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Properties

Label 1568.3.g.h
Level $1568$
Weight $3$
Character orbit 1568.g
Analytic conductor $42.725$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 6x^{2} + 16 \) 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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - 4 \beta_1 - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 2) q^{3} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - 4 \beta_1 - 3) q^{9} + ( - 6 \beta_1 - 4) q^{11} + (\beta_{3} - \beta_{2}) q^{13} - \beta_{3} q^{15} + (4 \beta_1 - 18) q^{17} + (19 \beta_1 + 2) q^{19} + ( - \beta_{3} - 8 \beta_{2}) q^{23} + ( - 28 \beta_1 - 17) q^{25} + (4 \beta_1 - 16) q^{27} - 6 \beta_{2} q^{29} + ( - 6 \beta_{3} - 4 \beta_{2}) q^{31} + ( - 8 \beta_1 + 4) q^{33} + ( - 4 \beta_{3} - 10 \beta_{2}) q^{37} + \beta_{3} q^{39} + ( - 12 \beta_1 + 10) q^{41} + (2 \beta_1 + 20) q^{43} + (7 \beta_{3} - 11 \beta_{2}) q^{45} + ( - 4 \beta_{3} - 4 \beta_{2}) q^{47} + (26 \beta_1 - 44) q^{51} + (10 \beta_{3} + 12 \beta_{2}) q^{53} + (10 \beta_{3} - 16 \beta_{2}) q^{55} + (36 \beta_1 - 34) q^{57} + ( - 11 \beta_1 + 46) q^{59} + (5 \beta_{3} + 3 \beta_{2}) q^{61} + (28 \beta_1 + 42) q^{65} + (16 \beta_1 + 56) q^{67} + ( - 10 \beta_{3} - 18 \beta_{2}) q^{69} + ( - 8 \beta_{3} - 16 \beta_{2}) q^{71} + (8 \beta_1 - 58) q^{73} + ( - 39 \beta_1 + 22) q^{75} + (4 \beta_{3} - 16 \beta_{2}) q^{79} + (60 \beta_1 - 13) q^{81} + (13 \beta_1 + 22) q^{83} + (14 \beta_{3} - 10 \beta_{2}) q^{85} + ( - 6 \beta_{3} - 12 \beta_{2}) q^{87} + ( - 24 \beta_1 - 78) q^{89} + ( - 16 \beta_{3} - 20 \beta_{2}) q^{93} + ( - 21 \beta_{3} + 40 \beta_{2}) q^{95} + (92 \beta_1 + 34) q^{97} + (34 \beta_1 + 60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{3} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{3} - 12 q^{9} - 16 q^{11} - 72 q^{17} + 8 q^{19} - 68 q^{25} - 64 q^{27} + 16 q^{33} + 40 q^{41} + 80 q^{43} - 176 q^{51} - 136 q^{57} + 184 q^{59} + 168 q^{65} + 224 q^{67} - 232 q^{73} + 88 q^{75} - 52 q^{81} + 88 q^{83} - 312 q^{89} + 136 q^{97} + 240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 6x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 10\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 5\beta_1 \) 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
−0.707107 − 1.87083i
−0.707107 + 1.87083i
0.707107 + 1.87083i
0.707107 − 1.87083i
0 0.585786 0 − 9.03316i 0 0 0 −8.65685 0
687.2 0 0.585786 0 9.03316i 0 0 0 −8.65685 0
687.3 0 3.41421 0 − 1.54985i 0 0 0 2.65685 0
687.4 0 3.41421 0 1.54985i 0 0 0 2.65685 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
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.h 4
4.b odd 2 1 392.3.g.h 4
7.b odd 2 1 224.3.g.a 4
8.b even 2 1 392.3.g.h 4
8.d odd 2 1 inner 1568.3.g.h 4
21.c even 2 1 2016.3.g.a 4
28.d even 2 1 56.3.g.a ✓ 4
28.f even 6 2 392.3.k.i 8
28.g odd 6 2 392.3.k.j 8
56.e even 2 1 224.3.g.a 4
56.h odd 2 1 56.3.g.a ✓ 4
56.j odd 6 2 392.3.k.i 8
56.p even 6 2 392.3.k.j 8
84.h odd 2 1 504.3.g.a 4
112.j even 4 2 1792.3.d.g 8
112.l odd 4 2 1792.3.d.g 8
168.e odd 2 1 2016.3.g.a 4
168.i even 2 1 504.3.g.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.3.g.a ✓ 4 28.d even 2 1
56.3.g.a ✓ 4 56.h odd 2 1
224.3.g.a 4 7.b odd 2 1
224.3.g.a 4 56.e even 2 1
392.3.g.h 4 4.b odd 2 1
392.3.g.h 4 8.b even 2 1
392.3.k.i 8 28.f even 6 2
392.3.k.i 8 56.j odd 6 2
392.3.k.j 8 28.g odd 6 2
392.3.k.j 8 56.p even 6 2
504.3.g.a 4 84.h odd 2 1
504.3.g.a 4 168.i even 2 1
1568.3.g.h 4 1.a even 1 1 trivial
1568.3.g.h 4 8.d odd 2 1 inner
1792.3.d.g 8 112.j even 4 2
1792.3.d.g 8 112.l odd 4 2
2016.3.g.a 4 21.c even 2 1
2016.3.g.a 4 168.e odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 4T_{3} + 2 \) acting on \(S_{3}^{\mathrm{new}}(1568, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 84T^{2} + 196 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T - 56)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 84T^{2} + 196 \) Copy content Toggle raw display
$17$ \( (T^{2} + 36 T + 292)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 4 T - 718)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1848 T^{2} + 753424 \) Copy content Toggle raw display
$29$ \( (T^{2} + 504)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 2464 T^{2} + 614656 \) Copy content Toggle raw display
$37$ \( T^{4} + 3696 T^{2} + 906304 \) Copy content Toggle raw display
$41$ \( (T^{2} - 20 T - 188)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 40 T + 392)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 1344 T^{2} + 50176 \) Copy content Toggle raw display
$53$ \( T^{4} + 9632 T^{2} + 614656 \) Copy content Toggle raw display
$59$ \( (T^{2} - 92 T + 1874)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 1652 T^{2} + 329476 \) Copy content Toggle raw display
$67$ \( (T^{2} - 112 T + 2624)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 10752 T^{2} + 3211264 \) Copy content Toggle raw display
$73$ \( (T^{2} + 116 T + 3236)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 8064 T^{2} + 9834496 \) Copy content Toggle raw display
$83$ \( (T^{2} - 44 T + 146)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 156 T + 4932)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 68 T - 15772)^{2} \) Copy content Toggle raw display
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