Properties

 Label 201.4.e.b Level 201 Weight 4 Character orbit 201.e Analytic conductor 11.859 Analytic rank 0 Dimension 36 CM no Inner twists 2

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Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.e (of order $$3$$, degree $$2$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ over $$\Q(\zeta_{3})$$ Coefficient ring index: multiple of None 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 2q^{2} - 108q^{3} - 90q^{4} - 4q^{5} - 6q^{6} + 22q^{7} + 48q^{8} + 324q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 2q^{2} - 108q^{3} - 90q^{4} - 4q^{5} - 6q^{6} + 22q^{7} + 48q^{8} + 324q^{9} + 14q^{10} - 16q^{11} + 270q^{12} - 46q^{13} + 14q^{14} + 12q^{15} - 346q^{16} - 8q^{17} + 18q^{18} - 154q^{19} - 180q^{20} - 66q^{21} + 214q^{22} - 104q^{23} - 144q^{24} + 1032q^{25} - 333q^{26} - 972q^{27} - 473q^{28} + 76q^{29} - 42q^{30} + 498q^{31} - 285q^{32} + 48q^{33} + 26q^{34} - 392q^{35} - 810q^{36} - 124q^{37} + 20q^{38} + 138q^{39} + 638q^{40} - 508q^{41} - 42q^{42} - 1400q^{43} - 333q^{44} - 36q^{45} - 1372q^{46} + 18q^{47} + 1038q^{48} - 238q^{49} - 337q^{50} + 24q^{51} + 3640q^{52} + 724q^{53} - 54q^{54} - 178q^{55} - 829q^{56} + 462q^{57} - 1472q^{58} + 720q^{59} + 540q^{60} + 232q^{61} - 3882q^{62} + 198q^{63} + 3628q^{64} - 1428q^{65} - 642q^{66} - 1164q^{67} + 1634q^{68} + 312q^{69} + 2550q^{70} + 406q^{71} + 432q^{72} - 2120q^{73} + 1375q^{74} - 3096q^{75} + 4190q^{76} - 800q^{77} + 999q^{78} + 1306q^{79} - 1927q^{80} + 2916q^{81} - 794q^{82} - 1010q^{83} + 1419q^{84} + 472q^{85} + 737q^{86} - 228q^{87} - 1838q^{88} + 1904q^{89} + 126q^{90} + 7340q^{91} + 7368q^{92} - 1494q^{93} - 9862q^{94} + 1678q^{95} + 855q^{96} - 2358q^{97} - 2610q^{98} - 144q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1 −2.72652 4.72247i −3.00000 −10.8678 + 18.8236i 12.9793 8.17955 + 14.1674i −14.7538 + 25.5543i 74.9007 9.00000 −35.3882 61.2942i
37.2 −2.52313 4.37020i −3.00000 −8.73242 + 15.1250i 0.340792 7.56940 + 13.1106i 12.2513 21.2198i 47.7621 9.00000 −0.859864 1.48933i
37.3 −2.37227 4.10889i −3.00000 −7.25534 + 12.5666i −19.3810 7.11681 + 12.3267i 3.13332 5.42707i 30.8902 9.00000 45.9769 + 79.6344i
37.4 −1.83140 3.17208i −3.00000 −2.70808 + 4.69053i 12.9712 5.49421 + 9.51625i 1.64155 2.84325i −9.46413 9.00000 −23.7555 41.1456i
37.5 −1.56036 2.70262i −3.00000 −0.869446 + 1.50592i 10.0899 4.68108 + 8.10787i −2.32642 + 4.02948i −19.5392 9.00000 −15.7439 27.2693i
37.6 −1.55793 2.69841i −3.00000 −0.854281 + 1.47966i −13.0874 4.67378 + 8.09523i −12.0084 + 20.7992i −19.6032 9.00000 20.3893 + 35.3153i
37.7 −0.829978 1.43756i −3.00000 2.62227 4.54191i −7.05463 2.48993 + 4.31269i 18.3944 31.8600i −21.9854 9.00000 5.85519 + 10.1415i
37.8 −0.561904 0.973245i −3.00000 3.36853 5.83446i −9.21684 1.68571 + 2.91974i 2.79031 4.83295i −16.5616 9.00000 5.17897 + 8.97024i
37.9 0.0760461 + 0.131716i −3.00000 3.98843 6.90817i 2.82816 −0.228138 0.395147i −1.28326 + 2.22268i 2.42996 9.00000 0.215070 + 0.372513i
37.10 0.244294 + 0.423129i −3.00000 3.88064 6.72147i 20.3821 −0.732881 1.26939i 8.86889 15.3614i 7.70076 9.00000 4.97921 + 8.62424i
37.11 0.824514 + 1.42810i −3.00000 2.64035 4.57323i −20.6834 −2.47354 4.28430i 1.21474 2.10399i 21.9003 9.00000 −17.0538 29.5380i
37.12 1.20764 + 2.09170i −3.00000 1.08320 1.87615i −7.44326 −3.62293 6.27510i −10.2926 + 17.8273i 24.5547 9.00000 −8.98880 15.5691i
37.13 1.46577 + 2.53879i −3.00000 −0.296981 + 0.514386i 10.1733 −4.39732 7.61638i −10.4213 + 18.0502i 21.7111 9.00000 14.9117 + 25.8278i
37.14 1.63047 + 2.82406i −3.00000 −1.31688 + 2.28090i −0.0205579 −4.89142 8.47218i 2.93590 5.08512i 17.4990 9.00000 −0.0335191 0.0580568i
37.15 2.13007 + 3.68940i −3.00000 −5.07443 + 8.78916i 13.6463 −6.39022 11.0682i 12.7106 22.0154i −9.15443 9.00000 29.0676 + 50.3465i
37.16 2.19185 + 3.79640i −3.00000 −5.60845 + 9.71412i −15.2934 −6.57556 11.3892i 12.5287 21.7004i −14.1019 9.00000 −33.5209 58.0599i
37.17 2.50438 + 4.33772i −3.00000 −8.54385 + 14.7984i 13.1979 −7.51314 13.0131i −7.35263 + 12.7351i −45.5181 9.00000 33.0525 + 57.2486i
37.18 2.68845 + 4.65653i −3.00000 −10.4555 + 18.1094i −6.42823 −8.06534 13.9696i −7.03129 + 12.1786i −69.4209 9.00000 −17.2819 29.9332i
163.1 −2.72652 + 4.72247i −3.00000 −10.8678 18.8236i 12.9793 8.17955 14.1674i −14.7538 25.5543i 74.9007 9.00000 −35.3882 + 61.2942i
163.2 −2.52313 + 4.37020i −3.00000 −8.73242 15.1250i 0.340792 7.56940 13.1106i 12.2513 + 21.2198i 47.7621 9.00000 −0.859864 + 1.48933i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 163.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.e.b 36
67.c even 3 1 inner 201.4.e.b 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.e.b 36 1.a even 1 1 trivial
201.4.e.b 36 67.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database