# Properties

 Label 27.9.f.a Level $27$ Weight $9$ Character orbit 27.f Analytic conductor $10.999$ 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] = [27,9,Mod(2,27)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(27, 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("27.2");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$27 = 3^{3}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 27.f (of order $$18$$, degree $$6$$, 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: $$10.9992224717$$ Analytic rank: $$0$$ Dimension: $$138$$ Relative dimension: $$23$$ over $$\Q(\zeta_{18})$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9}+O(q^{10})$$ 138 * q - 6 * q^2 - 6 * q^3 - 6 * q^4 - 447 * q^5 - 774 * q^6 - 6 * q^7 - 9 * q^8 - 12960 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17} - 241317 q^{18} - 3 q^{19} + 137913 q^{20} + 1164372 q^{21} - 185478 q^{22} + 68376 q^{23} - 2624256 q^{24} + 507585 q^{25} + 3500568 q^{27} - 12 q^{28} - 943980 q^{29} - 1533627 q^{30} + 920739 q^{31} - 3005136 q^{32} - 1245285 q^{33} + 660474 q^{34} + 6225408 q^{35} + 4021128 q^{36} - 3 q^{37} - 23716884 q^{38} - 5699685 q^{39} - 975273 q^{40} + 16694382 q^{41} + 1594134 q^{42} + 4412514 q^{43} + 17341119 q^{44} + 12986721 q^{45} - 3 q^{46} - 11341869 q^{47} - 26229183 q^{48} + 11347482 q^{49} - 40948977 q^{50} - 16497396 q^{51} + 14465511 q^{52} + 34106454 q^{54} - 12 q^{55} + 52215771 q^{56} + 26766582 q^{57} - 19078611 q^{58} + 76116738 q^{59} - 24031170 q^{60} + 34059450 q^{61} - 223709616 q^{62} - 149069421 q^{63} + 100663293 q^{64} + 20396037 q^{65} + 83336805 q^{66} - 103603884 q^{67} + 101921427 q^{68} + 38038563 q^{69} + 135373629 q^{70} + 125718795 q^{71} + 183269700 q^{72} - 7632642 q^{73} - 66643887 q^{74} - 167459145 q^{75} - 203790342 q^{76} - 343269159 q^{77} - 344470788 q^{78} - 68767890 q^{79} + 151392924 q^{81} - 12 q^{82} + 383244663 q^{83} + 836800692 q^{84} + 170435619 q^{85} + 71426730 q^{86} - 145695735 q^{87} - 192774918 q^{88} + 135692730 q^{89} - 821329020 q^{90} + 77546796 q^{91} - 1343159175 q^{92} - 414573177 q^{93} - 44451609 q^{94} + 881099997 q^{95} + 1578184830 q^{96} + 31339344 q^{97} + 1293135102 q^{98} + 228876057 q^{99}+O(q^{100})$$ 138 * q - 6 * q^2 - 6 * q^3 - 6 * q^4 - 447 * q^5 - 774 * q^6 - 6 * q^7 - 9 * q^8 - 12960 * q^9 - 3 * q^10 + 28668 * q^11 + 77421 * q^12 - 6 * q^13 - 120975 * q^14 + 105507 * q^15 - 774 * q^16 - 9 * q^17 - 241317 * q^18 - 3 * q^19 + 137913 * q^20 + 1164372 * q^21 - 185478 * q^22 + 68376 * q^23 - 2624256 * q^24 + 507585 * q^25 + 3500568 * q^27 - 12 * q^28 - 943980 * q^29 - 1533627 * q^30 + 920739 * q^31 - 3005136 * q^32 - 1245285 * q^33 + 660474 * q^34 + 6225408 * q^35 + 4021128 * q^36 - 3 * q^37 - 23716884 * q^38 - 5699685 * q^39 - 975273 * q^40 + 16694382 * q^41 + 1594134 * q^42 + 4412514 * q^43 + 17341119 * q^44 + 12986721 * q^45 - 3 * q^46 - 11341869 * q^47 - 26229183 * q^48 + 11347482 * q^49 - 40948977 * q^50 - 16497396 * q^51 + 14465511 * q^52 + 34106454 * q^54 - 12 * q^55 + 52215771 * q^56 + 26766582 * q^57 - 19078611 * q^58 + 76116738 * q^59 - 24031170 * q^60 + 34059450 * q^61 - 223709616 * q^62 - 149069421 * q^63 + 100663293 * q^64 + 20396037 * q^65 + 83336805 * q^66 - 103603884 * q^67 + 101921427 * q^68 + 38038563 * q^69 + 135373629 * q^70 + 125718795 * q^71 + 183269700 * q^72 - 7632642 * q^73 - 66643887 * q^74 - 167459145 * q^75 - 203790342 * q^76 - 343269159 * q^77 - 344470788 * q^78 - 68767890 * q^79 + 151392924 * q^81 - 12 * q^82 + 383244663 * q^83 + 836800692 * q^84 + 170435619 * q^85 + 71426730 * q^86 - 145695735 * q^87 - 192774918 * q^88 + 135692730 * q^89 - 821329020 * q^90 + 77546796 * q^91 - 1343159175 * q^92 - 414573177 * q^93 - 44451609 * q^94 + 881099997 * q^95 + 1578184830 * q^96 + 31339344 * q^97 + 1293135102 * q^98 + 228876057 * 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   $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1 −30.9899 5.46435i −38.9279 + 71.0325i 689.952 + 251.122i −413.144 492.366i 1594.52 1988.57i −1524.27 + 554.788i −13032.8 7524.48i −3530.24 5530.29i 10112.8 + 17515.9i
