Properties

Label 1472.4.a.h
Level $1472$
Weight $4$
Character orbit 1472.a
Self dual yes
Analytic conductor $86.851$
Analytic rank $1$
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] = [1472,4,Mod(1,1472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1472.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1472 = 2^{6} \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1472.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: \(86.8508115285\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
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)
 
\(f(q)\) \(=\) \( q + 5 q^{3} + 6 q^{5} - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 5 q^{3} + 6 q^{5} - 8 q^{7} - 2 q^{9} - 34 q^{11} + 57 q^{13} + 30 q^{15} - 80 q^{17} + 70 q^{19} - 40 q^{21} + 23 q^{23} - 89 q^{25} - 145 q^{27} - 245 q^{29} + 103 q^{31} - 170 q^{33} - 48 q^{35} + 298 q^{37} + 285 q^{39} + 95 q^{41} - 88 q^{43} - 12 q^{45} - 357 q^{47} - 279 q^{49} - 400 q^{51} + 414 q^{53} - 204 q^{55} + 350 q^{57} + 408 q^{59} - 822 q^{61} + 16 q^{63} + 342 q^{65} - 926 q^{67} + 115 q^{69} + 335 q^{71} - 899 q^{73} - 445 q^{75} + 272 q^{77} - 1322 q^{79} - 671 q^{81} + 36 q^{83} - 480 q^{85} - 1225 q^{87} - 460 q^{89} - 456 q^{91} + 515 q^{93} + 420 q^{95} - 964 q^{97} + 68 q^{99}+O(q^{100}) \) 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 5.00000 0 6.00000 0 −8.00000 0 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(23\) \( -1 \)

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 1472.4.a.h 1
4.b odd 2 1 1472.4.a.c 1
8.b even 2 1 23.4.a.a 1
8.d odd 2 1 368.4.a.d 1
24.h odd 2 1 207.4.a.a 1
40.f even 2 1 575.4.a.g 1
40.i odd 4 2 575.4.b.b 2
56.h odd 2 1 1127.4.a.a 1
184.e odd 2 1 529.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
23.4.a.a 1 8.b even 2 1
207.4.a.a 1 24.h odd 2 1
368.4.a.d 1 8.d odd 2 1
529.4.a.a 1 184.e odd 2 1
575.4.a.g 1 40.f even 2 1
575.4.b.b 2 40.i odd 4 2
1127.4.a.a 1 56.h odd 2 1
1472.4.a.c 1 4.b odd 2 1
1472.4.a.h 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 5 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1472))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 5 \) Copy content Toggle raw display
$5$ \( T - 6 \) Copy content Toggle raw display
$7$ \( T + 8 \) Copy content Toggle raw display
$11$ \( T + 34 \) Copy content Toggle raw display
$13$ \( T - 57 \) Copy content Toggle raw display
$17$ \( T + 80 \) Copy content Toggle raw display
$19$ \( T - 70 \) Copy content Toggle raw display
$23$ \( T - 23 \) Copy content Toggle raw display
$29$ \( T + 245 \) Copy content Toggle raw display
$31$ \( T - 103 \) Copy content Toggle raw display
$37$ \( T - 298 \) Copy content Toggle raw display
$41$ \( T - 95 \) Copy content Toggle raw display
$43$ \( T + 88 \) Copy content Toggle raw display
$47$ \( T + 357 \) Copy content Toggle raw display
$53$ \( T - 414 \) Copy content Toggle raw display
$59$ \( T - 408 \) Copy content Toggle raw display
$61$ \( T + 822 \) Copy content Toggle raw display
$67$ \( T + 926 \) Copy content Toggle raw display
$71$ \( T - 335 \) Copy content Toggle raw display
$73$ \( T + 899 \) Copy content Toggle raw display
$79$ \( T + 1322 \) Copy content Toggle raw display
$83$ \( T - 36 \) Copy content Toggle raw display
$89$ \( T + 460 \) Copy content Toggle raw display
$97$ \( T + 964 \) Copy content Toggle raw display
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