↑

Properties

Label 873.1.br.a
Level $873$
Weight $1$
Character orbit 873.br
Analytic conductor $0.436$
Analytic rank $0$
Dimension $16$
Projective image $D_{32}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [873,1,Mod(19,873)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(873, base_ring=CyclotomicField(32))
 
chi = DirichletCharacter(H, H._module([0, 27]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("873.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 873 = 3^{2} \cdot 97 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 873.br (of order \(32\), degree \(16\), 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: \(0.435683756029\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{32}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{32} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{32}^{6} q^{4} + (\zeta_{32}^{4} + \zeta_{32}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{32}^{6} q^{4} + (\zeta_{32}^{4} + \zeta_{32}) q^{7} + (\zeta_{32}^{14} - \zeta_{32}^{5}) q^{13} + \zeta_{32}^{12} q^{16} + ( - \zeta_{32}^{8} - \zeta_{32}^{3}) q^{19} + \zeta_{32}^{11} q^{25} + (\zeta_{32}^{10} + \zeta_{32}^{7}) q^{28} + ( - \zeta_{32}^{7} + \zeta_{32}^{3}) q^{31} + ( - \zeta_{32}^{13} - \zeta_{32}^{12}) q^{37} + ( - \zeta_{32}^{15} + \zeta_{32}^{13}) q^{43} + (\zeta_{32}^{8} + \cdots + \zeta_{32}^{2}) q^{49}+ \cdots - \zeta_{32}^{11} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{79}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/873\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(389\)
\(\chi(n)\) \(-\zeta_{32}^{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
0.980785 + 0.195090i
−0.831470 − 0.555570i
0.980785 − 0.195090i
0.195090 + 0.980785i
0.195090 − 0.980785i
0.555570 + 0.831470i
−0.555570 + 0.831470i
−0.831470 + 0.555570i
0.831470 − 0.555570i
0.555570 − 0.831470i
−0.555570 − 0.831470i
−0.195090 + 0.980785i
−0.195090 − 0.980785i
−0.980785 + 0.195090i
0.831470 + 0.555570i
−0.980785 − 0.195090i
0 0 0.382683 + 0.923880i 0 0 1.68789 + 0.902197i 0 0 0
28.1 0 0 −0.923880 − 0.382683i 0 0 −1.53858 + 0.151537i 0 0 0
46.1 0 0 0.382683 − 0.923880i 0 0 1.68789 − 0.902197i 0 0 0
55.1 0 0 −0.382683 + 0.923880i 0 0 0.902197 + 0.273678i 0 0 0
127.1 0 0 −0.382683 − 0.923880i 0 0 0.902197 − 0.273678i 0 0 0
271.1 0 0 0.923880 − 0.382683i 0 0 −0.151537 + 0.124363i 0 0 0
325.1 0 0 0.923880 + 0.382683i 0 0 −1.26268 + 1.53858i 0 0 0
343.1 0 0 −0.923880 + 0.382683i 0 0 −1.53858 − 0.151537i 0 0 0
433.1 0 0 −0.923880 + 0.382683i 0 0 0.124363 − 1.26268i 0 0 0
451.1 0 0 0.923880 + 0.382683i 0 0 −0.151537 − 0.124363i 0 0 0
505.1 0 0 0.923880 − 0.382683i 0 0 −1.26268 − 1.53858i 0 0 0
649.1 0 0 −0.382683 − 0.923880i 0 0 0.512016 + 1.68789i 0 0 0
721.1 0 0 −0.382683 + 0.923880i 0 0 0.512016 − 1.68789i 0 0 0
730.1 0 0 0.382683 − 0.923880i 0 0 −0.273678 − 0.512016i 0 0 0
748.1 0 0 −0.923880 − 0.382683i 0 0 0.124363 + 1.26268i 0 0 0
757.1 0 0 0.382683 + 0.923880i 0 0 −0.273678 + 0.512016i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
97.j odd 32 1 inner
291.s even 32 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 873.1.br.a ✓ 16
3.b odd 2 1 CM 873.1.br.a ✓ 16
97.j odd 32 1 inner 873.1.br.a ✓ 16
291.s even 32 1 inner 873.1.br.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
873.1.br.a ✓ 16 1.a even 1 1 trivial
873.1.br.a ✓ 16 3.b odd 2 1 CM
873.1.br.a ✓ 16 97.j odd 32 1 inner
873.1.br.a ✓ 16 291.s even 32 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(873, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 4 T^{12} + \cdots + 2 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} + 40 T^{10} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} + 8 T^{14} + \cdots + 2 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} - 16 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{16} + 4 T^{12} + \cdots + 2 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} - 16 T^{10} + \cdots + 4 \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} \) Copy content Toggle raw display
$59$ \( T^{16} \) Copy content Toggle raw display
$61$ \( (T^{8} - 8 T^{6} + 20 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} - 32 T^{11} + \cdots + 2 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( (T^{8} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 8 T^{7} + 28 T^{6} + \cdots + 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + 1 \) Copy content Toggle raw display
show more
show less