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Properties

Label 1680.2.bg.o
Level $1680$
Weight $2$
Character orbit 1680.bg
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(961,1680)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.961");
 
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.bg (of order \(3\), 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 \(\Q(\zeta_{3})\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - \zeta_{12}^{2} q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (\zeta_{12}^{2} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{12}^{2} q^{3} + (\zeta_{12}^{2} - 1) q^{5} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{7} + (\zeta_{12}^{2} - 1) q^{9} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}) q^{11} + (\zeta_{12}^{3} - 2 \zeta_{12} + 4) q^{13} + q^{15} + (\zeta_{12}^{3} - 5 \zeta_{12}^{2} + \zeta_{12}) q^{17} + ( - 4 \zeta_{12}^{3} - \zeta_{12}^{2} + \cdots + 1) q^{19} + \cdots + (\zeta_{12}^{3} - 2 \zeta_{12} + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 2 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 2 q^{5} - 2 q^{9} - 2 q^{11} + 16 q^{13} + 4 q^{15} - 10 q^{17} + 2 q^{19} - 6 q^{23} - 2 q^{25} + 4 q^{27} + 4 q^{29} + 6 q^{31} - 2 q^{33} - 4 q^{37} - 8 q^{39} + 4 q^{41} + 8 q^{43} - 2 q^{45} + 4 q^{47} - 22 q^{49} - 10 q^{51} - 4 q^{53} + 4 q^{55} - 4 q^{57} + 10 q^{59} - 8 q^{61} - 8 q^{65} - 12 q^{67} + 12 q^{69} - 4 q^{71} - 8 q^{73} - 2 q^{75} + 12 q^{77} + 6 q^{79} - 2 q^{81} - 12 q^{83} + 20 q^{85} - 2 q^{87} - 6 q^{89} + 6 q^{91} + 6 q^{93} + 2 q^{95} + 32 q^{97} + 4 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
0.866025 + 0.500000i
−0.866025 − 0.500000i
0.866025 − 0.500000i
−0.866025 + 0.500000i
0 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 −0.866025 − 2.50000i 0 −0.500000 + 0.866025i 0
961.2 0 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 0.866025 + 2.50000i 0 −0.500000 + 0.866025i 0
1201.1 0 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 −0.866025 + 2.50000i 0 −0.500000 − 0.866025i 0
1201.2 0 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 0.866025 − 2.50000i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.bg.o 4
4.b odd 2 1 105.2.i.d ✓ 4
7.c even 3 1 inner 1680.2.bg.o 4
12.b even 2 1 315.2.j.c 4
20.d odd 2 1 525.2.i.f 4
20.e even 4 1 525.2.r.a 4
20.e even 4 1 525.2.r.f 4
28.d even 2 1 735.2.i.l 4
28.f even 6 1 735.2.a.h 2
28.f even 6 1 735.2.i.l 4
28.g odd 6 1 105.2.i.d ✓ 4
28.g odd 6 1 735.2.a.g 2
84.j odd 6 1 2205.2.a.ba 2
84.n even 6 1 315.2.j.c 4
84.n even 6 1 2205.2.a.z 2
140.p odd 6 1 525.2.i.f 4
140.p odd 6 1 3675.2.a.bg 2
140.s even 6 1 3675.2.a.be 2
140.w even 12 1 525.2.r.a 4
140.w even 12 1 525.2.r.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.d ✓ 4 4.b odd 2 1
105.2.i.d ✓ 4 28.g odd 6 1
315.2.j.c 4 12.b even 2 1
315.2.j.c 4 84.n even 6 1
525.2.i.f 4 20.d odd 2 1
525.2.i.f 4 140.p odd 6 1
525.2.r.a 4 20.e even 4 1
525.2.r.a 4 140.w even 12 1
525.2.r.f 4 20.e even 4 1
525.2.r.f 4 140.w even 12 1
735.2.a.g 2 28.g odd 6 1
735.2.a.h 2 28.f even 6 1
735.2.i.l 4 28.d even 2 1
735.2.i.l 4 28.f even 6 1
1680.2.bg.o 4 1.a even 1 1 trivial
1680.2.bg.o 4 7.c even 3 1 inner
2205.2.a.z 2 84.n even 6 1
2205.2.a.ba 2 84.j odd 6 1
3675.2.a.be 2 140.s even 6 1
3675.2.a.bg 2 140.p odd 6 1

Hecke kernels

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

\( T_{11}^{4} + 2T_{11}^{3} + 6T_{11}^{2} - 4T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 8T_{13} + 13 \) Copy content Toggle raw display
\( T_{17}^{4} + 10T_{17}^{3} + 78T_{17}^{2} + 220T_{17} + 484 \) 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
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 11T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( (T^{2} - 8 T + 13)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + \cdots + 484 \) Copy content Toggle raw display
$19$ \( T^{4} - 2 T^{3} + \cdots + 121 \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( (T^{2} - 2 T - 26)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$37$ \( T^{4} + 4 T^{3} + \cdots + 529 \) Copy content Toggle raw display
$41$ \( (T^{2} - 2 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 4 T - 23)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 4 T^{3} + \cdots + 10816 \) Copy content Toggle raw display
$59$ \( T^{4} - 10 T^{3} + \cdots + 4 \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T + 16)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 12 T^{3} + \cdots + 1521 \) Copy content Toggle raw display
$71$ \( (T^{2} + 2 T - 26)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 8 T^{3} + \cdots + 3481 \) Copy content Toggle raw display
$79$ \( T^{4} - 6 T^{3} + \cdots + 9801 \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 138)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 6 T^{3} + \cdots + 19044 \) Copy content Toggle raw display
$97$ \( (T^{2} - 16 T + 16)^{2} \) Copy content Toggle raw display
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