Properties

Label 1584.4.a.y
Level $1584$
Weight $4$
Character orbit 1584.a
Self dual yes
Analytic conductor $93.459$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1584,4,Mod(1,1584)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1584, 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("1584.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1584.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: \(93.4590254491\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{31}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 31 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 396)
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 \(\beta = 2\sqrt{31}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 6) q^{5} + ( - \beta + 12) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 6) q^{5} + ( - \beta + 12) q^{7} - 11 q^{11} + (5 \beta - 4) q^{13} + ( - 6 \beta + 14) q^{17} + (2 \beta + 34) q^{19} + ( - 7 \beta - 134) q^{23} + ( - 12 \beta + 35) q^{25} + (4 \beta + 130) q^{29} + ( - 14 \beta - 56) q^{31} + (18 \beta - 196) q^{35} + ( - 22 \beta + 158) q^{37} + (8 \beta + 62) q^{41} + 74 q^{43} + (11 \beta - 286) q^{47} + ( - 24 \beta - 75) q^{49} + (45 \beta + 38) q^{53} + ( - 11 \beta + 66) q^{55} + (6 \beta - 432) q^{59} + (51 \beta - 68) q^{61} + ( - 34 \beta + 644) q^{65} + ( - 40 \beta + 256) q^{67} + ( - 35 \beta - 362) q^{71} + ( - 12 \beta + 486) q^{73} + (11 \beta - 132) q^{77} + (5 \beta + 264) q^{79} + (22 \beta - 1056) q^{83} + (50 \beta - 828) q^{85} + ( - 112 \beta + 144) q^{89} + (64 \beta - 668) q^{91} + (22 \beta + 44) q^{95} + ( - 72 \beta + 250) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 12 q^{5} + 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 12 q^{5} + 24 q^{7} - 22 q^{11} - 8 q^{13} + 28 q^{17} + 68 q^{19} - 268 q^{23} + 70 q^{25} + 260 q^{29} - 112 q^{31} - 392 q^{35} + 316 q^{37} + 124 q^{41} + 148 q^{43} - 572 q^{47} - 150 q^{49} + 76 q^{53} + 132 q^{55} - 864 q^{59} - 136 q^{61} + 1288 q^{65} + 512 q^{67} - 724 q^{71} + 972 q^{73} - 264 q^{77} + 528 q^{79} - 2112 q^{83} - 1656 q^{85} + 288 q^{89} - 1336 q^{91} + 88 q^{95} + 500 q^{97}+O(q^{100}) \) 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
−5.56776
5.56776
0 0 0 −17.1355 0 23.1355 0 0 0
1.2 0 0 0 5.13553 0 0.864471 0 0 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 1584.4.a.y 2
3.b odd 2 1 1584.4.a.bi 2
4.b odd 2 1 396.4.a.h 2
12.b even 2 1 396.4.a.j yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
396.4.a.h 2 4.b odd 2 1
396.4.a.j yes 2 12.b even 2 1
1584.4.a.y 2 1.a even 1 1 trivial
1584.4.a.bi 2 3.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1584))\):

\( T_{5}^{2} + 12T_{5} - 88 \) Copy content Toggle raw display
\( T_{7}^{2} - 24T_{7} + 20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 12T - 88 \) Copy content Toggle raw display
$7$ \( T^{2} - 24T + 20 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 8T - 3084 \) Copy content Toggle raw display
$17$ \( T^{2} - 28T - 4268 \) Copy content Toggle raw display
$19$ \( T^{2} - 68T + 660 \) Copy content Toggle raw display
$23$ \( T^{2} + 268T + 11880 \) Copy content Toggle raw display
$29$ \( T^{2} - 260T + 14916 \) Copy content Toggle raw display
$31$ \( T^{2} + 112T - 21168 \) Copy content Toggle raw display
$37$ \( T^{2} - 316T - 35052 \) Copy content Toggle raw display
$41$ \( T^{2} - 124T - 4092 \) Copy content Toggle raw display
$43$ \( (T - 74)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 572T + 66792 \) Copy content Toggle raw display
$53$ \( T^{2} - 76T - 249656 \) Copy content Toggle raw display
$59$ \( T^{2} + 864T + 182160 \) Copy content Toggle raw display
$61$ \( T^{2} + 136T - 317900 \) Copy content Toggle raw display
$67$ \( T^{2} - 512T - 132864 \) Copy content Toggle raw display
$71$ \( T^{2} + 724T - 20856 \) Copy content Toggle raw display
$73$ \( T^{2} - 972T + 218340 \) Copy content Toggle raw display
$79$ \( T^{2} - 528T + 66596 \) Copy content Toggle raw display
$83$ \( T^{2} + 2112 T + 1055120 \) Copy content Toggle raw display
$89$ \( T^{2} - 288 T - 1534720 \) Copy content Toggle raw display
$97$ \( T^{2} - 500T - 580316 \) Copy content Toggle raw display
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