Properties

 Label 6.16.a.b Level $6$ Weight $16$ Character orbit 6.a Self dual yes Analytic conductor $8.562$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$16$$ Character orbit: $$[\chi]$$ $$=$$ 6.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$8.56161030600$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 128 q^{2} - 2187 q^{3} + 16384 q^{4} - 114810 q^{5} - 279936 q^{6} - 3034528 q^{7} + 2097152 q^{8} + 4782969 q^{9} + O(q^{10})$$ $$q + 128 q^{2} - 2187 q^{3} + 16384 q^{4} - 114810 q^{5} - 279936 q^{6} - 3034528 q^{7} + 2097152 q^{8} + 4782969 q^{9} - 14695680 q^{10} - 103451700 q^{11} - 35831808 q^{12} - 104365834 q^{13} - 388419584 q^{14} + 251089470 q^{15} + 268435456 q^{16} + 997689762 q^{17} + 612220032 q^{18} + 4934015444 q^{19} - 1881047040 q^{20} + 6636512736 q^{21} - 13241817600 q^{22} + 8324920200 q^{23} - 4586471424 q^{24} - 17336242025 q^{25} - 13358826752 q^{26} - 10460353203 q^{27} - 49717706752 q^{28} + 104128242846 q^{29} + 32139452160 q^{30} - 296696681512 q^{31} + 34359738368 q^{32} + 226248867900 q^{33} + 127704289536 q^{34} + 348394159680 q^{35} + 78364164096 q^{36} - 178337455666 q^{37} + 631553976832 q^{38} + 228248078958 q^{39} - 240774021120 q^{40} - 1790882416086 q^{41} + 849473630208 q^{42} - 2863459422772 q^{43} - 1694952652800 q^{44} - 549132670890 q^{45} + 1065589785600 q^{46} + 4332907521600 q^{47} - 587068342272 q^{48} + 4460798672841 q^{49} - 2219038979200 q^{50} - 2181947509494 q^{51} - 1709929824256 q^{52} + 9732317104422 q^{53} - 1338925209984 q^{54} + 11877289677000 q^{55} - 6363866464256 q^{56} - 10790691776028 q^{57} + 13328415084288 q^{58} - 13514837176500 q^{59} + 4113849876480 q^{60} + 5352663511190 q^{61} - 37977175233536 q^{62} - 14514053353632 q^{63} + 4398046511104 q^{64} + 11982241401540 q^{65} + 28959855091200 q^{66} - 53233909720108 q^{67} + 16346149060608 q^{68} - 18206600477400 q^{69} + 44594452439040 q^{70} - 20229661643400 q^{71} + 10030613004288 q^{72} + 26264166466106 q^{73} - 22827194325248 q^{74} + 37914361308675 q^{75} + 80838909034496 q^{76} + 313927080297600 q^{77} + 29215754106624 q^{78} - 339031361615128 q^{79} - 30819074703360 q^{80} + 22876792454961 q^{81} - 229232949259008 q^{82} + 131684771045076 q^{83} + 108732624666624 q^{84} - 114544761575220 q^{85} - 366522806114816 q^{86} - 227728467104202 q^{87} - 216953939558400 q^{88} - 39352148322678 q^{89} - 70288981873920 q^{90} + 316701045516352 q^{91} + 136395492556800 q^{92} + 648875642466744 q^{93} + 554612162764800 q^{94} - 566474313125640 q^{95} - 75144747810816 q^{96} + 1128750908801474 q^{97} + 570982230123648 q^{98} - 494806274097300 q^{99} + O(q^{100})$$

Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
128.000 −2187.00 16384.0 −114810. −279936. −3.03453e6 2.09715e6 4.78297e6 −1.46957e7
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.16.a.b 1
3.b odd 2 1 18.16.a.b 1
4.b odd 2 1 48.16.a.d 1
5.b even 2 1 150.16.a.f 1
5.c odd 4 2 150.16.c.a 2
12.b even 2 1 144.16.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.16.a.b 1 1.a even 1 1 trivial
18.16.a.b 1 3.b odd 2 1
48.16.a.d 1 4.b odd 2 1
144.16.a.j 1 12.b even 2 1
150.16.a.f 1 5.b even 2 1
150.16.c.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 114810$$ acting on $$S_{16}^{\mathrm{new}}(\Gamma_0(6))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-128 + T$$
$3$ $$2187 + T$$
$5$ $$114810 + T$$
$7$ $$3034528 + T$$
$11$ $$103451700 + T$$
$13$ $$104365834 + T$$
$17$ $$-997689762 + T$$
$19$ $$-4934015444 + T$$
$23$ $$-8324920200 + T$$
$29$ $$-104128242846 + T$$
$31$ $$296696681512 + T$$
$37$ $$178337455666 + T$$
$41$ $$1790882416086 + T$$
$43$ $$2863459422772 + T$$
$47$ $$-4332907521600 + T$$
$53$ $$-9732317104422 + T$$
$59$ $$13514837176500 + T$$
$61$ $$-5352663511190 + T$$
$67$ $$53233909720108 + T$$
$71$ $$20229661643400 + T$$
$73$ $$-26264166466106 + T$$
$79$ $$339031361615128 + T$$
$83$ $$-131684771045076 + T$$
$89$ $$39352148322678 + T$$
$97$ $$-1128750908801474 + T$$