# Properties

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

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ x^2 - 7 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 200) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} - 4 q^{7} + 4 q^{9}+O(q^{10})$$ q + b * q^3 - 4 * q^7 + 4 * q^9 $$q + \beta q^{3} - 4 q^{7} + 4 q^{9} + \beta q^{11} - 3 q^{17} - \beta q^{19} - 4 \beta q^{21} - 4 q^{23} + \beta q^{27} + 4 q^{31} + 7 q^{33} - 4 \beta q^{37} + 5 q^{41} - 2 \beta q^{43} - 8 q^{47} + 9 q^{49} - 3 \beta q^{51} + 4 \beta q^{53} - 7 q^{57} + 2 \beta q^{59} - 4 \beta q^{61} - 16 q^{63} - 3 \beta q^{67} - 4 \beta q^{69} - 8 q^{71} + 7 q^{73} - 4 \beta q^{77} + 4 q^{79} - 5 q^{81} - 3 \beta q^{83} + q^{89} + 4 \beta q^{93} - 2 q^{97} + 4 \beta q^{99} +O(q^{100})$$ q + b * q^3 - 4 * q^7 + 4 * q^9 + b * q^11 - 3 * q^17 - b * q^19 - 4*b * q^21 - 4 * q^23 + b * q^27 + 4 * q^31 + 7 * q^33 - 4*b * q^37 + 5 * q^41 - 2*b * q^43 - 8 * q^47 + 9 * q^49 - 3*b * q^51 + 4*b * q^53 - 7 * q^57 + 2*b * q^59 - 4*b * q^61 - 16 * q^63 - 3*b * q^67 - 4*b * q^69 - 8 * q^71 + 7 * q^73 - 4*b * q^77 + 4 * q^79 - 5 * q^81 - 3*b * q^83 + q^89 + 4*b * q^93 - 2 * q^97 + 4*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{7} + 8 q^{9}+O(q^{10})$$ 2 * q - 8 * q^7 + 8 * q^9 $$2 q - 8 q^{7} + 8 q^{9} - 6 q^{17} - 8 q^{23} + 8 q^{31} + 14 q^{33} + 10 q^{41} - 16 q^{47} + 18 q^{49} - 14 q^{57} - 32 q^{63} - 16 q^{71} + 14 q^{73} + 8 q^{79} - 10 q^{81} + 2 q^{89} - 4 q^{97}+O(q^{100})$$ 2 * q - 8 * q^7 + 8 * q^9 - 6 * q^17 - 8 * q^23 + 8 * q^31 + 14 * q^33 + 10 * q^41 - 16 * q^47 + 18 * q^49 - 14 * q^57 - 32 * q^63 - 16 * q^71 + 14 * q^73 + 8 * q^79 - 10 * q^81 + 2 * q^89 - 4 * q^97

## 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
 −2.64575 2.64575
0 −2.64575 0 0 0 −4.00000 0 4.00000 0
1.2 0 2.64575 0 0 0 −4.00000 0 4.00000 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
8.b 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.bg 2
4.b odd 2 1 6400.2.a.cb 2
5.b even 2 1 6400.2.a.cc 2
8.b even 2 1 inner 6400.2.a.bg 2
8.d odd 2 1 6400.2.a.cb 2
16.e even 4 2 200.2.d.c yes 2
16.f odd 4 2 800.2.d.a 2
20.d odd 2 1 6400.2.a.bh 2
40.e odd 2 1 6400.2.a.bh 2
40.f even 2 1 6400.2.a.cc 2
48.i odd 4 2 1800.2.k.d 2
48.k even 4 2 7200.2.k.b 2
80.i odd 4 2 200.2.f.d 4
80.j even 4 2 800.2.f.d 4
80.k odd 4 2 800.2.d.d 2
80.q even 4 2 200.2.d.b 2
80.s even 4 2 800.2.f.d 4
80.t odd 4 2 200.2.f.d 4
240.t even 4 2 7200.2.k.i 2
240.z odd 4 2 7200.2.d.m 4
240.bb even 4 2 1800.2.d.m 4
240.bd odd 4 2 7200.2.d.m 4
240.bf even 4 2 1800.2.d.m 4
240.bm odd 4 2 1800.2.k.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.d.b 2 80.q even 4 2
200.2.d.c yes 2 16.e even 4 2
200.2.f.d 4 80.i odd 4 2
200.2.f.d 4 80.t odd 4 2
800.2.d.a 2 16.f odd 4 2
800.2.d.d 2 80.k odd 4 2
800.2.f.d 4 80.j even 4 2
800.2.f.d 4 80.s even 4 2
1800.2.d.m 4 240.bb even 4 2
1800.2.d.m 4 240.bf even 4 2
1800.2.k.d 2 48.i odd 4 2
1800.2.k.f 2 240.bm odd 4 2
6400.2.a.bg 2 1.a even 1 1 trivial
6400.2.a.bg 2 8.b even 2 1 inner
6400.2.a.bh 2 20.d odd 2 1
6400.2.a.bh 2 40.e odd 2 1
6400.2.a.cb 2 4.b odd 2 1
6400.2.a.cb 2 8.d odd 2 1
6400.2.a.cc 2 5.b even 2 1
6400.2.a.cc 2 40.f even 2 1
7200.2.d.m 4 240.z odd 4 2
7200.2.d.m 4 240.bd odd 4 2
7200.2.k.b 2 48.k even 4 2
7200.2.k.i 2 240.t even 4 2

## Hecke kernels

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

 $$T_{3}^{2} - 7$$ T3^2 - 7 $$T_{7} + 4$$ T7 + 4 $$T_{11}^{2} - 7$$ T11^2 - 7 $$T_{13}$$ T13 $$T_{17} + 3$$ T17 + 3 $$T_{29}$$ T29 $$T_{31} - 4$$ T31 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 7$$
$5$ $$T^{2}$$
$7$ $$(T + 4)^{2}$$
$11$ $$T^{2} - 7$$
$13$ $$T^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} - 7$$
$23$ $$(T + 4)^{2}$$
$29$ $$T^{2}$$
$31$ $$(T - 4)^{2}$$
$37$ $$T^{2} - 112$$
$41$ $$(T - 5)^{2}$$
$43$ $$T^{2} - 28$$
$47$ $$(T + 8)^{2}$$
$53$ $$T^{2} - 112$$
$59$ $$T^{2} - 28$$
$61$ $$T^{2} - 112$$
$67$ $$T^{2} - 63$$
$71$ $$(T + 8)^{2}$$
$73$ $$(T - 7)^{2}$$
$79$ $$(T - 4)^{2}$$
$83$ $$T^{2} - 63$$
$89$ $$(T - 1)^{2}$$
$97$ $$(T + 2)^{2}$$