# Properties

 Label 320.6.f.b Level 320 Weight 6 Character orbit 320.f Analytic conductor 51.323 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 320.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.73499483897856.45 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{10}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{3} -5 \beta_{5} q^{5} -11 \beta_{2} q^{7} + 123 q^{9} +O(q^{10})$$ $$q + \beta_{4} q^{3} -5 \beta_{5} q^{5} -11 \beta_{2} q^{7} + 123 q^{9} -149 \beta_{1} q^{11} + ( 44 \beta_{5} + 22 \beta_{6} ) q^{13} + ( 15 \beta_{2} - 305 \beta_{3} ) q^{15} -11 \beta_{7} q^{17} -1287 \beta_{1} q^{19} -671 \beta_{6} q^{21} -65 \beta_{2} q^{23} + ( 2975 + 25 \beta_{7} ) q^{25} -120 \beta_{4} q^{27} -1078 \beta_{6} q^{29} + 1798 \beta_{3} q^{31} -149 \beta_{7} q^{33} + ( -3355 \beta_{1} + 55 \beta_{4} ) q^{35} + ( 518 \beta_{5} + 259 \beta_{6} ) q^{37} + 2684 \beta_{3} q^{39} + 11396 q^{41} + 1001 \beta_{4} q^{43} -615 \beta_{5} q^{45} + 207 \beta_{2} q^{47} + 2045 q^{49} -4026 \beta_{1} q^{51} + ( 1560 \beta_{5} + 780 \beta_{6} ) q^{53} + ( -1490 \beta_{2} - 745 \beta_{3} ) q^{55} -1287 \beta_{7} q^{57} + 8189 \beta_{1} q^{59} + 14069 \beta_{6} q^{61} -1353 \beta_{2} q^{63} + ( -26840 - 110 \beta_{7} ) q^{65} + 439 \beta_{4} q^{67} -3965 \beta_{6} q^{69} -8398 \beta_{3} q^{71} -11 \beta_{7} q^{73} + ( 9150 \beta_{1} + 2975 \beta_{4} ) q^{75} + ( 3278 \beta_{5} + 1639 \beta_{6} ) q^{77} -8998 \beta_{3} q^{79} -73809 q^{81} + 5555 \beta_{4} q^{83} + ( -330 \beta_{5} - 6875 \beta_{6} ) q^{85} -6468 \beta_{2} q^{87} -99362 q^{89} + 29524 \beta_{1} q^{91} + ( 10788 \beta_{5} + 5394 \beta_{6} ) q^{93} + ( -12870 \beta_{2} - 6435 \beta_{3} ) q^{95} -4713 \beta_{7} q^{97} -18327 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 984q^{9} + O(q^{10})$$ $$8q + 984q^{9} + 23800q^{25} + 91168q^{41} + 16360q^{49} - 214720q^{65} - 590472q^{81} - 794896q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3721 x^{4} + 13845841$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{6}$$$$/226981$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 61 \nu^{3} + 3721 \nu$$$$)/3721$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{6} + 14884 \nu^{2}$$$$)/226981$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{7} + 61 \nu^{5} + 3721 \nu^{3} + 226981 \nu$$$$)/226981$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 2 \nu^{4} + 61 \nu^{3} + 3721 \nu - 3721$$$$)/3721$$ $$\beta_{6}$$ $$=$$ $$($$$$-4 \nu^{4} + 7442$$$$)/3721$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{7} + 122 \nu^{5} - 7442 \nu^{3} + 453962 \nu$$$$)/226981$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} + 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{2}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$61 \beta_{3} + 61 \beta_{1}$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$61 \beta_{6} + 122 \beta_{5} - 122 \beta_{2}$$$$)/4$$ $$\nu^{4}$$ $$=$$ $$($$$$-3721 \beta_{6} + 7442$$$$)/4$$ $$\nu^{5}$$ $$=$$ $$($$$$3721 \beta_{7} - 3721 \beta_{6} - 7442 \beta_{5} + 7442 \beta_{4} - 7442 \beta_{2}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$226981 \beta_{1}$$$$/2$$ $$\nu^{7}$$ $$=$$ $$($$$$226981 \beta_{7} + 226981 \beta_{6} + 453962 \beta_{5} - 453962 \beta_{4} - 453962 \beta_{2}$$$$)/8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 −2.02144 + 7.54412i −2.02144 − 7.54412i −7.54412 − 2.02144i −7.54412 + 2.02144i 7.54412 + 2.02144i 7.54412 − 2.02144i 2.02144 − 7.54412i 2.02144 + 7.54412i
