↑

Properties

Label 288.3.b.b
Level $288$
Weight $3$
Character orbit 288.b
Analytic conductor $7.847$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,3,Mod(271,288)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.271");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 288.b (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: \(7.84743161358\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} - 6x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + ( - \beta_{2} - \beta_1) q^{7} - 8 q^{11} - 2 \beta_{2} q^{13} + (2 \beta_{3} + 2) q^{17} + (\beta_{3} - 8) q^{19} + ( - 2 \beta_{2} - 2 \beta_1) q^{23} + (4 \beta_{3} - 11) q^{25} + ( - 4 \beta_{2} - 3 \beta_1) q^{29} + ( - 3 \beta_{2} + 5 \beta_1) q^{31} + (\beta_{3} + 24) q^{35} + ( - 2 \beta_{2} + 2 \beta_1) q^{37} + ( - 6 \beta_{3} - 10) q^{41} + (3 \beta_{3} - 8) q^{43} + ( - 2 \beta_{2} + 6 \beta_1) q^{47} + ( - 8 \beta_{3} - 11) q^{49} + (4 \beta_{2} - \beta_1) q^{53} - 8 \beta_1 q^{55} + (3 \beta_{3} - 32) q^{59} + (2 \beta_{2} - 6 \beta_1) q^{61} + (10 \beta_{3} - 24) q^{65} + ( - 3 \beta_{3} + 64) q^{67} + (10 \beta_{2} + 2 \beta_1) q^{71} + (8 \beta_{3} + 50) q^{73} + (8 \beta_{2} + 8 \beta_1) q^{77} + (9 \beta_{2} + \beta_1) q^{79} + ( - 4 \beta_{3} + 40) q^{83} + (8 \beta_{2} - 6 \beta_1) q^{85} + ( - 12 \beta_{3} + 50) q^{89} + ( - 14 \beta_{3} - 72) q^{91} + (4 \beta_{2} - 12 \beta_1) q^{95} + ( - 12 \beta_{3} + 14) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 32 q^{11} + 8 q^{17} - 32 q^{19} - 44 q^{25} + 96 q^{35} - 40 q^{41} - 32 q^{43} - 44 q^{49} - 128 q^{59} - 96 q^{65} + 256 q^{67} + 200 q^{73} + 160 q^{83} + 200 q^{89} - 288 q^{91} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 3x^{2} - 6x + 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + \nu^{2} - 8\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{3} - 5\nu^{2} - 8\nu + 6 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{3} + \nu^{2} + 6 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} - 2\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta _1 - 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} - \beta_{2} + \beta _1 + 18 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
−0.866025 + 1.99551i
0.866025 + 0.719687i
0.866025 − 0.719687i
−0.866025 − 1.99551i
0 0 0 − 7.98203i 0 2.13878i 0 0 0
271.2 0 0 0 − 2.87875i 0 10.7436i 0 0 0
271.3 0 0 0 2.87875i 0 − 10.7436i 0 0 0
271.4 0 0 0 7.98203i 0 − 2.13878i 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
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 288.3.b.b 4
3.b odd 2 1 96.3.b.a 4
4.b odd 2 1 72.3.b.b 4
8.b even 2 1 72.3.b.b 4
8.d odd 2 1 inner 288.3.b.b 4
12.b even 2 1 24.3.b.a ✓ 4
15.d odd 2 1 2400.3.g.a 4
15.e even 4 2 2400.3.p.a 8
16.e even 4 2 2304.3.g.z 8
16.f odd 4 2 2304.3.g.z 8
24.f even 2 1 96.3.b.a 4
24.h odd 2 1 24.3.b.a ✓ 4
48.i odd 4 2 768.3.g.h 8
48.k even 4 2 768.3.g.h 8
60.h even 2 1 600.3.g.a 4
60.l odd 4 2 600.3.p.a 8
120.i odd 2 1 600.3.g.a 4
120.m even 2 1 2400.3.g.a 4
120.q odd 4 2 2400.3.p.a 8
120.w even 4 2 600.3.p.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.3.b.a ✓ 4 12.b even 2 1
24.3.b.a ✓ 4 24.h odd 2 1
72.3.b.b 4 4.b odd 2 1
72.3.b.b 4 8.b even 2 1
96.3.b.a 4 3.b odd 2 1
96.3.b.a 4 24.f even 2 1
288.3.b.b 4 1.a even 1 1 trivial
288.3.b.b 4 8.d odd 2 1 inner
600.3.g.a 4 60.h even 2 1
600.3.g.a 4 120.i odd 2 1
600.3.p.a 8 60.l odd 4 2
600.3.p.a 8 120.w even 4 2
768.3.g.h 8 48.i odd 4 2
768.3.g.h 8 48.k even 4 2
2304.3.g.z 8 16.e even 4 2
2304.3.g.z 8 16.f odd 4 2
2400.3.g.a 4 15.d odd 2 1
2400.3.g.a 4 120.m even 2 1
2400.3.p.a 8 15.e even 4 2
2400.3.p.a 8 120.q odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 72T_{5}^{2} + 528 \) acting on \(S_{3}^{\mathrm{new}}(288, [\chi])\). 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
$5$ \( T^{4} + 72T^{2} + 528 \) Copy content Toggle raw display
$7$ \( T^{4} + 120T^{2} + 528 \) Copy content Toggle raw display
$11$ \( (T + 8)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 384 T^{2} + 33792 \) Copy content Toggle raw display
$17$ \( (T^{2} - 4 T - 188)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16 T + 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 480T^{2} + 8448 \) Copy content Toggle raw display
$29$ \( T^{4} + 1608T^{2} + 528 \) Copy content Toggle raw display
$31$ \( T^{4} + 3384 T^{2} + 279312 \) Copy content Toggle raw display
$37$ \( T^{4} + 864 T^{2} + 76032 \) Copy content Toggle raw display
$41$ \( (T^{2} + 20 T - 1628)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16 T - 368)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 3552 T^{2} + 8448 \) Copy content Toggle raw display
$53$ \( T^{4} + 1800 T^{2} + 803088 \) Copy content Toggle raw display
$59$ \( (T^{2} + 64 T + 592)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 3552 T^{2} + 8448 \) Copy content Toggle raw display
$67$ \( (T^{2} - 128 T + 3664)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 8928 T^{2} + 12849408 \) Copy content Toggle raw display
$73$ \( (T^{2} - 100 T - 572)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 7416 T^{2} + 10797072 \) Copy content Toggle raw display
$83$ \( (T^{2} - 80 T + 832)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 100 T - 4412)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 28 T - 6716)^{2} \) Copy content Toggle raw display
show more
show less