↑

Properties

Label 723.1.bc.a
Level $723$
Weight $1$
Character orbit 723.bc
Analytic conductor $0.361$
Analytic rank $0$
Dimension $16$
Projective image $D_{40}$
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] = [723,1,Mod(5,723)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(723, base_ring=CyclotomicField(40))
 
chi = DirichletCharacter(H, H._module([20, 23]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("723.5");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 723 = 3 \cdot 241 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 723.bc (of order \(40\), 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: \(0.360824004134\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\Q(\zeta_{40})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{12} + x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{40}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{40} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{40}^{13} q^{3} + \zeta_{40}^{10} q^{4} + ( - \zeta_{40}^{2} - \zeta_{40}) q^{7} - \zeta_{40}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{40}^{13} q^{3} + \zeta_{40}^{10} q^{4} + ( - \zeta_{40}^{2} - \zeta_{40}) q^{7} - \zeta_{40}^{6} q^{9} - \zeta_{40}^{3} q^{12} + (\zeta_{40}^{15} + \zeta_{40}^{6}) q^{13} - q^{16} + ( - \zeta_{40}^{18} + \zeta_{40}^{17}) q^{19} + ( - \zeta_{40}^{15} - \zeta_{40}^{14}) q^{21} + \zeta_{40}^{14} q^{25} - \zeta_{40}^{19} q^{27} + ( - \zeta_{40}^{12} - \zeta_{40}^{11}) q^{28} + ( - \zeta_{40}^{9} - \zeta_{40}^{4}) q^{31} - \zeta_{40}^{16} q^{36} + (\zeta_{40}^{18} + \zeta_{40}) q^{37} + (\zeta_{40}^{19} - \zeta_{40}^{8}) q^{39} + (\zeta_{40}^{12} + \zeta_{40}^{5}) q^{43} - \zeta_{40}^{13} q^{48} + (\zeta_{40}^{4} + \cdots + \zeta_{40}^{2}) q^{49} + \cdots + (\zeta_{40}^{19} - \zeta_{40}^{13}) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{16} - 4 q^{28} - 4 q^{31} + 4 q^{36} + 4 q^{39} + 4 q^{43} + 4 q^{49} - 4 q^{52} - 4 q^{63} - 4 q^{73} - 4 q^{76} + 4 q^{81} + 4 q^{84} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/723\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(242\)
\(\chi(n)\) \(\zeta_{40}^{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−0.987688 − 0.156434i
−0.453990 − 0.891007i
0.453990 − 0.891007i
0.891007 + 0.453990i
−0.891007 − 0.453990i
−0.453990 + 0.891007i
0.453990 + 0.891007i
0.987688 + 0.156434i
−0.156434 − 0.987688i
0.156434 − 0.987688i
−0.987688 + 0.156434i
0.891007 − 0.453990i
−0.891007 + 0.453990i
0.987688 − 0.156434i
−0.156434 + 0.987688i
0.156434 + 0.987688i
0 0.453990 − 0.891007i 1.00000i 0 0 0.0366318 − 0.152583i 0 −0.587785 − 0.809017i 0
41.1 0 0.156434 − 0.987688i − 1.00000i 0 0 1.04178 + 0.0819895i 0 −0.951057 − 0.309017i 0
47.1 0 −0.156434 − 0.987688i 1.00000i 0 0 0.133795 + 1.70002i 0 −0.951057 + 0.309017i 0
116.1 0 0.987688 − 0.156434i − 1.00000i 0 0 −1.47879 − 1.26301i 0 0.951057 − 0.309017i 0
125.1 0 −0.987688 + 0.156434i − 1.00000i 0 0 0.303221 − 0.355026i 0 0.951057 − 0.309017i 0
194.1 0 0.156434 + 0.987688i 1.00000i 0 0 1.04178 − 0.0819895i 0 −0.951057 + 0.309017i 0
200.1 0 −0.156434 + 0.987688i − 1.00000i 0 0 0.133795 − 1.70002i 0 −0.951057 − 0.309017i 0
236.1 0 −0.453990 + 0.891007i 1.00000i 0 0 −1.93874 − 0.465451i 0 −0.587785 − 0.809017i 0
302.1 0 −0.891007 + 0.453990i 1.00000i 0 0 1.10749 + 0.678671i 0 0.587785 − 0.809017i 0
320.1 0 0.891007 + 0.453990i − 1.00000i 0 0 0.794622 + 1.29671i 0 0.587785 + 0.809017i 0
434.1 0 0.453990 + 0.891007i − 1.00000i 0 0 0.0366318 + 0.152583i 0 −0.587785 + 0.809017i 0
455.1 0 0.987688 + 0.156434i 1.00000i 0 0 −1.47879 + 1.26301i 0 0.951057 + 0.309017i 0
509.1 0 −0.987688 − 0.156434i 1.00000i 0 0 0.303221 + 0.355026i 0 0.951057 + 0.309017i 0
530.1 0 −0.453990 − 0.891007i − 1.00000i 0 0 −1.93874 + 0.465451i 0 −0.587785 + 0.809017i 0
644.1 0 −0.891007 − 0.453990i − 1.00000i 0 0 1.10749 − 0.678671i 0 0.587785 + 0.809017i 0
662.1 0 0.891007 − 0.453990i 1.00000i 0 0 0.794622 − 1.29671i 0 0.587785 − 0.809017i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
241.o even 40 1 inner
723.bc odd 40 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 723.1.bc.a ✓ 16
3.b odd 2 1 CM 723.1.bc.a ✓ 16
241.o even 40 1 inner 723.1.bc.a ✓ 16
723.bc odd 40 1 inner 723.1.bc.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
723.1.bc.a ✓ 16 1.a even 1 1 trivial
723.1.bc.a ✓ 16 3.b odd 2 1 CM
723.1.bc.a ✓ 16 241.o even 40 1 inner
723.1.bc.a ✓ 16 723.bc odd 40 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(723, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} - T^{12} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{16} \) Copy content Toggle raw display
$13$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} \) Copy content Toggle raw display
$31$ \( T^{16} + 4 T^{15} + \cdots + 16 \) Copy content Toggle raw display
$37$ \( T^{16} - 2 T^{14} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} - 4 T^{15} + \cdots + 1 \) 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
$61$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$67$ \( T^{16} - 20 T^{12} + \cdots + 625 \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{16} - 11 T^{12} + \cdots + 1 \) Copy content Toggle raw display
$83$ \( T^{16} \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} - 4 T^{14} + \cdots + 1 \) Copy content Toggle raw display
show more
show less