Properties

Label 43.5.h.a
Level 43
Weight 5
Character orbit 43.h
Analytic conductor 4.445
Analytic rank 0
Dimension 168
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 43 \)
Weight: \( k \) = \( 5 \)
Character orbit: \([\chi]\) = 43.h (of order \(42\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.44490841261\)
Analytic rank: \(0\)
Dimension: \(168\)
Relative dimension: \(14\) over \(\Q(\zeta_{42})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 168q - 14q^{2} - 20q^{3} + 220q^{4} - 11q^{5} - 22q^{6} - 150q^{7} + 322q^{8} - 660q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 168q - 14q^{2} - 20q^{3} + 220q^{4} - 11q^{5} - 22q^{6} - 150q^{7} + 322q^{8} - 660q^{9} - 105q^{10} + 362q^{11} + 1124q^{12} - 166q^{13} - 876q^{14} + 1157q^{15} - 2752q^{16} - 37q^{17} + 988q^{18} - 1004q^{19} + 1909q^{20} - 2265q^{21} + 2338q^{22} - 976q^{23} + 3016q^{24} - 2199q^{25} + 1957q^{26} - 14q^{27} + 2022q^{28} + 571q^{29} - 7896q^{30} - 7476q^{31} + 21616q^{32} - 2900q^{33} - 1559q^{34} - 1623q^{35} - 9123q^{36} - 11949q^{37} - 19394q^{38} + 175q^{39} - 10317q^{40} - 4162q^{41} + 1005q^{43} + 744q^{44} + 16632q^{45} + 34415q^{46} + 8486q^{47} + 37831q^{48} + 20918q^{49} + 14433q^{50} + 25270q^{51} - 40864q^{52} - 7522q^{53} + 23330q^{54} - 30896q^{55} - 38943q^{56} + 14884q^{57} - 2644q^{58} - 2743q^{59} - 15231q^{60} - 164q^{61} + 15097q^{62} - 22771q^{63} - 7274q^{64} + 19138q^{65} + 16631q^{66} - 31599q^{67} - 17923q^{68} + 49515q^{69} + 69209q^{70} - 2057q^{71} + 62621q^{72} - 19678q^{73} - 86988q^{74} - 28525q^{75} - 38694q^{76} - 65721q^{77} - 112636q^{78} - 293q^{79} - 99738q^{80} - 38860q^{81} - 40460q^{82} - 45580q^{83} - 28041q^{84} + 6292q^{86} + 2598q^{87} + 48902q^{88} + 83608q^{89} + 180383q^{90} + 62161q^{91} + 39390q^{92} + 203670q^{93} + 199878q^{94} - 20897q^{95} + 59662q^{96} + 93627q^{97} - 40724q^{98} + 26734q^{99} + O(q^{100}) \)

