Properties

Label 1568.2.i.k
Level $1568$
Weight $2$
Character orbit 1568.i
Analytic conductor $12.521$
Analytic rank $0$
Dimension $2$
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] = [1568,2,Mod(961,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([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("1568.961");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.i (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: \(12.5205430369\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 224)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} - \zeta_{6} q^{9} + ( - 4 \zeta_{6} + 4) q^{11} - 4 q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + 6 \zeta_{6} q^{19} - 8 \zeta_{6} q^{23} + ( - 5 \zeta_{6} + 5) q^{25} + 4 q^{27} + 2 q^{29} + ( - 4 \zeta_{6} + 4) q^{31} - 8 \zeta_{6} q^{33} - 10 \zeta_{6} q^{37} + (8 \zeta_{6} - 8) q^{39} - 10 q^{41} + 4 q^{43} - 4 \zeta_{6} q^{47} - 4 \zeta_{6} q^{51} + ( - 2 \zeta_{6} + 2) q^{53} + 12 q^{57} + (10 \zeta_{6} - 10) q^{59} + 8 \zeta_{6} q^{61} + ( - 8 \zeta_{6} + 8) q^{67} - 16 q^{69} + ( - 6 \zeta_{6} + 6) q^{73} - 10 \zeta_{6} q^{75} + 16 \zeta_{6} q^{79} + ( - 11 \zeta_{6} + 11) q^{81} + 2 q^{83} + ( - 4 \zeta_{6} + 4) q^{87} - 18 \zeta_{6} q^{89} - 8 \zeta_{6} q^{93} - 2 q^{97} - 4 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - q^{9} + 4 q^{11} - 8 q^{13} + 2 q^{17} + 6 q^{19} - 8 q^{23} + 5 q^{25} + 8 q^{27} + 4 q^{29} + 4 q^{31} - 8 q^{33} - 10 q^{37} - 8 q^{39} - 20 q^{41} + 8 q^{43} - 4 q^{47} - 4 q^{51} + 2 q^{53} + 24 q^{57} - 10 q^{59} + 8 q^{61} + 8 q^{67} - 32 q^{69} + 6 q^{73} - 10 q^{75} + 16 q^{79} + 11 q^{81} + 4 q^{83} + 4 q^{87} - 18 q^{89} - 8 q^{93} - 4 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 + 1.73205i 0 0 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 1.00000 1.73205i 0 0 0 0 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 1568.2.i.k 2
4.b odd 2 1 1568.2.i.b 2
7.b odd 2 1 1568.2.i.c 2
7.c even 3 1 224.2.a.a 1
7.c even 3 1 inner 1568.2.i.k 2
7.d odd 6 1 1568.2.a.h 1
7.d odd 6 1 1568.2.i.c 2
21.h odd 6 1 2016.2.a.e 1
28.d even 2 1 1568.2.i.j 2
28.f even 6 1 1568.2.a.b 1
28.f even 6 1 1568.2.i.j 2
28.g odd 6 1 224.2.a.b yes 1
28.g odd 6 1 1568.2.i.b 2
35.j even 6 1 5600.2.a.t 1
56.j odd 6 1 3136.2.a.f 1
56.k odd 6 1 448.2.a.b 1
56.m even 6 1 3136.2.a.y 1
56.p even 6 1 448.2.a.f 1
84.n even 6 1 2016.2.a.g 1
112.u odd 12 2 1792.2.b.b 2
112.w even 12 2 1792.2.b.f 2
140.p odd 6 1 5600.2.a.c 1
168.s odd 6 1 4032.2.a.p 1
168.v even 6 1 4032.2.a.z 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 7.c even 3 1
224.2.a.b yes 1 28.g odd 6 1
448.2.a.b 1 56.k odd 6 1
448.2.a.f 1 56.p even 6 1
1568.2.a.b 1 28.f even 6 1
1568.2.a.h 1 7.d odd 6 1
1568.2.i.b 2 4.b odd 2 1
1568.2.i.b 2 28.g odd 6 1
1568.2.i.c 2 7.b odd 2 1
1568.2.i.c 2 7.d odd 6 1
1568.2.i.j 2 28.d even 2 1
1568.2.i.j 2 28.f even 6 1
1568.2.i.k 2 1.a even 1 1 trivial
1568.2.i.k 2 7.c even 3 1 inner
1792.2.b.b 2 112.u odd 12 2
1792.2.b.f 2 112.w even 12 2
2016.2.a.e 1 21.h odd 6 1
2016.2.a.g 1 84.n even 6 1
3136.2.a.f 1 56.j odd 6 1
3136.2.a.y 1 56.m even 6 1
4032.2.a.p 1 168.s odd 6 1
4032.2.a.z 1 168.v even 6 1
5600.2.a.c 1 140.p odd 6 1
5600.2.a.t 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1568, [\chi])\):

\( T_{3}^{2} - 2T_{3} + 4 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} - 4T_{11} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$13$ \( (T + 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$53$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$61$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$83$ \( (T - 2)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$97$ \( (T + 2)^{2} \) Copy content Toggle raw display
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