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Properties

Label 640.2.x.a
Level $640$
Weight $2$
Character orbit 640.x
Analytic conductor $5.110$
Analytic rank $0$
Dimension $64$
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] = [640,2,Mod(81,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 3, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.x (of order \(8\), degree \(4\), not 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: \(5.11042572936\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(16\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 160)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 64 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 64 q + 16 q^{23} + 48 q^{27} + 48 q^{39} + 16 q^{43} - 16 q^{51} - 32 q^{53} - 32 q^{59} - 32 q^{61} - 80 q^{63} - 80 q^{67} - 32 q^{69} - 32 q^{71} - 32 q^{77} + 80 q^{83} + 96 q^{91} + 64 q^{95} + 64 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1 0 −3.06141 − 1.26808i 0 −0.382683 − 0.923880i 0 1.47530 − 1.47530i 0 5.64289 + 5.64289i 0
81.2 0 −2.31924 − 0.960659i 0 0.382683 + 0.923880i 0 0.669347 − 0.669347i 0 2.33467 + 2.33467i 0
81.3 0 −2.00190 − 0.829213i 0 −0.382683 − 0.923880i 0 −0.773883 + 0.773883i 0 1.19868 + 1.19868i 0
81.4 0 −1.90039 − 0.787169i 0 0.382683 + 0.923880i 0 −2.82718 + 2.82718i 0 0.870540 + 0.870540i 0
81.5 0 −1.36989 − 0.567426i 0 0.382683 + 0.923880i 0 0.427148 − 0.427148i 0 −0.566703 − 0.566703i 0
81.6 0 −1.21680 − 0.504015i 0 −0.382683 − 0.923880i 0 −2.78045 + 2.78045i 0 −0.894749 − 0.894749i 0
81.7 0 −0.951096 − 0.393957i 0 −0.382683 − 0.923880i 0 2.40285 − 2.40285i 0 −1.37194 − 1.37194i 0
81.8 0 0.133682 + 0.0553729i 0 0.382683 + 0.923880i 0 0.648440 − 0.648440i 0 −2.10652 − 2.10652i 0
81.9 0 0.293011 + 0.121369i 0 −0.382683 − 0.923880i 0 −1.60956 + 1.60956i 0 −2.05020 − 2.05020i 0
81.10 0 0.332460 + 0.137709i 0 −0.382683 − 0.923880i 0 −1.44201 + 1.44201i 0 −2.02975 − 2.02975i 0
81.11 0 0.943398 + 0.390768i 0 0.382683 + 0.923880i 0 3.31153 − 3.31153i 0 −1.38402 − 1.38402i 0
81.12 0 0.950262 + 0.393611i 0 0.382683 + 0.923880i 0 −1.84130 + 1.84130i 0 −1.37325 − 1.37325i 0
81.13 0 2.25023 + 0.932074i 0 −0.382683 − 0.923880i 0 1.83918 − 1.83918i 0 2.07343 + 2.07343i 0
81.14 0 2.50775 + 1.03874i 0 −0.382683 − 0.923880i 0 −0.525640 + 0.525640i 0 3.08849 + 3.08849i 0
81.15 0 2.67164 + 1.10663i 0 0.382683 + 0.923880i 0 −3.58127 + 3.58127i 0 3.79172 + 3.79172i 0
81.16 0 2.73829 + 1.13424i 0 0.382683 + 0.923880i 0 1.77906 − 1.77906i 0 4.09041 + 4.09041i 0
241.1 0 −1.24322 − 3.00140i 0 −0.923880 − 0.382683i 0 1.78484 + 1.78484i 0 −5.34146 + 5.34146i 0
241.2 0 −1.14037 − 2.75309i 0 0.923880 + 0.382683i 0 1.26936 + 1.26936i 0 −4.15773 + 4.15773i 0
241.3 0 −1.03105 − 2.48917i 0 0.923880 + 0.382683i 0 −1.75057 − 1.75057i 0 −3.01160 + 3.01160i 0
241.4 0 −0.755932 − 1.82498i 0 0.923880 + 0.382683i 0 1.73757 + 1.73757i 0 −0.637806 + 0.637806i 0
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.x.a 64
4.b odd 2 1 160.2.x.a ✓ 64
20.d odd 2 1 800.2.y.c 64
20.e even 4 1 800.2.ba.e 64
20.e even 4 1 800.2.ba.g 64
32.g even 8 1 inner 640.2.x.a 64
32.h odd 8 1 160.2.x.a ✓ 64
160.u even 8 1 800.2.ba.g 64
160.y odd 8 1 800.2.y.c 64
160.ba even 8 1 800.2.ba.e 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.x.a ✓ 64 4.b odd 2 1
160.2.x.a ✓ 64 32.h odd 8 1
640.2.x.a 64 1.a even 1 1 trivial
640.2.x.a 64 32.g even 8 1 inner
800.2.y.c 64 20.d odd 2 1
800.2.y.c 64 160.y odd 8 1
800.2.ba.e 64 20.e even 4 1
800.2.ba.e 64 160.ba even 8 1
800.2.ba.g 64 20.e even 4 1
800.2.ba.g 64 160.u even 8 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(640, [\chi])\).