# Properties

 Label 320.3.h.f Level $320$ Weight $3$ Character orbit 320.h Analytic conductor $8.719$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$8.71936845953$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.1827904.1 Defining polynomial: $$x^{6} + 9 x^{4} + 14 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{9}$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{1} ) q^{3} + \beta_{3} q^{5} + ( 2 + \beta_{3} - \beta_{4} ) q^{7} + ( 4 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{1} ) q^{3} + \beta_{3} q^{5} + ( 2 + \beta_{3} - \beta_{4} ) q^{7} + ( 4 + 3 \beta_{1} - \beta_{3} + \beta_{4} ) q^{9} + ( -2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} ) q^{11} + \beta_{5} q^{13} + ( -5 + \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} ) q^{15} + ( 1 + \beta_{1} - 4 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{17} + ( -\beta_{2} + 2 \beta_{5} ) q^{19} + ( -1 + \beta_{1} + \beta_{3} - \beta_{4} ) q^{21} + ( -12 - 2 \beta_{1} + 3 \beta_{3} - 3 \beta_{4} ) q^{23} + ( -5 - 6 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{25} + ( -32 - 4 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{27} + ( -10 - 8 \beta_{1} ) q^{29} + ( -2 - 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{31} + ( 3 + 3 \beta_{1} + 4 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{33} + ( 15 - 5 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{35} + ( 1 + \beta_{1} + 8 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{37} + ( -4 - 4 \beta_{1} - 6 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} ) q^{39} + ( -13 - 5 \beta_{1} - \beta_{3} + \beta_{4} ) q^{41} + ( 13 + \beta_{1} + 6 \beta_{3} - 6 \beta_{4} ) q^{43} + ( 2 \beta_{1} - 8 \beta_{2} - \beta_{3} + 6 \beta_{4} - 3 \beta_{5} ) q^{45} + ( 42 - 8 \beta_{1} + \beta_{3} - \beta_{4} ) q^{47} + ( -12 - 5 \beta_{1} + 7 \beta_{3} - 7 \beta_{4} ) q^{49} + ( 6 + 6 \beta_{1} + 4 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{51} + ( 8 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} - 8 \beta_{4} + 3 \beta_{5} ) q^{53} + ( -55 - 11 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{55} + ( -7 - 7 \beta_{1} - 12 \beta_{2} + 7 \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{57} + ( -15 \beta_{2} - 2 \beta_{5} ) q^{59} + ( 17 + \beta_{1} - 7 \beta_{3} + 7 \beta_{4} ) q^{61} + ( -28 + 2 \beta_{1} - 7 \beta_{3} + 7 \beta_{4} ) q^{63} + ( 5 + 13 \beta_{1} + 8 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{65} + ( -47 + 13 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{67} + ( 39 + 25 \beta_{1} + \beta_{3} - \beta_{4} ) q^{69} + ( 6 + 6 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -1 - \beta_{1} + 4 \beta_{2} + \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{73} + ( 55 + 15 \beta_{1} + 10 \beta_{2} - 8 \beta_{3} ) q^{75} + ( -11 - 11 \beta_{1} + 24 \beta_{2} + 11 \beta_{3} + 11 \beta_{4} + \beta_{5} ) q^{77} + ( 