Properties

Label 532.2.a.b
Level $532$
Weight $2$
Character orbit 532.a
Self dual yes
Analytic conductor $4.248$
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] = [532,2,Mod(1,532)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(532, 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("532.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 532 = 2^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 532.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: \(4.24804138753\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{3} - q^{5} + q^{7} + (3 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{3} - q^{5} + q^{7} + (3 \beta - 1) q^{9} + (5 \beta - 2) q^{11} + ( - 2 \beta + 1) q^{13} + (\beta + 1) q^{15} + ( - \beta - 6) q^{17} - q^{19} + ( - \beta - 1) q^{21} + ( - 6 \beta + 1) q^{23} - 4 q^{25} + ( - 2 \beta + 1) q^{27} + (3 \beta - 4) q^{29} + (\beta - 4) q^{31} + ( - 8 \beta - 3) q^{33} - q^{35} + ( - 6 \beta + 3) q^{37} + (3 \beta + 1) q^{39} + (5 \beta - 8) q^{41} + (4 \beta + 2) q^{43} + ( - 3 \beta + 1) q^{45} + (6 \beta - 5) q^{47} + q^{49} + (8 \beta + 7) q^{51} + ( - 7 \beta - 3) q^{53} + ( - 5 \beta + 2) q^{55} + (\beta + 1) q^{57} + ( - 6 \beta - 1) q^{59} + (4 \beta - 1) q^{61} + (3 \beta - 1) q^{63} + (2 \beta - 1) q^{65} + ( - 5 \beta + 6) q^{67} + (11 \beta + 5) q^{69} + ( - 10 \beta + 5) q^{71} + (5 \beta - 9) q^{73} + (4 \beta + 4) q^{75} + (5 \beta - 2) q^{77} + ( - 8 \beta + 6) q^{79} + ( - 6 \beta + 4) q^{81} + (\beta + 13) q^{83} + (\beta + 6) q^{85} + ( - 2 \beta + 1) q^{87} + ( - 10 \beta + 4) q^{89} + ( - 2 \beta + 1) q^{91} + (2 \beta + 3) q^{93} + q^{95} + (6 \beta - 1) q^{97} + (4 \beta + 17) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} - 2 q^{5} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{3} - 2 q^{5} + 2 q^{7} + q^{9} + q^{11} + 3 q^{15} - 13 q^{17} - 2 q^{19} - 3 q^{21} - 4 q^{23} - 8 q^{25} - 5 q^{29} - 7 q^{31} - 14 q^{33} - 2 q^{35} + 5 q^{39} - 11 q^{41} + 8 q^{43} - q^{45} - 4 q^{47} + 2 q^{49} + 22 q^{51} - 13 q^{53} - q^{55} + 3 q^{57} - 8 q^{59} + 2 q^{61} + q^{63} + 7 q^{67} + 21 q^{69} - 13 q^{73} + 12 q^{75} + q^{77} + 4 q^{79} + 2 q^{81} + 27 q^{83} + 13 q^{85} - 2 q^{89} + 8 q^{93} + 2 q^{95} + 4 q^{97} + 38 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
0 −2.61803 0 −1.00000 0 1.00000 0 3.85410 0
1.2 0 −0.381966 0 −1.00000 0 1.00000 0 −2.85410 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( -1 \)
\(19\) \( +1 \)

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 532.2.a.b 2
3.b odd 2 1 4788.2.a.l 2
4.b odd 2 1 2128.2.a.m 2
7.b odd 2 1 3724.2.a.g 2
8.b even 2 1 8512.2.a.bg 2
8.d odd 2 1 8512.2.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
532.2.a.b 2 1.a even 1 1 trivial
2128.2.a.m 2 4.b odd 2 1
3724.2.a.g 2 7.b odd 2 1
4788.2.a.l 2 3.b odd 2 1
8512.2.a.k 2 8.d odd 2 1
8512.2.a.bg 2 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(532))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - T - 31 \) Copy content Toggle raw display
$13$ \( T^{2} - 5 \) Copy content Toggle raw display
$17$ \( T^{2} + 13T + 41 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$29$ \( T^{2} + 5T - 5 \) Copy content Toggle raw display
$31$ \( T^{2} + 7T + 11 \) Copy content Toggle raw display
$37$ \( T^{2} - 45 \) Copy content Toggle raw display
$41$ \( T^{2} + 11T - 1 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 4T - 41 \) Copy content Toggle raw display
$53$ \( T^{2} + 13T - 19 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 29 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$67$ \( T^{2} - 7T - 19 \) Copy content Toggle raw display
$71$ \( T^{2} - 125 \) Copy content Toggle raw display
$73$ \( T^{2} + 13T + 11 \) Copy content Toggle raw display
$79$ \( T^{2} - 4T - 76 \) Copy content Toggle raw display
$83$ \( T^{2} - 27T + 181 \) Copy content Toggle raw display
$89$ \( T^{2} + 2T - 124 \) Copy content Toggle raw display
$97$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
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