# Properties

 Label 320.6.c.h Level 320 Weight 6 Character orbit 320.c Analytic conductor 51.323 Analytic rank 0 Dimension 4 CM discriminant -20 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 5 \beta_{1} + \beta_{2} ) q^{3} -5 \beta_{3} q^{5} + ( -46 \beta_{1} - \beta_{2} ) q^{7} + ( -243 + 38 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 5 \beta_{1} + \beta_{2} ) q^{3} -5 \beta_{3} q^{5} + ( -46 \beta_{1} - \beta_{2} ) q^{7} + ( -243 + 38 \beta_{3} ) q^{9} + ( 180 \beta_{1} + 95 \beta_{2} ) q^{15} + ( 3602 - 202 \beta_{3} ) q^{21} + ( -664 \beta_{1} - 237 \beta_{2} ) q^{23} + 3125 q^{25} + ( -1368 \beta_{1} - 722 \beta_{2} ) q^{27} -1686 q^{29} + ( -385 \beta_{1} - 915 \beta_{2} ) q^{35} -1882 \beta_{3} q^{41} + ( 1123 \beta_{1} - 1439 \beta_{2} ) q^{43} + ( -23750 + 1215 \beta_{3} ) q^{45} + ( -3866 \beta_{1} + 1203 \beta_{2} ) q^{47} + ( -16807 + 366 \beta_{3} ) q^{49} -4674 \beta_{3} q^{61} + ( 14104 \beta_{1} + 7197 \beta_{2} ) q^{63} + ( -12853 \beta_{1} - 1195 \beta_{2} ) q^{67} + ( 75586 - 6922 \beta_{3} ) q^{69} + ( 15625 \beta_{1} + 3125 \beta_{2} ) q^{75} + ( 121451 - 9234 \beta_{3} ) q^{81} + ( 21187 \beta_{1} + 3117 \beta_{2} ) q^{83} + ( -8430 \beta_{1} - 1686 \beta_{2} ) q^{87} -149286 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 972q^{9} + O(q^{10})$$ $$4q - 972q^{9} + 14408q^{21} + 12500q^{25} - 6744q^{29} - 95000q^{45} - 67228q^{49} + 302344q^{69} + 485804q^{81} - 597144q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-4 \nu^{3} - 8 \nu$$ $$\beta_{2}$$ $$=$$ $$6 \nu^{3} + 22 \nu$$ $$\beta_{3}$$ $$=$$ $$10 \nu^{2} + 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$2 \beta_{2} + 3 \beta_{1}$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 15$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-4 \beta_{2} - 11 \beta_{1}$$$$)/20$$

## 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
 − 1.61803i 0.618034i − 0.618034i 1.61803i
0 30.1803i 0 55.9017 0 194.180i 0 −667.853 0
129.2 0 7.81966i 0 −55.9017 0 171.820i 0 181.853 0
129.3 0 7.81966i 0 −55.9017 0 171.820i 0 181.853 0
129.4 0 30.1803i 0 55.9017 0 194.180i 0 −667.853 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
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.h 4
4.b odd 2 1 inner 320.6.c.h 4
5.b even 2 1 inner 320.6.c.h 4
8.b even 2 1 160.6.c.b 4
8.d odd 2 1 160.6.c.b 4
20.d odd 2 1 CM 320.6.c.h 4
40.e odd 2 1 160.6.c.b 4
40.f even 2 1 160.6.c.b 4
40.i odd 4 1 800.6.a.f 2
40.i odd 4 1 800.6.a.m 2
40.k even 4 1 800.6.a.f 2
40.k even 4 1 800.6.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.6.c.b 4 8.b even 2 1
160.6.c.b 4 8.d odd 2 1
160.6.c.b 4 40.e odd 2 1
160.6.c.b 4 40.f even 2 1
320.6.c.h 4 1.a even 1 1 trivial
320.6.c.h 4 4.b odd 2 1 inner
320.6.c.h 4 5.b even 2 1 inner
320.6.c.h 4 20.d odd 2 1 CM
800.6.a.f 2 40.i odd 4 1
800.6.a.f 2 40.k even 4 1
800.6.a.m 2 40.i odd 4 1
800.6.a.m 2 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 $$S_{6}^{\mathrm{new}}(320, [\chi])$$:

 $$T_{3}^{4} + 972 T_{3}^{2} + 55696$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 38 T + 722 T^{2} - 9234 T^{3} + 59049 T^{4} )( 1 + 38 T + 722 T^{2} + 9234 T^{3} + 59049 T^{4} )$$
$5$ $$( 1 - 3125 T^{2} )^{2}$$
$7$ $$( 1 - 366 T + 66978 T^{2} - 6151362 T^{3} + 282475249 T^{4} )( 1 + 366 T + 66978 T^{2} + 6151362 T^{3} + 282475249 T^{4} )$$
$11$ $$( 1 + 161051 T^{2} )^{4}$$
$13$ $$( 1 - 371293 T^{2} )^{4}$$
$17$ $$( 1 - 1419857 T^{2} )^{4}$$
$19$ $$( 1 + 2476099 T^{2} )^{4}$$
$23$ $$( 1 - 4838 T + 11703122 T^{2} - 31139027434 T^{3} + 41426511213649 T^{4} )( 1 + 4838 T + 11703122 T^{2} + 31139027434 T^{3} + 41426511213649 T^{4} )$$
$29$ $$( 1 + 1686 T + 20511149 T^{2} )^{4}$$
$31$ $$( 1 + 28629151 T^{2} )^{4}$$
$37$ $$( 1 - 69343957 T^{2} )^{4}$$
$41$ $$( 1 - 211028098 T^{2} + 13422659310152401 T^{4} )^{2}$$
$43$ $$( 1 - 11862 T + 70353522 T^{2} - 1743814150866 T^{3} + 21611482313284249 T^{4} )( 1 + 11862 T + 70353522 T^{2} + 1743814150866 T^{3} + 21611482313284249 T^{4} )$$
$47$ $$( 1 - 33334 T + 555577778 T^{2} - 7644986463338 T^{3} + 52599132235830049 T^{4} )( 1 + 33334 T + 555577778 T^{2} + 7644986463338 T^{3} + 52599132235830049 T^{4} )$$
$53$ $$( 1 - 418195493 T^{2} )^{4}$$
$59$ $$( 1 + 714924299 T^{2} )^{4}$$
$61$ $$( 1 - 1041591898 T^{2} + 713342911662882601 T^{4} )^{2}$$
$67$ $$( 1 - 100434 T + 5043494178 T^{2} - 135598464996438 T^{3} + 1822837804551761449 T^{4} )( 1 + 100434 T + 5043494178 T^{2} + 135598464996438 T^{3} + 1822837804551761449 T^{4} )$$
$71$ $$( 1 + 1804229351 T^{2} )^{4}$$
$73$ $$( 1 - 2073071593 T^{2} )^{4}$$
$79$ $$( 1 + 3077056399 T^{2} )^{4}$$
$83$ $$( 1 - 163262 T + 13327240322 T^{2} - 643095653457466 T^{3} + 15516041187205853449 T^{4} )( 1 + 163262 T + 13327240322 T^{2} + 643095653457466 T^{3} + 15516041187205853449 T^{4} )$$
$89$ $$( 1 + 149286 T + 5584059449 T^{2} )^{4}$$
$97$ $$( 1 - 8587340257 T^{2} )^{4}$$