# Properties

 Label 160.3.p.a Level $160$ Weight $3$ Character orbit 160.p Analytic conductor $4.360$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 160.p (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35968422976$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -4 + 3 i ) q^{5} + 9 i q^{9} +O(q^{10})$$ $$q + ( -4 + 3 i ) q^{5} + 9 i q^{9} + ( -17 + 17 i ) q^{13} + ( 7 + 7 i ) q^{17} + ( 7 - 24 i ) q^{25} + 40 i q^{29} + ( -47 - 47 i ) q^{37} + 80 q^{41} + ( -27 - 36 i ) q^{45} -49 i q^{49} + ( 17 - 17 i ) q^{53} + 120 q^{61} + ( 17 - 119 i ) q^{65} + ( -103 + 103 i ) q^{73} -81 q^{81} + ( -49 - 7 i ) q^{85} + 160 i q^{89} + ( -7 - 7 i ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} + O(q^{10})$$ $$2q - 8q^{5} - 34q^{13} + 14q^{17} + 14q^{25} - 94q^{37} + 160q^{41} - 54q^{45} + 34q^{53} + 240q^{61} + 34q^{65} - 206q^{73} - 162q^{81} - 98q^{85} - 14q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$101$$ $$\chi(n)$$ $$1$$ $$i$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
33.1
 − 1.00000i 1.00000i
0 0 0 −4.00000 3.00000i 0 0 0 9.00000i 0
97.1 0 0 0 −4.00000 + 3.00000i 0 0 0 9.00000i 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.3.p.a 2
4.b odd 2 1 CM 160.3.p.a 2
5.b even 2 1 800.3.p.b 2
5.c odd 4 1 inner 160.3.p.a 2
5.c odd 4 1 800.3.p.b 2
8.b even 2 1 320.3.p.e 2
8.d odd 2 1 320.3.p.e 2
20.d odd 2 1 800.3.p.b 2
20.e even 4 1 inner 160.3.p.a 2
20.e even 4 1 800.3.p.b 2
40.i odd 4 1 320.3.p.e 2
40.k even 4 1 320.3.p.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.p.a 2 1.a even 1 1 trivial
160.3.p.a 2 4.b odd 2 1 CM
160.3.p.a 2 5.c odd 4 1 inner
160.3.p.a 2 20.e even 4 1 inner
320.3.p.e 2 8.b even 2 1
320.3.p.e 2 8.d odd 2 1
320.3.p.e 2 40.i odd 4 1
320.3.p.e 2 40.k even 4 1
800.3.p.b 2 5.b even 2 1
800.3.p.b 2 5.c odd 4 1
800.3.p.b 2 20.d odd 2 1
800.3.p.b 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(160, [\chi])$$:

 $$T_{3}$$ $$T_{13}^{2} + 34 T_{13} + 578$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$25 + 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$578 + 34 T + T^{2}$$
$17$ $$98 - 14 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$1600 + T^{2}$$
$31$ $$T^{2}$$
$37$ $$4418 + 94 T + T^{2}$$
$41$ $$( -80 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$578 - 34 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -120 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$21218 + 206 T + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$25600 + T^{2}$$
$97$ $$98 + 14 T + T^{2}$$