Properties

Label 43.10.g.a
Level 43
Weight 10
Character orbit 43.g
Analytic conductor 22.147
Analytic rank 0
Dimension 384
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 43.g (of order \(21\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(22.1465409550\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(32\) over \(\Q(\zeta_{21})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 384q + 56q^{2} - 321q^{3} - 16052q^{4} - 695q^{5} + 1274q^{6} + 32650q^{7} - 50022q^{8} + 373611q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 384q + 56q^{2} - 321q^{3} - 16052q^{4} - 695q^{5} + 1274q^{6} + 32650q^{7} - 50022q^{8} + 373611q^{9} + 6223q^{10} + 180668q^{11} - 270774q^{12} + 11245q^{13} - 271206q^{14} - 835123q^{15} - 6169448q^{16} - 65194q^{17} - 466218q^{18} - 1417919q^{19} + 5877261q^{20} + 3242929q^{21} - 4296822q^{22} - 2544899q^{23} - 6043976q^{24} + 9428207q^{25} - 11482629q^{26} + 20117988q^{27} + 7310292q^{28} - 5289921q^{29} + 23977480q^{30} - 47539873q^{31} + 18747558q^{32} - 68064193q^{33} - 73743233q^{34} - 15006505q^{35} - 203628199q^{36} + 62563960q^{37} + 108303102q^{38} + 70931426q^{39} + 14772713q^{40} - 35636128q^{41} + 29601006q^{42} - 207110915q^{43} + 367809332q^{44} + 122079484q^{45} + 164249271q^{46} + 25529106q^{47} + 191670995q^{48} - 815972056q^{49} - 210640197q^{50} - 392161947q^{51} - 873472922q^{52} + 87568125q^{53} + 627150582q^{54} + 628156420q^{55} + 725546751q^{56} - 402646583q^{57} - 677654606q^{58} + 552834314q^{59} + 990764675q^{60} + 376844345q^{61} + 76529685q^{62} - 505887225q^{63} + 621236102q^{64} + 8885906q^{65} + 895281647q^{66} + 789662056q^{67} - 2265959151q^{68} - 2055955619q^{69} + 3236566615q^{70} + 635670789q^{71} - 1964730073q^{72} + 1333013552q^{73} - 5184014638q^{74} + 720042326q^{75} + 477329288q^{76} + 2282266995q^{77} + 5133145036q^{78} + 2903853142q^{79} + 6779574014q^{80} + 3699743739q^{81} - 1268184088q^{82} - 3074947634q^{83} - 1313116801q^{84} + 1946616146q^{85} - 10088814540q^{86} - 6085081326q^{87} - 8012630430q^{88} - 1632065068q^{89} + 3316166089q^{90} + 5386571784q^{91} + 2280339030q^{92} + 2459095364q^{93} - 5048567488q^{94} - 1597173221q^{95} - 10288936616q^{96} - 4337315288q^{97} - 14989446392q^{98} - 5338096561q^{99} + O(q^{100}) \)

