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Properties

Label 201.3.g.b
Level $201$
Weight $3$
Character orbit 201.g
Analytic conductor $5.477$
Analytic rank $0$
Dimension $84$
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] = [201,3,Mod(29,201)]
 
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, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("201.29");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 201 = 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 201.g (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: \(5.47685331364\)
Analytic rank: \(0\)
Dimension: \(84\)
Relative dimension: \(42\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{3} + 82 q^{4} + 3 q^{6} - 10 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 84 q - 6 q^{3} + 82 q^{4} + 3 q^{6} - 10 q^{7} + 2 q^{9} + 6 q^{10} - 58 q^{12} - 6 q^{13} + 78 q^{15} - 130 q^{16} + 11 q^{18} + 72 q^{19} - 36 q^{21} + 140 q^{22} + 72 q^{24} - 428 q^{25} + 138 q^{27} - 8 q^{28} + 29 q^{30} - 28 q^{31} - 30 q^{33} - 54 q^{34} + 105 q^{36} - 24 q^{37} - 103 q^{39} + 128 q^{40} + 252 q^{42} + 148 q^{43} + 276 q^{45} - 146 q^{46} + 137 q^{48} - 176 q^{49} + 15 q^{51} - 792 q^{52} + 72 q^{54} + 28 q^{55} - 205 q^{57} - 120 q^{58} - 64 q^{60} + 162 q^{61} - 106 q^{63} - 604 q^{64} - 462 q^{66} - 460 q^{67} + 101 q^{69} + 636 q^{70} + 212 q^{72} + 238 q^{73} + 628 q^{75} + 40 q^{76} + 96 q^{78} + 462 q^{79} - 726 q^{81} - 344 q^{82} + 385 q^{84} - 94 q^{85} - 91 q^{87} + 234 q^{88} + 482 q^{90} - 536 q^{91} + 281 q^{93} + 580 q^{94} + 40 q^{96} + 68 q^{97} + 379 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1 −3.31832 − 1.91583i −2.72368 + 1.25760i 5.34082 + 9.25058i 5.12708i 11.4474 + 1.04501i 5.45809 + 9.45369i − 25.6018i 5.83690 − 6.85059i 9.82263 − 17.0133i
29.2 −3.24022 − 1.87074i −1.87878 − 2.33884i 4.99934 + 8.65911i − 4.08787i 1.71228 + 11.0931i −3.91757 − 6.78542i − 22.4439i −1.94039 + 8.78834i −7.64734 + 13.2456i
29.3 −3.10861 − 1.79476i 1.01703 + 2.82235i 4.44231 + 7.69431i − 6.24319i 1.90387 − 10.5989i 0.163929 + 0.283933i − 17.5334i −6.93129 + 5.74084i −11.2050 + 19.4077i
29.4 −3.07823 − 1.77721i 2.92996 + 0.644471i 4.31698 + 7.47723i 4.16735i −7.87371 − 7.19099i −2.85784 − 4.94993i − 16.4711i 8.16931 + 3.77655i 7.40628 − 12.8280i
29.5 −2.93008 − 1.69168i 1.12713 − 2.78021i 3.72358 + 6.44943i 3.95114i −8.00582 + 6.23950i 1.40220 + 2.42868i − 11.6630i −6.45916 − 6.26732i 6.68407 − 11.5772i
29.6 −2.55999 − 1.47801i −2.72630 + 1.25192i 2.36902 + 4.10327i − 7.04577i 8.82964 + 0.824596i 0.952080 + 1.64905i − 2.18168i 5.86540 − 6.82621i −10.4137 + 18.0371i
29.7 −2.37828 − 1.37310i 0.338320 + 2.98086i 1.77082 + 3.06716i 5.91841i 3.28841 − 7.55389i 1.61899 + 2.80418i 1.25873i −8.77108 + 2.01697i 8.12658 − 14.0757i
