Properties

Label 1568.2.b.f
Level $1568$
Weight $2$
Character orbit 1568.b
Analytic conductor $12.521$
Analytic rank $0$
Dimension $6$
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(785,1568)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.785");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1568.b (of order \(2\), degree \(1\), 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: \(6\)
Coefficient field: 6.0.1142512.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - x^{4} + 5x^{3} - 2x^{2} - 4x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 56)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{5} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + ( - \beta_{3} - \beta_1) q^{5} + (\beta_{5} - \beta_{2}) q^{9} + ( - \beta_{4} - \beta_{3}) q^{11} + \beta_{4} q^{13} + ( - \beta_{5} + 2) q^{15} + ( - \beta_{5} - \beta_{2} + 1) q^{17} + ( - \beta_{4} - \beta_{3} - 2 \beta_1) q^{19} + \beta_{2} q^{23} + 2 \beta_{2} q^{25} + ( - \beta_{4} + \beta_{3}) q^{27} + ( - \beta_{4} - 2 \beta_1) q^{29} + ( - \beta_{5} + 2) q^{31} + ( - 4 \beta_{5} + 2 \beta_{2} + 3) q^{33} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{37} + (3 \beta_{5} - \beta_{2}) q^{39} + ( - 3 \beta_{5} + \beta_{2}) q^{41} + ( - 2 \beta_{3} - 4 \beta_1) q^{43} + (\beta_{4} - 2 \beta_1) q^{45} + ( - 3 \beta_{5} + 6) q^{47} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{51} + (\beta_{4} - 3 \beta_{3} + \beta_1) q^{53} + (2 \beta_{5} - 3 \beta_{2}) q^{55} + ( - 4 \beta_{5} + 1) q^{57} + ( - 3 \beta_{3} - 4 \beta_1) q^{59} + (\beta_{4} + 3 \beta_{3} - 3 \beta_1) q^{61} + ( - \beta_{5} + 3 \beta_{2} - 2) q^{65} + (\beta_{3} + 2 \beta_1) q^{67} + (\beta_{3} - \beta_1) q^{69} + ( - 4 \beta_{5} + 2 \beta_{2} - 2) q^{71} + ( - 2 \beta_{5} + 4 \beta_{2} + 1) q^{73} + (2 \beta_{3} - 2 \beta_1) q^{75} + (2 \beta_{5} - \beta_{2} - 4) q^{79} + (\beta_{5} - 3 \beta_{2} - 3) q^{81} + ( - 2 \beta_{3} - 2 \beta_1) q^{83} + (\beta_{4} - 3 \beta_{3} - 3 \beta_1) q^{85} + ( - 3 \beta_{5} - \beta_{2} - 2) q^{87} + ( - 4 \beta_{2} + 3) q^{89} + (\beta_{4} + 3 \beta_{3} + \beta_1) q^{93} + (\beta_{2} - 6) q^{95} + (3 \beta_{5} + \beta_{2} + 2) q^{97} + (\beta_{4} + 6 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 10 q^{15} + 2 q^{17} + 2 q^{23} + 4 q^{25} + 10 q^{31} + 14 q^{33} + 4 q^{39} - 4 q^{41} + 30 q^{47} - 2 q^{55} - 2 q^{57} - 8 q^{65} - 16 q^{71} + 10 q^{73} - 22 q^{79} - 22 q^{81} - 20 q^{87} + 10 q^{89} - 34 q^{95} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - x^{4} + 5x^{3} - 2x^{2} - 4x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{4} - \nu^{3} + \nu^{2} + 3\nu - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + \nu^{4} + \nu^{3} - 5\nu^{2} + 6\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + \nu^{4} + \nu^{3} - 5\nu^{2} - 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} + \nu^{4} + 5\nu^{3} - \nu^{2} + 12 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 2\nu^{3} - 2\nu^{2} - \nu + 4 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{3} - \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} + \beta_{4} - \beta_{3} - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{4} + 3\beta_{3} - 2\beta_{2} + 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} - 2\beta _1 - 2 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
785.1
0.697966 + 1.22998i
1.17445 + 0.787829i
−1.37241 + 0.341295i
−1.37241 0.341295i
1.17445 0.787829i
0.697966 1.22998i
0 2.45995i 0 1.48598i 0 0 0 −3.05137 0
785.2 0 1.57566i 0 0.549738i 0 0 0 0.517304 0
785.3 0 0.682591i 0 3.23877i 0 0 0 2.53407 0
785.4 0 0.682591i 0 3.23877i 0 0 0 2.53407 0
785.5 0 1.57566i 0 0.549738i 0 0 0 0.517304 0
785.6 0 2.45995i 0 1.48598i 0 0 0 −3.05137 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 785.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.b.f 6
4.b odd 2 1 392.2.b.e 6
7.b odd 2 1 1568.2.b.e 6
7.c even 3 2 224.2.t.a 12
7.d odd 6 2 1568.2.t.g 12
8.b even 2 1 inner 1568.2.b.f 6
8.d odd 2 1 392.2.b.e 6
21.h odd 6 2 2016.2.cr.c 12
28.d even 2 1 392.2.b.f 6
28.f even 6 2 392.2.p.g 12
28.g odd 6 2 56.2.p.a 12
56.e even 2 1 392.2.b.f 6
56.h odd 2 1 1568.2.b.e 6
56.j odd 6 2 1568.2.t.g 12
56.k odd 6 2 56.2.p.a 12
56.m even 6 2 392.2.p.g 12
56.p even 6 2 224.2.t.a 12
84.n even 6 2 504.2.cj.c 12
168.s odd 6 2 2016.2.cr.c 12
168.v even 6 2 504.2.cj.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.p.a 12 28.g odd 6 2
56.2.p.a 12 56.k odd 6 2
224.2.t.a 12 7.c even 3 2
224.2.t.a 12 56.p even 6 2
392.2.b.e 6 4.b odd 2 1
392.2.b.e 6 8.d odd 2 1
392.2.b.f 6 28.d even 2 1
392.2.b.f 6 56.e even 2 1
392.2.p.g 12 28.f even 6 2
392.2.p.g 12 56.m even 6 2
504.2.cj.c 12 84.n even 6 2
504.2.cj.c 12 168.v even 6 2
1568.2.b.e 6 7.b odd 2 1
1568.2.b.e 6 56.h odd 2 1
1568.2.b.f 6 1.a even 1 1 trivial
1568.2.b.f 6 8.b even 2 1 inner
1568.2.t.g 12 7.d odd 6 2
1568.2.t.g 12 56.j odd 6 2
2016.2.cr.c 12 21.h odd 6 2
2016.2.cr.c 12 168.s odd 6 2

