# Properties

 Label 320.6.f.a Level 320 Weight 6 Character orbit 320.f Analytic conductor 51.323 Analytic rank 0 Dimension 4 CM discriminant -40 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 5^{2}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + 5 \beta_{3} q^{5} -11 \beta_{2} q^{7} -243 q^{9} +O(q^{10})$$ $$q + 5 \beta_{3} q^{5} -11 \beta_{2} q^{7} -243 q^{9} + 401 \beta_{1} q^{11} -22 \beta_{3} q^{13} + 957 \beta_{1} q^{19} + 149 \beta_{2} q^{23} + 3125 q^{25} -6875 \beta_{1} q^{35} + 1418 \beta_{3} q^{37} + 15202 q^{41} -1215 \beta_{3} q^{45} -687 \beta_{2} q^{47} -43693 q^{49} + 1506 \beta_{3} q^{53} + 2005 \beta_{2} q^{55} + 13993 \beta_{1} q^{59} + 2673 \beta_{2} q^{63} -13750 q^{65} + 17644 \beta_{3} q^{77} + 59049 q^{81} + 128786 q^{89} + 30250 \beta_{1} q^{91} + 4785 \beta_{2} q^{95} -97443 \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 972q^{9} + O(q^{10})$$ $$4q - 972q^{9} + 12500q^{25} + 60808q^{41} - 174772q^{49} - 55000q^{65} + 236196q^{81} + 515144q^{89} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$10 \nu^{3} + 40 \nu$$ $$\beta_{3}$$ $$=$$ $$10 \nu^{2} + 15$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 5 \beta_{1}$$$$)/20$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 15$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 10 \beta_{1}$$$$)/10$$

## 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
 1.61803i − 1.61803i 0.618034i − 0.618034i
0 0 0 −55.9017 0 245.967i 0 −243.000 0
289.2 0 0 0 −55.9017 0 245.967i 0 −243.000 0
289.3 0 0 0 55.9017 0 245.967i 0 −243.000 0
289.4 0 0 0 55.9017 0 245.967i 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
40.e odd 2 1 CM by $$\Q(\sqrt{-10})$$
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.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.a 4
4.b odd 2 1 inner 320.6.f.a 4
5.b even 2 1 inner 320.6.f.a 4
8.b even 2 1 inner 320.6.f.a 4
8.d odd 2 1 inner 320.6.f.a 4
20.d odd 2 1 inner 320.6.f.a 4
40.e odd 2 1 CM 320.6.f.a 4
40.f even 2 1 inner 320.6.f.a 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 243 T^{2} )^{4}$$
$5$ $$( 1 - 3125 T^{2} )^{2}$$
$7$ $$( 1 + 26886 T^{2} + 282475249 T^{4} )^{2}$$
$11$ $$( 1 + 321102 T^{2} + 25937424601 T^{4} )^{2}$$
$13$ $$( 1 + 682086 T^{2} + 137858491849 T^{4} )^{2}$$
$17$ $$( 1 - 1419857 T^{2} )^{4}$$
$19$ $$( 1 - 1288802 T^{2} + 6131066257801 T^{4} )^{2}$$
$23$ $$( 1 - 1772186 T^{2} + 41426511213649 T^{4} )^{2}$$
$29$ $$( 1 - 20511149 T^{2} )^{4}$$
$31$ $$( 1 + 28629151 T^{2} )^{4}$$
$37$ $$( 1 - 112652586 T^{2} + 4808584372417849 T^{4} )^{2}$$
$41$ $$( 1 - 15202 T + 115856201 T^{2} )^{4}$$
$43$ $$( 1 + 147008443 T^{2} )^{4}$$
$47$ $$( 1 - 222705514 T^{2} + 52599132235830049 T^{4} )^{2}$$
$53$ $$( 1 + 552886486 T^{2} + 174887470365513049 T^{4} )^{2}$$
$59$ $$( 1 - 646632402 T^{2} + 511116753300641401 T^{4} )^{2}$$
$61$ $$( 1 - 844596301 T^{2} )^{4}$$
$67$ $$( 1 + 1350125107 T^{2} )^{4}$$
$71$ $$( 1 + 1804229351 T^{2} )^{4}$$
$73$ $$( 1 - 2073071593 T^{2} )^{4}$$
$79$ $$( 1 + 3077056399 T^{2} )^{4}$$
$83$ $$( 1 + 3939040643 T^{2} )^{4}$$
$89$ $$( 1 - 128786 T + 5584059449 T^{2} )^{4}$$
$97$ $$( 1 - 8587340257 T^{2} )^{4}$$