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Properties

Label 1680.2.di.a
Level $1680$
Weight $2$
Character orbit 1680.di
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(289,1680)]
 
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([0, 0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.289");
 
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.di (of order \(6\), 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: \(13.4148675396\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 210)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{3} + (2 \zeta_{12}^{2} + \zeta_{12} - 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + ( - \zeta_{12}^{2} + 1) q^{9} - 5 \zeta_{12}^{2} q^{11} + \zeta_{12}^{3} q^{13} + ( - 2 \zeta_{12}^{3} - 1) q^{15} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{17} + (7 \zeta_{12}^{2} - 7) q^{19} + ( - 3 \zeta_{12}^{2} + 2) q^{21} + 3 \zeta_{12} q^{23} + (4 \zeta_{12}^{3} + \cdots - 4 \zeta_{12}) q^{25} + \cdots - 5 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 2 q^{9} - 10 q^{11} - 4 q^{15} - 14 q^{19} + 2 q^{21} - 6 q^{25} - 12 q^{31} - 10 q^{35} - 2 q^{39} - 36 q^{41} + 4 q^{45} - 4 q^{49} - 4 q^{51} + 40 q^{55} - 8 q^{59} + 4 q^{61} - 2 q^{65} - 12 q^{69} + 8 q^{71} + 8 q^{75} + 28 q^{79} - 2 q^{81} + 8 q^{85} + 20 q^{89} - 8 q^{91} - 28 q^{95} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
289.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 −0.133975 + 2.23205i 0 −1.73205 + 2.00000i 0 0.500000 − 0.866025i 0
289.2 0 0.866025 − 0.500000i 0 −1.86603 + 1.23205i 0 1.73205 − 2.00000i 0 0.500000 − 0.866025i 0
529.1 0 −0.866025 − 0.500000i 0 −0.133975 − 2.23205i 0 −1.73205 − 2.00000i 0 0.500000 + 0.866025i 0
529.2 0 0.866025 + 0.500000i 0 −1.86603 − 1.23205i 0 1.73205 + 2.00000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j 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.di.a 4
4.b odd 2 1 210.2.n.a ✓ 4
5.b even 2 1 inner 1680.2.di.a 4
7.c even 3 1 inner 1680.2.di.a 4
12.b even 2 1 630.2.u.c 4
20.d odd 2 1 210.2.n.a ✓ 4
20.e even 4 1 1050.2.i.f 2
20.e even 4 1 1050.2.i.o 2
28.d even 2 1 1470.2.n.i 4
28.f even 6 1 1470.2.g.a 2
28.f even 6 1 1470.2.n.i 4
28.g odd 6 1 210.2.n.a ✓ 4
28.g odd 6 1 1470.2.g.f 2
35.j even 6 1 inner 1680.2.di.a 4
60.h even 2 1 630.2.u.c 4
84.n even 6 1 630.2.u.c 4
140.c even 2 1 1470.2.n.i 4
140.p odd 6 1 210.2.n.a ✓ 4
140.p odd 6 1 1470.2.g.f 2
140.s even 6 1 1470.2.g.a 2
140.s even 6 1 1470.2.n.i 4
140.w even 12 1 1050.2.i.f 2
140.w even 12 1 1050.2.i.o 2
140.w even 12 1 7350.2.a.t 1
140.w even 12 1 7350.2.a.bn 1
140.x odd 12 1 7350.2.a.b 1
140.x odd 12 1 7350.2.a.ch 1
420.ba even 6 1 630.2.u.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a ✓ 4 4.b odd 2 1
210.2.n.a ✓ 4 20.d odd 2 1
210.2.n.a ✓ 4 28.g odd 6 1
210.2.n.a ✓ 4 140.p odd 6 1
630.2.u.c 4 12.b even 2 1
630.2.u.c 4 60.h even 2 1
630.2.u.c 4 84.n even 6 1
630.2.u.c 4 420.ba even 6 1
1050.2.i.f 2 20.e even 4 1
1050.2.i.f 2 140.w even 12 1
1050.2.i.o 2 20.e even 4 1
1050.2.i.o 2 140.w even 12 1
1470.2.g.a 2 28.f even 6 1
1470.2.g.a 2 140.s even 6 1
1470.2.g.f 2 28.g odd 6 1
1470.2.g.f 2 140.p odd 6 1
1470.2.n.i 4 28.d even 2 1
1470.2.n.i 4 28.f even 6 1
1470.2.n.i 4 140.c even 2 1
1470.2.n.i 4 140.s even 6 1
1680.2.di.a 4 1.a even 1 1 trivial
1680.2.di.a 4 5.b even 2 1 inner
1680.2.di.a 4 7.c even 3 1 inner
1680.2.di.a 4 35.j even 6 1 inner
7350.2.a.b 1 140.x odd 12 1
7350.2.a.t 1 140.w even 12 1
7350.2.a.bn 1 140.w even 12 1
7350.2.a.ch 1 140.x odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{2} + 5T_{11} + 25 \) 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
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$41$ \( (T + 9)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 169 T^{2} + 28561 \) Copy content Toggle raw display
$53$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$71$ \( (T - 2)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$79$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 100)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
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