# Properties

 Label 43.9.h.a Level $43$ Weight $9$ Character orbit 43.h Analytic conductor $17.517$ Analytic rank $0$ Dimension $336$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 43.h (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.5172802326$$ Analytic rank: $$0$$ Dimension: $$336$$ Relative dimension: $$28$$ over $$\Q(\zeta_{42})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{42}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$336q - 14q^{2} + 70q^{3} + 6460q^{4} - 11q^{5} - 1798q^{6} + 5010q^{7} + 5362q^{8} - 40176q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$336q - 14q^{2} + 70q^{3} + 6460q^{4} - 11q^{5} - 1798q^{6} + 5010q^{7} + 5362q^{8} - 40176q^{9} - 12505q^{10} - 31942q^{11} - 54796q^{12} - 75596q^{13} + 111780q^{14} + 4037q^{15} - 688224q^{16} - 5977q^{17} - 280532q^{18} - 552290q^{19} - 494171q^{20} + 141291q^{21} - 432782q^{22} - 619546q^{23} - 298136q^{24} - 2392669q^{25} - 113675q^{26} - 14q^{27} - 1143258q^{28} + 4198855q^{29} - 795576q^{30} + 3391610q^{31} + 3704176q^{32} + 469240q^{33} + 3670777q^{34} - 2602743q^{35} - 32236515q^{36} + 7009521q^{37} + 1762126q^{38} - 573251q^{39} - 41379997q^{40} - 11003050q^{41} + 7113655q^{43} + 64459560q^{44} + 11644332q^{45} - 14793057q^{46} + 9244886q^{47} - 34730249q^{48} + 82140290q^{49} - 80542767q^{50} - 7776902q^{51} - 98888784q^{52} + 39659528q^{53} + 139217042q^{54} + 96182224q^{55} - 59364207q^{56} - 94760066q^{57} + 149491596q^{58} - 74665213q^{59} + 188098449q^{60} + 55854238q^{61} - 52173383q^{62} + 40219829q^{63} + 13742518q^{64} + 87064978q^{65} - 64965481q^{66} + 210933441q^{67} - 248367043q^{68} + 405756759q^{69} - 362301191q^{70} - 34975301q^{71} - 125412979q^{72} - 216767388q^{73} + 53341380q^{74} + 436840145q^{75} - 29249062q^{76} - 59286141q^{77} + 204702644q^{78} + 71493313q^{79} - 737037738q^{80} - 235365082q^{81} - 806662220q^{82} + 254450510q^{83} - 979300857q^{84} + 50436244q^{86} + 696609678q^{87} + 65867942q^{88} + 949059334q^{89} + 1083992303q^{90} - 184351501q^{91} + 379980750q^{92} - 271490850q^{93} - 369651898q^{94} - 1349611517q^{95} - 1044343058q^{96} - 850069303q^{97} - 271546964q^{98} + 148520290q^{99} + O(q^{100})$$

