# Properties

 Label 201.2.p.b Level 201 Weight 2 Character orbit 201.p Analytic conductor 1.605 Analytic rank 0 Dimension 400 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 201.p (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$400$$ Relative dimension: $$20$$ over $$\Q(\zeta_{66})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$400q - 22q^{3} - 20q^{4} - 20q^{6} - 38q^{7} - 30q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$400q - 22q^{3} - 20q^{4} - 20q^{6} - 38q^{7} - 30q^{9} - 38q^{10} + 2q^{12} - 38q^{13} + 4q^{15} - 8q^{16} - 28q^{18} - 60q^{19} - 106q^{21} - 8q^{22} - 62q^{24} - 84q^{25} - 22q^{27} + 116q^{28} - 90q^{30} - 52q^{31} - 19q^{33} - 32q^{34} - 24q^{36} - 34q^{37} + 33q^{39} - 4q^{40} - 22q^{42} - 22q^{43} - 132q^{45} - 162q^{46} - 54q^{48} - 38q^{49} - 10q^{51} - 44q^{52} + 101q^{54} + 126q^{55} + 77q^{57} - 80q^{60} + 146q^{61} - 13q^{63} + 172q^{64} - 4q^{67} - q^{69} + 264q^{70} + 88q^{72} - 6q^{73} - 11q^{75} + 124q^{76} + 20q^{78} - 246q^{79} - 42q^{81} - 16q^{82} - 235q^{84} + 34q^{85} - 100q^{87} + 150q^{88} + 88q^{90} + 120q^{91} + 57q^{93} - 88q^{94} + 196q^{96} + 24q^{97} - 54q^{99} + O(q^{100})$$

