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Properties

Label 180.3.m.c
Level $180$
Weight $3$
Character orbit 180.m
Analytic conductor $4.905$
Analytic rank $0$
Dimension $40$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [180,3,Mod(107,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 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("180.107");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 180.m (of order \(4\), 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: \(4.90464475849\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 40 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q + 16 q^{10} - 40 q^{13} + 184 q^{16} + 112 q^{22} + 32 q^{28} - 264 q^{37} + 40 q^{40} - 160 q^{46} - 328 q^{52} - 720 q^{58} - 128 q^{61} + 464 q^{70} - 664 q^{73} + 576 q^{76} + 320 q^{82} + 608 q^{85} + 544 q^{88} + 360 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −1.99255 + 0.172449i 0 3.94052 − 0.687228i 2.98856 + 4.00855i 0 5.60654 − 5.60654i −7.73318 + 2.04888i 0 −6.64614 − 7.47187i
107.2 −1.97401 − 0.321406i 0 3.79340 + 1.26891i −0.196879 − 4.99612i 0 −8.42491 + 8.42491i −7.08035 − 3.72406i 0 −1.21714 + 9.92565i
107.3 −1.92656 + 0.537013i 0 3.42323 − 2.06917i 2.15036 − 4.51397i 0 4.16469 − 4.16469i −5.48388 + 5.82470i 0 −1.71872 + 9.85119i
107.4 −1.88663 − 0.663804i 0 3.11873 + 2.50470i −4.97776 + 0.471047i 0 0.696286 − 0.696286i −4.22125 − 6.79566i 0 9.70387 + 2.41557i
107.5 −1.47830 − 1.34709i 0 0.370713 + 3.98278i 4.91198 + 0.934037i 0 −6.91072 + 6.91072i 4.81713 − 6.38711i 0 −6.00313 − 7.99765i
107.6 −1.34709 − 1.47830i 0 −0.370713 + 3.98278i −4.91198 − 0.934037i 0 6.91072 − 6.91072i 6.38711 − 4.81713i 0 5.23609 + 8.51959i
107.7 −0.663804 − 1.88663i 0 −3.11873 + 2.50470i 4.97776 − 0.471047i 0 −0.696286 + 0.696286i 6.79566 + 4.22125i 0 −4.19295 − 9.07850i
107.8 −0.537013 + 1.92656i 0 −3.42323 − 2.06917i 2.15036 − 4.51397i 0 −4.16469 + 4.16469i 5.82470 − 5.48388i 0 7.54165 + 6.56685i
107.9 −0.321406 − 1.97401i 0 −3.79340 + 1.26891i 0.196879 + 4.99612i 0 8.42491 − 8.42491i 3.72406 + 7.08035i 0 9.79910 − 1.99442i
107.10 −0.172449 + 1.99255i 0 −3.94052 − 0.687228i 2.98856 + 4.00855i 0 −5.60654 + 5.60654i 2.04888 − 7.73318i 0 −8.50262 + 5.26359i
107.11 0.172449 − 1.99255i 0 −3.94052 − 0.687228i −2.98856 − 4.00855i 0 −5.60654 + 5.60654i −2.04888 + 7.73318i 0 −8.50262 + 5.26359i
107.12 0.321406 + 1.97401i 0 −3.79340 + 1.26891i −0.196879 − 4.99612i 0 8.42491 − 8.42491i −3.72406 − 7.08035i 0 9.79910 − 1.99442i
107.13 0.537013 − 1.92656i 0 −3.42323 − 2.06917i −2.15036 + 4.51397i 0 −4.16469 + 4.16469i −5.82470 + 5.48388i 0 7.54165 + 6.56685i
107.14 0.663804 + 1.88663i 0 −3.11873 + 2.50470i −4.97776 + 0.471047i 0 −0.696286 + 0.696286i −6.79566 − 4.22125i 0 −4.19295 − 9.07850i
107.15 1.34709 + 1.47830i 0 −0.370713 + 3.98278i 4.91198 + 0.934037i 0 6.91072 − 6.91072i −6.38711 + 4.81713i 0 5.23609 + 8.51959i
107.16 1.47830 + 1.34709i 0 0.370713 + 3.98278i −4.91198 − 0.934037i 0 −6.91072 + 6.91072i −4.81713 + 6.38711i 0 −6.00313 − 7.99765i
107.17 1.88663 + 0.663804i 0 3.11873 + 2.50470i 4.97776 − 0.471047i 0 0.696286 − 0.696286i 4.22125 + 6.79566i 0 9.70387 + 2.41557i
107.18 1.92656 − 0.537013i 0 3.42323 − 2.06917i −2.15036 + 4.51397i 0 4.16469 − 4.16469i 5.48388 − 5.82470i 0 −1.71872 + 9.85119i
107.19 1.97401 + 0.321406i 0 3.79340 + 1.26891i 0.196879 + 4.99612i 0 −8.42491 + 8.42491i 7.08035 + 3.72406i 0 −1.21714 + 9.92565i
107.20 1.99255 − 0.172449i 0 3.94052 − 0.687228i −2.98856 − 4.00855i 0 5.60654 − 5.60654i 7.73318 − 2.04888i 0 −6.64614 − 7.47187i
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.20
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.3.m.c ✓ 40
3.b odd 2 1 inner 180.3.m.c ✓ 40
4.b odd 2 1 inner 180.3.m.c ✓ 40
5.c odd 4 1 inner 180.3.m.c ✓ 40
12.b even 2 1 inner 180.3.m.c ✓ 40
15.e even 4 1 inner 180.3.m.c ✓ 40
20.e even 4 1 inner 180.3.m.c ✓ 40
60.l odd 4 1 inner 180.3.m.c ✓ 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.3.m.c ✓ 40 1.a even 1 1 trivial
180.3.m.c ✓ 40 3.b odd 2 1 inner
180.3.m.c ✓ 40 4.b odd 2 1 inner
180.3.m.c ✓ 40 5.c odd 4 1 inner
180.3.m.c ✓ 40 12.b even 2 1 inner
180.3.m.c ✓ 40 15.e even 4 1 inner
180.3.m.c ✓ 40 20.e even 4 1 inner
180.3.m.c ✓ 40 60.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(180, [\chi])\):

\( T_{7}^{20} + 34432T_{7}^{16} + 339574784T_{7}^{12} + 1087424839680T_{7}^{8} + 875413530214400T_{7}^{4} + 822083584000000 \) Copy content Toggle raw display
\( T_{13}^{10} + 10 T_{13}^{9} + 50 T_{13}^{8} - 4496 T_{13}^{7} + 95208 T_{13}^{6} - 377232 T_{13}^{5} + \cdots + 8872185632 \) Copy content Toggle raw display