Properties

 Label 6400.2.a.cd 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 320) 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.

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

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 −1.26795 0 −2.46410 0
1.2 0 2.73205 0 0 0 −4.73205 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.cd 2
4.b odd 2 1 6400.2.a.bf 2
5.b even 2 1 1280.2.a.b 2
8.b even 2 1 6400.2.a.y 2
8.d odd 2 1 6400.2.a.ck 2
16.e even 4 2 1600.2.d.h 4
16.f odd 4 2 1600.2.d.b 4
20.d odd 2 1 1280.2.a.m 2
40.e odd 2 1 1280.2.a.c 2
40.f even 2 1 1280.2.a.p 2
80.i odd 4 2 1600.2.f.e 4
80.j even 4 2 1600.2.f.d 4
80.k odd 4 2 320.2.d.b yes 4
80.q even 4 2 320.2.d.a 4
80.s even 4 2 1600.2.f.h 4
80.t odd 4 2 1600.2.f.i 4
240.t even 4 2 2880.2.k.l 4
240.bm odd 4 2 2880.2.k.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.d.a 4 80.q even 4 2
320.2.d.b yes 4 80.k odd 4 2
1280.2.a.b 2 5.b even 2 1
1280.2.a.c 2 40.e odd 2 1
1280.2.a.m 2 20.d odd 2 1
1280.2.a.p 2 40.f even 2 1
1600.2.d.b 4 16.f odd 4 2
1600.2.d.h 4 16.e even 4 2
1600.2.f.d 4 80.j even 4 2
1600.2.f.e 4 80.i odd 4 2
1600.2.f.h 4 80.s even 4 2
1600.2.f.i 4 80.t odd 4 2
2880.2.k.e 4 240.bm odd 4 2
2880.2.k.l 4 240.t even 4 2
6400.2.a.y 2 8.b even 2 1
6400.2.a.bf 2 4.b odd 2 1
6400.2.a.cd 2 1.a even 1 1 trivial
6400.2.a.ck 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6400))$$:

 $$T_{3}^{2} - 2 T_{3} - 2$$ $$T_{7}^{2} + 6 T_{7} + 6$$ $$T_{11}^{2} - 12$$ $$T_{13}^{2} - 12$$ $$T_{17}^{2} - 12$$ $$T_{29}$$ $$T_{31}^{2} - 12 T_{31} + 24$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-2 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$6 + 6 T + T^{2}$$
$11$ $$-12 + T^{2}$$
$13$ $$-12 + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$-18 + 6 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$24 - 12 T + T^{2}$$
$37$ $$( 6 + T )^{2}$$
$41$ $$24 + 12 T + T^{2}$$
$43$ $$-2 + 10 T + T^{2}$$
$47$ $$-18 + 6 T + T^{2}$$
$53$ $$-108 + T^{2}$$
$59$ $$( -6 + T )^{2}$$
$61$ $$-12 + 12 T + T^{2}$$
$67$ $$-2 - 10 T + T^{2}$$
$71$ $$-72 - 12 T + T^{2}$$
$73$ $$-92 + 8 T + T^{2}$$
$79$ $$( -12 + T )^{2}$$
$83$ $$6 - 6 T + T^{2}$$
$89$ $$-12 - 12 T + T^{2}$$
$97$ $$-92 + 8 T + T^{2}$$