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Properties

Label 800.1.bo.a
Level $800$
Weight $1$
Character orbit 800.bo
Analytic conductor $0.399$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
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] = [800,1,Mod(33,800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(800, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("800.33");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 800 = 2^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 800.bo (of order \(20\), degree \(8\), 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: \(0.399252010106\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{20} - \cdots)\)

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{20}^{7} q^{5} + \zeta_{20} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{20}^{7} q^{5} + \zeta_{20} q^{9} + (\zeta_{20}^{9} - \zeta_{20}^{8}) q^{13} + (\zeta_{20}^{6} + \zeta_{20}^{3}) q^{17} - \zeta_{20}^{4} q^{25} + (\zeta_{20}^{4} + \zeta_{20}^{2}) q^{29} + (\zeta_{20}^{5} - \zeta_{20}^{2}) q^{37} + ( - \zeta_{20}^{9} - \zeta_{20}^{3}) q^{41} + \zeta_{20}^{8} q^{45} - \zeta_{20}^{5} q^{49} + ( - \zeta_{20}^{6} - \zeta_{20}^{5}) q^{53} + ( - \zeta_{20}^{7} - \zeta_{20}) q^{61} + ( - \zeta_{20}^{6} + \zeta_{20}^{5}) q^{65} + ( - \zeta_{20}^{2} - \zeta_{20}) q^{73} + \zeta_{20}^{2} q^{81} + ( - \zeta_{20}^{3} - 1) q^{85} + (\zeta_{20}^{8} - 1) q^{89} + ( - \zeta_{20}^{7} + \zeta_{20}^{4}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{13} + 2 q^{17} + 2 q^{25} - 2 q^{37} - 2 q^{45} - 2 q^{53} - 2 q^{65} - 2 q^{73} + 2 q^{81} - 8 q^{85} - 10 q^{89} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(351\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{20}^{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−0.951057 − 0.309017i
−0.951057 + 0.309017i
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 0 0 0.587785 − 0.809017i 0 0 0 −0.951057 − 0.309017i 0
97.1 0 0 0 0.587785 + 0.809017i 0 0 0 −0.951057 + 0.309017i 0
353.1 0 0 0 −0.587785 − 0.809017i 0 0 0 0.951057 − 0.309017i 0
417.1 0 0 0 −0.587785 + 0.809017i 0 0 0 0.951057 + 0.309017i 0
513.1 0 0 0 0.951057 + 0.309017i 0 0 0 0.587785 + 0.809017i 0
577.1 0 0 0 0.951057 − 0.309017i 0 0 0 0.587785 − 0.809017i 0
673.1 0 0 0 −0.951057 + 0.309017i 0 0 0 −0.587785 + 0.809017i 0
737.1 0 0 0 −0.951057 − 0.309017i 0 0 0 −0.587785 − 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.1
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 800.1.bo.a ✓ 8
4.b odd 2 1 CM 800.1.bo.a ✓ 8
5.b even 2 1 4000.1.bo.b 8
5.c odd 4 1 4000.1.bo.a 8
5.c odd 4 1 4000.1.bo.c 8
8.b even 2 1 1600.1.bw.a 8
8.d odd 2 1 1600.1.bw.a 8
20.d odd 2 1 4000.1.bo.b 8
20.e even 4 1 4000.1.bo.a 8
20.e even 4 1 4000.1.bo.c 8
25.d even 5 1 4000.1.bo.c 8
25.e even 10 1 4000.1.bo.a 8
25.f odd 20 1 inner 800.1.bo.a ✓ 8
25.f odd 20 1 4000.1.bo.b 8
100.h odd 10 1 4000.1.bo.a 8
100.j odd 10 1 4000.1.bo.c 8
100.l even 20 1 inner 800.1.bo.a ✓ 8
100.l even 20 1 4000.1.bo.b 8
200.v even 20 1 1600.1.bw.a 8
200.x odd 20 1 1600.1.bw.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
800.1.bo.a ✓ 8 1.a even 1 1 trivial
800.1.bo.a ✓ 8 4.b odd 2 1 CM
800.1.bo.a ✓ 8 25.f odd 20 1 inner
800.1.bo.a ✓ 8 100.l even 20 1 inner
1600.1.bw.a 8 8.b even 2 1
1600.1.bw.a 8 8.d odd 2 1
1600.1.bw.a 8 200.v even 20 1
1600.1.bw.a 8 200.x odd 20 1
4000.1.bo.a 8 5.c odd 4 1
4000.1.bo.a 8 20.e even 4 1
4000.1.bo.a 8 25.e even 10 1
4000.1.bo.a 8 100.h odd 10 1
4000.1.bo.b 8 5.b even 2 1
4000.1.bo.b 8 20.d odd 2 1
4000.1.bo.b 8 25.f odd 20 1
4000.1.bo.b 8 100.l even 20 1
4000.1.bo.c 8 5.c odd 4 1
4000.1.bo.c 8 20.e even 4 1
4000.1.bo.c 8 25.d even 5 1
4000.1.bo.c 8 100.j odd 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(800, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - T^{6} + T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( (T^{4} + 5 T + 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( T^{8} + 10 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 2 T^{7} + \cdots + 1 \) Copy content Toggle raw display
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