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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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,2,Mod(38,201)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(201, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.38");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 201.f (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.60499308063\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(\zeta_{6})\)
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.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 38.20
Significant digits:
Format:

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 \( T_{2}^{40} + 32 T_{2}^{38} + 597 T_{2}^{36} + 7484 T_{2}^{34} + 70216 T_{2}^{32} + 506956 T_{2}^{30} + \cdots + 73441 \) acting on \(S_{2}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display