Properties

Label 448.4.i.b
Level $448$
Weight $4$
Character orbit 448.i
Analytic conductor $26.433$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,4,Mod(65,448)]
 
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([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.i (of order \(3\), 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: \(1\)
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 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + (5 \zeta_{6} - 5) q^{3} - 9 \zeta_{6} q^{5} + (14 \zeta_{6} - 21) q^{7} + 2 \zeta_{6} q^{9} + (57 \zeta_{6} - 57) q^{11} + 70 q^{13} + 45 q^{15} + (51 \zeta_{6} - 51) q^{17} + 5 \zeta_{6} q^{19} + ( - 105 \zeta_{6} + 35) q^{21} + \cdots - 114 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{3} - 9 q^{5} - 28 q^{7} + 2 q^{9} - 57 q^{11} + 140 q^{13} + 90 q^{15} - 51 q^{17} + 5 q^{19} - 35 q^{21} - 69 q^{23} + 44 q^{25} - 290 q^{27} - 228 q^{29} - 23 q^{31} - 285 q^{33} + 315 q^{35}+ \cdots - 228 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
65.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −2.50000 4.33013i 0 −4.50000 + 7.79423i 0 −14.0000 12.1244i 0 1.00000 1.73205i 0
193.1 0 −2.50000 + 4.33013i 0 −4.50000 7.79423i 0 −14.0000 + 12.1244i 0 1.00000 + 1.73205i 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.i.b 2
4.b odd 2 1 448.4.i.e 2
7.c even 3 1 inner 448.4.i.b 2
8.b even 2 1 14.4.c.a 2
8.d odd 2 1 112.4.i.a 2
24.h odd 2 1 126.4.g.d 2
28.g odd 6 1 448.4.i.e 2
40.f even 2 1 350.4.e.e 2
40.i odd 4 2 350.4.j.b 4
56.h odd 2 1 98.4.c.a 2
56.j odd 6 1 98.4.a.f 1
56.j odd 6 1 98.4.c.a 2
56.k odd 6 1 112.4.i.a 2
56.k odd 6 1 784.4.a.p 1
56.m even 6 1 784.4.a.c 1
56.p even 6 1 14.4.c.a 2
56.p even 6 1 98.4.a.d 1
168.i even 2 1 882.4.g.u 2
168.s odd 6 1 126.4.g.d 2
168.s odd 6 1 882.4.a.f 1
168.ba even 6 1 882.4.a.c 1
168.ba even 6 1 882.4.g.u 2
280.bf even 6 1 350.4.e.e 2
280.bf even 6 1 2450.4.a.q 1
280.bk odd 6 1 2450.4.a.d 1
280.bt odd 12 2 350.4.j.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 8.b even 2 1
14.4.c.a 2 56.p even 6 1
98.4.a.d 1 56.p even 6 1
98.4.a.f 1 56.j odd 6 1
98.4.c.a 2 56.h odd 2 1
98.4.c.a 2 56.j odd 6 1
112.4.i.a 2 8.d odd 2 1
112.4.i.a 2 56.k odd 6 1
126.4.g.d 2 24.h odd 2 1
126.4.g.d 2 168.s odd 6 1
350.4.e.e 2 40.f even 2 1
350.4.e.e 2 280.bf even 6 1
350.4.j.b 4 40.i odd 4 2
350.4.j.b 4 280.bt odd 12 2
448.4.i.b 2 1.a even 1 1 trivial
448.4.i.b 2 7.c even 3 1 inner
448.4.i.e 2 4.b odd 2 1
448.4.i.e 2 28.g odd 6 1
784.4.a.c 1 56.m even 6 1
784.4.a.p 1 56.k odd 6 1
882.4.a.c 1 168.ba even 6 1
882.4.a.f 1 168.s odd 6 1
882.4.g.u 2 168.i even 2 1
882.4.g.u 2 168.ba even 6 1
2450.4.a.d 1 280.bk odd 6 1
2450.4.a.q 1 280.bf even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\):

\( T_{3}^{2} + 5T_{3} + 25 \) Copy content Toggle raw display
\( T_{11}^{2} + 57T_{11} + 3249 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T - 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T + 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} + 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T + 42)^{2} \) Copy content Toggle raw display
$43$ \( (T - 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} + 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} - 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} + 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} - 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T + 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} + 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T - 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T + 1834)^{2} \) Copy content Toggle raw display
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