Properties

Label 722.3.b.f
Level $722$
Weight $3$
Character orbit 722.b
Analytic conductor $19.673$
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] = [722,3,Mod(721,722)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(722, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("722.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 722 = 2 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 722.b (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: \(19.6730750868\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 38)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{4} - 12 q^{5} + 36 q^{7} - 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 48 q^{4} - 12 q^{5} + 36 q^{7} - 108 q^{9} - 60 q^{11} + 96 q^{16} - 48 q^{17} + 24 q^{20} - 72 q^{23} + 132 q^{25} - 96 q^{26} - 72 q^{28} - 36 q^{35} + 216 q^{36} + 120 q^{39} - 240 q^{42} - 24 q^{43} + 120 q^{44} + 288 q^{45} - 240 q^{47} - 108 q^{49} + 456 q^{54} - 444 q^{55} + 48 q^{58} + 84 q^{61} - 120 q^{62} - 216 q^{63} - 192 q^{64} - 48 q^{66} + 96 q^{68} - 456 q^{73} + 720 q^{74} - 144 q^{77} - 48 q^{80} + 360 q^{81} - 96 q^{82} + 252 q^{83} + 984 q^{85} + 648 q^{87} + 144 q^{92} - 528 q^{93} + 1404 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
721.1 1.41421i 5.84230i −2.00000 −5.64041 −8.26227 1.98059 2.82843i −25.1325 7.97674i
721.2 1.41421i 5.05892i −2.00000 3.90971 −7.15439 −3.55148 2.82843i −16.5927 5.52916i
721.3 1.41421i 4.14954i −2.00000 −1.13449 −5.86834 11.1120 2.82843i −8.21868 1.60442i
721.4 1.41421i 3.70454i −2.00000 0.388748 −5.23901 5.92863 2.82843i −4.72359 0.549773i
721.5 1.41421i 0.423806i −2.00000 −8.50146 −0.599352 5.17090 2.82843i 8.82039 12.0229i
721.6 1.41421i 0.175771i −2.00000 7.89135 −0.248578 7.65909 2.82843i 8.96910 11.1601i
721.7 1.41421i 1.64949i −2.00000 4.64041 2.33273 4.36176 2.82843i 6.27919 6.56253i
721.8 1.41421i 2.42931i −2.00000 0.134495 3.43557 −9.58683 2.82843i 3.09843 0.190204i
721.9 1.41421i 2.44574i −2.00000 −4.90971 3.45880 −1.16621 2.82843i 3.01834 6.94338i
721.10 1.41421i 3.63432i −2.00000 7.50146 5.13971 −7.59704 2.82843i −4.20829 10.6087i
721.11 1.41421i 3.88163i −2.00000 −8.89135 5.48946 −5.10658 2.82843i −6.06708 12.5743i
721.12 1.41421i 5.31438i −2.00000 −1.38875 7.51567 8.79517 2.82843i −19.2426 1.96399i
721.13 1.41421i 5.31438i −2.00000 −1.38875 7.51567 8.79517 2.82843i −19.2426 1.96399i
721.14 1.41421i 3.88163i −2.00000 −8.89135 5.48946 −5.10658 2.82843i −6.06708 12.5743i
721.15 1.41421i 3.63432i −2.00000 7.50146 5.13971 −7.59704 2.82843i −4.20829 10.6087i
721.16 1.41421i 2.44574i −2.00000 −4.90971 3.45880 −1.16621 2.82843i 3.01834 6.94338i
721.17 1.41421i 2.42931i −2.00000 0.134495 3.43557 −9.58683 2.82843i 3.09843 0.190204i
721.18 1.41421i 1.64949i −2.00000 4.64041 2.33273 4.36176 2.82843i 6.27919 6.56253i
721.19 1.41421i 0.175771i −2.00000 7.89135 −0.248578 7.65909 2.82843i 8.96910 11.1601i
721.20 1.41421i 0.423806i −2.00000 −8.50146 −0.599352 5.17090 2.82843i 8.82039 12.0229i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 722.3.b.f 24
19.b odd 2 1 inner 722.3.b.f 24
19.e even 9 1 38.3.f.a 24
19.f odd 18 1 38.3.f.a 24
57.j even 18 1 342.3.z.b 24
57.l odd 18 1 342.3.z.b 24
76.k even 18 1 304.3.z.c 24
76.l odd 18 1 304.3.z.c 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
38.3.f.a 24 19.e even 9 1
38.3.f.a 24 19.f odd 18 1
304.3.z.c 24 76.k even 18 1
304.3.z.c 24 76.l odd 18 1
342.3.z.b 24 57.j even 18 1
342.3.z.b 24 57.l odd 18 1
722.3.b.f 24 1.a even 1 1 trivial
722.3.b.f 24 19.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 162 T_{3}^{22} + 11331 T_{3}^{20} + 449416 T_{3}^{18} + 11159685 T_{3}^{16} + 180632898 T_{3}^{14} + 1921819843 T_{3}^{12} + 13207043862 T_{3}^{10} + 55876820409 T_{3}^{8} + \cdots + 618367689 \) acting on \(S_{3}^{\mathrm{new}}(722, [\chi])\). Copy content Toggle raw display