Properties

Label 3360.1.cv.c
Level $3360$
Weight $1$
Character orbit 3360.cv
Analytic conductor $1.677$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -56
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(1553,3360)]
 
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, 2, 2, 3, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3360.1553");
 
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.cv (of order \(4\), degree \(2\), not 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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 840)
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.1778112000000.2

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{16} q^{3} - \zeta_{16} q^{5} - \zeta_{16}^{6} q^{7} + \zeta_{16}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{16} q^{3} - \zeta_{16} q^{5} - \zeta_{16}^{6} q^{7} + \zeta_{16}^{2} q^{9} + ( - \zeta_{16}^{7} + \zeta_{16}^{5}) q^{13} + \zeta_{16}^{2} q^{15} + (\zeta_{16}^{5} + \zeta_{16}^{3}) q^{19} + \zeta_{16}^{7} q^{21} + (\zeta_{16}^{4} - 1) q^{23} + \zeta_{16}^{2} q^{25} - \zeta_{16}^{3} q^{27} + \zeta_{16}^{7} q^{35} + ( - \zeta_{16}^{6} - 1) q^{39} - \zeta_{16}^{3} q^{45} - \zeta_{16}^{4} q^{49} + ( - \zeta_{16}^{6} - \zeta_{16}^{4}) q^{57} + ( - \zeta_{16}^{7} + \zeta_{16}) q^{59} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{61} + q^{63} + ( - \zeta_{16}^{6} - 1) q^{65} + ( - \zeta_{16}^{5} + \zeta_{16}) q^{69} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{71} - \zeta_{16}^{3} q^{75} + (\zeta_{16}^{6} + \zeta_{16}^{2}) q^{79} + \zeta_{16}^{4} q^{81} + ( - \zeta_{16}^{7} - \zeta_{16}^{5}) q^{83} + ( - \zeta_{16}^{5} + \zeta_{16}^{3}) q^{91} + ( - \zeta_{16}^{6} - \zeta_{16}^{4}) q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{23} - 8 q^{39} + 8 q^{63} - 8 q^{65}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(-\zeta_{16}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1553.1
0.923880 + 0.382683i
0.382683 0.923880i
−0.382683 + 0.923880i
−0.923880 0.382683i
0.923880 0.382683i
0.382683 + 0.923880i
−0.382683 0.923880i
−0.923880 + 0.382683i
0 −0.923880 0.382683i 0 −0.923880 0.382683i 0 0.707107 0.707107i 0 0.707107 + 0.707107i 0
1553.2 0 −0.382683 + 0.923880i 0 −0.382683 + 0.923880i 0 −0.707107 + 0.707107i 0 −0.707107 0.707107i 0
1553.3 0 0.382683 0.923880i 0 0.382683 0.923880i 0 −0.707107 + 0.707107i 0 −0.707107 0.707107i 0
1553.4 0 0.923880 + 0.382683i 0 0.923880 + 0.382683i 0 0.707107 0.707107i 0 0.707107 + 0.707107i 0
2897.1 0 −0.923880 + 0.382683i 0 −0.923880 + 0.382683i 0 0.707107 + 0.707107i 0 0.707107 0.707107i 0
2897.2 0 −0.382683 0.923880i 0 −0.382683 0.923880i 0 −0.707107 0.707107i 0 −0.707107 + 0.707107i 0
2897.3 0 0.382683 + 0.923880i 0 0.382683 + 0.923880i 0 −0.707107 0.707107i 0 −0.707107 + 0.707107i 0
2897.4 0 0.923880 0.382683i 0 0.923880 0.382683i 0 0.707107 + 0.707107i 0 0.707107 0.707107i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1553.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
7.b odd 2 1 inner
8.b even 2 1 inner
15.e even 4 1 inner
105.k odd 4 1 inner
120.w even 4 1 inner
840.bp 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.cv.c 8
3.b odd 2 1 3360.1.cv.d 8
4.b odd 2 1 840.1.bp.c 8
5.c odd 4 1 3360.1.cv.d 8
7.b odd 2 1 inner 3360.1.cv.c 8
8.b even 2 1 inner 3360.1.cv.c 8
8.d odd 2 1 840.1.bp.c 8
12.b even 2 1 840.1.bp.d yes 8
15.e even 4 1 inner 3360.1.cv.c 8
20.e even 4 1 840.1.bp.d yes 8
21.c even 2 1 3360.1.cv.d 8
24.f even 2 1 840.1.bp.d yes 8
24.h odd 2 1 3360.1.cv.d 8
28.d even 2 1 840.1.bp.c 8
35.f even 4 1 3360.1.cv.d 8
40.i odd 4 1 3360.1.cv.d 8
40.k even 4 1 840.1.bp.d yes 8
56.e even 2 1 840.1.bp.c 8
56.h odd 2 1 CM 3360.1.cv.c 8
60.l odd 4 1 840.1.bp.c 8
84.h odd 2 1 840.1.bp.d yes 8
105.k odd 4 1 inner 3360.1.cv.c 8
120.q odd 4 1 840.1.bp.c 8
120.w even 4 1 inner 3360.1.cv.c 8
140.j odd 4 1 840.1.bp.d yes 8
168.e odd 2 1 840.1.bp.d yes 8
168.i even 2 1 3360.1.cv.d 8
280.s even 4 1 3360.1.cv.d 8
280.y odd 4 1 840.1.bp.d yes 8
420.w even 4 1 840.1.bp.c 8
840.bm even 4 1 840.1.bp.c 8
840.bp odd 4 1 inner 3360.1.cv.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
840.1.bp.c 8 4.b odd 2 1
840.1.bp.c 8 8.d odd 2 1
840.1.bp.c 8 28.d even 2 1
840.1.bp.c 8 56.e even 2 1
840.1.bp.c 8 60.l odd 4 1
840.1.bp.c 8 120.q odd 4 1
840.1.bp.c 8 420.w even 4 1
840.1.bp.c 8 840.bm even 4 1
840.1.bp.d yes 8 12.b even 2 1
840.1.bp.d yes 8 20.e even 4 1
840.1.bp.d yes 8 24.f even 2 1
840.1.bp.d yes 8 40.k even 4 1
840.1.bp.d yes 8 84.h odd 2 1
840.1.bp.d yes 8 140.j odd 4 1
840.1.bp.d yes 8 168.e odd 2 1
840.1.bp.d yes 8 280.y odd 4 1
3360.1.cv.c 8 1.a even 1 1 trivial
3360.1.cv.c 8 7.b odd 2 1 inner
3360.1.cv.c 8 8.b even 2 1 inner
3360.1.cv.c 8 15.e even 4 1 inner
3360.1.cv.c 8 56.h odd 2 1 CM
3360.1.cv.c 8 105.k odd 4 1 inner
3360.1.cv.c 8 120.w even 4 1 inner
3360.1.cv.c 8 840.bp odd 4 1 inner
3360.1.cv.d 8 3.b odd 2 1
3360.1.cv.d 8 5.c odd 4 1
3360.1.cv.d 8 21.c even 2 1
3360.1.cv.d 8 24.h odd 2 1
3360.1.cv.d 8 35.f even 4 1
3360.1.cv.d 8 40.i odd 4 1
3360.1.cv.d 8 168.i even 2 1
3360.1.cv.d 8 280.s 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
\( T_{73} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 1 \) Copy content Toggle raw display
$5$ \( T^{8} + 1 \) Copy content Toggle raw display
$7$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 2 T + 2)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{2} + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$83$ \( T^{8} + 12T^{4} + 4 \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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