# Properties

 Label 6400.2.a.cj Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 40) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{3} + ( 1 - \beta ) q^{7} + ( 1 + 2 \beta ) q^{9} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{3} + ( 1 - \beta ) q^{7} + ( 1 + 2 \beta ) q^{9} -2 q^{11} + 2 \beta q^{13} -2 \beta q^{17} + ( -4 + 2 \beta ) q^{19} -2 q^{21} + ( 1 + 3 \beta ) q^{23} + 4 q^{27} + 4 \beta q^{29} + ( 2 + 2 \beta ) q^{31} + ( -2 - 2 \beta ) q^{33} -2 q^{37} + ( 6 + 2 \beta ) q^{39} + ( 2 - 2 \beta ) q^{41} + ( 7 - \beta ) q^{43} + ( 5 - \beta ) q^{47} + ( -3 - 2 \beta ) q^{49} + ( -6 - 2 \beta ) q^{51} + ( 8 + 2 \beta ) q^{53} + ( 2 - 2 \beta ) q^{57} + ( -4 - 2 \beta ) q^{59} + ( -2 - 4 \beta ) q^{61} + ( -5 + \beta ) q^{63} + ( 9 + \beta ) q^{67} + ( 10 + 4 \beta ) q^{69} + ( 2 + 2 \beta ) q^{71} + ( 4 + 2 \beta ) q^{73} + ( -2 + 2 \beta ) q^{77} + ( 8 - 4 \beta ) q^{79} + ( 1 - 2 \beta ) q^{81} + ( 3 - \beta ) q^{83} + ( 12 + 4 \beta ) q^{87} + ( -2 - 4 \beta ) q^{89} + ( -6 + 2 \beta ) q^{91} + ( 8 + 4 \beta ) q^{93} + ( 4 + 6 \beta ) q^{97} + ( -2 - 4 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} + 2q^{7} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{3} + 2q^{7} + 2q^{9} - 4q^{11} - 8q^{19} - 4q^{21} + 2q^{23} + 8q^{27} + 4q^{31} - 4q^{33} - 4q^{37} + 12q^{39} + 4q^{41} + 14q^{43} + 10q^{47} - 6q^{49} - 12q^{51} + 16q^{53} + 4q^{57} - 8q^{59} - 4q^{61} - 10q^{63} + 18q^{67} + 20q^{69} + 4q^{71} + 8q^{73} - 4q^{77} + 16q^{79} + 2q^{81} + 6q^{83} + 24q^{87} - 4q^{89} - 12q^{91} + 16q^{93} + 8q^{97} - 4q^{99} + O(q^{100})$$

## 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
 −1.73205 1.73205
0 −0.732051 0 0 0 2.73205 0 −2.46410 0
1.2 0 2.73205 0 0 0 −0.732051 0 4.46410 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.cj 2
4.b odd 2 1 6400.2.a.z 2
5.b even 2 1 1280.2.a.d 2
8.b even 2 1 6400.2.a.be 2
8.d odd 2 1 6400.2.a.ce 2
16.e even 4 2 800.2.d.e 4
16.f odd 4 2 200.2.d.f 4
20.d odd 2 1 1280.2.a.o 2
40.e odd 2 1 1280.2.a.a 2
40.f even 2 1 1280.2.a.n 2
48.i odd 4 2 7200.2.k.j 4
48.k even 4 2 1800.2.k.j 4
80.i odd 4 2 800.2.f.c 4
80.j even 4 2 200.2.f.e 4
80.k odd 4 2 40.2.d.a 4
80.q even 4 2 160.2.d.a 4
80.s even 4 2 200.2.f.c 4
80.t odd 4 2 800.2.f.e 4
240.t even 4 2 360.2.k.e 4
240.z odd 4 2 1800.2.d.p 4
240.bb even 4 2 7200.2.d.o 4
240.bd odd 4 2 1800.2.d.l 4
240.bf even 4 2 7200.2.d.n 4
240.bm odd 4 2 1440.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.d.a 4 80.k odd 4 2
160.2.d.a 4 80.q even 4 2
200.2.d.f 4 16.f odd 4 2
200.2.f.c 4 80.s even 4 2
200.2.f.e 4 80.j even 4 2
360.2.k.e 4 240.t even 4 2
800.2.d.e 4 16.e even 4 2
800.2.f.c 4 80.i odd 4 2
800.2.f.e 4 80.t odd 4 2
1280.2.a.a 2 40.e odd 2 1
1280.2.a.d 2 5.b even 2 1
1280.2.a.n 2 40.f even 2 1
1280.2.a.o 2 20.d odd 2 1
1440.2.k.e 4 240.bm odd 4 2
1800.2.d.l 4 240.bd odd 4 2
1800.2.d.p 4 240.z odd 4 2
1800.2.k.j 4 48.k even 4 2
6400.2.a.z 2 4.b odd 2 1
6400.2.a.be 2 8.b even 2 1
6400.2.a.ce 2 8.d odd 2 1
6400.2.a.cj 2 1.a even 1 1 trivial
7200.2.d.n 4 240.bf even 4 2
7200.2.d.o 4 240.bb even 4 2
7200.2.k.j 4 48.i odd 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 2 T_{3} - 2$$ $$T_{7}^{2} - 2 T_{7} - 2$$ $$T_{11} + 2$$ $$T_{13}^{2} - 12$$ $$T_{17}^{2} - 12$$ $$T_{29}^{2} - 48$$ $$T_{31}^{2} - 4 T_{31} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$-2 - 2 T + T^{2}$$
$11$ $$( 2 + T )^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$4 + 8 T + T^{2}$$
$23$ $$-26 - 2 T + T^{2}$$
$29$ $$-48 + T^{2}$$
$31$ $$-8 - 4 T + T^{2}$$
$37$ $$( 2 + T )^{2}$$
$41$ $$-8 - 4 T + T^{2}$$
$43$ $$46 - 14 T + T^{2}$$
$47$ $$22 - 10 T + T^{2}$$
$53$ $$52 - 16 T + T^{2}$$
$59$ $$4 + 8 T + T^{2}$$
$61$ $$-44 + 4 T + T^{2}$$
$67$ $$78 - 18 T + T^{2}$$
$71$ $$-8 - 4 T + T^{2}$$
$73$ $$4 - 8 T + T^{2}$$
$79$ $$16 - 16 T + T^{2}$$
$83$ $$6 - 6 T + T^{2}$$
$89$ $$-44 + 4 T + T^{2}$$
$97$ $$-92 - 8 T + T^{2}$$