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Properties

Label 216.4.d.d
Level $216$
Weight $4$
Character orbit 216.d
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(109,216)]
 
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([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("216.109");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 216.d (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
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 12 q^{4} - 24 q^{10} - 288 q^{16} + 336 q^{22} - 600 q^{25} + 84 q^{28} + 528 q^{31} + 24 q^{34} + 648 q^{40} - 864 q^{46} + 1536 q^{49} - 636 q^{52} + 144 q^{55} + 96 q^{58} - 888 q^{64} - 384 q^{70} + 216 q^{73} - 444 q^{76} - 2352 q^{79} + 336 q^{82} + 1896 q^{88} + 1200 q^{94} + 696 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1 −2.78622 − 0.486831i 0 7.52599 + 2.71283i − 6.66243i 0 6.63758 −19.6483 − 11.2224i 0 −3.24348 + 18.5630i
109.2 −2.78622 + 0.486831i 0 7.52599 − 2.71283i 6.66243i 0 6.63758 −19.6483 + 11.2224i 0 −3.24348 − 18.5630i
109.3 −2.30947 − 1.63289i 0 2.66733 + 7.54224i 4.37724i 0 −27.3085 6.15555 − 21.7740i 0 7.14757 − 10.1091i
109.4 −2.30947 + 1.63289i 0 2.66733 − 7.54224i − 4.37724i 0 −27.3085 6.15555 + 21.7740i 0 7.14757 + 10.1091i
109.5 −2.07264 − 1.92462i 0 0.591659 + 7.97809i 19.8612i 0 0.905321 14.1285 − 17.6744i 0 38.2252 − 41.1650i
109.6 −2.07264 + 1.92462i 0 0.591659 − 7.97809i − 19.8612i 0 0.905321 14.1285 + 17.6744i 0 38.2252 + 41.1650i
109.7 −1.94701 − 2.05162i 0 −0.418326 + 7.98906i − 13.3613i 0 33.2593 17.2050 − 14.6965i 0 −27.4125 + 26.0146i
109.8 −1.94701 + 2.05162i 0 −0.418326 − 7.98906i 13.3613i 0 33.2593 17.2050 + 14.6965i 0 −27.4125 − 26.0146i
109.9 −1.09748 − 2.60682i 0 −5.59106 + 5.72190i − 14.8953i 0 −21.8159 21.0521 + 8.29520i 0 −38.8293 + 16.3473i
109.10 −1.09748 + 2.60682i 0 −5.59106 − 5.72190i 14.8953i 0 −21.8159 21.0521 − 8.29520i 0 −38.8293 − 16.3473i
109.11 −0.334967 − 2.80852i 0 −7.77559 + 1.88152i 6.44911i 0 8.32217 7.88887 + 21.2077i 0 18.1125 − 2.16024i
109.12 −0.334967 + 2.80852i 0 −7.77559 − 1.88152i − 6.44911i 0 8.32217 7.88887 − 21.2077i 0 18.1125 + 2.16024i
109.13 0.334967 − 2.80852i 0 −7.77559 − 1.88152i 6.44911i 0 8.32217 −7.88887 + 21.2077i 0 18.1125 + 2.16024i
109.14 0.334967 + 2.80852i 0 −7.77559 + 1.88152i − 6.44911i 0 8.32217 −7.88887 − 21.2077i 0 18.1125 − 2.16024i
109.15 1.09748 − 2.60682i 0 −5.59106 − 5.72190i − 14.8953i 0 −21.8159 −21.0521 + 8.29520i 0 −38.8293 − 16.3473i
109.16 1.09748 + 2.60682i 0 −5.59106 + 5.72190i 14.8953i 0 −21.8159 −21.0521 − 8.29520i 0 −38.8293 + 16.3473i
109.17 1.94701 − 2.05162i 0 −0.418326 − 7.98906i − 13.3613i 0 33.2593 −17.2050 − 14.6965i 0 −27.4125 − 26.0146i
109.18 1.94701 + 2.05162i 0 −0.418326 + 7.98906i 13.3613i 0 33.2593 −17.2050 + 14.6965i 0 −27.4125 + 26.0146i
109.19 2.07264 − 1.92462i 0 0.591659 − 7.97809i 19.8612i 0 0.905321 −14.1285 − 17.6744i 0 38.2252 + 41.1650i
109.20 2.07264 + 1.92462i 0 0.591659 + 7.97809i − 19.8612i 0 0.905321 −14.1285 + 17.6744i 0 38.2252 − 41.1650i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.24
Significant digits:
Format:

Inner twists

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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 900T_{5}^{10} + 284616T_{5}^{8} + 39207168T_{5}^{6} + 2361016896T_{5}^{4} + 61572485376T_{5}^{2} + 552678924800 \) acting on \(S_{4}^{\mathrm{new}}(216, [\chi])\). Copy content Toggle raw display