Properties

Label 3120.1.dq.e
Level $3120$
Weight $1$
Character orbit 3120.dq
Analytic conductor $1.557$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -39
Inner twists $8$

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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: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) 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 4.2.129792000.9

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{2} - q^{3} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{3} q^{5} + \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{2} - q^{3} + \zeta_{8}^{2} q^{4} + \zeta_{8}^{3} q^{5} + \zeta_{8} q^{6} - \zeta_{8}^{3} q^{8} + q^{9} + q^{10} - \zeta_{8}^{3} q^{11} - \zeta_{8}^{2} q^{12} + \zeta_{8}^{2} q^{13} - \zeta_{8}^{3} q^{15} - q^{16} - \zeta_{8} q^{18} - \zeta_{8} q^{20} - 2 q^{22} + \zeta_{8}^{3} q^{24} - \zeta_{8}^{2} q^{25} - \zeta_{8}^{3} q^{26} - q^{27} - q^{30} + \zeta_{8} q^{32} + 2 \zeta_{8}^{3} q^{33} + \zeta_{8}^{2} q^{36} - \zeta_{8}^{2} q^{39} + \zeta_{8}^{2} q^{40} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{41} + 2 \zeta_{8} q^{44} + \zeta_{8}^{3} q^{45} + q^{48} + \zeta_{8}^{2} q^{49} + \zeta_{8}^{3} q^{50} - q^{52} + \zeta_{8} q^{54} + 2 \zeta_{8}^{2} q^{55} - \zeta_{8} q^{59} + \zeta_{8} q^{60} + (\zeta_{8}^{2} - 1) q^{61} - \zeta_{8}^{2} q^{64} - \zeta_{8} q^{65} + 2 q^{66} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{71} - \zeta_{8}^{3} q^{72} + \zeta_{8}^{2} q^{75} + \zeta_{8}^{3} q^{78} + q^{79} - \zeta_{8}^{3} q^{80} + q^{81} + (\zeta_{8}^{2} - 1) q^{82} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{83} - 2 \zeta_{8}^{2} q^{88} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{89} + q^{90} - \zeta_{8} q^{96} - \zeta_{8}^{3} q^{98} - 2 \zeta_{8}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 4 q^{9} + 4 q^{10} - 4 q^{16} - 8 q^{22} - 4 q^{27} - 4 q^{30} + 4 q^{48} - 4 q^{52} - 4 q^{61} + 8 q^{66} + 8 q^{79} + 4 q^{81} - 4 q^{82} + 4 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1951\) \(2081\) \(2341\) \(2497\) \(2641\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{8}^{2}\) \(\zeta_{8}^{2}\) \(-1\)

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
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
−0.707107 0.707107i −1.00000 1.00000i −0.707107 + 0.707107i 0.707107 + 0.707107i 0 0.707107 0.707107i 1.00000 1.00000
2027.2 0.707107 + 0.707107i −1.00000 1.00000i 0.707107 0.707107i −0.707107 0.707107i 0 −0.707107 + 0.707107i 1.00000 1.00000
2963.1 −0.707107 + 0.707107i −1.00000 1.00000i −0.707107 0.707107i 0.707107 0.707107i 0 0.707107 + 0.707107i 1.00000 1.00000
2963.2 0.707107 0.707107i −1.00000 1.00000i 0.707107 + 0.707107i −0.707107 + 0.707107i 0 −0.707107 0.707107i 1.00000 1.00000
\(n\): 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 \(\Q(\sqrt{-39}) \)
3.b odd 2 1 inner
13.b even 2 1 inner
80.s even 4 1 inner
240.z odd 4 1 inner
1040.cq 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.e yes 4
3.b odd 2 1 inner 3120.1.dq.e yes 4
5.c odd 4 1 3120.1.dl.e 4
13.b even 2 1 inner 3120.1.dq.e yes 4
15.e even 4 1 3120.1.dl.e 4
16.f odd 4 1 3120.1.dl.e 4
39.d odd 2 1 CM 3120.1.dq.e yes 4
48.k even 4 1 3120.1.dl.e 4
65.h odd 4 1 3120.1.dl.e 4
80.s even 4 1 inner 3120.1.dq.e yes 4
195.s even 4 1 3120.1.dl.e 4
208.o odd 4 1 3120.1.dl.e 4
240.z odd 4 1 inner 3120.1.dq.e yes 4
624.v even 4 1 3120.1.dl.e 4
1040.cq even 4 1 inner 3120.1.dq.e yes 4
3120.dq odd 4 1 inner 3120.1.dq.e yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3120.1.dl.e 4 5.c odd 4 1
3120.1.dl.e 4 15.e even 4 1
3120.1.dl.e 4 16.f odd 4 1
3120.1.dl.e 4 48.k even 4 1
3120.1.dl.e 4 65.h odd 4 1
3120.1.dl.e 4 195.s even 4 1
3120.1.dl.e 4 208.o odd 4 1
3120.1.dl.e 4 624.v even 4 1
3120.1.dq.e yes 4 1.a even 1 1 trivial
3120.1.dq.e yes 4 3.b odd 2 1 inner
3120.1.dq.e yes 4 13.b even 2 1 inner
3120.1.dq.e yes 4 39.d odd 2 1 CM
3120.1.dq.e yes 4 80.s even 4 1 inner
3120.1.dq.e yes 4 240.z odd 4 1 inner
3120.1.dq.e yes 4 1040.cq even 4 1 inner
3120.1.dq.e yes 4 3120.dq odd 4 1 inner

Hecke kernels

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

\( T_{11}^{4} + 16 \) Copy content Toggle raw display
\( T_{41}^{2} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 16 \) Copy content Toggle raw display
$61$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T - 2)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} \) Copy content Toggle raw display
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