Properties

Label 1920.1.i.c
Level $1920$
Weight $1$
Character orbit 1920.i
Self dual yes
Analytic conductor $0.958$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -20, -120, 24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1920,1,Mod(449,1920)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1920, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1920.449");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1920.i (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(0.958204824255\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{6})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.38400.3
Stark unit: Root of $x^{4} - 8116x^{3} + 13350x^{2} - 8116x + 1$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{3} - q^{5} + q^{9} - q^{15} + 2 q^{23} + q^{25} + q^{27} + 2 q^{29} - 2 q^{43} - q^{45} - 2 q^{47} + q^{49} - 2 q^{67} + 2 q^{69} + q^{75} + q^{81} + 2 q^{87}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(641\) \(901\) \(1537\)
\(\chi(n)\) \(0\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0
0 1.00000 0 −1.00000 0 0 0 1.00000 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 \(\Q(\sqrt{-5}) \)
24.f even 2 1 RM by \(\Q(\sqrt{6}) \)
120.i odd 2 1 CM by \(\Q(\sqrt{-30}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1920.1.i.c yes 1
3.b odd 2 1 1920.1.i.d yes 1
4.b odd 2 1 1920.1.i.a 1
5.b even 2 1 1920.1.i.a 1
8.b even 2 1 1920.1.i.b yes 1
8.d odd 2 1 1920.1.i.d yes 1
12.b even 2 1 1920.1.i.b yes 1
15.d odd 2 1 1920.1.i.b yes 1
16.e even 4 2 3840.1.c.f 2
16.f odd 4 2 3840.1.c.g 2
20.d odd 2 1 CM 1920.1.i.c yes 1
24.f even 2 1 RM 1920.1.i.c yes 1
24.h odd 2 1 1920.1.i.a 1
40.e odd 2 1 1920.1.i.b yes 1
40.f even 2 1 1920.1.i.d yes 1
48.i odd 4 2 3840.1.c.g 2
48.k even 4 2 3840.1.c.f 2
60.h even 2 1 1920.1.i.d yes 1
80.k odd 4 2 3840.1.c.f 2
80.q even 4 2 3840.1.c.g 2
120.i odd 2 1 CM 1920.1.i.c yes 1
120.m even 2 1 1920.1.i.a 1
240.t even 4 2 3840.1.c.g 2
240.bm odd 4 2 3840.1.c.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1920.1.i.a 1 4.b odd 2 1
1920.1.i.a 1 5.b even 2 1
1920.1.i.a 1 24.h odd 2 1
1920.1.i.a 1 120.m even 2 1
1920.1.i.b yes 1 8.b even 2 1
1920.1.i.b yes 1 12.b even 2 1
1920.1.i.b yes 1 15.d odd 2 1
1920.1.i.b yes 1 40.e odd 2 1
1920.1.i.c yes 1 1.a even 1 1 trivial
1920.1.i.c yes 1 20.d odd 2 1 CM
1920.1.i.c yes 1 24.f even 2 1 RM
1920.1.i.c yes 1 120.i odd 2 1 CM
1920.1.i.d yes 1 3.b odd 2 1
1920.1.i.d yes 1 8.d odd 2 1
1920.1.i.d yes 1 40.f even 2 1
1920.1.i.d yes 1 60.h even 2 1
3840.1.c.f 2 16.e even 4 2
3840.1.c.f 2 48.k even 4 2
3840.1.c.f 2 80.k odd 4 2
3840.1.c.f 2 240.bm odd 4 2
3840.1.c.g 2 16.f odd 4 2
3840.1.c.g 2 48.i odd 4 2
3840.1.c.g 2 80.q even 4 2
3840.1.c.g 2 240.t even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1920, [\chi])\):

\( T_{7} \) Copy content Toggle raw display
\( T_{23} - 2 \) Copy content Toggle raw display
\( T_{29} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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