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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [43,10,Mod(9,43)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(43, base_ring=CyclotomicField(42))
 
chi = DirichletCharacter(H, H._module([2]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("43.9");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 43.g (of order \(21\), degree \(12\), 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: \(22.1465409550\)
Analytic rank: \(0\)
Dimension: \(384\)
Relative dimension: \(32\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 384 q + 56 q^{2} - 321 q^{3} - 16052 q^{4} - 695 q^{5} + 1274 q^{6} + 32650 q^{7} - 50022 q^{8} + 373611 q^{9} + 6223 q^{10} + 180668 q^{11} - 270774 q^{12} + 11245 q^{13} - 271206 q^{14} - 835123 q^{15} - 6169448 q^{16} - 65194 q^{17} - 466218 q^{18} - 1417919 q^{19} + 5877261 q^{20} + 3242929 q^{21} - 4296822 q^{22} - 2544899 q^{23} - 6043976 q^{24} + 9428207 q^{25} - 11482629 q^{26} + 20117988 q^{27} + 7310292 q^{28} - 5289921 q^{29} + 23977480 q^{30} - 47539873 q^{31} + 18747558 q^{32} - 68064193 q^{33} - 73743233 q^{34} - 15006505 q^{35} - 203628199 q^{36} + 62563960 q^{37} + 108303102 q^{38} + 70931426 q^{39} + 14772713 q^{40} - 35636128 q^{41} + 29601006 q^{42} - 207110915 q^{43} + 367809332 q^{44} + 122079484 q^{45} + 164249271 q^{46} + 25529106 q^{47} + 191670995 q^{48} - 815972056 q^{49} - 210640197 q^{50} - 392161947 q^{51} - 873472922 q^{52} + 87568125 q^{53} + 627150582 q^{54} + 628156420 q^{55} + 725546751 q^{56} - 402646583 q^{57} - 677654606 q^{58} + 552834314 q^{59} + 990764675 q^{60} + 376844345 q^{61} + 76529685 q^{62} - 505887225 q^{63} + 621236102 q^{64} + 8885906 q^{65} + 895281647 q^{66} + 789662056 q^{67} - 2265959151 q^{68} - 2055955619 q^{69} + 3236566615 q^{70} + 635670789 q^{71} - 1964730073 q^{72} + 1333013552 q^{73} - 5184014638 q^{74} + 720042326 q^{75} + 477329288 q^{76} + 2282266995 q^{77} + 5133145036 q^{78} + 2903853142 q^{79} + 6779574014 q^{80} + 3699743739 q^{81} - 1268184088 q^{82} - 3074947634 q^{83} - 1313116801 q^{84} + 1946616146 q^{85} - 10088814540 q^{86} - 6085081326 q^{87} - 8012630430 q^{88} - 1632065068 q^{89} + 3316166089 q^{90} + 5386571784 q^{91} + 2280339030 q^{92} + 2459095364 q^{93} - 5048567488 q^{94} - 1597173221 q^{95} - 10288936616 q^{96} - 4337315288 q^{97} - 14989446392 q^{98} - 5338096561 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
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 9.32
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 \(S_{10}^{\mathrm{new}}(43, [\chi])\).