0 −19.1311 0 −55.2268 8.66025i 0 121.499i 0 123.000 0
289.2 0 −19.1311 0 −55.2268 + 8.66025i 0 121.499i 0 123.000 0
289.3 0 −19.1311 0 55.2268 8.66025i 0 121.499i 0 123.000 0
289.4 0 −19.1311 0 55.2268 + 8.66025i 0 121.499i 0 123.000 0
289.5 0 19.1311 0 −55.2268 8.66025i 0 121.499i 0 123.000 0
289.6 0 19.1311 0 −55.2268 + 8.66025i 0 121.499i 0 123.000 0
289.7 0 19.1311 0 55.2268 8.66025i 0 121.499i 0 123.000 0
289.8 0 19.1311 0 55.2268 + 8.66025i 0 121.499i 0 123.000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 289.8 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
8.b even 2 1 inner
8.d odd 2 1 inner
20.d odd 2 1 inner
40.e odd 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.f.b 8
4.b odd 2 1 inner 320.6.f.b 8
5.b even 2 1 inner 320.6.f.b 8
8.b even 2 1 inner 320.6.f.b 8
8.d odd 2 1 inner 320.6.f.b 8
20.d odd 2 1 inner 320.6.f.b 8
40.e odd 2 1 inner 320.6.f.b 8
40.f even 2 1 inner 320.6.f.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.f.b 8 1.a even 1 1 trivial
320.6.f.b 8 4.b odd 2 1 inner
320.6.f.b 8 5.b even 2 1 inner
320.6.f.b 8 8.b even 2 1 inner
320.6.f.b 8 8.d odd 2 1 inner
320.6.f.b 8 20.d odd 2 1 inner
320.6.f.b 8 40.e odd 2 1 inner
320.6.f.b 8 40.f even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 366$$ acting on $$S_{6}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 120 T^{2} + 59049 T^{4} )^{4}$$
$5$ $$( 1 - 5950 T^{2} + 9765625 T^{4} )^{2}$$
$7$ $$( 1 - 18852 T^{2} + 282475249 T^{4} )^{4}$$
$11$ $$( 1 - 233298 T^{2} + 25937424601 T^{4} )^{4}$$
$13$ $$( 1 + 506394 T^{2} + 137858491849 T^{4} )^{4}$$
$17$ $$( 1 - 2662570 T^{2} + 2015993900449 T^{4} )^{4}$$
$19$ $$( 1 + 1673278 T^{2} + 6131066257801 T^{4} )^{4}$$
$23$ $$( 1 - 12357236 T^{2} + 41426511213649 T^{4} )^{4}$$
$29$ $$( 1 - 27077290 T^{2} + 420707233300201 T^{4} )^{4}$$
$31$ $$( 1 + 18464654 T^{2} + 819628286980801 T^{4} )^{4}$$
$37$ $$( 1 + 105952386 T^{2} + 4808584372417849 T^{4} )^{4}$$
$41$ $$( 1 - 11396 T + 115856201 T^{2} )^{8}$$
$43$ $$( 1 - 72715480 T^{2} + 21611482313284249 T^{4} )^{4}$$
$47$ $$( 1 - 453462436 T^{2} + 52599132235830049 T^{4} )^{4}$$
$53$ $$( 1 + 539491786 T^{2} + 174887470365513049 T^{4} )^{4}$$
$59$ $$( 1 - 1161609714 T^{2} + 511116753300641401 T^{4} )^{4}$$
$61$ $$( 1 + 686048530 T^{2} + 713342911662882601 T^{4} )^{4}$$
$67$ $$( 1 + 2629714328 T^{2} + 1822837804551761449 T^{4} )^{4}$$
$71$ $$( 1 + 2762141854 T^{2} + 3255243551009881201 T^{4} )^{4}$$
$73$ $$( 1 - 4145966042 T^{2} + 4297625829703557649 T^{4} )^{4}$$
$79$ $$( 1 + 5182544750 T^{2} + 9468276082626847201 T^{4} )^{4}$$
$83$ $$( 1 - 3415955864 T^{2} + 15516041187205853449 T^{4} )^{4}$$
$89$ $$( 1 + 99362 T + 5584059449 T^{2} )^{8}$$
$97$ $$( 1 + 15344227702 T^{2} + 73742412689492826049 T^{4} )^{4}$$