# Properties

 Label 320.2.c.d Level $320$ Weight $2$ Character orbit 320.c Analytic conductor $2.555$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 320.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.55521286468$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{3} q^{7} + ( -3 + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} -\beta_{3} q^{7} + ( -3 + 2 \beta_{2} ) q^{9} + ( -2 \beta_{1} + \beta_{3} ) q^{15} + ( 2 + 2 \beta_{2} ) q^{21} + ( 2 \beta_{1} + \beta_{3} ) q^{23} + 5 q^{25} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{27} -6 q^{29} + ( -\beta_{1} - 2 \beta_{3} ) q^{35} + 2 \beta_{2} q^{41} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{43} + ( 10 - 3 \beta_{2} ) q^{45} + ( 4 \beta_{1} - \beta_{3} ) q^{47} + ( -7 - 6 \beta_{2} ) q^{49} -6 \beta_{2} q^{61} + ( -2 \beta_{1} - \beta_{3} ) q^{63} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{67} + ( -14 + 2 \beta_{2} ) q^{69} + 5 \beta_{1} q^{75} + ( 11 - 6 \beta_{2} ) q^{81} + ( -5 \beta_{1} + 2 \beta_{3} ) q^{83} -6 \beta_{1} q^{87} -6 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 8q^{21} + 20q^{25} - 24q^{29} + 40q^{45} - 28q^{49} - 56q^{69} + 44q^{81} - 24q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{3} + 10 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} - 5 \beta_{1}$$$$)/4$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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
 − 1.61803i − 0.618034i 0.618034i 1.61803i
0 3.23607i 0 −2.23607 0 0.763932i 0 −7.47214 0
129.2 0 1.23607i 0 2.23607 0 5.23607i 0 1.47214 0
129.3 0 1.23607i 0 2.23607 0 5.23607i 0 1.47214 0
129.4 0 3.23607i 0 −2.23607 0 0.763932i 0 −7.47214 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.2.c.d 4
3.b odd 2 1 2880.2.f.w 4
4.b odd 2 1 inner 320.2.c.d 4
5.b even 2 1 inner 320.2.c.d 4
5.c odd 4 1 1600.2.a.z 2
5.c odd 4 1 1600.2.a.bd 2
8.b even 2 1 160.2.c.b 4
8.d odd 2 1 160.2.c.b 4
12.b even 2 1 2880.2.f.w 4
15.d odd 2 1 2880.2.f.w 4
16.e even 4 1 1280.2.f.g 4
16.e even 4 1 1280.2.f.h 4
16.f odd 4 1 1280.2.f.g 4
16.f odd 4 1 1280.2.f.h 4
20.d odd 2 1 CM 320.2.c.d 4
20.e even 4 1 1600.2.a.z 2
20.e even 4 1 1600.2.a.bd 2
24.f even 2 1 1440.2.f.i 4
24.h odd 2 1 1440.2.f.i 4
40.e odd 2 1 160.2.c.b 4
40.f even 2 1 160.2.c.b 4
40.i odd 4 1 800.2.a.j 2
40.i odd 4 1 800.2.a.n 2
40.k even 4 1 800.2.a.j 2
40.k even 4 1 800.2.a.n 2
60.h even 2 1 2880.2.f.w 4
80.k odd 4 1 1280.2.f.g 4
80.k odd 4 1 1280.2.f.h 4
80.q even 4 1 1280.2.f.g 4
80.q even 4 1 1280.2.f.h 4
120.i odd 2 1 1440.2.f.i 4
120.m even 2 1 1440.2.f.i 4
120.q odd 4 1 7200.2.a.cb 2
120.q odd 4 1 7200.2.a.cr 2
120.w even 4 1 7200.2.a.cb 2
120.w even 4 1 7200.2.a.cr 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 8.b even 2 1
160.2.c.b 4 8.d odd 2 1
160.2.c.b 4 40.e odd 2 1
160.2.c.b 4 40.f even 2 1
320.2.c.d 4 1.a even 1 1 trivial
320.2.c.d 4 4.b odd 2 1 inner
320.2.c.d 4 5.b even 2 1 inner
320.2.c.d 4 20.d odd 2 1 CM
800.2.a.j 2 40.i odd 4 1
800.2.a.j 2 40.k even 4 1
800.2.a.n 2 40.i odd 4 1
800.2.a.n 2 40.k even 4 1
1280.2.f.g 4 16.e even 4 1
1280.2.f.g 4 16.f odd 4 1
1280.2.f.g 4 80.k odd 4 1
1280.2.f.g 4 80.q even 4 1
1280.2.f.h 4 16.e even 4 1
1280.2.f.h 4 16.f odd 4 1
1280.2.f.h 4 80.k odd 4 1
1280.2.f.h 4 80.q even 4 1
1440.2.f.i 4 24.f even 2 1
1440.2.f.i 4 24.h odd 2 1
1440.2.f.i 4 120.i odd 2 1
1440.2.f.i 4 120.m even 2 1
1600.2.a.z 2 5.c odd 4 1
1600.2.a.z 2 20.e even 4 1
1600.2.a.bd 2 5.c odd 4 1
1600.2.a.bd 2 20.e even 4 1
2880.2.f.w 4 3.b odd 2 1
2880.2.f.w 4 12.b even 2 1
2880.2.f.w 4 15.d odd 2 1
2880.2.f.w 4 60.h even 2 1
7200.2.a.cb 2 120.q odd 4 1
7200.2.a.cb 2 120.w even 4 1
7200.2.a.cr 2 120.q odd 4 1
7200.2.a.cr 2 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(320, [\chi])$$:

 $$T_{3}^{4} + 12 T_{3}^{2} + 16$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 12 T^{2} + T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$16 + 28 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$1936 + 92 T^{2} + T^{4}$$
$29$ $$( 6 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -20 + T^{2} )^{2}$$
$43$ $$5776 + 172 T^{2} + T^{4}$$
$47$ $$16 + 188 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -180 + T^{2} )^{2}$$
$67$ $$13456 + 268 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$5776 + 332 T^{2} + T^{4}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$T^{4}$$