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Properties

Label 216.4.f.b
Level $216$
Weight $4$
Character orbit 216.f
Analytic conductor $12.744$
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] = [216,4,Mod(107,216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("216.107");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.f (of order \(2\), degree \(1\), 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: \(12.7444125612\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{4} + 96 q^{10} - 48 q^{16} - 48 q^{19} + 168 q^{22} + 600 q^{25} - 348 q^{28} - 192 q^{34} - 264 q^{40} - 432 q^{43} - 72 q^{46} - 816 q^{49} + 84 q^{52} - 48 q^{58} - 984 q^{64} + 1632 q^{67} - 2472 q^{70} - 216 q^{73} + 348 q^{76} - 1872 q^{82} + 4392 q^{88} + 3600 q^{91} + 2616 q^{94} + 2280 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1 −2.79380 − 0.441249i 0 7.61060 + 2.46552i −5.34627 0 − 31.3735i −20.1746 − 10.2463i 0 14.9364 + 2.35904i
107.2 −2.79380 + 0.441249i 0 7.61060 − 2.46552i −5.34627 0 31.3735i −20.1746 + 10.2463i 0 14.9364 − 2.35904i
107.3 −2.61666 − 1.07383i 0 5.69377 + 5.61969i 9.32272 0 − 5.38753i −8.86405 − 20.8190i 0 −24.3943 − 10.0110i
107.4 −2.61666 + 1.07383i 0 5.69377 − 5.61969i 9.32272 0 5.38753i −8.86405 + 20.8190i 0 −24.3943 + 10.0110i
107.5 −2.34682 − 1.57875i 0 3.01510 + 7.41007i −21.0227 0 25.8057i 4.62275 − 22.1502i 0 49.3365 + 33.1896i
107.6 −2.34682 + 1.57875i 0 3.01510 − 7.41007i −21.0227 0 − 25.8057i 4.62275 + 22.1502i 0 49.3365 − 33.1896i
107.7 −1.97843 − 2.02134i 0 −0.171650 + 7.99816i 12.8173 0 12.3152i 16.5066 − 15.4768i 0 −25.3581 − 25.9081i
107.8 −1.97843 + 2.02134i 0 −0.171650 − 7.99816i 12.8173 0 − 12.3152i 16.5066 + 15.4768i 0 −25.3581 + 25.9081i
107.9 −0.910261 − 2.67795i 0 −6.34285 + 4.87527i 0.898373 0 − 20.7150i 18.8294 + 12.5481i 0 −0.817754 − 2.40580i
107.10 −0.910261 + 2.67795i 0 −6.34285 − 4.87527i 0.898373 0 20.7150i 18.8294 − 12.5481i 0 −0.817754 + 2.40580i
107.11 −0.772990 − 2.72075i 0 −6.80497 + 4.20623i −13.3214 0 1.40229i 16.7043 + 15.2633i 0 10.2973 + 36.2443i
107.12 −0.772990 + 2.72075i 0 −6.80497 − 4.20623i −13.3214 0 − 1.40229i 16.7043 − 15.2633i 0 10.2973 − 36.2443i
107.13 0.772990 − 2.72075i 0 −6.80497 − 4.20623i 13.3214 0 − 1.40229i −16.7043 + 15.2633i 0 10.2973 − 36.2443i
107.14 0.772990 + 2.72075i 0 −6.80497 + 4.20623i 13.3214 0 1.40229i −16.7043 − 15.2633i 0 10.2973 + 36.2443i
107.15 0.910261 − 2.67795i 0 −6.34285 − 4.87527i −0.898373 0 20.7150i −18.8294 + 12.5481i 0 −0.817754 + 2.40580i
107.16 0.910261 + 2.67795i 0 −6.34285 + 4.87527i −0.898373 0 − 20.7150i −18.8294 − 12.5481i 0 −0.817754 − 2.40580i
107.17 1.97843 − 2.02134i 0 −0.171650 − 7.99816i −12.8173 0 − 12.3152i −16.5066 − 15.4768i 0 −25.3581 + 25.9081i
107.18 1.97843 + 2.02134i 0 −0.171650 + 7.99816i −12.8173 0 12.3152i −16.5066 + 15.4768i 0 −25.3581 − 25.9081i
107.19 2.34682 − 1.57875i 0 3.01510 − 7.41007i 21.0227 0 − 25.8057i −4.62275 − 22.1502i 0 49.3365 − 33.1896i
107.20 2.34682 + 1.57875i 0 3.01510 + 7.41007i 21.0227 0 25.8057i −4.62275 + 22.1502i 0 49.3365 + 33.1896i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 216.4.f.b ✓ 24
3.b odd 2 1 inner 216.4.f.b ✓ 24
4.b odd 2 1 864.4.f.b 24
8.b even 2 1 864.4.f.b 24
8.d odd 2 1 inner 216.4.f.b ✓ 24
12.b even 2 1 864.4.f.b 24
24.f even 2 1 inner 216.4.f.b ✓ 24
24.h odd 2 1 864.4.f.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.4.f.b ✓ 24 1.a even 1 1 trivial
216.4.f.b ✓ 24 3.b odd 2 1 inner
216.4.f.b ✓ 24 8.d odd 2 1 inner
216.4.f.b ✓ 24 24.f even 2 1 inner
864.4.f.b 24 4.b odd 2 1
864.4.f.b 24 8.b even 2 1
864.4.f.b 24 12.b even 2 1
864.4.f.b 24 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 900T_{5}^{10} + 273912T_{5}^{8} - 35862912T_{5}^{6} + 1964503872T_{5}^{4} - 33570167040T_{5}^{2} + 25832756736 \) acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display