# Properties

 Label 38.9.f.a Level $38$ Weight $9$ Character orbit 38.f Analytic conductor $15.480$ Analytic rank $0$ Dimension $84$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$38 = 2 \cdot 19$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 38.f (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.4803871823$$ Analytic rank: $$0$$ Dimension: $$84$$ Relative dimension: $$14$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$84q - 168q^{3} - 2688q^{6} - 6570q^{7} - 2856q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$84q - 168q^{3} - 2688q^{6} - 6570q^{7} - 2856q^{9} - 20970q^{11} + 62208q^{12} - 101310q^{13} - 142848q^{14} + 331518q^{15} + 29178q^{17} - 319212q^{19} - 428544q^{20} - 186138q^{21} + 1238016q^{22} + 1989360q^{23} + 344064q^{24} - 272916q^{25} - 695808q^{26} - 6975648q^{27} - 489984q^{28} + 1676430q^{29} - 3498768q^{31} + 6273750q^{33} + 2740224q^{34} + 10942362q^{35} + 365568q^{36} + 5830272q^{38} - 23638704q^{39} + 430488q^{41} + 4230144q^{42} - 20999118q^{43} + 700416q^{44} + 49897728q^{45} + 12003840q^{46} - 22431150q^{47} - 5505024q^{48} - 59138004q^{49} - 24993792q^{50} + 6061860q^{51} - 2857728q^{52} + 69029982q^{53} + 63262080q^{54} + 20438670q^{55} - 50408076q^{57} - 31266816q^{58} - 85956678q^{59} - 9106176q^{60} + 29908896q^{61} - 39755520q^{62} - 29288274q^{63} + 88080384q^{64} + 199314810q^{65} + 117652224q^{66} + 64371642q^{67} + 25820928q^{68} - 186304482q^{69} - 187792896q^{70} - 168095268q^{71} + 23396352q^{72} + 104437176q^{73} + 33624576q^{74} + 6617088q^{76} - 277281288q^{77} + 30777600q^{78} + 199146114q^{79} + 245855040q^{81} - 69199872q^{82} - 14372262q^{83} - 348399360q^{84} - 386680068q^{85} - 155904768q^{86} + 36917496q^{87} + 292036248q^{89} + 694869504q^{90} + 285753588q^{91} + 95143680q^{92} + 529006800q^{93} - 166014666q^{95} + 251818254q^{97} - 595464192q^{98} - 1180902678q^{99} + O(q^{100})$$

