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Properties

Label 1568.3.g.l
Level $1568$
Weight $3$
Character orbit 1568.g
Analytic conductor $42.725$
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] = [1568,3,Mod(687,1568)]
 
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, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1568.687");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1568.g (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: \(42.7249054517\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.15582448.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{5} + (\beta_{2} - 4 \beta_1 + 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} - \beta_1 + 1) q^{3} + ( - \beta_{5} + \beta_{4} - \beta_{3}) q^{5} + (\beta_{2} - 4 \beta_1 + 5) q^{9} + (3 \beta_1 + 6) q^{11} + (2 \beta_{5} - 2 \beta_{4} - 5 \beta_{3}) q^{13} + ( - 3 \beta_{5} - 2 \beta_{3}) q^{15} + ( - \beta_{2} + 2 \beta_1 + 6) q^{17} + ( - 4 \beta_{2} - \beta_1 - 12) q^{19} + (7 \beta_{5} - 5 \beta_{3}) q^{23} + (2 \beta_{2} - 2 \beta_1 + 14) q^{25} + (2 \beta_{2} - 13 \beta_1 + 8) q^{27} + (2 \beta_{5} - 8 \beta_{4} + 3 \beta_{3}) q^{29} + ( - 9 \beta_{5} - 6 \beta_{3}) q^{31} + ( - 6 \beta_{2} + 6 \beta_1 - 9) q^{33} + ( - \beta_{5} - 9 \beta_{4} + 4 \beta_{3}) q^{37} + ( - 8 \beta_{5} - 14 \beta_{4} - 17 \beta_{3}) q^{39} + (\beta_{2} + 18 \beta_1 + 25) q^{41} + (10 \beta_{2} - 10 \beta_1 + 10) q^{43} + (2 \beta_{5} - 4 \beta_{4} - 9 \beta_{3}) q^{45} + (5 \beta_{5} + 16 \beta_{4} - 2 \beta_{3}) q^{47} + ( - 4 \beta_{2} + 3 \beta_1 + 4) q^{51} + ( - 7 \beta_{5} + 3 \beta_{4} - 20 \beta_{3}) q^{53} + ( - 9 \beta_{5} + 12 \beta_{4} - 3 \beta_{3}) q^{55} + (20 \beta_{2} + 12 \beta_1 + 25) q^{57} + ( - 5 \beta_{2} - 9 \beta_1 + 17) q^{59} + (25 \beta_{5} - 15 \beta_{4} + 10 \beta_{3}) q^{61} + ( - 11 \beta_{2} + 18 \beta_1 + 15) q^{65} + ( - 3 \beta_{2} - \beta_1 + 73) q^{67} + ( - 3 \beta_{5} - 31 \beta_{4} + 13 \beta_{3}) q^{69} + (8 \beta_{5} - 6 \beta_{4} + 22 \beta_{3}) q^{71} + ( - 12 \beta_{2} + 18 \beta_1 + 27) q^{73} + ( - 18 \beta_{2} - 24 \beta_1 + 8) q^{75} + ( - 25 \beta_{5} + 16 \beta_{4} + 5 \beta_{3}) q^{79} + ( - 21 \beta_{2} - 26 \beta_1 + 12) q^{81} + ( - 14 \beta_{2} + 12 \beta_1 - 36) q^{83} + ( - 11 \beta_{5} + 11 \beta_{4} - 4 \beta_{3}) q^{85} + (8 \beta_{5} + 8 \beta_{4} - 23 \beta_{3}) q^{87} + ( - 36 \beta_{2} + 4 \beta_1 + 49) q^{89} + ( - 21 \beta_{5} + 15 \beta_{4} - 54 \beta_{3}) q^{93} + (\beta_{5} - 10 \beta_{4} + 11 \beta_{3}) q^{95} + ( - 15 \beta_{2} - 8 \beta_1 + 15) q^{97} + (21 \beta_{2} + 12 \beta_1 - 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 40 q^{9} + 30 q^{11} + 30 q^{17} - 78 q^{19} + 92 q^{25} + 78 q^{27} - 78 q^{33} + 116 q^{41} + 100 q^{43} + 10 q^{51} + 166 q^{57} + 110 q^{59} + 32 q^{65} + 434 q^{67} + 102 q^{73} + 60 q^{75} + 82 q^{81} - 268 q^{83} + 214 q^{89} + 76 q^{97} - 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} + 13x^{4} - 21x^{3} + 20x^{2} - 10x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{4} - 2\nu^{3} + 10\nu^{2} - 9\nu + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 2\nu^{3} + 11\nu^{2} - 10\nu + 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{5} - 5\nu^{4} + 24\nu^{3} - 31\nu^{2} + 30\nu - 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -6\nu^{5} + 15\nu^{4} - 70\nu^{3} + 90\nu^{2} - 69\nu + 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -6\nu^{5} + 15\nu^{4} - 70\nu^{3} + 90\nu^{2} - 71\nu + 21 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} + 2\beta_{2} - 2\beta _1 - 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 9\beta_{5} - 8\beta_{4} + 3\beta_{3} + 3\beta_{2} - 3\beta _1 - 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{5} - 17\beta_{4} + 6\beta_{3} - 14\beta_{2} + 16\beta _1 + 51 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -61\beta_{5} + 54\beta_{4} - 20\beta_{3} - 40\beta_{2} + 45\beta _1 + 146 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
