# Properties

 Label 201.4.j.b Level 201 Weight 4 Character orbit 201.j Analytic conductor 11.859 Analytic rank 0 Dimension 640 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.j (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$640$$ Relative dimension: $$64$$ over $$\Q(\zeta_{22})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$640q - 11q^{3} - 290q^{4} + 35q^{6} - 22q^{7} - 33q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$640q - 11q^{3} - 290q^{4} + 35q^{6} - 22q^{7} - 33q^{9} + 14q^{10} - 143q^{12} - 22q^{13} - 31q^{15} - 578q^{16} - 11q^{18} - 18q^{19} - 1133q^{21} - 382q^{22} + 365q^{24} - 2102q^{25} - 11q^{27} - 1870q^{28} - 1210q^{31} - 743q^{33} - 22q^{34} + 1563q^{36} - 1048q^{37} + 2475q^{39} + 626q^{40} - 11q^{42} - 1738q^{43} - 4521q^{45} + 6578q^{46} - 3531q^{48} + 1198q^{49} - 11q^{51} + 10274q^{52} + 1948q^{54} + 3170q^{55} - 8921q^{57} + 418q^{58} - 649q^{60} - 2002q^{61} - 5951q^{63} - 14918q^{64} + 2038q^{67} - 11q^{69} - 8734q^{70} - 10681q^{72} + 9870q^{73} + 8426q^{75} + 5230q^{76} - 11q^{78} + 10934q^{79} + 2919q^{81} + 7442q^{82} - 4997q^{84} - 22q^{85} + 7480q^{87} + 8630q^{88} - 757q^{90} + 13286q^{91} - 6585q^{93} + 7106q^{94} + 16316q^{96} + 2739q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −3.59996 + 4.15458i 1.46644 + 4.98493i −3.16228 21.9941i 8.44887 2.48081i −25.9894 11.8531i 9.23706 + 8.00396i 65.7634 + 42.2636i −22.6991 + 14.6202i −20.1089 + 44.0324i
5.2 −3.53793 + 4.08299i −4.87975 + 1.78552i −3.01532 20.9720i −9.44479 + 2.77324i 9.97393 26.2410i −4.84382 4.19720i 59.9371 + 38.5193i 20.6238 17.4258i 22.0919 48.3745i
5.3 −3.44200 + 3.97228i 3.74600 3.60104i −2.79312 19.4266i 2.64211 0.775793i 1.41059 + 27.2749i −19.3454 16.7629i 51.4083 + 33.0381i 1.06506 26.9790i −6.01247 + 13.1655i
5.4 −3.30878 + 3.81854i −2.72588 4.42375i −2.49468 17.3509i 19.1622 5.62653i 25.9116 + 4.22837i −2.23659 1.93802i 40.5048 + 26.0309i −12.1392 + 24.1172i −41.9184 + 91.7886i
5.5 −3.23057 + 3.72827i 1.66163 4.92331i −2.32494 16.1703i −2.17468 + 0.638544i 12.9874 + 22.1001i 19.5744 + 16.9613i 34.5976 + 22.2345i −21.4780 16.3614i 4.64478 10.1707i
5.6 −3.20732 + 3.70144i 5.16210 + 0.593913i −2.27527 15.8248i −8.51736 + 2.50092i −18.7548 + 17.2024i 14.3505 + 12.4348i 32.9106 + 21.1503i 26.2945 + 6.13167i 18.0609 39.5478i
5.7 −3.03287 + 3.50012i 2.67965 + 4.45190i −1.91401 13.3122i −16.4593 + 4.83290i −23.7092 4.12294i −19.2826 16.7085i 21.2303 + 13.6439i −12.6389 + 23.8591i 33.0033 72.2672i
5.8 −3.01708 + 3.48189i −1.52278 4.96801i −1.88230 13.0917i −16.5234 + 4.85171i 21.8924 + 9.68669i −15.7610 13.6570i 20.2562 + 13.0179i −22.3623 + 15.1304i 32.9593 72.1707i
