↑

Properties

Label 1840.1.bq.a
Level $1840$
Weight $1$
Character orbit 1840.bq
Analytic conductor $0.918$
Analytic rank $0$
Dimension $20$
Projective image $D_{22}$
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] = [1840,1,Mod(239,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 0, 11, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.239");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1840.bq (of order \(22\), degree \(10\), 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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{22})\)
Coefficient field: \(\Q(\zeta_{44})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{22}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{22} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + ( - \zeta_{44}^{9} - \zeta_{44}^{7}) q^{3} + \zeta_{44}^{6} q^{5} + ( - \zeta_{44}^{19} + \zeta_{44}^{11}) q^{7} + (\zeta_{44}^{18} + \cdots + \zeta_{44}^{14}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{44}^{9} - \zeta_{44}^{7}) q^{3} + \zeta_{44}^{6} q^{5} + ( - \zeta_{44}^{19} + \zeta_{44}^{11}) q^{7} + (\zeta_{44}^{18} + \cdots + \zeta_{44}^{14}) q^{9}+ \cdots + (\zeta_{44}^{20} - \zeta_{44}^{18}) q^{89}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 2 q^{5} + 2 q^{9} - 2 q^{25} - 4 q^{29} + 4 q^{41} - 24 q^{45} - 20 q^{49} + 4 q^{61} - 2 q^{81} - 4 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{44}^{12}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1
−0.281733 − 0.959493i
0.281733 + 0.959493i
0.989821 + 0.142315i
−0.989821 − 0.142315i
−0.281733 + 0.959493i
0.281733 − 0.959493i
−0.909632 − 0.415415i
0.909632 + 0.415415i
0.755750 + 0.654861i
−0.755750 − 0.654861i
−0.909632 + 0.415415i
0.909632 − 0.415415i
0.989821 − 0.142315i
−0.989821 + 0.142315i
−0.540641 + 0.841254i
0.540641 − 0.841254i
0.755750 − 0.654861i
−0.755750 + 0.654861i
−0.540641 − 0.841254i
0.540641 + 0.841254i
0 −0.368991 − 0.425839i 0 0.142315 + 0.989821i 0 0.755750 − 1.65486i 0 0.0971309 − 0.675560i 0
239.2 0 0.368991 + 0.425839i 0 0.142315 + 0.989821i 0 −0.755750 + 1.65486i 0 0.0971309 − 0.675560i 0
399.1 0 −0.822373 − 1.80075i 0 0.654861 + 0.755750i 0 0.909632 + 0.584585i 0 −1.91153 + 2.20602i 0
399.2 0 0.822373 + 1.80075i 0 0.654861 + 0.755750i 0 −0.909632 − 0.584585i 0 −1.91153 + 2.20602i 0
639.1 0 −0.368991 + 0.425839i 0 0.142315 − 0.989821i 0 0.755750 + 1.65486i 0 0.0971309 + 0.675560i 0
639.2 0 0.368991 − 0.425839i 0 0.142315 − 0.989821i 0 −0.755750 − 1.65486i 0 0.0971309 + 0.675560i 0
719.1 0 −1.74557 − 0.512546i 0 −0.841254 + 0.540641i 0 −0.281733 + 1.95949i 0 1.94306 + 1.24873i 0
719.2 0 1.74557 + 0.512546i 0 −0.841254 + 0.540641i 0 0.281733 − 1.95949i 0 1.94306 + 1.24873i 0
959.1 0 −1.27155 + 0.817178i 0 −0.415415 − 0.909632i 0 −0.540641 + 0.158746i 0 0.533654 − 1.16854i 0
959.2 0 1.27155 − 0.817178i 0 −0.415415 − 0.909632i 0 0.540641 − 0.158746i 0 0.533654 − 1.16854i 0
1039.1 0 −1.74557 + 0.512546i 0 −0.841254 − 0.540641i 0 −0.281733 − 1.95949i 0 1.94306 − 1.24873i 0
1039.2 0 1.74557 − 0.512546i 0 −0.841254 − 0.540641i 0 0.281733 + 1.95949i 0 1.94306 − 1.24873i 0
