# Properties

 Label 1600.2.a.bc Level $1600$ Weight $2$ Character orbit 1600.a Self dual yes Analytic conductor $12.776$ Analytic rank $0$ Dimension $2$ CM no 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: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ 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: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + \beta q^{7} + 5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + \beta q^{7} + 5 q^{9} + 2 \beta q^{11} -2 q^{13} -2 q^{17} + 8 q^{21} -\beta q^{23} + 2 \beta q^{27} -6 q^{29} + 2 \beta q^{31} + 16 q^{33} -10 q^{37} -2 \beta q^{39} + 2 q^{41} -3 \beta q^{43} + \beta q^{47} + q^{49} -2 \beta q^{51} + 6 q^{53} -4 \beta q^{59} + 2 q^{61} + 5 \beta q^{63} -\beta q^{67} -8 q^{69} + 2 \beta q^{71} + 6 q^{73} + 16 q^{77} + 4 \beta q^{79} + q^{81} + \beta q^{83} -6 \beta q^{87} + 10 q^{89} -2 \beta q^{91} + 16 q^{93} -2 q^{97} + 10 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 10q^{9} + O(q^{10})$$ $$2q + 10q^{9} - 4q^{13} - 4q^{17} + 16q^{21} - 12q^{29} + 32q^{33} - 20q^{37} + 4q^{41} + 2q^{49} + 12q^{53} + 4q^{61} - 16q^{69} + 12q^{73} + 32q^{77} + 2q^{81} + 20q^{89} + 32q^{93} - 4q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −2.82843 0 0 0 −2.82843 0 5.00000 0
1.2 0 2.82843 0 0 0 2.82843 0 5.00000 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1600.2.a.bc 2
4.b odd 2 1 inner 1600.2.a.bc 2
5.b even 2 1 320.2.a.g 2
5.c odd 4 2 1600.2.c.n 4
8.b even 2 1 800.2.a.m 2
8.d odd 2 1 800.2.a.m 2
15.d odd 2 1 2880.2.a.bk 2
20.d odd 2 1 320.2.a.g 2
20.e even 4 2 1600.2.c.n 4
24.f even 2 1 7200.2.a.cm 2
24.h odd 2 1 7200.2.a.cm 2
40.e odd 2 1 160.2.a.c 2
40.f even 2 1 160.2.a.c 2
40.i odd 4 2 800.2.c.f 4
40.k even 4 2 800.2.c.f 4
60.h even 2 1 2880.2.a.bk 2
80.k odd 4 2 1280.2.d.l 4
80.q even 4 2 1280.2.d.l 4
120.i odd 2 1 1440.2.a.o 2
120.m even 2 1 1440.2.a.o 2
120.q odd 4 2 7200.2.f.bh 4
120.w even 4 2 7200.2.f.bh 4
280.c odd 2 1 7840.2.a.bf 2
280.n even 2 1 7840.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.a.c 2 40.e odd 2 1
160.2.a.c 2 40.f even 2 1
320.2.a.g 2 5.b even 2 1
320.2.a.g 2 20.d odd 2 1
800.2.a.m 2 8.b even 2 1
800.2.a.m 2 8.d odd 2 1
800.2.c.f 4 40.i odd 4 2
800.2.c.f 4 40.k even 4 2
1280.2.d.l 4 80.k odd 4 2
1280.2.d.l 4 80.q even 4 2
1440.2.a.o 2 120.i odd 2 1
1440.2.a.o 2 120.m even 2 1
1600.2.a.bc 2 1.a even 1 1 trivial
1600.2.a.bc 2 4.b odd 2 1 inner
1600.2.c.n 4 5.c odd 4 2
1600.2.c.n 4 20.e even 4 2
2880.2.a.bk 2 15.d odd 2 1
2880.2.a.bk 2 60.h even 2 1
7200.2.a.cm 2 24.f even 2 1
7200.2.a.cm 2 24.h odd 2 1
7200.2.f.bh 4 120.q odd 4 2
7200.2.f.bh 4 120.w even 4 2
7840.2.a.bf 2 280.c odd 2 1
7840.2.a.bf 2 280.n even 2 1

## Hecke kernels

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

 $$T_{3}^{2} - 8$$ $$T_{7}^{2} - 8$$ $$T_{11}^{2} - 32$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-8 + T^{2}$$
$11$ $$-32 + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$-32 + T^{2}$$
$37$ $$( 10 + T )^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$-72 + T^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$-128 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$-8 + T^{2}$$
$71$ $$-32 + T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$-128 + T^{2}$$
$83$ $$-8 + T^{2}$$
$89$ $$( -10 + T )^{2}$$
$97$ $$( 2 + T )^{2}$$