# Properties

 Label 8624.2.a.ce 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 77) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{3} + 2 q^{5} + (2 \beta + 3) q^{9}+O(q^{10})$$ q + (b + 1) * q^3 + 2 * q^5 + (2*b + 3) * q^9 $$q + (\beta + 1) q^{3} + 2 q^{5} + (2 \beta + 3) q^{9} + q^{11} + (\beta - 1) q^{13} + (2 \beta + 2) q^{15} + ( - \beta + 1) q^{17} + ( - 2 \beta + 2) q^{19} + (2 \beta + 2) q^{23} - q^{25} + (2 \beta + 10) q^{27} + ( - 2 \beta + 4) q^{29} + ( - \beta - 5) q^{31} + (\beta + 1) q^{33} + (2 \beta - 4) q^{37} + 4 q^{39} + ( - \beta + 9) q^{41} - 8 q^{43} + (4 \beta + 6) q^{45} + (\beta + 5) q^{47} - 4 q^{51} + (2 \beta + 4) q^{53} + 2 q^{55} - 8 q^{57} + (\beta + 1) q^{59} + ( - \beta + 5) q^{61} + (2 \beta - 2) q^{65} + (2 \beta - 10) q^{67} + (4 \beta + 12) q^{69} + ( - 2 \beta + 6) q^{71} + (\beta + 3) q^{73} + ( - \beta - 1) q^{75} - 4 \beta q^{79} + (6 \beta + 11) q^{81} + (6 \beta + 2) q^{83} + ( - 2 \beta + 2) q^{85} + (2 \beta - 6) q^{87} - 2 q^{89} + ( - 6 \beta - 10) q^{93} + ( - 4 \beta + 4) q^{95} + (6 \beta - 4) q^{97} + (2 \beta + 3) q^{99} +O(q^{100})$$ q + (b + 1) * q^3 + 2 * q^5 + (2*b + 3) * q^9 + q^11 + (b - 1) * q^13 + (2*b + 2) * q^15 + (-b + 1) * q^17 + (-2*b + 2) * q^19 + (2*b + 2) * q^23 - q^25 + (2*b + 10) * q^27 + (-2*b + 4) * q^29 + (-b - 5) * q^31 + (b + 1) * q^33 + (2*b - 4) * q^37 + 4 * q^39 + (-b + 9) * q^41 - 8 * q^43 + (4*b + 6) * q^45 + (b + 5) * q^47 - 4 * q^51 + (2*b + 4) * q^53 + 2 * q^55 - 8 * q^57 + (b + 1) * q^59 + (-b + 5) * q^61 + (2*b - 2) * q^65 + (2*b - 10) * q^67 + (4*b + 12) * q^69 + (-2*b + 6) * q^71 + (b + 3) * q^73 + (-b - 1) * q^75 - 4*b * q^79 + (6*b + 11) * q^81 + (6*b + 2) * q^83 + (-2*b + 2) * q^85 + (2*b - 6) * q^87 - 2 * q^89 + (-6*b - 10) * q^93 + (-4*b + 4) * q^95 + (6*b - 4) * q^97 + (2*b + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 + 4 * q^5 + 6 * q^9 $$2 q + 2 q^{3} + 4 q^{5} + 6 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{23} - 2 q^{25} + 20 q^{27} + 8 q^{29} - 10 q^{31} + 2 q^{33} - 8 q^{37} + 8 q^{39} + 18 q^{41} - 16 q^{43} + 12 q^{45} + 10 q^{47} - 8 q^{51} + 8 q^{53} + 4 q^{55} - 16 q^{57} + 2 q^{59} + 10 q^{61} - 4 q^{65} - 20 q^{67} + 24 q^{69} + 12 q^{71} + 6 q^{73} - 2 q^{75} + 22 q^{81} + 4 q^{83} + 4 q^{85} - 12 q^{87} - 4 q^{89} - 20 q^{93} + 8 q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 + 4 * q^5 + 6 * q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 + 2 * q^17 + 4 * q^19 + 4 * q^23 - 2 * q^25 + 20 * q^27 + 8 * q^29 - 10 * q^31 + 2 * q^33 - 8 * q^37 + 8 * q^39 + 18 * q^41 - 16 * q^43 + 12 * q^45 + 10 * q^47 - 8 * q^51 + 8 * q^53 + 4 * q^55 - 16 * q^57 + 2 * q^59 + 10 * q^61 - 4 * q^65 - 20 * q^67 + 24 * q^69 + 12 * q^71 + 6 * q^73 - 2 * q^75 + 22 * q^81 + 4 * q^83 + 4 * q^85 - 12 * q^87 - 4 * q^89 - 20 * q^93 + 8 * q^95 - 8 * q^97 + 6 * q^99

## 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
 −0.618034 1.61803
0 −1.23607 0 2.00000 0 0 0 −1.47214 0
1.2 0 3.23607 0 2.00000 0 0 0 7.47214 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.ce 2
4.b odd 2 1 539.2.a.f 2
7.b odd 2 1 1232.2.a.m 2
12.b even 2 1 4851.2.a.y 2
28.d even 2 1 77.2.a.d 2
28.f even 6 2 539.2.e.i 4
28.g odd 6 2 539.2.e.j 4
44.c even 2 1 5929.2.a.m 2
56.e even 2 1 4928.2.a.bm 2
56.h odd 2 1 4928.2.a.bv 2
84.h odd 2 1 693.2.a.h 2
140.c even 2 1 1925.2.a.r 2
140.j odd 4 2 1925.2.b.h 4
308.g odd 2 1 847.2.a.f 2
308.s odd 10 2 847.2.f.b 4
308.s odd 10 2 847.2.f.m 4
308.t even 10 2 847.2.f.a 4
308.t even 10 2 847.2.f.n 4
924.n even 2 1 7623.2.a.bl 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.d 2 28.d even 2 1
539.2.a.f 2 4.b odd 2 1
539.2.e.i 4 28.f even 6 2
539.2.e.j 4 28.g odd 6 2
693.2.a.h 2 84.h odd 2 1
847.2.a.f 2 308.g odd 2 1
847.2.f.a 4 308.t even 10 2
847.2.f.b 4 308.s odd 10 2
847.2.f.m 4 308.s odd 10 2
847.2.f.n 4 308.t even 10 2
1232.2.a.m 2 7.b odd 2 1
1925.2.a.r 2 140.c even 2 1
1925.2.b.h 4 140.j odd 4 2
4851.2.a.y 2 12.b even 2 1
4928.2.a.bm 2 56.e even 2 1
4928.2.a.bv 2 56.h odd 2 1
5929.2.a.m 2 44.c even 2 1
7623.2.a.bl 2 924.n even 2 1
8624.2.a.ce 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3}^{2} - 2T_{3} - 4$$ T3^2 - 2*T3 - 4 $$T_{5} - 2$$ T5 - 2 $$T_{13}^{2} + 2T_{13} - 4$$ T13^2 + 2*T13 - 4 $$T_{17}^{2} - 2T_{17} - 4$$ T17^2 - 2*T17 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2T - 4$$
$5$ $$(T - 2)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 2T - 4$$
$17$ $$T^{2} - 2T - 4$$
$19$ $$T^{2} - 4T - 16$$
$23$ $$T^{2} - 4T - 16$$
$29$ $$T^{2} - 8T - 4$$
$31$ $$T^{2} + 10T + 20$$
$37$ $$T^{2} + 8T - 4$$
$41$ $$T^{2} - 18T + 76$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} - 10T + 20$$
$53$ $$T^{2} - 8T - 4$$
$59$ $$T^{2} - 2T - 4$$
$61$ $$T^{2} - 10T + 20$$
$67$ $$T^{2} + 20T + 80$$
$71$ $$T^{2} - 12T + 16$$
$73$ $$T^{2} - 6T + 4$$
$79$ $$T^{2} - 80$$
$83$ $$T^{2} - 4T - 176$$
$89$ $$(T + 2)^{2}$$
$97$ $$T^{2} + 8T - 164$$