1199.1 0 −0.822373 + 1.80075i 0 0.654861 − 0.755750i 0 0.909632 − 0.584585i 0 −1.91153 − 2.20602i 0
1199.2 0 0.822373 − 1.80075i 0 0.654861 − 0.755750i 0 −0.909632 + 0.584585i 0 −1.91153 − 2.20602i 0
1359.1 0 −0.153882 − 1.07028i 0 0.959493 + 0.281733i 0 0.989821 − 1.14231i 0 −0.162317 + 0.0476607i 0
1359.2 0 0.153882 + 1.07028i 0 0.959493 + 0.281733i 0 −0.989821 + 1.14231i 0 −0.162317 + 0.0476607i 0
1439.1 0 −1.27155 − 0.817178i 0 −0.415415 + 0.909632i 0 −0.540641 − 0.158746i 0 0.533654 + 1.16854i 0
1439.2 0 1.27155 + 0.817178i 0 −0.415415 + 0.909632i 0 0.540641 + 0.158746i 0 0.533654 + 1.16854i 0
1599.1 0 −0.153882 + 1.07028i 0 0.959493 − 0.281733i 0 0.989821 + 1.14231i 0 −0.162317 − 0.0476607i 0
1599.2 0 0.153882 − 1.07028i 0 0.959493 − 0.281733i 0 −0.989821 − 1.14231i 0 −0.162317 − 0.0476607i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.2
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
23.c even 11 1 inner
92.g odd 22 1 inner
115.j even 22 1 inner
460.n odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1840.1.bq.a ✓ 20
4.b odd 2 1 inner 1840.1.bq.a ✓ 20
5.b even 2 1 inner 1840.1.bq.a ✓ 20
20.d odd 2 1 CM 1840.1.bq.a ✓ 20
23.c even 11 1 inner 1840.1.bq.a ✓ 20
92.g odd 22 1 inner 1840.1.bq.a ✓ 20
115.j even 22 1 inner 1840.1.bq.a ✓ 20
460.n odd 22 1 inner 1840.1.bq.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1840.1.bq.a ✓ 20 1.a even 1 1 trivial
1840.1.bq.a ✓ 20 4.b odd 2 1 inner
1840.1.bq.a ✓ 20 5.b even 2 1 inner
1840.1.bq.a ✓ 20 20.d odd 2 1 CM
1840.1.bq.a ✓ 20 23.c even 11 1 inner
1840.1.bq.a ✓ 20 92.g odd 22 1 inner
1840.1.bq.a ✓ 20 115.j even 22 1 inner
1840.1.bq.a ✓ 20 460.n odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1840, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} - 22 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( (T^{10} - T^{9} + T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{20} + 11 T^{18} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( T^{20} \) Copy content Toggle raw display
$13$ \( T^{20} \) Copy content Toggle raw display
$17$ \( T^{20} \) Copy content Toggle raw display
$19$ \( T^{20} \) Copy content Toggle raw display
$23$ \( T^{20} - T^{18} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} \) Copy content Toggle raw display
$37$ \( T^{20} \) Copy content Toggle raw display
$41$ \( (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + 55 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$47$ \( (T^{10} - 11 T^{8} + \cdots - 11)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} \) Copy content Toggle raw display
$59$ \( T^{20} \) Copy content Toggle raw display
$61$ \( (T^{10} - 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + 55 T^{14} + \cdots + 121 \) Copy content Toggle raw display
$71$ \( T^{20} \) Copy content Toggle raw display
$73$ \( T^{20} \) Copy content Toggle raw display
$79$ \( T^{20} \) Copy content Toggle raw display
$83$ \( T^{20} + 22 T^{12} + \cdots + 121 \) Copy content Toggle raw display
$89$ \( (T^{10} + 2 T^{9} + 4 T^{8} + \cdots + 1)^{2} \) Copy content Toggle raw display
$97$ \( T^{20} \) Copy content Toggle raw display
show more
show less