Properties

Label 5586.2.a.ca
Level $5586$
Weight $2$
Character orbit 5586.a
Self dual yes
Analytic conductor $44.604$
Analytic rank $1$
Dimension $4$
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: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.202932.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 12x^{2} + 12x + 33 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 798)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. 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_{2} + \beta_1 - 1) q^{11} - q^{12} - 2 q^{13} + \beta_1 q^{15} + q^{16} + (\beta_1 - 3) q^{17} + q^{18} - q^{19} - \beta_1 q^{20} + ( - \beta_{2} + \beta_1 - 1) q^{22} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{23} - q^{24} + (\beta_{2} + \beta_1 + 2) q^{25} - 2 q^{26} - q^{27} + ( - \beta_{3} - 1) q^{29} + \beta_1 q^{30} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{31} + q^{32} + (\beta_{2} - \beta_1 + 1) q^{33} + (\beta_1 - 3) q^{34} + q^{36} + (\beta_1 - 1) q^{37} - q^{38} + 2 q^{39} - \beta_1 q^{40} + (2 \beta_{3} + \beta_{2} - 3) q^{41} + ( - \beta_{3} - \beta_{2} - 3) q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{44} - \beta_1 q^{45} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{46} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{47} - q^{48} + (\beta_{2} + \beta_1 + 2) q^{50} + ( - \beta_1 + 3) q^{51} - 2 q^{52} + ( - \beta_{3} - \beta_{2} + 3 \beta_1 + 1) q^{53} - q^{54} + (2 \beta_{3} + \beta_{2} - 6) q^{55} + q^{57} + ( - \beta_{3} - 1) q^{58} + (\beta_{3} + 2 \beta_{2} + 3 \beta_1 - 6) q^{59} + \beta_1 q^{60} + (\beta_{3} - 3 \beta_{2} - \beta_1 - 4) q^{61} + ( - \beta_{3} + \beta_{2} + \beta_1 + 1) q^{62} + q^{64} + 2 \beta_1 q^{65} + (\beta_{2} - \beta_1 + 1) q^{66} + (2 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{67} + (\beta_1 - 3) q^{68} + ( - \beta_{3} - 2 \beta_{2} + \beta_1 - 3) q^{69} + ( - 3 \beta_1 - 3) q^{71} + q^{72} + (\beta_{3} - 4) q^{73} + (\beta_1 - 1) q^{74} + ( - \beta_{2} - \beta_1 - 2) q^{75} - q^{76} + 2 q^{78} + (\beta_{3} + 2 \beta_{2} - 1) q^{79} - \beta_1 q^{80} + q^{81} + (2 \beta_{3} + \beta_{2} - 3) q^{82} + ( - 2 \beta_{2} - 5) q^{83} + ( - \beta_{2} + 2 \beta_1 - 7) q^{85} + ( - \beta_{3} - \beta_{2} - 3) q^{86} + (\beta_{3} + 1) q^{87} + ( - \beta_{2} + \beta_1 - 1) q^{88} + (\beta_{2} + 1) q^{89} - \beta_1 q^{90} + (\beta_{3} + 2 \beta_{2} - \beta_1 + 3) q^{92} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{93} + ( - 3 \beta_{3} - \beta_{2} + \beta_1 + 2) q^{94} + \beta_1 q^{95} - q^{96} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 - 4) q^{97} + ( - \beta_{2} + \beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 2 q^{5} - 4 q^{6} + 4 q^{8} + 4 q^{9} - 2 q^{10} - 4 q^{12} - 8 q^{13} + 2 q^{15} + 4 q^{16} - 10 q^{17} + 4 q^{18} - 4 q^{19} - 2 q^{20} + 7 q^{23} - 4 q^{24} + 8 q^{25} - 8 q^{26} - 4 q^{27} - 5 q^{29} + 2 q^{30} + 3 q^{31} + 4 q^{32} - 10 q^{34} + 4 q^{36} - 2 q^{37} - 4 q^{38} + 8 q^{39} - 2 q^{40} - 12 q^{41} - 11 q^{43} - 2 q^{45} + 7 q^{46} + 9 q^{47} - 4 q^{48} + 8 q^{50} + 10 q^{51} - 8 q^{52} + 11 q^{53} - 4 q^{54} - 24 q^{55} + 4 q^{57} - 5 q^{58} - 21 q^{59} + 2 q^{60} - 11 q^{61} + 3 q^{62} + 4 q^{64} + 4 q^{65} - 14 q^{67} - 10 q^{68} - 7 q^{69} - 18 q^{71} + 4 q^{72} - 15 q^{73} - 2 q^{74} - 