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Properties

Label 640.2.ba.a
Level $640$
Weight $2$
Character orbit 640.ba
Analytic conductor $5.110$
Analytic rank $0$
Dimension $88$
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(207,640)]
 
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([4, 5, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.207");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 640.ba (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: \(88\)
Relative dimension: \(22\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{3} - 4 q^{5} + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 88 q + 4 q^{3} - 4 q^{5} + 8 q^{7} + 8 q^{11} - 4 q^{13} + 8 q^{15} + 16 q^{19} - 8 q^{21} + 8 q^{23} - 4 q^{25} - 8 q^{27} - 8 q^{33} + 48 q^{35} - 4 q^{37} - 8 q^{41} - 28 q^{43} - 4 q^{45} + 8 q^{47} + 40 q^{49} - 8 q^{51} - 4 q^{53} - 28 q^{55} - 40 q^{61} + 56 q^{63} - 8 q^{65} + 28 q^{67} - 24 q^{69} - 24 q^{71} + 16 q^{75} - 32 q^{77} - 36 q^{83} - 4 q^{85} + 8 q^{91} + 8 q^{93} - 8 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
207.1 0 −2.68192 − 1.11089i 0 −0.603584 + 2.15306i 0 0.874514 0 3.83732 + 3.83732i 0
207.2 0 −2.54622 − 1.05468i 0 2.21873 + 0.277906i 0 3.70855 0 3.24959 + 3.24959i 0
207.3 0 −2.54348 − 1.05354i 0 −1.68821 − 1.46627i 0 −4.43630 0 3.23800 + 3.23800i 0
207.4 0 −2.50226 − 1.03647i 0 0.0688146 − 2.23501i 0 2.65674 0 3.06572 + 3.06572i 0
207.5 0 −1.68650 − 0.698571i 0 −2.19211 − 0.441177i 0 2.70081 0 0.234961 + 0.234961i 0
207.6 0 −1.39485 − 0.577765i 0 −1.78859 + 1.34199i 0 −1.62907 0 −0.509534 − 0.509534i 0
207.7 0 −1.22690 − 0.508197i 0 0.868914 + 2.06034i 0 −0.810621 0 −0.874309 − 0.874309i 0
207.8 0 −0.673021 − 0.278775i 0 1.32500 − 1.80122i 0 −0.467309 0 −1.74608 − 1.74608i 0
207.9 0 −0.616647 − 0.255424i 0 −0.915235 − 2.04018i 0 −2.27809 0 −1.80631 − 1.80631i 0
207.10 0 −0.608697 − 0.252131i 0 1.26769 + 1.84200i 0 1.49067 0 −1.81438 − 1.81438i 0
207.11 0 −0.532554 − 0.220591i 0 2.20626 + 0.363921i 0 −3.48272 0 −1.88637 − 1.88637i 0
207.12 0 0.237464 + 0.0983610i 0 0.189205 − 2.22805i 0 4.12414 0 −2.07461 − 2.07461i 0
207.13 0 0.528116 + 0.218753i 0 −2.13281 + 0.671666i 0 0.814088 0 −1.89027 − 1.89027i 0
207.14 0 1.10776 + 0.458849i 0 −1.41630 + 1.73035i 0 4.27741 0 −1.10473 − 1.10473i 0
207.15 0 1.11473 + 0.461737i 0 2.18433 − 0.478219i 0 −2.85280 0 −1.09189 − 1.09189i 0
207.16 0 1.22899 + 0.509063i 0 −1.94326 − 1.10623i 0 −2.73471 0 −0.870059 − 0.870059i 0
207.17 0 1.51557 + 0.627770i 0 −0.661993 + 2.13583i 0 −4.80429 0 −0.218458 − 0.218458i 0
207.18 0 2.09546 + 0.867966i 0 1.79374 − 1.33511i 0 1.82364 0 1.51625 + 1.51625i 0
207.19 0 2.16430 + 0.896482i 0 2.08728 + 0.802025i 0 −0.225996 0 1.75919 + 1.75919i 0
207.20 0 2.23011 + 0.923741i 0 0.881968 + 2.05478i 0 3.63945 0 1.99876 + 1.99876i 0
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 207.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
160.ba 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.ba.a 88
4.b odd 2 1 160.2.ba.a yes 88
5.c odd 4 1 640.2.u.a 88
20.d odd 2 1 800.2.bb.b 88
20.e even 4 1 160.2.u.a ✓ 88
20.e even 4 1 800.2.v.b 88
32.g even 8 1 160.2.u.a ✓ 88
32.h odd 8 1 640.2.u.a 88
160.v odd 8 1 160.2.ba.a yes 88
160.z even 8 1 800.2.v.b 88
160.ba even 8 1 inner 640.2.ba.a 88
160.bb odd 8 1 800.2.bb.b 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.u.a ✓ 88 20.e even 4 1
160.2.u.a ✓ 88 32.g even 8 1
160.2.ba.a yes 88 4.b odd 2 1
160.2.ba.a yes 88 160.v odd 8 1
640.2.u.a 88 5.c odd 4 1
640.2.u.a 88 32.h odd 8 1
640.2.ba.a 88 1.a even 1 1 trivial
640.2.ba.a 88 160.ba even 8 1 inner
800.2.v.b 88 20.e even 4 1
800.2.v.b 88 160.z even 8 1
800.2.bb.b 88 20.d odd 2 1
800.2.bb.b 88 160.bb odd 8 1

Hecke kernels

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