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Properties

Label 500.2.e.d
Level $500$
Weight $2$
Character orbit 500.e
Analytic conductor $3.993$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,2,Mod(307,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.307");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.e (of order \(4\), degree \(2\), 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: \(3.99252010106\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q + 12 q^{6} + 28 q^{16} - 16 q^{21} - 16 q^{26} - 80 q^{36} - 24 q^{41} - 108 q^{46} - 148 q^{56} + 96 q^{61} - 100 q^{66} + 20 q^{76} + 32 q^{81} + 172 q^{86} + 272 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1 −1.41217 + 0.0760781i −1.79019 − 1.79019i 1.98842 − 0.214870i 0 2.66424 + 2.39185i 2.15032 − 2.15032i −2.79164 + 0.454707i 3.40955i 0
307.2 −1.36320 − 0.376404i 2.13471 + 2.13471i 1.71664 + 1.02623i 0 −2.10653 − 3.71355i −1.05174 + 1.05174i −1.95385 − 2.04511i 6.11395i 0
307.3 −1.35194 + 0.415033i −1.15279 − 1.15279i 1.65550 − 1.12220i 0 2.03695 + 1.08006i −1.42563 + 1.42563i −1.77238 + 2.20424i − 0.342155i 0
307.4 −1.30832 − 0.536946i −0.337789 − 0.337789i 1.42338 + 1.40499i 0 0.260560 + 0.623309i 3.00872 − 3.00872i −1.10782 − 2.60245i − 2.77180i 0
307.5 −1.28783 + 0.584371i 0.468102 + 0.468102i 1.31702 − 1.50514i 0 −0.876382 − 0.329292i −0.445070 + 0.445070i −0.816541 + 2.70800i − 2.56176i 0
307.6 −1.13798 − 0.839643i 1.60502 + 1.60502i 0.589998 + 1.91100i 0 −0.478838 − 3.17414i 2.91326 − 2.91326i 0.933149 − 2.67006i 2.15221i 0
307.7 −0.839643 − 1.13798i −1.60502 − 1.60502i −0.589998 + 1.91100i 0 −0.478838 + 3.17414i −2.91326 + 2.91326i 2.67006 − 0.933149i 2.15221i 0
307.8 −0.584371 + 1.28783i 0.468102 + 0.468102i −1.31702 − 1.50514i 0 −0.876382 + 0.329292i −0.445070 + 0.445070i 2.70800 − 0.816541i − 2.56176i 0
307.9 −0.536946 − 1.30832i 0.337789 + 0.337789i −1.42338 + 1.40499i 0 0.260560 − 0.623309i −3.00872 + 3.00872i 2.60245 + 1.10782i − 2.77180i 0
307.10 −0.415033 + 1.35194i −1.15279 − 1.15279i −1.65550 − 1.12220i 0 2.03695 − 1.08006i −1.42563 + 1.42563i 2.20424 − 1.77238i − 0.342155i 0
307.11 −0.376404 − 1.36320i −2.13471 − 2.13471i −1.71664 + 1.02623i 0 −2.10653 + 3.71355i 1.05174 − 1.05174i 2.04511 + 1.95385i 6.11395i 0
307.12 −0.0760781 + 1.41217i −1.79019 − 1.79019i −1.98842 − 0.214870i 0 2.66424 − 2.39185i 2.15032 − 2.15032i 0.454707 − 2.79164i 3.40955i 0
307.13 0.0760781 − 1.41217i 1.79019 + 1.79019i −1.98842 − 0.214870i 0 2.66424 − 2.39185i −2.15032 + 2.15032i −0.454707 + 2.79164i 3.40955i 0
307.14 0.376404 + 1.36320i 2.13471 + 2.13471i −1.71664 + 1.02623i 0 −2.10653 + 3.71355i −1.05174 + 1.05174i −2.04511 − 1.95385i 6.11395i 0
307.15 0.415033 − 1.35194i 1.15279 + 1.15279i −1.65550 − 1.12220i 0 2.03695 − 1.08006i 1.42563 − 1.42563i −2.20424 + 1.77238i − 0.342155i 0
307.16 0.536946 + 1.30832i −0.337789 − 0.337789i −1.42338 + 1.40499i 0 0.260560 − 0.623309i 3.00872 − 3.00872i −2.60245 − 1.10782i − 2.77180i 0
307.17 0.584371 − 1.28783i −0.468102 − 0.468102i −1.31702 − 1.50514i 0 −0.876382 + 0.329292i 0.445070 − 0.445070i −2.70800 + 0.816541i − 2.56176i 0
307.18 0.839643 + 1.13798i 1.60502 + 1.60502i −0.589998 + 1.91100i 0 −0.478838 + 3.17414i 2.91326 − 2.91326i −2.67006 + 0.933149i 2.15221i 0
307.19 1.13798 + 0.839643i −1.60502 − 1.60502i 0.589998 + 1.91100i 0 −0.478838 − 3.17414i −2.91326 + 2.91326i −0.933149 + 2.67006i 2.15221i 0
307.20 1.28783 − 0.584371i −0.468102 − 0.468102i 1.31702 − 1.50514i 0 −0.876382 − 0.329292i 0.445070 − 0.445070i 0.816541 − 2.70800i − 2.56176i 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.e.d ✓ 48
4.b odd 2 1 inner 500.2.e.d ✓ 48
5.b even 2 1 inner 500.2.e.d ✓ 48
5.c odd 4 2 inner 500.2.e.d ✓ 48
20.d odd 2 1 inner 500.2.e.d ✓ 48
20.e even 4 2 inner 500.2.e.d ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.2.e.d ✓ 48 1.a even 1 1 trivial
500.2.e.d ✓ 48 4.b odd 2 1 inner
500.2.e.d ✓ 48 5.b even 2 1 inner
500.2.e.d ✓ 48 5.c odd 4 2 inner
500.2.e.d ✓ 48 20.d odd 2 1 inner
500.2.e.d ✓ 48 20.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 158T_{3}^{20} + 7811T_{3}^{16} + 139870T_{3}^{12} + 673665T_{3}^{8} + 157600T_{3}^{4} + 6400 \) acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display