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

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 27.f (of order \(18\) and degree \(6\))

Newform invariants

Self dual: No
Analytic conductor: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
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.

\(\operatorname{Tr}(f)(q) = \) \(138q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 447q^{5} \) \(\mathstrut -\mathstrut 774q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut -\mathstrut 12960q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \(138q \) \(\mathstrut -\mathstrut 6q^{2} \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 447q^{5} \) \(\mathstrut -\mathstrut 774q^{6} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut -\mathstrut 9q^{8} \) \(\mathstrut -\mathstrut 12960q^{9} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 28668q^{11} \) \(\mathstrut +\mathstrut 77421q^{12} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut -\mathstrut 120975q^{14} \) \(\mathstrut +\mathstrut 105507q^{15} \) \(\mathstrut -\mathstrut 774q^{16} \) \(\mathstrut -\mathstrut 9q^{17} \) \(\mathstrut -\mathstrut 241317q^{18} \) \(\mathstrut -\mathstrut 3q^{19} \) \(\mathstrut +\mathstrut 137913q^{20} \) \(\mathstrut +\mathstrut 1164372q^{21} \) \(\mathstrut -\mathstrut 185478q^{22} \) \(\mathstrut +\mathstrut 68376q^{23} \) \(\mathstrut -\mathstrut 2624256q^{24} \) \(\mathstrut +\mathstrut 507585q^{25} \) \(\mathstrut +\mathstrut 3500568q^{27} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut -\mathstrut 943980q^{29} \) \(\mathstrut -\mathstrut 1533627q^{30} \) \(\mathstrut +\mathstrut 920739q^{31} \) \(\mathstrut -\mathstrut 3005136q^{32} \) \(\mathstrut -\mathstrut 1245285q^{33} \) \(\mathstrut +\mathstrut 660474q^{34} \) \(\mathstrut +\mathstrut 6225408q^{35} \) \(\mathstrut +\mathstrut 4021128q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 23716884q^{38} \) \(\mathstrut -\mathstrut 5699685q^{39} \) \(\mathstrut -\mathstrut 975273q^{40} \) \(\mathstrut +\mathstrut 16694382q^{41} \) \(\mathstrut +\mathstrut 1594134q^{42} \) \(\mathstrut +\mathstrut 4412514q^{43} \) \(\mathstrut +\mathstrut 17341119q^{44} \) \(\mathstrut +\mathstrut 12986721q^{45} \) \(\mathstrut -\mathstrut 3q^{46} \) \(\mathstrut -\mathstrut 11341869q^{47} \) \(\mathstrut -\mathstrut 26229183q^{48} \) \(\mathstrut +\mathstrut 11347482q^{49} \) \(\mathstrut -\mathstrut 40948977q^{50} \) \(\mathstrut -\mathstrut 16497396q^{51} \) \(\mathstrut +\mathstrut 14465511q^{52} \) \(\mathstrut +\mathstrut 34106454q^{54} \) \(\mathstrut -\mathstrut 12q^{55} \) \(\mathstrut +\mathstrut 52215771q^{56} \) \(\mathstrut +\mathstrut 26766582q^{57} \) \(\mathstrut -\mathstrut 19078611q^{58} \) \(\mathstrut +\mathstrut 76116738q^{59} \) \(\mathstrut -\mathstrut 24031170q^{60} \) \(\mathstrut +\mathstrut 34059450q^{61} \) \(\mathstrut -\mathstrut 223709616q^{62} \) \(\mathstrut -\mathstrut 149069421q^{63} \) \(\mathstrut +\mathstrut 100663293q^{64} \) \(\mathstrut +\mathstrut 20396037q^{65} \) \(\mathstrut +\mathstrut 83336805q^{66} \) \(\mathstrut -\mathstrut 103603884q^{67} \) \(\mathstrut +\mathstrut 101921427q^{68} \) \(\mathstrut +\mathstrut 38038563q^{69} \) \(\mathstrut +\mathstrut 135373629q^{70} \) \(\mathstrut +\mathstrut 125718795q^{71} \) \(\mathstrut +\mathstrut 183269700q^{72} \) \(\mathstrut -\mathstrut 7632642q^{73} \) \(\mathstrut -\mathstrut 66643887q^{74} \) \(\mathstrut -\mathstrut 167459145q^{75} \) \(\mathstrut -\mathstrut 203790342q^{76} \) \(\mathstrut -\mathstrut 343269159q^{77} \) \(\mathstrut -\mathstrut 344470788q^{78} \) \(\mathstrut -\mathstrut 68767890q^{79} \) \(\mathstrut +\mathstrut 151392924q^{81} \) \(\mathstrut -\mathstrut 12q^{82} \) \(\mathstrut +\mathstrut 383244663q^{83} \) \(\mathstrut +\mathstrut 836800692q^{84} \) \(\mathstrut +\mathstrut 170435619q^{85} \) \(\mathstrut +\mathstrut 71426730q^{86} \) \(\mathstrut -\mathstrut 145695735q^{87} \) \(\mathstrut -\mathstrut 192774918q^{88} \) \(\mathstrut +\mathstrut 135692730q^{89} \) \(\mathstrut -\mathstrut 821329020q^{90} \) \(\mathstrut +\mathstrut 77546796q^{91} \) \(\mathstrut -\mathstrut 1343159175q^{92} \) \(\mathstrut -\mathstrut 414573177q^{93} \) \(\mathstrut -\mathstrut 44451609q^{94} \) \(\mathstrut +\mathstrut 881099997q^{95} \) \(\mathstrut +\mathstrut 1578184830q^{96} \) \(\mathstrut +\mathstrut 31339344q^{97} \) \(\mathstrut +\mathstrut 1293135102q^{98} \) \(\mathstrut +\mathstrut 228876057q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 23.23
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

There are no other newforms in \(S_{9}^{\mathrm{new}}(27, [\chi])\).