Properties

Label 16.12.a
Level 16
Weight 12
Character orbit a
Rep. character \(\chi_{16}(1,\cdot)\)
Character field \(\Q\)
Dimension 5
Newform subspaces 4
Sturm bound 24
Trace bound 3

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Defining parameters

Level: \( N \) = \( 16 = 2^{4} \)
Weight: \( k \) = \( 12 \)
Character orbit: \([\chi]\) = 16.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(16))\).

Total New Old
Modular forms 25 6 19
Cusp forms 19 5 14
Eisenstein series 6 1 5

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

\(2\)Dim.
\(+\)\(3\)
\(-\)\(2\)

Trace form

\( 5q + 244q^{3} - 1322q^{5} - 68152q^{7} + 339817q^{9} + O(q^{10}) \) \( 5q + 244q^{3} - 1322q^{5} - 68152q^{7} + 339817q^{9} + 212732q^{11} + 123022q^{13} + 3718552q^{15} + 2386298q^{17} + 13952132q^{19} + 12388896q^{21} - 15109672q^{23} + 61783619q^{25} - 40065848q^{27} - 183066498q^{29} - 103299040q^{31} - 298469776q^{33} + 498296688q^{35} + 346926262q^{37} - 1570233992q^{39} + 62633826q^{41} + 1794010652q^{43} + 651613582q^{45} - 2108993232q^{47} + 1621590093q^{49} + 10554130856q^{51} - 1021104218q^{53} - 15975322488q^{55} - 4106770544q^{57} + 23266819244q^{59} + 1262097502q^{61} - 42744151704q^{63} + 4538759716q^{65} + 33281496884q^{67} - 11476529056q^{69} - 60900642296q^{71} + 3416691010q^{73} + 119699811116q^{75} - 9841925280q^{77} - 99564430000q^{79} - 16614818003q^{81} + 111732583108q^{83} + 76865658060q^{85} - 115244963016q^{87} + 27045197490q^{89} + 68128310320q^{91} + 27673840768q^{93} + 7826480248q^{95} - 19977510998q^{97} - 24971649364q^{99} + O(q^{100}) \)

