# Properties

 Label 81.9.f.a Level $81$ Weight $9$ Character orbit 81.f Analytic conductor $32.998$ Analytic rank $0$ Dimension $138$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,9,Mod(8,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("81.8");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.f (of order $$18$$, degree $$6$$, not 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: $$32.9976674150$$ Analytic rank: $$0$$ Dimension: $$138$$ Relative dimension: $$23$$ over $$\Q(\zeta_{18})$$ Twist minimal: no (minimal twist has level 27) Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8}+O(q^{10})$$ 138 * q + 6 * q^2 - 6 * q^4 + 447 * q^5 - 6 * q^7 + 9 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25} - 12 q^{28} + 943980 q^{29} + 920739 q^{31} + 3005136 q^{32} + 660474 q^{34} - 6225408 q^{35} - 3 q^{37} + 23716884 q^{38} - 975273 q^{40} - 16694382 q^{41} + 4412514 q^{43} - 17341119 q^{44} - 3 q^{46} + 11341869 q^{47} + 11347482 q^{49} + 40948977 q^{50} + 14465511 q^{52} - 12 q^{55} - 52215771 q^{56} - 19078611 q^{58} - 76116738 q^{59} + 34059450 q^{61} + 223709616 q^{62} + 100663293 q^{64} - 20396037 q^{65} - 103603884 q^{67} - 101921427 q^{68} + 135373629 q^{70} - 125718795 q^{71} - 7632642 q^{73} + 66643887 q^{74} - 203790342 q^{76} + 343269159 q^{77} - 68767890 q^{79} - 12 q^{82} - 383244663 q^{83} + 170435619 q^{85} - 71426730 q^{86} - 192774918 q^{88} - 135692730 q^{89} + 77546796 q^{91} + 1343159175 q^{92} - 44451609 q^{94} - 881099997 q^{95} + 31339344 q^{97} - 1293135102 q^{98}+O(q^{100})$$ 138 * q + 6 * q^2 - 6 * q^4 + 447 * q^5 - 6 * q^7 + 9 * q^8 - 3 * q^10 - 28668 * q^11 - 6 * q^13 + 120975 * q^14 - 774 * q^16 + 9 * q^17 - 3 * q^19 - 137913 * q^20 - 185478 * q^22 - 68376 * q^23 + 507585 * q^25 - 12 * q^28 + 943980 * q^29 + 920739 * q^31 + 3005136 * q^32 + 660474 * q^34 - 6225408 * q^35 - 3 * q^37 + 23716884 * q^38 - 975273 * q^40 - 16694382 * q^41 + 4412514 * q^43 - 17341119 * q^44 - 3 * q^46 + 11341869 * q^47 + 11347482 * q^49 + 40948977 * q^50 + 14465511 * q^52 - 12 * q^55 - 52215771 * q^56 - 19078611 * q^58 - 76116738 * q^59 + 34059450 * q^61 + 223709616 * q^62 + 100663293 * q^64 - 20396037 * q^65 - 103603884 * q^67 - 101921427 * q^68 + 135373629 * q^70 - 125718795 * q^71 - 7632642 * q^73 + 66643887 * q^74 - 203790342 * q^76 + 343269159 * q^77 - 68767890 * q^79 - 12 * q^82 - 383244663 * q^83 + 170435619 * q^85 - 71426730 * q^86 - 192774918 * q^88 - 135692730 * q^89 + 77546796 * q^91 + 1343159175 * q^92 - 44451609 * q^94 - 881099997 * q^95 + 31339344 * q^97 - 1293135102 * q^98

