# Properties

 Label 16.11.f.a Level 16 Weight 11 Character orbit 16.f Analytic conductor 10.166 Analytic rank 0 Dimension 38 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 16.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.1657160428$$ Analytic rank: $$0$$ Dimension: $$38$$ Relative dimension: $$19$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$38q - 2q^{2} - 2q^{3} - 1256q^{4} - 2q^{5} - 17216q^{6} - 4q^{7} + 69124q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$38q - 2q^{2} - 2q^{3} - 1256q^{4} - 2q^{5} - 17216q^{6} - 4q^{7} + 69124q^{8} - 147364q^{10} - 45906q^{11} - 10988q^{12} - 2q^{13} + 757836q^{14} - 469032q^{16} - 4q^{17} + 2066926q^{18} - 5107042q^{19} + 5301516q^{20} - 118100q^{21} - 20568924q^{22} - 8279748q^{23} + 58519728q^{24} - 18451512q^{26} - 25871552q^{27} - 36459784q^{28} - 29652306q^{29} + 132781252q^{30} - 52661752q^{32} - 4q^{33} + 68474900q^{34} + 14674460q^{35} - 15810676q^{36} - 94618130q^{37} + 156046000q^{38} - 279841732q^{39} - 125381720q^{40} + 107395416q^{42} + 172486862q^{43} + 73663988q^{44} + 19649346q^{45} - 437620836q^{46} + 625190376q^{48} + 1049193778q^{49} - 875760314q^{50} - 72924900q^{51} - 771392444q^{52} + 607072398q^{53} + 913650464q^{54} - 1427102468q^{55} - 1385623464q^{56} + 37561720q^{58} + 1543683854q^{59} + 318869600q^{60} + 1673280158q^{61} - 983062992q^{62} - 867678656q^{64} + 1098715212q^{65} + 768197860q^{66} - 4830427746q^{67} - 855827792q^{68} + 1698121420q^{69} - 5103882496q^{70} + 7572888316q^{71} + 6888513460q^{72} + 2513658844q^{74} - 11656678318q^{75} - 2250244316q^{76} - 1336459028q^{77} + 14243324268q^{78} - 13115458328q^{80} - 8523250762q^{81} + 2724305648q^{82} + 16141605438q^{83} - 1633924184q^{84} + 4779968748q^{85} - 4025865500q^{86} - 31832612676q^{87} + 1023457912q^{88} + 18960485240q^{90} + 19874217404q^{91} + 28060187160q^{92} - 15001617056q^{93} - 14829002256q^{94} + 23525947312q^{96} - 4q^{97} - 43134832310q^{98} - 25616384098q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −31.9996 + 0.164143i −221.985 + 221.985i 1023.95 10.5050i 2579.78 2579.78i 7066.97 7139.85i 4017.27 −32764.1 + 504.230i 39505.3i −82128.4 + 82975.3i
3.2 −30.1099 10.8349i 127.462 127.462i 789.210 + 652.475i −1086.76 + 1086.76i −5218.91 + 2456.83i −8881.75 −16693.5 28197.0i 26555.9i 44497.0 20947.2i
3.3 −29.1906 + 13.1115i 32.7265 32.7265i 680.178 765.463i −1645.29 + 1645.29i −526.211 + 1384.40i 16644.7 −9818.42 + 31262.4i 56907.0i 26454.7 69599.0i
3.4 −25.9865 + 18.6735i 280.546 280.546i 326.598 970.520i 3102.01 3102.01i −2051.63 + 12529.2i −27408.1 9635.89 + 31319.2i 98363.4i −22685.0 + 138536.i
3.5 −19.3334 + 25.4994i −222.022 + 222.022i −276.439 985.981i −2711.76 + 2711.76i −1368.99 9953.89i −24891.4 30486.4 + 12013.3i 39538.9i −16720.7 121576.i
3.6 −19.3314 25.5009i −196.215 + 196.215i −276.596 + 985.936i −1311.50 + 1311.50i 8796.79 + 1210.56i −1120.35 30489.3 12006.1i 17952.0i 58797.6 + 8091.39i
