# Properties

 Label 475.1.c.a Level $475$ Weight $1$ Character orbit 475.c Self dual yes Analytic conductor $0.237$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -19, -95, 5 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 475.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.237055881001$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(\sqrt{5}, \sqrt{-19})$$ Artin image: $D_4$ Artin field: Galois closure of 4.2.2375.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{4} + q^{9} + O(q^{10})$$ $$q + q^{4} + q^{9} - 2 q^{11} + q^{16} - q^{19} + q^{36} - 2 q^{44} - q^{49} - 2 q^{61} + q^{64} - q^{76} + q^{81} - 2 q^{99} + O(q^{100})$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 0
0 0 1.00000 0 0 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
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$
95.d odd 2 1 CM by $$\Q(\sqrt{-95})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.1.c.a 1
5.b even 2 1 RM 475.1.c.a 1
5.c odd 4 2 95.1.d.a 1
15.e even 4 2 855.1.g.a 1
19.b odd 2 1 CM 475.1.c.a 1
20.e even 4 2 1520.1.m.a 1
95.d odd 2 1 CM 475.1.c.a 1
95.g even 4 2 95.1.d.a 1
95.l even 12 4 1805.1.h.a 2
95.m odd 12 4 1805.1.h.a 2
95.q odd 36 12 1805.1.o.a 6
95.r even 36 12 1805.1.o.a 6
285.j odd 4 2 855.1.g.a 1
380.j odd 4 2 1520.1.m.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.1.d.a 1 5.c odd 4 2
95.1.d.a 1 95.g even 4 2
475.1.c.a 1 1.a even 1 1 trivial
475.1.c.a 1 5.b even 2 1 RM
475.1.c.a 1 19.b odd 2 1 CM
475.1.c.a 1 95.d odd 2 1 CM
855.1.g.a 1 15.e even 4 2
855.1.g.a 1 285.j odd 4 2
1520.1.m.a 1 20.e even 4 2
1520.1.m.a 1 380.j odd 4 2
1805.1.h.a 2 95.l even 12 4
1805.1.h.a 2 95.m odd 12 4
1805.1.o.a 6 95.q odd 36 12
1805.1.o.a 6 95.r even 36 12

## Hecke kernels

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

## Hecke characteristic polynomials

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