# Properties

 Label 160.4.c.b Level $160$ Weight $4$ Character orbit 160.c Analytic conductor $9.440$ Analytic rank $0$ Dimension $4$ CM no 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{29})$$ Defining polynomial: $$x^{4} + 15 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + ( -3 + \beta_{1} ) q^{5} + \beta_{2} q^{7} + 11 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + ( -3 + \beta_{1} ) q^{5} + \beta_{2} q^{7} + 11 q^{9} + \beta_{3} q^{11} + 2 \beta_{1} q^{13} + ( -3 \beta_{2} + \beta_{3} ) q^{15} -4 \beta_{1} q^{17} + 3 \beta_{3} q^{19} -16 q^{21} + 13 \beta_{2} q^{23} + ( -107 - 6 \beta_{1} ) q^{25} + 38 \beta_{2} q^{27} + 158 q^{29} + 4 \beta_{3} q^{31} -16 \beta_{1} q^{33} + ( -3 \beta_{2} + \beta_{3} ) q^{35} + 26 \beta_{1} q^{37} + 2 \beta_{3} q^{39} -170 q^{41} -79 \beta_{2} q^{43} + ( -33 + 11 \beta_{1} ) q^{45} + 61 \beta_{2} q^{47} + 327 q^{49} -4 \beta_{3} q^{51} + 46 \beta_{1} q^{53} + ( -116 \beta_{2} - 3 \beta_{3} ) q^{55} -48 \beta_{1} q^{57} -15 \beta_{3} q^{59} + 82 q^{61} + 11 \beta_{2} q^{63} + ( -232 - 6 \beta_{1} ) q^{65} + 173 \beta_{2} q^{67} -208 q^{69} -22 \beta_{3} q^{71} -40 \beta_{1} q^{73} + ( -107 \beta_{2} - 6 \beta_{3} ) q^{75} -16 \beta_{1} q^{77} -8 \beta_{3} q^{79} -311 q^{81} -235 \beta_{2} q^{83} + ( 464 + 12 \beta_{1} ) q^{85} + 158 \beta_{2} q^{87} -6 q^{89} + 2 \beta_{3} q^{91} -64 \beta_{1} q^{93} + ( -348 \beta_{2} - 9 \beta_{3} ) q^{95} + 100 \beta_{1} q^{97} + 11 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{5} + 44q^{9} + O(q^{10})$$ $$4q - 12q^{5} + 44q^{9} - 64q^{21} - 428q^{25} + 632q^{29} - 680q^{41} - 132q^{45} + 1308q^{49} + 328q^{61} - 928q^{65} - 832q^{69} - 1244q^{81} + 1856q^{85} - 24q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 15 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{3} + 44 \nu$$$$)/7$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{3} - 32 \nu$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$16 \nu^{2} + 120$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 2 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 120$$$$)/16$$ $$\nu^{3}$$ $$=$$ $$($$$$-11 \beta_{2} - 8 \beta_{1}$$$$)/4$$

## 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
 − 3.19258i 2.19258i − 2.19258i 3.19258i
0 4.00000i 0 −3.00000 10.7703i 0 4.00000i 0 11.0000 0
129.2 0 4.00000i 0 −3.00000 + 10.7703i 0 4.00000i 0 11.0000 0
129.3 0 4.00000i 0 −3.00000 10.7703i 0 4.00000i 0 11.0000 0
129.4 0 4.00000i 0 −3.00000 + 10.7703i 0 4.00000i 0 11.0000 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 inner
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.b 4
3.b odd 2 1 1440.4.f.h 4
4.b odd 2 1 inner 160.4.c.b 4
5.b even 2 1 inner 160.4.c.b 4
5.c odd 4 1 800.4.a.n 2
5.c odd 4 1 800.4.a.r 2
8.b even 2 1 320.4.c.i 4
8.d odd 2 1 320.4.c.i 4
12.b even 2 1 1440.4.f.h 4
15.d odd 2 1 1440.4.f.h 4
20.d odd 2 1 inner 160.4.c.b 4
20.e even 4 1 800.4.a.n 2
20.e even 4 1 800.4.a.r 2
40.e odd 2 1 320.4.c.i 4
40.f even 2 1 320.4.c.i 4
40.i odd 4 1 1600.4.a.cc 2
40.i odd 4 1 1600.4.a.co 2
40.k even 4 1 1600.4.a.cc 2
40.k even 4 1 1600.4.a.co 2
60.h even 2 1 1440.4.f.h 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 16$$ acting on $$S_{4}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 16 + T^{2} )^{2}$$
$5$ $$( 125 + 6 T + T^{2} )^{2}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$( -1856 + T^{2} )^{2}$$
$13$ $$( 464 + T^{2} )^{2}$$
$17$ $$( 1856 + T^{2} )^{2}$$
$19$ $$( -16704 + T^{2} )^{2}$$
$23$ $$( 2704 + T^{2} )^{2}$$
$29$ $$( -158 + T )^{4}$$
$31$ $$( -29696 + T^{2} )^{2}$$
$37$ $$( 78416 + T^{2} )^{2}$$
$41$ $$( 170 + T )^{4}$$
$43$ $$( 99856 + T^{2} )^{2}$$
$47$ $$( 59536 + T^{2} )^{2}$$
$53$ $$( 245456 + T^{2} )^{2}$$
$59$ $$( -417600 + T^{2} )^{2}$$
$61$ $$( -82 + T )^{4}$$
$67$ $$( 478864 + T^{2} )^{2}$$
$71$ $$( -898304 + T^{2} )^{2}$$
$73$ $$( 185600 + T^{2} )^{2}$$
$79$ $$( -118784 + T^{2} )^{2}$$
$83$ $$( 883600 + T^{2} )^{2}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$( 1160000 + T^{2} )^{2}$$
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