Properties

Label 288.3.e.b
Level $288$
Weight $3$
Character orbit 288.e
Analytic conductor $7.847$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,3,Mod(161,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, 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("288.161");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.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: \(7.84743161358\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 7 \beta q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + 7 \beta q^{5} - 24 q^{13} - 7 \beta q^{17} - 73 q^{25} + 41 \beta q^{29} + 70 q^{37} + 31 \beta q^{41} - 49 q^{49} + 17 \beta q^{53} - 22 q^{61} - 168 \beta q^{65} + 96 q^{73} + 98 q^{85} - 41 \beta q^{89} + 144 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 48 q^{13} - 146 q^{25} + 140 q^{37} - 98 q^{49} - 44 q^{61} + 192 q^{73} + 196 q^{85} + 288 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.41421i
1.41421i
0 0 0 9.89949i 0 0 0 0 0
161.2 0 0 0 9.89949i 0 0 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 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.3.e.b 2
3.b odd 2 1 inner 288.3.e.b 2
4.b odd 2 1 CM 288.3.e.b 2
8.b even 2 1 576.3.e.e 2
8.d odd 2 1 576.3.e.e 2
12.b even 2 1 inner 288.3.e.b 2
16.e even 4 2 2304.3.h.e 4
16.f odd 4 2 2304.3.h.e 4
24.f even 2 1 576.3.e.e 2
24.h odd 2 1 576.3.e.e 2
48.i odd 4 2 2304.3.h.e 4
48.k even 4 2 2304.3.h.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.3.e.b 2 1.a even 1 1 trivial
288.3.e.b 2 3.b odd 2 1 inner
288.3.e.b 2 4.b odd 2 1 CM
288.3.e.b 2 12.b even 2 1 inner
576.3.e.e 2 8.b even 2 1
576.3.e.e 2 8.d odd 2 1
576.3.e.e 2 24.f even 2 1
576.3.e.e 2 24.h odd 2 1
2304.3.h.e 4 16.e even 4 2
2304.3.h.e 4 16.f odd 4 2
2304.3.h.e 4 48.i odd 4 2
2304.3.h.e 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 \(S_{3}^{\mathrm{new}}(288, [\chi])\):

\( T_{5}^{2} + 98 \) Copy content Toggle raw display
\( T_{7} \) 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
$5$ \( T^{2} + 98 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T + 24)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 98 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 3362 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 70)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 1922 \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 578 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 22)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 96)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 3362 \) Copy content Toggle raw display
$97$ \( (T - 144)^{2} \) Copy content Toggle raw display
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