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 −3.21040 + 6.66647i 13.4095 + 1.00491i −24.1593 30.2948i −6.13495 + 19.8890i −49.7491 + 86.1681i −69.5021 + 40.1271i 164.101 37.4550i 98.7103 + 14.8782i −112.894 104.750i
3.2 −2.89615 + 6.01392i −13.0489 0.977880i −17.8037 22.3251i −7.18026 + 23.2778i 43.6724 75.6429i 24.2992 14.0292i 81.7020 18.6479i 89.2222 + 13.4481i −119.196 110.598i
3.3 −2.82198 + 5.85989i −0.311439 0.0233391i −16.3990 20.5636i 13.6957 44.4004i 1.01564 1.75914i 19.2501 11.1141i 65.3235 14.9097i −79.9988 12.0579i 221.533 + 205.552i
3.4 −2.02166 + 4.19802i −0.581783 0.0435986i −3.56045 4.46466i −5.63046 + 18.2535i 1.35920 2.35420i −16.3870 + 9.46103i −46.7413 + 10.6684i −79.7587 12.0217i −65.2457 60.5392i
3.5 −1.59415 + 3.31028i 12.0593 + 0.903718i 1.55919 + 1.95516i −2.23527 + 7.24655i −22.2158 + 38.4789i 79.2542 45.7575i −66.2699 + 15.1257i 64.5140 + 9.72393i −20.4248 18.9514i
3.6 −0.833611 + 1.73101i −5.15016 0.385951i 7.67434 + 9.62332i 0.609730 1.97669i 4.96132 8.59325i −32.4341 + 18.7258i −53.0252 + 12.1027i −53.7201 8.09700i 2.91340 + 2.70324i
3.7 −0.410655 + 0.852734i −16.8171 1.26027i 9.41732 + 11.8089i 6.39035 20.7170i 7.98071 13.8230i 31.1693 17.9956i −28.7009 + 6.55080i 201.132 + 30.3157i 15.0419 + 13.9568i
3.8 −0.317475 + 0.659244i 14.0677 + 1.05423i 9.64202 + 12.0907i 13.2386 42.9184i −5.16114 + 8.93936i −57.8093 + 33.3762i −22.4456 + 5.12306i 116.694 + 17.5888i 24.0908 + 22.3530i
3.9 0.597906 1.24156i 8.51855 + 0.638377i 8.79185 + 11.0246i −9.70706 + 31.4695i 5.88588 10.1946i −17.8087 + 10.2819i 40.4402 9.23021i −7.93707 1.19632i 33.2675 + 30.8677i
3.10 1.19998 2.49178i −0.581777 0.0435981i 5.20683 + 6.52916i 5.37775 17.4342i −0.806755 + 1.39734i 61.1701 35.3166i 65.6584 14.9861i −79.7587 12.0217i −36.9890 34.3208i
3.11 1.67806 3.48454i −11.6032 0.869542i 0.649739 + 0.814747i −10.6959 + 34.6753i −22.5009 + 38.9727i −13.1758 + 7.60705i 64.2586 14.6666i 53.7836 + 8.10658i 102.879 + 95.4577i
3.12 2.49210 5.17490i −7.43652 0.557291i −10.5932 13.2835i 9.68683 31.4039i −21.4165 + 37.0945i −66.6650 + 38.4891i −5.54483 + 1.26557i −25.1040 3.78381i −138.372 128.390i
3.13 2.55695 5.30956i 11.5931 + 0.868784i −11.6776 14.6433i 0.622954 2.01957i 34.2559 59.3329i −3.58135 + 2.06769i −15.6817 + 3.57925i 53.5501 + 8.07138i −9.13015 8.47154i
3.14 3.43387 7.13051i −5.05054 0.378485i −29.0769 36.4612i −5.90976 + 19.1590i −20.0417 + 34.7132i 59.2015 34.1800i −236.380 + 53.9522i −54.7306 8.24932i 116.320 + 107.929i
5.1 −6.99583 1.59675i 1.41459 4.58598i 31.9766 + 15.3991i −39.9449 + 15.6772i −17.2189 + 29.8240i 11.2347 6.48633i −109.351 87.2042i 47.8952 + 32.6544i 304.481 45.8931i
5.2 −6.62097 1.51119i −2.79970 + 9.07640i 27.1380 + 13.0690i 20.2385 7.94303i 32.2529 55.8637i −41.2058 + 23.7902i −74.9765 59.7917i −7.61736 5.19343i −146.002 + 22.0063i
5.3 −6.02577 1.37534i 4.35150 14.1072i 20.0029 + 9.63287i 39.0145 15.3121i −45.6234 + 79.0220i 34.4295 19.8779i −29.9675 23.8983i −113.152 77.1459i −256.152 + 38.6087i
5.4 −3.91193 0.892874i −5.06803 + 16.4302i 0.0905063 + 0.0435855i −31.2456 + 12.2630i 34.4959 59.7486i 70.0200 40.4261i 49.8789 + 39.7771i −177.340 120.908i 133.180 20.0737i
5.5 −3.69887 0.844242i −0.654121 + 2.12061i −1.44664 0.696665i 5.40819 2.12256i 4.20981 7.29161i −10.8272 + 6.25106i 52.2229 + 41.6464i 62.8562 + 42.8547i −21.7961 + 3.28523i
5.6 −2.83316 0.646651i 2.81726 9.13334i −6.80686 3.27801i −12.8396 + 5.03918i −13.8878 + 24.0544i −12.3907 + 7.15376i 53.5175 + 42.6788i −8.55552 5.83306i 39.6353 5.97406i
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 34.14
Significant digits:
Format:

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 43.5.h.a 168
43.h odd 42 1 inner 43.5.h.a 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.5.h.a 168 1.a even 1 1 trivial
43.5.h.a 168 43.h odd 42 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(43, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database