687.1
0.500000 + 0.148124i
0.500000 − 0.148124i
0.500000 − 2.94141i
0.500000 + 2.94141i
0.500000 + 0.759064i
0.500000 − 0.759064i
0 −3.98103 0 − 1.88252i 0 0 0 6.84860 0
687.2 0 −3.98103 0 1.88252i 0 0 0 6.84860 0
687.3 0 1.64878 0 − 4.56111i 0 0 0 −6.28154 0
687.4 0 1.64878 0 4.56111i 0 0 0 −6.28154 0
687.5 0 5.33225 0 − 2.15693i 0 0 0 19.4329 0
687.6 0 5.33225 0 2.15693i 0 0 0 19.4329 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 687.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d 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.g.l 6
4.b odd 2 1 392.3.g.i 6
7.b odd 2 1 1568.3.g.j 6
7.d odd 6 2 224.3.o.d 12
8.b even 2 1 392.3.g.i 6
8.d odd 2 1 inner 1568.3.g.l 6
28.d even 2 1 392.3.g.j 6
28.f even 6 2 56.3.k.d ✓ 12
28.g odd 6 2 392.3.k.l 12
56.e even 2 1 1568.3.g.j 6
56.h odd 2 1 392.3.g.j 6
56.j odd 6 2 56.3.k.d ✓ 12
56.m even 6 2 224.3.o.d 12
56.p even 6 2 392.3.k.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.k.d ✓ 12 28.f even 6 2
56.3.k.d ✓ 12 56.j odd 6 2
224.3.o.d 12 7.d odd 6 2
224.3.o.d 12 56.m even 6 2
392.3.g.i 6 4.b odd 2 1
392.3.g.i 6 8.b even 2 1
392.3.g.j 6 28.d even 2 1
392.3.g.j 6 56.h odd 2 1
392.3.k.l 12 28.g odd 6 2
392.3.k.l 12 56.p even 6 2
1568.3.g.j 6 7.b odd 2 1
1568.3.g.j 6 56.e even 2 1
1568.3.g.l 6 1.a even 1 1 trivial
1568.3.g.l 6 8.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 3T_{3}^{2} - 19T_{3} + 35 \) acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T^{3} - 3 T^{2} - 19 T + 35)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 29 T^{4} + \cdots + 343 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 15 T^{2} + \cdots + 513)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 928 T^{4} + \cdots + 20420848 \) Copy content Toggle raw display
$17$ \( (T^{3} - 15 T^{2} + \cdots + 35)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + 39 T^{2} + \cdots + 539)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + 2481 T^{4} + \cdots + 103165447 \) Copy content Toggle raw display
$29$ \( T^{6} + 1384 T^{4} + \cdots + 35753200 \) Copy content Toggle raw display
$31$ \( T^{6} + 2205 T^{4} + \cdots + 306307575 \) Copy content Toggle raw display
$37$ \( T^{6} + 2429 T^{4} + \cdots + 64563583 \) Copy content Toggle raw display
$41$ \( (T^{3} - 58 T^{2} + \cdots + 106036)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 50 T^{2} + \cdots + 77000)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 5967253663 \) Copy content Toggle raw display
$53$ \( T^{6} + 6293 T^{4} + \cdots + 91073143 \) Copy content Toggle raw display
$59$ \( (T^{3} - 55 T^{2} + \cdots + 6559)^{2} \) Copy content Toggle raw display
$61$ \( T^{6} + \cdots + 4507234375 \) Copy content Toggle raw display
$67$ \( (T^{3} - 217 T^{2} + \cdots - 368695)^{2} \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 1372000000 \) Copy content Toggle raw display
$73$ \( (T^{3} - 51 T^{2} + \cdots - 46305)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 41185599175 \) Copy content Toggle raw display
$83$ \( (T^{3} + 134 T^{2} + \cdots - 185080)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 107 T^{2} + \cdots + 908663)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 38 T^{2} + \cdots + 117740)^{2} \) Copy content Toggle raw display
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