# Properties

 Label 6400.2.a.cm Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $1$ Dimension $4$ CM no Inner twists $4$

# 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: $$4$$ Coefficient field: $$\Q(\zeta_{24})^+$$ Defining polynomial: $$x^{4} - 4 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} - q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{7} - q^{9} -\beta_{3} q^{11} -2 \beta_{2} q^{17} + \beta_{3} q^{19} + \beta_{3} q^{21} + \beta_{2} q^{23} -4 \beta_{1} q^{27} -4 q^{31} -2 \beta_{2} q^{33} -6 \beta_{1} q^{37} + 3 \beta_{1} q^{43} -3 \beta_{2} q^{47} - q^{49} -2 \beta_{3} q^{51} -4 \beta_{1} q^{53} + 2 \beta_{2} q^{57} + 3 \beta_{3} q^{59} + \beta_{3} q^{61} -\beta_{2} q^{63} -3 \beta_{1} q^{67} + \beta_{3} q^{69} -12 q^{71} -2 \beta_{2} q^{73} -6 \beta_{1} q^{77} -4 q^{79} -5 q^{81} + 7 \beta_{1} q^{83} + 6 q^{89} -4 \beta_{1} q^{93} + 2 \beta_{2} q^{97} + \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 16q^{31} - 4q^{49} - 48q^{71} - 16q^{79} - 20q^{81} + 24q^{89} + O(q^{100})$$

## 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
 −1.93185 0.517638 −0.517638 1.93185
0 −1.41421 0 0 0 −2.44949 0 −1.00000 0
1.2 0 −1.41421 0 0 0 2.44949 0 −1.00000 0
1.3 0 1.41421 0 0 0 −2.44949 0 −1.00000 0
1.4 0 1.41421 0 0 0 2.44949 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6400.2.a.cm 4
4.b odd 2 1 6400.2.a.co 4
5.b even 2 1 inner 6400.2.a.cm 4
5.c odd 4 2 1280.2.c.k 4
8.b even 2 1 inner 6400.2.a.cm 4
8.d odd 2 1 6400.2.a.co 4
16.e even 4 2 800.2.d.f 4
16.f odd 4 2 200.2.d.e 4
20.d odd 2 1 6400.2.a.co 4
20.e even 4 2 1280.2.c.i 4
40.e odd 2 1 6400.2.a.co 4
40.f even 2 1 inner 6400.2.a.cm 4
40.i odd 4 2 1280.2.c.k 4
40.k even 4 2 1280.2.c.i 4
48.i odd 4 2 7200.2.k.l 4
48.k even 4 2 1800.2.k.m 4
80.i odd 4 2 160.2.f.a 4
80.j even 4 2 40.2.f.a 4
80.k odd 4 2 200.2.d.e 4
80.q even 4 2 800.2.d.f 4
80.s even 4 2 40.2.f.a 4
80.t odd 4 2 160.2.f.a 4
240.t even 4 2 1800.2.k.m 4
240.z odd 4 2 360.2.d.b 4
240.bb even 4 2 1440.2.d.c 4
240.bd odd 4 2 360.2.d.b 4
240.bf even 4 2 1440.2.d.c 4
240.bm odd 4 2 7200.2.k.l 4

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

## Hecke kernels

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

 $$T_{3}^{2} - 2$$ $$T_{7}^{2} - 6$$ $$T_{11}^{2} - 12$$ $$T_{13}$$ $$T_{17}^{2} - 24$$ $$T_{29}$$ $$T_{31} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -6 + T^{2} )^{2}$$
$11$ $$( -12 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( -12 + T^{2} )^{2}$$
$23$ $$( -6 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$( -72 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -18 + T^{2} )^{2}$$
$47$ $$( -54 + T^{2} )^{2}$$
$53$ $$( -32 + T^{2} )^{2}$$
$59$ $$( -108 + T^{2} )^{2}$$
$61$ $$( -12 + T^{2} )^{2}$$
$67$ $$( -18 + T^{2} )^{2}$$
$71$ $$( 12 + T )^{4}$$
$73$ $$( -24 + T^{2} )^{2}$$
$79$ $$( 4 + T )^{4}$$
$83$ $$( -98 + T^{2} )^{2}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( -24 + T^{2} )^{2}$$