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Properties

Label 3360.1.ds.a
Level $3360$
Weight $1$
Character orbit 3360.ds
Analytic conductor $1.677$
Analytic rank $0$
Dimension $8$
Projective image $D_{12}$
CM discriminant -20
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(1409,3360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3360, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 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("3360.1409");
 
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.ds (of order \(6\), 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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{12}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-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.

\(f(q)\) \(=\) \( q + \zeta_{24}^{9} q^{3} + \zeta_{24}^{2} q^{5} + \zeta_{24}^{7} q^{7} - \zeta_{24}^{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{24}^{9} q^{3} + \zeta_{24}^{2} q^{5} + \zeta_{24}^{7} q^{7} - \zeta_{24}^{6} q^{9} + \zeta_{24}^{11} q^{15} - \zeta_{24}^{4} q^{21} + (\zeta_{24}^{3} - \zeta_{24}) q^{23} + \zeta_{24}^{4} q^{25} + \zeta_{24}^{3} q^{27} + (\zeta_{24}^{8} + \zeta_{24}^{4}) q^{29} + \zeta_{24}^{9} q^{35} - \zeta_{24}^{6} q^{41} + (\zeta_{24}^{7} + \zeta_{24}^{5}) q^{43} - \zeta_{24}^{8} q^{45} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{47} - \zeta_{24}^{2} q^{49} + (\zeta_{24}^{10} + \zeta_{24}^{6}) q^{61} + \zeta_{24} q^{63} + ( - \zeta_{24}^{5} + \zeta_{24}^{3}) q^{67} + ( - \zeta_{24}^{10} - 1) q^{69} - \zeta_{24} q^{75} - q^{81} + (\zeta_{24}^{11} - \zeta_{24}) q^{83} + ( - \zeta_{24}^{5} - \zeta_{24}) q^{87} + ( - \zeta_{24}^{4} - 1) q^{89} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{21} + 4 q^{25} + 4 q^{45} - 8 q^{69} - 8 q^{81} - 12 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

\(n\) \(421\) \(1121\) \(1471\) \(1921\) \(2017\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(\zeta_{24}^{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1409.1
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 − 0.258819i
0.258819 − 0.965926i
0.965926 − 0.258819i
−0.258819 − 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i
0 −0.707107 − 0.707107i 0 −0.866025 − 0.500000i 0 0.965926 + 0.258819i 0 1.00000i 0
1409.2 0 −0.707107 + 0.707107i 0 0.866025 + 0.500000i 0 −0.258819 + 0.965926i 0 − 1.00000i 0
1409.3 0 0.707107 − 0.707107i 0 0.866025 + 0.500000i 0 0.258819 − 0.965926i 0 − 1.00000i 0
1409.4 0 0.707107 + 0.707107i 0 −0.866025 − 0.500000i 0 −0.965926 − 0.258819i 0 1.00000i 0
3329.1 0 −0.707107 − 0.707107i 0 0.866025 − 0.500000i 0 −0.258819 − 0.965926i 0 1.00000i 0
3329.2 0 −0.707107 + 0.707107i 0 −0.866025 + 0.500000i 0 0.965926 − 0.258819i 0 − 1.00000i 0
3329.3 0 0.707107 − 0.707107i 0 −0.866025 + 0.500000i 0 −0.965926 + 0.258819i 0 − 1.00000i 0
3329.4 0 0.707107 + 0.707107i 0 0.866025 − 0.500000i 0 0.258819 + 0.965926i 0 1.00000i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1409.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
4.b odd 2 1 inner
5.b even 2 1 inner
21.h odd 6 1 inner
84.n even 6 1 inner
105.o odd 6 1 inner
420.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3360.1.ds.a ✓ 8
3.b odd 2 1 3360.1.ds.b yes 8
4.b odd 2 1 inner 3360.1.ds.a ✓ 8
5.b even 2 1 inner 3360.1.ds.a ✓ 8
7.c even 3 1 3360.1.ds.b yes 8
12.b even 2 1 3360.1.ds.b yes 8
15.d odd 2 1 3360.1.ds.b yes 8
20.d odd 2 1 CM 3360.1.ds.a ✓ 8
21.h odd 6 1 inner 3360.1.ds.a ✓ 8
28.g odd 6 1 3360.1.ds.b yes 8
35.j even 6 1 3360.1.ds.b yes 8
60.h even 2 1 3360.1.ds.b yes 8
84.n even 6 1 inner 3360.1.ds.a ✓ 8
105.o odd 6 1 inner 3360.1.ds.a ✓ 8
140.p odd 6 1 3360.1.ds.b yes 8
420.ba even 6 1 inner 3360.1.ds.a ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3360.1.ds.a ✓ 8 1.a even 1 1 trivial
3360.1.ds.a ✓ 8 4.b odd 2 1 inner
3360.1.ds.a ✓ 8 5.b even 2 1 inner
3360.1.ds.a ✓ 8 20.d odd 2 1 CM
3360.1.ds.a ✓ 8 21.h odd 6 1 inner
3360.1.ds.a ✓ 8 84.n even 6 1 inner
3360.1.ds.a ✓ 8 105.o odd 6 1 inner
3360.1.ds.a ✓ 8 420.ba even 6 1 inner
3360.1.ds.b yes 8 3.b odd 2 1
3360.1.ds.b yes 8 7.c even 3 1
3360.1.ds.b yes 8 12.b even 2 1
3360.1.ds.b yes 8 15.d odd 2 1
3360.1.ds.b yes 8 28.g odd 6 1
3360.1.ds.b yes 8 35.j even 6 1
3360.1.ds.b yes 8 60.h even 2 1
3360.1.ds.b yes 8 140.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{89}^{2} + 3T_{89} + 3 \) acting on \(S_{1}^{\mathrm{new}}(3360, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} + 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 3 T^{2} + 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 4 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 4 T^{2} + 1)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 3 T + 3)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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