Properties

Label 448.2.e.a
Level $448$
Weight $2$
Character orbit 448.e
Analytic conductor $3.577$
Analytic rank $0$
Dimension $8$
CM discriminant -56
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(223,448)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.223");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.e (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: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.629407744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{7} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + \beta_1 q^{5} - \beta_{4} q^{7} + ( - \beta_{7} - 3) q^{9} + (\beta_{6} - \beta_1) q^{13} + ( - 2 \beta_{4} - \beta_{2}) q^{15} + \beta_{5} q^{19} + ( - \beta_{6} - \beta_1) q^{21} + 3 \beta_{2} q^{23} + ( - \beta_{7} + 5) q^{25} + ( - \beta_{5} + 5 \beta_{3}) q^{27} + ( - \beta_{5} - 2 \beta_{3}) q^{35} + ( - 2 \beta_{4} - 5 \beta_{2}) q^{39} + ( - 3 \beta_{6} + \beta_1) q^{45} - 7 q^{49} + ( - \beta_{7} + 2) q^{57} + (\beta_{5} - 4 \beta_{3}) q^{59} + (2 \beta_{6} + \beta_1) q^{61} + (3 \beta_{4} + 7 \beta_{2}) q^{63} + (3 \beta_{7} - 6) q^{65} + 3 \beta_{6} q^{69} + 6 \beta_{4} q^{71} - \beta_{5} q^{75} - 2 \beta_{4} q^{79} + (3 \beta_{7} + 19) q^{81} + (2 \beta_{5} + \beta_{3}) q^{83} + (2 \beta_{5} - 3 \beta_{3}) q^{91} + (6 \beta_{4} - 9 \beta_{2}) q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} + 40 q^{25} - 56 q^{49} + 16 q^{57} - 48 q^{65} + 152 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} - 10\nu^{3} + 28\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 2\nu^{4} - 2\nu^{2} - 4 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 2\nu^{3} - 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - \nu^{4} + 4\nu^{2} - 7 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 4\nu^{5} + 2\nu^{3} + 4\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 2\nu^{3} + 10\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{6} + 2\nu^{4} + 2\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{3} + 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + 2\beta_{4} - \beta_{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{6} + \beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{6} - 3\beta_{5} + 5\beta_{3} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -\beta_{7} + 2\beta_{4} + 5\beta_{2} + 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -\beta_{6} + \beta_{5} + 7\beta_{3} + 4\beta_1 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
223.1
−1.38255 + 0.297594i
1.38255 + 0.297594i
−0.767178 + 1.18804i
0.767178 + 1.18804i
−0.767178 1.18804i
0.767178 1.18804i
−1.38255 0.297594i
1.38255 0.297594i
0 3.36028i 0 −2.16991 0 2.64575i 0 −8.29150 0
223.2 0 3.36028i 0 2.16991 0 2.64575i 0 −8.29150 0
223.3 0 0.841723i 0 −3.91044 0 2.64575i 0 2.29150 0
223.4 0 0.841723i 0 3.91044 0 2.64575i 0 2.29150 0
223.5 0 0.841723i 0 −3.91044 0 2.64575i 0 2.29150 0
223.6 0 0.841723i 0 3.91044 0 2.64575i 0 2.29150 0
223.7 0 3.36028i 0 −2.16991 0 2.64575i 0 −8.29150 0
223.8 0 3.36028i 0 2.16991 0 2.64575i 0 −8.29150 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 223.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
4.b odd 2 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.d even 2 1 inner
56.e 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.e.a 8
3.b odd 2 1 4032.2.p.h 8
4.b odd 2 1 inner 448.2.e.a 8
7.b odd 2 1 inner 448.2.e.a 8
8.b even 2 1 inner 448.2.e.a 8
8.d odd 2 1 inner 448.2.e.a 8
12.b even 2 1 4032.2.p.h 8
16.e even 4 2 1792.2.f.k 8
16.f odd 4 2 1792.2.f.k 8
21.c even 2 1 4032.2.p.h 8
24.f even 2 1 4032.2.p.h 8
24.h odd 2 1 4032.2.p.h 8
28.d even 2 1 inner 448.2.e.a 8
56.e even 2 1 inner 448.2.e.a 8
56.h odd 2 1 CM 448.2.e.a 8
84.h odd 2 1 4032.2.p.h 8
112.j even 4 2 1792.2.f.k 8
112.l odd 4 2 1792.2.f.k 8
168.e odd 2 1 4032.2.p.h 8
168.i even 2 1 4032.2.p.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.2.e.a 8 1.a even 1 1 trivial
448.2.e.a 8 4.b odd 2 1 inner
448.2.e.a 8 7.b odd 2 1 inner
448.2.e.a 8 8.b even 2 1 inner
448.2.e.a 8 8.d odd 2 1 inner
448.2.e.a 8 28.d even 2 1 inner
448.2.e.a 8 56.e even 2 1 inner
448.2.e.a 8 56.h odd 2 1 CM
1792.2.f.k 8 16.e even 4 2
1792.2.f.k 8 16.f odd 4 2
1792.2.f.k 8 112.j even 4 2
1792.2.f.k 8 112.l odd 4 2
4032.2.p.h 8 3.b odd 2 1
4032.2.p.h 8 12.b even 2 1
4032.2.p.h 8 21.c even 2 1
4032.2.p.h 8 24.f even 2 1
4032.2.p.h 8 24.h odd 2 1
4032.2.p.h 8 84.h odd 2 1
4032.2.p.h 8 168.e odd 2 1
4032.2.p.h 8 168.i even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 12T_{3}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} + 12 T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 20 T^{2} + 72)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 52 T^{2} + 648)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} + 76 T^{2} + 72)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 236 T^{2} + 5832)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 244 T^{2} + 72)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 252)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( (T^{2} + 28)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 332 T^{2} + 648)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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