Properties

Label 201.4.e.a
Level 201
Weight 4
Character orbit 201.e
Analytic conductor 11.859
Analytic rank 0
Dimension 32
CM No

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Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{3})\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 2q^{2} + 96q^{3} - 66q^{4} + 4q^{5} + 6q^{6} - 14q^{7} + 108q^{8} + 288q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 2q^{2} + 96q^{3} - 66q^{4} + 4q^{5} + 6q^{6} - 14q^{7} + 108q^{8} + 288q^{9} - 2q^{10} + 16q^{11} - 198q^{12} + 88q^{13} + 214q^{14} + 12q^{15} - 298q^{16} + 52q^{17} + 18q^{18} - 2q^{19} + 164q^{20} - 42q^{21} - 506q^{22} + 160q^{23} + 324q^{24} + 572q^{25} + 353q^{26} + 864q^{27} - 433q^{28} + 48q^{29} - 6q^{30} + 292q^{31} - 525q^{32} + 48q^{33} + 138q^{34} - 328q^{35} - 594q^{36} - 616q^{37} - 194q^{38} + 264q^{39} - 1794q^{40} + 124q^{41} + 642q^{42} - 292q^{43} - 179q^{44} + 36q^{45} + 1324q^{46} + 402q^{47} - 894q^{48} + 172q^{49} + 171q^{50} + 156q^{51} - 3344q^{52} + 852q^{53} + 54q^{54} + 1238q^{55} - 47q^{56} - 6q^{57} - 3320q^{58} + 1200q^{59} + 492q^{60} - 454q^{61} - 5810q^{62} - 126q^{63} + 2340q^{64} - 24q^{65} - 1518q^{66} + 110q^{67} + 906q^{68} + 480q^{69} - 10q^{70} + 406q^{71} + 972q^{72} + 1274q^{73} - 1945q^{74} + 1716q^{75} - 2698q^{76} + 1436q^{77} + 1059q^{78} + 1236q^{79} + 6697q^{80} + 2592q^{81} + 2950q^{82} + 2190q^{83} - 1299q^{84} + 2032q^{85} + 273q^{86} + 144q^{87} + 1938q^{88} - 2160q^{89} - 18q^{90} - 3020q^{91} - 3020q^{92} + 876q^{93} - 2886q^{94} - 102q^{95} - 1575q^{96} + 1860q^{97} + 2612q^{98} + 144q^{99} + O(q^{100}) \)

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 −2.81607 4.87758i 3.00000 −11.8605 + 20.5430i −14.5837 −8.44821 14.6327i −2.46134 + 4.26316i 88.5430 9.00000 41.0687 + 71.1331i
37.2 −2.32403 4.02534i 3.00000 −6.80224 + 11.7818i 1.94047 −6.97209 12.0760i −1.41925 + 2.45821i 26.0500 9.00000 −4.50971 7.81105i
37.3 −2.07306 3.59065i 3.00000 −4.59517 + 7.95907i 0.952655 −6.21918 10.7719i 4.18709 7.25224i 4.93529 9.00000 −1.97491 3.42065i
37.4 −2.00589 3.47431i 3.00000 −4.04720 + 7.00996i 19.8930 −6.01767 10.4229i −14.1596 + 24.5251i 0.378714 9.00000 −39.9031 69.1143i
37.5 −1.18372 2.05026i 3.00000 1.19762 2.07434i −11.7568 −3.55115 6.15078i −6.15324 + 10.6577i −24.6101 9.00000 13.9167 + 24.1045i
37.6 −0.970745 1.68138i 3.00000 2.11531 3.66382i 11.3693 −2.91224 5.04414i 14.7963 25.6279i −23.7456 9.00000 −11.0367 19.1161i
37.7 −0.406448 0.703989i 3.00000 3.66960 6.35593i −6.19665 −1.21934 2.11197i −7.99071 + 13.8403i −12.4692 9.00000 2.51862 + 4.36237i
37.8 0.252285 + 0.436970i 3.00000 3.87271 6.70772i −15.4844 0.756854 + 1.31091i 10.1103 17.5115i 7.94465 9.00000 −3.90648 6.76622i
37.9 0.320316 + 0.554804i 3.00000 3.79479 6.57278i 9.63085 0.960949 + 1.66441i 2.30594 3.99400i 9.98720 9.00000 3.08492 + 5.34324i
37.10 0.660757 + 1.14447i 3.00000 3.12680 5.41578i 15.1026 1.98227 + 3.43340i −6.71331 + 11.6278i 18.8363 9.00000 9.97913 + 17.2844i
37.11 1.18517 + 2.05278i 3.00000 1.19074 2.06242i −8.96032 3.55551 + 6.15833i −14.9139 + 25.8316i 24.6077 9.00000 −10.6195 18.3935i
37.12 1.48985 + 2.58049i 3.00000 −0.439276 + 0.760849i −11.6804 4.46954 + 7.74146i 9.08077 15.7283i 21.2197 9.00000 −17.4019 30.1410i
37.13 1.70037 + 2.94513i 3.00000 −1.78253 + 3.08744i 10.4196 5.10112 + 8.83539i 9.10470 15.7698i 15.0821 9.00000 17.7172 + 30.6871i
37.14 2.16603 + 3.75167i 3.00000 −5.38337 + 9.32426i 6.71011 6.49809 + 11.2550i 9.19807 15.9315i −11.9857 9.00000 14.5343 + 25.1741i
37.15 2.46601 + 4.27125i 3.00000 −8.16236 + 14.1376i 11.8437 7.39802 + 12.8137i −12.5285 + 21.6999i −41.0576 9.00000 29.2066 + 50.5874i
37.16 2.53918 + 4.39800i 3.00000 −8.89492 + 15.4064i −17.1999 7.61755 + 13.1940i 0.556661 0.964165i −49.7164 9.00000 −43.6738 75.6453i
163.1 −2.81607 + 4.87758i 3.00000 −11.8605 20.5430i −14.5837 −8.44821 + 14.6327i −2.46134 4.26316i 88.5430 9.00000 41.0687 71.1331i
163.2 −2.32403 + 4.02534i 3.00000 −6.80224 11.7818i 1.94047 −6.97209 + 12.0760i −1.41925 2.45821i 26.0500 9.00000 −4.50971 + 7.81105i
163.3 −2.07306 + 3.59065i 3.00000 −4.59517 7.95907i 0.952655 −6.21918 + 10.7719i 4.18709 + 7.25224i 4.93529 9.00000 −1.97491 + 3.42065i
163.4 −2.00589 + 3.47431i 3.00000 −4.04720 7.00996i 19.8930 −6.01767 + 10.4229i −14.1596 24.5251i 0.378714 9.00000 −39.9031 + 69.1143i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 163.16
Significant digits:
Format:

Inner twists

This newform does not have CM; other inner twists have not been computed.

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2}^{32} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(201, [\chi])\).