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Properties

Label 84.4.o.a
Level $84$
Weight $4$
Character orbit 84.o
Analytic conductor $4.956$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,4,Mod(19,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.19");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.o (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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{2} - 36 q^{3} - 5 q^{4} + 6 q^{6} + 10 q^{7} - 76 q^{8} - 108 q^{9} - 19 q^{10} + 6 q^{11} - 15 q^{12} - 62 q^{14} - 89 q^{16} - 9 q^{18} + 42 q^{19} - 134 q^{20} - 60 q^{21} + 14 q^{22} - 21 q^{24}+ \cdots + 7795 q^{98}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −2.75199 + 0.653091i −1.50000 − 2.59808i 7.14694 − 3.59460i 9.22104 + 5.32377i 5.82477 + 6.17025i −18.0489 − 4.15182i −17.3207 + 14.5599i −4.50000 + 7.79423i −28.8532 − 8.62881i
19.2 −2.65324 − 0.979961i −1.50000 − 2.59808i 6.07935 + 5.20014i −3.65498 − 2.11020i 1.43384 + 8.36326i 4.33605 − 18.0055i −11.0340 − 19.7547i −4.50000 + 7.79423i 7.62961 + 9.18060i
19.3 −2.51510 + 1.29394i −1.50000 − 2.59808i 4.65146 − 6.50876i −13.3774 − 7.72344i 7.13439 + 4.59352i 10.8861 + 14.9831i −3.27697 + 22.3889i −4.50000 + 7.79423i 43.6391 + 2.11574i
19.4 −1.80305 − 2.17922i −1.50000 − 2.59808i −1.49801 + 7.85850i 8.01713 + 4.62869i −2.95720 + 7.95330i 11.9290 + 14.1668i 19.8264 − 10.9048i −4.50000 + 7.79423i −4.36836 − 25.8169i
19.5 −0.746666 + 2.72809i −1.50000 − 2.59808i −6.88498 − 4.07395i 13.2084 + 7.62590i 8.20779 − 2.15224i −10.0891 + 15.5309i 16.2549 − 15.7410i −4.50000 + 7.79423i −30.6665 + 30.3399i
19.6 −0.199616 − 2.82137i −1.50000 − 2.59808i −7.92031 + 1.12638i −0.0819989 − 0.0473421i −7.03072 + 4.75068i −18.4054 − 2.05975i 4.75896 + 22.1213i −4.50000 + 7.79423i −0.117202 + 0.240800i
19.7 −0.0644263 + 2.82769i −1.50000 − 2.59808i −7.99170 − 0.364356i −4.32707 − 2.49824i 7.44320 − 4.07416i 15.5037 − 10.1309i 1.54516 − 22.5746i −4.50000 + 7.79423i 7.34303 − 12.0747i
19.8 1.49919 + 2.39842i −1.50000 − 2.59808i −3.50484 + 7.19139i −11.8351 − 6.83298i 3.98249 − 7.49265i −16.3232 + 8.74950i −22.5024 + 2.37520i −4.50000 + 7.79423i −1.35469 − 38.6294i
19.9 1.63466 − 2.30822i −1.50000 − 2.59808i −2.65580 − 7.54631i −12.0209 − 6.94026i −8.44893 − 0.784626i 13.4721 + 12.7083i −21.7599 − 6.20544i −4.50000 + 7.79423i −35.6697 + 16.4020i
19.10 2.12158 − 1.87053i −1.50000 − 2.59808i 1.00225 − 7.93697i 18.9410 + 10.9356i −8.04215 − 2.70625i 2.66595 − 18.3274i −12.7200 − 18.7137i −4.50000 + 7.79423i 60.6402 − 12.2289i
19.11 2.22073 + 1.75167i −1.50000 − 2.59808i 1.86331 + 7.77998i 7.82897 + 4.52006i 1.21987 − 8.39714i 17.6230 − 5.69464i −9.49004 + 20.5412i −4.50000 + 7.79423i 9.46840 + 23.7516i
19.12 2.75793 − 0.627568i −1.50000 − 2.59808i 7.21232 − 3.46157i −11.9192 − 6.88154i −5.76736 − 6.22395i −8.54943 − 16.4289i 17.7187 − 14.0730i −4.50000 + 7.79423i −37.1908 − 11.4987i
31.1 −2.75199 − 0.653091i −1.50000 + 2.59808i 7.14694 + 3.59460i 9.22104 − 5.32377i 5.82477 − 6.17025i −18.0489 + 4.15182i −17.3207 − 14.5599i −4.50000 − 7.79423i −28.8532 + 8.62881i
31.2 −2.65324 + 0.979961i −1.50000 + 2.59808i 6.07935 − 5.20014i −3.65498 + 2.11020i 1.43384 − 8.36326i 4.33605 + 18.0055i −11.0340 + 19.7547i −4.50000 − 7.79423i 7.62961 − 9.18060i
31.3 −2.51510 − 1.29394i −1.50000 + 2.59808i 4.65146 + 6.50876i −13.3774 + 7.72344i 7.13439 − 4.59352i 10.8861 − 14.9831i −3.27697 − 22.3889i −4.50000 − 7.79423i 43.6391 − 2.11574i
31.4 −1.80305 + 2.17922i −1.50000 + 2.59808i −1.49801 − 7.85850i 8.01713 − 4.62869i −2.95720 − 7.95330i 11.9290 − 14.1668i 19.8264 + 10.9048i −4.50000 − 7.79423i −4.36836 + 25.8169i
31.5 −0.746666 − 2.72809i −1.50000 + 2.59808i −6.88498 + 4.07395i 13.2084 − 7.62590i 8.20779 + 2.15224i −10.0891 − 15.5309i 16.2549 + 15.7410i −4.50000 − 7.79423i −30.6665 − 30.3399i
31.6 −0.199616 + 2.82137i −1.50000 + 2.59808i −7.92031 − 1.12638i −0.0819989 + 0.0473421i −7.03072 − 4.75068i −18.4054 + 2.05975i 4.75896 − 22.1213i −4.50000 − 7.79423i −0.117202 − 0.240800i
31.7 −0.0644263 − 2.82769i −1.50000 + 2.59808i −7.99170 + 0.364356i −4.32707 + 2.49824i 7.44320 + 4.07416i 15.5037 + 10.1309i 1.54516 + 22.5746i −4.50000 − 7.79423i 7.34303 + 12.0747i
31.8 1.49919 − 2.39842i −1.50000 + 2.59808i −3.50484 − 7.19139i −11.8351 + 6.83298i 3.98249 + 7.49265i −16.3232 − 8.74950i −22.5024 − 2.37520i −4.50000 − 7.79423i −1.35469 + 38.6294i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.4.o.a ✓ 24
4.b odd 2 1 84.4.o.b yes 24
7.d odd 6 1 84.4.o.b yes 24
28.f even 6 1 inner 84.4.o.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.o.a ✓ 24 1.a even 1 1 trivial
84.4.o.a ✓ 24 28.f even 6 1 inner
84.4.o.b yes 24 4.b odd 2 1
84.4.o.b yes 24 7.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{24} - 6 T_{11}^{23} - 8863 T_{11}^{22} + 53250 T_{11}^{21} + 51135286 T_{11}^{20} + \cdots + 32\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display