8 q^{75} - 4 q^{76} + 8 q^{78} - 7 q^{79} - 2 q^{80} + 4 q^{81} - 12 q^{82} - 16 q^{83} - 22 q^{85} - 11 q^{86} + 5 q^{87} + 2 q^{89} - 2 q^{90} + 7 q^{92} - 3 q^{93} + 9 q^{94} + 2 q^{95} - 4 q^{96} - 19 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} - 12x^{2} + 12x + 33 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 3\nu^{2} - 5\nu + 13 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 3\beta_{2} + 8\beta _1 + 8 \) 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
3.71730
2.32952
−1.49111
−2.55571
1.00000 −1.00000 1.00000 −3.71730 −1.00000 0 1.00000 1.00000 −3.71730
1.2 1.00000 −1.00000 1.00000 −2.32952 −1.00000 0 1.00000 1.00000 −2.32952
1.3 1.00000 −1.00000 1.00000 1.49111 −1.00000 0 1.00000 1.00000 1.49111
1.4 1.00000 −1.00000 1.00000 2.55571 −1.00000 0 1.00000 1.00000 2.55571
\(n\): 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.ca 4
7.b odd 2 1 5586.2.a.cb 4
7.d odd 6 2 798.2.j.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
798.2.j.k 8 7.d odd 6 2
5586.2.a.ca 4 1.a even 1 1 trivial
5586.2.a.cb 4 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5586))\):

\( T_{5}^{4} + 2T_{5}^{3} - 12T_{5}^{2} - 12T_{5} + 33 \) Copy content Toggle raw display
\( T_{11}^{4} - 30T_{11}^{2} + 12T_{11} + 9 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{4} + 10T_{17}^{3} + 24T_{17}^{2} - 6T_{17} - 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{4} \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + 2 T^{3} - 12 T^{2} - 12 T + 33 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 30 T^{2} + 12 T + 9 \) Copy content Toggle raw display
$13$ \( (T + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 10 T^{3} + 24 T^{2} - 6 T - 12 \) Copy content Toggle raw display
$19$ \( (T + 1)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 7 T^{3} - 54 T^{2} + 492 T - 894 \) Copy content Toggle raw display
$29$ \( T^{4} + 5 T^{3} - 21 T^{2} - 81 T + 12 \) Copy content Toggle raw display
$31$ \( T^{4} - 3 T^{3} - 69 T^{2} + 335 T - 168 \) Copy content Toggle raw display
$37$ \( T^{4} + 2 T^{3} - 12 T^{2} - 14 T + 32 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} - 54 T^{2} + \cdots + 1944 \) Copy content Toggle raw display
$43$ \( T^{4} + 11 T^{3} + 12 T^{2} - 86 T + 14 \) Copy content Toggle raw display
$47$ \( T^{4} - 9 T^{3} - 192 T^{2} + \cdots + 7056 \) Copy content Toggle raw display
$53$ \( T^{4} - 11 T^{3} - 69 T^{2} + \cdots + 1704 \) Copy content Toggle raw display
$59$ \( T^{4} + 21 T^{3} - 81 T^{2} + \cdots - 18738 \) Copy content Toggle raw display
$61$ \( T^{4} + 11 T^{3} - 222 T^{2} + \cdots + 10264 \) Copy content Toggle raw display
$67$ \( T^{4} + 14 T^{3} - 216 T^{2} + \cdots - 9088 \) Copy content Toggle raw display
$71$ \( T^{4} + 18 T^{3} - 702 T + 972 \) Copy content Toggle raw display
$73$ \( T^{4} + 15 T^{3} + 54 T^{2} + \cdots - 108 \) Copy content Toggle raw display
$79$ \( T^{4} + 7 T^{3} - 57 T^{2} - 311 T - 356 \) Copy content Toggle raw display
$83$ \( T^{4} + 16 T^{3} + 18 T^{2} + \cdots + 453 \) Copy content Toggle raw display
$89$ \( T^{4} - 2 T^{3} - 18 T^{2} + 18 T + 84 \) Copy content Toggle raw display
$97$ \( T^{4} + 19 T^{3} - 15 T^{2} + \cdots + 1556 \) Copy content Toggle raw display
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