Properties

Label 2888.1.l.a
Level $2888$
Weight $1$
Character orbit 2888.l
Analytic conductor $1.441$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -152
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2888,1,Mod(69,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2888.69");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.l (of order \(6\), degree \(2\), 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: \(1.44129975648\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 152)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.152.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.1267762688.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{7} + q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{6}^{2} q^{2} - \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} + \zeta_{6} q^{6} - q^{7} + q^{8} - q^{12} + \zeta_{6} q^{13} - \zeta_{6}^{2} q^{14} + \zeta_{6}^{2} q^{16} - \zeta_{6}^{2} q^{17} + \zeta_{6}^{2} q^{21} + \zeta_{6} q^{23} - \zeta_{6}^{2} q^{24} - \zeta_{6} q^{25} - q^{26} + q^{27} + \zeta_{6} q^{28} + \zeta_{6} q^{29} - \zeta_{6} q^{32} + \zeta_{6} q^{34} + q^{37} + q^{39} - \zeta_{6} q^{42} - q^{46} - \zeta_{6} q^{47} + \zeta_{6} q^{48} + q^{50} - \zeta_{6} q^{51} - \zeta_{6}^{2} q^{52} + \zeta_{6} q^{53} + \zeta_{6}^{2} q^{54} - q^{56} - q^{58} - \zeta_{6}^{2} q^{59} + q^{64} + \zeta_{6} q^{67} - q^{68} + q^{69} - \zeta_{6}^{2} q^{73} + 2 \zeta_{6}^{2} q^{74} - q^{75} + \zeta_{6}^{2} q^{78} - \zeta_{6}^{2} q^{81} + q^{84} + q^{87} - \zeta_{6} q^{91} - \zeta_{6}^{2} q^{92} + 2 q^{94} - q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{2} + q^{3} - q^{4} + q^{6} - 2 q^{7} + 2 q^{8} - 2 q^{12} + q^{13} + q^{14} - q^{16} + q^{17} - q^{21} + q^{23} + q^{24} - q^{25} - 2 q^{26} + 2 q^{27} + q^{28} + q^{29} - q^{32} + q^{34} + 4 q^{37} + 2 q^{39} - q^{42} - 2 q^{46} - 2 q^{47} + q^{48} + 2 q^{50} - q^{51} + q^{52} + q^{53} - q^{54} - 2 q^{56} - 2 q^{58} + q^{59} + 2 q^{64} + q^{67} - 2 q^{68} + 2 q^{69} + q^{73} - 2 q^{74} - 2 q^{75} - q^{78} + q^{81} + 2 q^{84} + 2 q^{87} - q^{91} + q^{92} + 4 q^{94} - 2 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
69.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 1.00000 0 0
293.1 −0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 1.00000 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
152.g odd 2 1 CM by \(\Q(\sqrt{-38}) \)
19.c even 3 1 inner
152.l odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.l.a 2
8.b even 2 1 2888.1.l.b 2
19.b odd 2 1 2888.1.l.b 2
19.c even 3 1 152.1.g.b yes 1
19.c even 3 1 inner 2888.1.l.a 2
19.d odd 6 1 152.1.g.a 1
19.d odd 6 1 2888.1.l.b 2
19.e even 9 6 2888.1.s.a 6
19.f odd 18 6 2888.1.s.b 6
57.f even 6 1 1368.1.i.b 1
57.h odd 6 1 1368.1.i.a 1
76.f even 6 1 608.1.g.a 1
76.g odd 6 1 608.1.g.b 1
95.h odd 6 1 3800.1.o.b 1
95.i even 6 1 3800.1.o.a 1
95.l even 12 2 3800.1.b.a 2
95.m odd 12 2 3800.1.b.b 2
152.g odd 2 1 CM 2888.1.l.a 2
152.k odd 6 1 608.1.g.a 1
152.l odd 6 1 152.1.g.b yes 1
152.l odd 6 1 inner 2888.1.l.a 2
152.o even 6 1 608.1.g.b 1
152.p even 6 1 152.1.g.a 1
152.p even 6 1 2888.1.l.b 2
152.s odd 18 6 2888.1.s.a 6
152.t even 18 6 2888.1.s.b 6
456.v even 6 1 1368.1.i.a 1
456.x odd 6 1 1368.1.i.b 1
760.z even 6 1 3800.1.o.b 1
760.bh odd 6 1 3800.1.o.a 1
760.bp even 12 2 3800.1.b.b 2
760.br odd 12 2 3800.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.g.a 1 19.d odd 6 1
152.1.g.a 1 152.p even 6 1
152.1.g.b yes 1 19.c even 3 1
152.1.g.b yes 1 152.l odd 6 1
608.1.g.a 1 76.f even 6 1
608.1.g.a 1 152.k odd 6 1
608.1.g.b 1 76.g odd 6 1
608.1.g.b 1 152.o even 6 1
1368.1.i.a 1 57.h odd 6 1
1368.1.i.a 1 456.v even 6 1
1368.1.i.b 1 57.f even 6 1
1368.1.i.b 1 456.x odd 6 1
2888.1.l.a 2 1.a even 1 1 trivial
2888.1.l.a 2 19.c even 3 1 inner
2888.1.l.a 2 152.g odd 2 1 CM
2888.1.l.a 2 152.l odd 6 1 inner
2888.1.l.b 2 8.b even 2 1
2888.1.l.b 2 19.b odd 2 1
2888.1.l.b 2 19.d odd 6 1
2888.1.l.b 2 152.p even 6 1
2888.1.s.a 6 19.e even 9 6
2888.1.s.a 6 152.s odd 18 6
2888.1.s.b 6 19.f odd 18 6
2888.1.s.b 6 152.t even 18 6
3800.1.b.a 2 95.l even 12 2
3800.1.b.a 2 760.br odd 12 2
3800.1.b.b 2 95.m odd 12 2
3800.1.b.b 2 760.bp even 12 2
3800.1.o.a 1 95.i even 6 1
3800.1.o.a 1 760.bh odd 6 1
3800.1.o.b 1 95.h odd 6 1
3800.1.o.b 1 760.z even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2888, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$29$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( (T - 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$53$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$59$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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