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Properties

Label 975.2.bz.a
Level $975$
Weight $2$
Character orbit 975.bz
Analytic conductor $7.785$
Analytic rank $0$
Dimension $64$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [975,2,Mod(38,975)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(975, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 19, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("975.38");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.bz (of order \(20\), degree \(8\), 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: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(64\)
Relative dimension: \(8\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

$q$-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
\(\operatorname{Tr}(f)(q) = \) \( 64 q - 8 q^{10} - 24 q^{12} + 64 q^{16} + 56 q^{22} + 72 q^{30} - 96 q^{36} - 88 q^{40} - 32 q^{43} + 48 q^{48} - 104 q^{52} - 400 q^{64} + 144 q^{81} - 544 q^{82} + 456 q^{88} - 200 q^{94}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
38.1 −2.64659 + 0.419179i −0.786335 + 1.54327i 4.92662 − 1.60076i 2.13343 + 0.669677i 1.43420 − 4.41402i 0 −7.59270 + 3.86867i −1.76336 − 2.42705i −5.92704 − 0.878071i
38.2 −1.99836 + 0.316509i 0.786335 − 1.54327i 1.99115 − 0.646963i −2.00241 + 0.995174i −1.08292 + 3.33289i 0 −0.168768 + 0.0859918i −1.76336 − 2.42705i 3.68654 − 2.62250i
38.3 −1.95212 + 0.309186i −0.786335 + 1.54327i 1.81308 − 0.589105i −0.995174 − 2.00241i 1.05787 − 3.25577i 0 0.164864 − 0.0840023i −1.76336 − 2.42705i 2.56182 + 3.60125i
38.4 −0.894306 + 0.141644i 0.786335 − 1.54327i −1.12239 + 0.364688i −0.669677 + 2.13343i −0.484629 + 1.49153i 0 2.56564 − 1.30726i −1.76336 − 2.42705i 0.296708 − 2.00280i
38.5 0.894306 − 0.141644i 0.786335 − 1.54327i −1.12239 + 0.364688i 0.669677 − 2.13343i 0.484629 − 1.49153i 0 −2.56564 + 1.30726i −1.76336 − 2.42705i 0.296708 − 2.00280i
38.6 1.95212 − 0.309186i −0.786335 + 1.54327i 1.81308 − 0.589105i 0.995174 + 2.00241i −1.05787 + 3.25577i 0 −0.164864 + 0.0840023i −1.76336 − 2.42705i 2.56182 + 3.60125i
38.7 1.99836 − 0.316509i 0.786335 − 1.54327i 1.99115 − 0.646963i 2.00241 − 0.995174i 1.08292 − 3.33289i 0 0.168768 − 0.0859918i −1.76336 − 2.42705i 3.68654 − 2.62250i
38.8 2.64659 − 0.419179i −0.786335 + 1.54327i 4.92662 − 1.60076i −2.13343 − 0.669677i −1.43420 + 4.41402i 0 7.59270 − 3.86867i −1.76336 − 2.42705i −5.92704 − 0.878071i
77.1 −2.64659 − 0.419179i −0.786335 − 1.54327i 4.92662 + 1.60076i 2.13343 − 0.669677i 1.43420 + 4.41402i 0 −7.59270 − 3.86867i −1.76336 + 2.42705i −5.92704 + 0.878071i
77.2 −1.99836 − 0.316509i 0.786335 + 1.54327i 1.99115 + 0.646963i −2.00241 − 0.995174i −1.08292 − 3.33289i 0 −0.168768 − 0.0859918i −1.76336 + 2.42705i 3.68654 + 2.62250i
77.3 −1.95212 − 0.309186i −0.786335 − 1.54327i 1.81308 + 0.589105i −0.995174 + 2.00241i 1.05787 + 3.25577i 0 0.164864 + 0.0840023i −1.76336 + 2.42705i 2.56182 − 3.60125i
77.4 −0.894306 − 0.141644i 0.786335 + 1.54327i −1.12239 − 0.364688i −0.669677 − 2.13343i −0.484629 − 1.49153i 0 2.56564 + 1.30726i −1.76336 + 2.42705i 0.296708 + 2.00280i
77.5 0.894306 + 0.141644i 0.786335 + 1.54327i −1.12239 − 0.364688i 0.669677 + 2.13343i 0.484629 + 1.49153i 0 −2.56564 − 1.30726i −1.76336 + 2.42705i 0.296708 + 2.00280i
77.6 1.95212 + 0.309186i −0.786335 − 1.54327i 1.81308 + 0.589105i 0.995174 − 2.00241i −1.05787 − 3.25577i 0 −0.164864 − 0.0840023i −1.76336 + 2.42705i 2.56182 − 3.60125i
77.7 1.99836 + 0.316509i 0.786335 + 1.54327i 1.99115 + 0.646963i 2.00241 + 0.995174i 1.08292 + 3.33289i 0 0.168768 + 0.0859918i −1.76336 + 2.42705i 3.68654 + 2.62250i
77.8 2.64659 + 0.419179i −0.786335 − 1.54327i 4.92662 + 1.60076i −2.13343 + 0.669677i −1.43420 − 4.41402i 0 7.59270 + 3.86867i −1.76336 + 2.42705i −5.92704 + 0.878071i
233.1 −2.49356 − 1.27053i 1.71073 + 0.270952i 3.42803 + 4.71828i 0.327690 + 2.21193i −3.92155 − 2.84917i 0 −1.67768 − 10.5925i 2.85317 + 0.927051i 1.99321 − 5.93192i
233.2 −2.40575 − 1.22579i −1.71073 − 0.270952i 3.10951 + 4.27987i 1.29617 − 1.82207i 3.78345 + 2.74884i 0 −1.38972 − 8.77432i 2.85317 + 0.927051i −5.35174 + 2.79462i
233.3 −0.750665 − 0.382483i 1.71073 + 0.270952i −0.758366 − 1.04380i −1.82207 − 1.29617i −1.18055 − 0.857718i 0 0.433632 + 2.73784i 2.85317 + 0.927051i 0.872005 + 1.66990i
233.4 −0.365081 − 0.186018i −1.71073 − 0.270952i −1.07689 − 1.48221i −2.21193 + 0.327690i 0.574151 + 0.417145i 0 0.245629 + 1.55084i 2.85317 + 0.927051i 0.868488 + 0.291825i
See all 64 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 38.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner
325.bc odd 20 1 inner
975.bz even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 975.2.bz.a ✓ 64
3.b odd 2 1 inner 975.2.bz.a ✓ 64
13.b even 2 1 inner 975.2.bz.a ✓ 64
25.f odd 20 1 inner 975.2.bz.a ✓ 64
39.d odd 2 1 CM 975.2.bz.a ✓ 64
75.l even 20 1 inner 975.2.bz.a ✓ 64
325.bc odd 20 1 inner 975.2.bz.a ✓ 64
975.bz even 20 1 inner 975.2.bz.a ✓ 64
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
975.2.bz.a ✓ 64 1.a even 1 1 trivial
975.2.bz.a ✓ 64 3.b odd 2 1 inner
975.2.bz.a ✓ 64 13.b even 2 1 inner
975.2.bz.a ✓ 64 25.f odd 20 1 inner
975.2.bz.a ✓ 64 39.d odd 2 1 CM
975.2.bz.a ✓ 64 75.l even 20 1 inner
975.2.bz.a ✓ 64 325.bc odd 20 1 inner
975.2.bz.a ✓ 64 975.bz even 20 1 inner