# Properties

 Label 43.8.g.a Level $43$ Weight $8$ Character orbit 43.g Analytic conductor $13.433$ Analytic rank $0$ Dimension $300$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 43.g (of order $$21$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4325560958$$ Analytic rank: $$0$$ Dimension: $$300$$ Relative dimension: $$25$$ over $$\Q(\zeta_{21})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{21}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$300q - 40q^{2} - 12q^{3} - 3028q^{4} + 235q^{5} + 1274q^{6} - 4122q^{7} + 762q^{8} + 8097q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$300q - 40q^{2} - 12q^{3} - 3028q^{4} + 235q^{5} + 1274q^{6} - 4122q^{7} + 762q^{8} + 8097q^{9} - 1681q^{10} + 4580q^{11} + 9330q^{12} - 20444q^{13} - 45798q^{14} + 50477q^{15} - 74504q^{16} - 25312q^{17} + 2046q^{18} + 4046q^{19} + 140781q^{20} + 100555q^{21} + 95810q^{22} + 1648q^{23} - 292592q^{24} + 419068q^{25} + 393435q^{26} + 274140q^{27} + 233644q^{28} + 605697q^{29} - 1216016q^{30} + 358580q^{31} - 19194q^{32} + 1280291q^{33} + 1137255q^{34} + 652739q^{35} - 8980975q^{36} - 2360047q^{37} - 308490q^{38} + 536339q^{39} + 3057993q^{40} - 1376320q^{41} + 8417790q^{42} + 7173000q^{43} - 3856780q^{44} + 2761486q^{45} - 7073993q^{46} + 1437090q^{47} - 9543469q^{48} - 16053170q^{49} - 3046893q^{50} + 2972901q^{51} + 17415142q^{52} + 3320604q^{53} + 6000486q^{54} + 13427750q^{55} - 5450793q^{56} + 3333100q^{57} - 7876190q^{58} - 7084765q^{59} + 7356443q^{60} + 3354606q^{61} - 719427q^{62} + 14716815q^{63} - 16086234q^{64} - 19914646q^{65} - 8293345q^{66} + 13025470q^{67} - 22191927q^{68} - 33964295q^{69} + 8094727q^{70} + 30843495q^{71} + 93841607q^{72} - 3067637q^{73} + 18179570q^{74} - 41508571q^{75} - 19751872q^{76} - 15169179q^{77} - 69509396q^{78} - 12090149q^{79} - 69662530q^{80} + 3387954q^{81} + 39226888q^{82} + 16408777q^{83} + 98603231q^{84} - 13423424q^{85} + 111906660q^{86} + 49022010q^{87} + 76213954q^{88} - 12540205q^{89} - 92608607q^{90} - 59091937q^{91} - 50489370q^{92} - 52218334q^{93} - 3441600q^{94} - 79947551q^{95} - 23449784q^{96} - 116467587q^{97} + 73909552q^{98} - 20061115q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −13.3861 16.7857i 9.91222 + 1.49403i −74.0876 + 324.599i 247.262 + 168.580i −107.608 186.382i −560.464 + 970.753i 3964.38 1909.15i −1993.82 615.011i −480.145 6407.09i
9.2 −12.0734 15.1396i −73.9419 11.1450i −54.9571 + 240.783i −26.8606 18.3132i 724.002 + 1254.01i 53.2365 92.2083i 2075.71 999.609i 3253.36 + 1003.53i 47.0442 + 627.761i
9.3 −11.9571 14.9937i 22.8341 + 3.44169i −53.3566 + 233.771i −375.006 255.675i −221.425 383.520i 97.1595 168.285i 1931.43 930.128i −1580.29 487.454i 650.465 + 8679.85i
9.4 −10.7744 13.5107i 85.1508 + 12.8344i −37.9683 + 166.350i 153.575 + 104.706i −744.050 1288.73i 118.145 204.633i 663.700 319.621i 4996.09 + 1541.09i −240.036 3203.05i
9.5 −9.86658 12.3723i −10.7092 1.61415i −27.2418 + 119.354i 180.379 + 122.981i 85.6924 + 148.424i 797.705 1381.66i −79.5102 + 38.2901i −1977.76 610.057i −258.175 3445.11i
