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Properties

Label 1680.2.dx.d
Level $1680$
Weight $2$
Character orbit 1680.dx
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(31,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.dx (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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{2} + 1) q^{3} + (\zeta_{12}^{3} - \zeta_{12}) q^{5} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{7} - \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{2} + 1) q^{3} + (\zeta_{12}^{3} - \zeta_{12}) q^{5} + (\zeta_{12}^{3} - 3 \zeta_{12}) q^{7} - \zeta_{12}^{2} q^{9} + (\zeta_{12}^{2} + 3 \zeta_{12} + 1) q^{11} - \zeta_{12}^{3} q^{13} + \zeta_{12}^{3} q^{15} + (3 \zeta_{12}^{2} + \zeta_{12} + 3) q^{17} + ( - 2 \zeta_{12}^{3} + \cdots - 2 \zeta_{12}) q^{19} + \cdots + ( - 3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 2 q^{9} + 6 q^{11} + 18 q^{17} - 2 q^{19} + 6 q^{23} + 2 q^{25} - 4 q^{27} + 4 q^{29} + 6 q^{31} + 6 q^{33} + 10 q^{35} + 20 q^{37} - 12 q^{47} + 26 q^{49} + 18 q^{51} - 4 q^{53} - 12 q^{55} - 4 q^{57} + 18 q^{59} + 12 q^{61} + 2 q^{65} + 12 q^{67} + 24 q^{73} - 2 q^{75} - 24 q^{77} + 6 q^{79} - 2 q^{81} + 28 q^{83} - 4 q^{85} + 2 q^{87} - 6 q^{89} - 2 q^{91} - 6 q^{93} + 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 0.500000 − 0.866025i 0 −0.866025 + 0.500000i 0 −2.59808 − 0.500000i 0 −0.500000 − 0.866025i 0
31.2 0 0.500000 − 0.866025i 0 0.866025 − 0.500000i 0 2.59808 + 0.500000i 0 −0.500000 − 0.866025i 0
271.1 0 0.500000 + 0.866025i 0 −0.866025 − 0.500000i 0 −2.59808 + 0.500000i 0 −0.500000 + 0.866025i 0
271.2 0 0.500000 + 0.866025i 0 0.866025 + 0.500000i 0 2.59808 − 0.500000i 0 −0.500000 + 0.866025i 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
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.dx.d yes 4
4.b odd 2 1 1680.2.dx.a ✓ 4
7.d odd 6 1 1680.2.dx.a ✓ 4
28.f even 6 1 inner 1680.2.dx.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1680.2.dx.a ✓ 4 4.b odd 2 1
1680.2.dx.a ✓ 4 7.d odd 6 1
1680.2.dx.d yes 4 1.a even 1 1 trivial
1680.2.dx.d yes 4 28.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} - 6T_{11}^{3} + 6T_{11}^{2} + 36T_{11} + 36 \) acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 13T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 18 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{4} - 6 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( T^{4} - 20 T^{3} + \cdots + 9409 \) Copy content Toggle raw display
$41$ \( T^{4} + 56T^{2} + 676 \) Copy content Toggle raw display
$43$ \( T^{4} + 74T^{2} + 169 \) Copy content Toggle raw display
$47$ \( T^{4} + 12 T^{3} + \cdots + 144 \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$59$ \( T^{4} - 18 T^{3} + \cdots + 2916 \) Copy content Toggle raw display
$61$ \( T^{4} - 12 T^{3} + \cdots + 2704 \) Copy content Toggle raw display
$67$ \( T^{4} - 12 T^{3} + \cdots + 4761 \) Copy content Toggle raw display
$71$ \( T^{4} + 248T^{2} + 676 \) Copy content Toggle raw display
$73$ \( T^{4} - 24 T^{3} + \cdots + 2209 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + \cdots + 1089 \) Copy content Toggle raw display
$83$ \( (T^{2} - 14 T + 46)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$97$ \( T^{4} + 104T^{2} + 1936 \) Copy content Toggle raw display
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