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Properties

Label 539.2.m.b
Level $539$
Weight $2$
Character orbit 539.m
Analytic conductor $4.304$
Analytic rank $0$
Dimension $96$
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] = [539,2,Mod(195,539)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(539, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("539.195");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.m (of order \(10\), degree \(4\), 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: \(4.30393666895\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(24\) over \(\Q(\zeta_{10})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 16 q^{4} - 40 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 16 q^{4} - 40 q^{8} - 24 q^{15} - 88 q^{16} + 100 q^{18} + 32 q^{22} + 16 q^{23} + 72 q^{25} - 48 q^{36} + 16 q^{37} - 160 q^{39} + 92 q^{44} - 140 q^{46} + 160 q^{51} - 48 q^{53} + 100 q^{58} - 112 q^{60} + 48 q^{64} + 112 q^{67} + 88 q^{71} - 100 q^{72} - 64 q^{78} - 200 q^{79} - 16 q^{81} - 160 q^{85} - 36 q^{86} + 104 q^{88} + 44 q^{92} - 240 q^{93} + 160 q^{95} - 184 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
195.1 −1.49408 − 2.05642i −0.854073 + 0.277505i −1.37857 + 4.24281i 0.927683 − 1.27685i 1.84672 + 1.34172i 0 5.94976 − 1.93319i −1.77462 + 1.28934i −4.01177
195.2 −1.49408 − 2.05642i 0.854073 − 0.277505i −1.37857 + 4.24281i −0.927683 + 1.27685i −1.84672 − 1.34172i 0 5.94976 − 1.93319i −1.77462 + 1.28934i 4.01177
195.3 −1.29910 − 1.78806i −1.32776 + 0.431415i −0.891458 + 2.74363i 1.03893 − 1.42997i 2.49629 + 1.81366i 0 1.85988 − 0.604312i −0.850226 + 0.617725i −3.90654
195.4 −1.29910 − 1.78806i 1.32776 − 0.431415i −0.891458 + 2.74363i −1.03893 + 1.42997i −2.49629 − 1.81366i 0 1.85988 − 0.604312i −0.850226 + 0.617725i 3.90654
195.5 −0.957444 − 1.31781i −1.40972 + 0.458047i −0.201886 + 0.621340i 1.71401 − 2.35914i 1.95335 + 1.41919i 0 −2.08625 + 0.677864i −0.649537 + 0.471916i −4.74996
195.6 −0.957444 − 1.31781i 1.40972 − 0.458047i −0.201886 + 0.621340i −1.71401 + 2.35914i −1.95335 − 1.41919i 0 −2.08625 + 0.677864i −0.649537 + 0.471916i 4.74996
195.7 −0.407816 − 0.561311i −2.90213 + 0.942959i 0.469278 − 1.44429i −0.277827 + 0.382396i 1.71283 + 1.24444i 0 −2.32180 + 0.754397i 5.10613 − 3.70982i 0.327946
195.8 −0.407816 − 0.561311i 2.90213 − 0.942959i 0.469278 − 1.44429i 0.277827 − 0.382396i −1.71283 − 1.24444i 0 −2.32180 + 0.754397i 5.10613 − 3.70982i −0.327946
195.9 −0.306536 − 0.421911i −0.719219 + 0.233688i 0.533990 − 1.64345i −0.911014 + 1.25390i 0.319062 + 0.231812i 0 −1.84905 + 0.600793i −1.96439 + 1.42721i 0.808295
195.10 −0.306536 − 0.421911i 0.719219 − 0.233688i 0.533990 − 1.64345i 0.911014 − 1.25390i −0.319062 − 0.231812i 0 −1.84905 + 0.600793i −1.96439 + 1.42721i −0.808295
195.11 0.0564687 + 0.0777226i −2.10647 + 0.684434i 0.615182 − 1.89334i −1.81378 + 2.49645i −0.172146 − 0.125071i 0 0.364630 − 0.118476i 1.54172 − 1.12012i −0.296452
195.12 0.0564687 + 0.0777226i 2.10647 − 0.684434i 0.615182 − 1.89334i 1.81378 − 2.49645i 0.172146 + 0.125071i 0 0.364630 − 0.118476i 1.54172 − 1.12012i 0.296452
195.13 0.519198 + 0.714615i −0.456391 + 0.148291i 0.376926 − 1.16006i 2.15024 − 2.95955i −0.342928 − 0.249152i 0 2.70486 − 0.878861i −2.24075 + 1.62800i 3.23134
195.14 0.519198 + 0.714615i 0.456391 − 0.148291i 0.376926 − 1.16006i −2.15024 + 2.95955i 0.342928 + 0.249152i 0 2.70486 − 0.878861i −2.24075 + 1.62800i −3.23134
195.15 0.552656 + 0.760665i −1.46683 + 0.476603i 0.344851 − 1.06134i −0.516362 + 0.710711i −1.17319 − 0.852372i 0 2.78634 − 0.905337i −0.502600 + 0.365160i −0.825984
195.16 0.552656 + 0.760665i 1.46683 − 0.476603i 0.344851 − 1.06134i 0.516362 − 0.710711i 1.17319 + 0.852372i 0 2.78634 − 0.905337i −0.502600 + 0.365160i 0.825984
195.17 1.20951 + 1.66474i −1.79105 + 0.581946i −0.690431 + 2.12493i −1.43298 + 1.97233i −3.13507 − 2.27776i 0 −0.458498 + 0.148975i 0.442131 − 0.321227i −5.01663
195.18 1.20951 + 1.66474i 1.79105 − 0.581946i −0.690431 + 2.12493i 1.43298 − 1.97233i 3.13507 + 2.27776i 0 −0.458498 + 0.148975i 0.442131 − 0.321227i 5.01663
195.19 1.23952 + 1.70605i −2.81867 + 0.915841i −0.756172 + 2.32726i 2.03740 − 2.80423i −5.05627 − 3.67359i 0 −0.896547 + 0.291306i 4.67908 − 3.39955i 7.30957
195.20 1.23952 + 1.70605i 2.81867 − 0.915841i −0.756172 + 2.32726i −2.03740 + 2.80423i 5.05627 + 3.67359i 0 −0.896547 + 0.291306i 4.67908 − 3.39955i −7.30957
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 195.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
11.d odd 10 1 inner
77.l even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 539.2.m.b ✓ 96
7.b odd 2 1 inner 539.2.m.b ✓ 96
7.c even 3 2 539.2.s.e 192
7.d odd 6 2 539.2.s.e 192
11.d odd 10 1 inner 539.2.m.b ✓ 96
77.l even 10 1 inner 539.2.m.b ✓ 96
77.n even 30 2 539.2.s.e 192
77.o odd 30 2 539.2.s.e 192
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
539.2.m.b ✓ 96 1.a even 1 1 trivial
539.2.m.b ✓ 96 7.b odd 2 1 inner
539.2.m.b ✓ 96 11.d odd 10 1 inner
539.2.m.b ✓ 96 77.l even 10 1 inner
539.2.s.e 192 7.c even 3 2
539.2.s.e 192 7.d odd 6 2
539.2.s.e 192 77.n even 30 2
539.2.s.e 192 77.o odd 30 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{48} - 16 T_{2}^{46} + 20 T_{2}^{45} + 190 T_{2}^{44} - 320 T_{2}^{43} - 1740 T_{2}^{42} + \cdots + 625 \) acting on \(S_{2}^{\mathrm{new}}(539, [\chi])\). Copy content Toggle raw display