9.6 −7.47390 9.37197i −35.4123 5.33754i −3.49202 + 15.2995i −37.7951 25.7682i 214.645 + 371.775i −223.831 + 387.687i −1212.93 + 584.115i −864.297 266.600i 40.9773 + 546.804i
9.7 −6.81088 8.54058i 54.1377 + 8.15994i 1.92934 8.45297i −122.186 83.3051i −299.035 517.944i −199.852 + 346.154i −1345.11 + 647.772i 774.466 + 238.891i 120.722 + 1610.92i
9.8 −6.35832 7.97308i −51.9023 7.82301i 5.34089 23.4000i −119.077 81.1857i 267.638 + 463.562i −639.426 + 1107.52i −1396.60 + 672.566i 542.807 + 167.434i 109.833 + 1465.62i
9.9 −5.72725 7.18175i 9.19934 + 1.38658i 9.70661 42.5274i 445.851 + 303.976i −42.7289 74.0086i −356.405 + 617.311i −1420.36 + 684.008i −2007.13 619.118i −370.422 4942.94i
9.10 −2.70223 3.38849i −50.7403 7.64788i 24.3029 106.478i −388.722 265.026i 111.197 + 192.599i 811.865 1406.19i −926.290 + 446.078i 426.254 + 131.482i 152.378 + 2033.34i
9.11 −2.56596 3.21761i −90.2338 13.6006i 24.7138 108.278i 394.907 + 269.243i 187.775 + 325.235i 265.754 460.300i −886.424 + 426.880i 5867.33 + 1809.83i −146.996 1961.52i
9.12 −1.23932 1.55406i 45.7182 + 6.89092i 27.6035 120.939i 82.5181 + 56.2599i −45.9507 79.5890i 577.502 1000.26i −451.388 + 217.377i −47.1653 14.5486i −14.8352 197.962i
9.13 −0.913875 1.14596i 31.0397 + 4.67849i 28.0046 122.696i −325.645 222.021i −23.0051 39.8460i −420.198 + 727.804i −335.233 + 161.440i −1148.26 354.191i 43.1710 + 576.078i
9.14 2.22112 + 2.78519i −37.3010 5.62222i 25.6587 112.418i 39.7972 + 27.1333i −67.1910 116.378i −76.5793 + 132.639i 780.928 376.075i −730.081 225.200i 12.8229 + 171.109i
9.15 3.07492 + 3.85582i 78.3361 + 11.8073i 23.0704 101.078i 247.532 + 168.764i 195.350 + 338.356i −868.409 + 1504.13i 1029.43 495.748i 3907.29 + 1205.24i 110.414 + 1473.37i
9.16 3.34377 + 4.19295i −17.0374 2.56798i 22.0826 96.7502i 150.846 + 102.845i −46.2018 80.0239i −185.160 + 320.707i 1097.99 528.764i −1806.16 557.126i 73.1696 + 976.380i
9.17 5.94589 + 7.45591i 86.0795 + 12.9744i 8.24571 36.1268i −256.150 174.640i 415.083 + 718.945i 380.244 658.602i 1418.17 682.954i 5151.52 + 1589.03i −220.939 2948.23i
9.18 6.51857 + 8.17403i −88.4724 13.3351i 4.15968 18.2247i −356.783 243.251i −467.712 810.101i −739.305 + 1280.51i 1381.79 665.437i 5559.70 + 1714.94i −337.379 4502.01i
9.19 7.72335 + 9.68478i 15.1267 + 2.27999i −5.66207 + 24.8071i −308.593 210.395i 94.7479 + 164.108i 75.0831 130.048i 1144.57 551.196i −1866.22 575.652i −345.743 4613.62i
9.20 8.65759 + 10.8563i 35.8811 + 5.40821i −14.4222 + 63.1876i 369.067 + 251.626i 251.931 + 436.357i 471.340 816.384i 790.511 380.690i −831.633 256.525i 463.514 + 6185.17i
See next 80 embeddings (of 300 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 40.25 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.g even 21 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.8.g.a 300
43.g even 21 1 inner 43.8.g.a 300

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.8.g.a 300 1.a even 1 1 trivial
43.8.g.a 300 43.g even 21 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{8}^{\mathrm{new}}(43, [\chi])$$.