Properties

Label 288.4.d.c
Level $288$
Weight $4$
Character orbit 288.d
Analytic conductor $16.993$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.145");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(16.9925500817\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-10}, \sqrt{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: no (minimal twist has level 72)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} - 10 q^{7} - 6 \beta_1 q^{11} + \beta_{3} q^{13} - \beta_{2} q^{17} - 2 \beta_{3} q^{19} - 2 \beta_{2} q^{23} + 85 q^{25} - 39 \beta_1 q^{29} - 62 q^{31} - 10 \beta_1 q^{35} + \beta_{3} q^{37} - 5 \beta_{2} q^{41} + 2 \beta_{3} q^{43} - 6 \beta_{2} q^{47} - 243 q^{49} - 21 \beta_1 q^{53} + 240 q^{55} - 116 \beta_1 q^{59} - 9 \beta_{3} q^{61} - 5 \beta_{2} q^{65} + 12 \beta_{3} q^{67} + 30 q^{73} + 60 \beta_1 q^{77} - 94 q^{79} + 106 \beta_1 q^{83} - 8 \beta_{3} q^{85} + 10 \beta_{2} q^{89} - 10 \beta_{3} q^{91} + 10 \beta_{2} q^{95} + 130 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 40 q^{7} + 340 q^{25} - 248 q^{31} - 972 q^{49} + 960 q^{55} + 120 q^{73} - 376 q^{79} + 520 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 2\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{3} + 28\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 8\nu^{2} - 24 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 8\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 24 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{2} + 56\beta_1 ) / 16 \) 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
2.34521 1.58114i
−2.34521 1.58114i
−2.34521 + 1.58114i
2.34521 + 1.58114i
0 0 0 6.32456i 0 −10.0000 0 0 0
145.2 0 0 0 6.32456i 0 −10.0000 0 0 0
145.3 0 0 0 6.32456i 0 −10.0000 0 0 0
145.4 0 0 0 6.32456i 0 −10.0000 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
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 288.4.d.c 4
3.b odd 2 1 inner 288.4.d.c 4
4.b odd 2 1 72.4.d.c 4
8.b even 2 1 inner 288.4.d.c 4
8.d odd 2 1 72.4.d.c 4
12.b even 2 1 72.4.d.c 4
16.e even 4 2 2304.4.a.cc 4
16.f odd 4 2 2304.4.a.bx 4
24.f even 2 1 72.4.d.c 4
24.h odd 2 1 inner 288.4.d.c 4
48.i odd 4 2 2304.4.a.cc 4
48.k even 4 2 2304.4.a.bx 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.4.d.c 4 4.b odd 2 1
72.4.d.c 4 8.d odd 2 1
72.4.d.c 4 12.b even 2 1
72.4.d.c 4 24.f even 2 1
288.4.d.c 4 1.a even 1 1 trivial
288.4.d.c 4 3.b odd 2 1 inner
288.4.d.c 4 8.b even 2 1 inner
288.4.d.c 4 24.h odd 2 1 inner
2304.4.a.bx 4 16.f odd 4 2
2304.4.a.bx 4 48.k even 4 2
2304.4.a.cc 4 16.e even 4 2
2304.4.a.cc 4 48.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 40 \) acting on \(S_{4}^{\mathrm{new}}(288, [\chi])\). 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
$5$ \( (T^{2} + 40)^{2} \) Copy content Toggle raw display
$7$ \( (T + 10)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1440)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3520)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 5632)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 14080)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 22528)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 60840)^{2} \) Copy content Toggle raw display
$31$ \( (T + 62)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3520)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 140800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 14080)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 202752)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 17640)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 538240)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 285120)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 506880)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 30)^{4} \) Copy content Toggle raw display
$79$ \( (T + 94)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 449440)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 563200)^{2} \) Copy content Toggle raw display
$97$ \( (T - 130)^{4} \) Copy content Toggle raw display
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