# Properties

 Label 1600.2.a.bd Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM discriminant -20 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1600 = 2^{6} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1600.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$12.7760643234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$-1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( -3 + \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( -3 + \beta ) q^{7} + ( 3 + 2 \beta ) q^{9} + ( 2 - 2 \beta ) q^{21} + ( -1 + 3 \beta ) q^{23} + ( 10 + 2 \beta ) q^{27} + 6 q^{29} -2 \beta q^{41} + ( 9 + \beta ) q^{43} + ( -7 - 3 \beta ) q^{47} + ( 7 - 6 \beta ) q^{49} + 6 \beta q^{61} + ( 1 - 3 \beta ) q^{63} + ( 3 - 5 \beta ) q^{67} + ( 14 + 2 \beta ) q^{69} + ( 11 + 6 \beta ) q^{81} + ( -11 - 3 \beta ) q^{83} + ( 6 + 6 \beta ) q^{87} + 6 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - 6q^{7} + 6q^{9} + O(q^{10})$$ $$2q + 2q^{3} - 6q^{7} + 6q^{9} + 4q^{21} - 2q^{23} + 20q^{27} + 12q^{29} + 18q^{43} - 14q^{47} + 14q^{49} + 2q^{63} + 6q^{67} + 28q^{69} + 22q^{81} - 22q^{83} + 12q^{87} + 12q^{89} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −0.618034 1.61803
0 −1.23607 0 0 0 −5.23607 0 −1.47214 0
1.2 0 3.23607 0 0 0 −0.763932 0 7.47214 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.bd 2
4.b odd 2 1 1600.2.a.z 2
5.b even 2 1 1600.2.a.z 2
5.c odd 4 2 320.2.c.d 4
8.b even 2 1 800.2.a.j 2
8.d odd 2 1 800.2.a.n 2
15.e even 4 2 2880.2.f.w 4
20.d odd 2 1 CM 1600.2.a.bd 2
20.e even 4 2 320.2.c.d 4
24.f even 2 1 7200.2.a.cr 2
24.h odd 2 1 7200.2.a.cb 2
40.e odd 2 1 800.2.a.j 2
40.f even 2 1 800.2.a.n 2
40.i odd 4 2 160.2.c.b 4
40.k even 4 2 160.2.c.b 4
60.l odd 4 2 2880.2.f.w 4
80.i odd 4 2 1280.2.f.g 4
80.j even 4 2 1280.2.f.g 4
80.s even 4 2 1280.2.f.h 4
80.t odd 4 2 1280.2.f.h 4
120.i odd 2 1 7200.2.a.cr 2
120.m even 2 1 7200.2.a.cb 2
120.q odd 4 2 1440.2.f.i 4
120.w even 4 2 1440.2.f.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.b 4 40.i odd 4 2
160.2.c.b 4 40.k even 4 2
320.2.c.d 4 5.c odd 4 2
320.2.c.d 4 20.e even 4 2
800.2.a.j 2 8.b even 2 1
800.2.a.j 2 40.e odd 2 1
800.2.a.n 2 8.d odd 2 1
800.2.a.n 2 40.f even 2 1
1280.2.f.g 4 80.i odd 4 2
1280.2.f.g 4 80.j even 4 2
1280.2.f.h 4 80.s even 4 2
1280.2.f.h 4 80.t odd 4 2
1440.2.f.i 4 120.q odd 4 2
1440.2.f.i 4 120.w even 4 2
1600.2.a.z 2 4.b odd 2 1
1600.2.a.z 2 5.b even 2 1
1600.2.a.bd 2 1.a even 1 1 trivial
1600.2.a.bd 2 20.d odd 2 1 CM
2880.2.f.w 4 15.e even 4 2
2880.2.f.w 4 60.l odd 4 2
7200.2.a.cb 2 24.h odd 2 1
7200.2.a.cb 2 120.m even 2 1
7200.2.a.cr 2 24.f even 2 1
7200.2.a.cr 2 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1600))$$:

 $$T_{3}^{2} - 2 T_{3} - 4$$ $$T_{7}^{2} + 6 T_{7} + 4$$ $$T_{11}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + 6 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$-44 + 2 T + T^{2}$$
$29$ $$( -6 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2}$$
$41$ $$-20 + T^{2}$$
$43$ $$76 - 18 T + T^{2}$$
$47$ $$4 + 14 T + T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$-180 + T^{2}$$
$67$ $$-116 - 6 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$T^{2}$$
$83$ $$76 + 22 T + T^{2}$$
$89$ $$( -6 + T )^{2}$$
$97$ $$T^{2}$$