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Properties

Label 144.4.l.a
Level $144$
Weight $4$
Character orbit 144.l
Analytic conductor $8.496$
Analytic rank $0$
Dimension $48$
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] = [144,4,Mod(35,144)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(144, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.35");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.l (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: \(8.49627504083\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 120 q^{10} - 144 q^{16} - 48 q^{19} + 72 q^{22} + 72 q^{28} - 984 q^{34} - 1272 q^{40} + 864 q^{43} - 1416 q^{46} + 2352 q^{49} - 648 q^{52} - 576 q^{55} + 1128 q^{58} + 1824 q^{61} + 3024 q^{64} + 816 q^{67} + 2664 q^{70} + 1920 q^{76} + 1200 q^{82} - 480 q^{85} - 4560 q^{88} - 3600 q^{91} - 11304 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
35.1 −2.81788 − 0.244009i 0 7.88092 + 1.37518i 15.2065 + 15.2065i 0 24.4971 −21.8719 − 5.79811i 0 −39.1395 − 46.5605i
35.2 −2.79016 + 0.463706i 0 7.56995 − 2.58762i −9.40057 − 9.40057i 0 −3.57327 −19.9215 + 10.7301i 0 30.5882 + 21.8700i
35.3 −2.78125 − 0.514448i 0 7.47069 + 2.86161i −3.43656 − 3.43656i 0 8.14652 −19.3057 − 11.8021i 0 7.79000 + 11.3259i
35.4 −2.44348 + 1.42457i 0 3.94119 − 6.96183i 11.7679 + 11.7679i 0 −11.9978 0.287399 + 22.6256i 0 −45.5190 − 11.9904i
35.5 −2.19729 − 1.78098i 0 1.65621 + 7.82668i 3.62348 + 3.62348i 0 −33.7361 10.3000 − 20.1472i 0 −1.50850 − 14.4152i
35.6 −1.89081 − 2.10353i 0 −0.849709 + 7.95475i 11.2962 + 11.2962i 0 −19.1985 18.3397 − 13.2535i 0 2.40303 − 45.1210i
35.7 −1.80581 + 2.17694i 0 −1.47812 − 7.86226i −6.31601 − 6.31601i 0 16.2772 19.7849 + 10.9799i 0 25.1551 − 2.34407i
35.8 −1.73660 + 2.23254i 0 −1.96847 − 7.75404i 3.22357 + 3.22357i 0 −13.1030 20.7296 + 9.07094i 0 −12.7948 + 1.59871i
35.9 −1.64440 − 2.30129i 0 −2.59187 + 7.56850i −7.28730 − 7.28730i 0 29.9574 21.6794 − 6.48105i 0 −4.78691 + 28.7535i
35.10 −1.05126 − 2.62580i 0 −5.78970 + 5.52081i −11.8474 − 11.8474i 0 −12.7523 20.5831 + 9.39881i 0 −18.6543 + 43.5638i
35.11 −0.272518 + 2.81527i 0 −7.85147 − 1.53442i −6.30986 − 6.30986i 0 27.2034 6.45947 − 21.6858i 0 19.4835 − 16.0444i
35.12 −0.0720364 + 2.82751i 0 −7.98962 − 0.407367i −2.40838 − 2.40838i 0 −11.7205 1.72738 − 22.5614i 0 6.98322 − 6.63623i
35.13 0.0720364 − 2.82751i 0 −7.98962 − 0.407367i 2.40838 + 2.40838i 0 −11.7205 −1.72738 + 22.5614i 0 6.98322 − 6.63623i
35.14 0.272518 − 2.81527i 0 −7.85147 − 1.53442i 6.30986 + 6.30986i 0 27.2034 −6.45947 + 21.6858i 0 19.4835 − 16.0444i
35.15 1.05126 + 2.62580i 0 −5.78970 + 5.52081i 11.8474 + 11.8474i 0 −12.7523 −20.5831 − 9.39881i 0 −18.6543 + 43.5638i
35.16 1.64440 + 2.30129i 0 −2.59187 + 7.56850i 7.28730 + 7.28730i 0 29.9574 −21.6794 + 6.48105i 0 −4.78691 + 28.7535i
35.17 1.73660 − 2.23254i 0 −1.96847 − 7.75404i −3.22357 − 3.22357i 0 −13.1030 −20.7296 − 9.07094i 0 −12.7948 + 1.59871i
35.18 1.80581 − 2.17694i 0 −1.47812 − 7.86226i 6.31601 + 6.31601i 0 16.2772 −19.7849 − 10.9799i 0 25.1551 − 2.34407i
35.19 1.89081 + 2.10353i 0 −0.849709 + 7.95475i −11.2962 − 11.2962i 0 −19.1985 −18.3397 + 13.2535i 0 2.40303 − 45.1210i
35.20 2.19729 + 1.78098i 0 1.65621 + 7.82668i −3.62348 − 3.62348i 0 −33.7361 −10.3000 + 20.1472i 0 −1.50850 − 14.4152i
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 35.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.l.a ✓ 48
3.b odd 2 1 inner 144.4.l.a ✓ 48
4.b odd 2 1 576.4.l.a 48
8.b even 2 1 1152.4.l.b 48
8.d odd 2 1 1152.4.l.a 48
12.b even 2 1 576.4.l.a 48
16.e even 4 1 576.4.l.a 48
16.e even 4 1 1152.4.l.a 48
16.f odd 4 1 inner 144.4.l.a ✓ 48
16.f odd 4 1 1152.4.l.b 48
24.f even 2 1 1152.4.l.a 48
24.h odd 2 1 1152.4.l.b 48
48.i odd 4 1 576.4.l.a 48
48.i odd 4 1 1152.4.l.a 48
48.k even 4 1 inner 144.4.l.a ✓ 48
48.k even 4 1 1152.4.l.b 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.4.l.a ✓ 48 1.a even 1 1 trivial
144.4.l.a ✓ 48 3.b odd 2 1 inner
144.4.l.a ✓ 48 16.f odd 4 1 inner
144.4.l.a ✓ 48 48.k even 4 1 inner
576.4.l.a 48 4.b odd 2 1
576.4.l.a 48 12.b even 2 1
576.4.l.a 48 16.e even 4 1
576.4.l.a 48 48.i odd 4 1
1152.4.l.a 48 8.d odd 2 1
1152.4.l.a 48 16.e even 4 1
1152.4.l.a 48 24.f even 2 1
1152.4.l.a 48 48.i odd 4 1
1152.4.l.b 48 8.b even 2 1
1152.4.l.b 48 16.f odd 4 1
1152.4.l.b 48 24.h odd 2 1
1152.4.l.b 48 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(144, [\chi])\).