Properties

Label 5586.2.a.bq
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $0$
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] = [5586,2,Mod(1,5586)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5586, 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("5586.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5586 = 2 \cdot 3 \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5586.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: \(44.6044345691\)
Analytic rank: \(0\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} + (\beta + 1) q^{5} + q^{6} + q^{8} + q^{9} + (\beta + 1) q^{10} + (\beta - 3) q^{11} + q^{12} + (\beta + 1) q^{13} + (\beta + 1) q^{15} + q^{16} + (\beta + 5) q^{17} + q^{18} - q^{19} + (\beta + 1) q^{20} + (\beta - 3) q^{22} + ( - \beta + 3) q^{23} + q^{24} + (2 \beta + 1) q^{25} + (\beta + 1) q^{26} + q^{27} - 2 \beta q^{29} + (\beta + 1) q^{30} + ( - 2 \beta - 2) q^{31} + q^{32} + (\beta - 3) q^{33} + (\beta + 5) q^{34} + q^{36} + 6 q^{37} - q^{38} + (\beta + 1) q^{39} + (\beta + 1) q^{40} + 4 \beta q^{41} + ( - 2 \beta + 2) q^{43} + (\beta - 3) q^{44} + (\beta + 1) q^{45} + ( - \beta + 3) q^{46} + ( - 4 \beta + 4) q^{47} + q^{48} + (2 \beta + 1) q^{50} + (\beta + 5) q^{51} + (\beta + 1) q^{52} - 6 \beta q^{53} + q^{54} + ( - 2 \beta + 2) q^{55} - q^{57} - 2 \beta q^{58} - 4 \beta q^{59} + (\beta + 1) q^{60} + ( - 2 \beta - 2) q^{61} + ( - 2 \beta - 2) q^{62} + q^{64} + (2 \beta + 6) q^{65} + (\beta - 3) q^{66} + ( - \beta + 7) q^{67} + (\beta + 5) q^{68} + ( - \beta + 3) q^{69} + ( - 2 \beta - 2) q^{71} + q^{72} + 6 q^{74} + (2 \beta + 1) q^{75} - q^{76} + (\beta + 1) q^{78} + ( - 3 \beta + 5) q^{79} + (\beta + 1) q^{80} + q^{81} + 4 \beta q^{82} + (2 \beta - 10) q^{83} + (6 \beta + 10) q^{85} + ( - 2 \beta + 2) q^{86} - 2 \beta q^{87} + (\beta - 3) q^{88} + 4 \beta q^{89} + (\beta + 1) q^{90} + ( - \beta + 3) q^{92} + ( - 2 \beta - 2) q^{93} + ( - 4 \beta + 4) q^{94} + ( - \beta - 1) q^{95} + q^{96} + (3 \beta - 1) q^{97} + (\beta - 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{10} - 6 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{15} + 2 q^{16} + 10 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{20} - 6 q^{22} + 6 q^{23} + 2 q^{24} + 2 q^{25} + 2 q^{26} + 2 q^{27} + 2 q^{30} - 4 q^{31} + 2 q^{32} - 6 q^{33} + 10 q^{34} + 2 q^{36} + 12 q^{37} - 2 q^{38} + 2 q^{39} + 2 q^{40} + 4 q^{43} - 6 q^{44} + 2 q^{45} + 6 q^{46} + 8 q^{47} + 2 q^{48} + 2 q^{50} + 10 q^{51} + 2 q^{52} + 2 q^{54} + 4 q^{55} - 2 q^{57} + 2 q^{60} - 4 q^{61} - 4 q^{62} + 2 q^{64} + 12 q^{65} - 6 q^{66} + 14 q^{67} + 10 q^{68} + 6 q^{69} - 4 q^{71} + 2 q^{72} + 12 q^{74} + 2 q^{75} - 2 q^{76} + 2 q^{78} + 10 q^{79} + 2 q^{80} + 2 q^{81} - 20 q^{83} + 20 q^{85} + 4 q^{86} - 6 q^{88} + 2 q^{90} + 6 q^{92} - 4 q^{93} + 8 q^{94} - 2 q^{95} + 2 q^{96} - 2 q^{97} - 6 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
1.00000 1.00000 1.00000 −1.23607 1.00000 0 1.00000 1.00000 −1.23607
1.2 1.00000 1.00000 1.00000 3.23607 1.00000 0 1.00000 1.00000 3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(-1\)
\(19\) \(1\)

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 5586.2.a.bq yes 2
7.b odd 2 1 5586.2.a.bh 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5586.2.a.bh 2 7.b odd 2 1
5586.2.a.bq 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{2} - 2T_{5} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} - 10T_{17} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$29$ \( T^{2} - 20 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$37$ \( (T - 6)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 80 \) Copy content Toggle raw display
$43$ \( T^{2} - 4T - 16 \) Copy content Toggle raw display
$47$ \( T^{2} - 8T - 64 \) Copy content Toggle raw display
$53$ \( T^{2} - 180 \) Copy content Toggle raw display
$59$ \( T^{2} - 80 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$67$ \( T^{2} - 14T + 44 \) Copy content Toggle raw display
$71$ \( T^{2} + 4T - 16 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 80 \) Copy content Toggle raw display
$89$ \( T^{2} - 80 \) Copy content Toggle raw display
$97$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
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