# Properties

 Label 775.1.d.b Level $775$ Weight $1$ Character orbit 775.d Self dual yes Analytic conductor $0.387$ 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] = [775,1,Mod(526,775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("775.526");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$775 = 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 775.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: $$0.386775384791$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 31) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.31.1 Artin image: $D_6$ Artin field: Galois closure of 6.2.120125.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/775\mathbb{Z}\right)^\times$$.

 $$n$$ $$251$$ $$652$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
526.1
 0
1.00000 0 0 0 0 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
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 775.1.d.b 1
5.b even 2 1 31.1.b.a 1
5.c odd 4 2 775.1.c.a 2
15.d odd 2 1 279.1.d.b 1
20.d odd 2 1 496.1.e.a 1
31.b odd 2 1 CM 775.1.d.b 1
35.c odd 2 1 1519.1.c.a 1
35.i odd 6 2 1519.1.n.a 2
35.j even 6 2 1519.1.n.b 2
40.e odd 2 1 1984.1.e.b 1
40.f even 2 1 1984.1.e.a 1
45.h odd 6 2 2511.1.m.a 2
45.j even 6 2 2511.1.m.e 2
55.d odd 2 1 3751.1.d.b 1
55.h odd 10 4 3751.1.t.a 4
55.j even 10 4 3751.1.t.c 4
155.c odd 2 1 31.1.b.a 1
155.f even 4 2 775.1.c.a 2
155.i odd 6 2 961.1.e.a 2
155.j even 6 2 961.1.e.a 2
155.m odd 10 4 961.1.f.a 4
155.n even 10 4 961.1.f.a 4
155.u even 30 8 961.1.h.a 8
155.v odd 30 8 961.1.h.a 8
465.g even 2 1 279.1.d.b 1
620.e even 2 1 496.1.e.a 1
1085.b even 2 1 1519.1.c.a 1
1085.bh odd 6 2 1519.1.n.b 2
1085.bn even 6 2 1519.1.n.a 2
1240.e odd 2 1 1984.1.e.a 1
1240.o even 2 1 1984.1.e.b 1
1395.ba even 6 2 2511.1.m.a 2
1395.bm odd 6 2 2511.1.m.e 2
1705.h even 2 1 3751.1.d.b 1
1705.bi even 10 4 3751.1.t.a 4
1705.cb odd 10 4 3751.1.t.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 5.b even 2 1
31.1.b.a 1 155.c odd 2 1
279.1.d.b 1 15.d odd 2 1
279.1.d.b 1 465.g even 2 1
496.1.e.a 1 20.d odd 2 1
496.1.e.a 1 620.e 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 1.a even 1 1 trivial
775.1.d.b 1 31.b odd 2 1 CM
961.1.e.a 2 155.i odd 6 2
961.1.e.a 2 155.j even 6 2
961.1.f.a 4 155.m odd 10 4
961.1.f.a 4 155.n even 10 4
961.1.h.a 8 155.u even 30 8
961.1.h.a 8 155.v odd 30 8
1519.1.c.a 1 35.c odd 2 1
1519.1.c.a 1 1085.b even 2 1
1519.1.n.a 2 35.i odd 6 2
1519.1.n.a 2 1085.bn even 6 2
1519.1.n.b 2 35.j even 6 2
1519.1.n.b 2 1085.bh odd 6 2
1984.1.e.a 1 40.f even 2 1
1984.1.e.a 1 1240.e odd 2 1
1984.1.e.b 1 40.e odd 2 1
1984.1.e.b 1 1240.o even 2 1
2511.1.m.a 2 45.h odd 6 2
2511.1.m.a 2 1395.ba even 6 2
2511.1.m.e 2 45.j even 6 2
2511.1.m.e 2 1395.bm odd 6 2
3751.1.d.b 1 55.d odd 2 1
3751.1.d.b 1 1705.h even 2 1
3751.1.t.a 4 55.h odd 10 4
3751.1.t.a 4 1705.bi even 10 4
3751.1.t.c 4 55.j even 10 4
3751.1.t.c 4 1705.cb odd 10 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{1}^{\mathrm{new}}(775, [\chi])$$.

## Hecke characteristic polynomials

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