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Properties

Label 144.3.j.a
Level $144$
Weight $3$
Character orbit 144.j
Analytic conductor $3.924$
Analytic rank $0$
Dimension $32$
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] = [144,3,Mod(53,144)]
 
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([0, 1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.53");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 144.j (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: \(3.92371580679\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 40 q^{10} + 48 q^{16} + 64 q^{19} - 88 q^{22} - 120 q^{28} - 248 q^{34} - 184 q^{40} + 128 q^{43} + 24 q^{46} - 224 q^{49} + 632 q^{52} + 456 q^{58} + 64 q^{61} - 48 q^{64} - 64 q^{67} - 312 q^{70} - 576 q^{76} - 512 q^{79} - 720 q^{82} + 320 q^{85} - 400 q^{88} - 192 q^{91} + 696 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −1.99710 − 0.107698i 0 3.97680 + 0.430168i −1.92848 + 1.92848i 0 2.43162i −7.89574 − 1.28738i 0 4.05906 − 3.64367i
53.2 −1.95704 − 0.412300i 0 3.66002 + 1.61377i 5.52702 − 5.52702i 0 7.79421i −6.49745 − 4.66725i 0 −13.0954 + 8.53782i
53.3 −1.76559 − 0.939508i 0 2.23465 + 3.31758i −6.08688 + 6.08688i 0 − 9.40026i −0.828591 − 7.95697i 0 16.4656 − 5.02829i
53.4 −1.56339 + 1.24732i 0 0.888382 − 3.90010i −1.84570 + 1.84570i 0 0.226665i 3.47579 + 7.20548i 0 0.583370 − 5.18772i
53.5 −1.13812 − 1.64459i 0 −1.40936 + 3.74349i 3.90343 − 3.90343i 0 0.778757i 7.76053 − 1.94271i 0 −10.8621 − 1.97697i
53.6 −1.11574 + 1.65986i 0 −1.51025 − 3.70394i −0.900179 + 0.900179i 0 − 7.66460i 7.83305 + 1.62583i 0 −0.489803 − 2.49853i
53.7 −0.252839 − 1.98395i 0 −3.87214 + 1.00324i −1.66372 + 1.66372i 0 13.3224i 2.96942 + 7.42850i 0 3.72140 + 2.88010i
53.8 −0.126306 + 1.99601i 0 −3.96809 − 0.504216i 5.14405 − 5.14405i 0 − 7.48880i 1.50761 − 7.85666i 0 9.61783 + 10.9173i
53.9 0.126306 − 1.99601i 0 −3.96809 − 0.504216i −5.14405 + 5.14405i 0 − 7.48880i −1.50761 + 7.85666i 0 9.61783 + 10.9173i
53.10 0.252839 + 1.98395i 0 −3.87214 + 1.00324i 1.66372 − 1.66372i 0 13.3224i −2.96942 − 7.42850i 0 3.72140 + 2.88010i
53.11 1.11574 − 1.65986i 0 −1.51025 − 3.70394i 0.900179 − 0.900179i 0 − 7.66460i −7.83305 − 1.62583i 0 −0.489803 − 2.49853i
53.12 1.13812 + 1.64459i 0 −1.40936 + 3.74349i −3.90343 + 3.90343i 0 0.778757i −7.76053 + 1.94271i 0 −10.8621 − 1.97697i
53.13 1.56339 − 1.24732i 0 0.888382 − 3.90010i 1.84570 − 1.84570i 0 0.226665i −3.47579 − 7.20548i 0 0.583370 − 5.18772i
53.14 1.76559 + 0.939508i 0 2.23465 + 3.31758i 6.08688 − 6.08688i 0 − 9.40026i 0.828591 + 7.95697i 0 16.4656 − 5.02829i
53.15 1.95704 + 0.412300i 0 3.66002 + 1.61377i −5.52702 + 5.52702i 0 7.79421i 6.49745 + 4.66725i 0 −13.0954 + 8.53782i
53.16 1.99710 + 0.107698i 0 3.97680 + 0.430168i 1.92848 − 1.92848i 0 2.43162i 7.89574 + 1.28738i 0 4.05906 − 3.64367i
125.1 −1.99710 + 0.107698i 0 3.97680 − 0.430168i −1.92848 − 1.92848i 0 − 2.43162i −7.89574 + 1.28738i 0 4.05906 + 3.64367i
125.2 −1.95704 + 0.412300i 0 3.66002 − 1.61377i 5.52702 + 5.52702i 0 − 7.79421i −6.49745 + 4.66725i 0 −13.0954 − 8.53782i
125.3 −1.76559 + 0.939508i 0 2.23465 − 3.31758i −6.08688 − 6.08688i 0 9.40026i −0.828591 + 7.95697i 0 16.4656 + 5.02829i
125.4 −1.56339 − 1.24732i 0 0.888382 + 3.90010i −1.84570 − 1.84570i 0 − 0.226665i 3.47579 − 7.20548i 0 0.583370 + 5.18772i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.3.j.a ✓ 32
3.b odd 2 1 inner 144.3.j.a ✓ 32
4.b odd 2 1 576.3.j.a 32
8.b even 2 1 1152.3.j.a 32
8.d odd 2 1 1152.3.j.b 32
12.b even 2 1 576.3.j.a 32
16.e even 4 1 inner 144.3.j.a ✓ 32
16.e even 4 1 1152.3.j.a 32
16.f odd 4 1 576.3.j.a 32
16.f odd 4 1 1152.3.j.b 32
24.f even 2 1 1152.3.j.b 32
24.h odd 2 1 1152.3.j.a 32
48.i odd 4 1 inner 144.3.j.a ✓ 32
48.i odd 4 1 1152.3.j.a 32
48.k even 4 1 576.3.j.a 32
48.k even 4 1 1152.3.j.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.3.j.a ✓ 32 1.a even 1 1 trivial
144.3.j.a ✓ 32 3.b odd 2 1 inner
144.3.j.a ✓ 32 16.e even 4 1 inner
144.3.j.a ✓ 32 48.i odd 4 1 inner
576.3.j.a 32 4.b odd 2 1
576.3.j.a 32 12.b even 2 1
576.3.j.a 32 16.f odd 4 1
576.3.j.a 32 48.k even 4 1
1152.3.j.a 32 8.b even 2 1
1152.3.j.a 32 16.e even 4 1
1152.3.j.a 32 24.h odd 2 1
1152.3.j.a 32 48.i odd 4 1
1152.3.j.b 32 8.d odd 2 1
1152.3.j.b 32 16.f odd 4 1
1152.3.j.b 32 24.f even 2 1
1152.3.j.b 32 48.k even 4 1

Hecke kernels

This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(144, [\chi])\).