Properties

Label 400.2.n.c
Level $400$
Weight $2$
Character orbit 400.n
Analytic conductor $3.194$
Analytic rank $0$
Dimension $4$
CM discriminant -20
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,2,Mod(143,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.143");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.n (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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_{2} q^{7} - 7 \beta_1 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_{2} q^{7} - 7 \beta_1 q^{9} + 10 q^{21} + 3 \beta_{3} q^{23} - 4 \beta_{2} q^{27} - 6 \beta_1 q^{29} - 12 q^{41} + \beta_{3} q^{43} + 3 \beta_{2} q^{47} + 3 \beta_1 q^{49} + 8 q^{61} - 7 \beta_{3} q^{63} + 5 \beta_{2} q^{67} + 30 \beta_1 q^{69} - 19 q^{81} - 3 \beta_{3} q^{83} - 6 \beta_{2} q^{87} - 6 \beta_1 q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 40 q^{21} - 48 q^{41} + 32 q^{61} - 76 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + 2\nu^{2} + 4\nu + 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 2\nu^{2} + 4\nu - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + \beta_{2} - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + 4\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(-\beta_{1}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
1.61803i
0.618034i
1.61803i
0.618034i
0 −2.23607 + 2.23607i 0 0 0 −2.23607 2.23607i 0 7.00000i 0
143.2 0 2.23607 2.23607i 0 0 0 2.23607 + 2.23607i 0 7.00000i 0
207.1 0 −2.23607 2.23607i 0 0 0 −2.23607 + 2.23607i 0 7.00000i 0
207.2 0 2.23607 + 2.23607i 0 0 0 2.23607 2.23607i 0 7.00000i 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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.n.c 4
3.b odd 2 1 3600.2.x.f 4
4.b odd 2 1 inner 400.2.n.c 4
5.b even 2 1 inner 400.2.n.c 4
5.c odd 4 2 inner 400.2.n.c 4
8.b even 2 1 1600.2.n.q 4
8.d odd 2 1 1600.2.n.q 4
12.b even 2 1 3600.2.x.f 4
15.d odd 2 1 3600.2.x.f 4
15.e even 4 2 3600.2.x.f 4
20.d odd 2 1 CM 400.2.n.c 4
20.e even 4 2 inner 400.2.n.c 4
40.e odd 2 1 1600.2.n.q 4
40.f even 2 1 1600.2.n.q 4
40.i odd 4 2 1600.2.n.q 4
40.k even 4 2 1600.2.n.q 4
60.h even 2 1 3600.2.x.f 4
60.l odd 4 2 3600.2.x.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.n.c 4 1.a even 1 1 trivial
400.2.n.c 4 4.b odd 2 1 inner
400.2.n.c 4 5.b even 2 1 inner
400.2.n.c 4 5.c odd 4 2 inner
400.2.n.c 4 20.d odd 2 1 CM
400.2.n.c 4 20.e even 4 2 inner
1600.2.n.q 4 8.b even 2 1
1600.2.n.q 4 8.d odd 2 1
1600.2.n.q 4 40.e odd 2 1
1600.2.n.q 4 40.f even 2 1
1600.2.n.q 4 40.i odd 4 2
1600.2.n.q 4 40.k even 4 2
3600.2.x.f 4 3.b odd 2 1
3600.2.x.f 4 12.b even 2 1
3600.2.x.f 4 15.d odd 2 1
3600.2.x.f 4 15.e even 4 2
3600.2.x.f 4 60.h even 2 1
3600.2.x.f 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 100 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 100 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 100 \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 8100 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T + 12)^{4} \) Copy content Toggle raw display
$43$ \( T^{4} + 100 \) Copy content Toggle raw display
$47$ \( T^{4} + 8100 \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T - 8)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 62500 \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} + 8100 \) Copy content Toggle raw display
$89$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
show more
show less