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Properties

Label 150.2.l.a
Level $150$
Weight $2$
Character orbit 150.l
Analytic conductor $1.198$
Analytic rank $0$
Dimension $80$
Inner twists $4$

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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] = [150,2,Mod(17,150)] 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("150.17"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(150, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 13])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.19775603032\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(10\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 4 q^{3} + 4 q^{7} - 4 q^{10} - 4 q^{12} - 8 q^{15} + 20 q^{16} - 8 q^{18} - 40 q^{19} - 36 q^{22} - 104 q^{25} + 4 q^{27} - 16 q^{28} + 12 q^{30} + 4 q^{33} - 40 q^{34} - 24 q^{37} - 40 q^{39} - 8 q^{40}+ \cdots - 32 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −0.891007 − 0.453990i −1.64867 − 0.530925i 0.587785 + 0.809017i 1.35266 + 1.78054i 1.22794 + 1.22154i −0.152718 − 0.152718i −0.156434 − 0.987688i 2.43624 + 1.75064i −0.396880 − 2.20056i
17.2 −0.891007 − 0.453990i −0.873670 + 1.49556i 0.587785 + 0.809017i −2.21391 + 0.314018i 1.45742 − 0.935916i −2.72680 − 2.72680i −0.156434 − 0.987688i −1.47340 − 2.61325i 2.11517 + 0.725301i
17.3 −0.891007 − 0.453990i −0.322239 − 1.70181i 0.587785 + 0.809017i −0.0831552 − 2.23452i −0.485490 + 1.66262i 0.0556476 + 0.0556476i −0.156434 − 0.987688i −2.79232 + 1.09678i −0.940360 + 2.02872i
17.4 −0.891007 − 0.453990i 0.983013 + 1.42607i 0.587785 + 0.809017i 1.04600 − 1.97633i −0.228447 − 1.71692i 0.462249 + 0.462249i −0.156434 − 0.987688i −1.06737 + 2.80370i −1.82923 + 1.28605i
17.5 −0.891007 − 0.453990i 1.71953 + 0.207905i 0.587785 + 0.809017i −1.62298 + 1.53816i −1.43772 − 0.965894i 2.58285 + 2.58285i −0.156434 − 0.987688i 2.91355 + 0.714995i 2.14440 − 0.633691i
17.6 0.891007 + 0.453990i −1.69961 − 0.333634i 0.587785 + 0.809017i 1.62298 − 1.53816i −1.36290 − 1.06888i 2.58285 + 2.58285i 0.156434 + 0.987688i 2.77738 + 1.13410i 2.14440 − 0.633691i
17.7 0.891007 + 0.453990i −1.37558 + 1.05251i 0.587785 + 0.809017i −1.04600 + 1.97633i −1.70348 + 0.313291i 0.462249 + 0.462249i 0.156434 + 0.987688i 0.784450 − 2.89562i −1.82923 + 1.28605i
17.8 0.891007 + 0.453990i 0.368756 + 1.69234i 0.587785 + 0.809017i 2.21391 − 0.314018i −0.439743 + 1.67530i −2.72680 − 2.72680i 0.156434 + 0.987688i −2.72804 + 1.24812i 2.11517 + 0.725301i
17.9 0.891007 + 0.453990i 0.832356 − 1.51894i 0.587785 + 0.809017i 0.0831552 + 2.23452i 1.43122 − 0.975505i 0.0556476 + 0.0556476i 0.156434 + 0.987688i −1.61437 − 2.52860i −0.940360 + 2.02872i
17.10 0.891007 + 0.453990i 1.73204 + 0.00452789i 0.587785 + 0.809017i −1.35266 − 1.78054i 1.54121 + 0.790366i −0.152718 − 0.152718i 0.156434 + 0.987688i 2.99996 + 0.0156850i −0.396880 − 2.20056i
23.1 −0.987688 − 0.156434i −1.58737 + 0.693000i 0.951057 + 0.309017i −0.197931 − 2.22729i 1.67624 − 0.436149i −1.43195 + 1.43195i −0.891007 − 0.453990i 2.03950 − 2.20010i −0.152931 + 2.23083i