Hecke kernels

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

\( T_{3}^{6} + 9T_{3}^{4} + 19T_{3}^{2} + 7 \) Copy content Toggle raw display
\( T_{17}^{3} - T_{17}^{2} - 17T_{17} + 21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 9 T^{4} + 19 T^{2} + 7 \) Copy content Toggle raw display
$5$ \( T^{6} + 13 T^{4} + 27 T^{2} + 7 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 37 T^{4} + 327 T^{2} + \cdots + 847 \) Copy content Toggle raw display
$13$ \( T^{6} + 32 T^{4} + 320 T^{2} + \cdots + 1008 \) Copy content Toggle raw display
$17$ \( (T^{3} - T^{2} - 17 T + 21)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 61 T^{4} + 815 T^{2} + \cdots + 3087 \) Copy content Toggle raw display
$23$ \( (T^{3} - T^{2} - 7 T + 9)^{2} \) Copy content Toggle raw display
$29$ \( T^{6} + 72 T^{4} + 304 T^{2} + \cdots + 112 \) Copy content Toggle raw display
$31$ \( (T^{3} - 5 T^{2} + 3 T + 7)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 61 T^{4} + 899 T^{2} + \cdots + 63 \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} - 40 T - 84)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 164 T^{4} + 4016 T^{2} + \cdots + 4032 \) Copy content Toggle raw display
$47$ \( (T^{3} - 15 T^{2} + 27 T + 189)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} + 157 T^{4} + 4899 T^{2} + \cdots + 26047 \) Copy content Toggle raw display
$59$ \( T^{6} + 177 T^{4} + 4147 T^{2} + \cdots + 26047 \) Copy content Toggle raw display
$61$ \( T^{6} + 293 T^{4} + 24947 T^{2} + \cdots + 642663 \) Copy content Toggle raw display
$67$ \( T^{6} + 41 T^{4} + 251 T^{2} + \cdots + 63 \) Copy content Toggle raw display
$71$ \( (T^{3} + 8 T^{2} - 56 T - 432)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 5 T^{2} - 93 T + 441)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + 11 T^{2} + 21 T + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 52 T^{4} + 432 T^{2} + \cdots + 448 \) Copy content Toggle raw display
$89$ \( (T^{3} - 5 T^{2} - 109 T - 231)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 10 T^{2} - 36 T - 28)^{2} \) Copy content Toggle raw display
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