Properties

Label 2888.1.u.f
Level $2888$
Weight $1$
Character orbit 2888.u
Analytic conductor $1.441$
Analytic rank $0$
Dimension $6$
Projective image $D_{9}$
CM discriminant -8
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(99,2888)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2888, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 9, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2888.99");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2888 = 2^{3} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2888.u (of order \(18\), degree \(6\), 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: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{3} + 1 \) 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: \(D_{9}\)
Projective 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)
 

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

\(f(q)\) \(=\) \( q + \zeta_{18}^{5} q^{2} + (\zeta_{18}^{3} + \zeta_{18}) q^{3} - \zeta_{18} q^{4} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{6} - \zeta_{18}^{6} q^{8} + (\zeta_{18}^{6} + \cdots + \zeta_{18}^{2}) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \zeta_{18}^{5} q^{2} + (\zeta_{18}^{3} + \zeta_{18}) q^{3} - \zeta_{18} q^{4} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{6} - \zeta_{18}^{6} q^{8} + (\zeta_{18}^{6} + \cdots + \zeta_{18}^{2}) q^{9} + \cdots + (\zeta_{18}^{8} + \cdots - \zeta_{18}^{3}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 3 q^{6} + 3 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 3 q^{6} + 3 q^{8} - 3 q^{9} - 6 q^{18} + 3 q^{22} + 6 q^{24} - 3 q^{27} + 3 q^{33} - 3 q^{36} - 6 q^{41} - 3 q^{44} + 3 q^{48} - 3 q^{49} + 3 q^{50} - 3 q^{51} - 3 q^{54} - 6 q^{59} - 3 q^{64} + 6 q^{66} + 3 q^{67} + 3 q^{68} + 3 q^{72} - 3 q^{73} - 6 q^{81} + 6 q^{82} + 3 q^{97} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1445\) \(2167\) \(2529\)
\(\chi(n)\) \(-1\) \(-1\) \(-\zeta_{18}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.766044 0.642788i
−0.173648 + 0.984808i
−0.173648 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
0.939693 0.342020i
0.939693 + 0.342020i −0.266044 1.50881i 0.766044 + 0.642788i 0 0.266044 1.50881i 0 0.500000 + 0.866025i −1.26604 + 0.460802i 0
595.1 −0.766044 + 0.642788i 0.326352 + 0.118782i 0.173648 0.984808i 0 −0.326352 + 0.118782i 0 0.500000 + 0.866025i −0.673648 0.565258i 0
1859.1 −0.766044 0.642788i 0.326352 0.118782i 0.173648 + 0.984808i 0 −0.326352 0.118782i 0 0.500000 0.866025i −0.673648 + 0.565258i 0
1867.1 0.939693 0.342020i −0.266044 + 1.50881i 0.766044 0.642788i 0 0.266044 + 1.50881i 0 0.500000 0.866025i −1.26604 0.460802i 0
2411.1 −0.173648 + 0.984808i 1.43969 + 1.20805i −0.939693 0.342020i 0 −1.43969 + 1.20805i 0 0.500000 0.866025i 0.439693 + 2.49362i 0
2555.1 −0.173648 0.984808i 1.43969 1.20805i −0.939693 + 0.342020i 0 −1.43969 1.20805i 0 0.500000 + 0.866025i 0.439693 2.49362i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
19.e even 9 1 inner
152.u odd 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2888.1.u.f 6
8.d odd 2 1 CM 2888.1.u.f 6
19.b odd 2 1 2888.1.u.b 6
19.c even 3 1 2888.1.u.a 6
19.c even 3 1 2888.1.u.e 6
19.d odd 6 1 152.1.u.a 6
19.d odd 6 1 2888.1.u.g 6
19.e even 9 1 2888.1.f.c 3
19.e even 9 2 2888.1.k.c 6
19.e even 9 1 2888.1.u.a 6
19.e even 9 1 2888.1.u.e 6
19.e even 9 1 inner 2888.1.u.f 6
19.f odd 18 1 152.1.u.a 6
19.f odd 18 1 2888.1.f.d 3
19.f odd 18 2 2888.1.k.b 6
19.f odd 18 1 2888.1.u.b 6
19.f odd 18 1 2888.1.u.g 6
