Properties

Label 3360.1.bn.b
Level $3360$
Weight $1$
Character orbit 3360.bn
Analytic conductor $1.677$
Analytic rank $0$
Dimension $4$
Projective image $D_{4}$
CM discriminant -84
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3360,1,Mod(1217,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("3360.1217");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3360 = 2^{5} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3360.bn (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: \(1.67685844245\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.294000.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{8} q^{3} + \zeta_{8} q^{5} - \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{8} q^{3} + \zeta_{8} q^{5} - \zeta_{8}^{3} q^{7} + \zeta_{8}^{2} q^{9} - \zeta_{8}^{2} q^{15} - \zeta_{8} q^{17} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{19} - q^{21} + ( - \zeta_{8}^{2} - 1) q^{23} + \zeta_{8}^{2} q^{25} - \zeta_{8}^{3} q^{27} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{31} + q^{35} + (\zeta_{8}^{2} + 1) q^{37} + ( - \zeta_{8}^{3} + \zeta_{8}) q^{41} + \zeta_{8}^{3} q^{45} - \zeta_{8}^{2} q^{49} + 2 \zeta_{8}^{2} q^{51} + ( - \zeta_{8}^{2} - 1) q^{57} + \zeta_{8} q^{63} + (\zeta_{8}^{3} + \zeta_{8}) q^{69} + \zeta_{8}^{2} q^{71} - \zeta_{8}^{3} q^{75} - q^{81} - 2 \zeta_{8}^{2} q^{85} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{89} + (\zeta_{8}^{2} - 1) q^{93} + (\zeta_{8}^{2} + 1) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{21} - 4 q^{23} + 4 q^{35} + 4 q^{37} - 4 q^{57} - 4 q^{81} - 4 q^{93} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3360\mathbb{Z}\right)^\times\).

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(-1\) \(\zeta_{8}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1217.1
0.707107 + 0.707107i
−0.707107 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 −0.707107 0.707107i 0 0.707107 + 0.707107i 0 0.707107 0.707107i 0 1.00000i 0
1217.2 0 0.707107 + 0.707107i 0 −0.707107 0.707107i 0 −0.707107 + 0.707107i 0 1.00000i 0
3233.1 0 −0.707107 + 0.707107i 0 0.707107 0.707107i 0 0.707107 + 0.707107i 0 1.00000i 0
3233.2 0 0.707107 0.707107i 0 −0.707107 + 0.707107i 0 −0.707107 0.707107i 0 1.00000i 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
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
7.b odd 2 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
105.k odd 4 1 inner
140.j odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.bn.b 4
3.b odd 2 1 3360.1.bn.c yes 4
4.b odd 2 1 3360.1.bn.c yes 4
5.c odd 4 1 3360.1.bn.c yes 4
7.b odd 2 1 inner 3360.1.bn.b 4
12.b even 2 1 inner 3360.1.bn.b 4
15.e even 4 1 inner 3360.1.bn.b 4
20.e even 4 1 inner 3360.1.bn.b 4
21.c even 2 1 3360.1.bn.c yes 4
28.d even 2 1 3360.1.bn.c yes 4
35.f even 4 1 3360.1.bn.c yes 4
60.l odd 4 1 3360.1.bn.c yes 4
84.h odd 2 1 CM 3360.1.bn.b 4
105.k odd 4 1 inner 3360.1.bn.b 4
140.j odd 4 1 inner 3360.1.bn.b 4
420.w even 4 1 3360.1.bn.c yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.1.bn.b 4 1.a even 1 1 trivial
3360.1.bn.b 4 7.b odd 2 1 inner
3360.1.bn.b 4 12.b even 2 1 inner
3360.1.bn.b 4 15.e even 4 1 inner
3360.1.bn.b 4 20.e even 4 1 inner
3360.1.bn.b 4 84.h odd 2 1 CM
3360.1.bn.b 4 105.k odd 4 1 inner
3360.1.bn.b 4 140.j odd 4 1 inner
3360.1.bn.c yes 4 3.b odd 2 1
3360.1.bn.c yes 4 4.b odd 2 1
3360.1.bn.c yes 4 5.c odd 4 1
3360.1.bn.c yes 4 21.c even 2 1
3360.1.bn.c yes 4 28.d even 2 1
3360.1.bn.c yes 4 35.f even 4 1
3360.1.bn.c yes 4 60.l odd 4 1
3360.1.bn.c yes 4 420.w even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3360, [\chi])\):

\( T_{11} \) Copy content Toggle raw display
\( T_{23}^{2} + 2T_{23} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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