\(12\!\cdots\!11\)\( T^{6} \))
$73$ (\( 1 + 28875844262 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 24521574348 T + \)\(76\!\cdots\!54\)\( T^{2} + \)\(76\!\cdots\!96\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))(\( 1 - 25879583268 T + \)\(42\!\cdots\!94\)\( T^{2} - \)\(81\!\cdots\!36\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))(\( 1 - 40752219934 T + \)\(12\!\cdots\!95\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + \)\(38\!\cdots\!15\)\( T^{4} - \)\(40\!\cdots\!86\)\( T^{5} + \)\(30\!\cdots\!33\)\( T^{6} \))
$79$ (\( 1 - 27100302240 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 79243055560 T + \)\(30\!\cdots\!58\)\( T^{2} + \)\(59\!\cdots\!40\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))(\( 1 - 22324995440 T + \)\(14\!\cdots\!58\)\( T^{2} - \)\(16\!\cdots\!60\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))(\( 1 - 113159960 T + \)\(18\!\cdots\!37\)\( T^{2} + \)\(13\!\cdots\!20\)\( T^{3} + \)\(13\!\cdots\!23\)\( T^{4} - \)\(63\!\cdots\!60\)\( T^{5} + \)\(41\!\cdots\!39\)\( T^{6} \))
$83$ (\( 1 + 34365255132 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 - 9245226696 T - \)\(48\!\cdots\!98\)\( T^{2} - \)\(11\!\cdots\!32\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))(\( 1 - 48014508984 T + \)\(30\!\cdots\!22\)\( T^{2} - \)\(61\!\cdots\!28\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))(\( 1 - 6259660308 T + \)\(37\!\cdots\!21\)\( T^{2} - \)\(16\!\cdots\!88\)\( T^{3} + \)\(48\!\cdots\!07\)\( T^{4} - \)\(10\!\cdots\!12\)\( T^{5} + \)\(21\!\cdots\!63\)\( T^{6} \))
$89$ (\( 1 + 63500412630 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 22117321236 T + \)\(55\!\cdots\!78\)\( T^{2} - \)\(61\!\cdots\!04\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))(\( 1 - 79209683076 T + \)\(58\!\cdots\!98\)\( T^{2} - \)\(21\!\cdots\!64\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))(\( 1 + 59972401554 T + \)\(29\!\cdots\!27\)\( T^{2} + \)\(84\!\cdots\!12\)\( T^{3} + \)\(83\!\cdots\!03\)\( T^{4} + \)\(46\!\cdots\!34\)\( T^{5} + \)\(21\!\cdots\!69\)\( T^{6} \))
$97$ (\( 1 - 19634495234 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 160363673468 T + \)\(20\!\cdots\!62\)\( T^{2} + \)\(11\!\cdots\!04\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))(\( 1 + 37075227452 T + \)\(14\!\cdots\!82\)\( T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))(\( 1 + 207831285882 T + \)\(28\!\cdots\!67\)\( T^{2} + \)\(26\!\cdots\!76\)\( T^{3} + \)\(20\!\cdots\!51\)\( T^{4} + \)\(10\!\cdots\!38\)\( T^{5} + \)\(36\!\cdots\!77\)\( T^{6} \))
show more
show less