57.f even 6 1 1368.1.eh.a 6
57.j even 18 1 1368.1.eh.a 6
76.f even 6 1 608.1.bg.a 6
76.k even 18 1 608.1.bg.a 6
95.h odd 6 1 3800.1.cv.c 6
95.l even 12 2 3800.1.cq.b 12
95.o odd 18 1 3800.1.cv.c 6
95.r even 36 2 3800.1.cq.b 12
152.b even 2 1 2888.1.u.b 6
152.k odd 6 1 2888.1.u.a 6
152.k odd 6 1 2888.1.u.e 6
152.l odd 6 1 608.1.bg.a 6
152.o even 6 1 152.1.u.a 6
152.o even 6 1 2888.1.u.g 6
152.s odd 18 1 608.1.bg.a 6
152.u odd 18 1 2888.1.f.c 3
152.u odd 18 2 2888.1.k.c 6
152.u odd 18 1 2888.1.u.a 6
152.u odd 18 1 2888.1.u.e 6
152.u odd 18 1 inner 2888.1.u.f 6
152.v even 18 1 152.1.u.a 6
152.v even 18 1 2888.1.f.d 3
152.v even 18 2 2888.1.k.b 6
152.v even 18 1 2888.1.u.b 6
152.v even 18 1 2888.1.u.g 6
456.s odd 6 1 1368.1.eh.a 6
456.bt odd 18 1 1368.1.eh.a 6
760.bf even 6 1 3800.1.cv.c 6
760.bu odd 12 2 3800.1.cq.b 12
760.bx even 18 1 3800.1.cv.c 6
760.cn odd 36 2 3800.1.cq.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.1.u.a 6 19.d odd 6 1
152.1.u.a 6 19.f odd 18 1
152.1.u.a 6 152.o even 6 1
152.1.u.a 6 152.v even 18 1
608.1.bg.a 6 76.f even 6 1
608.1.bg.a 6 76.k even 18 1
608.1.bg.a 6 152.l odd 6 1
608.1.bg.a 6 152.s odd 18 1
1368.1.eh.a 6 57.f even 6 1
1368.1.eh.a 6 57.j even 18 1
1368.1.eh.a 6 456.s odd 6 1
1368.1.eh.a 6 456.bt odd 18 1
2888.1.f.c 3 19.e even 9 1
2888.1.f.c 3 152.u odd 18 1
2888.1.f.d 3 19.f odd 18 1
2888.1.f.d 3 152.v even 18 1
2888.1.k.b 6 19.f odd 18 2
2888.1.k.b 6 152.v even 18 2
2888.1.k.c 6 19.e even 9 2
2888.1.k.c 6 152.u odd 18 2
2888.1.u.a 6 19.c even 3 1
2888.1.u.a 6 19.e even 9 1
2888.1.u.a 6 152.k odd 6 1
2888.1.u.a 6 152.u odd 18 1
2888.1.u.b 6 19.b odd 2 1
2888.1.u.b 6 19.f odd 18 1
2888.1.u.b 6 152.b even 2 1
2888.1.u.b 6 152.v even 18 1
2888.1.u.e 6 19.c even 3 1
2888.1.u.e 6 19.e even 9 1
2888.1.u.e 6 152.k odd 6 1
2888.1.u.e 6 152.u odd 18 1
2888.1.u.f 6 1.a even 1 1 trivial
2888.1.u.f 6 8.d odd 2 1 CM
2888.1.u.f 6 19.e even 9 1 inner
2888.1.u.f 6 152.u odd 18 1 inner
2888.1.u.g 6 19.d odd 6 1
2888.1.u.g 6 19.f odd 18 1
2888.1.u.g 6 152.o even 6 1
2888.1.u.g 6 152.v even 18 1
3800.1.cq.b 12 95.l even 12 2
3800.1.cq.b 12 95.r even 36 2
3800.1.cq.b 12 760.bu odd 12 2
3800.1.cq.b 12 760.cn odd 36 2
3800.1.cv.c 6 95.h odd 6 1
3800.1.cv.c 6 95.o odd 18 1
3800.1.cv.c 6 760.bf even 6 1
3800.1.cv.c 6 760.bx even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3T_{3}^{5} + 6T_{3}^{4} - 8T_{3}^{3} + 12T_{3}^{2} - 6T_{3} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2888, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} \) Copy content Toggle raw display
$29$ \( T^{6} \) Copy content Toggle raw display
$31$ \( T^{6} \) Copy content Toggle raw display
$37$ \( T^{6} \) Copy content Toggle raw display
$41$ \( T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$43$ \( T^{6} - T^{3} + 1 \) Copy content Toggle raw display
$47$ \( T^{6} \) Copy content Toggle raw display
$53$ \( T^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 6 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$61$ \( T^{6} \) Copy content Toggle raw display
$67$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$71$ \( T^{6} \) Copy content Toggle raw display
$73$ \( T^{6} + 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$79$ \( T^{6} \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$89$ \( T^{6} + T^{3} + 1 \) Copy content Toggle raw display
$97$ \( T^{6} - 3 T^{5} + \cdots + 1 \) Copy content Toggle raw display
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