# Properties

 Label 8624.2.a.bi Level $8624$ Weight $2$ Character orbit 8624.a Self dual yes Analytic conductor $68.863$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 616) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{3} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{9} +O(q^{10})$$ $$q -\beta q^{3} + ( 2 - \beta ) q^{5} + ( 1 + \beta ) q^{9} - q^{11} + ( 2 - 2 \beta ) q^{13} + ( 4 - \beta ) q^{15} + 2 q^{17} -2 \beta q^{19} + ( 4 - \beta ) q^{23} + ( 3 - 3 \beta ) q^{25} + ( -4 + \beta ) q^{27} -2 q^{29} + ( -8 - \beta ) q^{31} + \beta q^{33} + ( 2 + \beta ) q^{37} + 8 q^{39} + 10 q^{41} -4 q^{43} -2 q^{45} + ( 4 - 4 \beta ) q^{47} -2 \beta q^{51} + ( 6 - 4 \beta ) q^{53} + ( -2 + \beta ) q^{55} + ( 8 + 2 \beta ) q^{57} + ( -8 + \beta ) q^{59} + ( -6 + 4 \beta ) q^{61} + ( 12 - 4 \beta ) q^{65} + \beta q^{67} + ( 4 - 3 \beta ) q^{69} + ( 4 - 3 \beta ) q^{71} + ( -6 + 4 \beta ) q^{73} + 12 q^{75} -2 \beta q^{79} -7 q^{81} -8 q^{83} + ( 4 - 2 \beta ) q^{85} + 2 \beta q^{87} + ( 10 + \beta ) q^{89} + ( 4 + 9 \beta ) q^{93} + ( 8 - 2 \beta ) q^{95} + ( -6 - \beta ) q^{97} + ( -1 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + 3 q^{5} + 3 q^{9} + O(q^{10})$$ $$2 q - q^{3} + 3 q^{5} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 17 q^{31} + q^{33} + 5 q^{37} + 16 q^{39} + 20 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} - 2 q^{51} + 8 q^{53} - 3 q^{55} + 18 q^{57} - 15 q^{59} - 8 q^{61} + 20 q^{65} + q^{67} + 5 q^{69} + 5 q^{71} - 8 q^{73} + 24 q^{75} - 2 q^{79} - 14 q^{81} - 16 q^{83} + 6 q^{85} + 2 q^{87} + 21 q^{89} + 17 q^{93} + 14 q^{95} - 13 q^{97} - 3 q^{99} + O(q^{100})$$

## 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
 2.56155 −1.56155
0 −2.56155 0 −0.561553 0 0 0 3.56155 0
1.2 0 1.56155 0 3.56155 0 0 0 −0.561553 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8624.2.a.bi 2
4.b odd 2 1 4312.2.a.t 2
7.b odd 2 1 1232.2.a.o 2
28.d even 2 1 616.2.a.f 2
56.e even 2 1 4928.2.a.bs 2
56.h odd 2 1 4928.2.a.bo 2
84.h odd 2 1 5544.2.a.bf 2
308.g odd 2 1 6776.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.a.f 2 28.d even 2 1
1232.2.a.o 2 7.b odd 2 1
4312.2.a.t 2 4.b odd 2 1
4928.2.a.bo 2 56.h odd 2 1
4928.2.a.bs 2 56.e even 2 1
5544.2.a.bf 2 84.h odd 2 1
6776.2.a.l 2 308.g odd 2 1
8624.2.a.bi 2 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{2} + T_{3} - 4$$ $$T_{5}^{2} - 3 T_{5} - 2$$ $$T_{13}^{2} - 2 T_{13} - 16$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-4 + T + T^{2}$$
$5$ $$-2 - 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$-16 - 2 T + T^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$-16 + 2 T + T^{2}$$
$23$ $$8 - 7 T + T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$68 + 17 T + T^{2}$$
$37$ $$2 - 5 T + T^{2}$$
$41$ $$( -10 + T )^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-64 - 4 T + T^{2}$$
$53$ $$-52 - 8 T + T^{2}$$
$59$ $$52 + 15 T + T^{2}$$
$61$ $$-52 + 8 T + T^{2}$$
$67$ $$-4 - T + T^{2}$$
$71$ $$-32 - 5 T + T^{2}$$
$73$ $$-52 + 8 T + T^{2}$$
$79$ $$-16 + 2 T + T^{2}$$
$83$ $$( 8 + T )^{2}$$
$89$ $$106 - 21 T + T^{2}$$
$97$ $$38 + 13 T + T^{2}$$