Properties

Label 1568.2.i.f
Level $1568$
Weight $2$
Character orbit 1568.i
Analytic conductor $12.521$
Analytic rank $1$
Dimension $2$
CM discriminant -4
Inner twists $4$

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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: \(1\)
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 32)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} - 6 q^{13} + ( - 2 \zeta_{6} + 2) q^{17} + ( - \zeta_{6} + 1) q^{25} - 10 q^{29} + 2 \zeta_{6} q^{37} - 10 q^{41} + ( - 6 \zeta_{6} + 6) q^{45} + (14 \zeta_{6} - 14) q^{53} + \cdots - 18 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 3 q^{9} - 12 q^{13} + 2 q^{17} + q^{25} - 20 q^{29} + 2 q^{37} - 20 q^{41} + 6 q^{45} - 14 q^{53} - 10 q^{61} + 12 q^{65} - 6 q^{73} - 9 q^{81} - 8 q^{85} + 10 q^{89} - 36 q^{97}+O(q^{100}) \) 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 0 0 −1.00000 + 1.73205i 0 0 0 1.50000 2.59808i 0
1537.1 0 0 0 −1.00000 1.73205i 0 0 0 1.50000 + 2.59808i 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
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
7.c even 3 1 inner
28.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.f 2
4.b odd 2 1 CM 1568.2.i.f 2
7.b odd 2 1 1568.2.i.g 2
7.c even 3 1 1568.2.a.e 1
7.c even 3 1 inner 1568.2.i.f 2
7.d odd 6 1 32.2.a.a 1
7.d odd 6 1 1568.2.i.g 2
21.g even 6 1 288.2.a.d 1
28.d even 2 1 1568.2.i.g 2
28.f even 6 1 32.2.a.a 1
28.f even 6 1 1568.2.i.g 2
28.g odd 6 1 1568.2.a.e 1
28.g odd 6 1 inner 1568.2.i.f 2
35.i odd 6 1 800.2.a.d 1
35.k even 12 2 800.2.c.e 2
56.j odd 6 1 64.2.a.a 1
56.k odd 6 1 3136.2.a.m 1
56.m even 6 1 64.2.a.a 1
56.p even 6 1 3136.2.a.m 1
63.i even 6 1 2592.2.i.e 2
63.k odd 6 1 2592.2.i.t 2
63.s even 6 1 2592.2.i.e 2
63.t odd 6 1 2592.2.i.t 2
77.i even 6 1 3872.2.a.f 1
84.j odd 6 1 288.2.a.d 1
91.s odd 6 1 5408.2.a.g 1
105.p even 6 1 7200.2.a.v 1
105.w odd 12 2 7200.2.f.m 2
112.v even 12 2 256.2.b.b 2
112.x odd 12 2 256.2.b.b 2
119.h odd 6 1 9248.2.a.f 1
140.s even 6 1 800.2.a.d 1
140.x odd 12 2 800.2.c.e 2
168.ba even 6 1 576.2.a.c 1
168.be odd 6 1 576.2.a.c 1
224.bc odd 24 4 1024.2.e.j 4
224.be even 24 4 1024.2.e.j 4
252.n even 6 1 2592.2.i.t 2
252.r odd 6 1 2592.2.i.e 2
252.bj even 6 1 2592.2.i.t 2
252.bn odd 6 1 2592.2.i.e 2
280.ba even 6 1 1600.2.a.n 1
280.bk odd 6 1 1600.2.a.n 1
280.bp odd 12 2 1600.2.c.l 2
280.bv even 12 2 1600.2.c.l 2
308.m odd 6 1 3872.2.a.f 1
336.bo even 12 2 2304.2.d.j 2
336.br odd 12 2 2304.2.d.j 2
364.x even 6 1 5408.2.a.g 1
420.be odd 6 1 7200.2.a.v 1
420.br even 12 2 7200.2.f.m 2
476.q even 6 1 9248.2.a.f 1
616.s even 6 1 7744.2.a.v 1
616.z odd 6 1 7744.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.a.a 1 7.d odd 6 1
32.2.a.a 1 28.f even 6 1
64.2.a.a 1 56.j odd 6 1
64.2.a.a 1 56.m even 6 1
256.2.b.b 2 112.v even 12 2
256.2.b.b 2 112.x odd 12 2
288.2.a.d 1 21.g even 6 1
288.2.a.d 1 84.j odd 6 1
576.2.a.c 1 168.ba even 6 1
576.2.a.c 1 168.be odd 6 1
800.2.a.d 1 35.i odd 6 1
800.2.a.d 1 140.s even 6 1
800.2.c.e 2 35.k even 12 2
800.2.c.e 2 140.x odd 12 2
1024.2.e.j 4 224.bc odd 24 4
1024.2.e.j 4 224.be even 24 4
1568.2.a.e 1 7.c even 3 1
1568.2.a.e 1 28.g odd 6 1
1568.2.i.f 2 1.a even 1 1 trivial
1568.2.i.f 2 4.b odd 2 1 CM
1568.2.i.f 2 7.c even 3 1 inner
1568.2.i.f 2 28.g odd 6 1 inner
1568.2.i.g 2 7.b odd 2 1
1568.2.i.g 2 7.d odd 6 1
1568.2.i.g 2 28.d even 2 1
1568.2.i.g 2 28.f even 6 1
1600.2.a.n 1 280.ba even 6 1
1600.2.a.n 1 280.bk odd 6 1
1600.2.c.l 2 280.bp odd 12 2
1600.2.c.l 2 280.bv even 12 2
2304.2.d.j 2 336.bo even 12 2
2304.2.d.j 2 336.br odd 12 2
2592.2.i.e 2 63.i even 6 1
2592.2.i.e 2 63.s even 6 1
2592.2.i.e 2 252.r odd 6 1
2592.2.i.e 2 252.bn odd 6 1
2592.2.i.t 2 63.k odd 6 1
2592.2.i.t 2 63.t odd 6 1
2592.2.i.t 2 252.n even 6 1
2592.2.i.t 2 252.bj even 6 1
3136.2.a.m 1 56.k odd 6 1
3136.2.a.m 1 56.p even 6 1
3872.2.a.f 1 77.i even 6 1
3872.2.a.f 1 308.m odd 6 1
5408.2.a.g 1 91.s odd 6 1
5408.2.a.g 1 364.x even 6 1
7200.2.a.v 1 105.p even 6 1
7200.2.a.v 1 420.be odd 6 1
7200.2.f.m 2 105.w odd 12 2
7200.2.f.m 2 420.br even 12 2
7744.2.a.v 1 616.s even 6 1
7744.2.a.v 1 616.z odd 6 1
9248.2.a.f 1 119.h odd 6 1
9248.2.a.f 1 476.q even 6 1

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 6)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$41$ \( (T + 10)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$97$ \( (T + 18)^{2} \) Copy content Toggle raw display
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