Properties

Label 529.4.a.e
Level $529$
Weight $4$
Character orbit 529.a
Self dual yes
Analytic conductor $31.212$
Analytic rank $1$
Dimension $2$
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] = [529,4,Mod(1,529)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(529, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("529.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 529 = 23^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 529.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: \(31.2120103930\)
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: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 2) q^{2} + (3 \beta - 1) q^{3} + ( - 4 \beta - 1) q^{4} + ( - 7 \beta + 8) q^{5} + ( - 7 \beta + 11) q^{6} + ( - \beta + 11) q^{7} + ( - \beta + 6) q^{8} + ( - 6 \beta + 1) q^{9} + (22 \beta - 37) q^{10}+ \cdots + ( - 97 \beta + 141) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} - 2 q^{3} - 2 q^{4} + 16 q^{5} + 22 q^{6} + 22 q^{7} + 12 q^{8} + 2 q^{9} - 74 q^{10} + 30 q^{11} - 70 q^{12} - 22 q^{13} - 50 q^{14} - 142 q^{15} - 14 q^{16} - 104 q^{17} - 40 q^{18} + 10 q^{19}+ \cdots + 282 q^{99}+O(q^{100}) \) 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
−1.73205
1.73205
−3.73205 −6.19615 5.92820 20.1244 23.1244 12.7321 7.73205 11.3923 −75.1051
1.2 −0.267949 4.19615 −7.92820 −4.12436 −1.12436 9.26795 4.26795 −9.39230 1.10512
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 529.4.a.e yes 2
23.b odd 2 1 529.4.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
529.4.a.d 2 23.b odd 2 1
529.4.a.e yes 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(529))\):

\( T_{2}^{2} + 4T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} + 2T_{3} - 26 \) Copy content Toggle raw display
\( T_{5}^{2} - 16T_{5} - 83 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 2T - 26 \) Copy content Toggle raw display
$5$ \( T^{2} - 16T - 83 \) Copy content Toggle raw display
$7$ \( T^{2} - 22T + 118 \) Copy content Toggle raw display
$11$ \( T^{2} - 30T + 78 \) Copy content Toggle raw display
$13$ \( T^{2} + 22T - 3767 \) Copy content Toggle raw display
$17$ \( T^{2} + 104T + 676 \) Copy content Toggle raw display
$19$ \( T^{2} - 10T - 7178 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 146T - 9371 \) Copy content Toggle raw display
$31$ \( T^{2} + 172T + 3508 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 110588 \) Copy content Toggle raw display
$41$ \( T^{2} + 270T + 13893 \) Copy content Toggle raw display
$43$ \( T^{2} + 500T + 48628 \) Copy content Toggle raw display
$47$ \( T^{2} + 258T - 9306 \) Copy content Toggle raw display
$53$ \( T^{2} + 196T - 331103 \) Copy content Toggle raw display
$59$ \( T^{2} + 614T - 163298 \) Copy content Toggle raw display
$61$ \( T^{2} + 100T - 34463 \) Copy content Toggle raw display
$67$ \( T^{2} - 36T - 245064 \) Copy content Toggle raw display
$71$ \( T^{2} + 290T - 627650 \) Copy content Toggle raw display
$73$ \( T^{2} - 318T - 342219 \) Copy content Toggle raw display
$79$ \( T^{2} - 212T - 564296 \) Copy content Toggle raw display
$83$ \( T^{2} - 1476 T + 434052 \) Copy content Toggle raw display
$89$ \( T^{2} + 1308 T - 516447 \) Copy content Toggle raw display
$97$ \( T^{2} - 200T - 230267 \) Copy content Toggle raw display
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