# Properties

 Label 320.6.c.c Level 320 Weight 6 Character orbit 320.c Analytic conductor 51.323 Analytic rank 0 Dimension 2 CM discriminant -4 Inner twists 4

# 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: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 160) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -41 - 38 i ) q^{5} + 243 q^{9} +O(q^{10})$$ $$q + ( -41 - 38 i ) q^{5} + 243 q^{9} + 244 i q^{13} + 808 i q^{17} + ( 237 + 3116 i ) q^{25} -2950 q^{29} -11292 i q^{37} + 20950 q^{41} + ( -9963 - 9234 i ) q^{45} + 16807 q^{49} -40244 i q^{53} + 18950 q^{61} + ( 9272 - 10004 i ) q^{65} + 20144 i q^{73} + 59049 q^{81} + ( 30704 - 33128 i ) q^{85} + 51050 q^{89} -160808 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 82q^{5} + 486q^{9} + O(q^{10})$$ $$2q - 82q^{5} + 486q^{9} + 474q^{25} - 5900q^{29} + 41900q^{41} - 19926q^{45} + 33614q^{49} + 37900q^{61} + 18544q^{65} + 118098q^{81} + 61408q^{85} + 102100q^{89} + O(q^{100})$$

## 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
 1.00000i − 1.00000i
0 0 0 −41.0000 38.0000i 0 0 0 243.000 0
129.2 0 0 0 −41.0000 + 38.0000i 0 0 0 243.000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
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 320.6.c.c 2
4.b odd 2 1 CM 320.6.c.c 2
5.b even 2 1 inner 320.6.c.c 2
8.b even 2 1 160.6.c.a 2
8.d odd 2 1 160.6.c.a 2
20.d odd 2 1 inner 320.6.c.c 2
40.e odd 2 1 160.6.c.a 2
40.f even 2 1 160.6.c.a 2
40.i odd 4 1 800.6.a.b 1
40.i odd 4 1 800.6.a.c 1
40.k even 4 1 800.6.a.b 1
40.k even 4 1 800.6.a.c 1

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

## Hecke kernels

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

 $$T_{3}$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 243 T^{2} )^{2}$$
$5$ $$1 + 82 T + 3125 T^{2}$$
$7$ $$( 1 - 16807 T^{2} )^{2}$$
$11$ $$( 1 + 161051 T^{2} )^{2}$$
$13$ $$( 1 - 1194 T + 371293 T^{2} )( 1 + 1194 T + 371293 T^{2} )$$
$17$ $$( 1 - 2242 T + 1419857 T^{2} )( 1 + 2242 T + 1419857 T^{2} )$$
$19$ $$( 1 + 2476099 T^{2} )^{2}$$
$23$ $$( 1 - 6436343 T^{2} )^{2}$$
$29$ $$( 1 + 2950 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 + 28629151 T^{2} )^{2}$$
$37$ $$( 1 - 12242 T + 69343957 T^{2} )( 1 + 12242 T + 69343957 T^{2} )$$
$41$ $$( 1 - 20950 T + 115856201 T^{2} )^{2}$$
$43$ $$( 1 - 147008443 T^{2} )^{2}$$
$47$ $$( 1 - 229345007 T^{2} )^{2}$$
$53$ $$( 1 - 7294 T + 418195493 T^{2} )( 1 + 7294 T + 418195493 T^{2} )$$
$59$ $$( 1 + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 18950 T + 844596301 T^{2} )^{2}$$
$67$ $$( 1 - 1350125107 T^{2} )^{2}$$
$71$ $$( 1 + 1804229351 T^{2} )^{2}$$
$73$ $$( 1 - 88806 T + 2073071593 T^{2} )( 1 + 88806 T + 2073071593 T^{2} )$$
$79$ $$( 1 + 3077056399 T^{2} )^{2}$$
$83$ $$( 1 - 3939040643 T^{2} )^{2}$$
$89$ $$( 1 - 51050 T + 5584059449 T^{2} )^{2}$$
$97$ $$( 1 - 92142 T + 8587340257 T^{2} )( 1 + 92142 T + 8587340257 T^{2} )$$