## 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 −12.8635 + 26.7113i −30.3876 2.27723i −388.410 487.050i 136.958 444.007i 451.717 782.398i 1583.71 914.357i 10606.6 2420.89i −5569.50 839.467i 10098.2 + 9369.79i
3.2 −12.1501 + 25.2300i 111.026 + 8.32026i −329.314 412.947i −217.435 + 704.908i −1558.90 + 2700.10i 939.408 542.367i 7430.79 1696.03i 5769.85 + 869.665i −15143.0 14050.6i
3.3 −11.8000 + 24.5029i −134.895 10.1090i −301.539 378.118i −246.719 + 799.843i 1839.46 3186.03i 1956.96 1129.85i 6035.48 1377.56i 11606.8 + 1749.44i −16687.2 15483.4i
3.4 −11.2950 + 23.4544i 75.5829 + 5.66415i −262.917 329.688i 125.810 407.865i −986.561 + 1708.77i −2233.56 + 1289.54i 4205.09 959.784i −807.029 121.640i 8145.21 + 7557.65i
3.5 −10.6351 + 22.0841i −39.8410 2.98567i −214.987 269.585i −185.498 + 601.371i 489.650 848.099i −2744.41 + 1584.49i 2122.34 484.410i −4909.33 739.962i −11307.9 10492.2i
3.6 −9.45062 + 19.6244i −134.063 10.0466i −136.190 170.777i 301.256 976.647i 1464.14 2535.96i −1417.77 + 818.552i −797.764 + 182.084i 11384.2 + 1715.90i 16319.1 + 15141.9i
3.7 −7.77760 + 16.1503i 155.893 + 11.6826i −40.7292 51.0728i 221.849 719.217i −1401.15 + 2426.87i 890.825 514.318i −3332.26 + 760.567i 17678.6 + 2664.61i 9890.15 + 9176.71i
3.8 −7.23745 + 15.0287i −0.134245 0.0100603i −13.8685 17.3905i −60.7674 + 197.003i 1.12279 1.94472i 4135.93 2387.88i −3801.45 + 867.655i −6487.70 977.864i −2520.90 2339.06i
3.9 −6.93493 + 14.4005i 26.0262 + 1.95039i 0.331155 + 0.415255i 249.037 807.358i −208.577 + 361.265i 1041.77 601.469i −3997.44 + 912.389i −5814.16 876.344i 9899.34 + 9185.24i
3.10 −5.92801 + 12.3096i −79.6637 5.96997i 43.2276 + 54.2057i −75.1797 + 243.727i 545.735 945.241i −916.434 + 529.103i −4333.45 + 989.083i −177.047 26.6855i −2554.52 2370.25i
3.11 −5.27963 + 10.9633i 85.9978 + 6.44465i 67.2948 + 84.3850i −236.441 + 766.523i −524.691 + 908.791i −59.3437 + 34.2621i −4317.41 + 985.420i 866.371 + 130.584i −7155.27 6639.12i
3.12 −1.75641 + 3.64723i 68.2937 + 5.11790i 149.396 + 187.337i 59.6815 193.483i −138.618 + 240.093i −4069.77 + 2349.69i −1956.00 + 446.443i −1849.89 278.826i 600.850 + 557.507i
3.13 −1.25301 + 2.60190i −146.817 11.0024i 154.414 + 193.628i −60.6007 + 196.463i 212.590 368.217i 3053.32 1762.84i −1418.05 + 323.660i 14946.6 + 2252.83i −435.242 403.846i
3.14 −0.385973 + 0.801481i −82.9635 6.21726i 159.120 + 199.530i 153.970 499.159i 37.0047 64.0940i −701.273 + 404.880i −443.358 + 101.194i 356.569 + 53.7442i 340.638 + 316.066i
3.15 0.864743 1.79566i 1.37697 + 0.103190i 157.137 + 197.043i 286.537 928.929i 1.37602 2.38334i 941.195 543.399i 987.128 225.306i −6485.83 977.582i −1420.26 1317.81i
3.16 1.44533 3.00125i −54.6425 4.09489i 152.695 + 191.473i −330.560 + 1071.65i −91.2660 + 158.077i −764.563 + 441.421i 1626.74 371.294i −3518.69 530.357i 2738.52 + 2540.98i
3.17 1.83542 3.81129i 121.748 + 9.12376i 148.456 + 186.158i 21.4535 69.5504i 258.232 447.271i 1381.02 797.333i 2037.77 465.107i 8251.65 + 1243.74i −225.701 209.420i
3.18 3.70483 7.69316i 20.6613 + 1.54835i 114.154 + 143.145i −181.686 + 589.011i 88.4583 153.214i 2375.79 1371.67i 3655.28 834.293i −6063.23 913.884i 3858.24 + 3579.93i
3.19 6.51335 13.5251i −137.776 10.3249i 19.1085 + 23.9613i −3.15223 + 10.2193i −1037.03 + 1796.19i −2870.44 + 1657.25i 4195.19 957.525i 12387.9 + 1867.18i 117.685 + 109.196i
3.20 7.08079 14.7034i −17.7574 1.33074i −6.43935 8.07469i 52.6500 170.687i −145.303 + 251.672i −2020.79 + 1166.70i 3908.74 892.144i −6174.16 930.605i −2136.88 1982.73i
See next 80 embeddings (of 336 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 34.28 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.h odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.9.h.a 336
43.h odd 42 1 inner 43.9.h.a 336

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.9.h.a 336 1.a even 1 1 trivial
43.9.h.a 336 43.h odd 42 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(43, [\chi])$$.