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Properties

Label 501.2.c.a
Level $501$
Weight $2$
Character orbit 501.c
Analytic conductor $4.001$
Analytic rank $0$
Dimension $22$
CM discriminant -167
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [501,2,Mod(500,501)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(501, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("501.500");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 501 = 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 501.c (of order \(2\), degree \(1\), 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: \(4.00050514127\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 44 q^{4}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 22 q - 44 q^{4} + 88 q^{16} - 110 q^{25} + 11 q^{42} - 55 q^{48} + 154 q^{49} + 77 q^{54} - 176 q^{64} - 121 q^{72} + 143 q^{84}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
500.1 − 2.82841i −0.605482 − 1.62277i −5.99993 0 −4.58987 + 1.71255i −2.21228 11.3135i −2.26678 + 1.96512i 0
500.2 − 2.71625i −1.72765 + 0.123359i −5.37799 0 0.335073 + 4.69273i 5.07277 9.17544i 2.96957 − 0.426242i 0
500.3 − 2.71144i 1.22460 + 1.22489i −5.35192 0 3.32122 − 3.32043i −4.44309 9.08853i −0.000709735 3.00000i 0
500.4 − 2.38402i −0.829904 + 1.52028i −3.68356 0 3.62439 + 1.97851i 0.768419 4.01365i −1.62252 − 2.52338i 0
500.5 − 2.37480i 1.62292 + 0.605098i −3.63970 0 1.43699 − 3.85411i 3.47691 3.89396i 2.26771 + 1.96405i 0
500.6 − 1.85866i 1.03814 − 1.38646i −1.45462 0 −2.57695 − 1.92956i −5.29148 − 1.01368i −0.844517 − 2.87868i 0
500.7 − 1.84578i 0.123767 − 1.72762i −1.40689 0 −3.18880 − 0.228447i 3.45345 − 1.09475i −2.96936 − 0.427647i 0
500.8 − 1.18272i 1.69242 − 0.368374i 0.601174 0 −0.435683 − 2.00166i 0.737694 − 3.07646i 2.72860 − 1.24689i 0
500.9 − 1.16721i −1.52009 + 0.830263i 0.637617 0 0.969093 + 1.77426i −4.45987 − 3.07866i 1.62133 − 2.52414i 0
500.10 − 0.410963i 0.367974 + 1.69251i 1.83111 0 0.695559 − 0.151224i 5.08151 − 1.57444i −2.72919 + 1.24560i 0
500.11 − 0.394088i −1.38670 + 1.03782i 1.84469 0 0.408990 + 0.546482i −2.18404 − 1.51515i 0.845879 − 2.87828i 0
500.12 0.394088i −1.38670 − 1.03782i 1.84469 0 0.408990 − 0.546482i −2.18404 1.51515i 0.845879 + 2.87828i 0
500.13 0.410963i 0.367974 − 1.69251i 1.83111 0 0.695559 + 0.151224i 5.08151 1.57444i −2.72919 − 1.24560i 0
500.14 1.16721i −1.52009 − 0.830263i 0.637617 0 0.969093 − 1.77426i −4.45987 3.07866i 1.62133 + 2.52414i 0
500.15 1.18272i 1.69242 + 0.368374i 0.601174 0 −0.435683 + 2.00166i 0.737694 3.07646i 2.72860 + 1.24689i 0
500.16 1.84578i 0.123767 + 1.72762i −1.40689 0 −3.18880 + 0.228447i 3.45345 1.09475i −2.96936 + 0.427647i 0
500.17 1.85866i 1.03814 + 1.38646i −1.45462 0 −2.57695 + 1.92956i −5.29148 1.01368i −0.844517 + 2.87868i 0
500.18 2.37480i 1.62292 − 0.605098i −3.63970 0 1.43699 + 3.85411i 3.47691 − 3.89396i 2.26771 − 1.96405i 0
500.19 2.38402i −0.829904 − 1.52028i −3.68356 0 3.62439 − 1.97851i 0.768419 − 4.01365i −1.62252 + 2.52338i 0
500.20 2.71144i 1.22460 − 1.22489i −5.35192 0 3.32122 + 3.32043i −4.44309 − 9.08853i −0.000709735 − 3.00000i 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 500.22
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.b odd 2 1 CM by \(\Q(\sqrt{-167}) \)
3.b odd 2 1 inner
501.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 501.2.c.a ✓ 22
3.b odd 2 1 inner 501.2.c.a ✓ 22
167.b odd 2 1 CM 501.2.c.a ✓ 22
501.c even 2 1 inner 501.2.c.a ✓ 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.c.a ✓ 22 1.a even 1 1 trivial
501.2.c.a ✓ 22 3.b odd 2 1 inner
501.2.c.a ✓ 22 167.b odd 2 1 CM
501.2.c.a ✓ 22 501.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 44 T_{2}^{20} + 836 T_{2}^{18} + 8976 T_{2}^{16} + 59840 T_{2}^{14} + 256256 T_{2}^{12} + \cdots + 8183 \) acting on \(S_{2}^{\mathrm{new}}(501, [\chi])\). Copy content Toggle raw display