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Properties

Label 171.3.ba.e
Level $171$
Weight $3$
Character orbit 171.ba
Analytic conductor $4.659$
Analytic rank $0$
Dimension $36$
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] = [171,3,Mod(10,171)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(171, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 17]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("171.10");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.ba (of order \(18\), degree \(6\), 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: \(4.65941252056\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(6\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{4} + 18 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 6 q^{4} + 18 q^{7} + 24 q^{10} - 6 q^{16} - 72 q^{19} - 96 q^{22} + 96 q^{25} + 624 q^{28} + 216 q^{31} + 150 q^{34} - 426 q^{40} - 564 q^{43} - 1170 q^{46} - 156 q^{49} + 462 q^{52} - 174 q^{55} + 120 q^{58} - 246 q^{61} + 570 q^{64} - 60 q^{67} - 516 q^{70} + 138 q^{73} - 696 q^{76} - 84 q^{79} + 1146 q^{82} + 942 q^{85} + 2016 q^{88} + 360 q^{91} - 282 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1 −3.49948 + 0.617053i 0 8.10685 − 2.95065i 2.51930 + 0.916949i 0 3.24636 − 5.62286i −14.2395 + 8.22119i 0 −9.38204 − 1.65431i
10.2 −2.46240 + 0.434187i 0 2.11612 − 0.770203i −8.80142 − 3.20346i 0 −3.40652 + 5.90027i 3.78528 − 2.18543i 0 23.0635 + 4.06672i
10.3 −1.24135 + 0.218883i 0 −2.26574 + 0.824660i 4.28103 + 1.55817i 0 −2.69100 + 4.66096i 6.99855 − 4.04062i 0 −5.65531 − 0.997183i
10.4 1.24135 − 0.218883i 0 −2.26574 + 0.824660i −4.28103 − 1.55817i 0 −2.69100 + 4.66096i −6.99855 + 4.04062i 0 −5.65531 − 0.997183i
10.5 2.46240 − 0.434187i 0 2.11612 − 0.770203i 8.80142 + 3.20346i 0 −3.40652 + 5.90027i −3.78528 + 2.18543i 0 23.0635 + 4.06672i
10.6 3.49948 − 0.617053i 0 8.10685 − 2.95065i −2.51930 − 0.916949i 0 3.24636 − 5.62286i 14.2395 − 8.22119i 0 −9.38204 − 1.65431i
91.1 −1.26501 + 3.47558i 0 −7.41522 − 6.22211i 6.55613 − 5.50125i 0 −1.65449 + 2.86566i 18.1933 − 10.5039i 0 10.8265 + 29.7455i
91.2 −0.896722 + 2.46372i 0 −2.20165 − 1.84740i −4.18839 + 3.51448i 0 0.355886 − 0.616413i −2.55656 + 1.47603i 0 −4.90288 − 13.4706i
91.3 −0.358514 + 0.985009i 0 2.22247 + 1.86487i 4.99505 − 4.19135i 0 4.74944 − 8.22627i −6.26486 + 3.61702i 0 2.33772 + 6.42283i
91.4 0.358514 − 0.985009i 0 2.22247 + 1.86487i −4.99505 + 4.19135i 0 4.74944 − 8.22627i 6.26486 − 3.61702i 0 2.33772 + 6.42283i
91.5 0.896722 − 2.46372i 0 −2.20165 − 1.84740i 4.18839 − 3.51448i 0 0.355886 − 0.616413i 2.55656 − 1.47603i 0 −4.90288 − 13.4706i
91.6 1.26501 − 3.47558i 0 −7.41522 − 6.22211i −6.55613 + 5.50125i 0 −1.65449 + 2.86566i −18.1933 + 10.5039i 0 10.8265 + 29.7455i
109.1 −1.26501 − 3.47558i 0 −7.41522 + 6.22211i 6.55613 + 5.50125i 0 −1.65449 − 2.86566i 18.1933 + 10.5039i 0 10.8265 − 29.7455i
109.2 −0.896722 − 2.46372i 0 −2.20165 + 1.84740i −4.18839 − 3.51448i 0 0.355886 + 0.616413i −2.55656 − 1.47603i 0 −4.90288 + 13.4706i
109.3 −0.358514 − 0.985009i 0 2.22247 − 1.86487i 4.99505 + 4.19135i 0 4.74944 + 8.22627i −6.26486 − 3.61702i 0 2.33772 − 6.42283i
109.4 0.358514 + 0.985009i 0 2.22247 − 1.86487i −4.99505 − 4.19135i 0 4.74944 + 8.22627i 6.26486 + 3.61702i 0 2.33772 − 6.42283i
109.5 0.896722 + 2.46372i 0 −2.20165 + 1.84740i 4.18839 + 3.51448i 0 0.355886 + 0.616413i 2.55656 + 1.47603i 0 −4.90288 + 13.4706i
109.6 1.26501 + 3.47558i 0 −7.41522 + 6.22211i −6.55613 − 5.50125i 0 −1.65449 − 2.86566i −18.1933 − 10.5039i 0 10.8265 − 29.7455i
127.1 −2.41686 − 2.88031i 0 −1.76034 + 9.98338i −1.11426 − 6.31928i 0 6.05127 − 10.4811i 19.9848 − 11.5382i 0 −15.5084 + 18.4822i
127.2 −1.43532 − 1.71055i 0 −0.171238 + 0.971138i 1.16146 + 6.58695i 0 3.16793 − 5.48701i −5.82825 + 3.36494i 0 9.60023 − 11.4411i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
19.f odd 18 1 inner
57.j even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.ba.e ✓ 36
3.b odd 2 1 inner 171.3.ba.e ✓ 36
19.f odd 18 1 inner 171.3.ba.e ✓ 36
57.j even 18 1 inner 171.3.ba.e ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.3.ba.e ✓ 36 1.a even 1 1 trivial
171.3.ba.e ✓ 36 3.b odd 2 1 inner
171.3.ba.e ✓ 36 19.f odd 18 1 inner
171.3.ba.e ✓ 36 57.j even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} + 3 T_{2}^{34} + 42 T_{2}^{32} - 2618 T_{2}^{30} - 7899 T_{2}^{28} - 102447 T_{2}^{26} + \cdots + 18877191454521 \) acting on \(S_{3}^{\mathrm{new}}(171, [\chi])\). Copy content Toggle raw display