# Properties

 Label 201.4.f.a Level 201 Weight 4 Character orbit 201.f Analytic conductor 11.859 Analytic rank 0 Dimension 132 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$132$$ Relative dimension: $$66$$ over $$\Q(\zeta_{6})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$132q - 258q^{4} - 23q^{6} - 66q^{7} - 70q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$132q - 258q^{4} - 23q^{6} - 66q^{7} - 70q^{9} - 18q^{10} + 114q^{12} - 180q^{13} - 188q^{15} - 738q^{16} + 159q^{18} - 208q^{19} + 96q^{21} - 324q^{22} + 736q^{24} + 2508q^{25} + 1704q^{28} - 843q^{30} + 612q^{31} + 146q^{33} - 762q^{34} + 221q^{36} - 238q^{37} - 394q^{39} - 864q^{40} - 3462q^{46} + 951q^{48} + 2316q^{49} - 309q^{51} - 376q^{54} - 96q^{55} - 1113q^{57} + 122q^{60} + 1728q^{61} - 534q^{63} + 900q^{64} - 1214q^{67} - 372q^{69} + 578q^{73} - 184q^{76} - 4686q^{78} + 4476q^{79} + 666q^{81} + 1368q^{82} + 1161q^{84} - 1908q^{85} - 462q^{87} + 2562q^{88} - 1160q^{90} - 3636q^{91} - 1828q^{93} - 3900q^{96} + 1074q^{97} + 906q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
38.1 −2.76011 + 4.78064i 4.75868 + 2.08685i −11.2364 19.4620i 5.35805 −23.1110 + 16.9896i −6.62568 + 3.82534i 79.8926 18.2901 + 19.8613i −14.7888 + 25.6149i
38.2 −2.64802 + 4.58651i 2.08276 4.76047i −10.0240 17.3621i −19.9728 16.3187 + 22.1584i −22.9190 + 13.2323i 63.8072 −18.3242 19.8299i 52.8883 91.6052i
38.3 −2.60270 + 4.50801i −3.38102 3.94572i −9.54810 16.5378i 1.95562 26.5871 4.97217i 6.62765 3.82647i 57.7601 −4.13736 + 26.6811i −5.08989 + 8.81595i
38.4 −2.57563 + 4.46113i −4.36337 + 2.82152i −9.26778 16.0523i −10.1201 −1.34870 26.7328i 9.96980 5.75607i 54.2715 11.0781 24.6227i 26.0657 45.1471i
38.5 −2.52337 + 4.37060i −2.19586 + 4.70937i −8.73479 15.1291i 8.19911 −15.0419 21.4807i −30.0864 + 17.3704i 47.7905 −17.3564 20.6822i −20.6894 + 35.8351i
38.6 −2.47054 + 4.27911i 3.74376 3.60336i −8.20717 14.2152i 12.2051 6.17004 + 24.9222i 13.3204 7.69055i 41.5760 1.03155 26.9803i −30.1533 + 52.2270i
38.7 −2.32472 + 4.02652i −0.204978 + 5.19211i −6.80860 11.7928i 14.3621 −20.4296 12.8955i 25.5981 14.7791i 26.1168 −26.9160 2.12853i −33.3879 + 57.8295i
38.8 −2.18859 + 3.79075i 5.19172 0.214507i −5.57986 9.66459i −14.7915 −10.5494 + 20.1500i 21.3930 12.3512i 13.8306 26.9080 2.22732i 32.3725 56.0708i
38.9 −2.17777 + 3.77200i −5.01884 1.34584i −5.48532 9.50086i 15.8201 16.0064 16.0001i −22.0161 + 12.7110i 12.9387 23.3774 + 13.5091i −34.4525 + 59.6735i
38.10 −2.13016 + 3.68954i 1.62492 + 4.93555i −5.07514 8.79039i −3.57613 −21.6712 4.51829i 10.1310 5.84914i 9.16083 −21.7193 + 16.0397i 7.61773 13.1943i
38.11 −2.07228 + 3.58929i 3.94053 + 3.38707i −4.58865 7.94778i −6.98175 −20.3230 + 7.12474i −12.0511 + 6.95772i 4.87942 4.05552 + 26.6937i 14.4681 25.0595i
38.12 −2.03028 + 3.51655i 0.528423 5.16921i −4.24406 7.35093i 13.6688 17.1049 + 12.3532i −13.4260 + 7.75148i 1.98205 −26.4415 5.46307i −27.7515 + 48.0670i
38.13 −2.01114 + 3.48339i −5.13832 0.773071i −4.08936 7.08298i −12.8586 13.0268 16.3441i −3.82711 + 2.20958i 0.718856 25.8047 + 7.94458i 25.8605 44.7917i
38.14 −1.87376 + 3.24545i 0.464052 5.17539i −3.02198 5.23422i −9.19120 15.9270 + 11.2035i 22.1449 12.7853i −7.33032 −26.5693 4.80330i 17.2221 29.8296i
38.15 −1.75566 + 3.04090i 4.85215 1.85920i −2.16472 3.74940i −0.266766 −2.86511 + 18.0190i −25.8891 + 14.9471i −12.8886 20.0868 18.0422i 0.468351 0.811208i
38.16 −1.71781 + 2.97534i −0.924450 + 5.11326i −1.90177 3.29396i −18.5280 −13.6256 11.5342i −15.8917 + 9.17505i −14.4175 −25.2908 9.45390i 31.8277 55.1273i
38.17 −1.63784 + 2.83682i −5.10653 + 0.960933i −1.36502 2.36428i 5.86600 5.63767 16.0601i 8.89123 5.13335i −17.2627 25.1532 9.81406i −9.60754 + 16.6408i
38.18 −1.59031 + 2.75450i 4.84662 + 1.87357i −1.05816 1.83279i 22.1315 −12.8684 + 10.3704i 2.99805 1.73092i −18.7137 19.9794 + 18.1610i −35.1959 + 60.9611i
38.19 −1.41509 + 2.45101i −3.33481 3.98485i −0.00495393 0.00858046i −13.4714 14.4859 2.53472i −13.0426 + 7.53016i −22.6134 −4.75808 + 26.5774i 19.0632 33.0185i
38.20 −1.38906 + 2.40592i −3.61547 + 3.73207i 0.141034 + 0.244278i 10.9301 −3.95695 13.8826i 2.00045 1.15496i −23.0086 −0.856687 26.9864i −15.1826 + 26.2970i
See next 80 embeddings (of 132 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 164.66 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.d odd 6 1 inner
201.f even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.f.a 132
3.b odd 2 1 inner 201.4.f.a 132
67.d odd 6 1 inner 201.4.f.a 132
201.f even 6 1 inner 201.4.f.a 132

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.f.a 132 1.a even 1 1 trivial
201.4.f.a 132 3.b odd 2 1 inner
201.4.f.a 132 67.d odd 6 1 inner
201.4.f.a 132 201.f even 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database