6 + 6 \beta_{1} - 24 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 46 + 19 \beta_{1} + 7 \beta_{3} - 7 \beta_{4} ) q^{81} + ( 33 - 3 \beta_{1} - 14 \beta_{3} + 14 \beta_{4} ) q^{83} + ( 5 - 11 \beta_{1} - 16 \beta_{2} + 3 \beta_{3} - 13 \beta_{4} - \beta_{5} ) q^{85} + ( 106 + 26 \beta_{1} - 8 \beta_{3} + 8 \beta_{4} ) q^{87} + ( 10 - 8 \beta_{1} - 16 \beta_{3} + 16 \beta_{4} ) q^{89} + ( -2 - 2 \beta_{1} + 10 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{91} + ( -8 - 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} + 8 \beta_{4} ) q^{93} + ( 5 + 25 \beta_{1} + 15 \beta_{2} - \beta_{3} - 5 \beta_{4} - 5 \beta_{5} ) q^{95} + ( 5 + 5 \beta_{1} - 12 \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - \beta_{5} ) q^{97} + ( -4 - 4 \beta_{1} - 33 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 10 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 4 q^{3} + 2 q^{5} + 12 q^{7} + 18 q^{9} + O(q^{10})$$ $$6 q - 4 q^{3} + 2 q^{5} + 12 q^{7} + 18 q^{9} - 36 q^{15} - 8 q^{21} - 68 q^{23} - 10 q^{25} - 184 q^{27} - 44 q^{29} + 108 q^{35} - 68 q^{41} + 76 q^{43} + 6 q^{45} + 268 q^{47} - 62 q^{49} - 288 q^{55} + 100 q^{61} - 172 q^{63} - 308 q^{67} + 184 q^{69} + 284 q^{75} + 238 q^{81} + 204 q^{83} + 32 q^{85} + 584 q^{87} + 76 q^{89} - 32 q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 9 x^{4} + 14 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$2 \nu^{4} + 20 \nu^{2} + 23$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{5} + 40 \nu^{3} + 76 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$($$$$-8 \nu^{5} - 2 \nu^{4} - 70 \nu^{3} - 10 \nu^{2} - 92 \nu + 7$$$$)/5$$ $$\beta_{4}$$ $$=$$ $$($$$$-8 \nu^{5} + 4 \nu^{4} - 70 \nu^{3} + 30 \nu^{2} - 92 \nu + 21$$$$)/5$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{5} + 140 \nu^{3} + 224 \nu$$$$)/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + \beta_{3} - \beta_{1} - 1$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{4} + \beta_{3} + 3 \beta_{1} - 11$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_{1} + 2$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{4} - 5 \beta_{3} - 10 \beta_{1} + 32$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$41 \beta_{5} + 21 \beta_{4} + 21 \beta_{3} - 70 \beta_{2} - 21 \beta_{1} - 21$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 0.273891i − 0.273891i − 1.37720i 1.37720i − 2.65109i 2.65109i
0 −5.30219 0 1.54778 4.75441i 0 −0.206625 0 19.1132 0
319.2 0 −5.30219 0 1.54778 + 4.75441i 0 −0.206625 0 19.1132 0
319.3 0 0.547781 0 3.75441 3.30219i 0 10.0566 0 −8.69994 0
319.4 0 0.547781 0 3.75441 + 3.30219i 0 10.0566 0 −8.69994 0
319.5 0 2.75441 0 −4.30219 2.54778i 0 −3.84997 0 −1.41325 0
319.6 0 2.75441 0 −4.30219 + 2.54778i 0 −3.84997 0 −1.41325 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 319.6 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.3.h.f 6
4.b odd 2 1 320.3.h.g 6
