Properties

Label 600.2.bu.a
Level $600$
Weight $2$
Character orbit 600.bu
Analytic conductor $4.791$
Analytic rank $0$
Dimension $240$
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] = [600,2,Mod(17,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 10, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.17");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bu (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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(30\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 4 q^{7} - 8 q^{13} + 12 q^{15} + 8 q^{25} + 24 q^{27} + 12 q^{33} + 32 q^{37} + 40 q^{39} + 64 q^{45} - 28 q^{55} + 40 q^{57} - 4 q^{63} - 40 q^{67} - 20 q^{73} - 72 q^{85} + 20 q^{87} + 24 q^{93} - 16 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 0 −1.68774 + 0.389291i 0 −1.62862 1.53218i 0 −0.785661 0.785661i 0 2.69691 1.31404i 0
17.2 0 −1.65974 + 0.495252i 0 1.48459 + 1.67212i 0 3.63505 + 3.63505i 0 2.50945 1.64398i 0
17.3 0 −1.62090 0.610467i 0 2.16992 0.539844i 0 0.548883 + 0.548883i 0 2.25466 + 1.97902i 0
17.4 0 −1.59112 0.684344i 0 −1.02804 + 1.98573i 0 −1.26254 1.26254i 0 2.06335 + 2.17775i 0
17.5 0 −1.56705 0.737803i 0 −2.23532 0.0579740i 0 1.32422 + 1.32422i 0 1.91129 + 2.31235i 0
17.6 0 −1.48940 + 0.884135i 0 −1.32812 + 1.79892i 0 1.66646 + 1.66646i 0 1.43661 2.63366i 0
17.7 0 −1.43901 + 0.963981i 0 1.87829 + 1.21326i 0 −2.44789 2.44789i 0 1.14148 2.77435i 0
17.8 0 −1.24488 + 1.20427i 0 0.125563 2.23254i 0 0.631710 + 0.631710i 0 0.0994587 2.99835i 0
17.9 0 −1.16124 1.28512i 0 1.95784 1.08022i 0 −2.10037 2.10037i 0 −0.303061 + 2.98465i 0
17.10 0 −1.04119 1.38417i 0 −0.137473 2.23184i 0 2.98357 + 2.98357i 0 −0.831844 + 2.88237i 0
17.11 0 −0.705878 + 1.58169i 0 1.35766 1.77673i 0 0.131544 + 0.131544i 0 −2.00347 2.23296i 0
17.12 0 −0.619322 1.61754i 0 −0.132259 + 2.23215i 0 −3.14250 3.14250i 0 −2.23288 + 2.00356i 0
17.13 0 −0.450732 + 1.67238i 0 1.67263 + 1.48402i 0 −2.25950 2.25950i 0 −2.59368 1.50759i 0
17.14 0 −0.128084 1.72731i 0 −1.29140 1.82546i 0 −0.526415 0.526415i 0 −2.96719 + 0.442481i 0
17.15 0 −0.0881207 + 1.72981i 0 −1.67263 1.48402i 0 −2.25950 2.25950i 0 −2.98447 0.304864i 0
17.16 0 0.0788384 1.73026i 0 −2.06252 + 0.863731i 0 1.38221 + 1.38221i 0 −2.98757 0.272821i 0
17.17 0 0.182562 + 1.72240i 0 −1.35766 + 1.77673i 0 0.131544 + 0.131544i 0 −2.93334 + 0.628890i 0
17.18 0 0.459699 1.66993i 0 2.06252 0.863731i 0 1.38221 + 1.38221i 0 −2.57735 1.53533i 0
17.19 0 0.655583 1.60319i 0 1.29140 + 1.82546i 0 −0.526415 0.526415i 0 −2.14042 2.10204i 0
17.20 0 0.811812 + 1.53002i 0 −0.125563 + 2.23254i 0 0.631710 + 0.631710i 0 −1.68192 + 2.48418i 0
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.30
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.f odd 20 1 inner
75.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.bu.a 240
3.b odd 2 1 inner 600.2.bu.a 240
25.f odd 20 1 inner 600.2.bu.a 240
75.l even 20 1 inner 600.2.bu.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bu.a 240 1.a even 1 1 trivial
600.2.bu.a 240 3.b odd 2 1 inner
600.2.bu.a 240 25.f odd 20 1 inner
600.2.bu.a 240 75.l even 20 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(600, [\chi])\).