2.2 −29.3275 5.17124i 80.7896 5.83408i 592.802 + 215.762i 641.275 + 764.242i −2399.53 246.683i 548.101 199.492i −9667.37 5581.46i 6492.93 942.667i −14855.0 25729.5i
2.3 −24.3796 4.29877i −14.9563 79.6072i 335.322 + 122.047i −12.1292 14.4550i 22.4138 + 2005.08i −3223.30 + 1173.19i −2161.96 1248.21i −6113.62 + 2381.25i 233.565 + 404.547i
2.4 −23.0508 4.06447i 55.5291 58.9705i 274.257 + 99.8213i −685.642 817.116i −1519.67 + 1133.62i 4384.71 1595.90i −726.861 419.653i −394.048 6549.16i 12483.4 + 21621.9i
2.5 −21.5826 3.80559i −80.3561 10.1929i 210.764 + 76.7119i 129.757 + 154.639i 1695.50 + 525.792i 1479.28 538.412i 601.827 + 347.465i 6353.21 + 1638.13i −2212.01 3831.31i
2.6 −18.2801 3.22328i 61.0463 + 53.2386i 83.2125 + 30.2869i −223.403 266.242i −944.332 1169.98i −1438.86 + 523.701i 2691.76 + 1554.09i 892.309 + 6500.04i 3225.67 + 5587.03i
2.7 −18.1654 3.20306i −0.948605 + 80.9944i 79.1620 + 28.8126i 462.775 + 551.514i 276.662 1468.26i 1642.82 597.939i 2743.73 + 1584.09i −6559.20 153.664i −6639.98 11500.8i
2.8 −8.24786 1.45432i 12.3580 80.0517i −174.649 63.5671i 723.153 + 861.820i −218.348 + 642.283i 2191.55 797.660i 3204.82 + 1850.30i −6255.56 1978.56i −4711.10 8159.87i
2.9 −7.92414 1.39724i −47.7313 + 65.4425i −179.722 65.4133i −625.377 745.295i 469.668 451.883i 845.976 307.910i 3116.64 + 1799.40i −2004.44 6247.31i 3914.22 + 6779.62i
2.10 −7.66670 1.35185i 72.7671 35.5802i −183.611 66.8288i −24.0534 28.6657i −605.982 + 174.413i −2598.90 + 945.921i 3043.29 + 1757.05i 4029.09 5178.14i 145.659 + 252.288i
2.11 −3.24198 0.571649i −71.7460 + 37.5966i −230.378 83.8506i 387.667 + 462.003i 254.092 80.8741i −3862.15 + 1405.71i 1428.79 + 824.913i 3733.99 5394.82i −992.706 1719.42i
2.12 −2.46646 0.434904i −52.6972 61.5143i −234.667 85.4118i −411.949 490.942i 103.223 + 174.641i 243.334 88.5664i 1096.91 + 633.301i −1007.01 + 6483.26i 802.546 + 1390.05i
2.13 1.99728 + 0.352174i 77.6955 + 22.9000i −236.696 86.1504i 67.5893 + 80.5497i 147.115 + 73.1000i 3385.55 1232.24i −892.041 515.020i 5512.18 + 3558.45i 106.627 + 184.683i
2.14 7.08460 + 1.24921i 14.6630 + 79.6618i −191.930 69.8569i 349.559 + 416.588i 4.36750 + 582.689i 22.4234 8.16143i −2867.39 1655.49i −6130.99 + 2336.16i 1956.08 + 3388.03i
2.15 13.3375 + 2.35175i 34.9288 + 73.0820i −68.2042 24.8243i −636.805 758.914i 293.990 + 1056.87i −2502.00 + 910.653i −3853.85 2225.02i −4120.96 + 5105.33i −6708.58 11619.6i
2.16 14.0349 + 2.47473i 37.9905 71.5383i −49.7072 18.0920i −245.194 292.211i 710.230 910.016i −336.792 + 122.582i −3812.44 2201.11i −3674.45 5435.54i −2718.13 4707.95i
2.17 14.2805 + 2.51803i −64.8590 48.5213i −42.9702 15.6399i 501.169 + 597.270i −804.038 856.223i −473.173 + 172.221i −3789.11 2187.64i 1852.37 + 6294.08i 5652.98 + 9791.25i
2.18 14.5831 + 2.57139i −73.7172 + 33.5675i −34.5070 12.5595i −51.5257 61.4059i −1161.34 + 299.961i 3577.75 1302.19i −3753.91 2167.32i 4307.45 4949.00i −593.505 1027.98i
2.19 21.0020 + 3.70323i 80.9924 + 1.11237i 186.811 + 67.9935i 674.968 + 804.395i 1696.89 + 323.295i −2905.52 + 1057.52i −1056.43 609.927i 6558.53 + 180.188i 11196.8 + 19393.5i
2.20 25.1387 + 4.43262i −80.5339 8.67746i 371.742 + 135.303i −546.503 651.297i −1986.05 575.116i −3663.86 + 1333.53i 3086.08 + 1781.75i 6410.40 + 1397.66i −10851.4 18795.2i
See next 80 embeddings (of 138 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.23 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 27.9.f.a 138
3.b odd 2 1 81.9.f.a 138
27.e even 9 1 81.9.f.a 138
27.f odd 18 1 inner 27.9.f.a 138

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(27, [\chi])$$.