5.b even 2 1 320.3.h.g 6
5.c odd 4 1 1600.3.b.v 6
5.c odd 4 1 1600.3.b.w 6
8.b even 2 1 160.3.h.b yes 6
8.d odd 2 1 160.3.h.a 6
16.e even 4 1 1280.3.e.f 6
16.e even 4 1 1280.3.e.h 6
16.f odd 4 1 1280.3.e.g 6
16.f odd 4 1 1280.3.e.i 6
20.d odd 2 1 inner 320.3.h.f 6
20.e even 4 1 1600.3.b.v 6
20.e even 4 1 1600.3.b.w 6
24.f even 2 1 1440.3.j.a 6
24.h odd 2 1 1440.3.j.b 6
40.e odd 2 1 160.3.h.b yes 6
40.f even 2 1 160.3.h.a 6
40.i odd 4 1 800.3.b.h 6
40.i odd 4 1 800.3.b.i 6
40.k even 4 1 800.3.b.h 6
40.k even 4 1 800.3.b.i 6
80.k odd 4 1 1280.3.e.f 6
80.k odd 4 1 1280.3.e.h 6
80.q even 4 1 1280.3.e.g 6
80.q even 4 1 1280.3.e.i 6
120.i odd 2 1 1440.3.j.a 6
120.m even 2 1 1440.3.j.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.h.a 6 8.d odd 2 1
160.3.h.a 6 40.f even 2 1
160.3.h.b yes 6 8.b even 2 1
160.3.h.b yes 6 40.e odd 2 1
320.3.h.f 6 1.a even 1 1 trivial
320.3.h.f 6 20.d odd 2 1 inner
320.3.h.g 6 4.b odd 2 1
320.3.h.g 6 5.b even 2 1
800.3.b.h 6 40.i odd 4 1
800.3.b.h 6 40.k even 4 1
800.3.b.i 6 40.i odd 4 1
800.3.b.i 6 40.k even 4 1
1280.3.e.f 6 16.e even 4 1
1280.3.e.f 6 80.k odd 4 1
1280.3.e.g 6 16.f odd 4 1
1280.3.e.g 6 80.q even 4 1
1280.3.e.h 6 16.e even 4 1
1280.3.e.h 6 80.k odd 4 1
1280.3.e.i 6 16.f odd 4 1
1280.3.e.i 6 80.q even 4 1
1440.3.j.a 6 24.f even 2 1
1440.3.j.a 6 120.i odd 2 1
1440.3.j.b 6 24.h odd 2 1
1440.3.j.b 6 120.m even 2 1
1600.3.b.v 6 5.c odd 4 1
1600.3.b.v 6 20.e even 4 1
1600.3.b.w 6 5.c odd 4 1
1600.3.b.w 6 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{3} + 2 T_{3}^{2} - 16 T_{3} + 8$$ acting on $$S_{3}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$( 8 - 16 T + 2 T^{2} + T^{3} )^{2}$$
$5$ $$15625 - 1250 T + 175 T^{2} + 100 T^{3} + 7 T^{4} - 2 T^{5} + T^{6}$$
$7$ $$( -8 - 40 T - 6 T^{2} + T^{3} )^{2}$$
$11$ $$2560000 + 86784 T^{2} + 560 T^{4} + T^{6}$$
$13$ $$692224 + 43264 T^{2} + 416 T^{4} + T^{6}$$
$17$ $$6553600 + 270336 T^{2} + 1088 T^{4} + T^{6}$$
$19$ $$11505664 + 533248 T^{2} + 1712 T^{4} + T^{6}$$
$23$ $$( -9160 - 152 T + 34 T^{2} + T^{3} )^{2}$$
$29$ $$( 2120 - 948 T + 22 T^{2} + T^{3} )^{2}$$
$31$ $$419430400 + 1900544 T^{2} + 2560 T^{4} + T^{6}$$
$37$ $$432640000 + 2985216 T^{2} + 5408 T^{4} + T^{6}$$
$41$ $$( -5000 - 100 T + 34 T^{2} + T^{3} )^{2}$$
$43$ $$( 6280 - 1408 T - 38 T^{2} + T^{3} )^{2}$$
$47$ $$( -22216 + 4824 T - 134 T^{2} + T^{3} )^{2}$$
$53$ $$50319462400 + 49973504 T^{2} + 12960 T^{4} + T^{6}$$
$59$ $$25416011776 + 41968384 T^{2} + 12464 T^{4} + T^{6}$$
$61$ $$( 81544 - 1732 T - 50 T^{2} + T^{3} )^{2}$$
$67$ $$( -24760 + 4768 T + 154 T^{2} + T^{3} )^{2}$$
$71$ $$14231535616 + 19738624 T^{2} + 8384 T^{4} + T^{6}$$
$73$ $$3114532864 + 6447104 T^{2} + 4416 T^{4} + T^{6}$$
$79$ $$37060870144 + 159514624 T^{2} + 25856 T^{4} + T^{6}$$
$83$ $$( 483400 - 6880 T - 102 T^{2} + T^{3} )^{2}$$
$89$ $$( 155000 - 13940 T - 38 T^{2} + T^{3} )^{2}$$
$97$ $$44302336 + 1384448 T^{2} + 7488 T^{4} + T^{6}$$