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Properties

Label 144.4.k.c
Level $144$
Weight $4$
Character orbit 144.k
Analytic conductor $8.496$
Analytic rank $0$
Dimension $24$
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(37,144)]
 
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, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("144.37");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 144.k (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: \(24\)
Relative dimension: \(12\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 20 q^{4} + 48 q^{10} - 192 q^{16} + 24 q^{19} + 232 q^{22} + 416 q^{28} + 744 q^{31} + 296 q^{34} - 16 q^{37} + 104 q^{40} - 376 q^{43} - 32 q^{46} - 1176 q^{49} - 2088 q^{52} - 808 q^{58} - 912 q^{61} - 2968 q^{64} + 1440 q^{67} + 2408 q^{70} + 2408 q^{76} - 328 q^{79} + 5800 q^{82} - 240 q^{85} + 2160 q^{88} - 104 q^{91} - 2896 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1 −2.78466 + 0.495630i 0 7.50870 − 2.76033i 4.01270 + 4.01270i 0 − 25.9833i −19.5411 + 11.4081i 0 −13.1628 − 9.18520i
37.2 −2.49746 + 1.32767i 0 4.47457 − 6.63161i −9.19636 − 9.19636i 0 31.8982i −2.37041 + 22.5029i 0 35.1773 + 10.7577i
37.3 −2.36549 − 1.55063i 0 3.19111 + 7.33599i 8.32567 + 8.32567i 0 22.1273i 3.82683 − 22.3015i 0 −6.78431 − 32.6043i
37.4 −2.24328 − 1.72270i 0 2.06463 + 7.72899i −14.8202 − 14.8202i 0 − 10.4191i 8.68316 − 20.8950i 0 7.71515 + 58.7764i
37.5 −1.21292 + 2.55516i 0 −5.05765 − 6.19840i 5.80444 + 5.80444i 0 3.41531i 21.9724 − 5.40494i 0 −21.8716 + 7.79093i
37.6 −0.639782 − 2.75512i 0 −7.18136 + 3.52535i 5.16526 + 5.16526i 0 − 7.03833i 14.3073 + 17.5300i 0 10.9263 − 17.5356i
37.7 0.639782 + 2.75512i 0 −7.18136 + 3.52535i −5.16526 − 5.16526i 0 − 7.03833i −14.3073 − 17.5300i 0 10.9263 − 17.5356i
37.8 1.21292 − 2.55516i 0 −5.05765 − 6.19840i −5.80444 − 5.80444i 0 3.41531i −21.9724 + 5.40494i 0 −21.8716 + 7.79093i
37.9 2.24328 + 1.72270i 0 2.06463 + 7.72899i 14.8202 + 14.8202i 0 − 10.4191i −8.68316 + 20.8950i 0 7.71515 + 58.7764i
37.10 2.36549 + 1.55063i 0 3.19111 + 7.33599i −8.32567 − 8.32567i 0 22.1273i −3.82683 + 22.3015i 0 −6.78431 − 32.6043i
37.11 2.49746 − 1.32767i 0 4.47457 − 6.63161i 9.19636 + 9.19636i 0 31.8982i 2.37041 − 22.5029i 0 35.1773 + 10.7577i
37.12 2.78466 − 0.495630i 0 7.50870 − 2.76033i −4.01270 − 4.01270i 0 − 25.9833i 19.5411 − 11.4081i 0 −13.1628 − 9.18520i
109.1 −2.78466 − 0.495630i 0 7.50870 + 2.76033i 4.01270 − 4.01270i 0 25.9833i −19.5411 − 11.4081i 0 −13.1628 + 9.18520i
109.2 −2.49746 − 1.32767i 0 4.47457 + 6.63161i −9.19636 + 9.19636i 0 − 31.8982i −2.37041 − 22.5029i 0 35.1773 − 10.7577i
109.3 −2.36549 + 1.55063i 0 3.19111 − 7.33599i 8.32567 − 8.32567i 0 − 22.1273i 3.82683 + 22.3015i 0 −6.78431 + 32.6043i
109.4 −2.24328 + 1.72270i 0 2.06463 − 7.72899i −14.8202 + 14.8202i 0 10.4191i 8.68316 + 20.8950i 0 7.71515 − 58.7764i
109.5 −1.21292 − 2.55516i 0 −5.05765 + 6.19840i 5.80444 − 5.80444i 0 − 3.41531i 21.9724 + 5.40494i 0 −21.8716 − 7.79093i
109.6 −0.639782 + 2.75512i 0 −7.18136 − 3.52535i 5.16526 − 5.16526i 0 7.03833i 14.3073 − 17.5300i 0 10.9263 + 17.5356i
109.7 0.639782 − 2.75512i 0 −7.18136 − 3.52535i −5.16526 + 5.16526i 0 7.03833i −14.3073 + 17.5300i 0 10.9263 + 17.5356i
109.8 1.21292 + 2.55516i 0 −5.05765 + 6.19840i −5.80444 + 5.80444i 0 − 3.41531i −21.9724 − 5.40494i 0 −21.8716 − 7.79093i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.12
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 144.4.k.c ✓ 24
3.b odd 2 1 inner 144.4.k.c ✓ 24
4.b odd 2 1 576.4.k.c 24
12.b even 2 1 576.4.k.c 24
16.e even 4 1 inner 144.4.k.c ✓ 24
16.f odd 4 1 576.4.k.c 24
48.i odd 4 1 inner 144.4.k.c ✓ 24
48.k even 4 1 576.4.k.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
144.4.k.c ✓ 24 1.a even 1 1 trivial
144.4.k.c ✓ 24 3.b odd 2 1 inner
144.4.k.c ✓ 24 16.e even 4 1 inner
144.4.k.c ✓ 24 48.i odd 4 1 inner
576.4.k.c 24 4.b odd 2 1
576.4.k.c 24 12.b even 2 1
576.4.k.c 24 16.f odd 4 1
576.4.k.c 24 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} + 249216 T_{5}^{20} + 11828373696 T_{5}^{16} + 193462878781440 T_{5}^{12} + \cdots + 14\!\cdots\!00 \) acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display