# Properties

 Label 171.3.c.b Level $171$ Weight $3$ Character orbit 171.c Analytic conductor $4.659$ Analytic rank $0$ Dimension $2$ CM no 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: no Analytic conductor: $$4.65941252056$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-13})$$ Defining polynomial: $$x^{2} + 13$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-13}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} -9 q^{4} -4 q^{5} -5 q^{7} -5 \beta q^{8} +O(q^{10})$$ $$q + \beta q^{2} -9 q^{4} -4 q^{5} -5 q^{7} -5 \beta q^{8} -4 \beta q^{10} + 10 q^{11} -\beta q^{13} -5 \beta q^{14} + 29 q^{16} -15 q^{17} + ( -6 - 5 \beta ) q^{19} + 36 q^{20} + 10 \beta q^{22} -35 q^{23} -9 q^{25} + 13 q^{26} + 45 q^{28} + 5 \beta q^{29} + 10 \beta q^{31} + 9 \beta q^{32} -15 \beta q^{34} + 20 q^{35} + 6 \beta q^{37} + ( 65 - 6 \beta ) q^{38} + 20 \beta q^{40} + 10 \beta q^{41} -20 q^{43} -90 q^{44} -35 \beta q^{46} -10 q^{47} -24 q^{49} -9 \beta q^{50} + 9 \beta q^{52} -21 \beta q^{53} -40 q^{55} + 25 \beta q^{56} -65 q^{58} + 5 \beta q^{59} -40 q^{61} -130 q^{62} - q^{64} + 4 \beta q^{65} -11 \beta q^{67} + 135 q^{68} + 20 \beta q^{70} + 30 \beta q^{71} + 105 q^{73} -78 q^{74} + ( 54 + 45 \beta ) q^{76} -50 q^{77} + 10 \beta q^{79} -116 q^{80} -130 q^{82} + 40 q^{83} + 60 q^{85} -20 \beta q^{86} -50 \beta q^{88} + 5 \beta q^{91} + 315 q^{92} -10 \beta q^{94} + ( 24 + 20 \beta ) q^{95} -34 \beta q^{97} -24 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{4} - 8q^{5} - 10q^{7} + O(q^{10})$$ $$2q - 18q^{4} - 8q^{5} - 10q^{7} + 20q^{11} + 58q^{16} - 30q^{17} - 12q^{19} + 72q^{20} - 70q^{23} - 18q^{25} + 26q^{26} + 90q^{28} + 40q^{35} + 130q^{38} - 40q^{43} - 180q^{44} - 20q^{47} - 48q^{49} - 80q^{55} - 130q^{58} - 80q^{61} - 260q^{62} - 2q^{64} + 270q^{68} + 210q^{73} - 156q^{74} + 108q^{76} - 100q^{77} - 232q^{80} - 260q^{82} + 80q^{83} + 120q^{85} + 630q^{92} + 48q^{95} + O(q^{100})$$

## 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
 − 3.60555i 3.60555i
3.60555i 0 −9.00000 −4.00000 0 −5.00000 18.0278i 0 14.4222i
37.2 3.60555i 0 −9.00000 −4.00000 0 −5.00000 18.0278i 0 14.4222i
 $$n$$: 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 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 3.b odd 2 1
19.3.b.b 2 57.d even 2 1
171.3.c.b 2 1.a even 1 1 trivial
171.3.c.b 2 19.b odd 2 1 inner
304.3.e.d 2 12.b even 2 1
304.3.e.d 2 228.b odd 2 1
361.3.d.b 4 57.f even 6 2
361.3.d.b 4 57.h odd 6 2
361.3.f.d 12 57.j even 18 6
361.3.f.d 12 57.l odd 18 6
475.3.c.b 2 15.d odd 2 1
475.3.c.b 2 285.b even 2 1
475.3.d.b 4 15.e even 4 2
475.3.d.b 4 285.j odd 4 2
1216.3.e.g 2 24.h odd 2 1
1216.3.e.g 2 456.p even 2 1
1216.3.e.h 2 24.f even 2 1
1216.3.e.h 2 456.l odd 2 1
2736.3.o.d 2 4.b odd 2 1
2736.3.o.d 2 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 $$S_{3}^{\mathrm{new}}(171, [\chi])$$:

 $$T_{2}^{2} + 13$$ $$T_{5} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$13 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 4 + T )^{2}$$
$7$ $$( 5 + T )^{2}$$
$11$ $$( -10 + T )^{2}$$
$13$ $$13 + T^{2}$$
$17$ $$( 15 + T )^{2}$$
$19$ $$361 + 12 T + T^{2}$$
$23$ $$( 35 + T )^{2}$$
$29$ $$325 + T^{2}$$
$31$ $$1300 + T^{2}$$
$37$ $$468 + T^{2}$$
$41$ $$1300 + T^{2}$$
$43$ $$( 20 + T )^{2}$$
$47$ $$( 10 + T )^{2}$$
$53$ $$5733 + T^{2}$$
$59$ $$325 + T^{2}$$
$61$ $$( 40 + T )^{2}$$
$67$ $$1573 + T^{2}$$
$71$ $$11700 + T^{2}$$
$73$ $$( -105 + T )^{2}$$
$79$ $$1300 + T^{2}$$
$83$ $$( -40 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$15028 + T^{2}$$