Properties

Label 448.2.f.b
Level $448$
Weight $2$
Character orbit 448.f
Analytic conductor $3.577$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(447,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.447");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.f (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: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-7}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{7} - 3 q^{9} - 2 \beta q^{11} - 2 \beta q^{23} + 5 q^{25} + 2 q^{29} - 6 q^{37} - 2 \beta q^{43} - 7 q^{49} + 10 q^{53} + 3 \beta q^{63} + 6 \beta q^{67} - 2 \beta q^{71} - 14 q^{77} + 6 \beta q^{79} + 9 q^{81} + 6 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{9} + 10 q^{25} + 4 q^{29} - 12 q^{37} - 14 q^{49} + 20 q^{53} - 28 q^{77} + 18 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
447.1
0.500000 + 1.32288i
0.500000 1.32288i
0 0 0 0 0 2.64575i 0 −3.00000 0
447.2 0 0 0 0 0 2.64575i 0 −3.00000 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
4.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.f.b 2
3.b odd 2 1 4032.2.b.e 2
4.b odd 2 1 inner 448.2.f.b 2
7.b odd 2 1 CM 448.2.f.b 2
8.b even 2 1 28.2.d.a 2
8.d odd 2 1 28.2.d.a 2
12.b even 2 1 4032.2.b.e 2
16.e even 4 2 1792.2.e.b 4
16.f odd 4 2 1792.2.e.b 4
21.c even 2 1 4032.2.b.e 2
24.f even 2 1 252.2.b.a 2
24.h odd 2 1 252.2.b.a 2
28.d even 2 1 inner 448.2.f.b 2
40.e odd 2 1 700.2.g.a 2
40.f even 2 1 700.2.g.a 2
40.i odd 4 2 700.2.c.d 4
40.k even 4 2 700.2.c.d 4
56.e even 2 1 28.2.d.a 2
56.h odd 2 1 28.2.d.a 2
56.j odd 6 2 196.2.f.b 4
56.k odd 6 2 196.2.f.b 4
56.m even 6 2 196.2.f.b 4
56.p even 6 2 196.2.f.b 4
84.h odd 2 1 4032.2.b.e 2
112.j even 4 2 1792.2.e.b 4
112.l odd 4 2 1792.2.e.b 4
168.e odd 2 1 252.2.b.a 2
168.i even 2 1 252.2.b.a 2
280.c odd 2 1 700.2.g.a 2
280.n even 2 1 700.2.g.a 2
280.s even 4 2 700.2.c.d 4
280.y odd 4 2 700.2.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.d.a 2 8.b even 2 1
28.2.d.a 2 8.d odd 2 1
28.2.d.a 2 56.e even 2 1
28.2.d.a 2 56.h odd 2 1
196.2.f.b 4 56.j odd 6 2
196.2.f.b 4 56.k odd 6 2
196.2.f.b 4 56.m even 6 2
196.2.f.b 4 56.p even 6 2
252.2.b.a 2 24.f even 2 1
252.2.b.a 2 24.h odd 2 1
252.2.b.a 2 168.e odd 2 1
252.2.b.a 2 168.i even 2 1
448.2.f.b 2 1.a even 1 1 trivial
448.2.f.b 2 4.b odd 2 1 inner
448.2.f.b 2 7.b odd 2 1 CM
448.2.f.b 2 28.d even 2 1 inner
700.2.c.d 4 40.i odd 4 2
700.2.c.d 4 40.k even 4 2
700.2.c.d 4 280.s even 4 2
700.2.c.d 4 280.y odd 4 2
700.2.g.a 2 40.e odd 2 1
700.2.g.a 2 40.f even 2 1
700.2.g.a 2 280.c odd 2 1
700.2.g.a 2 280.n even 2 1
1792.2.e.b 4 16.e even 4 2
1792.2.e.b 4 16.f odd 4 2
1792.2.e.b 4 112.j even 4 2
1792.2.e.b 4 112.l odd 4 2
4032.2.b.e 2 3.b odd 2 1
4032.2.b.e 2 12.b even 2 1
4032.2.b.e 2 21.c even 2 1
4032.2.b.e 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\). 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 28 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 28 \) Copy content Toggle raw display
$29$ \( (T - 2)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T + 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 28 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 10)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 252 \) Copy content Toggle raw display
$71$ \( T^{2} + 28 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 252 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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