## 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 −0.131747 + 2.76571i −1.12721 1.31507i −5.64086 0.538637i −0.779337 + 0.899403i 3.78560 2.94428i −0.762358 1.47877i 1.44479 10.0487i −0.458797 + 2.96471i −2.38481 2.27392i
2.2 −0.111707 + 2.34502i 1.26924 1.17858i −3.49572 0.333801i 2.49839 2.88330i 2.62201 + 3.10804i −0.0230737 0.0447567i 0.505048 3.51269i 0.221918 2.99178i 6.48232 + 6.18088i
2.3 −0.106680 + 2.23948i −0.226858 + 1.71713i −3.01294 0.287701i −0.703208 + 0.811545i −3.82127 0.691227i −1.71157 3.31998i 0.327574 2.27833i −2.89707 0.779090i −1.74242 1.66139i
2.4 −0.0942093 + 1.97770i 1.35004 1.08508i −1.91146 0.182522i −2.22606 + 2.56901i 2.01877 + 2.77219i 0.731283 + 1.41849i −0.0224982 + 0.156478i 0.645203 2.92980i −4.87100 4.64449i
2.5 −0.0888646 + 1.86550i −1.52053 + 0.829459i −1.48124 0.141441i −1.38993 + 1.60406i −1.41223 2.91025i 1.29209 + 2.50631i −0.136090 + 0.946526i 1.62400 2.52243i −2.86886 2.73545i
2.6 −0.0824371 + 1.73057i 0.744203 + 1.56402i −0.997123 0.0952137i 2.18001 2.51587i −2.76799 + 1.15896i 0.930855 + 1.80561i −0.246156 + 1.71205i −1.89232 + 2.32790i 4.17416 + 3.98006i
2.7 −0.0765690 + 1.60738i −1.57501 0.720644i −0.586870 0.0560393i 0.869862 1.00387i 1.27895 2.47647i 0.664215 + 1.28840i −0.323015 + 2.24662i 1.96134 + 2.27005i 1.54700 + 1.47507i
2.8 −0.0485141 + 1.01844i 1.68562 + 0.398362i 0.956085 + 0.0912951i −0.397858 + 0.459152i −0.487483 + 1.69737i −1.04530 2.02760i −0.429568 + 2.98771i 2.68262 + 1.34297i −0.448316 0.427468i
2.9 −0.0407297 + 0.855022i −0.210665 1.71919i 1.26154 + 0.120462i 1.16294 1.34211i 1.47853 0.110101i −1.52832 2.96453i −0.398021 + 2.76830i −2.91124 + 0.724347i 1.10017 + 1.04901i
2.10 −0.00573694 + 0.120433i −1.18335 + 1.26478i 1.97647 + 0.188730i 1.79917 2.07635i −0.145533 0.149771i 0.198599 + 0.385227i −0.0683861 + 0.475635i −0.199345 2.99337i 0.239740 + 0.228592i
2.11 0.00573694 0.120433i 0.311708 + 1.70377i 1.97647 + 0.188730i −1.79917 + 2.07635i 0.206979 0.0277656i 0.198599 + 0.385227i 0.0683861 0.475635i −2.80568 + 1.06216i 0.239740 + 0.228592i
2.12 0.0407297 0.855022i 1.10669 1.33238i 1.26154 + 0.120462i −1.16294 + 1.34211i −1.09414 1.00051i −1.52832 2.96453i 0.398021 2.76830i −0.550484 2.94906i 1.10017 + 1.04901i
2.13 0.0485141 1.01844i −1.63340 0.576190i 0.956085 + 0.0912951i 0.397858 0.459152i −0.666056 + 1.63556i −1.04530 2.02760i 0.429568 2.98771i 2.33601 + 1.88230i −0.448316 0.427468i
2.14 0.0765690 1.60738i 1.71460 + 0.245273i −0.586870 0.0560393i −0.869862 + 1.00387i 0.525532 2.73723i 0.664215 + 1.28840i 0.323015 2.24662i 2.87968 + 0.841088i 1.54700 + 1.47507i
2.15 0.0824371 1.73057i −1.47164 + 0.913392i −0.997123 0.0952137i −2.18001 + 2.51587i 1.45937 + 2.62206i 0.930855 + 1.80561i 0.246156 1.71205i 1.33143 2.68836i 4.17416 + 3.98006i
2.16 0.0888646 1.86550i 0.830708 + 1.51984i −1.48124 0.141441i 1.38993 1.60406i 2.90908 1.41462i 1.29209 + 2.50631i 0.136090 0.946526i −1.61985 + 2.52509i −2.86886 2.73545i
2.17 0.0942093 1.97770i −0.549085 1.64271i −1.91146 0.182522i 2.22606 2.56901i −3.30051 + 0.931165i 0.731283 + 1.41849i 0.0224982 0.156478i −2.39701 + 1.80398i −4.87100 4.64449i
2.18 0.106680 2.23948i −0.737505 + 1.56719i −3.01294 0.287701i 0.703208 0.811545i 3.43101 + 1.81881i −1.71157 3.31998i −0.327574 + 2.27833i −1.91217 2.31162i −1.74242 1.66139i
2.19 0.111707 2.34502i −0.430563 1.67768i −3.49572 0.333801i −2.49839 + 2.88330i −3.98230 + 0.822271i −0.0230737 0.0447567i −0.505048 + 3.51269i −2.62923 + 1.44469i 6.48232 + 6.18088i
2.20 0.131747 2.76571i 1.65925 0.496888i −5.64086 0.538637i 0.779337 0.899403i −1.15565 4.65446i −0.762358 1.47877i −1.44479 + 10.0487i 2.50620 1.64892i −2.38481 2.27392i
See next 80 embeddings (of 400 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.20 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.h odd 66 1 inner
201.p even 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.p.b 400
3.b odd 2 1 inner 201.2.p.b 400
67.h odd 66 1 inner 201.2.p.b 400
201.p even 66 1 inner 201.2.p.b 400

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.p.b 400 1.a even 1 1 trivial
201.2.p.b 400 3.b odd 2 1 inner
201.2.p.b 400 67.h odd 66 1 inner
201.2.p.b 400 201.p even 66 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{400} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database