# Properties

 Label 8624.2.a.by Level $8624$ Weight $2$ Character orbit 8624.a Self dual yes Analytic conductor $68.863$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8624 = 2^{4} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8624.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$68.8629867032$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 1078) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + 5 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + 5 q^{9} + q^{11} + \beta q^{13} -\beta q^{17} + 2 \beta q^{19} -8 q^{23} -5 q^{25} + 2 \beta q^{27} + 2 q^{29} + 3 \beta q^{31} + \beta q^{33} + 2 q^{37} + 8 q^{39} -\beta q^{41} + 4 q^{43} + \beta q^{47} -8 q^{51} + 14 q^{53} + 16 q^{57} -3 \beta q^{59} + 3 \beta q^{61} -4 q^{67} -8 \beta q^{69} + 5 \beta q^{73} -5 \beta q^{75} + 16 q^{79} + q^{81} -6 \beta q^{83} + 2 \beta q^{87} + 4 \beta q^{89} + 24 q^{93} + 6 \beta q^{97} + 5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{9} + O(q^{10})$$ $$2 q + 10 q^{9} + 2 q^{11} - 16 q^{23} - 10 q^{25} + 4 q^{29} + 4 q^{37} + 16 q^{39} + 8 q^{43} - 16 q^{51} + 28 q^{53} + 32 q^{57} - 8 q^{67} + 32 q^{79} + 2 q^{81} + 48 q^{93} + 10 q^{99} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 −2.82843 0 0 0 0 0 5.00000 0
1.2 0 2.82843 0 0 0 0 0 5.00000 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
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8624.2.a.by 2
4.b odd 2 1 1078.2.a.r 2
7.b odd 2 1 inner 8624.2.a.by 2
12.b even 2 1 9702.2.a.dn 2
28.d even 2 1 1078.2.a.r 2
28.f even 6 2 1078.2.e.r 4
28.g odd 6 2 1078.2.e.r 4
84.h odd 2 1 9702.2.a.dn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1078.2.a.r 2 4.b odd 2 1
1078.2.a.r 2 28.d even 2 1
1078.2.e.r 4 28.f even 6 2
1078.2.e.r 4 28.g odd 6 2
8624.2.a.by 2 1.a even 1 1 trivial
8624.2.a.by 2 7.b odd 2 1 inner
9702.2.a.dn 2 12.b even 2 1
9702.2.a.dn 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3}^{2} - 8$$ $$T_{5}$$ $$T_{13}^{2} - 8$$ $$T_{17}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-8 + T^{2}$$
$17$ $$-8 + T^{2}$$
$19$ $$-32 + T^{2}$$
$23$ $$( 8 + T )^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$-72 + T^{2}$$
$37$ $$( -2 + T )^{2}$$
$41$ $$-8 + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$( -14 + T )^{2}$$
$59$ $$-72 + T^{2}$$
$61$ $$-72 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$-200 + T^{2}$$
$79$ $$( -16 + T )^{2}$$
$83$ $$-288 + T^{2}$$
$89$ $$-128 + T^{2}$$
$97$ $$-288 + T^{2}$$