# Properties

 Label 6400.2.a.ct 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 320) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{7} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{3} + \beta_{1} q^{7} + 3 q^{9} -2 q^{11} -4 \beta_{1} q^{13} -2 \beta_{2} q^{17} -6 q^{19} + \beta_{3} q^{21} + 5 \beta_{1} q^{23} -2 \beta_{3} q^{29} + 2 \beta_{3} q^{31} -2 \beta_{2} q^{33} + 2 \beta_{1} q^{37} -4 \beta_{3} q^{39} + 4 q^{41} -\beta_{2} q^{43} -3 \beta_{1} q^{47} -5 q^{49} -12 q^{51} -6 \beta_{2} q^{57} -2 q^{59} -\beta_{3} q^{61} + 3 \beta_{1} q^{63} + \beta_{2} q^{67} + 5 \beta_{3} q^{69} + 2 \beta_{3} q^{71} -2 \beta_{2} q^{73} -2 \beta_{1} q^{77} + 2 \beta_{3} q^{79} -9 q^{81} -5 \beta_{2} q^{83} -12 \beta_{1} q^{87} -2 q^{89} -8 q^{91} + 12 \beta_{1} q^{93} -6 \beta_{2} q^{97} -6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9} + O(q^{10})$$ $$4 q + 12 q^{9} - 8 q^{11} - 24 q^{19} + 16 q^{41} - 20 q^{49} - 48 q^{51} - 8 q^{59} - 36 q^{81} - 8 q^{89} - 32 q^{91} - 24 q^{99} + O(q^{100})$$

## 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
 −1.93185 −0.517638 0.517638 1.93185
0 −2.44949 0 0 0 −1.41421 0 3.00000 0
1.2 0 −2.44949 0 0 0 1.41421 0 3.00000 0
1.3 0 2.44949 0 0 0 −1.41421 0 3.00000 0
1.4 0 2.44949 0 0 0 1.41421 0 3.00000 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.d odd 2 1 inner
40.e odd 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.ct 4
4.b odd 2 1 6400.2.a.cu 4
5.b even 2 1 inner 6400.2.a.ct 4
5.c odd 4 2 1280.2.c.g 4
8.b even 2 1 6400.2.a.cu 4
8.d odd 2 1 inner 6400.2.a.ct 4
16.e even 4 2 1600.2.d.i 8
16.f odd 4 2 1600.2.d.i 8
20.d odd 2 1 6400.2.a.cu 4
20.e even 4 2 1280.2.c.h 4
40.e odd 2 1 inner 6400.2.a.ct 4
40.f even 2 1 6400.2.a.cu 4
40.i odd 4 2 1280.2.c.h 4
40.k even 4 2 1280.2.c.g 4
80.i odd 4 2 320.2.f.b 8
80.j even 4 2 320.2.f.b 8
80.k odd 4 2 1600.2.d.i 8
80.q even 4 2 1600.2.d.i 8
80.s even 4 2 320.2.f.b 8
80.t odd 4 2 320.2.f.b 8
240.z odd 4 2 2880.2.d.g 8
240.bb even 4 2 2880.2.d.g 8
240.bd odd 4 2 2880.2.d.g 8
240.bf even 4 2 2880.2.d.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.2.f.b 8 80.i odd 4 2
320.2.f.b 8 80.j even 4 2
320.2.f.b 8 80.s even 4 2
320.2.f.b 8 80.t odd 4 2
1280.2.c.g 4 5.c odd 4 2
1280.2.c.g 4 40.k even 4 2
1280.2.c.h 4 20.e even 4 2
1280.2.c.h 4 40.i odd 4 2
1600.2.d.i 8 16.e even 4 2
1600.2.d.i 8 16.f odd 4 2
1600.2.d.i 8 80.k odd 4 2
1600.2.d.i 8 80.q even 4 2
2880.2.d.g 8 240.z odd 4 2
2880.2.d.g 8 240.bb even 4 2
2880.2.d.g 8 240.bd odd 4 2
2880.2.d.g 8 240.bf even 4 2
6400.2.a.ct 4 1.a even 1 1 trivial
6400.2.a.ct 4 5.b even 2 1 inner
6400.2.a.ct 4 8.d odd 2 1 inner
6400.2.a.ct 4 40.e odd 2 1 inner
6400.2.a.cu 4 4.b odd 2 1
6400.2.a.cu 4 8.b even 2 1
6400.2.a.cu 4 20.d odd 2 1
6400.2.a.cu 4 40.f even 2 1

## Hecke kernels

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

 $$T_{3}^{2} - 6$$ $$T_{7}^{2} - 2$$ $$T_{11} + 2$$ $$T_{13}^{2} - 32$$ $$T_{17}^{2} - 24$$ $$T_{29}^{2} - 48$$ $$T_{31}^{2} - 48$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -6 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( -2 + T^{2} )^{2}$$
$11$ $$( 2 + T )^{4}$$
$13$ $$( -32 + T^{2} )^{2}$$
$17$ $$( -24 + T^{2} )^{2}$$
$19$ $$( 6 + T )^{4}$$
$23$ $$( -50 + T^{2} )^{2}$$
$29$ $$( -48 + T^{2} )^{2}$$
$31$ $$( -48 + T^{2} )^{2}$$
$37$ $$( -8 + T^{2} )^{2}$$
$41$ $$( -4 + T )^{4}$$
$43$ $$( -6 + T^{2} )^{2}$$
$47$ $$( -18 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$( 2 + T )^{4}$$
$61$ $$( -12 + T^{2} )^{2}$$
$67$ $$( -6 + T^{2} )^{2}$$
$71$ $$( -48 + T^{2} )^{2}$$
$73$ $$( -24 + T^{2} )^{2}$$
$79$ $$( -48 + T^{2} )^{2}$$
$83$ $$( -150 + T^{2} )^{2}$$
$89$ $$( 2 + T )^{4}$$
$97$ $$( -216 + T^{2} )^{2}$$