# Properties

 Label 3200.2.a.w Level $3200$ Weight $2$ Character orbit 3200.a Self dual yes Analytic conductor $25.552$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 640) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + q^{9} + O(q^{10})$$ $$q + 2 q^{3} + q^{9} - 2 q^{11} + 2 q^{13} - 6 q^{17} - 6 q^{19} - 4 q^{27} - 10 q^{29} - 8 q^{31} - 4 q^{33} + 2 q^{37} + 4 q^{39} - 6 q^{41} - 2 q^{43} + 12 q^{47} - 7 q^{49} - 12 q^{51} + 10 q^{53} - 12 q^{57} + 6 q^{59} + 6 q^{61} - 14 q^{67} - 4 q^{71} + 10 q^{73} + 8 q^{79} - 11 q^{81} + 10 q^{83} - 20 q^{87} + 14 q^{89} - 16 q^{93} - 6 q^{97} - 2 q^{99} + O(q^{100})$$

## 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 2.00000 0 0 0 0 0 1.00000 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.w 1
4.b odd 2 1 3200.2.a.g 1
5.b even 2 1 640.2.a.b yes 1
5.c odd 4 2 3200.2.c.h 2
8.b even 2 1 3200.2.a.f 1
8.d odd 2 1 3200.2.a.v 1
15.d odd 2 1 5760.2.a.m 1
20.d odd 2 1 640.2.a.h yes 1
20.e even 4 2 3200.2.c.j 2
40.e odd 2 1 640.2.a.a 1
40.f even 2 1 640.2.a.g yes 1
40.i odd 4 2 3200.2.c.i 2
40.k even 4 2 3200.2.c.g 2
60.h even 2 1 5760.2.a.k 1
80.k odd 4 2 1280.2.d.f 2
80.q even 4 2 1280.2.d.e 2
120.i odd 2 1 5760.2.a.bj 1
120.m even 2 1 5760.2.a.bl 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.2.a.a 1 40.e odd 2 1
640.2.a.b yes 1 5.b even 2 1
640.2.a.g yes 1 40.f even 2 1
640.2.a.h yes 1 20.d odd 2 1
1280.2.d.e 2 80.q even 4 2
1280.2.d.f 2 80.k odd 4 2
3200.2.a.f 1 8.b even 2 1
3200.2.a.g 1 4.b odd 2 1
3200.2.a.v 1 8.d odd 2 1
3200.2.a.w 1 1.a even 1 1 trivial
3200.2.c.g 2 40.k even 4 2
3200.2.c.h 2 5.c odd 4 2
3200.2.c.i 2 40.i odd 4 2
3200.2.c.j 2 20.e even 4 2
5760.2.a.k 1 60.h even 2 1
5760.2.a.m 1 15.d odd 2 1
5760.2.a.bj 1 120.i odd 2 1
5760.2.a.bl 1 120.m even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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