↑

Properties

Label 2888.1.f.d
Level $2888$
Weight $1$
Character orbit 2888.f
Self dual yes
Analytic conductor $1.441$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -8
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2888,1,Mod(723,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2888.723");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.f (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: yes
Analytic conductor: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.69564674215936.1
Artin image: $D_9$
Artin field: Galois closure of 9.1.69564674215936.1

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} - \beta_1 q^{6} + q^{8} + (\beta_{2} + 1) q^{9} + ( - \beta_{2} + \beta_1) q^{11} - \beta_1 q^{12} + q^{16} - q^{17} + (\beta_{2} + 1) q^{18} + ( - \beta_{2} + \beta_1) q^{22} - \beta_1 q^{24} + q^{25} + ( - \beta_1 - 1) q^{27} + q^{32} + ( - \beta_{2} + \beta_1 - 1) q^{33} - q^{34} + (\beta_{2} + 1) q^{36} + \beta_{2} q^{41} - q^{43} + ( - \beta_{2} + \beta_1) q^{44} - \beta_1 q^{48} + q^{49} + q^{50} + \beta_1 q^{51} + ( - \beta_1 - 1) q^{54} + ( - \beta_{2} + \beta_1) q^{59} + q^{64} + ( - \beta_{2} + \beta_1 - 1) q^{66} + \beta_{2} q^{67} - q^{68} + (\beta_{2} + 1) q^{72} + ( - \beta_{2} + \beta_1) q^{73} - \beta_1 q^{75} + (\beta_1 + 1) q^{81} + \beta_{2} q^{82} + \beta_{2} q^{83} - q^{86} + ( - \beta_{2} + \beta_1) q^{88} - q^{89} - \beta_1 q^{96} + \beta_{2} q^{97} + q^{98} + (\beta_1 - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{8} + 3 q^{9} + 3 q^{16} - 3 q^{17} + 3 q^{18} + 3 q^{25} - 3 q^{27} + 3 q^{32} - 3 q^{33} - 3 q^{34} + 3 q^{36} - 3 q^{43} + 3 q^{49} + 3 q^{50} - 3 q^{54} + 3 q^{64} - 3 q^{66} - 3 q^{68} + 3 q^{72} + 3 q^{81} - 3 q^{86} - 3 q^{89} + 3 q^{98} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2888\mathbb{Z}\right)^\times\).

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
723.1
1.87939
−0.347296
−1.53209
1.00000 −1.87939 1.00000 0 −1.87939 0 1.00000 2.53209 0
723.2 1.00000 0.347296 1.00000 0 0.347296 0 1.00000 −0.879385 0
723.3 1.00000 1.53209 1.00000 0 1.53209 0 1.00000 1.34730 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 CM by \(\Q(\sqrt{-2}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.f.d 3
8.d odd 2 1 CM 2888.1.f.d 3
19.b odd 2 1 2888.1.f.c 3
19.c even 3 2 2888.1.k.b 6
19.d odd 6 2 2888.1.k.c 6
19.e even 9 2 152.1.u.a ✓ 6
19.e even 9 2 2888.1.u.b 6
19.e even 9 2 2888.1.u.g 6
19.f odd 18 2 2888.1.u.a 6
19.f odd 18 2 2888.1.u.e 6
19.f odd 18 2 2888.1.u.f 6
57.l odd 18 2 1368.1.eh.a 6
76.l odd 18 2 608.1.bg.a 6
95.p even 18 2 3800.1.cv.c 6
95.q odd 36 4 3800.1.cq.b 12
152.b even 2 1 2888.1.f.c 3
152.k odd 6 2 2888.1.k.b 6
152.o even 6 2 2888.1.k.c 6
152.t even 18 2 608.1.bg.a 6
152.u odd 18 2 152.1.u.a ✓ 6
152.u odd 18 2 2888.1.u.b 6
152.u odd 18 2 2888.1.u.g 6
152.v even 18 2 2888.1.u.a 6
152.v even 18 2 2888.1.u.e 6
152.v even 18 2 2888.1.u.f 6
456.bu even 18 2 1368.1.eh.a 6
760.bz odd 18 2 3800.1.cv.c 6
760.cp even 36 4 3800.1.cq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a ✓ 6 19.e even 9 2
152.1.u.a ✓ 6 152.u odd 18 2
608.1.bg.a 6 76.l odd 18 2
608.1.bg.a 6 152.t even 18 2
1368.1.eh.a 6 57.l odd 18 2
1368.1.eh.a 6 456.bu even 18 2
2888.1.f.c 3 19.b odd 2 1
2888.1.f.c 3 152.b even 2 1
2888.1.f.d 3 1.a even 1 1 trivial
2888.1.f.d 3 8.d odd 2 1 CM
2888.1.k.b 6 19.c even 3 2
2888.1.k.b 6 152.k odd 6 2
2888.1.k.c 6 19.d odd 6 2
2888.1.k.c 6 152.o even 6 2
2888.1.u.a 6 19.f odd 18 2
2888.1.u.a 6 152.v even 18 2
2888.1.u.b 6 19.e even 9 2
2888.1.u.b 6 152.u odd 18 2
2888.1.u.e 6 19.f odd 18 2
2888.1.u.e 6 152.v even 18 2
2888.1.u.f 6 19.f odd 18 2
2888.1.u.f 6 152.v even 18 2
2888.1.u.g 6 19.e even 9 2
2888.1.u.g 6 152.u odd 18 2
3800.1.cq.b 12 95.q odd 36 4
3800.1.cq.b 12 760.cp even 36 4
3800.1.cv.c 6 95.p even 18 2
3800.1.cv.c 6 760.bz odd 18 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 3T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2888, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( (T + 1)^{3} \) Copy content Toggle raw display
$19$ \( T^{3} \) Copy content Toggle raw display
$23$ \( T^{3} \) Copy content Toggle raw display
$29$ \( T^{3} \) Copy content Toggle raw display
$31$ \( T^{3} \) Copy content Toggle raw display
$37$ \( T^{3} \) Copy content Toggle raw display
$41$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$43$ \( (T + 1)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} \) Copy content Toggle raw display
$53$ \( T^{3} \) Copy content Toggle raw display
$59$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$61$ \( T^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$71$ \( T^{3} \) Copy content Toggle raw display
$73$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$79$ \( T^{3} \) Copy content Toggle raw display
$83$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
$89$ \( (T + 1)^{3} \) Copy content Toggle raw display
$97$ \( T^{3} - 3T + 1 \) Copy content Toggle raw display
show more
show less