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Properties

Label 189.3.k.a
Level $189$
Weight $3$
Character orbit 189.k
Analytic conductor $5.150$
Analytic rank $0$
Dimension $28$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [189,3,Mod(10,189)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(189, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.k (of order \(6\), degree \(2\), not 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: \(5.14987699641\)
Analytic rank: \(0\)
Dimension: \(28\)
Relative dimension: \(14\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 63)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - q^{2} - 23 q^{4} + 16 q^{8} - 6 q^{10} + 14 q^{11} + 15 q^{13} + 11 q^{14} - 27 q^{16} + 33 q^{17} - 6 q^{19} - 108 q^{20} - 10 q^{22} + 68 q^{23} - 62 q^{25} - 54 q^{26} - 16 q^{28} - 70 q^{29}+ \cdots - 811 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −1.91801 + 3.32210i 0 −5.35756 − 9.27956i 0.216546i 0 6.73498 − 1.90790i 25.7594 0 −0.719386 − 0.415338i
10.2 −1.62718 + 2.81835i 0 −3.29541 − 5.70782i − 3.39483i 0 −6.91332 − 1.09817i 8.43145 0 9.56782 + 5.52398i
10.3 −1.41697 + 2.45427i 0 −2.01561 − 3.49114i − 2.39855i 0 0.0107242 + 6.99999i 0.0884848 0 5.88667 + 3.39867i
10.4 −0.902282 + 1.56280i 0 0.371774 + 0.643931i 5.75495i 0 6.44289 + 2.73664i −8.56004 0 −8.99383 − 5.19259i
10.5 −0.826674 + 1.43184i 0 0.633221 + 1.09677i 7.86923i 0 −5.81886 − 3.89113i −8.70726 0 −11.2675 − 6.50529i
10.6 −0.662399 + 1.14731i 0 1.12246 + 1.94415i − 7.23514i 0 −3.90816 − 5.80744i −8.27324 0 8.30093 + 4.79254i
10.7 −0.198068 + 0.343064i 0 1.92154 + 3.32820i − 2.97240i 0 2.98301 − 6.33259i −3.10693 0 1.01972 + 0.588737i
10.8 0.178911 − 0.309883i 0 1.93598 + 3.35322i 4.59004i 0 −6.01934 + 3.57317i 2.81677 0 1.42238 + 0.821210i
10.9 0.227576 − 0.394173i 0 1.89642 + 3.28469i − 4.37081i 0 5.22047 + 4.66334i 3.54692 0 −1.72285 − 0.994690i
10.10 0.840995 − 1.45665i 0 0.585454 + 1.01404i 2.34462i 0 −3.93446 + 5.78964i 8.69742 0 3.41529 + 1.97182i
10.11 1.12025 − 1.94033i 0 −0.509909 − 0.883189i − 1.93444i 0 3.87064 − 5.83251i 6.67708 0 −3.75345 − 2.16706i
10.12 1.32841 − 2.30087i 0 −1.52933 − 2.64888i 9.20400i 0 6.96620 + 0.687028i 2.50096 0 21.1772 + 12.2267i
10.13 1.67756 − 2.90562i 0 −3.62842 − 6.28461i − 0.888628i 0 −3.29335 − 6.17688i −10.9271 0 −2.58202 − 1.49073i
10.14 1.67789 − 2.90618i 0 −3.63061 − 6.28839i − 8.51666i 0 −2.34142 + 6.59680i −10.9439 0 −24.7510 − 14.2900i
19.1 −1.91801 − 3.32210i 0 −5.35756 + 9.27956i − 0.216546i 0 6.73498 + 1.90790i 25.7594 0 −0.719386 + 0.415338i
19.2 −1.62718 − 2.81835i 0 −3.29541 + 5.70782i 3.39483i 0 −6.91332 + 1.09817i 8.43145 0 9.56782 − 5.52398i
19.3 −1.41697 − 2.45427i 0 −2.01561 + 3.49114i 2.39855i 0 0.0107242 − 6.99999i 0.0884848 0 5.88667 − 3.39867i
19.4 −0.902282 − 1.56280i 0 0.371774 − 0.643931i − 5.75495i 0 6.44289 − 2.73664i −8.56004 0 −8.99383 + 5.19259i
19.5 −0.826674 − 1.43184i 0 0.633221 − 1.09677i − 7.86923i 0 −5.81886 + 3.89113i −8.70726 0 −11.2675 + 6.50529i
19.6 −0.662399 − 1.14731i 0 1.12246 − 1.94415i 7.23514i 0 −3.90816 + 5.80744i −8.27324 0 8.30093 − 4.79254i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.k odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.3.k.a 28
3.b odd 2 1 63.3.k.a ✓ 28
7.d odd 6 1 189.3.t.a 28
9.c even 3 1 189.3.t.a 28
9.d odd 6 1 63.3.t.a yes 28
21.c even 2 1 441.3.k.b 28
21.g even 6 1 63.3.t.a yes 28
21.g even 6 1 441.3.l.b 28
21.h odd 6 1 441.3.l.a 28
21.h odd 6 1 441.3.t.a 28
63.i even 6 1 441.3.l.a 28
63.j odd 6 1 441.3.l.b 28
63.k odd 6 1 inner 189.3.k.a 28
63.n odd 6 1 441.3.k.b 28
63.o even 6 1 441.3.t.a 28
63.s even 6 1 63.3.k.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.3.k.a ✓ 28 3.b odd 2 1
63.3.k.a ✓ 28 63.s even 6 1
63.3.t.a yes 28 9.d odd 6 1
63.3.t.a yes 28 21.g even 6 1
189.3.k.a 28 1.a even 1 1 trivial
189.3.k.a 28 63.k odd 6 1 inner
189.3.t.a 28 7.d odd 6 1
189.3.t.a 28 9.c even 3 1
441.3.k.b 28 21.c even 2 1
441.3.k.b 28 63.n odd 6 1
441.3.l.a 28 21.h odd 6 1
441.3.l.a 28 63.i even 6 1
441.3.l.b 28 21.g even 6 1
441.3.l.b 28 63.j odd 6 1
441.3.t.a 28 21.h odd 6 1
441.3.t.a 28 63.o even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(189, [\chi])\).