Properties

Label 400.1.x.a
Level $400$
Weight $1$
Character orbit 400.x
Analytic conductor $0.200$
Analytic rank $0$
Dimension $4$
Projective image $D_{10}$
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] = [400,1,Mod(79,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.79");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 400.x (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: \(0.199626005053\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{10}\)
Projective field: Galois closure of 10.2.195312500000000.4

$q$-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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
0 0 0 0.809017 0.587785i 0 0 0 −0.309017 0.951057i 0
159.1 0 0 0 −0.309017 + 0.951057i 0 0 0 0.809017 0.587785i 0
239.1 0 0 0 −0.309017 0.951057i 0 0 0 0.809017 + 0.587785i 0
319.1 0 0 0 0.809017 + 0.587785i 0 0 0 −0.309017 + 0.951057i 0
\(n\): e.g. 2-40 or 990-1000
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 400.1.x.a 4
3.b odd 2 1 3600.1.ct.a 4
4.b odd 2 1 CM 400.1.x.a 4
5.b even 2 1 2000.1.x.a 4
5.c odd 4 2 2000.1.z.a 8
8.b even 2 1 1600.1.bf.a 4
8.d odd 2 1 1600.1.bf.a 4
12.b even 2 1 3600.1.ct.a 4
20.d odd 2 1 2000.1.x.a 4
20.e even 4 2 2000.1.z.a 8
25.d even 5 1 2000.1.x.a 4
25.e even 10 1 inner 400.1.x.a 4
25.f odd 20 2 2000.1.z.a 8
75.h odd 10 1 3600.1.ct.a 4
100.h odd 10 1 inner 400.1.x.a 4
100.j odd 10 1 2000.1.x.a 4
100.l even 20 2 2000.1.z.a 8
200.o even 10 1 1600.1.bf.a 4
200.s odd 10 1 1600.1.bf.a 4
300.r even 10 1 3600.1.ct.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.1.x.a 4 1.a even 1 1 trivial
400.1.x.a 4 4.b odd 2 1 CM
400.1.x.a 4 25.e even 10 1 inner
400.1.x.a 4 100.h odd 10 1 inner
1600.1.bf.a 4 8.b even 2 1
1600.1.bf.a 4 8.d odd 2 1
1600.1.bf.a 4 200.o even 10 1
1600.1.bf.a 4 200.s odd 10 1
2000.1.x.a 4 5.b even 2 1
2000.1.x.a 4 20.d odd 2 1
2000.1.x.a 4 25.d even 5 1
2000.1.x.a 4 100.j odd 10 1
2000.1.z.a 8 5.c odd 4 2
2000.1.z.a 8 20.e even 4 2
2000.1.z.a 8 25.f odd 20 2
2000.1.z.a 8 100.l even 20 2
3600.1.ct.a 4 3.b odd 2 1
3600.1.ct.a 4 12.b even 2 1
3600.1.ct.a 4 75.h odd 10 1
3600.1.ct.a 4 300.r even 10 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(400, [\chi])\).

Hecke characteristic polynomials

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