Properties

Label 5328.2.a.bc
Level $5328$
Weight $2$
Character orbit 5328.a
Self dual yes
Analytic conductor $42.544$
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] = [5328,2,Mod(1,5328)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5328, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5328.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5328 = 2^{4} \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5328.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: \(42.5442941969\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
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.

\(f(q)\) \(=\) \( q + ( - 3 \beta + 1) q^{5} + 2 \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta + 1) q^{5} + 2 \beta q^{7} + (\beta - 3) q^{11} + ( - 3 \beta + 2) q^{13} + (4 \beta - 2) q^{17} + ( - 4 \beta + 2) q^{19} + (3 \beta - 2) q^{23} + (3 \beta + 5) q^{25} + (7 \beta - 2) q^{29} + (\beta - 9) q^{31} + ( - 4 \beta - 6) q^{35} - q^{37} + ( - \beta - 8) q^{41} + (2 \beta + 2) q^{43} + ( - 2 \beta + 2) q^{47} + (4 \beta - 3) q^{49} + ( - 4 \beta + 6) q^{53} + (7 \beta - 6) q^{55} + (2 \beta - 8) q^{59} + (\beta + 9) q^{61} + 11 q^{65} + ( - 5 \beta + 7) q^{67} + (8 \beta - 10) q^{71} + (5 \beta - 1) q^{73} + ( - 4 \beta + 2) q^{77} + (9 \beta - 6) q^{79} + ( - 4 \beta - 8) q^{83} + ( - 2 \beta - 14) q^{85} + ( - 4 \beta + 8) q^{89} + ( - 2 \beta - 6) q^{91} + (2 \beta + 14) q^{95} + ( - 4 \beta + 6) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{5} + 2 q^{7} - 5 q^{11} + q^{13} - q^{23} + 13 q^{25} + 3 q^{29} - 17 q^{31} - 16 q^{35} - 2 q^{37} - 17 q^{41} + 6 q^{43} + 2 q^{47} - 2 q^{49} + 8 q^{53} - 5 q^{55} - 14 q^{59} + 19 q^{61} + 22 q^{65} + 9 q^{67} - 12 q^{71} + 3 q^{73} - 3 q^{79} - 20 q^{83} - 30 q^{85} + 12 q^{89} - 14 q^{91} + 30 q^{95} + 8 q^{97}+O(q^{100}) \) 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
1.61803
−0.618034
0 0 0 −3.85410 0 3.23607 0 0 0
1.2 0 0 0 2.85410 0 −1.23607 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(37\) \( +1 \)

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 5328.2.a.bc 2
3.b odd 2 1 592.2.a.g 2
4.b odd 2 1 666.2.a.i 2
12.b even 2 1 74.2.a.b 2
24.f even 2 1 2368.2.a.y 2
24.h odd 2 1 2368.2.a.u 2
60.h even 2 1 1850.2.a.t 2
60.l odd 4 2 1850.2.b.j 4
84.h odd 2 1 3626.2.a.s 2
132.d odd 2 1 8954.2.a.j 2
444.g even 2 1 2738.2.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.a.b 2 12.b even 2 1
592.2.a.g 2 3.b odd 2 1
666.2.a.i 2 4.b odd 2 1
1850.2.a.t 2 60.h even 2 1
1850.2.b.j 4 60.l odd 4 2
2368.2.a.u 2 24.h odd 2 1
2368.2.a.y 2 24.f even 2 1
2738.2.a.g 2 444.g even 2 1
3626.2.a.s 2 84.h odd 2 1
5328.2.a.bc 2 1.a even 1 1 trivial
8954.2.a.j 2 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5328))\):

\( T_{5}^{2} + T_{5} - 11 \) Copy content Toggle raw display
\( T_{7}^{2} - 2T_{7} - 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 5T_{11} + 5 \) Copy content Toggle raw display
\( T_{13}^{2} - T_{13} - 11 \) 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
$5$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 5T + 5 \) Copy content Toggle raw display
$13$ \( T^{2} - T - 11 \) Copy content Toggle raw display
$17$ \( T^{2} - 20 \) Copy content Toggle raw display
$19$ \( T^{2} - 20 \) Copy content Toggle raw display
$23$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$31$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$37$ \( (T + 1)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$47$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$53$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$59$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$61$ \( T^{2} - 19T + 89 \) Copy content Toggle raw display
$67$ \( T^{2} - 9T - 11 \) Copy content Toggle raw display
$71$ \( T^{2} + 12T - 44 \) Copy content Toggle raw display
$73$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} + 3T - 99 \) Copy content Toggle raw display
$83$ \( T^{2} + 20T + 80 \) Copy content Toggle raw display
$89$ \( T^{2} - 12T + 16 \) Copy content Toggle raw display
$97$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
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