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Properties

Label 1152.3.e.e
Level $1152$
Weight $3$
Character orbit 1152.e
Analytic conductor $31.390$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1152,3,Mod(1025,1152)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.1025");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1152.e (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: \(31.3897264543\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_1) q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + \beta_1) q^{5} + 2 q^{7} + (2 \beta_{3} - 6 \beta_1) q^{11} + (\beta_{2} - 8) q^{13} + ( - 2 \beta_{3} - \beta_1) q^{17} + ( - 2 \beta_{2} - 8) q^{19} + ( - 4 \beta_{3} + 6 \beta_1) q^{23} + (2 \beta_{2} - 21) q^{25} + ( - \beta_{3} - 13 \beta_1) q^{29} + (4 \beta_{2} + 2) q^{31} + ( - 2 \beta_{3} + 2 \beta_1) q^{35} - 26 q^{37} + ( - 2 \beta_{3} - 11 \beta_1) q^{41} + (4 \beta_{2} - 28) q^{43} + (4 \beta_{3} + 38 \beta_1) q^{47} - 45 q^{49} + (3 \beta_{3} - 21 \beta_1) q^{53} + ( - 8 \beta_{2} + 100) q^{55} - 52 \beta_1 q^{59} + ( - 4 \beta_{2} - 30) q^{61} + (10 \beta_{3} - 52 \beta_1) q^{65} + ( - 2 \beta_{2} - 84) q^{67} + (8 \beta_{3} - 22 \beta_1) q^{71} + ( - 12 \beta_{2} - 28) q^{73} + (4 \beta_{3} - 12 \beta_1) q^{77} + ( - 4 \beta_{2} + 38) q^{79} + ( - 6 \beta_{3} + 6 \beta_1) q^{83} + (\beta_{2} - 86) q^{85} + (12 \beta_{3} - 47 \beta_1) q^{89} + (2 \beta_{2} - 16) q^{91} + (4 \beta_{3} + 80 \beta_1) q^{95} + (2 \beta_{2} - 88) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{7} - 32 q^{13} - 32 q^{19} - 84 q^{25} + 8 q^{31} - 104 q^{37} - 112 q^{43} - 180 q^{49} + 400 q^{55} - 120 q^{61} - 336 q^{67} - 112 q^{73} + 152 q^{79} - 344 q^{85} - 64 q^{91} - 352 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 11x^{2} - 10x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{3} - 3\nu^{2} + 19\nu - 9 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\nu^{2} - 2\nu + 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{3} - 12\nu^{2} + 88\nu - 42 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{3} - 4\beta _1 + 2 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 2\beta_{2} - 4\beta _1 - 18 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{3} + 3\beta_{2} + 38\beta _1 - 28 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1025.1
0.500000 + 3.07253i
0.500000 + 0.244099i
0.500000 − 0.244099i
0.500000 − 3.07253i
0 0 0 − 8.04746i 0 2.00000 0 0 0
1025.2 0 0 0 − 5.21904i 0 2.00000 0 0 0
1025.3 0 0 0 5.21904i 0 2.00000 0 0 0
1025.4 0 0 0 8.04746i 0 2.00000 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1152.3.e.e yes 4
3.b odd 2 1 inner 1152.3.e.e yes 4
4.b odd 2 1 1152.3.e.a ✓ 4
8.b even 2 1 1152.3.e.g yes 4
8.d odd 2 1 1152.3.e.c yes 4
12.b even 2 1 1152.3.e.a ✓ 4
16.e even 4 2 2304.3.h.j 8
16.f odd 4 2 2304.3.h.l 8
24.f even 2 1 1152.3.e.c yes 4
24.h odd 2 1 1152.3.e.g yes 4
48.i odd 4 2 2304.3.h.j 8
48.k even 4 2 2304.3.h.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.3.e.a ✓ 4 4.b odd 2 1
1152.3.e.a ✓ 4 12.b even 2 1
1152.3.e.c yes 4 8.d odd 2 1
1152.3.e.c yes 4 24.f even 2 1
1152.3.e.e yes 4 1.a even 1 1 trivial
1152.3.e.e yes 4 3.b odd 2 1 inner
1152.3.e.g yes 4 8.b even 2 1
1152.3.e.g yes 4 24.h odd 2 1
2304.3.h.j 8 16.e even 4 2
2304.3.h.j 8 48.i odd 4 2
2304.3.h.l 8 16.f odd 4 2
2304.3.h.l 8 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 \(S_{3}^{\mathrm{new}}(1152, [\chi])\):

\( T_{5}^{4} + 92T_{5}^{2} + 1764 \) Copy content Toggle raw display
\( T_{7} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} + 16T_{13} - 24 \) 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
$5$ \( T^{4} + 92T^{2} + 1764 \) Copy content Toggle raw display
$7$ \( (T - 2)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 496 T^{2} + 10816 \) Copy content Toggle raw display
$13$ \( (T^{2} + 16 T - 24)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 356 T^{2} + 30276 \) Copy content Toggle raw display
$19$ \( (T^{2} + 16 T - 288)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 1552 T^{2} + 399424 \) Copy content Toggle raw display
$29$ \( T^{4} + 764 T^{2} + 86436 \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 1404)^{2} \) Copy content Toggle raw display
$37$ \( (T + 26)^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 836T^{2} + 4356 \) Copy content Toggle raw display
$43$ \( (T^{2} + 56 T - 624)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 7184 T^{2} + 4769856 \) Copy content Toggle raw display
$53$ \( T^{4} + 2556 T^{2} + 236196 \) Copy content Toggle raw display
$59$ \( (T^{2} + 5408)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 60 T - 508)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 168 T + 6704)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 7568 T^{2} + 3415104 \) Copy content Toggle raw display
$73$ \( (T^{2} + 56 T - 11888)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 76 T + 36)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 3312 T^{2} + 2286144 \) Copy content Toggle raw display
$89$ \( T^{4} + 21508 T^{2} + 3678724 \) Copy content Toggle raw display
$97$ \( (T^{2} + 176 T + 7392)^{2} \) Copy content Toggle raw display
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