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 −27.2199 34.1327i −145.141 21.8765i −310.186 + 1359.01i −589.424 401.863i 3204.02 + 5549.53i 4835.78 8375.81i 34691.1 16706.4i 1778.80 + 548.686i 2327.42 + 31057.3i
9.2 −27.0878 33.9670i 238.836 + 35.9987i −306.078 + 1341.02i −1078.90 735.578i −5246.77 9087.66i −2254.62 + 3905.12i 33800.1 16277.3i 36938.2 + 11393.9i 4239.49 + 56572.0i
9.3 −24.7402 31.0232i 98.3493 + 14.8238i −236.433 + 1035.88i 1493.82 + 1018.47i −1973.30 3417.86i 2565.02 4442.74i 19681.4 9478.07i −9355.69 2885.85i −5361.22 71540.5i
9.4 −23.2329 29.1331i −34.9255 5.26417i −195.041 + 854.531i −829.320 565.421i 658.058 + 1139.79i −4562.90 + 7903.17i 12237.4 5893.22i −17616.5 5433.96i 2795.03 + 37297.1i
9.5 −22.7164 28.4855i −193.287 29.1333i −181.456 + 795.011i 1708.11 + 1164.57i 3560.90 + 6167.67i −3657.76 + 6335.43i 9961.31 4797.11i 17702.5 + 5460.49i −5628.79 75111.0i
9.6 −19.3029 24.2051i −19.7836 2.98190i −99.3527 + 435.293i −962.959 656.534i 309.704 + 536.424i 642.011 1112.00i −1827.39 + 880.025i −18426.0 5683.68i 2696.44 + 35981.5i
9.7 −18.0000 22.5713i 152.742 + 23.0222i −71.5330 + 313.406i 827.323 + 564.059i −2229.72 3861.99i −595.853 + 1032.05i −4955.94 + 2386.65i 3991.59 + 1231.24i −2160.28 28826.9i
9.8 −16.6235 20.8452i 172.616 + 26.0176i −44.2510 + 193.876i −1824.53 1243.94i −2327.13 4030.71i 5716.05 9900.48i −7522.08 + 3622.44i 10310.8 + 3180.45i 4399.80 + 58711.2i
9.9 −16.1357 20.2335i −257.419 38.7997i −35.1038 + 153.800i −1370.08 934.106i 3368.58 + 5834.56i 71.3754 123.626i −8259.84 + 3977.73i 45950.8 + 14173.9i 3206.97 + 42794.0i
9.10 −13.7971 17.3010i −134.920 20.3359i 4.96529 21.7544i 1094.58 + 746.274i 1509.67 + 2614.83i 2836.01 4912.12i −10652.8 + 5130.13i −1018.71 314.231i −2190.77 29233.8i
9.11 −10.1893 12.7770i 228.731 + 34.4757i 54.5013 238.786i −780.841 532.368i −1890.12 3273.78i −4940.14 + 8556.57i −11145.0 + 5367.13i 32320.8 + 9969.64i 1154.17 + 15401.3i
9.12 −9.77399 12.2562i −129.957 19.5879i 59.2473 259.579i 828.781 + 565.053i 1030.13 + 1784.23i 3211.01 5561.63i −10991.9 + 5293.44i −2303.38 710.499i −1175.09 15680.5i
9.13 −6.76674 8.48522i 43.2869 + 6.52445i 87.7205 384.328i 1337.26 + 911.727i −237.550 411.448i −3892.00 + 6741.15i −8861.15 + 4267.30i −16977.4 5236.82i −1312.67 17516.3i
9.14 −5.98413 7.50386i −64.3608 9.70083i 93.4326 409.355i −1644.31 1121.07i 312.350 + 541.005i −2499.13 + 4328.62i −8058.28 + 3880.66i −14760.3 4552.96i 1427.40 + 19047.3i
9.15 −4.32399 5.42212i 83.5118 + 12.5874i 103.228 452.273i −697.274 475.394i −292.854 507.238i 3015.21 5222.49i −6097.79 + 2936.54i −11992.8 3699.28i 437.370 + 5836.30i
9.16 −3.71937 4.66394i 263.417 + 39.7037i 106.012 464.469i 1636.60 + 1115.81i −794.568 1376.23i 3931.83 6810.12i −5312.37 + 2558.30i 49003.4 + 15115.6i −883.022 11783.1i
9.17 0.815943 + 1.02316i −202.451 30.5146i 113.550 497.493i 626.829 + 427.365i −133.967 232.038i −4732.25 + 8196.49i 1205.35 580.466i 21246.7 + 6553.73i 74.1941 + 990.052i
9.18 3.74995 + 4.70229i 29.2285 + 4.40549i 105.881 463.896i 328.917 + 224.252i 88.8898 + 153.962i 2917.98 5054.09i 5352.87 2577.81i −17973.6 5544.13i 178.926 + 2387.60i
9.19 5.76177 + 7.22503i −210.346 31.7046i 94.9276 415.905i −491.581 335.154i −982.900 1702.43i 3301.83 5718.94i 7814.79 3763.41i 24431.8 + 7536.22i −410.877 5482.77i
9.20 6.85017 + 8.58985i 188.212 + 28.3684i 87.0701 381.479i −1245.58 849.221i 1045.61 + 1811.04i −1101.80 + 1908.38i 8941.47 4305.99i 15810.5 + 4876.89i −1237.75 16516.6i
See next 80 embeddings (of 384 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 40.32
Significant digits:
Format:

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 43.10.g.a 384
43.g even 21 1 inner 43.10.g.a 384
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.10.g.a 384 1.a even 1 1 trivial
43.10.g.a 384 43.g even 21 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database