Properties

Label 43.9.f.a
Level 43
Weight 9
Character orbit 43.f
Analytic conductor 17.517
Analytic rank 0
Dimension 174
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 43.f (of order \(14\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.5172802326\)
Analytic rank: \(0\)
Dimension: \(174\)
Relative dimension: \(29\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 174q - 7q^{2} - 7q^{3} + 4021q^{4} - 7q^{5} + 1780q^{6} - 5383q^{8} + 56490q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 174q - 7q^{2} - 7q^{3} + 4021q^{4} - 7q^{5} + 1780q^{6} - 5383q^{8} + 56490q^{9} - 24989q^{10} - 14864q^{11} - 1799q^{12} + 133078q^{13} - 243363q^{14} - 7457q^{15} - 558511q^{16} + 55162q^{17} - 45934q^{18} + 619745q^{19} + 1207493q^{20} + 307272q^{21} + 432761q^{22} + 473962q^{23} + 179363q^{24} + 2991984q^{25} + 112889q^{26} - 7q^{27} + 458745q^{28} + 1369823q^{29} + 4208148q^{30} + 608540q^{31} - 15074437q^{32} - 10988257q^{33} + 1542905q^{34} - 4078926q^{35} + 53343072q^{36} - 15207265q^{38} - 16862587q^{39} - 537581q^{40} - 852935q^{41} + 2982351q^{43} - 10366284q^{44} + 52423812q^{45} + 51003036q^{46} + 5828578q^{47} - 32078599q^{48} - 145853506q^{49} - 30015937q^{51} + 71975797q^{52} - 7800770q^{53} + 43151245q^{54} - 53485957q^{55} + 25379079q^{56} - 21662515q^{57} - 191329035q^{58} + 69716890q^{59} - 106388277q^{60} - 49131775q^{61} + 67656953q^{62} + 355075q^{63} + 227160853q^{64} - 87064999q^{65} + 146686057q^{66} - 31581292q^{67} + 200919430q^{68} - 30179079q^{69} + 86518670q^{70} - 60255454q^{71} - 438784346q^{72} - 50897868q^{73} + 5087868q^{74} + 206284834q^{75} + 461519674q^{76} + 256281375q^{77} - 182130842q^{78} - 236682026q^{79} - 488977844q^{81} - 226181851q^{82} - 429023996q^{83} - 513228846q^{84} + 336855461q^{86} + 255103986q^{87} + 977894029q^{88} - 258201748q^{89} + 1399550866q^{90} + 246776320q^{91} + 363749100q^{92} - 637129955q^{94} + 60088655q^{95} - 1169196874q^{96} + 218930393q^{97} - 330217825q^{98} + 459482633q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2.1 −13.4582 27.9461i −47.5652 + 98.7701i −440.252 + 552.058i 167.318 + 38.1893i 3400.38 4550.37i 13611.4 + 3106.71i −3402.37 4266.43i −1184.55 5189.85i
2.2 −13.1485 27.3031i 64.2865 133.492i −412.962 + 517.838i −972.457 221.957i −4490.02 292.786i 12005.0 + 2740.07i −9596.72 12033.9i 6726.21 + 29469.4i
2.3 −12.3921 25.7324i 27.0409 56.1510i −348.980 + 437.608i 1129.38 + 257.774i −1780.00 11.8724i 8457.04 + 1930.26i 1668.99 + 2092.85i −7362.24 32256.1i
2.4 −11.5206 23.9228i −29.0150 + 60.2502i −279.962 + 351.061i −438.800 100.153i 1775.62 3131.86i 4996.73 + 1140.47i 1302.50 + 1633.28i 2659.29 + 11651.1i
2.5 −10.2670 21.3197i 22.6641 47.0626i −189.504 + 237.631i 67.6603 + 15.4430i −1236.05 643.549i 1106.00 + 252.436i 2389.49 + 2996.33i −365.429 1601.05i
2.6 −9.26628 19.2416i 5.00685 10.3968i −124.763 + 156.448i −758.071 173.025i −246.447 4635.90i −1163.81 265.632i 4007.69 + 5025.49i 3695.22 + 16189.8i
2.7 −7.83493 16.2694i −45.2846 + 94.0344i −43.6938 + 54.7902i 843.199 + 192.455i 1884.69 324.806i −3273.12 747.068i −2701.06 3387.03i −3475.28 15226.2i
2.8 −6.10369 12.6745i 64.4015 133.731i 36.2267 45.4269i 363.007 + 82.8539i −2088.06 1931.33i −4307.89 983.247i −9645.74 12095.4i −1165.55 5106.63i
2.9 −5.86394 12.1766i −67.7996 + 140.787i 45.7298 57.3434i −958.314 218.729i 2111.88 747.870i −4339.50 990.462i −11133.6 13961.0i 2956.12 + 12951.6i
2.10 −5.51717 11.4565i 38.3400 79.6139i 58.8006 73.7336i 121.203 + 27.6637i −1123.63 1965.00i −4342.77 991.209i −777.697 975.201i −351.766 1541.19i
2.11 −5.04009 10.4659i −24.0340 + 49.9071i 75.4818 94.6511i 11.0383 + 2.51943i 643.453 674.726i −4270.24 974.653i 2177.63 + 2730.67i −29.2663 128.224i
2.12 −3.74733 7.78141i 23.1343 48.0389i 113.106 141.830i −1134.95 259.044i −460.503 3469.08i −3683.04 840.631i 2318.17 + 2906.90i 2237.29 + 9802.22i
2.13 −1.08134 2.24542i −24.6440 + 51.1739i 155.741 195.293i −234.966 53.6295i 141.556 271.708i −1228.94 280.497i 2079.28 + 2607.33i 133.657 + 585.590i
2.14 −0.911670 1.89310i 5.18580 10.7684i 156.861 196.697i 618.914 + 141.263i −25.1134 3955.82i −1039.79 237.325i 4001.65 + 5017.91i −296.820 1300.45i
2.15 −0.397118 0.824624i 14.6893 30.5027i 159.091 199.494i 1129.77 + 257.863i −30.9866 4070.52i −456.118 104.106i 3376.08 + 4233.47i −236.012 1034.04i
2.16 1.53552 + 3.18853i 57.5019 119.404i 151.804 190.357i −620.261 141.571i 469.018 2752.04i 1723.33 + 393.338i −6860.10 8602.29i −501.019 2195.11i
2.17 3.22586 + 6.69857i −59.4708 + 123.492i 125.149 156.932i 267.347 + 61.0203i −1019.07 4439.62i 3310.53 + 755.607i −7622.86 9558.77i 453.677 + 1987.69i
2.18 3.66053 + 7.60117i −21.3391 + 44.3111i 115.235 144.500i −661.934 151.082i −414.929 1496.55i 3625.83 + 827.571i 2582.60 + 3238.48i −1274.63 5584.51i
2.19 3.66094 + 7.60201i −55.7168 + 115.697i 115.225 144.488i 474.386 + 108.275i −1083.51 3609.61i 3626.10 + 827.633i −6190.76 7762.96i 913.584 + 4002.67i
2.20 4.47023 + 9.28252i 45.7755 95.0539i 93.4311 117.159i 498.592 + 113.800i 1086.97 1744.42i 4076.58 + 930.453i −2849.12 3572.69i 1172.46 + 5136.90i
See next 80 embeddings (of 174 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 39.29
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.f odd 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.9.f.a 174
43.f odd 14 1 inner 43.9.f.a 174
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.9.f.a 174 1.a even 1 1 trivial
43.9.f.a 174 43.f odd 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(43, [\chi])\).

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database