# Properties

 Label 16.9.f.a Level $16$ Weight $9$ Character orbit 16.f Analytic conductor $6.518$ Analytic rank $0$ Dimension $30$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,9,Mod(3,16)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("16.3");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 16.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$6.51805776098$$ Analytic rank: $$0$$ Dimension: $$30$$ Relative dimension: $$15$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 2 q^{2} - 2 q^{3} + 184 q^{4} - 2 q^{5} + 3232 q^{6} - 4 q^{7} - 8732 q^{8}+O(q^{10})$$ 30 * q - 2 * q^2 - 2 * q^3 + 184 * q^4 - 2 * q^5 + 3232 * q^6 - 4 * q^7 - 8732 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$30 q - 2 q^{2} - 2 q^{3} + 184 q^{4} - 2 q^{5} + 3232 q^{6} - 4 q^{7} - 8732 q^{8} - 1860 q^{10} - 19778 q^{11} + 14068 q^{12} - 2 q^{13} - 58900 q^{14} + 245336 q^{16} - 4 q^{17} - 223730 q^{18} + 167550 q^{19} - 135380 q^{20} - 13124 q^{21} + 985700 q^{22} - 845572 q^{23} - 800592 q^{24} - 184760 q^{26} + 38656 q^{27} + 94136 q^{28} + 1066174 q^{29} + 1881700 q^{30} - 3395192 q^{32} - 4 q^{33} + 1634900 q^{34} + 426620 q^{35} + 1877324 q^{36} + 2360254 q^{37} - 3010352 q^{38} - 7650052 q^{39} + 3256936 q^{40} - 3731304 q^{42} + 6314814 q^{43} - 10360940 q^{44} + 794370 q^{45} + 1510780 q^{46} + 4591848 q^{48} + 14823770 q^{49} + 14752710 q^{50} - 28969860 q^{51} + 4335652 q^{52} + 2679358 q^{53} + 18667040 q^{54} + 46326780 q^{55} - 36346408 q^{56} - 1034696 q^{58} - 46004162 q^{59} - 20687200 q^{60} - 24476034 q^{61} - 64043472 q^{62} + 50827456 q^{64} - 14970820 q^{65} + 65035780 q^{66} + 59474558 q^{67} + 74339312 q^{68} + 8623420 q^{69} + 39082496 q^{70} - 79832068 q^{71} + 929812 q^{72} - 168043460 q^{74} + 88519490 q^{75} - 166219388 q^{76} + 35952572 q^{77} + 5059788 q^{78} - 12904856 q^{80} - 66961570 q^{81} + 261305296 q^{82} - 34471682 q^{83} + 456781480 q^{84} - 107741252 q^{85} + 9841444 q^{86} + 149712636 q^{87} - 173775752 q^{88} - 650366536 q^{90} - 76429060 q^{91} - 449169832 q^{92} + 138269056 q^{93} - 239278032 q^{94} + 378359728 q^{96} - 4 q^{97} + 630040554 q^{98} + 184838270 q^{99}+O(q^{100})$$ 30 * q - 2 * q^2 - 2 * q^3 + 184 * q^4 - 2 * q^5 + 3232 * q^6 - 4 * q^7 - 8732 * q^8 - 1860 * q^10 - 19778 * q^11 + 14068 * q^12 - 2 * q^13 - 58900 * q^14 + 245336 * q^16 - 4 * q^17 - 223730 * q^18 + 167550 * q^19 - 135380 * q^20 - 13124 * q^21 + 985700 * q^22 - 845572 * q^23 - 800592 * q^24 - 184760 * q^26 + 38656 * q^27 + 94136 * q^28 + 1066174 * q^29 + 1881700 * q^30 - 3395192 * q^32 - 4 * q^33 + 1634900 * q^34 + 426620 * q^35 + 1877324 * q^36 + 2360254 * q^37 - 3010352 * q^38 - 7650052 * q^39 + 3256936 * q^40 - 3731304 * q^42 + 6314814 * q^43 - 10360940 * q^44 + 794370 * q^45 + 1510780 * q^46 + 4591848 * q^48 + 14823770 * q^49 + 14752710 * q^50 - 28969860 * q^51 + 4335652 * q^52 + 2679358 * q^53 + 18667040 * q^54 + 46326780 * q^55 - 36346408 * q^56 - 1034696 * q^58 - 46004162 * q^59 - 20687200 * q^60 - 24476034 * q^61 - 64043472 * q^62 + 50827456 * q^64 - 14970820 * q^65 + 65035780 * q^66 + 59474558 * q^67 + 74339312 * q^68 + 8623420 * q^69 + 39082496 * q^70 - 79832068 * q^71 + 929812 * q^72 - 168043460 * q^74 + 88519490 * q^75 - 166219388 * q^76 + 35952572 * q^77 + 5059788 * q^78 - 12904856 * q^80 - 66961570 * q^81 + 261305296 * q^82 - 34471682 * q^83 + 456781480 * q^84 - 107741252 * q^85 + 9841444 * q^86 + 149712636 * q^87 - 173775752 * q^88 - 650366536 * q^90 - 76429060 * q^91 - 449169832 * q^92 + 138269056 * q^93 - 239278032 * q^94 + 378359728 * q^96 - 4 * q^97 + 630040554 * q^98 + 184838270 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1 −15.9265 1.53176i 110.050 110.050i 251.307 + 48.7911i −128.245 + 128.245i −1921.29 + 1584.14i 3057.02 −3927.71 1162.01i 17661.1i 2238.94 1846.06i