23.2 −0.987688 − 0.156434i −1.36724 − 1.06332i 0.951057 + 0.309017i −1.48885 + 1.66833i 1.18407 + 1.26411i −1.08662 + 1.08662i −0.891007 − 0.453990i 0.738706 + 2.90763i 1.73150 − 1.41488i
23.3 −0.987688 − 0.156434i 0.487767 − 1.66195i 0.951057 + 0.309017i 2.22772 − 0.193062i −0.741748 + 1.56519i −0.104631 + 0.104631i −0.891007 − 0.453990i −2.52417 − 1.62129i −2.23049 − 0.157807i
23.4 −0.987688 − 0.156434i 1.10553 + 1.33334i 0.951057 + 0.309017i −0.334535 − 2.21090i −0.883341 − 1.48987i 2.78015 − 2.78015i −0.891007 − 0.453990i −0.555594 + 2.94810i −0.0154450 + 2.23601i
23.5 −0.987688 − 0.156434i 1.64008 + 0.556890i 0.951057 + 0.309017i 0.545143 + 2.16860i −1.53277 − 0.806599i −1.41702 + 1.41702i −0.891007 − 0.453990i 2.37975 + 1.82669i −0.199188 − 2.22718i
23.6 0.987688 + 0.156434i −1.49368 + 0.876876i 0.951057 + 0.309017i 0.197931 + 2.22729i −1.61247 + 0.632416i −1.43195 + 1.43195i 0.891007 + 0.453990i 1.46218 − 2.61955i −0.152931 + 2.23083i
23.7 0.987688 + 0.156434i −0.428879 − 1.67811i 0.951057 + 0.309017i 0.334535 + 2.21090i −0.161084 − 1.72454i 2.78015 − 2.78015i 0.891007 + 0.453990i −2.63213 + 1.43942i −0.0154450 + 2.23601i
23.8 0.987688 + 0.156434i 0.0565979 + 1.73113i 0.951057 + 0.309017i 1.48885 − 1.66833i −0.214907 + 1.71867i −1.08662 + 1.08662i 0.891007 + 0.453990i −2.99359 + 0.195956i 1.73150 − 1.41488i
23.9 0.987688 + 0.156434i 0.513483 − 1.65419i 0.951057 + 0.309017i −0.545143 − 2.16860i 0.765933 − 1.55349i −1.41702 + 1.41702i 0.891007 + 0.453990i −2.47267 − 1.69880i −0.199188 − 2.22718i
23.10 0.987688 + 0.156434i 1.63125 + 0.582259i 0.951057 + 0.309017i −2.22772 + 0.193062i 1.52008 + 0.830274i −0.104631 + 0.104631i 0.891007 + 0.453990i 2.32195 + 1.89962i −2.23049 − 0.157807i
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.10
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
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 150.2.l.a ✓ 80
3.b odd 2 1 inner 150.2.l.a ✓ 80
5.b even 2 1 750.2.l.b 80
5.c odd 4 1 750.2.l.a 80
5.c odd 4 1 750.2.l.c 80
15.d odd 2 1 750.2.l.b 80
15.e even 4 1 750.2.l.a 80
15.e even 4 1 750.2.l.c 80
25.d even 5 1 750.2.l.c 80
25.e even 10 1 750.2.l.a 80
25.f odd 20 1 inner 150.2.l.a ✓ 80
25.f odd 20 1 750.2.l.b 80
75.h odd 10 1 750.2.l.a 80
75.j odd 10 1 750.2.l.c 80
75.l even 20 1 inner 150.2.l.a ✓ 80
75.l even 20 1 750.2.l.b 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.l.a ✓ 80 1.a even 1 1 trivial
150.2.l.a ✓ 80 3.b odd 2 1 inner
150.2.l.a ✓ 80 25.f odd 20 1 inner
150.2.l.a ✓ 80 75.l even 20 1 inner
750.2.l.a 80 5.c odd 4 1
750.2.l.a 80 15.e even 4 1
750.2.l.a 80 25.e even 10 1
750.2.l.a 80 75.h odd 10 1
750.2.l.b 80 5.b even 2 1
750.2.l.b 80 15.d odd 2 1
750.2.l.b 80 25.f odd 20 1
750.2.l.b 80 75.l even 20 1
750.2.l.c 80 5.c odd 4 1
750.2.l.c 80 15.e even 4 1
750.2.l.c 80 25.d even 5 1
750.2.l.c 80 75.j odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(150, [\chi])\).