# Properties

 Label 3200.2.a.z Level $3200$ Weight $2$ Character orbit 3200.a Self dual yes Analytic conductor $25.552$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3200 = 2^{7} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3200.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.5521286468$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{3} - 4 q^{7} + 6 q^{9} + O(q^{10})$$ $$q + 3 q^{3} - 4 q^{7} + 6 q^{9} + 3 q^{11} + 2 q^{13} - 3 q^{17} + 7 q^{19} - 12 q^{21} - 6 q^{23} + 9 q^{27} + 6 q^{29} + 10 q^{31} + 9 q^{33} + 6 q^{39} - 11 q^{41} + 4 q^{43} + 2 q^{47} + 9 q^{49} - 9 q^{51} + 14 q^{53} + 21 q^{57} + 4 q^{59} + 8 q^{61} - 24 q^{63} + 5 q^{67} - 18 q^{69} - 2 q^{71} - 9 q^{73} - 12 q^{77} - 12 q^{79} + 9 q^{81} + q^{83} + 18 q^{87} + 3 q^{89} - 8 q^{91} + 30 q^{93} + 2 q^{97} + 18 q^{99} + O(q^{100})$$

## 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
0 3.00000 0 0 0 −4.00000 0 6.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3200.2.a.z yes 1
4.b odd 2 1 3200.2.a.c yes 1
5.b even 2 1 3200.2.a.d yes 1
5.c odd 4 2 3200.2.c.c 2
8.b even 2 1 3200.2.a.a 1
8.d odd 2 1 3200.2.a.bb yes 1
20.d odd 2 1 3200.2.a.y yes 1
20.e even 4 2 3200.2.c.a 2
40.e odd 2 1 3200.2.a.b yes 1
40.f even 2 1 3200.2.a.ba yes 1
40.i odd 4 2 3200.2.c.b 2
40.k even 4 2 3200.2.c.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.a.a 1 8.b even 2 1
3200.2.a.b yes 1 40.e odd 2 1
3200.2.a.c yes 1 4.b odd 2 1
3200.2.a.d yes 1 5.b even 2 1
3200.2.a.y yes 1 20.d odd 2 1
3200.2.a.z yes 1 1.a even 1 1 trivial
3200.2.a.ba yes 1 40.f even 2 1
3200.2.a.bb yes 1 8.d odd 2 1
3200.2.c.a 2 20.e even 4 2
3200.2.c.b 2 40.i odd 4 2
3200.2.c.c 2 5.c odd 4 2
3200.2.c.d 2 40.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3200))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$-3 + T$$
$13$ $$-2 + T$$
$17$ $$3 + T$$
$19$ $$-7 + T$$
$23$ $$6 + T$$
$29$ $$-6 + T$$
$31$ $$-10 + T$$
$37$ $$T$$
$41$ $$11 + T$$
$43$ $$-4 + T$$
$47$ $$-2 + T$$
$53$ $$-14 + T$$
$59$ $$-4 + T$$
$61$ $$-8 + T$$
$67$ $$-5 + T$$
$71$ $$2 + T$$
$73$ $$9 + T$$
$79$ $$12 + T$$
$83$ $$-1 + T$$
$89$ $$-3 + T$$
$97$ $$-2 + T$$