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Properties

Label 500.2.l.c
Level $500$
Weight $2$
Character orbit 500.l
Analytic conductor $3.993$
Analytic rank $0$
Dimension $8$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,2,Mod(7,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 500.l (of order \(20\), degree \(8\), not 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: \(8\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 100)
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{20}^{6} + \zeta_{20}^{4} + \cdots + 1) q^{2}+ \cdots + 3 \zeta_{20}^{7} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{20}^{6} + \zeta_{20}^{4} + \cdots + 1) q^{2}+ \cdots + (7 \zeta_{20}^{6} - 7 \zeta_{20}^{4} + \cdots - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 4 q^{8} + 2 q^{13} + 8 q^{16} - 6 q^{17} - 24 q^{18} - 4 q^{26} + 32 q^{32} - 50 q^{34} - 12 q^{36} + 4 q^{37} + 16 q^{41} + 4 q^{52} + 52 q^{53} - 8 q^{58} - 24 q^{61} + 12 q^{68} + 12 q^{72} + 22 q^{73} + 18 q^{81} - 16 q^{82} + 50 q^{89} - 26 q^{97} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(\zeta_{20}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.587785 − 0.809017i
0.951057 − 0.309017i
−0.587785 − 0.809017i
0.587785 + 0.809017i
−0.951057 + 0.309017i
−0.587785 + 0.809017i
−0.951057 − 0.309017i
0.951057 + 0.309017i
−1.26007 + 0.642040i 0 1.17557 − 1.61803i 0 0 0 −0.442463 + 2.79360i 2.85317 − 0.927051i 0
43.1 1.39680 − 0.221232i 0 1.90211 − 0.618034i 0 0 0 2.52015 − 1.28408i −1.76336 − 2.42705i 0
107.1 0.642040 − 1.26007i 0 −1.17557 − 1.61803i 0 0 0 −2.79360 + 0.442463i −2.85317 − 0.927051i 0
143.1 −1.26007 − 0.642040i 0 1.17557 + 1.61803i 0 0 0 −0.442463 − 2.79360i 2.85317 + 0.927051i 0
207.1 0.221232 + 1.39680i 0 −1.90211 + 0.618034i 0 0 0 −1.28408 − 2.52015i 1.76336 + 2.42705i 0
243.1 0.642040 + 1.26007i 0 −1.17557 + 1.61803i 0 0 0 −2.79360 − 0.442463i −2.85317 + 0.927051i 0
343.1 0.221232 − 1.39680i 0 −1.90211 − 0.618034i 0 0 0 −1.28408 + 2.52015i 1.76336 − 2.42705i 0
407.1 1.39680 + 0.221232i 0 1.90211 + 0.618034i 0 0 0 2.52015 + 1.28408i −1.76336 + 2.42705i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.l.c 8
4.b odd 2 1 CM 500.2.l.c 8
5.b even 2 1 500.2.l.a 8
5.c odd 4 1 100.2.l.a ✓ 8
5.c odd 4 1 500.2.l.b 8
15.e even 4 1 900.2.bj.a 8
20.d odd 2 1 500.2.l.a 8
20.e even 4 1 100.2.l.a ✓ 8
20.e even 4 1 500.2.l.b 8
25.d even 5 1 100.2.l.a ✓ 8
25.e even 10 1 500.2.l.b 8
25.f odd 20 1 500.2.l.a 8
25.f odd 20 1 inner 500.2.l.c 8
60.l odd 4 1 900.2.bj.a 8
75.j odd 10 1 900.2.bj.a 8
100.h odd 10 1 500.2.l.b 8
100.j odd 10 1 100.2.l.a ✓ 8
100.l even 20 1 500.2.l.a 8
100.l even 20 1 inner 500.2.l.c 8
300.n even 10 1 900.2.bj.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
100.2.l.a ✓ 8 5.c odd 4 1
100.2.l.a ✓ 8 20.e even 4 1
100.2.l.a ✓ 8 25.d even 5 1
100.2.l.a ✓ 8 100.j odd 10 1
500.2.l.a 8 5.b even 2 1
500.2.l.a 8 20.d odd 2 1
500.2.l.a 8 25.f odd 20 1
500.2.l.a 8 100.l even 20 1
500.2.l.b 8 5.c odd 4 1
500.2.l.b 8 20.e even 4 1
500.2.l.b 8 25.e even 10 1
500.2.l.b 8 100.h odd 10 1
500.2.l.c 8 1.a even 1 1 trivial
500.2.l.c 8 4.b odd 2 1 CM
500.2.l.c 8 25.f odd 20 1 inner
500.2.l.c 8 100.l even 20 1 inner
900.2.bj.a 8 15.e even 4 1
900.2.bj.a 8 60.l odd 4 1
900.2.bj.a 8 75.j odd 10 1
900.2.bj.a 8 300.n even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{13}^{8} - 2T_{13}^{7} + 2T_{13}^{6} - 390T_{13}^{5} - 784T_{13}^{4} + 910T_{13}^{3} + 54673T_{13}^{2} + 277534T_{13} + 516961 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 516961 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 57121 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 16 T^{6} + \cdots + 4583881 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 4 T^{7} + \cdots + 2825761 \) Copy content Toggle raw display
$41$ \( T^{8} - 16 T^{7} + \cdots + 383161 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} - 52 T^{7} + \cdots + 6974881 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 24 T^{7} + \cdots + 20967241 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 22 T^{7} + \cdots + 9740641 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} - 50 T^{7} + \cdots + 77070841 \) Copy content Toggle raw display
$97$ \( T^{8} + 26 T^{7} + \cdots + 6974881 \) Copy content Toggle raw display
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