# Properties

 Label 640.2.d.d Level $640$ Weight $2$ Character orbit 640.d Analytic conductor $5.110$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$5.11042572936$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + q^{9} +O(q^{10})$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{5} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{7} + q^{9} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{11} + 2 \zeta_{8}^{2} q^{13} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{15} + 6 q^{17} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{19} + 6 \zeta_{8}^{2} q^{21} + ( 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{23} - q^{25} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} + 4 \zeta_{8}^{2} q^{29} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{31} + 8 q^{33} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{35} + 2 \zeta_{8}^{2} q^{37} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{39} -8 q^{41} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{43} + \zeta_{8}^{2} q^{45} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{47} + 11 q^{49} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{51} -2 \zeta_{8}^{2} q^{53} + ( -4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{55} + 4 q^{57} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{59} + 14 \zeta_{8}^{2} q^{61} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{63} -2 q^{65} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{67} -10 \zeta_{8}^{2} q^{69} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{71} -6 q^{73} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{75} -24 \zeta_{8}^{2} q^{77} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{79} -5 q^{81} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{83} + 6 \zeta_{8}^{2} q^{85} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{87} -6 q^{89} + ( -6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{91} + 4 \zeta_{8}^{2} q^{93} + ( -2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{95} + 10 q^{97} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} + 24q^{17} - 4q^{25} + 32q^{33} - 32q^{41} + 44q^{49} + 16q^{57} - 8q^{65} - 24q^{73} - 20q^{81} - 24q^{89} + 40q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
0 1.41421i 0 1.00000i 0 4.24264 0 1.00000 0
321.2 0 1.41421i 0 1.00000i 0 −4.24264 0 1.00000 0
321.3 0 1.41421i 0 1.00000i 0 −4.24264 0 1.00000 0
321.4 0 1.41421i 0 1.00000i 0 4.24264 0 1.00000 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
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.2.d.d 4
3.b odd 2 1 5760.2.k.o 4
4.b odd 2 1 inner 640.2.d.d 4
5.b even 2 1 3200.2.d.s 4
5.c odd 4 1 3200.2.f.i 4
5.c odd 4 1 3200.2.f.j 4
8.b even 2 1 inner 640.2.d.d 4
8.d odd 2 1 inner 640.2.d.d 4
12.b even 2 1 5760.2.k.o 4
16.e even 4 1 1280.2.a.f 2
16.e even 4 1 1280.2.a.j 2
16.f odd 4 1 1280.2.a.f 2
16.f odd 4 1 1280.2.a.j 2
20.d odd 2 1 3200.2.d.s 4
20.e even 4 1 3200.2.f.i 4
20.e even 4 1 3200.2.f.j 4
24.f even 2 1 5760.2.k.o 4
24.h odd 2 1 5760.2.k.o 4
40.e odd 2 1 3200.2.d.s 4
40.f even 2 1 3200.2.d.s 4
40.i odd 4 1 3200.2.f.i 4
40.i odd 4 1 3200.2.f.j 4
40.k even 4 1 3200.2.f.i 4
40.k even 4 1 3200.2.f.j 4
80.k odd 4 1 6400.2.a.bl 2
80.k odd 4 1 6400.2.a.bn 2
80.q even 4 1 6400.2.a.bl 2
80.q even 4 1 6400.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.d.d 4 1.a even 1 1 trivial
640.2.d.d 4 4.b odd 2 1 inner
640.2.d.d 4 8.b even 2 1 inner
640.2.d.d 4 8.d odd 2 1 inner
1280.2.a.f 2 16.e even 4 1
1280.2.a.f 2 16.f odd 4 1
1280.2.a.j 2 16.e even 4 1
1280.2.a.j 2 16.f odd 4 1
3200.2.d.s 4 5.b even 2 1
3200.2.d.s 4 20.d odd 2 1
3200.2.d.s 4 40.e odd 2 1
3200.2.d.s 4 40.f even 2 1
3200.2.f.i 4 5.c odd 4 1
3200.2.f.i 4 20.e even 4 1
3200.2.f.i 4 40.i odd 4 1
3200.2.f.i 4 40.k even 4 1
3200.2.f.j 4 5.c odd 4 1
3200.2.f.j 4 20.e even 4 1
3200.2.f.j 4 40.i odd 4 1
3200.2.f.j 4 40.k even 4 1
5760.2.k.o 4 3.b odd 2 1
5760.2.k.o 4 12.b even 2 1
5760.2.k.o 4 24.f even 2 1
5760.2.k.o 4 24.h odd 2 1
6400.2.a.bl 2 80.k odd 4 1
6400.2.a.bl 2 80.q even 4 1
6400.2.a.bn 2 80.k odd 4 1
6400.2.a.bn 2 80.q even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 2$$ $$T_{7}^{2} - 18$$ $$T_{11}^{2} + 32$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 2 + T^{2} )^{2}$$
$5$ $$( 1 + T^{2} )^{2}$$
$7$ $$( -18 + T^{2} )^{2}$$
$11$ $$( 32 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( -6 + T )^{4}$$
$19$ $$( 8 + T^{2} )^{2}$$
$23$ $$( -50 + T^{2} )^{2}$$
$29$ $$( 16 + T^{2} )^{2}$$
$31$ $$( -8 + T^{2} )^{2}$$
$37$ $$( 4 + T^{2} )^{2}$$
$41$ $$( 8 + T )^{4}$$
$43$ $$( 2 + T^{2} )^{2}$$
$47$ $$( -2 + T^{2} )^{2}$$
$53$ $$( 4 + T^{2} )^{2}$$
$59$ $$( 8 + T^{2} )^{2}$$
$61$ $$( 196 + T^{2} )^{2}$$
$67$ $$( 18 + T^{2} )^{2}$$
$71$ $$( -8 + T^{2} )^{2}$$
$73$ $$( 6 + T )^{4}$$
$79$ $$( -288 + T^{2} )^{2}$$
$83$ $$( 162 + T^{2} )^{2}$$
$89$ $$( 6 + T )^{4}$$
$97$ $$( -10 + T )^{4}$$