Properties

Label 1152.4.d.l
Level $1152$
Weight $4$
Character orbit 1152.d
Analytic conductor $67.970$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,4,Mod(577,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.577");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1152.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(67.9702003266\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{5} + \beta_{2} q^{7} + \beta_1 q^{11} + 17 q^{25} - 21 \beta_{3} q^{29} + 23 \beta_{2} q^{31} + 27 \beta_1 q^{35} - 127 q^{49} + 49 \beta_{3} q^{53} - 4 \beta_{2} q^{55} + 98 \beta_1 q^{59} + 322 q^{73} - 8 \beta_{3} q^{77} - 21 \beta_{2} q^{79} + 217 \beta_1 q^{83} - 574 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 68 q^{25} - 508 q^{49} + 1288 q^{73} - 2296 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( -2\nu^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -3\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 6\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{2} - 3\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 0 0 10.3923i 0 −14.6969 0 0 0
577.2 0 0 0 10.3923i 0 14.6969 0 0 0
577.3 0 0 0 10.3923i 0 −14.6969 0 0 0
577.4 0 0 0 10.3923i 0 14.6969 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.4.d.l 4
3.b odd 2 1 inner 1152.4.d.l 4
4.b odd 2 1 inner 1152.4.d.l 4
8.b even 2 1 inner 1152.4.d.l 4
8.d odd 2 1 inner 1152.4.d.l 4
12.b even 2 1 inner 1152.4.d.l 4
16.e even 4 2 2304.4.a.ca 4
16.f odd 4 2 2304.4.a.ca 4
24.f even 2 1 inner 1152.4.d.l 4
24.h odd 2 1 CM 1152.4.d.l 4
48.i odd 4 2 2304.4.a.ca 4
48.k even 4 2 2304.4.a.ca 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.4.d.l 4 1.a even 1 1 trivial
1152.4.d.l 4 3.b odd 2 1 inner
1152.4.d.l 4 4.b odd 2 1 inner
1152.4.d.l 4 8.b even 2 1 inner
1152.4.d.l 4 8.d odd 2 1 inner
1152.4.d.l 4 12.b even 2 1 inner
1152.4.d.l 4 24.f even 2 1 inner
1152.4.d.l 4 24.h odd 2 1 CM
2304.4.a.ca 4 16.e even 4 2
2304.4.a.ca 4 16.f odd 4 2
2304.4.a.ca 4 48.i odd 4 2
2304.4.a.ca 4 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{2} + 108 \) Copy content Toggle raw display
\( T_{7}^{2} - 216 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 108)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 216)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 47628)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 114264)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 259308)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 307328)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 322)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 95256)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 1506848)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T + 574)^{4} \) Copy content Toggle raw display
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