Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(209,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.209");
 
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.k (of order \(2\), degree \(1\), 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: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{3} - 1) q^{7} + (2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} + (\beta_{2} + \beta_1) q^{5} + (\beta_{3} - 1) q^{7} + (2 \beta_1 - 1) q^{9} - 2 \beta_1 q^{11} - 4 q^{13} + (\beta_{3} + \beta_{2} + \beta_1 - 2) q^{15} + 2 \beta_1 q^{17} + (\beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{21} - 2 \beta_{2} q^{23} + (2 \beta_{3} + 1) q^{25} + (\beta_1 - 5) q^{27} - 4 \beta_1 q^{29} + 4 \beta_{3} q^{31} + ( - 2 \beta_1 + 4) q^{33} + ( - 3 \beta_{2} + 2 \beta_1) q^{35} + ( - 4 \beta_1 - 4) q^{39} + 2 \beta_{2} q^{41} + 2 \beta_{3} q^{43} + (2 \beta_{3} - \beta_{2} - \beta_1 - 4) q^{45} - 2 \beta_1 q^{47} + ( - 2 \beta_{3} - 5) q^{49} + (2 \beta_1 - 4) q^{51} + ( - 2 \beta_{3} + 4) q^{55} + 4 \beta_{2} q^{59} + 4 \beta_{3} q^{61} + ( - \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 1) q^{63} + ( - 4 \beta_{2} - 4 \beta_1) q^{65} - 2 \beta_{3} q^{67} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{69} - 2 \beta_1 q^{71} + 8 q^{73} + (2 \beta_{3} - 4 \beta_{2} + \beta_1 + 1) q^{75} + (4 \beta_{2} + 2 \beta_1) q^{77} - 8 q^{79} + ( - 4 \beta_1 - 7) q^{81} - 2 \beta_1 q^{83} + (2 \beta_{3} - 4) q^{85} + ( - 4 \beta_1 + 8) q^{87} - 6 \beta_{2} q^{89} + ( - 4 \beta_{3} + 4) q^{91} + (4 \beta_{3} - 8 \beta_{2}) q^{93} + 8 q^{97} + (2 \beta_1 + 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} - 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} - 4 q^{7} - 4 q^{9} - 16 q^{13} - 8 q^{15} - 4 q^{21} + 4 q^{25} - 20 q^{27} + 16 q^{33} - 16 q^{39} - 16 q^{45} - 20 q^{49} - 16 q^{51} + 16 q^{55} + 4 q^{63} + 32 q^{73} + 4 q^{75} - 32 q^{79} - 28 q^{81} - 16 q^{85} + 32 q^{87} + 16 q^{91} + 32 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 5\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 5\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

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

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
1.93185i
0.517638i
1.93185i
0.517638i
0 1.00000 1.41421i 0 −1.73205 1.41421i 0 −1.00000 + 2.44949i 0 −1.00000 2.82843i 0
209.2 0 1.00000 1.41421i 0 1.73205 1.41421i 0 −1.00000 2.44949i 0 −1.00000 2.82843i 0
209.3 0 1.00000 + 1.41421i 0 −1.73205 + 1.41421i 0 −1.00000 2.44949i 0 −1.00000 + 2.82843i 0
209.4 0 1.00000 + 1.41421i 0 1.73205 + 1.41421i 0 −1.00000 + 2.44949i 0 −1.00000 + 2.82843i 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
3.b odd 2 1 inner
35.c odd 2 1 inner
105.g even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.k.c 4
3.b odd 2 1 inner 1680.2.k.c 4
4.b odd 2 1 105.2.g.a 4
5.b even 2 1 1680.2.k.a 4
7.b odd 2 1 1680.2.k.a 4
12.b even 2 1 105.2.g.a 4
15.d odd 2 1 1680.2.k.a 4
20.d odd 2 1 105.2.g.c yes 4
20.e even 4 2 525.2.b.j 8
21.c even 2 1 1680.2.k.a 4
28.d even 2 1 105.2.g.c yes 4
28.f even 6 2 735.2.p.a 8
28.g odd 6 2 735.2.p.c 8
35.c odd 2 1 inner 1680.2.k.c 4
60.h even 2 1 105.2.g.c yes 4
60.l odd 4 2 525.2.b.j 8
84.h odd 2 1 105.2.g.c yes 4
84.j odd 6 2 735.2.p.a 8
84.n even 6 2 735.2.p.c 8
105.g even 2 1 inner 1680.2.k.c 4
140.c even 2 1 105.2.g.a 4
140.j odd 4 2 525.2.b.j 8
140.p odd 6 2 735.2.p.a 8
140.s even 6 2 735.2.p.c 8
420.o odd 2 1 105.2.g.a 4
420.w even 4 2 525.2.b.j 8
420.ba even 6 2 735.2.p.a 8
420.be odd 6 2 735.2.p.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.g.a 4 4.b odd 2 1
105.2.g.a 4 12.b even 2 1
105.2.g.a 4 140.c even 2 1
105.2.g.a 4 420.o odd 2 1
105.2.g.c yes 4 20.d odd 2 1
105.2.g.c yes 4 28.d even 2 1
105.2.g.c yes 4 60.h even 2 1
105.2.g.c yes 4 84.h odd 2 1
525.2.b.j 8 20.e even 4 2
525.2.b.j 8 60.l odd 4 2
525.2.b.j 8 140.j odd 4 2
525.2.b.j 8 420.w even 4 2
735.2.p.a 8 28.f even 6 2
735.2.p.a 8 84.j odd 6 2
735.2.p.a 8 140.p odd 6 2
735.2.p.a 8 420.ba even 6 2
735.2.p.c 8 28.g odd 6 2
735.2.p.c 8 84.n even 6 2
735.2.p.c 8 140.s even 6 2
735.2.p.c 8 420.be odd 6 2
1680.2.k.a 4 5.b even 2 1
1680.2.k.a 4 7.b odd 2 1
1680.2.k.a 4 15.d odd 2 1
1680.2.k.a 4 21.c even 2 1
1680.2.k.c 4 1.a even 1 1 trivial
1680.2.k.c 4 3.b odd 2 1 inner
1680.2.k.c 4 35.c odd 2 1 inner
1680.2.k.c 4 105.g even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1680, [\chi])\):

\( T_{11}^{2} + 8 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display
\( T_{23}^{2} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 2 T + 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 2T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 7)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$73$ \( (T - 8)^{4} \) Copy content Toggle raw display
$79$ \( (T + 8)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 108)^{2} \) Copy content Toggle raw display
$97$ \( (T - 8)^{4} \) Copy content Toggle raw display
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