# 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 $4$

# Related objects

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: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{5} q^{2} + (\zeta_{18}^{3} + \zeta_{18}) q^{3} - \zeta_{18} q^{4} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{6} - \zeta_{18}^{6} q^{8} + (\zeta_{18}^{6} + \zeta_{18}^{4} + \zeta_{18}^{2}) q^{9} +O(q^{10})$$ q + z^5 * q^2 + (z^3 + z) * q^3 - z * q^4 + (z^8 + z^6) * q^6 - z^6 * q^8 + (z^6 + z^4 + z^2) * q^9 $$q + \zeta_{18}^{5} q^{2} + (\zeta_{18}^{3} + \zeta_{18}) q^{3} - \zeta_{18} q^{4} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{6} - \zeta_{18}^{6} q^{8} + (\zeta_{18}^{6} + \zeta_{18}^{4} + \zeta_{18}^{2}) q^{9} + (\zeta_{18}^{2} - \zeta_{18}) q^{11} + ( - \zeta_{18}^{4} - \zeta_{18}^{2}) q^{12} + \zeta_{18}^{2} q^{16} + \zeta_{18}^{5} q^{17} + (\zeta_{18}^{7} - \zeta_{18}^{2} - 1) q^{18} + (\zeta_{18}^{7} - \zeta_{18}^{6}) q^{22} + ( - \zeta_{18}^{7} + 1) q^{24} - \zeta_{18}^{7} q^{25} + (\zeta_{18}^{7} + \zeta_{18}^{5} + \zeta_{18}^{3} - 1) q^{27} + \zeta_{18}^{7} q^{32} + (\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{3} - \zeta_{18}^{2}) q^{33} - \zeta_{18} q^{34} + ( - \zeta_{18}^{7} - \zeta_{18}^{5} - \zeta_{18}^{3}) q^{36} + ( - \zeta_{18}^{4} - 1) q^{41} - \zeta_{18}^{8} q^{43} + ( - \zeta_{18}^{3} + \zeta_{18}^{2}) q^{44} + (\zeta_{18}^{5} + \zeta_{18}^{3}) q^{48} + \zeta_{18}^{6} q^{49} + \zeta_{18}^{3} q^{50} + (\zeta_{18}^{8} + \zeta_{18}^{6}) q^{51} + (\zeta_{18}^{8} - \zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18}) q^{54} + (\zeta_{18} - 1) q^{59} - \zeta_{18}^{3} q^{64} + (\zeta_{18}^{8} - \zeta_{18}^{7} - \zeta_{18} + 1) q^{66} + ( - \zeta_{18}^{6} - \zeta_{18}^{2}) q^{67} - \zeta_{18}^{6} q^{68} + ( - \zeta_{18}^{8} + \zeta_{18}^{3} + \zeta_{18}) q^{72} + ( - \zeta_{18}^{7} + \zeta_{18}^{6}) q^{73} + ( - \zeta_{18}^{8} + \zeta_{18}) q^{75} + (\zeta_{18}^{8} + \zeta_{18}^{6} + \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}) q^{81} + ( - \zeta_{18}^{5} + 1) q^{82} + ( - \zeta_{18}^{5} - \zeta_{18}) q^{83} + \zeta_{18}^{4} q^{86} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{88} - \zeta_{18}^{7} q^{89} + (\zeta_{18}^{8} - \zeta_{18}) q^{96} + (\zeta_{18}^{7} + \zeta_{18}^{3}) q^{97} - \zeta_{18}^{2} q^{98} + (\zeta_{18}^{8} - \zeta_{18}^{7} + \zeta_{18}^{6} - \zeta_{18}^{5} + \zeta_{18}^{4} - \zeta_{18}^{3}) q^{99} +O(q^{100})$$ q + z^5 * q^2 + (z^3 + z) * q^3 - z * q^4 + (z^8 + z^6) * q^6 - z^6 * q^8 + (z^6 + z^4 + z^2) * q^9 + (z^2 - z) * q^11 + (-z^4 - z^2) * q^12 + z^2 * q^16 + z^5 * q^17 + (z^7 - z^2 - 1) * q^18 + (z^7 - z^6) * q^22 + (-z^7 + 1) * q^24 - z^7 * q^25 + (z^7 + z^5 + z^3 - 1) * q^27 + z^7 * q^32 + (z^5 - z^4 + z^3 - z^2) * q^33 - z * q^34 + (-z^7 - z^5 - z^3) * q^36 + (-z^4 - 1) * q^41 - z^8 * q^43 + (-z^3 + z^2) * q^44 + (z^5 + z^3) * q^48 + z^6 * q^49 + z^3 * q^50 + (z^8 + z^6) * q^51 + (z^8 - z^5 - z^3 - z) * q^54 + (z - 1) * q^59 - z^3 * q^64 + (z^8 - z^7 - z + 1) * q^66 + (-z^6 - z^2) * q^67 - z^6 * q^68 + (-z^8 + z^3 + z) * q^72 + (-z^7 + z^6) * q^73 + (-z^8 + z) * q^75 + (z^8 + z^6 + z^4 - z^3 + z) * q^81 + (-z^5 + 1) * q^82 + (-z^5 - z) * q^83 + z^4 * q^86 + (-z^8 + z^7) * q^88 - z^7 * q^89 + (z^8 - z) * q^96 + (z^7 + z^3) * q^97 - z^2 * q^98 + (z^8 - z^7 + z^6 - z^5 + z^4 - z^3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 3 q^{3} - 3 q^{6} + 3 q^{8} - 3 q^{9}+O(q^{10})$$ 6 * q + 3 * q^3 - 3 * q^6 + 3 * q^8 - 3 * q^9 $$6 q + 3 q^{3} - 3 q^{6} + 3 q^{8} - 3 q^{9} - 6 q^{18} + 3 q^{22} + 6 q^{24} - 3 q^{27} + 3 q^{33} - 3 q^{36} - 6 q^{41} - 3 q^{44} + 3 q^{48} - 3 q^{49} + 3 q^{50} - 3 q^{51} - 3 q^{54} - 6 q^{59} - 3 q^{64} + 6 q^{66} + 3 q^{67} + 3 q^{68} + 3 q^{72} - 3 q^{73} - 6 q^{81} + 6 q^{82} + 3 q^{97} - 6 q^{99}+O(q^{100})$$ 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{1}^{\mathrm{new}}(2888, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - T^{3} + 1$$
$3$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$
$5$ $$T^{6}$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$13$ $$T^{6}$$
$17$ $$T^{6} - T^{3} + 1$$
$19$ $$T^{6}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$T^{6}$$
$37$ $$T^{6}$$
$41$ $$T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1$$
$43$ $$T^{6} - T^{3} + 1$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1$$
$61$ $$T^{6}$$
$67$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$
$71$ $$T^{6}$$
$73$ $$T^{6} + 3 T^{5} + 6 T^{4} + 8 T^{3} + \cdots + 1$$
$79$ $$T^{6}$$
$83$ $$T^{6} + 3 T^{4} + 2 T^{3} + 9 T^{2} + \cdots + 1$$
$89$ $$T^{6} + T^{3} + 1$$
$97$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$