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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 $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [201,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.38");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 4 \)
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: \(11.8593839112\)
Analytic rank: \(0\)
Dimension: \(132\)
Relative dimension: \(66\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 132 q - 258 q^{4} - 23 q^{6} - 66 q^{7} - 70 q^{9} - 18 q^{10} + 114 q^{12} - 180 q^{13} - 188 q^{15} - 738 q^{16} + 159 q^{18} - 208 q^{19} + 96 q^{21} - 324 q^{22} + 736 q^{24} + 2508 q^{25} + 1704 q^{28} - 843 q^{30} + 612 q^{31} + 146 q^{33} - 762 q^{34} + 221 q^{36} - 238 q^{37} - 394 q^{39} - 864 q^{40} - 3462 q^{46} + 951 q^{48} + 2316 q^{49} - 309 q^{51} - 376 q^{54} - 96 q^{55} - 1113 q^{57} + 122 q^{60} + 1728 q^{61} - 534 q^{63} + 900 q^{64} - 1214 q^{67} - 372 q^{69} + 578 q^{73} - 184 q^{76} - 4686 q^{78} + 4476 q^{79} + 666 q^{81} + 1368 q^{82} + 1161 q^{84} - 1908 q^{85} - 462 q^{87} + 2562 q^{88} - 1160 q^{90} - 3636 q^{91} - 1828 q^{93} - 3900 q^{96} + 1074 q^{97} + 906 q^{99}+O(q^{100}) \) 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 −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 38.66
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 201.4.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 \(S_{4}^{\mathrm{new}}(201, [\chi])\).