Properties

Label 528.4.a.f
Level $528$
Weight $4$
Character orbit 528.a
Self dual yes
Analytic conductor $31.153$
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] = [528,4,Mod(1,528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(528, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("528.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 528 = 2^{4} \cdot 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 528.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: \(31.1530084830\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 264)
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)
 
\(f(q)\) \(=\) \( q + 3 q^{3} - 18 q^{5} + 28 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} - 18 q^{5} + 28 q^{7} + 9 q^{9} - 11 q^{11} - 18 q^{13} - 54 q^{15} - 34 q^{17} - 80 q^{19} + 84 q^{21} - 128 q^{23} + 199 q^{25} + 27 q^{27} + 162 q^{29} + 312 q^{31} - 33 q^{33} - 504 q^{35} - 290 q^{37} - 54 q^{39} - 146 q^{41} - 256 q^{43} - 162 q^{45} - 432 q^{47} + 441 q^{49} - 102 q^{51} - 490 q^{53} + 198 q^{55} - 240 q^{57} - 836 q^{59} + 230 q^{61} + 252 q^{63} + 324 q^{65} - 900 q^{67} - 384 q^{69} - 520 q^{71} - 798 q^{73} + 597 q^{75} - 308 q^{77} + 484 q^{79} + 81 q^{81} + 812 q^{83} + 612 q^{85} + 486 q^{87} + 74 q^{89} - 504 q^{91} + 936 q^{93} + 1440 q^{95} - 1790 q^{97} - 99 q^{99}+O(q^{100}) \) 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 3.00000 0 −18.0000 0 28.0000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(11\) \(1\)

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 528.4.a.f 1
3.b odd 2 1 1584.4.a.u 1
4.b odd 2 1 264.4.a.a 1
8.b even 2 1 2112.4.a.m 1
8.d odd 2 1 2112.4.a.z 1
12.b even 2 1 792.4.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.4.a.a 1 4.b odd 2 1
528.4.a.f 1 1.a even 1 1 trivial
792.4.a.f 1 12.b even 2 1
1584.4.a.u 1 3.b odd 2 1
2112.4.a.m 1 8.b even 2 1
2112.4.a.z 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(528))\):

\( T_{5} + 18 \) Copy content Toggle raw display
\( T_{7} - 28 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 3 \) Copy content Toggle raw display
$5$ \( T + 18 \) Copy content Toggle raw display
$7$ \( T - 28 \) Copy content Toggle raw display
$11$ \( T + 11 \) Copy content Toggle raw display
$13$ \( T + 18 \) Copy content Toggle raw display
$17$ \( T + 34 \) Copy content Toggle raw display
$19$ \( T + 80 \) Copy content Toggle raw display
$23$ \( T + 128 \) Copy content Toggle raw display
$29$ \( T - 162 \) Copy content Toggle raw display
$31$ \( T - 312 \) Copy content Toggle raw display
$37$ \( T + 290 \) Copy content Toggle raw display
$41$ \( T + 146 \) Copy content Toggle raw display
$43$ \( T + 256 \) Copy content Toggle raw display
$47$ \( T + 432 \) Copy content Toggle raw display
$53$ \( T + 490 \) Copy content Toggle raw display
$59$ \( T + 836 \) Copy content Toggle raw display
$61$ \( T - 230 \) Copy content Toggle raw display
$67$ \( T + 900 \) Copy content Toggle raw display
$71$ \( T + 520 \) Copy content Toggle raw display
$73$ \( T + 798 \) Copy content Toggle raw display
$79$ \( T - 484 \) Copy content Toggle raw display
$83$ \( T - 812 \) Copy content Toggle raw display
$89$ \( T - 74 \) Copy content Toggle raw display
$97$ \( T + 1790 \) Copy content Toggle raw display
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