# Properties

 Label 6400.2.a.cc Level $6400$ Weight $2$ Character orbit 6400.a Self dual yes Analytic conductor $51.104$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

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## 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: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ 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.

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

## 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
 −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.cc 2
4.b odd 2 1 6400.2.a.bh 2
5.b even 2 1 6400.2.a.bg 2
8.b even 2 1 inner 6400.2.a.cc 2
8.d odd 2 1 6400.2.a.bh 2
16.e even 4 2 200.2.d.b 2
16.f odd 4 2 800.2.d.d 2
20.d odd 2 1 6400.2.a.cb 2
40.e odd 2 1 6400.2.a.cb 2
40.f even 2 1 6400.2.a.bg 2
48.i odd 4 2 1800.2.k.f 2
48.k even 4 2 7200.2.k.i 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.a 2
80.q even 4 2 200.2.d.c yes 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.b 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.d 2

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

## Hecke kernels

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

 $$T_{3}^{2} - 7$$ $$T_{7} - 4$$ $$T_{11}^{2} - 7$$ $$T_{13}$$ $$T_{17} - 3$$ $$T_{29}$$ $$T_{31} - 4$$

## Hecke characteristic polynomials

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