Properties

Label 144.4.k.b
Level $144$
Weight $4$
Character orbit 144.k
Analytic conductor $8.496$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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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: no (minimal twist has level 48)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} - 84 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 20 q^{4} - 84 q^{8} + 72 q^{10} + 40 q^{11} + 348 q^{14} - 192 q^{16} + 24 q^{19} - 80 q^{20} + 704 q^{22} + 20 q^{26} - 344 q^{28} - 400 q^{29} - 744 q^{31} + 960 q^{32} - 704 q^{34} + 456 q^{35} + 16 q^{37} - 1256 q^{38} + 1744 q^{40} + 1240 q^{43} + 200 q^{44} - 1432 q^{46} - 1176 q^{49} - 708 q^{50} + 1008 q^{52} - 752 q^{53} - 1344 q^{56} + 1936 q^{58} + 1376 q^{59} - 912 q^{61} + 996 q^{62} - 56 q^{64} - 976 q^{65} - 2256 q^{67} + 1568 q^{68} - 1760 q^{70} + 2740 q^{74} - 1880 q^{76} - 1904 q^{77} + 5992 q^{79} - 712 q^{80} - 40 q^{82} - 2680 q^{83} - 240 q^{85} + 1712 q^{86} - 3936 q^{88} - 3496 q^{91} - 5296 q^{92} + 5272 q^{94} + 7728 q^{95} - 6760 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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 −2.70307 + 0.832707i 0 6.61320 4.50173i −11.7911 11.7911i 0 12.5754i −14.1273 + 17.6754i 0 41.6906 + 22.0536i
37.2 −2.59717 1.12013i 0 5.49064 + 5.81833i −0.706564 0.706564i 0 4.44122i −7.74288 21.2614i 0 1.04363 + 2.62651i
37.3 −2.07099 + 1.92640i 0 0.577966 7.97909i −0.644922 0.644922i 0 7.13926i 14.1740 + 17.6380i 0 2.57800 + 0.0932465i
37.4 −1.94824 2.05046i 0 −0.408732 + 7.98955i −2.24191 2.24191i 0 9.00196i 17.1785 14.7275i 0 −0.229160 + 8.96471i
37.5 −0.954009 + 2.66268i 0 −6.17974 5.08044i 8.83384 + 8.83384i 0 29.4760i 19.4231 11.6079i 0 −31.9493 + 15.0941i
37.6 0.220074 2.81985i 0 −7.90313 1.24115i −10.2951 10.2951i 0 32.8369i −5.23914 + 22.0125i 0 −31.2964 + 26.7650i
37.7 0.716137 2.73627i 0 −6.97430 3.91908i 11.7719 + 11.7719i 0 14.7089i −15.7182 + 16.2769i 0 40.6415 23.7808i
37.8 0.987020 + 2.65062i 0 −6.05158 + 5.23243i 11.8955 + 11.8955i 0 0.485059i −19.8422 10.8759i 0 −19.7893 + 43.2714i
37.9 1.40656 + 2.45389i 0 −4.04315 + 6.90311i −3.22588 3.22588i 0 24.6080i −22.6264 0.211795i 0 3.37855 12.4534i
37.10 1.92738 2.07008i 0 −0.570442 7.97964i 7.29121 + 7.29121i 0 22.1610i −17.6179 14.1989i 0 29.1463 1.04047i
37.11 2.24080 + 1.72593i 0 2.04234 + 7.73491i −14.6111 14.6111i 0 26.8889i −8.77342 + 20.8573i 0 −7.52282 57.9584i
37.12 2.77551 + 0.544550i 0 7.40693 + 3.02281i 3.72414 + 3.72414i 0 20.2675i 18.9120 + 12.4233i 0 8.30842 + 12.3644i
109.1 −2.70307 0.832707i 0 6.61320 + 4.50173i −11.7911 + 11.7911i 0 12.5754i −14.1273 17.6754i 0 41.6906 22.0536i
109.2 −2.59717 + 1.12013i 0 5.49064 5.81833i −0.706564 + 0.706564i 0 4.44122i −7.74288 + 21.2614i 0 1.04363 2.62651i
109.3 −2.07099 1.92640i 0 0.577966 + 7.97909i −0.644922 + 0.644922i 0 7.13926i 14.1740 17.6380i 0 2.57800 0.0932465i
109.4 −1.94824 + 2.05046i 0 −0.408732 7.98955i −2.24191 + 2.24191i 0 9.00196i 17.1785 + 14.7275i 0 −0.229160 8.96471i
109.5 −0.954009 2.66268i 0 −6.17974 + 5.08044i 8.83384 8.83384i 0 29.4760i 19.4231 + 11.6079i 0 −31.9493 15.0941i
109.6 0.220074 + 2.81985i 0 −7.90313 + 1.24115i −10.2951 + 10.2951i 0 32.8369i −5.23914 22.0125i 0 −31.2964 26.7650i
109.7 0.716137 + 2.73627i 0 −6.97430 + 3.91908i 11.7719 11.7719i 0 14.7089i −15.7182 16.2769i 0 40.6415 + 23.7808i
109.8 0.987020 2.65062i 0 −6.05158 5.23243i 11.8955 11.8955i 0 0.485059i −19.8422 + 10.8759i 0 −19.7893 43.2714i
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
16.e 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.k.b 24
3.b odd 2 1 48.4.j.a 24
4.b odd 2 1 576.4.k.b 24
12.b even 2 1 192.4.j.a 24
16.e even 4 1 inner 144.4.k.b 24
16.f odd 4 1 576.4.k.b 24
24.f even 2 1 384.4.j.a 24
24.h odd 2 1 384.4.j.b 24
48.i odd 4 1 48.4.j.a 24
48.i odd 4 1 384.4.j.b 24
48.k even 4 1 192.4.j.a 24
48.k even 4 1 384.4.j.a 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.4.j.a 24 3.b odd 2 1
48.4.j.a 24 48.i odd 4 1
144.4.k.b 24 1.a even 1 1 trivial
144.4.k.b 24 16.e even 4 1 inner
192.4.j.a 24 12.b even 2 1
192.4.j.a 24 48.k even 4 1
384.4.j.a 24 24.f even 2 1
384.4.j.a 24 48.k even 4 1
384.4.j.b 24 24.h odd 2 1
384.4.j.b 24 48.i odd 4 1
576.4.k.b 24 4.b odd 2 1
576.4.k.b 24 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{24} - 1936 T_{5}^{21} + 249216 T_{5}^{20} - 752832 T_{5}^{19} + 1874048 T_{5}^{18} - 217848704 T_{5}^{17} + 18944520384 T_{5}^{16} - 94398676992 T_{5}^{15} + \cdots + 15\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(144, [\chi])\). Copy content Toggle raw display