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(16))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2
16.12.a.a \(1\) \(12.293\) \(\Q\) None \(0\) \(-252\) \(4830\) \(16744\) \(-\) \(q-252q^{3}+4830q^{5}+16744q^{7}+\cdots\)
16.12.a.b \(1\) \(12.293\) \(\Q\) None \(0\) \(36\) \(-3490\) \(55464\) \(+\) \(q+6^{2}q^{3}-3490q^{5}+55464q^{7}-175851q^{9}+\cdots\)
16.12.a.c \(1\) \(12.293\) \(\Q\) None \(0\) \(516\) \(-10530\) \(-49304\) \(-\) \(q+516q^{3}-10530q^{5}-49304q^{7}+\cdots\)
16.12.a.d \(2\) \(12.293\) \(\Q(\sqrt{109}) \) None \(0\) \(-56\) \(7868\) \(-91056\) \(+\) \(q+(-28-\beta )q^{3}+(3934-12\beta )q^{5}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(16)) \cong \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 5}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( \))(\( \))(\( \))(\( \))
$3$ (\( 1 + 252 T + 177147 T^{2} \))(\( 1 - 36 T + 177147 T^{2} \))(\( 1 - 516 T + 177147 T^{2} \))(\( 1 + 56 T - 91386 T^{2} + 9920232 T^{3} + 31381059609 T^{4} \))
$5$ (\( 1 - 4830 T + 48828125 T^{2} \))(\( 1 + 3490 T + 48828125 T^{2} \))(\( 1 + 10530 T + 48828125 T^{2} \))(\( 1 - 7868 T + 48841790 T^{2} - 384179687500 T^{3} + 2384185791015625 T^{4} \))
$7$ (\( 1 - 16744 T + 1977326743 T^{2} \))(\( 1 - 55464 T + 1977326743 T^{2} \))(\( 1 + 49304 T + 1977326743 T^{2} \))(\( 1 + 91056 T + 5239889774 T^{2} + 180047463910608 T^{3} + 3909821048582988049 T^{4} \))
$11$ (\( 1 + 534612 T + 285311670611 T^{2} \))(\( 1 - 597004 T + 285311670611 T^{2} \))(\( 1 - 309420 T + 285311670611 T^{2} \))(\( 1 + 159080 T + 362536063286 T^{2} + 45387380560797880 T^{3} + \)\(81\!\cdots\!21\)\( T^{4} \))
$13$ (\( 1 + 577738 T + 1792160394037 T^{2} \))(\( 1 - 1373878 T + 1792160394037 T^{2} \))(\( 1 + 1723594 T + 1792160394037 T^{2} \))(\( 1 - 1050476 T + 3458956762062 T^{2} - 1882621482086411612 T^{3} + \)\(32\!\cdots\!69\)\( T^{4} \))
$17$ (\( 1 + 6905934 T + 34271896307633 T^{2} \))(\( 1 - 10140850 T + 34271896307633 T^{2} \))(\( 1 + 2279502 T + 34271896307633 T^{2} \))(\( 1 - 1430884 T + 16748384809766 T^{2} - 49039108076251137572 T^{3} + \)\(11\!\cdots\!89\)\( T^{4} \))
$19$ (\( 1 + 10661420 T + 116490258898219 T^{2} \))(\( 1 - 7297396 T + 116490258898219 T^{2} \))(\( 1 + 4550444 T + 116490258898219 T^{2} \))(\( 1 - 21866600 T + 343123710088422 T^{2} - \)\(25\!\cdots\!00\)\( T^{3} + \)\(13\!\cdots\!61\)\( T^{4} \))
$23$ (\( 1 + 18643272 T + 952809757913927 T^{2} \))(\( 1 - 32057464 T + 952809757913927 T^{2} \))(\( 1 - 7282872 T + 952809757913927 T^{2} \))(\( 1 + 35806736 T + 1952928132896462 T^{2} + \)\(34\!\cdots\!72\)\( T^{3} + \)\(90\!\cdots\!29\)\( T^{4} \))
$29$ (\( 1 - 128406630 T + 12200509765705829 T^{2} \))(\( 1 + 13605402 T + 12200509765705829 T^{2} \))(\( 1 + 69040026 T + 12200509765705829 T^{2} \))(\( 1 + 228827700 T + 35907977791027054 T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!41\)\( T^{4} \))
$31$ (\( 1 - 52843168 T + 25408476896404831 T^{2} \))(\( 1 + 233160800 T + 25408476896404831 T^{2} \))(\( 1 - 141740704 T + 25408476896404831 T^{2} \))(\( 1 + 64722112 T + 51498184379920062 T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + \)\(64\!\cdots\!61\)\( T^{4} \))
$37$ (\( 1 + 182213314 T + 177917621779460413 T^{2} \))(\( 1 + 257786178 T + 177917621779460413 T^{2} \))(\( 1 - 711366974 T + 177917621779460413 T^{2} \))(\( 1 - 75558780 T + 354292341022528382 T^{2} - \)\(13\!\cdots\!40\)\( T^{3} + \)\(31\!\cdots\!69\)\( T^{4} \))
$41$ (\( 1 - 308120442 T + 550329031716248441 T^{2} \))(\( 1 + 221438598 T + 550329031716248441 T^{2} \))(\( 1 + 1225262214 T + 550329031716248441 T^{2} \))(\( 1 - 1201214196 T + 1274442257818453270 T^{2} - \)\(66\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!81\)\( T^{4} \))
