# Properties

 Label 171.3.c.a Level $171$ Weight $3$ Character orbit 171.c Self dual yes Analytic conductor $4.659$ Analytic rank $0$ Dimension $1$ CM discriminant -19 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$171 = 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 171.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.65941252056$$ 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

 $$f(q)$$ $$=$$ $$q + 4 q^{4} + 9 q^{5} - 5 q^{7} + O(q^{10})$$ $$q + 4 q^{4} + 9 q^{5} - 5 q^{7} - 3 q^{11} + 16 q^{16} - 15 q^{17} - 19 q^{19} + 36 q^{20} + 30 q^{23} + 56 q^{25} - 20 q^{28} - 45 q^{35} - 85 q^{43} - 12 q^{44} - 75 q^{47} - 24 q^{49} - 27 q^{55} + 103 q^{61} + 64 q^{64} - 60 q^{68} - 25 q^{73} - 76 q^{76} + 15 q^{77} + 144 q^{80} - 90 q^{83} - 135 q^{85} + 120 q^{92} - 171 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$$.

 $$n$$ $$20$$ $$154$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
37.1
 0
0 0 4.00000 9.00000 0 −5.00000 0 0 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 171.3.c.a 1
3.b odd 2 1 19.3.b.a 1
4.b odd 2 1 2736.3.o.a 1
12.b even 2 1 304.3.e.a 1
15.d odd 2 1 475.3.c.a 1
15.e even 4 2 475.3.d.a 2
19.b odd 2 1 CM 171.3.c.a 1
24.f even 2 1 1216.3.e.b 1
24.h odd 2 1 1216.3.e.a 1
57.d even 2 1 19.3.b.a 1
57.f even 6 2 361.3.d.a 2
57.h odd 6 2 361.3.d.a 2
57.j even 18 6 361.3.f.a 6
57.l odd 18 6 361.3.f.a 6
76.d even 2 1 2736.3.o.a 1
228.b odd 2 1 304.3.e.a 1
285.b even 2 1 475.3.c.a 1
285.j odd 4 2 475.3.d.a 2
456.l odd 2 1 1216.3.e.b 1
456.p even 2 1 1216.3.e.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 3.b odd 2 1
19.3.b.a 1 57.d even 2 1
171.3.c.a 1 1.a even 1 1 trivial
171.3.c.a 1 19.b odd 2 1 CM
304.3.e.a 1 12.b even 2 1
304.3.e.a 1 228.b odd 2 1
361.3.d.a 2 57.f even 6 2
361.3.d.a 2 57.h odd 6 2
361.3.f.a 6 57.j even 18 6
361.3.f.a 6 57.l odd 18 6
475.3.c.a 1 15.d odd 2 1
475.3.c.a 1 285.b even 2 1
475.3.d.a 2 15.e even 4 2
475.3.d.a 2 285.j odd 4 2
1216.3.e.a 1 24.h odd 2 1
1216.3.e.a 1 456.p even 2 1
1216.3.e.b 1 24.f even 2 1
1216.3.e.b 1 456.l odd 2 1
2736.3.o.a 1 4.b odd 2 1
2736.3.o.a 1 76.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(171, [\chi])$$:

 $$T_{2}$$ $$T_{5} - 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-9 + T$$
$7$ $$5 + T$$
$11$ $$3 + T$$
$13$ $$T$$
$17$ $$15 + T$$
$19$ $$19 + T$$
$23$ $$-30 + T$$
$29$ $$T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$85 + T$$
$47$ $$75 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$-103 + T$$
$67$ $$T$$
$71$ $$T$$
$73$ $$25 + T$$
$79$ $$T$$
$83$ $$90 + T$$
$89$ $$T$$
$97$ $$T$$