Properties

Label 1152.4.d.m
Level $1152$
Weight $4$
Character orbit 1152.d
Analytic conductor $67.970$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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: \(\Q(i, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + 3 \beta_1 q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 \beta_1 q^{5} - \beta_{2} q^{7} - \beta_{3} q^{11} + 10 \beta_1 q^{13} + 8 q^{17} - 2 \beta_{3} q^{19} + 8 \beta_{2} q^{23} + 89 q^{25} + 23 \beta_1 q^{29} + \beta_{2} q^{31} - 3 \beta_{3} q^{35} + 82 \beta_1 q^{37} - 312 q^{41} + 10 \beta_{3} q^{43} - 8 \beta_{2} q^{47} + 105 q^{49} + 133 \beta_1 q^{53} + 12 \beta_{2} q^{55} + 6 \beta_{3} q^{59} - 66 \beta_1 q^{61} - 120 q^{65} - 12 \beta_{3} q^{67} - 32 \beta_{2} q^{71} - 246 q^{73} + 448 \beta_1 q^{77} - 11 \beta_{2} q^{79} + 23 \beta_{3} q^{83} + 24 \beta_1 q^{85} - 1392 q^{89} - 10 \beta_{3} q^{91} + 24 \beta_{2} q^{95} - 302 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 32 q^{17} + 356 q^{25} - 1248 q^{41} + 420 q^{49} - 480 q^{65} - 984 q^{73} - 5568 q^{89} - 1208 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu^{3} - \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -4\nu^{3} + 20\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 32\nu^{2} - 48 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 4\beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 48 ) / 32 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 20\beta_1 ) / 16 \) 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
1.32288 0.500000i
−1.32288 0.500000i
1.32288 + 0.500000i
−1.32288 + 0.500000i
0 0 0 6.00000i 0 −21.1660 0 0 0
577.2 0 0 0 6.00000i 0 21.1660 0 0 0
577.3 0 0 0 6.00000i 0 −21.1660 0 0 0
577.4 0 0 0 6.00000i 0 21.1660 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 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.m yes 4
3.b odd 2 1 1152.4.d.k 4
4.b odd 2 1 inner 1152.4.d.m yes 4
8.b even 2 1 inner 1152.4.d.m yes 4
8.d odd 2 1 inner 1152.4.d.m yes 4
12.b even 2 1 1152.4.d.k 4
16.e even 4 1 2304.4.a.r 2
16.e even 4 1 2304.4.a.br 2
16.f odd 4 1 2304.4.a.r 2
16.f odd 4 1 2304.4.a.br 2
24.f even 2 1 1152.4.d.k 4
24.h odd 2 1 1152.4.d.k 4
48.i odd 4 1 2304.4.a.q 2
48.i odd 4 1 2304.4.a.bs 2
48.k even 4 1 2304.4.a.q 2
48.k even 4 1 2304.4.a.bs 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.4.d.k 4 3.b odd 2 1
1152.4.d.k 4 12.b even 2 1
1152.4.d.k 4 24.f even 2 1
1152.4.d.k 4 24.h odd 2 1
1152.4.d.m yes 4 1.a even 1 1 trivial
1152.4.d.m yes 4 4.b odd 2 1 inner
1152.4.d.m yes 4 8.b even 2 1 inner
1152.4.d.m yes 4 8.d odd 2 1 inner
2304.4.a.q 2 48.i odd 4 1
2304.4.a.q 2 48.k even 4 1
2304.4.a.r 2 16.e even 4 1
2304.4.a.r 2 16.f odd 4 1
2304.4.a.br 2 16.e even 4 1
2304.4.a.br 2 16.f odd 4 1
2304.4.a.bs 2 48.i odd 4 1
2304.4.a.bs 2 48.k even 4 1

Hecke kernels

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

\( T_{5}^{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{2} - 448 \) Copy content Toggle raw display
\( T_{17} - 8 \) 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
$5$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 448)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1792)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 400)^{2} \) Copy content Toggle raw display
$17$ \( (T - 8)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 7168)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 28672)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2116)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 448)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 26896)^{2} \) Copy content Toggle raw display
$41$ \( (T + 312)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 179200)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 28672)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 70756)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 64512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 17424)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 258048)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 458752)^{2} \) Copy content Toggle raw display
$73$ \( (T + 246)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 54208)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 947968)^{2} \) Copy content Toggle raw display
$89$ \( (T + 1392)^{4} \) Copy content Toggle raw display
$97$ \( (T + 302)^{4} \) Copy content Toggle raw display
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