# Properties

 Label 59.1.b.a Level $59$ Weight $1$ Character orbit 59.b Self dual yes Analytic conductor $0.029$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -59 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [59,1,Mod(58,59)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(59, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("59.58");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$59$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 59.b (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: $$0.0294448357453$$ 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 3.1.59.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.59.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{4} - q^{5} - q^{7}+O(q^{10})$$ q - q^3 + q^4 - q^5 - q^7 $$q - q^{3} + q^{4} - q^{5} - q^{7} - q^{12} + q^{15} + q^{16} + 2 q^{17} - q^{19} - q^{20} + q^{21} + q^{27} - q^{28} - q^{29} + q^{35} - q^{41} - q^{48} - 2 q^{51} - q^{53} + q^{57} + q^{59} + q^{60} + q^{64} + 2 q^{68} + 2 q^{71} - q^{76} - q^{79} - q^{80} - q^{81} + q^{84} - 2 q^{85} + q^{87} + q^{95}+O(q^{100})$$ q - q^3 + q^4 - q^5 - q^7 - q^12 + q^15 + q^16 + 2 * q^17 - q^19 - q^20 + q^21 + q^27 - q^28 - q^29 + q^35 - q^41 - q^48 - 2 * q^51 - q^53 + q^57 + q^59 + q^60 + q^64 + 2 * q^68 + 2 * q^71 - q^76 - q^79 - q^80 - q^81 + q^84 - 2 * q^85 + q^87 + q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/59\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
58.1
 0
0 −1.00000 1.00000 −1.00000 0 −1.00000 0 0 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
59.b odd 2 1 CM by $$\Q(\sqrt{-59})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 59.1.b.a 1
3.b odd 2 1 531.1.c.a 1
4.b odd 2 1 944.1.h.a 1
5.b even 2 1 1475.1.c.b 1
5.c odd 4 2 1475.1.d.a 2
7.b odd 2 1 2891.1.c.e 1
7.c even 3 2 2891.1.g.d 2
7.d odd 6 2 2891.1.g.b 2
8.b even 2 1 3776.1.h.b 1
8.d odd 2 1 3776.1.h.a 1
59.b odd 2 1 CM 59.1.b.a 1
59.c even 29 28 3481.1.d.a 28
59.d odd 58 28 3481.1.d.a 28
177.d even 2 1 531.1.c.a 1
236.c even 2 1 944.1.h.a 1
295.d odd 2 1 1475.1.c.b 1
295.e even 4 2 1475.1.d.a 2
413.b even 2 1 2891.1.c.e 1
413.g odd 6 2 2891.1.g.d 2
413.h even 6 2 2891.1.g.b 2
472.c odd 2 1 3776.1.h.b 1
472.f even 2 1 3776.1.h.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
59.1.b.a 1 1.a even 1 1 trivial
59.1.b.a 1 59.b odd 2 1 CM
531.1.c.a 1 3.b odd 2 1
531.1.c.a 1 177.d even 2 1
944.1.h.a 1 4.b odd 2 1
944.1.h.a 1 236.c even 2 1
1475.1.c.b 1 5.b even 2 1
1475.1.c.b 1 295.d odd 2 1
1475.1.d.a 2 5.c odd 4 2
1475.1.d.a 2 295.e even 4 2
2891.1.c.e 1 7.b odd 2 1
2891.1.c.e 1 413.b even 2 1
2891.1.g.b 2 7.d odd 6 2
2891.1.g.b 2 413.h even 6 2
2891.1.g.d 2 7.c even 3 2
2891.1.g.d 2 413.g odd 6 2
3481.1.d.a 28 59.c even 29 28
3481.1.d.a 28 59.d odd 58 28
3776.1.h.a 1 8.d odd 2 1
3776.1.h.a 1 472.f even 2 1
3776.1.h.b 1 8.b even 2 1
3776.1.h.b 1 472.c odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(59, [\chi])$$.

## Hecke characteristic polynomials

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