# Properties

 Label 31.1.b.a Level $31$ Weight $1$ Character orbit 31.b Self dual yes Analytic conductor $0.015$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -31 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [31,1,Mod(30,31)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(31, 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("31.30");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 31.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: $$0.0154710153916$$ 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 3.1.31.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.31.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{5} - q^{7} + q^{8} + q^{9}+O(q^{10})$$ q - q^2 - q^5 - q^7 + q^8 + q^9 $$q - q^{2} - q^{5} - q^{7} + q^{8} + q^{9} + q^{10} + q^{14} - q^{16} - q^{18} - q^{19} + q^{31} + q^{35} + q^{38} - q^{40} - q^{41} - q^{45} + 2 q^{47} - q^{56} - q^{59} - q^{62} - q^{63} + q^{64} + 2 q^{67} - q^{70} - q^{71} + q^{72} + q^{80} + q^{81} + q^{82} + q^{90} - 2 q^{94} + q^{95} - q^{97}+O(q^{100})$$ q - q^2 - q^5 - q^7 + q^8 + q^9 + q^10 + q^14 - q^16 - q^18 - q^19 + q^31 + q^35 + q^38 - q^40 - q^41 - q^45 + 2 * q^47 - q^56 - q^59 - q^62 - q^63 + q^64 + 2 * q^67 - q^70 - q^71 + q^72 + q^80 + q^81 + q^82 + q^90 - 2 * q^94 + q^95 - q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/31\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
30.1
 0
−1.00000 0 0 −1.00000 0 −1.00000 1.00000 1.00000 1.00000
 $$n$$: 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
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.1.b.a 1
3.b odd 2 1 279.1.d.b 1
4.b odd 2 1 496.1.e.a 1
5.b even 2 1 775.1.d.b 1
5.c odd 4 2 775.1.c.a 2
7.b odd 2 1 1519.1.c.a 1
7.c even 3 2 1519.1.n.b 2
7.d odd 6 2 1519.1.n.a 2
8.b even 2 1 1984.1.e.a 1
8.d odd 2 1 1984.1.e.b 1
9.c even 3 2 2511.1.m.e 2
9.d odd 6 2 2511.1.m.a 2
11.b odd 2 1 3751.1.d.b 1
11.c even 5 4 3751.1.t.c 4
11.d odd 10 4 3751.1.t.a 4
31.b odd 2 1 CM 31.1.b.a 1
31.c even 3 2 961.1.e.a 2
31.d even 5 4 961.1.f.a 4
31.e odd 6 2 961.1.e.a 2
31.f odd 10 4 961.1.f.a 4
31.g even 15 8 961.1.h.a 8
31.h odd 30 8 961.1.h.a 8
93.c even 2 1 279.1.d.b 1
124.d even 2 1 496.1.e.a 1
155.c odd 2 1 775.1.d.b 1
155.f even 4 2 775.1.c.a 2
217.d even 2 1 1519.1.c.a 1
217.m even 6 2 1519.1.n.a 2
217.n odd 6 2 1519.1.n.b 2
248.b even 2 1 1984.1.e.b 1
248.g odd 2 1 1984.1.e.a 1
279.m odd 6 2 2511.1.m.e 2
279.s even 6 2 2511.1.m.a 2
341.b even 2 1 3751.1.d.b 1
341.t odd 10 4 3751.1.t.c 4
341.ba even 10 4 3751.1.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 1.a even 1 1 trivial
31.1.b.a 1 31.b odd 2 1 CM
279.1.d.b 1 3.b odd 2 1
279.1.d.b 1 93.c even 2 1
496.1.e.a 1 4.b odd 2 1
496.1.e.a 1 124.d even 2 1
775.1.c.a 2 5.c odd 4 2
775.1.c.a 2 155.f even 4 2
775.1.d.b 1 5.b even 2 1
775.1.d.b 1 155.c odd 2 1
961.1.e.a 2 31.c even 3 2
961.1.e.a 2 31.e odd 6 2
961.1.f.a 4 31.d even 5 4
961.1.f.a 4 31.f odd 10 4
961.1.h.a 8 31.g even 15 8
961.1.h.a 8 31.h odd 30 8
1519.1.c.a 1 7.b odd 2 1
1519.1.c.a 1 217.d even 2 1
1519.1.n.a 2 7.d odd 6 2
1519.1.n.a 2 217.m even 6 2
1519.1.n.b 2 7.c even 3 2
1519.1.n.b 2 217.n odd 6 2
1984.1.e.a 1 8.b even 2 1
1984.1.e.a 1 248.g odd 2 1
1984.1.e.b 1 8.d odd 2 1
1984.1.e.b 1 248.b even 2 1
2511.1.m.a 2 9.d odd 6 2
2511.1.m.a 2 279.s even 6 2
2511.1.m.e 2 9.c even 3 2
2511.1.m.e 2 279.m odd 6 2
3751.1.d.b 1 11.b odd 2 1
3751.1.d.b 1 341.b even 2 1
3751.1.t.a 4 11.d odd 10 4
3751.1.t.a 4 341.ba even 10 4
3751.1.t.c 4 11.c even 5 4
3751.1.t.c 4 341.t odd 10 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(31, [\chi])$$.

## Hecke characteristic polynomials

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