3.7 −16.8116 27.2281i 183.793 183.793i −458.741 + 915.496i 3496.01 3496.01i −8094.18 1914.49i 26135.6 32639.4 2900.27i 8510.59i −153963. 36416.3i
3.8 −9.61264 + 30.5221i −107.144 + 107.144i −839.194 586.796i 3434.17 3434.17i −2240.31 4300.18i 8771.09 25977.1 19973.3i 36089.4i 71806.6 + 137829.i
3.9 −6.25659 + 31.3824i 196.669 196.669i −945.710 392.694i −1812.92 + 1812.92i 4941.46 + 7402.41i 22339.6 18240.6 27221.7i 18308.2i −45551.0 68236.4i
3.10 −1.21444 31.9769i 287.146 287.146i −1021.05 + 77.6681i −3974.49 + 3974.49i −9530.78 8833.33i −11755.9 3723.59 + 32555.7i 105857.i 131919. + 122265.i
3.11 2.46002 31.9053i −46.0373 + 46.0373i −1011.90 156.975i 1933.37 1933.37i 1355.58 + 1582.09i −25927.9 −7497.63 + 31898.7i 54810.1i −56928.5 66440.7i
3.12 12.2776 29.5510i −126.065 + 126.065i −722.523 725.628i −1655.31 + 1655.31i 2177.58 + 5273.11i 29862.3 −30313.9 + 12442.3i 27264.3i 28593.0 + 69239.3i
3.13 12.5336 + 29.4433i 44.0645 44.0645i −709.815 + 738.064i −126.270 + 126.270i 1849.69 + 745.115i −19594.2 −30627.6 11648.7i 55165.6i −5300.42 2135.18i
3.14 17.5699 + 26.7451i −286.288 + 286.288i −406.599 + 939.816i −1338.28 + 1338.28i −12686.8 2626.75i 19657.4 −32279.3 + 5637.94i 104873.i −59305.9 12279.0i
3.15 26.0928 18.5248i 164.886 164.886i 337.664 966.726i 897.573 897.573i 1247.85 7356.80i 594.817 −9097.83 31479.7i 4674.41i 6792.79 40047.5i
3.16 27.4041 + 16.5231i 305.982 305.982i 477.973 + 905.603i 155.264 155.264i 13440.9 3329.39i 8523.80 −1864.95 + 32714.9i 128200.i 6820.32 1689.43i
3.17 28.0563 15.3897i −317.338 + 317.338i 550.315 863.556i 1564.48 1564.48i −4019.61 + 13787.1i −15496.4 2149.97 32697.4i 142358.i 19816.8 67970.6i
3.18 30.4523 + 9.83157i −61.4496 + 61.4496i 830.681 + 598.787i 2595.70 2595.70i −2475.42 + 1267.13i 6393.57 19409.1 + 26401.3i 51496.9i 104565. 53525.2i
3.19 32.0000 + 0.0155982i −39.7294 + 39.7294i 1024.00 + 0.998283i −4096.79 + 4096.79i −1271.96 + 1270.72i −7866.10 32768.0 + 47.9176i 55892.1i −131161. + 131033.i
11.1 −31.9996 0.164143i −221.985 221.985i 1023.95 + 10.5050i 2579.78 + 2579.78i 7066.97 + 7139.85i 4017.27 −32764.1 504.230i 39505.3i −82128.4 82975.3i
See all 38 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.19 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 16.11.f.a 38
4.b odd 2 1 64.11.f.a 38
8.b even 2 1 128.11.f.b 38
8.d odd 2 1 128.11.f.a 38
16.e even 4 1 64.11.f.a 38
16.e even 4 1 128.11.f.a 38
16.f odd 4 1 inner 16.11.f.a 38
16.f odd 4 1 128.11.f.b 38

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.11.f.a 38 1.a even 1 1 trivial
16.11.f.a 38 16.f odd 4 1 inner
64.11.f.a 38 4.b odd 2 1
64.11.f.a 38 16.e even 4 1
128.11.f.a 38 8.d odd 2 1
128.11.f.a 38 16.e even 4 1
128.11.f.b 38 8.b even 2 1
128.11.f.b 38 16.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{11}^{\mathrm{new}}(16, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database