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Properties

Label 221.2.m.c
Level $221$
Weight $2$
Character orbit 221.m
Analytic conductor $1.765$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [221,2,Mod(69,221)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("221.69"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(221, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 221 = 13 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 221.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.76469388467\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 3 q^{3} + 14 q^{4} - 9 q^{6} - 3 q^{7} - 14 q^{9} + q^{10} + 6 q^{11} - 18 q^{12} + q^{13} - 12 q^{14} + 3 q^{15} - 24 q^{16} - 11 q^{17} + 3 q^{19} + 12 q^{20} - 11 q^{22} + 3 q^{23} + 57 q^{24}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1 −2.39781 − 1.38438i −0.0844000 + 0.146185i 2.83300 + 4.90689i − 2.08714i 0.404751 − 0.233683i −2.66656 + 1.53954i − 10.1502i 1.48575 + 2.57340i −2.88938 + 5.00456i
69.2 −2.10061 − 1.21279i −1.10523 + 1.91432i 1.94171 + 3.36314i 3.27693i 4.64332 − 2.68082i 0.415698 − 0.240003i − 4.56836i −0.943072 − 1.63345i 3.97422 − 6.88355i
69.3 −1.60408 − 0.926118i 0.634338 − 1.09871i 0.715389 + 1.23909i − 4.17850i −2.03506 + 1.17494i 3.61684 − 2.08819i 1.05433i 0.695230 + 1.20417i −3.86978 + 6.70266i
69.4 −0.990812 − 0.572046i 0.440946 − 0.763741i −0.345527 − 0.598471i 4.00280i −0.873790 + 0.504483i 2.20607 − 1.27368i 3.07881i 1.11113 + 1.92454i 2.28979 − 3.96603i
69.5 −0.544554 − 0.314398i 1.63897 − 2.83878i −0.802307 − 1.38964i − 0.315403i −1.78502 + 1.03058i −0.424278 + 0.244957i 2.26657i −3.87245 − 6.70728i −0.0991621 + 0.171754i
69.6 0.445025 + 0.256935i −1.43985 + 2.49389i −0.867969 − 1.50337i − 3.95509i −1.28154 + 0.739896i −2.83913 + 1.63917i − 1.91979i −2.64633 − 4.58358i 1.01620 − 1.76011i
69.7 0.486078 + 0.280637i −0.531774 + 0.921059i −0.842486 − 1.45923i 2.54634i −0.516967 + 0.298471i −4.09319 + 2.36320i − 2.06828i 0.934433 + 1.61849i −0.714598 + 1.23772i
69.8 0.720682 + 0.416086i −0.251133 + 0.434974i −0.653745 − 1.13232i − 1.03543i −0.361974 + 0.208986i 2.58322 − 1.49142i − 2.75240i 1.37386 + 2.37960i 0.430827 − 0.746214i
69.9 1.79032 + 1.03364i −1.71618 + 2.97252i 1.13683 + 1.96904i 0.855003i −6.14503 + 3.54783i 2.31756 − 1.33804i 0.565718i −4.39057 − 7.60469i −0.883766 + 1.53073i
69.10 1.98509 + 1.14609i 1.31039 − 2.26966i 1.62706 + 2.81816i 1.31703i 5.20249 − 3.00366i −3.37658 + 1.94947i 2.87469i −1.93424 − 3.35020i −1.50944 + 2.61442i
69.11 2.21067 + 1.27633i −0.396074 + 0.686021i 2.25805 + 3.91106i − 2.15860i −1.75118 + 1.01105i 0.760332 − 0.438978i 6.42278i 1.18625 + 2.05465i 2.75509 − 4.77196i
205.1 −2.39781 + 1.38438i −0.0844000 − 0.146185i 2.83300 − 4.90689i 2.08714i 0.404751 + 0.233683i −2.66656 − 1.53954i 10.1502i 1.48575 − 2.57340i −2.88938 − 5.00456i
205.2 −2.10061 + 1.21279i −1.10523 − 1.91432i 1.94171 − 3.36314i − 3.27693i 4.64332 + 2.68082i 0.415698 + 0.240003i 4.56836i −0.943072 + 1.63345i 3.97422 + 6.88355i
205.3 −1.60408 + 0.926118i 0.634338 + 1.09871i 0.715389 − 1.23909i 4.17850i −2.03506 − 1.17494i 3.61684 + 2.08819i − 1.05433i 0.695230 − 1.20417i −3.86978 − 6.70266i
205.4 −0.990812 + 0.572046i 0.440946 + 0.763741i −0.345527 + 0.598471i − 4.00280i −0.873790 − 0.504483i 2.20607 + 1.27368i − 3.07881i 1.11113 − 1.92454i 2.28979 + 3.96603i
205.5 −0.544554 + 0.314398i 1.63897 + 2.83878i −0.802307 + 1.38964i 0.315403i −1.78502 − 1.03058i −0.424278 − 0.244957i − 2.26657i −3.87245 + 6.70728i −0.0991621 − 0.171754i
205.6 0.445025 − 0.256935i −1.43985 − 2.49389i −0.867969 + 1.50337i 3.95509i −1.28154 − 0.739896i −2.83913 − 1.63917i 1.91979i −2.64633 + 4.58358i 1.01620 + 1.76011i
205.7 0.486078 − 0.280637i −0.531774 − 0.921059i −0.842486 + 1.45923i − 2.54634i −0.516967 − 0.298471i −4.09319 − 2.36320i 2.06828i 0.934433 − 1.61849i −0.714598 − 1.23772i
205.8 0.720682 − 0.416086i −0.251133 − 0.434974i −0.653745 + 1.13232i 1.03543i −0.361974 − 0.208986i 2.58322 + 1.49142i 2.75240i 1.37386 − 2.37960i 0.430827 + 0.746214i
205.9 1.79032 − 1.03364i −1.71618 − 2.97252i 1.13683 − 1.96904i − 0.855003i −6.14503 − 3.54783i 2.31756 + 1.33804i − 0.565718i −4.39057 + 7.60469i −0.883766 − 1.53073i
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 69.11
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 221.2.m.c ✓ 22
13.e even 6 1 inner 221.2.m.c ✓ 22
13.f odd 12 2 2873.2.a.x 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
221.2.m.c ✓ 22 1.a even 1 1 trivial
221.2.m.c ✓ 22 13.e even 6 1 inner
2873.2.a.x 22 13.f odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} - 18 T_{2}^{20} + 211 T_{2}^{18} - 21 T_{2}^{17} - 1455 T_{2}^{16} + 225 T_{2}^{15} + \cdots + 675 \) acting on \(S_{2}^{\mathrm{new}}(221, [\chi])\). Copy content Toggle raw display