# Properties

 Label 475.3.c.a Level $475$ Weight $3$ Character orbit 475.c Self dual yes Analytic conductor $12.943$ Analytic rank $0$ Dimension $1$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 475.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.9428125571$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 4q^{4} + 5q^{7} + 9q^{9} + O(q^{10})$$ $$q + 4q^{4} + 5q^{7} + 9q^{9} + 3q^{11} + 16q^{16} - 15q^{17} - 19q^{19} + 30q^{23} + 20q^{28} + 36q^{36} + 85q^{43} + 12q^{44} - 75q^{47} - 24q^{49} + 103q^{61} + 45q^{63} + 64q^{64} - 60q^{68} + 25q^{73} - 76q^{76} + 15q^{77} + 81q^{81} - 90q^{83} + 120q^{92} + 27q^{99} + O(q^{100})$$

## 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 4.00000 0 0 5.00000 0 9.00000 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
19.b odd 2 1 CM by $$\Q(\sqrt{-19})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.3.c.a 1
5.b even 2 1 19.3.b.a 1
5.c odd 4 2 475.3.d.a 2
15.d odd 2 1 171.3.c.a 1
19.b odd 2 1 CM 475.3.c.a 1
20.d odd 2 1 304.3.e.a 1
40.e odd 2 1 1216.3.e.b 1
40.f even 2 1 1216.3.e.a 1
60.h even 2 1 2736.3.o.a 1
95.d odd 2 1 19.3.b.a 1
95.g even 4 2 475.3.d.a 2
95.h odd 6 2 361.3.d.a 2
95.i even 6 2 361.3.d.a 2
95.o odd 18 6 361.3.f.a 6
95.p even 18 6 361.3.f.a 6
285.b even 2 1 171.3.c.a 1
380.d even 2 1 304.3.e.a 1
760.b odd 2 1 1216.3.e.a 1
760.p even 2 1 1216.3.e.b 1
1140.p odd 2 1 2736.3.o.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 5.b even 2 1
19.3.b.a 1 95.d odd 2 1
171.3.c.a 1 15.d odd 2 1
171.3.c.a 1 285.b even 2 1
304.3.e.a 1 20.d odd 2 1
304.3.e.a 1 380.d even 2 1
361.3.d.a 2 95.h odd 6 2
361.3.d.a 2 95.i even 6 2
361.3.f.a 6 95.o odd 18 6
361.3.f.a 6 95.p even 18 6
475.3.c.a 1 1.a even 1 1 trivial
475.3.c.a 1 19.b odd 2 1 CM
475.3.d.a 2 5.c odd 4 2
475.3.d.a 2 95.g even 4 2
1216.3.e.a 1 40.f even 2 1
1216.3.e.a 1 760.b odd 2 1
1216.3.e.b 1 40.e odd 2 1
1216.3.e.b 1 760.p even 2 1
2736.3.o.a 1 60.h even 2 1
2736.3.o.a 1 1140.p odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(475, [\chi])$$:

 $$T_{2}$$ $$T_{7} - 5$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-5 + T$$
$11$ $$-3 + T$$
$13$ $$T$$
$17$ $$15 + T$$
$19$ $$19 + T$$
$23$ $$-30 + T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$-85 + T$$
$47$ $$75 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$-103 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$-25 + T$$
$79$ $$T$$
$83$ $$90 + T$$
$89$ $$T$$
$97$ $$T$$