# Properties

 Label 160.4.c.a Level $160$ Weight $4$ Character orbit 160.c Analytic conductor $9.440$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 160.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.44030560092$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -11 + 2 i ) q^{5} + 27 q^{9} +O(q^{10})$$ $$q + ( -11 + 2 i ) q^{5} + 27 q^{9} -92 i q^{13} -104 i q^{17} + ( 117 - 44 i ) q^{25} -130 q^{29} -396 i q^{37} + 230 q^{41} + ( -297 + 54 i ) q^{45} + 343 q^{49} + 572 i q^{53} -830 q^{61} + ( 184 + 1012 i ) q^{65} + 592 i q^{73} + 729 q^{81} + ( 208 + 1144 i ) q^{85} -1670 q^{89} -1816 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 22q^{5} + 54q^{9} + O(q^{10})$$ $$2q - 22q^{5} + 54q^{9} + 234q^{25} - 260q^{29} + 460q^{41} - 594q^{45} + 686q^{49} - 1660q^{61} + 368q^{65} + 1458q^{81} + 416q^{85} - 3340q^{89} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 0 0 −11.0000 2.00000i 0 0 0 27.0000 0
129.2 0 0 0 −11.0000 + 2.00000i 0 0 0 27.0000 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.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.c.a 2
3.b odd 2 1 1440.4.f.e 2
4.b odd 2 1 CM 160.4.c.a 2
5.b even 2 1 inner 160.4.c.a 2
5.c odd 4 1 800.4.a.e 1
5.c odd 4 1 800.4.a.g 1
8.b even 2 1 320.4.c.e 2
8.d odd 2 1 320.4.c.e 2
12.b even 2 1 1440.4.f.e 2
15.d odd 2 1 1440.4.f.e 2
20.d odd 2 1 inner 160.4.c.a 2
20.e even 4 1 800.4.a.e 1
20.e even 4 1 800.4.a.g 1
40.e odd 2 1 320.4.c.e 2
40.f even 2 1 320.4.c.e 2
40.i odd 4 1 1600.4.a.z 1
40.i odd 4 1 1600.4.a.bb 1
40.k even 4 1 1600.4.a.z 1
40.k even 4 1 1600.4.a.bb 1
60.h even 2 1 1440.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.a 2 1.a even 1 1 trivial
160.4.c.a 2 4.b odd 2 1 CM
160.4.c.a 2 5.b even 2 1 inner
160.4.c.a 2 20.d odd 2 1 inner
320.4.c.e 2 8.b even 2 1
320.4.c.e 2 8.d odd 2 1
320.4.c.e 2 40.e odd 2 1
320.4.c.e 2 40.f even 2 1
800.4.a.e 1 5.c odd 4 1
800.4.a.e 1 20.e even 4 1
800.4.a.g 1 5.c odd 4 1
800.4.a.g 1 20.e even 4 1
1440.4.f.e 2 3.b odd 2 1
1440.4.f.e 2 12.b even 2 1
1440.4.f.e 2 15.d odd 2 1
1440.4.f.e 2 60.h even 2 1
1600.4.a.z 1 40.i odd 4 1
1600.4.a.z 1 40.k even 4 1
1600.4.a.bb 1 40.i odd 4 1
1600.4.a.bb 1 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}$$ acting on $$S_{4}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$125 + 22 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$8464 + T^{2}$$
$17$ $$10816 + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 130 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$156816 + T^{2}$$
$41$ $$( -230 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$327184 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$( 830 + T )^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$350464 + T^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$( 1670 + T )^{2}$$
$97$ $$3297856 + T^{2}$$
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