## 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 −7.27231 + 8.66680i −41.5037 114.031i −22.2270 126.055i −20.6862 + 117.317i 1290.11 + 469.561i 2332.01 + 4039.17i 1254.14 + 724.077i −6254.38 + 5248.05i −866.329 1032.45i
3.2 −7.27231 + 8.66680i −27.0631 74.3551i −22.2270 126.055i 182.228 1033.47i 841.232 + 306.183i −805.065 1394.41i 1254.14 + 724.077i 229.742 192.776i 7631.64 + 9095.03i
3.3 −7.27231 + 8.66680i −6.31575 17.3524i −22.2270 126.055i −119.553 + 678.019i 196.320 + 71.4546i 116.087 + 201.069i 1254.14 + 724.077i 4764.80 3998.14i −5006.83 5966.91i
3.4 −7.27231 + 8.66680i 6.44442 + 17.7059i −22.2270 126.055i −26.7224 + 151.550i −200.319 72.9103i −536.717 929.622i 1254.14 + 724.077i 4754.05 3989.12i −1119.12 1333.72i
3.5 −7.27231 + 8.66680i 18.2825 + 50.2307i −22.2270 126.055i 35.6515 202.190i −568.295 206.843i 500.998 + 867.755i 1254.14 + 724.077i 2837.15 2380.65i 1493.07 + 1779.37i
3.6 −7.27231 + 8.66680i 47.5951 + 130.767i −22.2270 126.055i 122.997 697.551i −1479.45 538.478i 2061.80 + 3571.14i 1254.14 + 724.077i −9808.58 + 8230.38i 5151.07 + 6138.80i
3.7 −7.27231 + 8.66680i 51.6161 + 141.814i −22.2270 126.055i −125.467 + 711.560i −1604.44 583.970i −2203.20 3816.05i 1254.14 + 724.077i −12421.0 + 10422.5i −5254.51 6262.08i
3.8 7.27231 8.66680i −42.9893 118.112i −22.2270 126.055i −164.500 + 932.925i −1336.29 486.369i 512.177 + 887.116i −1254.14 724.077i −7076.39 + 5937.80i 6889.19 + 8210.21i
3.9 7.27231 8.66680i −36.7655 101.012i −22.2270 126.055i 63.9854 362.879i −1142.82 415.954i −925.290 1602.65i −1254.14 724.077i −3825.78 + 3210.21i −2679.68 3193.52i
3.10 7.27231 8.66680i −12.4163 34.1135i −22.2270 126.055i 148.158 840.244i −385.951 140.475i −14.1611 24.5277i −1254.14 724.077i 4016.45 3370.20i −6204.78 7394.57i
3.11 7.27231 8.66680i 6.75740 + 18.5658i −22.2270 126.055i −174.645 + 990.461i 210.048 + 76.4513i −1967.20 3407.30i −1254.14 724.077i 4726.99 3966.42i 7314.05 + 8716.55i
3.12 7.27231 8.66680i 11.8688 + 32.6091i −22.2270 126.055i 32.5420 184.555i 368.930 + 134.280i 1327.43 + 2299.17i −1254.14 724.077i 4103.53 3443.27i −1362.85 1624.18i
3.13 7.27231 8.66680i 31.3279 + 86.0726i −22.2270 126.055i −59.6685 + 338.397i 973.800 + 354.434i 321.792 + 557.360i −1254.14 724.077i −1401.04 + 1175.61i 2498.89 + 2978.06i
3.14 7.27231 8.66680i 49.7737 + 136.752i −22.2270 126.055i 202.576 1148.87i 1547.17 + 563.125i −1213.32 2101.54i −1254.14 724.077i −11197.7 + 9395.96i −8483.80 10110.6i
13.1 −7.27231 8.66680i −41.5037 + 114.031i −22.2270 + 126.055i −20.6862 117.317i 1290.11 469.561i 2332.01 4039.17i 1254.14 724.077i −6254.38 5248.05i −866.329 + 1032.45i
13.2 −7.27231 8.66680i −27.0631 + 74.3551i −22.2270 + 126.055i 182.228 + 1033.47i 841.232 306.183i −805.065 + 1394.41i 1254.14 724.077i 229.742 + 192.776i 7631.64 9095.03i
13.3 −7.27231 8.66680i −6.31575 + 17.3524i −22.2270 + 126.055i −119.553 678.019i 196.320 71.4546i 116.087 201.069i 1254.14 724.077i 4764.80 + 3998.14i −5006.83 + 5966.91i
13.4 −7.27231 8.66680i 6.44442 17.7059i −22.2270 + 126.055i −26.7224 151.550i −200.319 + 72.9103i −536.717 + 929.622i 1254.14 724.077i 4754.05 + 3989.12i −1119.12 + 1333.72i
13.5 −7.27231 8.66680i 18.2825 50.2307i −22.2270 + 126.055i 35.6515 + 202.190i −568.295 + 206.843i 500.998 867.755i 1254.14 724.077i 2837.15 + 2380.65i 1493.07 1779.37i
13.6 −7.27231 8.66680i 47.5951 130.767i −22.2270 + 126.055i 122.997 + 697.551i −1479.45 + 538.478i 2061.80 3571.14i 1254.14 724.077i −9808.58 8230.38i 5151.07 6138.80i
See all 84 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 33.14 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 38.9.f.a 84
19.f odd 18 1 inner 38.9.f.a 84

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.9.f.a 84 1.a even 1 1 trivial
38.9.f.a 84 19.f odd 18 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(38, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database