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Properties

Label 43.8.c.a
Level $43$
Weight $8$
Character orbit 43.c
Analytic conductor $13.433$
Analytic rank $0$
Dimension $50$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [43,8,Mod(6,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.6");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.c (of order \(3\), degree \(2\), 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: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(50\)
Relative dimension: \(25\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q + 26 q^{2} - 2 q^{3} + 3014 q^{4} - 249 q^{5} - 1281 q^{6} - 687 q^{7} + 5496 q^{8} - 13389 q^{9} + 1667 q^{10} - 4594 q^{11} - 8448 q^{12} + 2972 q^{13} - 15928 q^{14} - 2485 q^{15} + 185846 q^{16} + 25298 q^{17} + 13249 q^{18} + 3269 q^{19} + 2705 q^{20} - 85260 q^{21} + 150968 q^{22} - 19120 q^{23} - 55070 q^{24} - 296876 q^{25} - 139545 q^{26} - 274154 q^{27} - 118970 q^{28} - 231057 q^{29} + 240447 q^{30} - 177637 q^{31} + 1254274 q^{32} - 344986 q^{33} + 104923 q^{34} - 518850 q^{35} + 816168 q^{36} + 885140 q^{37} + 200956 q^{38} - 59884 q^{39} + 160425 q^{40} + 2202852 q^{41} - 349786 q^{42} + 1009342 q^{43} - 1012518 q^{44} + 775768 q^{45} - 1277532 q^{46} - 2784982 q^{47} - 2238049 q^{48} - 1241240 q^{49} - 5126146 q^{50} - 2477574 q^{51} - 268404 q^{52} - 398258 q^{53} + 6491644 q^{54} - 2551752 q^{55} - 4471819 q^{56} + 2094777 q^{57} - 2294852 q^{58} + 5664766 q^{59} + 1374965 q^{60} + 2652290 q^{61} - 5621243 q^{62} - 3210173 q^{63} + 5395148 q^{64} + 22093214 q^{65} - 284525 q^{66} + 5696184 q^{67} + 2422800 q^{68} + 4199245 q^{69} - 8874170 q^{70} - 16652990 q^{71} + 5916954 q^{72} + 108058 q^{73} + 8260200 q^{74} - 3594466 q^{75} - 7386456 q^{76} - 8874183 q^{77} - 27627308 q^{78} - 7378370 q^{79} + 7523523 q^{80} - 14862361 q^{81} + 7898428 q^{82} + 18099816 q^{83} + 47917640 q^{84} + 5843026 q^{85} + 14578342 q^{86} - 24244628 q^{87} + 23110740 q^{88} - 20405960 q^{89} + 32640076 q^{90} + 10283016 q^{91} - 27186571 q^{92} - 19267997 q^{93} - 52636002 q^{94} + 1708621 q^{95} + 8113092 q^{96} + 53505940 q^{97} - 3752059 q^{98} + 10929650 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −21.6932 −19.2928 + 33.4160i 342.597 52.6531 − 91.1979i 418.523 − 724.902i 19.7967 + 34.2889i −4655.31 349.079 + 604.622i −1142.22 + 1978.38i
6.2 −18.9187 27.7018 − 47.9809i 229.916 −117.718 + 203.893i −524.081 + 907.736i −543.713 − 941.739i −1928.13 −441.279 − 764.318i 2227.07 − 3857.40i
6.3 −17.9067 33.1840 − 57.4764i 192.651 49.7971 − 86.2510i −594.217 + 1029.21i 775.650 + 1343.47i −1157.68 −1108.86 − 1920.59i −891.702 + 1544.47i
6.4 −17.3293 −16.5117 + 28.5992i 172.305 −197.750 + 342.513i 286.137 − 495.604i 401.179 + 694.863i −767.772 548.224 + 949.553i 3426.87 − 5935.51i
