# Properties

 Label 6400.2.a.q Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6400 = 2^{8} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6400.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$51.1042572936$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3200) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} - 4q^{7} - 2q^{9} + O(q^{10})$$ $$q + q^{3} - 4q^{7} - 2q^{9} + 3q^{11} - q^{17} + 7q^{19} - 4q^{21} - 4q^{23} - 5q^{27} + 8q^{29} - 4q^{31} + 3q^{33} + 4q^{37} - 3q^{41} + 8q^{43} + 9q^{49} - q^{51} - 12q^{53} + 7q^{57} + 8q^{59} - 4q^{61} + 8q^{63} - 9q^{67} - 4q^{69} - 16q^{71} - 11q^{73} - 12q^{77} - 4q^{79} + q^{81} - q^{83} + 8q^{87} - 13q^{89} - 4q^{93} - 14q^{97} - 6q^{99} + O(q^{100})$$

## 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 1.00000 0 0 0 −4.00000 0 −2.00000 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 6400.2.a.q 1
4.b odd 2 1 6400.2.a.h 1
5.b even 2 1 6400.2.a.i 1
8.b even 2 1 6400.2.a.b 1
8.d odd 2 1 6400.2.a.w 1
16.e even 4 2 3200.2.d.g yes 2
16.f odd 4 2 3200.2.d.a 2
20.d odd 2 1 6400.2.a.p 1
40.e odd 2 1 6400.2.a.c 1
40.f even 2 1 6400.2.a.v 1
80.i odd 4 2 3200.2.f.a 2
80.j even 4 2 3200.2.f.b 2
80.k odd 4 2 3200.2.d.h yes 2
80.q even 4 2 3200.2.d.b yes 2
80.s even 4 2 3200.2.f.f 2
80.t odd 4 2 3200.2.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3200.2.d.a 2 16.f odd 4 2
3200.2.d.b yes 2 80.q even 4 2
3200.2.d.g yes 2 16.e even 4 2
3200.2.d.h yes 2 80.k odd 4 2
3200.2.f.a 2 80.i odd 4 2
3200.2.f.b 2 80.j even 4 2
3200.2.f.e 2 80.t odd 4 2
3200.2.f.f 2 80.s even 4 2
6400.2.a.b 1 8.b even 2 1
6400.2.a.c 1 40.e odd 2 1
6400.2.a.h 1 4.b odd 2 1
6400.2.a.i 1 5.b even 2 1
6400.2.a.p 1 20.d odd 2 1
6400.2.a.q 1 1.a even 1 1 trivial
6400.2.a.v 1 40.f even 2 1
6400.2.a.w 1 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3} - 1$$ $$T_{7} + 4$$ $$T_{11} - 3$$ $$T_{13}$$ $$T_{17} + 1$$ $$T_{29} - 8$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$T$$
$7$ $$4 + T$$
$11$ $$-3 + T$$
$13$ $$T$$
$17$ $$1 + T$$
$19$ $$-7 + T$$
$23$ $$4 + T$$
$29$ $$-8 + T$$
$31$ $$4 + T$$
$37$ $$-4 + T$$
$41$ $$3 + T$$
$43$ $$-8 + T$$
$47$ $$T$$
$53$ $$12 + T$$
$59$ $$-8 + T$$
$61$ $$4 + T$$
$67$ $$9 + T$$
$71$ $$16 + T$$
$73$ $$11 + T$$
$79$ $$4 + T$$
$83$ $$1 + T$$
$89$ $$13 + T$$
$97$ $$14 + T$$