Properties

Label 3120.1.dq.b
Level $3120$
Weight $1$
Character orbit 3120.dq
Analytic conductor $1.557$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
CM discriminant -39
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3120,1,Mod(2027,3120)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3120, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 2, 1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3120.2027");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3120.dq (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: \(1.55708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.9984000.1

$q$-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.

\(f(q)\) \(=\) \( q + i q^{2} + q^{3} - q^{4} - q^{5} + i q^{6} - i q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} + q^{3} - q^{4} - q^{5} + i q^{6} - i q^{8} + q^{9} - i q^{10} + (i + 1) q^{11} - q^{12} - i q^{13} - q^{15} + q^{16} + i q^{18} + q^{20} + (i - 1) q^{22} - i q^{24} + q^{25} + q^{26} + q^{27} - i q^{30} + i q^{32} + (i + 1) q^{33} - q^{36} - i q^{39} + i q^{40} + i q^{43} + ( - i - 1) q^{44} - q^{45} + (i + 1) q^{47} + q^{48} - i q^{49} + i q^{50} + i q^{52} + i q^{54} + ( - i - 1) q^{55} + ( - i + 1) q^{59} + q^{60} + (i + 1) q^{61} - q^{64} + i q^{65} + (i - 1) q^{66} - i q^{72} + q^{75} + q^{78} - q^{80} + q^{81} - q^{83} - 2 q^{86} + ( - i + 1) q^{88} - i q^{90} + (i - 1) q^{94} + i q^{96} + q^{98} + (i + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{4} - 2 q^{5} + 2 q^{9} + 2 q^{11} - 2 q^{12} - 2 q^{15} + 2 q^{16} + 2 q^{20} - 2 q^{22} + 2 q^{25} + 2 q^{26} + 2 q^{27} + 2 q^{33} - 2 q^{36} - 2 q^{44} - 2 q^{45} + 2 q^{47} + 2 q^{48} - 2 q^{55} + 2 q^{59} + 2 q^{60} + 2 q^{61} - 2 q^{64} - 2 q^{66} + 2 q^{75} + 2 q^{78} - 2 q^{80} + 2 q^{81} - 4 q^{83} - 4 q^{86} + 2 q^{88} - 2 q^{94} + 2 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3120\mathbb{Z}\right)^\times\).

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(-1\) \(-1\) \(-i\) \(-i\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2027.1
1.00000i
1.00000i
1.00000i 1.00000 −1.00000 −1.00000 1.00000i 0 1.00000i 1.00000 1.00000i
2963.1 1.00000i 1.00000 −1.00000 −1.00000 1.00000i 0 1.00000i 1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
39.d odd 2 1 CM by \(\Q(\sqrt{-39}) \)
80.s even 4 1 inner
3120.dq odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3120.1.dq.b yes 2
3.b odd 2 1 3120.1.dq.c yes 2
5.c odd 4 1 3120.1.dl.a 2
13.b even 2 1 3120.1.dq.c yes 2
15.e even 4 1 3120.1.dl.d yes 2
16.f odd 4 1 3120.1.dl.a 2
39.d odd 2 1 CM 3120.1.dq.b yes 2
48.k even 4 1 3120.1.dl.d yes 2
65.h odd 4 1 3120.1.dl.d yes 2
80.s even 4 1 inner 3120.1.dq.b yes 2
195.s even 4 1 3120.1.dl.a 2
208.o odd 4 1 3120.1.dl.d yes 2
240.z odd 4 1 3120.1.dq.c yes 2
624.v even 4 1 3120.1.dl.a 2
1040.cq even 4 1 3120.1.dq.c yes 2
3120.dq odd 4 1 inner 3120.1.dq.b yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3120.1.dl.a 2 5.c odd 4 1
3120.1.dl.a 2 16.f odd 4 1
3120.1.dl.a 2 195.s even 4 1
3120.1.dl.a 2 624.v even 4 1
3120.1.dl.d yes 2 15.e even 4 1
3120.1.dl.d yes 2 48.k even 4 1
3120.1.dl.d yes 2 65.h odd 4 1
3120.1.dl.d yes 2 208.o odd 4 1
3120.1.dq.b yes 2 1.a even 1 1 trivial
3120.1.dq.b yes 2 39.d odd 2 1 CM
3120.1.dq.b yes 2 80.s even 4 1 inner
3120.1.dq.b yes 2 3120.dq odd 4 1 inner
3120.1.dq.c yes 2 3.b odd 2 1
3120.1.dq.c yes 2 13.b even 2 1
3120.1.dq.c yes 2 240.z odd 4 1
3120.1.dq.c yes 2 1040.cq even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3120, [\chi])\):

\( T_{11}^{2} - 2T_{11} + 2 \) Copy content Toggle raw display
\( T_{41} \) Copy content Toggle raw display

Hecke characteristic polynomials

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