6.5 −16.2915 8.91610 − 15.4431i 137.411 221.893 − 384.331i −145.256 + 251.591i −242.621 − 420.232i −153.326 934.506 + 1618.61i −3614.96 + 6261.30i
6.6 −13.8355 −40.6999 + 70.4944i 63.4217 −8.95282 + 15.5067i 563.105 − 975.327i −449.623 − 778.770i 893.474 −2219.47 − 3844.24i 123.867 − 214.544i
6.7 −10.8748 5.90741 − 10.2319i −9.73883 −205.317 + 355.620i −64.2419 + 111.270i −205.037 − 355.135i 1497.88 1023.71 + 1773.11i 2232.78 − 3867.29i
6.8 −9.69651 −18.8428 + 32.6367i −33.9777 119.656 − 207.250i 182.709 − 316.462i 493.624 + 854.982i 1570.62 383.397 + 664.064i −1160.25 + 2009.60i
6.9 −6.76607 25.9136 − 44.8837i −82.2203 −60.8352 + 105.370i −175.333 + 303.686i 496.526 + 860.008i 1422.37 −249.529 − 432.197i 411.615 − 712.939i
6.10 −6.04595 46.4623 − 80.4751i −91.4465 84.4107 − 146.204i −280.909 + 486.549i −519.092 − 899.094i 1326.76 −3224.00 − 5584.13i −510.343 + 883.940i
6.11 −4.76761 2.16046 − 3.74202i −105.270 48.3823 − 83.8006i −10.3002 + 17.8405i −545.058 − 944.068i 1112.14 1084.16 + 1877.83i −230.668 + 399.529i
6.12 1.06697 −34.1249 + 59.1060i −126.862 −137.791 + 238.662i −36.4102 + 63.0644i 809.683 + 1402.41i −271.930 −1235.52 − 2139.98i −147.019 + 254.645i
6.13 1.13767 −21.7565 + 37.6833i −126.706 −195.707 + 338.975i −24.7517 + 42.8713i −513.808 − 889.941i −289.771 146.811 + 254.285i −222.650 + 385.642i
6.14 2.62126 −33.9622 + 58.8242i −121.129 214.398 − 371.348i −89.0236 + 154.193i −153.973 − 266.688i −653.032 −1213.36 − 2101.60i 561.992 − 973.398i
6.15 4.85637 12.5310 − 21.7043i −104.416 −27.5688 + 47.7505i 60.8550 − 105.404i 312.159 + 540.675i −1128.70 779.450 + 1350.05i −133.884 + 231.894i
6.16 5.12015 23.6605 − 40.9812i −101.784 246.931 − 427.696i 121.145 − 209.830i 159.926 + 276.999i −1176.53 −26.1400 − 45.2757i 1264.32 − 2189.87i
6.17 7.44937 37.3503 − 64.6927i −72.5069 −250.945 + 434.650i 278.237 − 481.920i −47.3845 − 82.0723i −1493.65 −1696.60 − 2938.59i −1869.39 + 3237.87i
6.18 11.0451 −18.5527 + 32.1342i −6.00588 52.2320 − 90.4684i −204.916 + 354.925i −372.866 − 645.822i −1480.11 405.097 + 701.648i 576.907 − 999.232i
6.19 14.0250 −5.97086 + 10.3418i 68.7000 −104.947 + 181.774i −83.7412 + 145.044i 163.247 + 282.752i −831.681 1022.20 + 1770.50i −1471.88 + 2549.38i
6.20 14.6604 24.9634 − 43.2379i 86.9282 29.8218 − 51.6529i 365.974 − 633.886i −872.174 − 1510.65i −602.130 −152.844 − 264.734i 437.201 − 757.254i
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.25
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.8.c.a ✓ 50
43.c even 3 1 inner 43.8.c.a ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.8.c.a ✓ 50 1.a even 1 1 trivial
43.8.c.a ✓ 50 43.c even 3 1 inner

Hecke kernels

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