Properties

 Label 201.2.f.b Level 201 Weight 2 Character orbit 201.f Analytic conductor 1.605 Analytic rank 0 Dimension 40 CM no Inner twists 4

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 24q^{4} - 2q^{6} - 6q^{7} + 8q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 24q^{4} - 2q^{6} - 6q^{7} + 8q^{9} - 6q^{10} + 9q^{12} - 6q^{13} - 26q^{15} - 36q^{16} + 6q^{18} - 6q^{19} + 18q^{21} + 52q^{22} - 26q^{24} + 40q^{25} - 72q^{28} + 57q^{30} - 36q^{31} + 8q^{33} - 12q^{34} + 2q^{36} + 12q^{37} + 11q^{39} - 40q^{40} + 30q^{46} - 78q^{48} - 6q^{49} - 12q^{51} + 20q^{54} - 16q^{55} + 80q^{60} + 30q^{61} + 24q^{63} + 136q^{64} + 4q^{67} - 21q^{69} - 16q^{73} + 52q^{76} - 42q^{78} - 18q^{79} - 24q^{81} + 104q^{82} + 15q^{84} - 78q^{85} - 21q^{87} - 62q^{88} - 110q^{90} - 32q^{91} - 13q^{93} - 9q^{96} - 90q^{97} + 87q^{99} + O(q^{100})$$

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 −1.35950 + 2.35473i 0.494616 + 1.65993i −2.69650 4.67047i −3.45591 −4.58111 1.09199i 0.770900 0.445079i 9.22558 −2.51071 + 1.64205i 4.69833 8.13774i
38.2 −1.30392 + 2.25846i −1.69795 + 0.341983i −2.40042 4.15766i 3.98713 1.44164 4.28068i 2.31524 1.33671i 7.30417 2.76609 1.16134i −5.19891 + 9.00477i
38.3 −1.22221 + 2.11693i 1.40479 1.01319i −1.98761 3.44264i 0.554650 0.427913 + 4.21219i 0.273771 0.158062i 4.82826 0.946874 2.84665i −0.677901 + 1.17416i
38.4 −1.03804 + 1.79794i −0.401302 1.68492i −1.15505 2.00061i −0.547134 3.44595 + 1.02750i −1.46018 + 0.843037i 0.643812 −2.67791 + 1.35232i 0.567947 0.983712i
38.5 −0.834649 + 1.44565i −1.24177 + 1.20748i −0.393279 0.681179i −0.655150 −0.709154 2.80299i −1.66309 + 0.960187i −2.02560 0.0839915 2.99882i 0.546821 0.947121i
38.6 −0.725905 + 1.25730i 1.60534 + 0.650289i −0.0538753 0.0933147i 0.670871 −1.98294 + 1.54635i 4.01999 2.32094i −2.74719 2.15425 + 2.08787i −0.486988 + 0.843488i
38.7 −0.577158 + 0.999667i 1.56100 + 0.750518i 0.333777 + 0.578119i −2.29211 −1.65121 + 1.12731i −3.24552 + 1.87380i −3.07920 1.87345 + 2.34312i 1.32291 2.29135i
38.8 −0.451684 + 0.782339i −1.52537 0.820518i 0.591964 + 1.02531i 1.39831 1.33091 0.822741i −0.0330810 + 0.0190993i −2.87626 1.65350 + 2.50319i −0.631592 + 1.09395i
38.9 −0.314267 + 0.544326i 1.35150 1.08326i 0.802472 + 1.38992i 3.91059 0.164916 + 1.07609i −3.42502 + 1.97744i −2.26583 0.653093 2.92805i −1.22897 + 2.12864i
38.10 −0.144006 + 0.249426i −0.169378 + 1.72375i 0.958525 + 1.66021i 2.85857 −0.405556 0.290477i 0.946995 0.546748i −1.12816 −2.94262 0.583929i −0.411651 + 0.713000i
38.11 0.144006 0.249426i 0.169378 + 1.72375i 0.958525 + 1.66021i −2.85857 0.454338 + 0.205983i 0.946995 0.546748i 1.12816 −2.94262 + 0.583929i −0.411651 + 0.713000i
38.12 0.314267 0.544326i −1.35150 1.08326i 0.802472 + 1.38992i −3.91059 −1.01438 + 0.395223i −3.42502 + 1.97744i 2.26583 0.653093 + 2.92805i −1.22897 + 2.12864i
38.13 0.451684 0.782339i 1.52537 0.820518i 0.591964 + 1.02531i −1.39831 0.0470610 1.56397i −0.0330810 + 0.0190993i 2.87626 1.65350 2.50319i −0.631592 + 1.09395i
38.14 0.577158 0.999667i −1.56100 + 0.750518i 0.333777 + 0.578119i 2.29211 −0.150676 + 1.99365i −3.24552 + 1.87380i 3.07920 1.87345 2.34312i 1.32291 2.29135i
38.15 0.725905 1.25730i −1.60534 + 0.650289i −0.0538753 0.0933147i −0.670871 −0.347714 + 2.49045i 4.01999 2.32094i 2.74719 2.15425 2.08787i −0.486988 + 0.843488i
38.16 0.834649 1.44565i 1.24177 + 1.20748i −0.393279 0.681179i 0.655150 2.78204 0.787352i −1.66309 + 0.960187i 2.02560 0.0839915 + 2.99882i 0.546821 0.947121i
38.17 1.03804 1.79794i 0.401302 1.68492i −1.15505 2.00061i 0.547134 −2.61282 2.47053i −1.46018 + 0.843037i −0.643812 −2.67791 1.35232i 0.567947 0.983712i
38.18 1.22221 2.11693i −1.40479 1.01319i −1.98761 3.44264i −0.554650 −3.86182 + 1.73551i 0.273771 0.158062i −4.82826 0.946874 + 2.84665i −0.677901 + 1.17416i
38.19 1.30392 2.25846i 1.69795 + 0.341983i −2.40042 4.15766i −3.98713 2.98635 3.38884i 2.31524 1.33671i −7.30417 2.76609 + 1.16134i −5.19891 + 9.00477i
38.20 1.35950 2.35473i −0.494616 + 1.65993i −2.69650 4.67047i 3.45591 3.23624 + 3.42136i 0.770900 0.445079i −9.22558 −2.51071 1.64205i 4.69833 8.13774i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 164.20 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.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 201.2.f.b 40
3.b odd 2 1 inner 201.2.f.b 40
67.d odd 6 1 inner 201.2.f.b 40
201.f even 6 1 inner 201.2.f.b 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.f.b 40 1.a even 1 1 trivial
201.2.f.b 40 3.b odd 2 1 inner
201.2.f.b 40 67.d odd 6 1 inner
201.2.f.b 40 201.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database