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Properties

Label 500.4.c.b
Level $500$
Weight $4$
Character orbit 500.c
Analytic conductor $29.501$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,4,Mod(249,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.249");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 500.c (of order \(2\), degree \(1\), 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: \(29.5009550029\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 36x^{6} + 431x^{4} + 2016x^{2} + 2896 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{5} + \beta_1) q^{7} + (\beta_{6} - \beta_{4} - 17) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{5} + \beta_1) q^{7} + (\beta_{6} - \beta_{4} - 17) q^{9} + ( - \beta_{7} - \beta_{4} + 5) q^{11} + ( - \beta_{3} + 2 \beta_{2} - 5 \beta_1) q^{13} + (\beta_{5} - 2 \beta_{3} + \cdots + 7 \beta_1) q^{17}+ \cdots + (21 \beta_{7} - \beta_{6} + 225 \beta_{4} - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 132 q^{9} + 40 q^{11} + 128 q^{19} + 328 q^{21} - 452 q^{29} + 88 q^{31} - 1712 q^{39} + 1748 q^{41} - 2324 q^{49} + 2456 q^{51} - 664 q^{59} + 2316 q^{61} - 3256 q^{69} + 624 q^{71} - 2136 q^{79} + 3272 q^{81} - 2628 q^{89} + 2552 q^{91} - 880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 36x^{6} + 431x^{4} + 2016x^{2} + 2896 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} + 32\nu^{5} + 295\nu^{3} + 668\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} - 24\nu^{5} - 55\nu^{3} + 732\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -11\nu^{7} - 288\nu^{5} - 1805\nu^{3} - 1428\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{6} - 140\nu^{4} - 1055\nu^{2} - 2012 ) / 24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{7} + 57\nu^{5} + 440\nu^{3} + 891\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{6} + 100\nu^{4} + 921\nu^{2} + 2056 ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -27\nu^{6} - 740\nu^{4} - 5169\nu^{2} - 7772 ) / 24 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{3} - \beta_{2} - 6\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - \beta_{6} - 12\beta_{4} - 187 ) / 20 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -11\beta_{5} - 13\beta_{3} + 27\beta_{2} + 60\beta_1 ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -36\beta_{7} + 33\beta_{6} + 234\beta_{4} + 2305 ) / 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31\beta_{5} + 43\beta_{3} - 103\beta_{2} - 126\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 586\beta_{7} - 713\beta_{6} - 4116\beta_{4} - 33131 ) / 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -2383\beta_{5} - 3713\beta_{3} + 9183\beta_{2} + 7428\beta_1 ) / 20 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(251\) \(377\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
249.1
− 2.75002i
− 2.95548i
1.59904i
− 4.14072i
4.14072i
− 1.59904i
2.95548i
2.75002i
0 − 9.47155i 0 0 0 28.5382i 0 −62.7103 0
249.2 0 − 8.23253i 0 0 0 − 23.1625i 0 −40.7746 0
249.3 0 − 3.53797i 0 0 0 31.5688i 0 14.4828 0
249.4 0 − 1.99947i 0 0 0 − 13.6557i 0 23.0021 0
249.5 0 1.99947i 0 0 0 13.6557i 0 23.0021 0
249.6 0 3.53797i 0 0 0 − 31.5688i 0 14.4828 0
249.7 0 8.23253i 0 0 0 23.1625i 0 −40.7746 0
249.8 0 9.47155i 0 0 0 − 28.5382i 0 −62.7103 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 249.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.4.c.b 8
5.b even 2 1 inner 500.4.c.b 8
5.c odd 4 2 500.4.a.d ✓ 8
20.e even 4 2 2000.4.a.r 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.4.a.d ✓ 8 5.c odd 4 2
500.4.c.b 8 1.a even 1 1 trivial
500.4.c.b 8 5.b even 2 1 inner
2000.4.a.r 8 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 174T_{3}^{6} + 8731T_{3}^{4} + 108294T_{3}^{2} + 304261 \) acting on \(S_{4}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 174 T^{6} + \cdots + 304261 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 81202891741 \) Copy content Toggle raw display
$11$ \( (T^{4} - 20 T^{3} + \cdots + 1300400)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 14465800842496 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 64359701335296 \) Copy content Toggle raw display
$19$ \( (T^{4} - 64 T^{3} + \cdots + 131296176)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 21\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{4} + 226 T^{3} + \cdots - 204520219)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 44 T^{3} + \cdots + 2972016)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 46\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( (T^{4} - 874 T^{3} + \cdots - 8912828419)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 17\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 31\!\cdots\!21 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 27\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( (T^{4} + 332 T^{3} + \cdots + 545761136)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 1158 T^{3} + \cdots + 790667001)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 56\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{4} - 312 T^{3} + \cdots + 5296879216)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 10\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + 1068 T^{3} + \cdots + 38890097136)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 18\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1314 T^{3} + \cdots + 22542093001)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 84\!\cdots\!36 \) Copy content Toggle raw display
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