29.8 −2.32976 − 1.34509i 2.41314 − 1.78235i 1.61852 + 2.80337i − 6.52133i −8.01945 + 0.906562i −2.84365 − 4.92535i 2.05247i 2.64647 − 8.60210i −8.77176 + 15.1931i
29.9 −2.30980 − 1.33356i −2.73908 − 1.22370i 1.55678 + 2.69642i 8.02052i 4.69483 + 6.47924i −4.24579 − 7.35393i 2.36426i 6.00510 + 6.70364i 10.6959 − 18.5258i
29.10 −2.22999 − 1.28749i −2.28852 + 1.93976i 1.31525 + 2.27807i − 0.0457250i 7.60080 − 1.37922i −4.62513 − 8.01096i 3.52645i 1.47465 − 8.87837i −0.0588703 + 0.101966i
29.11 −2.19394 − 1.26667i −1.19967 − 2.74969i 1.20891 + 2.09389i − 2.77685i −0.850943 + 7.55223i 5.01703 + 8.68975i 4.00822i −6.12157 + 6.59745i −3.51736 + 6.09224i
29.12 −1.76138 − 1.01693i 2.83847 + 0.971138i 0.0683121 + 0.118320i − 2.80344i −4.01204 − 4.59708i 5.21582 + 9.03406i 7.85760i 7.11378 + 5.51309i −2.85091 + 4.93792i
29.13 −1.42247 − 0.821262i 2.43078 − 1.75821i −0.651057 − 1.12766i 8.34621i −4.90166 + 0.504680i 1.86864 + 3.23657i 8.70885i 2.81742 − 8.54764i 6.85443 − 11.8722i
29.14 −1.34106 − 0.774260i −2.89057 − 0.802859i −0.801042 − 1.38745i 1.39955i 3.25481 + 3.31474i 3.37843 + 5.85161i 8.67494i 7.71083 + 4.64145i 1.08362 − 1.87688i
29.15 −1.29938 − 0.750199i 2.00560 + 2.23105i −0.874403 − 1.51451i − 2.91605i −0.932314 − 4.40358i −6.24309 − 10.8134i 8.62550i −0.955134 + 8.94917i −2.18761 + 3.78906i
29.16 −1.03426 − 0.597131i −1.13621 + 2.77651i −1.28687 − 2.22892i 0.439293i 2.83308 − 2.19317i −0.664310 − 1.15062i 7.85076i −6.41804 − 6.30942i 0.262316 − 0.454344i
29.17 −0.922015 − 0.532326i −0.226863 − 2.99141i −1.43326 − 2.48248i 3.49902i −1.38323 + 2.87889i −3.66463 − 6.34732i 7.31045i −8.89707 + 1.35728i 1.86262 − 3.22615i
29.18 −0.718547 − 0.414853i 2.98530 − 0.296617i −1.65579 − 2.86792i 4.81931i −2.26813 − 1.02533i −2.30167 − 3.98661i 6.06647i 8.82404 − 1.77098i 1.99931 − 3.46290i
29.19 −0.671554 − 0.387722i −2.19554 − 2.04441i −1.69934 − 2.94335i − 9.75594i 0.681761 + 2.22419i −3.81946 − 6.61550i 5.73727i 0.640779 + 8.97716i −3.78259 + 6.55164i
29.20 −0.481246 − 0.277848i −1.17888 + 2.75867i −1.84560 − 3.19668i − 8.73533i 1.33382 − 1.00005i 5.21194 + 9.02735i 4.27397i −6.22049 − 6.50427i −2.42709 + 4.20384i
See all 84 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.42
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
67.c even 3 1 inner
201.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.g.b ✓ 84
3.b odd 2 1 inner 201.3.g.b ✓ 84
67.c even 3 1 inner 201.3.g.b ✓ 84
201.g odd 6 1 inner 201.3.g.b ✓ 84
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.g.b ✓ 84 1.a even 1 1 trivial
201.3.g.b ✓ 84 3.b odd 2 1 inner
201.3.g.b ✓ 84 67.c even 3 1 inner
201.3.g.b ✓ 84 201.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{84} - 125 T_{2}^{82} + 8427 T_{2}^{80} - 392950 T_{2}^{78} + 14032566 T_{2}^{76} + \cdots + 52\!\cdots\!89 \) acting on \(S_{3}^{\mathrm{new}}(201, [\chi])\). Copy content Toggle raw display