# Properties

 Label 87.1.d.b Level $87$ Weight $1$ Character orbit 87.d Self dual yes Analytic conductor $0.043$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -87 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [87,1,Mod(86,87)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(87, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("87.86");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$87 = 3 \cdot 29$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 87.d (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: $$0.0434186560991$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.87.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.22707.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} - q^{6} - q^{7} - q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 - q^6 - q^7 - q^8 + q^9 $$q + q^{2} - q^{3} - q^{6} - q^{7} - q^{8} + q^{9} + q^{11} - q^{13} - q^{14} - q^{16} + q^{17} + q^{18} + q^{21} + q^{22} + q^{24} + q^{25} - q^{26} - q^{27} - q^{29} - q^{33} + q^{34} + q^{39} - 2 q^{41} + q^{42} + q^{47} + q^{48} + q^{50} - q^{51} - q^{54} + q^{56} - q^{58} - q^{63} + q^{64} - q^{66} - q^{67} - q^{72} - q^{75} - q^{77} + q^{78} + q^{81} - 2 q^{82} + q^{87} - q^{88} + q^{89} + q^{91} + q^{94} + q^{99}+O(q^{100})$$ q + q^2 - q^3 - q^6 - q^7 - q^8 + q^9 + q^11 - q^13 - q^14 - q^16 + q^17 + q^18 + q^21 + q^22 + q^24 + q^25 - q^26 - q^27 - q^29 - q^33 + q^34 + q^39 - 2 * q^41 + q^42 + q^47 + q^48 + q^50 - q^51 - q^54 + q^56 - q^58 - q^63 + q^64 - q^66 - q^67 - q^72 - q^75 - q^77 + q^78 + q^81 - 2 * q^82 + q^87 - q^88 + q^89 + q^91 + q^94 + q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$59$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
86.1
 0
1.00000 −1.00000 0 0 −1.00000 −1.00000 −1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
87.d odd 2 1 CM by $$\Q(\sqrt{-87})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.1.d.b yes 1
3.b odd 2 1 87.1.d.a 1
4.b odd 2 1 1392.1.i.b 1
5.b even 2 1 2175.1.h.a 1
5.c odd 4 2 2175.1.b.b 2
9.c even 3 2 2349.1.h.a 2
9.d odd 6 2 2349.1.h.b 2
12.b even 2 1 1392.1.i.a 1
15.d odd 2 1 2175.1.h.b 1
15.e even 4 2 2175.1.b.a 2
29.b even 2 1 87.1.d.a 1
29.c odd 4 2 2523.1.b.b 2
29.d even 7 6 2523.1.h.a 6
29.e even 14 6 2523.1.h.b 6
29.f odd 28 12 2523.1.j.b 12
87.d odd 2 1 CM 87.1.d.b yes 1
87.f even 4 2 2523.1.b.b 2
87.h odd 14 6 2523.1.h.a 6
87.j odd 14 6 2523.1.h.b 6
87.k even 28 12 2523.1.j.b 12
116.d odd 2 1 1392.1.i.a 1
145.d even 2 1 2175.1.h.b 1
145.h odd 4 2 2175.1.b.a 2
261.h odd 6 2 2349.1.h.a 2
261.i even 6 2 2349.1.h.b 2
348.b even 2 1 1392.1.i.b 1
435.b odd 2 1 2175.1.h.a 1
435.p even 4 2 2175.1.b.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.1.d.a 1 3.b odd 2 1
87.1.d.a 1 29.b even 2 1
87.1.d.b yes 1 1.a even 1 1 trivial
87.1.d.b yes 1 87.d odd 2 1 CM
1392.1.i.a 1 12.b even 2 1
1392.1.i.a 1 116.d odd 2 1
1392.1.i.b 1 4.b odd 2 1
1392.1.i.b 1 348.b even 2 1
2175.1.b.a 2 15.e even 4 2
2175.1.b.a 2 145.h odd 4 2
2175.1.b.b 2 5.c odd 4 2
2175.1.b.b 2 435.p even 4 2
2175.1.h.a 1 5.b even 2 1
2175.1.h.a 1 435.b odd 2 1
2175.1.h.b 1 15.d odd 2 1
2175.1.h.b 1 145.d even 2 1
2349.1.h.a 2 9.c even 3 2
2349.1.h.a 2 261.h odd 6 2
2349.1.h.b 2 9.d odd 6 2
2349.1.h.b 2 261.i even 6 2
2523.1.b.b 2 29.c odd 4 2
2523.1.b.b 2 87.f even 4 2
2523.1.h.a 6 29.d even 7 6
2523.1.h.a 6 87.h odd 14 6
2523.1.h.b 6 29.e even 14 6
2523.1.h.b 6 87.j odd 14 6
2523.1.j.b 12 29.f odd 28 12
2523.1.j.b 12 87.k even 28 12

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{1}^{\mathrm{new}}(87, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T + 1$$
$11$ $$T - 1$$
$13$ $$T + 1$$
$17$ $$T - 1$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 1$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T + 2$$
$43$ $$T$$
$47$ $$T - 1$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T + 1$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 1$$
$97$ $$T$$