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Properties

Label 189.3.t.a
Level $189$
Weight $3$
Character orbit 189.t
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(73,189)]
 
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([4, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("189.73");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 189.t (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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} + 46 q^{4} + 3 q^{5} + 16 q^{8} - 6 q^{10} - 7 q^{11} - 15 q^{13} - 10 q^{14} + 54 q^{16} + 33 q^{17} - 6 q^{19} + 108 q^{20} - 10 q^{22} - 34 q^{23} + 31 q^{25} - 54 q^{26} - 16 q^{28} - 70 q^{29}+ \cdots - 811 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1 −3.35577 0 7.26121 7.37564 + 4.25833i 0 6.88370 − 1.27067i −10.9439 0 −24.7510 − 14.2900i
73.2 −3.35512 0 7.25684 0.769575 + 0.444314i 0 −3.70266 + 5.94056i −10.9271 0 −2.58202 − 1.49073i
73.3 −2.65681 0 3.05866 −7.97090 − 4.60200i 0 −2.88812 − 6.37642i 2.50096 0 21.1772 + 12.2267i
73.4 −2.24050 0 1.01982 1.67528 + 0.967222i 0 −6.98642 − 0.435817i 6.67708 0 −3.75345 − 2.16706i
73.5 −1.68199 0 −1.17091 −2.03050 − 1.17231i 0 6.98121 + 0.512524i 8.69742 0 3.41529 + 1.97182i
73.6 −0.455152 0 −3.79284 3.78523 + 2.18540i 0 1.42833 − 6.85273i 3.54692 0 −1.72285 − 0.994690i
73.7 −0.357823 0 −3.87196 −3.97509 − 2.29502i 0 6.10412 + 3.42632i 2.81677 0 1.42238 + 0.821210i
73.8 0.396136 0 −3.84308 2.57417 + 1.48620i 0 −6.97569 + 0.582933i −3.10693 0 1.01972 + 0.588737i
73.9 1.32480 0 −2.24491 6.26581 + 3.61757i 0 −3.07531 + 6.28828i −8.27324 0 8.30093 + 4.79254i
73.10 1.65335 0 −1.26644 −6.81496 − 3.93462i 0 −0.460386 + 6.98484i −8.70726 0 −11.2675 − 6.50529i
73.11 1.80456 0 −0.743548 −4.98393 − 2.87747i 0 −0.851442 − 6.94802i −8.56004 0 −8.99383 − 5.19259i
73.12 2.83394 0 4.03122 2.07720 + 1.19927i 0 6.05681 − 3.50928i 0.0884848 0 5.88667 + 3.39867i
73.13 3.25435 0 6.59082 2.94001 + 1.69741i 0 2.50562 + 6.53620i 8.43145 0 9.56782 + 5.52398i
73.14 3.83603 0 10.7151 −0.187534 − 0.108273i 0 −5.01978 − 4.87871i 25.7594 0 −0.719386 − 0.415338i
145.1 −3.35577 0 7.26121 7.37564 − 4.25833i 0 6.88370 + 1.27067i −10.9439 0 −24.7510 + 14.2900i
145.2 −3.35512 0 7.25684 0.769575 − 0.444314i 0 −3.70266 − 5.94056i −10.9271 0 −2.58202 + 1.49073i
145.3 −2.65681 0 3.05866 −7.97090 + 4.60200i 0 −2.88812 + 6.37642i 2.50096 0 21.1772 − 12.2267i
145.4 −2.24050 0 1.01982 1.67528 − 0.967222i 0 −6.98642 + 0.435817i 6.67708 0 −3.75345 + 2.16706i
145.5 −1.68199 0 −1.17091 −2.03050 + 1.17231i 0 6.98121 − 0.512524i 8.69742 0 3.41529 − 1.97182i
145.6 −0.455152 0 −3.79284 3.78523 − 2.18540i 0 1.42833 + 6.85273i 3.54692 0 −1.72285 + 0.994690i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 73.14
Significant digits:
Format:

Inner twists

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

Hecke kernels

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