# Properties

 Label 320.3.b.c Level $320$ Weight $3$ Character orbit 320.b Analytic conductor $8.719$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 320.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.71936845953$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 20) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{5} + ( -2 + 4 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{7} + ( 1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -2 + 4 \zeta_{10} - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{3} + ( 1 + 2 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{5} + ( -2 + 4 \zeta_{10} - 6 \zeta_{10}^{2} - 2 \zeta_{10}^{3} ) q^{7} + ( 1 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{9} + ( 4 - 8 \zeta_{10} - 4 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{11} + ( 6 + 4 \zeta_{10}^{2} - 4 \zeta_{10}^{3} ) q^{13} + ( 2 - 4 \zeta_{10} + 6 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{15} + ( 2 + 16 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{17} + ( 8 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{19} + ( 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{21} + ( -6 + 12 \zeta_{10} - 10 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{23} + 5 q^{25} + ( -12 + 24 \zeta_{10} - 4 \zeta_{10}^{2} + 20 \zeta_{10}^{3} ) q^{27} + ( 6 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{29} + ( 20 - 40 \zeta_{10} + 28 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{31} + ( 32 + 24 \zeta_{10}^{2} - 24 \zeta_{10}^{3} ) q^{33} + ( 10 - 20 \zeta_{10} + 10 \zeta_{10}^{2} - 10 \zeta_{10}^{3} ) q^{35} + ( 6 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{37} + ( -4 + 8 \zeta_{10} + 4 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{39} + ( -22 + 12 \zeta_{10}^{2} - 12 \zeta_{10}^{3} ) q^{41} + ( 6 - 12 \zeta_{10} - 2 \zeta_{10}^{2} - 14 \zeta_{10}^{3} ) q^{43} + ( 9 - 2 \zeta_{10}^{2} + 2 \zeta_{10}^{3} ) q^{45} + ( -26 + 52 \zeta_{10} - 14 \zeta_{10}^{2} + 38 \zeta_{10}^{3} ) q^{47} + ( 9 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{49} + ( 28 - 56 \zeta_{10} + 60 \zeta_{10}^{2} + 4 \zeta_{10}^{3} ) q^{51} + ( 54 + 20 \zeta_{10}^{2} - 20 \zeta_{10}^{3} ) q^{53} + ( 12 - 24 \zeta_{10} - 4 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{55} + ( -16 - 32 \zeta_{10}^{2} + 32 \zeta_{10}^{3} ) q^{57} + ( -24 + 48 \zeta_{10} - 48 \zeta_{10}^{2} ) q^{59} + ( -58 - 52 \zeta_{10}^{2} + 52 \zeta_{10}^{3} ) q^{61} + ( 22 - 44 \zeta_{10} + 26 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{63} + ( 14 + 8 \zeta_{10}^{2} - 8 \zeta_{10}^{3} ) q^{65} + ( -22 + 44 \zeta_{10} - 62 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{67} + ( -16 + 28 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{69} + ( 4 - 8 \zeta_{10} - 36 \zeta_{10}^{2} - 44 \zeta_{10}^{3} ) q^{71} + ( 34 - 64 \zeta_{10}^{2} + 64 \zeta_{10}^{3} ) q^{73} + ( -10 + 20 \zeta_{10} - 10 \zeta_{10}^{2} + 10 \zeta_{10}^{3} ) q^{75} + ( -80 - 40 \zeta_{10}^{2} + 40 \zeta_{10}^{3} ) q^{77} + ( -40 + 80 \zeta_{10} - 72 \zeta_{10}^{2} + 8 \zeta_{10}^{3} ) q^{79} + ( -55 + 28 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{81} + ( -26 + 52 \zeta_{10} - 2 \zeta_{10}^{2} + 50 \zeta_{10}^{3} ) q^{83} + ( 34 - 12 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{85} + ( 4 - 8 \zeta_{10} + 20 \zeta_{10}^{2} + 12 \zeta_{10}^{3} ) q^{87} + ( -62 - 80 \zeta_{10}^{2} + 80 \zeta_{10}^{3} ) q^{89} + ( 12 - 24 \zeta_{10} - 4 \zeta_{10}^{2} - 28 \zeta_{10}^{3} ) q^{91} + ( 64 - 72 \zeta_{10}^{2} + 72 \zeta_{10}^{3} ) q^{93} + ( -16 + 32 \zeta_{10} - 8 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{95} + ( -78 - 24 \zeta_{10}^{2} + 24 \zeta_{10}^{3} ) q^{97} + ( 20 - 40 \zeta_{10} - 4 \zeta_{10}^{2} - 44 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 16q^{13} - 24q^{17} - 40q^{21} + 20q^{25} + 8q^{29} + 80q^{33} - 16q^{37} - 112q^{41} + 40q^{45} - 4q^{49} + 176q^{53} - 128q^{61} + 40q^{65} - 120q^{69} + 264q^{73} - 240q^{77} - 276q^{81} + 160q^{85} - 88q^{89} + 400q^{93} - 264q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
191.1
 −0.309017 − 0.951057i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.309017 + 0.951057i
0 3.80423i 0 −2.23607 0 8.50651i 0 −5.47214 0
191.2 0 2.35114i 0 2.23607 0 5.25731i 0 3.47214 0
191.3 0 2.35114i 0 2.23607 0 5.25731i 0 3.47214 0
191.4 0 3.80423i 0 −2.23607 0 8.50651i 0 −5.47214 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.b.c 4
3.b odd 2 1 2880.3.e.e 4
4.b odd 2 1 inner 320.3.b.c 4
5.b even 2 1 1600.3.b.s 4
5.c odd 4 2 1600.3.h.n 8
8.b even 2 1 20.3.b.a 4
8.d odd 2 1 20.3.b.a 4
12.b even 2 1 2880.3.e.e 4
16.e even 4 2 1280.3.g.e 8
16.f odd 4 2 1280.3.g.e 8
20.d odd 2 1 1600.3.b.s 4
20.e even 4 2 1600.3.h.n 8
24.f even 2 1 180.3.c.a 4
24.h odd 2 1 180.3.c.a 4
40.e odd 2 1 100.3.b.f 4
40.f even 2 1 100.3.b.f 4
40.i odd 4 2 100.3.d.b 8
40.k even 4 2 100.3.d.b 8
120.i odd 2 1 900.3.c.k 4
120.m even 2 1 900.3.c.k 4
120.q odd 4 2 900.3.f.e 8
120.w even 4 2 900.3.f.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.3.b.a 4 8.b even 2 1
20.3.b.a 4 8.d odd 2 1
100.3.b.f 4 40.e odd 2 1
100.3.b.f 4 40.f even 2 1
100.3.d.b 8 40.i odd 4 2
100.3.d.b 8 40.k even 4 2
180.3.c.a 4 24.f even 2 1
180.3.c.a 4 24.h odd 2 1
320.3.b.c 4 1.a even 1 1 trivial
320.3.b.c 4 4.b odd 2 1 inner
900.3.c.k 4 120.i odd 2 1
900.3.c.k 4 120.m even 2 1
900.3.f.e 8 120.q odd 4 2
900.3.f.e 8 120.w even 4 2
1280.3.g.e 8 16.e even 4 2
1280.3.g.e 8 16.f odd 4 2
1600.3.b.s 4 5.b even 2 1
1600.3.b.s 4 20.d odd 2 1
1600.3.h.n 8 5.c odd 4 2
1600.3.h.n 8 20.e even 4 2
2880.3.e.e 4 3.b odd 2 1
2880.3.e.e 4 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 20 T_{3}^{2} + 80$$ acting on $$S_{3}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$80 + 20 T^{2} + T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$2000 + 100 T^{2} + T^{4}$$
$11$ $$1280 + 400 T^{2} + T^{4}$$
$13$ $$( -4 - 8 T + T^{2} )^{2}$$
$17$ $$( -284 + 12 T + T^{2} )^{2}$$
$19$ $$20480 + 320 T^{2} + T^{4}$$
$23$ $$80 + 260 T^{2} + T^{4}$$
$29$ $$( -76 - 4 T + T^{2} )^{2}$$
$31$ $$154880 + 2320 T^{2} + T^{4}$$
$37$ $$( -484 + 8 T + T^{2} )^{2}$$
$41$ $$( 604 + 56 T + T^{2} )^{2}$$
$43$ $$2000 + 500 T^{2} + T^{4}$$
$47$ $$3561680 + 4100 T^{2} + T^{4}$$
$53$ $$( 1436 - 88 T + T^{2} )^{2}$$
$59$ $$1658880 + 5760 T^{2} + T^{4}$$
$61$ $$( -2356 + 64 T + T^{2} )^{2}$$
$67$ $$19920080 + 10420 T^{2} + T^{4}$$
$71$ $$10138880 + 8080 T^{2} + T^{4}$$
$73$ $$( -764 - 132 T + T^{2} )^{2}$$
$79$ $$2478080 + 13120 T^{2} + T^{4}$$
$83$ $$2620880 + 6260 T^{2} + T^{4}$$
$89$ $$( -7516 + 44 T + T^{2} )^{2}$$
$97$ $$( 3636 + 132 T + T^{2} )^{2}$$