Properties

Label 1568.3.d.b
Level $1568$
Weight $3$
Character orbit 1568.d
Analytic conductor $42.725$
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] = [1568,3,Mod(1471,1568)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1568, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.1471");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.d (of order \(2\), degree \(1\), 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: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 32)
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 \(\beta = 4i\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} - 2 q^{5} - 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{3} - 2 q^{5} - 7 q^{9} + \beta q^{11} + 14 q^{13} - 2 \beta q^{15} - 18 q^{17} - 3 \beta q^{19} - 10 \beta q^{23} - 21 q^{25} + 2 \beta q^{27} - 14 q^{29} - 8 \beta q^{31} - 16 q^{33} - 30 q^{37} + 14 \beta q^{39} + 14 q^{41} - 7 \beta q^{43} + 14 q^{45} + 4 \beta q^{47} - 18 \beta q^{51} + 66 q^{53} - 2 \beta q^{55} + 48 q^{57} - 13 \beta q^{59} - 82 q^{61} - 28 q^{65} - \beta q^{67} + 160 q^{69} - 14 \beta q^{71} - 66 q^{73} - 21 \beta q^{75} + 4 \beta q^{79} - 95 q^{81} - 35 \beta q^{83} + 36 q^{85} - 14 \beta q^{87} + 30 q^{89} + 128 q^{93} + 6 \beta q^{95} + 14 q^{97} - 7 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 14 q^{9} + 28 q^{13} - 36 q^{17} - 42 q^{25} - 28 q^{29} - 32 q^{33} - 60 q^{37} + 28 q^{41} + 28 q^{45} + 132 q^{53} + 96 q^{57} - 164 q^{61} - 56 q^{65} + 320 q^{69} - 132 q^{73} - 190 q^{81} + 72 q^{85} + 60 q^{89} + 256 q^{93} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(197\) \(1471\) \(1473\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1471.1
1.00000i
1.00000i
0 4.00000i 0 −2.00000 0 0 0 −7.00000 0
1471.2 0 4.00000i 0 −2.00000 0 0 0 −7.00000 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 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.3.d.b 2
4.b odd 2 1 inner 1568.3.d.b 2
7.b odd 2 1 32.3.c.a 2
21.c even 2 1 288.3.g.b 2
28.d even 2 1 32.3.c.a 2
35.c odd 2 1 800.3.b.a 2
35.f even 4 1 800.3.h.a 2
35.f even 4 1 800.3.h.b 2
56.e even 2 1 64.3.c.b 2
56.h odd 2 1 64.3.c.b 2
84.h odd 2 1 288.3.g.b 2
112.j even 4 1 256.3.d.a 2
112.j even 4 1 256.3.d.c 2
112.l odd 4 1 256.3.d.a 2
112.l odd 4 1 256.3.d.c 2
140.c even 2 1 800.3.b.a 2
140.j odd 4 1 800.3.h.a 2
140.j odd 4 1 800.3.h.b 2
168.e odd 2 1 576.3.g.g 2
168.i even 2 1 576.3.g.g 2
280.c odd 2 1 1600.3.b.e 2
280.n even 2 1 1600.3.b.e 2
280.s even 4 1 1600.3.h.a 2
280.s even 4 1 1600.3.h.c 2
280.y odd 4 1 1600.3.h.a 2
280.y odd 4 1 1600.3.h.c 2
336.v odd 4 1 2304.3.b.c 2
336.v odd 4 1 2304.3.b.g 2
336.y even 4 1 2304.3.b.c 2
336.y even 4 1 2304.3.b.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.3.c.a 2 7.b odd 2 1
32.3.c.a 2 28.d even 2 1
64.3.c.b 2 56.e even 2 1
64.3.c.b 2 56.h odd 2 1
256.3.d.a 2 112.j even 4 1
256.3.d.a 2 112.l odd 4 1
256.3.d.c 2 112.j even 4 1
256.3.d.c 2 112.l odd 4 1
288.3.g.b 2 21.c even 2 1
288.3.g.b 2 84.h odd 2 1
576.3.g.g 2 168.e odd 2 1
576.3.g.g 2 168.i even 2 1
800.3.b.a 2 35.c odd 2 1
800.3.b.a 2 140.c even 2 1
800.3.h.a 2 35.f even 4 1
800.3.h.a 2 140.j odd 4 1
800.3.h.b 2 35.f even 4 1
800.3.h.b 2 140.j odd 4 1
1568.3.d.b 2 1.a even 1 1 trivial
1568.3.d.b 2 4.b odd 2 1 inner
1600.3.b.e 2 280.c odd 2 1
1600.3.b.e 2 280.n even 2 1
1600.3.h.a 2 280.s even 4 1
1600.3.h.a 2 280.y odd 4 1
1600.3.h.c 2 280.s even 4 1
1600.3.h.c 2 280.y odd 4 1
2304.3.b.c 2 336.v odd 4 1
2304.3.b.c 2 336.y even 4 1
2304.3.b.g 2 336.v odd 4 1
2304.3.b.g 2 336.y even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 16 \) Copy content Toggle raw display
$13$ \( (T - 14)^{2} \) Copy content Toggle raw display
$17$ \( (T + 18)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 144 \) Copy content Toggle raw display
$23$ \( T^{2} + 1600 \) Copy content Toggle raw display
$29$ \( (T + 14)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 1024 \) Copy content Toggle raw display
$37$ \( (T + 30)^{2} \) Copy content Toggle raw display
$41$ \( (T - 14)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 784 \) Copy content Toggle raw display
$47$ \( T^{2} + 256 \) Copy content Toggle raw display
$53$ \( (T - 66)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 2704 \) Copy content Toggle raw display
$61$ \( (T + 82)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 16 \) Copy content Toggle raw display
$71$ \( T^{2} + 3136 \) Copy content Toggle raw display
$73$ \( (T + 66)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 256 \) Copy content Toggle raw display
$83$ \( T^{2} + 19600 \) Copy content Toggle raw display
$89$ \( (T - 30)^{2} \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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