$43$ (\( 1 - 17125708 T + 929293739471222707 T^{2} \))(\( 1 - 1697758892 T + 929293739471222707 T^{2} \))(\( 1 - 33606220 T + 929293739471222707 T^{2} \))(\( 1 - 45519832 T + 1812222731951247606 T^{2} - \)\(42\!\cdots\!24\)\( T^{3} + \)\(86\!\cdots\!49\)\( T^{4} \))
$47$ (\( 1 + 2687348496 T + 2472159215084012303 T^{2} \))(\( 1 + 527509392 T + 2472159215084012303 T^{2} \))(\( 1 + 123214608 T + 2472159215084012303 T^{2} \))(\( 1 - 1229079264 T + 5024390553242310430 T^{2} - \)\(30\!\cdots\!92\)\( T^{3} + \)\(61\!\cdots\!09\)\( T^{4} \))
$53$ (\( 1 + 1596055698 T + 9269035929372191597 T^{2} \))(\( 1 - 3277379822 T + 9269035929372191597 T^{2} \))(\( 1 - 1106121582 T + 9269035929372191597 T^{2} \))(\( 1 + 3808549924 T + 16648234587904299038 T^{2} + \)\(35\!\cdots\!28\)\( T^{3} + \)\(85\!\cdots\!09\)\( T^{4} \))
$59$ (\( 1 - 5189203740 T + 30155888444737842659 T^{2} \))(\( 1 - 3001908988 T + 30155888444737842659 T^{2} \))(\( 1 - 9062779932 T + 30155888444737842659 T^{2} \))(\( 1 - 6012926584 T + 61066799326040273366 T^{2} - \)\(18\!\cdots\!56\)\( T^{3} + \)\(90\!\cdots\!81\)\( T^{4} \))
$61$ (\( 1 - 6956478662 T + 43513917611435838661 T^{2} \))(\( 1 + 11630023610 T + 43513917611435838661 T^{2} \))(\( 1 + 3854150458 T + 43513917611435838661 T^{2} \))(\( 1 - 9789792908 T + \)\(10\!\cdots\!42\)\( T^{2} - \)\(42\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!21\)\( T^{4} \))
$67$ (\( 1 - 15481826884 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 - 17189000548 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 - 15313764676 T + \)\(12\!\cdots\!83\)\( T^{2} \))(\( 1 + 14703095224 T + 95414710392374126214 T^{2} + \)\(17\!\cdots\!92\)\( T^{3} + \)\(14\!\cdots\!89\)\( T^{4} \))
$71$ (\( 1 + 9791485272 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 + 26169539608 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 + 20619626328 T + \)\(23\!\cdots\!71\)\( T^{2} \))(\( 1 + 4319991088 T + \)\(28\!\cdots\!82\)\( T^{2} + \)\(99\!\cdots\!48\)\( T^{3} + \)\(53\!\cdots\!41\)\( T^{4} \))
$73$ (\( 1 - 1463791322 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 7039021094 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 + 2063718694 T + \)\(31\!\cdots\!77\)\( T^{2} \))(\( 1 - 11055639476 T + \)\(65\!\cdots\!62\)\( T^{2} - \)\(34\!\cdots\!52\)\( T^{3} + \)\(98\!\cdots\!29\)\( T^{4} \))
$79$ (\( 1 + 38116845680 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 - 4199910416 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 13689871472 T + \)\(74\!\cdots\!79\)\( T^{2} \))(\( 1 + 51957623264 T + \)\(14\!\cdots\!78\)\( T^{2} + \)\(38\!\cdots\!56\)\( T^{3} + \)\(55\!\cdots\!41\)\( T^{4} \))
$83$ (\( 1 - 29335099668 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 - 39739936436 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 + 65570428908 T + \)\(12\!\cdots\!67\)\( T^{2} \))(\( 1 - 108227975912 T + \)\(54\!\cdots\!06\)\( T^{2} - \)\(13\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!89\)\( T^{4} \))
$89$ (\( 1 + 24992917110 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 10565331594 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 + 29715508854 T + \)\(27\!\cdots\!89\)\( T^{2} \))(\( 1 - 71188291860 T + \)\(31\!\cdots\!82\)\( T^{2} - \)\(19\!\cdots\!40\)\( T^{3} + \)\(77\!\cdots\!21\)\( T^{4} \))
$97$ (\( 1 - 75013568546 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 69851645662 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 23439626206 T + \)\(71\!\cdots\!53\)\( T^{2} \))(\( 1 + 1699807676 T + \)\(12\!\cdots\!50\)\( T^{2} + \)\(12\!\cdots\!28\)\( T^{3} + \)\(51\!\cdots\!09\)\( T^{4} \))
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