# Properties

 Label 475.1.c.b Level $475$ Weight $1$ Character orbit 475.c Analytic conductor $0.237$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -95 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,1,Mod(151,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, 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("475.151");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 475.c (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: no Analytic conductor: $$0.237055881001$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.475.1 Artin image: $\SD_{16}$ Artin field: Galois closure of 8.2.107171875.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{2} - \beta q^{3} - q^{4} - 2 q^{6} - q^{9} +O(q^{10})$$ q - b * q^2 - b * q^3 - q^4 - 2 * q^6 - q^9 $$q - \beta q^{2} - \beta q^{3} - q^{4} - 2 q^{6} - q^{9} + \beta q^{12} + \beta q^{13} - q^{16} + \beta q^{18} + q^{19} + 2 q^{26} + \beta q^{32} + q^{36} + \beta q^{37} - \beta q^{38} + 2 q^{39} + \beta q^{48} - q^{49} - \beta q^{52} - \beta q^{53} - \beta q^{57} + q^{64} - \beta q^{67} + 2 q^{74} - q^{76} - 2 \beta q^{78} - q^{81} + 2 q^{96} + \beta q^{97} + \beta q^{98} +O(q^{100})$$ q - b * q^2 - b * q^3 - q^4 - 2 * q^6 - q^9 + b * q^12 + b * q^13 - q^16 + b * q^18 + q^19 + 2 * q^26 + b * q^32 + q^36 + b * q^37 - b * q^38 + 2 * q^39 + b * q^48 - q^49 - b * q^52 - b * q^53 - b * q^57 + q^64 - b * q^67 + 2 * q^74 - q^76 - 2*b * q^78 - q^81 + 2 * q^96 + b * q^97 + b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 $$2 q - 2 q^{4} - 4 q^{6} - 2 q^{9} - 2 q^{16} + 2 q^{19} + 4 q^{26} + 2 q^{36} + 4 q^{39} - 2 q^{49} + 2 q^{64} + 4 q^{74} - 2 q^{76} - 2 q^{81} + 4 q^{96}+O(q^{100})$$ 2 * q - 2 * q^4 - 4 * q^6 - 2 * q^9 - 2 * q^16 + 2 * q^19 + 4 * q^26 + 2 * q^36 + 4 * q^39 - 2 * q^49 + 2 * q^64 + 4 * q^74 - 2 * q^76 - 2 * q^81 + 4 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 1.41421i − 1.41421i
1.41421i 1.41421i −1.00000 0 −2.00000 0 0 −1.00000 0
151.2 1.41421i 1.41421i −1.00000 0 −2.00000 0 0 −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
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$
5.b even 2 1 inner
19.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.1.c.b 2
5.b even 2 1 inner 475.1.c.b 2
5.c odd 4 2 95.1.d.b 2
15.e even 4 2 855.1.g.c 2
19.b odd 2 1 inner 475.1.c.b 2
20.e even 4 2 1520.1.m.b 2
95.d odd 2 1 CM 475.1.c.b 2
95.g even 4 2 95.1.d.b 2
95.l even 12 4 1805.1.h.b 4
95.m odd 12 4 1805.1.h.b 4
95.q odd 36 12 1805.1.o.b 12
95.r even 36 12 1805.1.o.b 12
285.j odd 4 2 855.1.g.c 2
380.j odd 4 2 1520.1.m.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.b 2 5.c odd 4 2
95.1.d.b 2 95.g even 4 2
475.1.c.b 2 1.a even 1 1 trivial
475.1.c.b 2 5.b even 2 1 inner
475.1.c.b 2 19.b odd 2 1 inner
475.1.c.b 2 95.d odd 2 1 CM
855.1.g.c 2 15.e even 4 2
855.1.g.c 2 285.j odd 4 2
1520.1.m.b 2 20.e even 4 2
1520.1.m.b 2 380.j odd 4 2
1805.1.h.b 4 95.l even 12 4
1805.1.h.b 4 95.m odd 12 4
1805.1.o.b 12 95.q odd 36 12
1805.1.o.b 12 95.r even 36 12

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2$$ acting on $$S_{1}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

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