Properties

Label 288.2.d.a
Level $288$
Weight $2$
Character orbit 288.d
Analytic conductor $2.300$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,2,Mod(145,288)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.145");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 288.d (of order \(2\), degree \(1\), not 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: \(2.29969157821\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 72)
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 \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - 2 q^{7} + 2 \beta q^{11} - 3 q^{25} + \beta q^{29} + 10 q^{31} - 2 \beta q^{35} - 3 q^{49} - 5 \beta q^{53} - 16 q^{55} - 4 \beta q^{59} + 14 q^{73} - 4 \beta q^{77} + 10 q^{79} + 2 \beta q^{83} + 2 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 6 q^{25} + 20 q^{31} - 6 q^{49} - 32 q^{55} + 28 q^{73} + 20 q^{79} + 4 q^{97}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
145.1
1.41421i
1.41421i
0 0 0 2.82843i 0 −2.00000 0 0 0
145.2 0 0 0 2.82843i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.2.d.a 2
3.b odd 2 1 inner 288.2.d.a 2
4.b odd 2 1 72.2.d.a 2
5.b even 2 1 7200.2.k.h 2
5.c odd 4 2 7200.2.d.p 4
8.b even 2 1 inner 288.2.d.a 2
8.d odd 2 1 72.2.d.a 2
9.c even 3 2 2592.2.r.i 4
9.d odd 6 2 2592.2.r.i 4
12.b even 2 1 72.2.d.a 2
15.d odd 2 1 7200.2.k.h 2
15.e even 4 2 7200.2.d.p 4
16.e even 4 2 2304.2.a.y 2
16.f odd 4 2 2304.2.a.q 2
20.d odd 2 1 1800.2.k.e 2
20.e even 4 2 1800.2.d.n 4
24.f even 2 1 72.2.d.a 2
24.h odd 2 1 CM 288.2.d.a 2
36.f odd 6 2 648.2.n.h 4
36.h even 6 2 648.2.n.h 4
40.e odd 2 1 1800.2.k.e 2
40.f even 2 1 7200.2.k.h 2
40.i odd 4 2 7200.2.d.p 4
40.k even 4 2 1800.2.d.n 4
48.i odd 4 2 2304.2.a.y 2
48.k even 4 2 2304.2.a.q 2
60.h even 2 1 1800.2.k.e 2
60.l odd 4 2 1800.2.d.n 4
72.j odd 6 2 2592.2.r.i 4
72.l even 6 2 648.2.n.h 4
72.n even 6 2 2592.2.r.i 4
72.p odd 6 2 648.2.n.h 4
120.i odd 2 1 7200.2.k.h 2
120.m even 2 1 1800.2.k.e 2
120.q odd 4 2 1800.2.d.n 4
120.w even 4 2 7200.2.d.p 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.d.a 2 4.b odd 2 1
72.2.d.a 2 8.d odd 2 1
72.2.d.a 2 12.b even 2 1
72.2.d.a 2 24.f even 2 1
288.2.d.a 2 1.a even 1 1 trivial
288.2.d.a 2 3.b odd 2 1 inner
288.2.d.a 2 8.b even 2 1 inner
288.2.d.a 2 24.h odd 2 1 CM
648.2.n.h 4 36.f odd 6 2
648.2.n.h 4 36.h even 6 2
648.2.n.h 4 72.l even 6 2
648.2.n.h 4 72.p odd 6 2
1800.2.d.n 4 20.e even 4 2
1800.2.d.n 4 40.k even 4 2
1800.2.d.n 4 60.l odd 4 2
1800.2.d.n 4 120.q odd 4 2
1800.2.k.e 2 20.d odd 2 1
1800.2.k.e 2 40.e odd 2 1
1800.2.k.e 2 60.h even 2 1
1800.2.k.e 2 120.m even 2 1
2304.2.a.q 2 16.f odd 4 2
2304.2.a.q 2 48.k even 4 2
2304.2.a.y 2 16.e even 4 2
2304.2.a.y 2 48.i odd 4 2
2592.2.r.i 4 9.c even 3 2
2592.2.r.i 4 9.d odd 6 2
2592.2.r.i 4 72.j odd 6 2
2592.2.r.i 4 72.n even 6 2
7200.2.d.p 4 5.c odd 4 2
7200.2.d.p 4 15.e even 4 2
7200.2.d.p 4 40.i odd 4 2
7200.2.d.p 4 120.w even 4 2
7200.2.k.h 2 5.b even 2 1
7200.2.k.h 2 15.d odd 2 1
7200.2.k.h 2 40.f even 2 1
7200.2.k.h 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 8 \) acting on \(S_{2}^{\mathrm{new}}(288, [\chi])\). 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
$5$ \( T^{2} + 8 \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 32 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 8 \) Copy content Toggle raw display
$31$ \( (T - 10)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 200 \) Copy content Toggle raw display
$59$ \( T^{2} + 128 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( (T - 10)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 32 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 2)^{2} \) Copy content Toggle raw display
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