# Properties

 Label 320.6.c.f Level 320 Weight 6 Character orbit 320.c Analytic conductor 51.323 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$51.3228223402$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-11})$$ Defining polynomial: $$x^{2} - x + 3$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 5) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-11}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 \beta q^{3} + ( 45 + 5 \beta ) q^{5} + 9 \beta q^{7} -153 q^{9} +O(q^{10})$$ $$q -3 \beta q^{3} + ( 45 + 5 \beta ) q^{5} + 9 \beta q^{7} -153 q^{9} -252 q^{11} -18 \beta q^{13} + ( 660 - 135 \beta ) q^{15} + 104 \beta q^{17} + 220 q^{19} + 1188 q^{21} -367 \beta q^{23} + ( 925 + 450 \beta ) q^{25} -270 \beta q^{27} + 6930 q^{29} + 6752 q^{31} + 756 \beta q^{33} + ( -1980 + 405 \beta ) q^{35} + 2106 \beta q^{37} -2376 q^{39} -198 q^{41} -63 \beta q^{43} + ( -6885 - 765 \beta ) q^{45} + 1589 \beta q^{47} + 13243 q^{49} + 13728 q^{51} -878 \beta q^{53} + ( -11340 - 1260 \beta ) q^{55} -660 \beta q^{57} + 24660 q^{59} + 5698 q^{61} -1377 \beta q^{63} + ( 3960 - 810 \beta ) q^{65} -6579 \beta q^{67} -48444 q^{69} + 53352 q^{71} -10692 \beta q^{73} + ( 59400 - 2775 \beta ) q^{75} -2268 \beta q^{77} + 51920 q^{79} -72819 q^{81} -9323 \beta q^{83} + ( -22880 + 4680 \beta ) q^{85} -20790 \beta q^{87} -9990 q^{89} + 7128 q^{91} -20256 \beta q^{93} + ( 9900 + 1100 \beta ) q^{95} + 15264 \beta q^{97} + 38556 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 90q^{5} - 306q^{9} + O(q^{10})$$ $$2q + 90q^{5} - 306q^{9} - 504q^{11} + 1320q^{15} + 440q^{19} + 2376q^{21} + 1850q^{25} + 13860q^{29} + 13504q^{31} - 3960q^{35} - 4752q^{39} - 396q^{41} - 13770q^{45} + 26486q^{49} + 27456q^{51} - 22680q^{55} + 49320q^{59} + 11396q^{61} + 7920q^{65} - 96888q^{69} + 106704q^{71} + 118800q^{75} + 103840q^{79} - 145638q^{81} - 45760q^{85} - 19980q^{89} + 14256q^{91} + 19800q^{95} + 77112q^{99} + O(q^{100})$$

## Character values

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

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

## 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
 0.5 + 1.65831i 0.5 − 1.65831i
0 19.8997i 0 45.0000 + 33.1662i 0 59.6992i 0 −153.000 0
129.2 0 19.8997i 0 45.0000 33.1662i 0 59.6992i 0 −153.000 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
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 320.6.c.f 2
4.b odd 2 1 320.6.c.g 2
5.b even 2 1 inner 320.6.c.f 2
8.b even 2 1 5.6.b.a 2
8.d odd 2 1 80.6.c.a 2
20.d odd 2 1 320.6.c.g 2
24.f even 2 1 720.6.f.f 2
24.h odd 2 1 45.6.b.b 2
40.e odd 2 1 80.6.c.a 2
40.f even 2 1 5.6.b.a 2
40.i odd 4 2 25.6.a.c 2
40.k even 4 2 400.6.a.t 2
56.h odd 2 1 245.6.b.a 2
120.i odd 2 1 45.6.b.b 2
120.m even 2 1 720.6.f.f 2
120.w even 4 2 225.6.a.n 2
280.c odd 2 1 245.6.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.6.b.a 2 8.b even 2 1
5.6.b.a 2 40.f even 2 1
25.6.a.c 2 40.i odd 4 2
45.6.b.b 2 24.h odd 2 1
45.6.b.b 2 120.i odd 2 1
80.6.c.a 2 8.d odd 2 1
80.6.c.a 2 40.e odd 2 1
225.6.a.n 2 120.w even 4 2
245.6.b.a 2 56.h odd 2 1
245.6.b.a 2 280.c odd 2 1
320.6.c.f 2 1.a even 1 1 trivial
320.6.c.f 2 5.b even 2 1 inner
320.6.c.g 2 4.b odd 2 1
320.6.c.g 2 20.d odd 2 1
400.6.a.t 2 40.k even 4 2
720.6.f.f 2 24.f even 2 1
720.6.f.f 2 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(320, [\chi])$$:

 $$T_{3}^{2} + 396$$ $$T_{11} + 252$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 24 T + 243 T^{2} )( 1 + 24 T + 243 T^{2} )$$
$5$ $$1 - 90 T + 3125 T^{2}$$
$7$ $$1 - 30050 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 252 T + 161051 T^{2} )^{2}$$
$13$ $$1 - 728330 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 2363810 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 - 220 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 6946370 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 6930 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 6752 T + 28629151 T^{2} )^{2}$$
$37$ $$1 + 56462470 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 198 T + 115856201 T^{2} )^{2}$$
$43$ $$1 - 293842250 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 347593490 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 802472090 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 24660 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 5698 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 795787610 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 53352 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 + 883886830 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 51920 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 4053674810 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 9990 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 6923133890 T^{2} + 73742412689492826049 T^{4}$$