Properties

Label 1472.4.a.f
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 46)
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)
 
\(f(q)\) \(=\) \( q + q^{3} + 10 q^{5} - 12 q^{7} - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 10 q^{5} - 12 q^{7} - 26 q^{9} + 42 q^{11} - 7 q^{13} + 10 q^{15} + 20 q^{17} - 106 q^{19} - 12 q^{21} + 23 q^{23} - 25 q^{25} - 53 q^{27} + 227 q^{29} + 67 q^{31} + 42 q^{33} - 120 q^{35} - 74 q^{37} - 7 q^{39} - 497 q^{41} + 88 q^{43} - 260 q^{45} + 215 q^{47} - 199 q^{49} + 20 q^{51} - 314 q^{53} + 420 q^{55} - 106 q^{57} - 176 q^{59} + 298 q^{61} + 312 q^{63} - 70 q^{65} - 266 q^{67} + 23 q^{69} - 981 q^{71} - 411 q^{73} - 25 q^{75} - 504 q^{77} + 806 q^{79} + 649 q^{81} + 952 q^{83} + 200 q^{85} + 227 q^{87} - 1332 q^{89} + 84 q^{91} + 67 q^{93} - 1060 q^{95} - 1328 q^{97} - 1092 q^{99}+O(q^{100}) \) 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 1.00000 0 10.0000 0 −12.0000 0 −26.0000 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.f 1
4.b odd 2 1 1472.4.a.e 1
8.b even 2 1 46.4.a.a 1
8.d odd 2 1 368.4.a.b 1
24.h odd 2 1 414.4.a.d 1
40.f even 2 1 1150.4.a.g 1
40.i odd 4 2 1150.4.b.e 2
56.h odd 2 1 2254.4.a.a 1
184.e odd 2 1 1058.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
46.4.a.a 1 8.b even 2 1
368.4.a.b 1 8.d odd 2 1
414.4.a.d 1 24.h odd 2 1
1058.4.a.a 1 184.e odd 2 1
1150.4.a.g 1 40.f even 2 1
1150.4.b.e 2 40.i odd 4 2
1472.4.a.e 1 4.b odd 2 1
1472.4.a.f 1 1.a even 1 1 trivial
2254.4.a.a 1 56.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 1 \) 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
$3$ \( T - 1 \) Copy content Toggle raw display
$5$ \( T - 10 \) Copy content Toggle raw display
$7$ \( T + 12 \) Copy content Toggle raw display
$11$ \( T - 42 \) Copy content Toggle raw display
$13$ \( T + 7 \) Copy content Toggle raw display
$17$ \( T - 20 \) Copy content Toggle raw display
$19$ \( T + 106 \) Copy content Toggle raw display
$23$ \( T - 23 \) Copy content Toggle raw display
$29$ \( T - 227 \) Copy content Toggle raw display
$31$ \( T - 67 \) Copy content Toggle raw display
$37$ \( T + 74 \) Copy content Toggle raw display
$41$ \( T + 497 \) Copy content Toggle raw display
$43$ \( T - 88 \) Copy content Toggle raw display
$47$ \( T - 215 \) Copy content Toggle raw display
$53$ \( T + 314 \) Copy content Toggle raw display
$59$ \( T + 176 \) Copy content Toggle raw display
$61$ \( T - 298 \) Copy content Toggle raw display
$67$ \( T + 266 \) Copy content Toggle raw display
$71$ \( T + 981 \) Copy content Toggle raw display
$73$ \( T + 411 \) Copy content Toggle raw display
$79$ \( T - 806 \) Copy content Toggle raw display
$83$ \( T - 952 \) Copy content Toggle raw display
$89$ \( T + 1332 \) Copy content Toggle raw display
$97$ \( T + 1328 \) Copy content Toggle raw display
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