# Properties

 Label 640.2.f.d Level $640$ Weight $2$ Character orbit 640.f Analytic conductor $5.110$ Analytic rank $0$ Dimension $4$ CM discriminant -40 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$640 = 2^{7} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 640.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.11042572936$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{5})$$ Defining polynomial: $$x^{4} + 6 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{5} -\beta_{1} q^{7} -3 q^{9} +O(q^{10})$$ $$q + \beta_{3} q^{5} -\beta_{1} q^{7} -3 q^{9} -\beta_{2} q^{11} -2 \beta_{3} q^{13} -\beta_{2} q^{19} + 3 \beta_{1} q^{23} + 5 q^{25} -\beta_{2} q^{35} + 2 \beta_{3} q^{37} -2 q^{41} -3 \beta_{3} q^{45} -\beta_{1} q^{47} - q^{49} + 6 \beta_{3} q^{53} -5 \beta_{1} q^{55} -\beta_{2} q^{59} + 3 \beta_{1} q^{63} -10 q^{65} -8 \beta_{3} q^{77} + 9 q^{81} + 14 q^{89} + 2 \beta_{2} q^{91} -5 \beta_{1} q^{95} + 3 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{9} + O(q^{10})$$ $$4q - 12q^{9} + 20q^{25} - 8q^{41} - 4q^{49} - 40q^{65} + 36q^{81} + 56q^{89} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 6 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} + 4 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} + 8 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 3$$ $$\nu^{3}$$ $$=$$ $$-\beta_{2} + 2 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$$.

 $$n$$ $$257$$ $$261$$ $$511$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 2.28825i 2.28825i 0.874032i − 0.874032i
0 0 0 −2.23607 0 2.82843i 0 −3.00000 0
449.2 0 0 0 −2.23607 0 2.82843i 0 −3.00000 0
449.3 0 0 0 2.23607 0 2.82843i 0 −3.00000 0
449.4 0 0 0 2.23607 0 2.82843i 0 −3.00000 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 640.2.f.d 4
4.b odd 2 1 inner 640.2.f.d 4
5.b even 2 1 inner 640.2.f.d 4
5.c odd 4 2 3200.2.d.w 4
8.b even 2 1 inner 640.2.f.d 4
8.d odd 2 1 inner 640.2.f.d 4
16.e even 4 2 1280.2.c.l 4
16.f odd 4 2 1280.2.c.l 4
20.d odd 2 1 inner 640.2.f.d 4
20.e even 4 2 3200.2.d.w 4
40.e odd 2 1 CM 640.2.f.d 4
40.f even 2 1 inner 640.2.f.d 4
40.i odd 4 2 3200.2.d.w 4
40.k even 4 2 3200.2.d.w 4
80.i odd 4 2 6400.2.a.cl 4
80.j even 4 2 6400.2.a.cl 4
80.k odd 4 2 1280.2.c.l 4
80.q even 4 2 1280.2.c.l 4
80.s even 4 2 6400.2.a.cl 4
80.t odd 4 2 6400.2.a.cl 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.f.d 4 1.a even 1 1 trivial
640.2.f.d 4 4.b odd 2 1 inner
640.2.f.d 4 5.b even 2 1 inner
640.2.f.d 4 8.b even 2 1 inner
640.2.f.d 4 8.d odd 2 1 inner
640.2.f.d 4 20.d odd 2 1 inner
640.2.f.d 4 40.e odd 2 1 CM
640.2.f.d 4 40.f even 2 1 inner
1280.2.c.l 4 16.e even 4 2
1280.2.c.l 4 16.f odd 4 2
1280.2.c.l 4 80.k odd 4 2
1280.2.c.l 4 80.q even 4 2
3200.2.d.w 4 5.c odd 4 2
3200.2.d.w 4 20.e even 4 2
3200.2.d.w 4 40.i odd 4 2
3200.2.d.w 4 40.k even 4 2
6400.2.a.cl 4 80.i odd 4 2
6400.2.a.cl 4 80.j even 4 2
6400.2.a.cl 4 80.s even 4 2
6400.2.a.cl 4 80.t odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(640, [\chi])$$:

 $$T_{3}$$ $$T_{7}^{2} + 8$$ $$T_{13}^{2} - 20$$ $$T_{37}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$( 8 + T^{2} )^{2}$$
$11$ $$( 40 + T^{2} )^{2}$$
$13$ $$( -20 + T^{2} )^{2}$$
$17$ $$T^{4}$$
$19$ $$( 40 + T^{2} )^{2}$$
$23$ $$( 72 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$( -20 + T^{2} )^{2}$$
$41$ $$( 2 + T )^{4}$$
$43$ $$T^{4}$$
$47$ $$( 8 + T^{2} )^{2}$$
$53$ $$( -180 + T^{2} )^{2}$$
$59$ $$( 40 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( -14 + T )^{4}$$
$97$ $$T^{4}$$