Properties

Label 448.4.p.a
Level $448$
Weight $4$
Character orbit 448.p
Analytic conductor $26.433$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,4,Mod(255,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.255");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.p (of order \(6\), degree \(2\), 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: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (7 \zeta_{6} - 7) q^{3} + (9 \zeta_{6} - 18) q^{5} + ( - 14 \zeta_{6} - 7) q^{7} - 22 \zeta_{6} q^{9} + ( - 7 \zeta_{6} - 7) q^{11} + ( - 80 \zeta_{6} + 40) q^{13} + ( - 126 \zeta_{6} + 63) q^{15} + \cdots + (308 \zeta_{6} - 154) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 7 q^{3} - 27 q^{5} - 28 q^{7} - 22 q^{9} - 21 q^{11} - 57 q^{17} + 119 q^{19} + 245 q^{21} + 231 q^{23} + 118 q^{25} - 70 q^{27} - 420 q^{29} - 301 q^{31} + 147 q^{33} + 567 q^{35} - 77 q^{37} + 840 q^{39}+ \cdots - 3213 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(127\) \(129\) \(197\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
255.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −3.50000 + 6.06218i 0 −13.5000 + 7.79423i 0 −14.0000 12.1244i 0 −11.0000 19.0526i 0
383.1 0 −3.50000 6.06218i 0 −13.5000 7.79423i 0 −14.0000 + 12.1244i 0 −11.0000 + 19.0526i 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
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.p.a 2
4.b odd 2 1 448.4.p.d 2
7.d odd 6 1 448.4.p.d 2
8.b even 2 1 112.4.p.d yes 2
8.d odd 2 1 112.4.p.a 2
28.f even 6 1 inner 448.4.p.a 2
56.j odd 6 1 112.4.p.a 2
56.j odd 6 1 784.4.f.d 2
56.k odd 6 1 784.4.f.d 2
56.m even 6 1 112.4.p.d yes 2
56.m even 6 1 784.4.f.a 2
56.p even 6 1 784.4.f.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.4.p.a 2 8.d odd 2 1
112.4.p.a 2 56.j odd 6 1
112.4.p.d yes 2 8.b even 2 1
112.4.p.d yes 2 56.m even 6 1
448.4.p.a 2 1.a even 1 1 trivial
448.4.p.a 2 28.f even 6 1 inner
448.4.p.d 2 4.b odd 2 1
448.4.p.d 2 7.d odd 6 1
784.4.f.a 2 56.m even 6 1
784.4.f.a 2 56.p even 6 1
784.4.f.d 2 56.j odd 6 1
784.4.f.d 2 56.k odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 7T_{3} + 49 \) acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$5$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$7$ \( T^{2} + 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 21T + 147 \) Copy content Toggle raw display
$13$ \( T^{2} + 4800 \) Copy content Toggle raw display
$17$ \( T^{2} + 57T + 1083 \) Copy content Toggle raw display
$19$ \( T^{2} - 119T + 14161 \) Copy content Toggle raw display
$23$ \( T^{2} - 231T + 17787 \) Copy content Toggle raw display
$29$ \( (T + 210)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 301T + 90601 \) Copy content Toggle raw display
$37$ \( T^{2} + 77T + 5929 \) Copy content Toggle raw display
$41$ \( T^{2} + 13872 \) Copy content Toggle raw display
$43$ \( T^{2} + 14700 \) Copy content Toggle raw display
$47$ \( T^{2} - 357T + 127449 \) Copy content Toggle raw display
$53$ \( T^{2} - 327T + 106929 \) Copy content Toggle raw display
$59$ \( T^{2} - 609T + 370881 \) Copy content Toggle raw display
$61$ \( T^{2} + 1191 T + 472827 \) Copy content Toggle raw display
$67$ \( T^{2} + 273T + 24843 \) Copy content Toggle raw display
$71$ \( T^{2} + 47628 \) Copy content Toggle raw display
$73$ \( T^{2} - 99T + 3267 \) Copy content Toggle raw display
$79$ \( T^{2} - 1407 T + 659883 \) Copy content Toggle raw display
$83$ \( (T - 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1275 T + 541875 \) Copy content Toggle raw display
$97$ \( T^{2} + 97200 \) Copy content Toggle raw display
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