# Properties

 Label 320.2.s.a Level $320$ Weight $2$ Character orbit 320.s Analytic conductor $2.555$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 320.s (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.55521286468$$ 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: $$1$$ Twist minimal: no (minimal twist has level 80) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + ( -2 + i ) q^{5} + ( 3 + 3 i ) q^{7} + q^{9} +O(q^{10})$$ $$q + 2 q^{3} + ( -2 + i ) q^{5} + ( 3 + 3 i ) q^{7} + q^{9} + ( 1 - i ) q^{11} + 2 i q^{13} + ( -4 + 2 i ) q^{15} + ( 1 + i ) q^{17} + ( 3 - 3 i ) q^{19} + ( 6 + 6 i ) q^{21} + ( 1 - i ) q^{23} + ( 3 - 4 i ) q^{25} -4 q^{27} + ( -7 - 7 i ) q^{29} + 2 i q^{31} + ( 2 - 2 i ) q^{33} + ( -9 - 3 i ) q^{35} -6 i q^{37} + 4 i q^{39} + 4 i q^{41} + 4 i q^{43} + ( -2 + i ) q^{45} + ( 7 - 7 i ) q^{47} + 11 i q^{49} + ( 2 + 2 i ) q^{51} -8 q^{53} + ( -1 + 3 i ) q^{55} + ( 6 - 6 i ) q^{57} + ( -3 - 3 i ) q^{59} + ( -1 + i ) q^{61} + ( 3 + 3 i ) q^{63} + ( -2 - 4 i ) q^{65} -4 i q^{67} + ( 2 - 2 i ) q^{69} + ( 3 + 3 i ) q^{73} + ( 6 - 8 i ) q^{75} + 6 q^{77} -8 q^{79} -11 q^{81} + 2 q^{83} + ( -3 - i ) q^{85} + ( -14 - 14 i ) q^{87} -6 q^{89} + ( -6 + 6 i ) q^{91} + 4 i q^{93} + ( -3 + 9 i ) q^{95} + ( -11 - 11 i ) q^{97} + ( 1 - i ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{3} - 4q^{5} + 6q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 4q^{3} - 4q^{5} + 6q^{7} + 2q^{9} + 2q^{11} - 8q^{15} + 2q^{17} + 6q^{19} + 12q^{21} + 2q^{23} + 6q^{25} - 8q^{27} - 14q^{29} + 4q^{33} - 18q^{35} - 4q^{45} + 14q^{47} + 4q^{51} - 16q^{53} - 2q^{55} + 12q^{57} - 6q^{59} - 2q^{61} + 6q^{63} - 4q^{65} + 4q^{69} + 6q^{73} + 12q^{75} + 12q^{77} - 16q^{79} - 22q^{81} + 4q^{83} - 6q^{85} - 28q^{87} - 12q^{89} - 12q^{91} - 6q^{95} - 22q^{97} + 2q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
207.1
 1.00000i − 1.00000i
0 2.00000 0 −2.00000 + 1.00000i 0 3.00000 + 3.00000i 0 1.00000 0
303.1 0 2.00000 0 −2.00000 1.00000i 0 3.00000 3.00000i 0 1.00000 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
80.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.2.s.a 2
4.b odd 2 1 80.2.s.a yes 2
5.b even 2 1 1600.2.s.a 2
5.c odd 4 1 320.2.j.a 2
5.c odd 4 1 1600.2.j.a 2
8.b even 2 1 640.2.s.a 2
8.d odd 2 1 640.2.s.b 2
12.b even 2 1 720.2.z.d 2
16.e even 4 1 80.2.j.a 2
16.e even 4 1 640.2.j.a 2
16.f odd 4 1 320.2.j.a 2
16.f odd 4 1 640.2.j.b 2
20.d odd 2 1 400.2.s.a 2
20.e even 4 1 80.2.j.a 2
20.e even 4 1 400.2.j.a 2
40.i odd 4 1 640.2.j.b 2
40.k even 4 1 640.2.j.a 2
48.i odd 4 1 720.2.bd.a 2
60.l odd 4 1 720.2.bd.a 2
80.i odd 4 1 80.2.s.a yes 2
80.j even 4 1 640.2.s.a 2
80.j even 4 1 1600.2.s.a 2
80.k odd 4 1 1600.2.j.a 2
80.q even 4 1 400.2.j.a 2
80.s even 4 1 inner 320.2.s.a 2
80.t odd 4 1 400.2.s.a 2
80.t odd 4 1 640.2.s.b 2
240.bb even 4 1 720.2.z.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.j.a 2 16.e even 4 1
80.2.j.a 2 20.e even 4 1
80.2.s.a yes 2 4.b odd 2 1
80.2.s.a yes 2 80.i odd 4 1
320.2.j.a 2 5.c odd 4 1
320.2.j.a 2 16.f odd 4 1
320.2.s.a 2 1.a even 1 1 trivial
320.2.s.a 2 80.s even 4 1 inner
400.2.j.a 2 20.e even 4 1
400.2.j.a 2 80.q even 4 1
400.2.s.a 2 20.d odd 2 1
400.2.s.a 2 80.t odd 4 1
640.2.j.a 2 16.e even 4 1
640.2.j.a 2 40.k even 4 1
640.2.j.b 2 16.f odd 4 1
640.2.j.b 2 40.i odd 4 1
640.2.s.a 2 8.b even 2 1
640.2.s.a 2 80.j even 4 1
640.2.s.b 2 8.d odd 2 1
640.2.s.b 2 80.t odd 4 1
720.2.z.d 2 12.b even 2 1
720.2.z.d 2 240.bb even 4 1
720.2.bd.a 2 48.i odd 4 1
720.2.bd.a 2 60.l odd 4 1
1600.2.j.a 2 5.c odd 4 1
1600.2.j.a 2 80.k odd 4 1
1600.2.s.a 2 5.b even 2 1
1600.2.s.a 2 80.j even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 2$$ acting on $$S_{2}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -2 + T )^{2}$$
$5$ $$5 + 4 T + T^{2}$$
$7$ $$18 - 6 T + T^{2}$$
$11$ $$2 - 2 T + T^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$2 - 2 T + T^{2}$$
$19$ $$18 - 6 T + T^{2}$$
$23$ $$2 - 2 T + T^{2}$$
$29$ $$98 + 14 T + T^{2}$$
$31$ $$4 + T^{2}$$
$37$ $$36 + T^{2}$$
$41$ $$16 + T^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$98 - 14 T + T^{2}$$
$53$ $$( 8 + T )^{2}$$
$59$ $$18 + 6 T + T^{2}$$
$61$ $$2 + 2 T + T^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$T^{2}$$
$73$ $$18 - 6 T + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$( -2 + T )^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$242 + 22 T + T^{2}$$