## 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
8.1 −28.3033 4.99063i 0 535.607 + 194.945i −187.097 222.974i 0 1603.17 583.507i −7814.83 4511.89i 0 4182.68 + 7244.61i
8.2 −26.8116 4.72761i 0 455.951 + 165.953i −232.968 277.641i 0 45.0827 16.4088i −5404.32 3120.19i 0 4933.68 + 8545.38i
8.3 −26.8100 4.72733i 0 455.867 + 165.922i 418.443 + 498.682i 0 2557.31 930.784i −5401.88 3118.78i 0 −8861.04 15347.8i
8.4 −25.1387 4.43262i 0 371.742 + 135.303i 546.503 + 651.297i 0 −3663.86 + 1333.53i −3086.08 1781.75i 0 −10851.4 18795.2i
8.5 −21.0020 3.70323i 0 186.811 + 67.9935i −674.968 804.395i 0 −2905.52 + 1057.52i 1056.43 + 609.927i 0 11196.8 + 19393.5i
8.6 −14.5831 2.57139i 0 −34.5070 12.5595i 51.5257 + 61.4059i 0 3577.75 1302.19i 3753.91 + 2167.32i 0 −593.505 1027.98i
8.7 −14.2805 2.51803i 0 −42.9702 15.6399i −501.169 597.270i 0 −473.173 + 172.221i 3789.11 + 2187.64i 0 5652.98 + 9791.25i
8.8 −14.0349 2.47473i 0 −49.7072 18.0920i 245.194 + 292.211i 0 −336.792 + 122.582i 3812.44 + 2201.11i 0 −2718.13 4707.95i
8.9 −13.3375 2.35175i 0 −68.2042 24.8243i 636.805 + 758.914i 0 −2502.00 + 910.653i 3853.85 + 2225.02i 0 −6708.58 11619.6i
8.10 −7.08460 1.24921i 0 −191.930 69.8569i −349.559 416.588i 0 22.4234 8.16143i 2867.39 + 1655.49i 0 1956.08 + 3388.03i
8.11 −1.99728 0.352174i 0 −236.696 86.1504i −67.5893 80.5497i 0 3385.55 1232.24i 892.041 + 515.020i 0 106.627 + 184.683i
8.12 2.46646 + 0.434904i 0 −234.667 85.4118i 411.949 + 490.942i 0 243.334 88.5664i −1096.91 633.301i 0 802.546 + 1390.05i
8.13 3.24198 + 0.571649i 0 −230.378 83.8506i −387.667 462.003i 0 −3862.15 + 1405.71i −1428.79 824.913i 0 −992.706 1719.42i
8.14 7.66670 + 1.35185i 0 −183.611 66.8288i 24.0534 + 28.6657i 0 −2598.90 + 945.921i −3043.29 1757.05i 0 145.659 + 252.288i
8.15 7.92414 + 1.39724i 0 −179.722 65.4133i 625.377 + 745.295i 0 845.976 307.910i −3116.64 1799.40i 0 3914.22 + 6779.62i
8.16 8.24786 + 1.45432i 0 −174.649 63.5671i −723.153 861.820i 0 2191.55 797.660i −3204.82 1850.30i 0 −4711.10 8159.87i
8.17 18.1654 + 3.20306i 0 79.1620 + 28.8126i −462.775 551.514i 0 1642.82 597.939i −2743.73 1584.09i 0 −6639.98 11500.8i
8.18 18.2801 + 3.22328i 0 83.2125 + 30.2869i 223.403 + 266.242i 0 −1438.86 + 523.701i −2691.76 1554.09i 0 3225.67 + 5587.03i
8.19 21.5826 + 3.80559i 0 210.764 + 76.7119i −129.757 154.639i 0 1479.28 538.412i −601.827 347.465i 0 −2212.01 3831.31i
8.20 23.0508 + 4.06447i 0 274.257 + 99.8213i 685.642 + 817.116i 0 4384.71 1595.90i 726.861 + 419.653i 0 12483.4 + 21621.9i
See next 80 embeddings (of 138 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 8.23 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
27.f odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 81.9.f.a 138
3.b odd 2 1 27.9.f.a 138
27.e even 9 1 27.9.f.a 138
27.f odd 18 1 inner 81.9.f.a 138

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
27.9.f.a 138 3.b odd 2 1
27.9.f.a 138 27.e even 9 1
81.9.f.a 138 1.a even 1 1 trivial
81.9.f.a 138 27.f odd 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(81, [\chi])$$.