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Properties

Label 500.2.e.a
Level $500$
Weight $2$
Character orbit 500.e
Analytic conductor $3.993$
Analytic rank $0$
Dimension $8$
CM discriminant -20
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(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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

$q$-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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_{2} q^{4} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + (2 \beta_{2} + 2) q^{8} + ( - \beta_{5} + 4 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} - 1) q^{2} - \beta_{3} q^{3} - 2 \beta_{2} q^{4} + (\beta_{4} + \beta_{3}) q^{6} + ( - \beta_{6} + \beta_{2} - 1) q^{7} + (2 \beta_{2} + 2) q^{8} + ( - \beta_{5} + 4 \beta_{2}) q^{9} - 2 \beta_{4} q^{12} + (\beta_{6} - 2 \beta_{2} + \beta_1) q^{14} - 4 q^{16} + ( - \beta_{7} + \beta_{5} - 4 \beta_{2} - 4) q^{18} + ( - 2 \beta_{6} + \beta_{4} + \beta_{3} + \cdots + 1) q^{21}+ \cdots + (\beta_{7} - \beta_{5} - 6 \beta_{3} + \cdots + 5) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 2 q^{3} - 4 q^{6} - 6 q^{7} + 16 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 2 q^{3} - 4 q^{6} - 6 q^{7} + 16 q^{8} + 4 q^{12} - 32 q^{16} - 32 q^{18} + 12 q^{21} + 2 q^{23} + 8 q^{27} + 12 q^{28} + 32 q^{32} + 64 q^{36} - 24 q^{41} - 12 q^{42} - 18 q^{43} - 4 q^{46} + 14 q^{47} - 8 q^{48} - 24 q^{56} + 12 q^{58} + 16 q^{61} - 4 q^{63} + 24 q^{67} - 64 q^{72} - 100 q^{81} + 24 q^{82} + 22 q^{83} + 36 q^{86} - 82 q^{87} + 4 q^{92} + 16 q^{96} + 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{20}^{4} + 2\zeta_{20}^{3} + 2\zeta_{20}^{2} + \zeta_{20} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{20}^{5} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{20}^{5} - 2\zeta_{20}^{4} + \zeta_{20}^{3} + \zeta_{20}^{2} - 2\zeta_{20} - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + 2\zeta_{20}^{3} - \zeta_{20}^{2} - 2\zeta_{20} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\zeta_{20}^{7} - 3\zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4} + 2\zeta_{20}^{3} - 4\zeta_{20}^{2} + 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 3\zeta_{20}^{7} + 3\zeta_{20}^{6} - \zeta_{20}^{5} - 2\zeta_{20}^{4} + \zeta_{20}^{3} + 2\zeta_{20}^{2} - \zeta_{20} - 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\zeta_{20}^{7} + 2\zeta_{20}^{6} + \zeta_{20}^{5} - 2\zeta_{20}^{4} + \zeta_{20}^{3} + 2\zeta_{20} - 1 \) Copy content Toggle raw display

Display \(\zeta_{20}^j\) in terms of \(\beta_i\)

\(\zeta_{20}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - 2\beta_{4} - 2\beta_{3} - 3\beta_{2} + \beta _1 - 1 ) / 10 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( ( -\beta_{7} - \beta_{5} - \beta_{4} + \beta_{3} + 2\beta _1 + 2 ) / 10 \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} + 2\beta_1 ) / 10 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} - \beta_{2} + \beta _1 - 3 ) / 10 \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 2\beta_{4} - 2\beta_{3} - \beta_{2} - \beta _1 + 3 ) / 10 \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 2\beta_{2} ) / 10 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{20}^j\)

Character values

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

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
307.1
−0.587785 − 0.809017i
−0.951057 + 0.309017i
0.587785 − 0.809017i
0.951057 + 0.309017i
−0.587785 + 0.809017i
−0.951057 − 0.309017i
0.587785 + 0.809017i
0.951057 − 0.309017i
−1.00000 + 1.00000i −2.43564 − 2.43564i − 2.00000i 0 4.87129 −1.11272 + 1.11272i 2.00000 + 2.00000i 8.86472i 0
307.2 −1.00000 + 1.00000i −0.505311 − 0.505311i − 2.00000i 0 1.01062 −1.19958 + 1.19958i 2.00000 + 2.00000i − 2.48932i 0
307.3 −1.00000 + 1.00000i 1.81761 + 1.81761i − 2.00000i 0 −3.63522 −3.74138 + 3.74138i 2.00000 + 2.00000i 3.60741i 0
307.4 −1.00000 + 1.00000i 2.12334 + 2.12334i − 2.00000i 0 −4.24669 3.05368 − 3.05368i 2.00000 + 2.00000i 6.01719i 0
443.1 −1.00000 − 1.00000i −2.43564 + 2.43564i 2.00000i 0 4.87129 −1.11272 − 1.11272i 2.00000 − 2.00000i − 8.86472i 0
443.2 −1.00000 − 1.00000i −0.505311 + 0.505311i 2.00000i 0 1.01062 −1.19958 − 1.19958i 2.00000 − 2.00000i 2.48932i 0
443.3 −1.00000 − 1.00000i 1.81761 − 1.81761i 2.00000i 0 −3.63522 −3.74138 − 3.74138i 2.00000 − 2.00000i − 3.60741i 0
443.4 −1.00000 − 1.00000i 2.12334 − 2.12334i 2.00000i 0 −4.24669 3.05368 + 3.05368i 2.00000 − 2.00000i − 6.01719i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 307.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
5.c odd 4 1 inner
20.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.2.e.a ✓ 8
4.b odd 2 1 500.2.e.b yes 8
5.b even 2 1 500.2.e.b yes 8
5.c odd 4 1 inner 500.2.e.a ✓ 8
5.c odd 4 1 500.2.e.b yes 8
20.d odd 2 1 CM 500.2.e.a ✓ 8
20.e even 4 1 inner 500.2.e.a ✓ 8
20.e even 4 1 500.2.e.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.2.e.a ✓ 8 1.a even 1 1 trivial
500.2.e.a ✓ 8 5.c odd 4 1 inner
500.2.e.a ✓ 8 20.d odd 2 1 CM
500.2.e.a ✓ 8 20.e even 4 1 inner
500.2.e.b yes 8 4.b odd 2 1
500.2.e.b yes 8 5.b even 2 1
500.2.e.b yes 8 5.c odd 4 1
500.2.e.b yes 8 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 2T_{3}^{6} + 131T_{3}^{4} - 300T_{3}^{3} + 338T_{3}^{2} + 494T_{3} + 361 \) acting on \(S_{2}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 361 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 6 T^{7} + \cdots + 3721 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 5851561 \) Copy content Toggle raw display
$29$ \( T^{8} + 254 T^{6} + \cdots + 78961 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} + 12 T^{3} + \cdots - 379)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 18 T^{7} + \cdots + 434281 \) Copy content Toggle raw display
$47$ \( T^{8} - 14 T^{7} + \cdots + 5669161 \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} - 8 T^{3} + \cdots + 3181)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 6 T + 18)^{4} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 22 T^{7} + \cdots + 55071241 \) Copy content Toggle raw display
$89$ \( T^{8} + 854 T^{6} + \cdots + 619064161 \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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