3.2 −15.9118 1.67767i −93.3844 + 93.3844i 250.371 + 53.3896i −695.406 + 695.406i 1642.58 1329.25i 805.482 −3894.28 1269.56i 10880.3i 12231.8 9898.49i
3.3 −14.5765 6.59745i −7.14041 + 7.14041i 168.947 + 192.335i 610.669 610.669i 151.190 56.9735i −3410.30 −1193.74 3918.19i 6459.03i −12930.3 + 4872.54i
3.4 −13.4426 + 8.67733i −4.15109 + 4.15109i 105.408 233.292i 178.569 178.569i 19.7811 91.8219i 678.560 607.393 + 4050.71i 6526.54i −850.931 + 3949.93i
3.5 −8.02121 13.8441i 17.4297 17.4297i −127.320 + 222.093i −396.274 + 396.274i −381.106 101.492i 432.698 4095.96 18.8169i 5953.41i 8664.66 + 2307.47i
3.6 −5.55129 + 15.0061i 65.0768 65.0768i −194.366 166.606i −526.870 + 526.870i 615.290 + 1337.81i −3638.93 3579.10 1991.80i 1908.99i −4981.46 10831.1i
3.7 −2.48211 15.8063i −100.879 + 100.879i −243.678 + 78.4659i 623.715 623.715i 1844.91 + 1344.13i 1797.37 1845.09 + 3656.89i 13792.1i −11406.8 8310.50i
3.8 −1.81651 + 15.8965i −71.2555 + 71.2555i −249.401 57.7524i −8.38358 + 8.38358i −1003.28 1262.15i 1087.59 1371.10 3859.70i 3593.69i −118.041 148.499i
3.9 4.89750 15.2320i 60.9999 60.9999i −208.029 149.198i 283.594 283.594i −630.404 1227.90i 502.933 −3291.41 + 2438.00i 880.969i −2930.81 5708.61i
3.10 5.41380 + 15.0563i 63.2572 63.2572i −197.381 + 163.023i 825.937 825.937i 1294.88 + 609.955i 711.293 −3523.10 2089.25i 1441.95i 16907.0 + 7964.05i
3.11 10.2604 12.2770i −49.7393 + 49.7393i −45.4477 251.934i −502.907 + 502.907i 100.302 + 1120.99i −2327.86 −3559.29 2026.98i 1613.00i 1014.13 + 11334.2i
3.12 11.2357 + 11.3912i 23.8972 23.8972i −3.51862 + 255.976i −695.590 + 695.590i 540.720 + 3.71617i 3167.79 −2955.40 + 2835.98i 5418.84i −15739.0 108.169i
3.13 13.3691 + 8.79011i −67.4818 + 67.4818i 101.468 + 235.033i 191.722 191.722i −1495.35 + 309.001i −3935.28 −709.426 + 4034.10i 2546.60i 4248.42 877.901i
3.14 15.6071 3.52385i −28.7960 + 28.7960i 231.165 109.995i 301.267 301.267i −347.951 + 550.897i 3827.65 3220.22 2531.29i 4902.58i 3640.29 5763.53i
3.15 15.9448 1.32739i 81.1164 81.1164i 252.476 42.3300i −62.7969 + 62.7969i 1185.72 1401.06i −2758.02 3969.50 1010.08i 6598.75i −917.931 + 1084.64i
11.1 −15.9265 + 1.53176i 110.050 + 110.050i 251.307 48.7911i −128.245 128.245i −1921.29 1584.14i 3057.02 −3927.71 + 1162.01i 17661.1i 2238.94 + 1846.06i
11.2 −15.9118 + 1.67767i −93.3844 93.3844i 250.371 53.3896i −695.406 695.406i 1642.58 + 1329.25i 805.482 −3894.28 + 1269.56i 10880.3i 12231.8 + 9898.49i
11.3 −14.5765 + 6.59745i −7.14041 7.14041i 168.947 192.335i 610.669 + 610.669i 151.190 + 56.9735i −3410.30 −1193.74 + 3918.19i 6459.03i −12930.3 4872.54i
11.4 −13.4426 8.67733i −4.15109 4.15109i 105.408 + 233.292i 178.569 + 178.569i 19.7811 + 91.8219i 678.560 607.393 4050.71i 6526.54i −850.931 3949.93i
11.5 −8.02121 + 13.8441i 17.4297 + 17.4297i −127.320 222.093i −396.274 396.274i −381.106 + 101.492i 432.698 4095.96 + 18.8169i 5953.41i 8664.66 2307.47i
See all 30 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3.15 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 16.9.f.a 30
4.b odd 2 1 64.9.f.a 30
8.b even 2 1 128.9.f.b 30
8.d odd 2 1 128.9.f.a 30
16.e even 4 1 64.9.f.a 30
16.e even 4 1 128.9.f.a 30
16.f odd 4 1 inner 16.9.f.a 30
16.f odd 4 1 128.9.f.b 30

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.f.a 30 1.a even 1 1 trivial
16.9.f.a 30 16.f odd 4 1 inner
64.9.f.a 30 4.b odd 2 1
64.9.f.a 30 16.e even 4 1
128.9.f.a 30 8.d odd 2 1
128.9.f.a 30 16.e even 4 1
128.9.f.b 30 8.b even 2 1
128.9.f.b 30 16.f odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(16, [\chi])$$.