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Properties

Label 531.2.i.d
Level $531$
Weight $2$
Character orbit 531.i
Analytic conductor $4.240$
Analytic rank $0$
Dimension $280$
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] = [531,2,Mod(19,531)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(531, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([0, 38]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("531.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 531.i (of order \(29\), degree \(28\), 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: \(4.24005634733\)
Analytic rank: \(0\)
Dimension: \(280\)
Relative dimension: \(10\) over \(\Q(\zeta_{29})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{29}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 280 q - 10 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 280 q - 10 q^{4} - 4 q^{7} + 4 q^{10} + 8 q^{13} - 2 q^{16} + 12 q^{19} - 4 q^{22} - 14 q^{25} + 4 q^{28} + 12 q^{31} - 8 q^{34} + 12 q^{40} + 8 q^{43} - 112 q^{46} - 106 q^{49} - 228 q^{52} - 194 q^{55} - 160 q^{58} - 88 q^{61} - 446 q^{64} - 68 q^{67} - 244 q^{70} - 54 q^{73} + 8 q^{76} + 16 q^{79} + 36 q^{82} - 20 q^{85} - 96 q^{88} + 48 q^{91} + 24 q^{94} - 40 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −1.16076 + 2.18942i 0 −2.32384 − 3.42741i −2.45508 + 0.540404i 0 −1.63504 + 1.24293i 5.27434 − 0.573619i 0 1.66658 − 6.00249i
19.2 −0.933078 + 1.75997i 0 −1.10449 − 1.62900i −0.540541 + 0.118982i 0 3.00225 − 2.28225i −0.0631056 + 0.00686314i 0 0.294962 − 1.06236i
19.3 −0.750600 + 1.41578i 0 −0.318663 − 0.469993i 1.76817 − 0.389204i 0 −4.02537 + 3.06001i −2.28151 + 0.248129i 0 −0.776162 + 2.79548i
19.4 −0.640926 + 1.20891i 0 0.0716850 + 0.105727i 3.80759 − 0.838115i 0 1.40428 − 1.06751i −2.89433 + 0.314777i 0 −1.42718 + 5.14023i
19.5 −0.314507 + 0.593223i 0 0.869375 + 1.28223i −2.44832 + 0.538916i 0 0.996291 − 0.757360i −2.36908 + 0.257653i 0 0.450316 − 1.62189i
19.6 0.314507 − 0.593223i 0 0.869375 + 1.28223i 2.44832 − 0.538916i 0 0.996291 − 0.757360i 2.36908 − 0.257653i 0 0.450316 − 1.62189i
19.7 0.640926 − 1.20891i 0 0.0716850 + 0.105727i −3.80759 + 0.838115i 0 1.40428 − 1.06751i 2.89433 − 0.314777i 0 −1.42718 + 5.14023i
19.8 0.750600 − 1.41578i 0 −0.318663 − 0.469993i −1.76817 + 0.389204i 0 −4.02537 + 3.06001i 2.28151 − 0.248129i 0 −0.776162 + 2.79548i
19.9 0.933078 − 1.75997i 0 −1.10449 − 1.62900i 0.540541 − 0.118982i 0 3.00225 − 2.28225i 0.0631056 − 0.00686314i 0 0.294962 − 1.06236i
19.10 1.16076 − 2.18942i 0 −2.32384 − 3.42741i 2.45508 − 0.540404i 0 −1.63504 + 1.24293i −5.27434 + 0.573619i 0 1.66658 − 6.00249i
28.1 −1.16076 − 2.18942i 0 −2.32384 + 3.42741i −2.45508 − 0.540404i 0 −1.63504 − 1.24293i 5.27434 + 0.573619i 0 1.66658 + 6.00249i
28.2 −0.933078 − 1.75997i 0 −1.10449 + 1.62900i −0.540541 − 0.118982i 0 3.00225 + 2.28225i −0.0631056 − 0.00686314i 0 0.294962 + 1.06236i
28.3 −0.750600 − 1.41578i 0 −0.318663 + 0.469993i 1.76817 + 0.389204i 0 −4.02537 − 3.06001i −2.28151 − 0.248129i 0 −0.776162 − 2.79548i
28.4 −0.640926 − 1.20891i 0 0.0716850 − 0.105727i 3.80759 + 0.838115i 0 1.40428 + 1.06751i −2.89433 − 0.314777i 0 −1.42718 − 5.14023i
28.5 −0.314507 − 0.593223i 0 0.869375 − 1.28223i −2.44832 − 0.538916i 0 0.996291 + 0.757360i −2.36908 − 0.257653i 0 0.450316 + 1.62189i
28.6 0.314507 + 0.593223i 0 0.869375 − 1.28223i 2.44832 + 0.538916i 0 0.996291 + 0.757360i 2.36908 + 0.257653i 0 0.450316 + 1.62189i
28.7 0.640926 + 1.20891i 0 0.0716850 − 0.105727i −3.80759 − 0.838115i 0 1.40428 + 1.06751i 2.89433 + 0.314777i 0 −1.42718 − 5.14023i
28.8 0.750600 + 1.41578i 0 −0.318663 + 0.469993i −1.76817 − 0.389204i 0 −4.02537 − 3.06001i 2.28151 + 0.248129i 0 −0.776162 − 2.79548i
28.9 0.933078 + 1.75997i 0 −1.10449 + 1.62900i 0.540541 + 0.118982i 0 3.00225 + 2.28225i 0.0631056 + 0.00686314i 0 0.294962 + 1.06236i
28.10 1.16076 + 2.18942i 0 −2.32384 + 3.42741i 2.45508 + 0.540404i 0 −1.63504 − 1.24293i −5.27434 − 0.573619i 0 1.66658 + 6.00249i
See next 80 embeddings (of 280 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
59.c even 29 1 inner
177.h odd 58 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 531.2.i.d ✓ 280
3.b odd 2 1 inner 531.2.i.d ✓ 280
59.c even 29 1 inner 531.2.i.d ✓ 280
177.h odd 58 1 inner 531.2.i.d ✓ 280
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
531.2.i.d ✓ 280 1.a even 1 1 trivial
531.2.i.d ✓ 280 3.b odd 2 1 inner
531.2.i.d ✓ 280 59.c even 29 1 inner
531.2.i.d ✓ 280 177.h odd 58 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{280} + 15 T_{2}^{278} + 148 T_{2}^{276} + 1686 T_{2}^{274} + 16196 T_{2}^{272} + \cdots + 61\!\cdots\!61 \) acting on \(S_{2}^{\mathrm{new}}(531, [\chi])\). Copy content Toggle raw display