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Properties

Label 800.2.bg.b
Level $800$
Weight $2$
Character orbit 800.bg
Analytic conductor $6.388$
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] = [800,2,Mod(129,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.129");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 800.bg (of order \(10\), degree \(4\), 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: \(6.38803216170\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
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: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + 3 \beta_{4} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} - \beta_{4} + \beta_{3} + \cdots - 1) q^{5}+ \cdots + ( - 18 \beta_{6} + 9 \beta_{4} + \cdots + 9) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{5} - 6 q^{9} + 6 q^{25} + 20 q^{29} + 10 q^{37} + 20 q^{41} - 6 q^{45} + 56 q^{49} - 70 q^{53} - 20 q^{61} + 6 q^{65} - 18 q^{81} + 38 q^{85} + 30 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

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

\(\zeta_{20}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{20}^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{4}\)\(=\) \( \beta_{4} \) Copy content Toggle raw display
\(\zeta_{20}^{5}\)\(=\) \( ( \beta_{5} ) / 2 \) Copy content Toggle raw display
\(\zeta_{20}^{6}\)\(=\) \( \beta_{6} \) Copy content Toggle raw display
\(\zeta_{20}^{7}\)\(=\) \( ( \beta_{7} ) / 2 \) 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 \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
−0.951057 − 0.309017i
0.951057 + 0.309017i
0.587785 + 0.809017i
−0.587785 − 0.809017i
0.587785 − 0.809017i
−0.587785 + 0.809017i
−0.951057 + 0.309017i
0.951057 − 0.309017i
0 0 0 −1.98459 − 1.03025i 0 0 0 0.927051 + 2.85317i 0
129.2 0 0 0 0.366554 + 2.20582i 0 0 0 0.927051 + 2.85317i 0
289.1 0 0 0 −1.59310 + 1.56909i 0 0 0 −2.42705 − 1.76336i 0
289.2 0 0 0 2.21113 + 0.333023i 0 0 0 −2.42705 − 1.76336i 0
609.1 0 0 0 −1.59310 − 1.56909i 0 0 0 −2.42705 + 1.76336i 0
609.2 0 0 0 2.21113 − 0.333023i 0 0 0 −2.42705 + 1.76336i 0
769.1 0 0 0 −1.98459 + 1.03025i 0 0 0 0.927051 − 2.85317i 0
769.2 0 0 0 0.366554 − 2.20582i 0 0 0 0.927051 − 2.85317i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.2
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.e even 10 1 inner
100.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 800.2.bg.b ✓ 8
4.b odd 2 1 CM 800.2.bg.b ✓ 8
25.e even 10 1 inner 800.2.bg.b ✓ 8
100.h odd 10 1 inner 800.2.bg.b ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.2.bg.b ✓ 8 1.a even 1 1 trivial
800.2.bg.b ✓ 8 4.b odd 2 1 CM
800.2.bg.b ✓ 8 25.e even 10 1 inner
800.2.bg.b ✓ 8 100.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 16 T^{6} + \cdots + 3721 \) Copy content Toggle raw display
$17$ \( T^{8} - 64 T^{6} + \cdots + 10201 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 20 T^{7} + \cdots + 87025 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 10 T^{7} + \cdots + 885481 \) Copy content Toggle raw display
$41$ \( T^{8} - 20 T^{7} + \cdots + 4389025 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 70 T^{7} + \cdots + 101223721 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 20 T^{7} + \cdots + 3591025 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} - 256 T^{6} + \cdots + 1585081 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( T^{8} - 30 T^{7} + \cdots + 26061025 \) Copy content Toggle raw display
$97$ \( T^{8} - 64 T^{6} + \cdots + 404050201 \) Copy content Toggle raw display
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