Properties

Label 448.2.a.h
Level $448$
Weight $2$
Character orbit 448.a
Self dual yes
Analytic conductor $3.577$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(1,448)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.a (trivial)

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: yes
Analytic conductor: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 2 q^{3} + 4 q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{3} + 4 q^{5} - q^{7} + q^{9} + 8 q^{15} - 2 q^{17} - 2 q^{19} - 2 q^{21} - 8 q^{23} + 11 q^{25} - 4 q^{27} - 2 q^{29} - 4 q^{31} - 4 q^{35} + 6 q^{37} - 2 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} + q^{49} - 4 q^{51} + 10 q^{53} - 4 q^{57} + 6 q^{59} - 4 q^{61} - q^{63} - 12 q^{67} - 16 q^{69} - 14 q^{73} + 22 q^{75} + 8 q^{79} - 11 q^{81} + 6 q^{83} - 8 q^{85} - 4 q^{87} + 10 q^{89} - 8 q^{93} - 8 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 2.00000 0 4.00000 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.a.h 1
3.b odd 2 1 4032.2.a.a 1
4.b odd 2 1 448.2.a.c 1
7.b odd 2 1 3136.2.a.c 1
8.b even 2 1 112.2.a.a 1
8.d odd 2 1 56.2.a.b 1
12.b even 2 1 4032.2.a.d 1
16.e even 4 2 1792.2.b.h 2
16.f odd 4 2 1792.2.b.a 2
24.f even 2 1 504.2.a.h 1
24.h odd 2 1 1008.2.a.m 1
28.d even 2 1 3136.2.a.w 1
40.e odd 2 1 1400.2.a.a 1
40.f even 2 1 2800.2.a.bd 1
40.i odd 4 2 2800.2.g.g 2
40.k even 4 2 1400.2.g.b 2
56.e even 2 1 392.2.a.b 1
56.h odd 2 1 784.2.a.i 1
56.j odd 6 2 784.2.i.b 2
56.k odd 6 2 392.2.i.a 2
56.m even 6 2 392.2.i.e 2
56.p even 6 2 784.2.i.j 2
88.g even 2 1 6776.2.a.h 1
104.h odd 2 1 9464.2.a.h 1
168.e odd 2 1 3528.2.a.b 1
168.i even 2 1 7056.2.a.c 1
168.v even 6 2 3528.2.s.a 2
168.be odd 6 2 3528.2.s.ba 2
280.n even 2 1 9800.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.b 1 8.d odd 2 1
112.2.a.a 1 8.b even 2 1
392.2.a.b 1 56.e even 2 1
392.2.i.a 2 56.k odd 6 2
392.2.i.e 2 56.m even 6 2
448.2.a.c 1 4.b odd 2 1
448.2.a.h 1 1.a even 1 1 trivial
504.2.a.h 1 24.f even 2 1
784.2.a.i 1 56.h odd 2 1
784.2.i.b 2 56.j odd 6 2
784.2.i.j 2 56.p even 6 2
1008.2.a.m 1 24.h odd 2 1
1400.2.a.a 1 40.e odd 2 1
1400.2.g.b 2 40.k even 4 2
1792.2.b.a 2 16.f odd 4 2
1792.2.b.h 2 16.e even 4 2
2800.2.a.bd 1 40.f even 2 1
2800.2.g.g 2 40.i odd 4 2
3136.2.a.c 1 7.b odd 2 1
3136.2.a.w 1 28.d even 2 1
3528.2.a.b 1 168.e odd 2 1
3528.2.s.a 2 168.v even 6 2
3528.2.s.ba 2 168.be odd 6 2
4032.2.a.a 1 3.b odd 2 1
4032.2.a.d 1 12.b even 2 1
6776.2.a.h 1 88.g even 2 1
7056.2.a.c 1 168.i even 2 1
9464.2.a.h 1 104.h odd 2 1
9800.2.a.bj 1 280.n even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(448))\):

\( T_{3} - 2 \) Copy content Toggle raw display
\( T_{5} - 4 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 2 \) Copy content Toggle raw display
$5$ \( T - 4 \) Copy content Toggle raw display
$7$ \( T + 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T + 2 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T + 8 \) Copy content Toggle raw display
$29$ \( T + 2 \) Copy content Toggle raw display
$31$ \( T + 4 \) Copy content Toggle raw display
$37$ \( T - 6 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T - 4 \) Copy content Toggle raw display
$53$ \( T - 10 \) Copy content Toggle raw display
$59$ \( T - 6 \) Copy content Toggle raw display
$61$ \( T + 4 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T - 8 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 10 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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