5.9 −2.89391 + 3.33975i −2.77933 + 4.39037i −1.64070 11.4113i 10.1030 2.96651i −6.61962 21.9876i −7.38800 6.40174i 13.1183 + 8.43060i −11.5507 24.4045i −19.3298 + 42.3263i
5.10 −2.88756 + 3.33242i −5.17395 0.479845i −1.62851 11.3266i 8.13953 2.38998i 16.5391 15.8562i 11.0496 + 9.57453i 12.7717 + 8.20789i 26.5395 + 4.96539i −15.5390 + 34.0256i
5.11 −2.83150 + 3.26773i 5.07100 + 1.13356i −1.52212 10.5866i 16.6951 4.90213i −18.0627 + 13.3610i −9.98954 8.65599i 9.80457 + 6.30101i 24.4301 + 11.4965i −31.2535 + 68.4356i
5.12 −2.79597 + 3.22672i −3.97281 3.34915i −1.45576 10.1250i −12.6136 + 3.70369i 21.9146 3.45504i 19.5475 + 16.9380i 8.00668 + 5.14558i 4.56644 + 26.6110i 23.3165 51.0560i
5.13 −2.41929 + 2.79201i −2.24694 + 4.68522i −0.803837 5.59081i −11.7964 + 3.46373i −7.64518 17.6084i 12.9935 + 11.2589i −7.30882 4.69710i −16.9025 21.0548i 18.8681 41.3155i
5.14 −2.39820 + 2.76767i −4.92182 1.66605i −0.770116 5.35628i 1.44052 0.422974i 16.4146 9.62645i −26.0066 22.5349i −7.97509 5.12528i 21.4486 + 16.4000i −2.28400 + 5.00125i
5.15 −2.31446 + 2.67103i 2.56918 + 4.51656i −0.639155 4.44542i −1.61194 + 0.473307i −18.0101 3.59106i −13.2348 11.4680i −10.4327 6.70466i −13.7987 + 23.2077i 2.46655 5.40098i
5.16 −2.08344 + 2.40442i 4.78249 2.03169i −0.301982 2.10033i −1.53336 + 0.450234i −5.07900 + 15.7320i −6.46221 5.59953i −15.7323 10.1106i 18.7445 19.4331i 2.11210 4.62486i
5.17 −2.00708 + 2.31629i 0.0979400 5.19523i −0.198329 1.37941i 4.53381 1.33125i 11.8371 + 10.6541i −2.62239 2.27231i −17.0337 10.9469i −26.9808 1.01764i −6.01617 + 13.1736i
5.18 −1.98603 + 2.29200i 4.03838 3.26978i −0.170435 1.18540i 19.1936 5.63576i −0.526014 + 15.7499i 24.7274 + 21.4264i −17.3551 11.1534i 5.61708 26.4092i −25.2020 + 55.1846i
5.19 −1.97915 + 2.28406i 1.50596 + 4.97314i −0.161385 1.12245i 9.47455 2.78198i −14.3395 6.40290i 23.8049 + 20.6270i −17.4567 11.2187i −22.4642 + 14.9787i −12.3974 + 27.1465i
5.20 −1.71809 + 1.98278i 3.65642 3.69196i 0.158929 + 1.10537i −18.6302 + 5.47032i 1.03829 + 13.5930i −6.55071 5.67622i −20.1216 12.9314i −0.261161 26.9987i 21.1619 46.3381i
See next 80 embeddings (of 640 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.64 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.f odd 22 1 inner
201.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.j.b 640
3.b odd 2 1 inner 201.4.j.b 640
67.f odd 22 1 inner 201.4.j.b 640
201.j even 22 1 inner 201.4.j.b 640

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.j.b 640 1.a even 1 1 trivial
201.4.j.b 640 3.b odd 2 1 inner
201.4.j.b 640 67.f odd 22 1 inner
201.4.j.b 640 201.j even 22 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{640} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database