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Properties

Label 444.2.g.d
Level $444$
Weight $2$
Character orbit 444.g
Analytic conductor $3.545$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [444,2,Mod(443,444)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("444.443"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(444, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 444 = 2^{2} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 444.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.54535784974\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 8 q^{4} + 20 q^{9} - 20 q^{10} + 14 q^{12} + 8 q^{16} + 8 q^{21} - 24 q^{25} + 40 q^{28} - 24 q^{30} - 40 q^{33} + 56 q^{34} + 10 q^{36} - 48 q^{37} + 60 q^{40} + 68 q^{46} - 54 q^{48} + 32 q^{49}+ \cdots - 34 q^{90}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
443.1 −1.38529 − 0.284535i −0.898596 + 1.48072i 1.83808 + 0.788329i −1.07517 1.66614 − 1.79555i − 0.486712i −2.32197 − 1.61507i −1.38505 − 2.66114i 1.48942 + 0.305922i
443.2 −1.38529 − 0.284535i 0.898596 + 1.48072i 1.83808 + 0.788329i −1.07517 −0.823504 − 2.30691i − 0.486712i −2.32197 − 1.61507i −1.38505 + 2.66114i 1.48942 + 0.305922i
443.3 −1.38529 + 0.284535i −0.898596 − 1.48072i 1.83808 − 0.788329i −1.07517 1.66614 + 1.79555i 0.486712i −2.32197 + 1.61507i −1.38505 + 2.66114i 1.48942 − 0.305922i
443.4 −1.38529 + 0.284535i 0.898596 − 1.48072i 1.83808 − 0.788329i −1.07517 −0.823504 + 2.30691i 0.486712i −2.32197 + 1.61507i −1.38505 − 2.66114i 1.48942 − 0.305922i
443.5 −1.33802 − 0.457941i −1.33116 − 1.10816i 1.58058 + 1.22547i 1.47820 1.27364 + 2.09233i 3.69880i −1.55365 − 2.36351i 0.543963 + 2.95027i −1.97785 − 0.676926i
443.6 −1.33802 − 0.457941i 1.33116 − 1.10816i 1.58058 + 1.22547i 1.47820 −2.28858 + 0.873146i 3.69880i −1.55365 − 2.36351i 0.543963 − 2.95027i −1.97785 − 0.676926i
443.7 −1.33802 + 0.457941i −1.33116 + 1.10816i 1.58058 − 1.22547i 1.47820 1.27364 − 2.09233i − 3.69880i −1.55365 + 2.36351i 0.543963 − 2.95027i −1.97785 + 0.676926i
443.8 −1.33802 + 0.457941i 1.33116 + 1.10816i 1.58058 − 1.22547i 1.47820 −2.28858 − 0.873146i − 3.69880i −1.55365 + 2.36351i 0.543963 + 2.95027i −1.97785 + 0.676926i
443.9 −1.19022 − 0.763797i −1.66944 − 0.461484i 0.833229 + 1.81817i −1.28804 1.63452 + 1.82438i − 3.76441i 0.396988 − 2.80043i 2.57407 + 1.54084i 1.53305 + 0.983803i
443.10 −1.19022 − 0.763797i 1.66944 − 0.461484i 0.833229 + 1.81817i −1.28804 −2.33948 − 0.725848i − 3.76441i 0.396988 − 2.80043i 2.57407 − 1.54084i 1.53305 + 0.983803i
443.11 −1.19022 + 0.763797i −1.66944 + 0.461484i 0.833229 − 1.81817i −1.28804 1.63452 − 1.82438i 3.76441i 0.396988 + 2.80043i 2.57407 − 1.54084i 1.53305 − 0.983803i
443.12 −1.19022 + 0.763797i 1.66944 + 0.461484i 0.833229 − 1.81817i −1.28804 −2.33948 + 0.725848i 3.76441i 0.396988 + 2.80043i 2.57407 + 1.54084i 1.53305 − 0.983803i
443.13 −1.03597 − 0.962690i −1.48753 + 0.887270i 0.146457 + 1.99463i 3.04955 2.39520 + 0.512849i 0.254601i 1.76849 − 2.20736i 1.42550 − 2.63968i −3.15924 − 2.93578i
443.14 −1.03597 − 0.962690i 1.48753 + 0.887270i 0.146457 + 1.99463i 3.04955 −0.686869 − 2.35121i 0.254601i 1.76849 − 2.20736i 1.42550 + 2.63968i −3.15924 − 2.93578i
443.15 −1.03597 + 0.962690i −1.48753 − 0.887270i 0.146457 − 1.99463i 3.04955 2.39520 − 0.512849i − 0.254601i 1.76849 + 2.20736i 1.42550 + 2.63968i −3.15924 + 2.93578i
443.16 −1.03597 + 0.962690i 1.48753 − 0.887270i 0.146457 − 1.99463i 3.04955 −0.686869 + 2.35121i − 0.254601i 1.76849 + 2.20736i 1.42550 − 2.63968i −3.15924 + 2.93578i
443.17 −0.528598 − 1.31171i −0.903125 − 1.47796i −1.44117 + 1.38673i −0.254924 −1.46127 + 1.96589i − 2.56483i 2.58079 + 1.15737i −1.36873 + 2.66957i 0.134752 + 0.334387i
443.18 −0.528598 − 1.31171i 0.903125 − 1.47796i −1.44117 + 1.38673i −0.254924 −2.41605 − 0.403393i − 2.56483i 2.58079 + 1.15737i −1.36873 − 2.66957i 0.134752 + 0.334387i
443.19 −0.528598 + 1.31171i −0.903125 + 1.47796i −1.44117 − 1.38673i −0.254924 −1.46127 − 1.96589i 2.56483i 2.58079 − 1.15737i −1.36873 − 2.66957i 0.134752 − 0.334387i
443.20 −0.528598 + 1.31171i 0.903125 + 1.47796i −1.44117 − 1.38673i −0.254924 −2.41605 + 0.403393i 2.56483i 2.58079 − 1.15737i −1.36873 + 2.66957i 0.134752 − 0.334387i
See all 48 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 443.48
Significant digits:
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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
12.b even 2 1 inner
37.b even 2 1 inner
111.d odd 2 1 inner
148.b odd 2 1 inner
444.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 444.2.g.d ✓ 48
3.b odd 2 1 inner 444.2.g.d ✓ 48
4.b odd 2 1 inner 444.2.g.d ✓ 48
12.b even 2 1 inner 444.2.g.d ✓ 48
37.b even 2 1 inner 444.2.g.d ✓ 48
111.d odd 2 1 inner 444.2.g.d ✓ 48
148.b odd 2 1 inner 444.2.g.d ✓ 48
444.g even 2 1 inner 444.2.g.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
444.2.g.d ✓ 48 1.a even 1 1 trivial
444.2.g.d ✓ 48 3.b odd 2 1 inner
444.2.g.d ✓ 48 4.b odd 2 1 inner
444.2.g.d ✓ 48 12.b even 2 1 inner
444.2.g.d ✓ 48 37.b even 2 1 inner
444.2.g.d ✓ 48 111.d odd 2 1 inner
444.2.g.d ✓ 48 148.b odd 2 1 inner
444.2.g.d ✓ 48 444.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 27T_{5}^{10} + 237T_{5}^{8} - 784T_{5}^{6} + 1090T_{5}^{4} - 560T_{5}^{2} + 32 \) acting on \(S_{2}^{\mathrm{new}}(444, [\chi])\). Copy content Toggle raw display