# Properties

 Label 8624.2.a.ca 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: $$\Q(\sqrt{7})$$ 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 154) 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( 1 + \beta ) q^{5} + 4 q^{9} +O(q^{10})$$ $$q + \beta q^{3} + ( 1 + \beta ) q^{5} + 4 q^{9} - q^{11} + 5 q^{13} + ( 7 + \beta ) q^{15} + 6 q^{17} + ( 3 - \beta ) q^{19} + ( 1 + \beta ) q^{23} + ( 3 + 2 \beta ) q^{25} + \beta q^{27} + ( 1 - 2 \beta ) q^{29} + 4 q^{31} -\beta q^{33} + ( 1 - \beta ) q^{37} + 5 \beta q^{39} + ( 3 - 3 \beta ) q^{41} + 4 q^{43} + ( 4 + 4 \beta ) q^{45} + ( -8 - 2 \beta ) q^{47} + 6 \beta q^{51} + ( -1 - \beta ) q^{53} + ( -1 - \beta ) q^{55} + ( -7 + 3 \beta ) q^{57} + ( -2 + \beta ) q^{59} + ( 9 - 2 \beta ) q^{61} + ( 5 + 5 \beta ) q^{65} + ( 4 - 3 \beta ) q^{67} + ( 7 + \beta ) q^{69} + ( -7 - \beta ) q^{71} + ( 3 + \beta ) q^{73} + ( 14 + 3 \beta ) q^{75} -\beta q^{79} -5 q^{81} + ( -8 - 2 \beta ) q^{83} + ( 6 + 6 \beta ) q^{85} + ( -14 + \beta ) q^{87} + ( -4 - 4 \beta ) q^{89} + 4 \beta q^{93} + ( -4 + 2 \beta ) q^{95} + ( -11 + 2 \beta ) q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 8 q^{9} + O(q^{10})$$ $$2 q + 2 q^{5} + 8 q^{9} - 2 q^{11} + 10 q^{13} + 14 q^{15} + 12 q^{17} + 6 q^{19} + 2 q^{23} + 6 q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{37} + 6 q^{41} + 8 q^{43} + 8 q^{45} - 16 q^{47} - 2 q^{53} - 2 q^{55} - 14 q^{57} - 4 q^{59} + 18 q^{61} + 10 q^{65} + 8 q^{67} + 14 q^{69} - 14 q^{71} + 6 q^{73} + 28 q^{75} - 10 q^{81} - 16 q^{83} + 12 q^{85} - 28 q^{87} - 8 q^{89} - 8 q^{95} - 22 q^{97} - 8 q^{99} + O(q^{100})$$

## 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
 −2.64575 2.64575
0 −2.64575 0 −1.64575 0 0 0 4.00000 0
1.2 0 2.64575 0 3.64575 0 0 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
$$2$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$1$$

## 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.ca 2
4.b odd 2 1 1078.2.a.s 2
7.b odd 2 1 8624.2.a.bk 2
7.c even 3 2 1232.2.q.g 4
12.b even 2 1 9702.2.a.cz 2
28.d even 2 1 1078.2.a.n 2
28.f even 6 2 1078.2.e.v 4
28.g odd 6 2 154.2.e.f 4
84.h odd 2 1 9702.2.a.dr 2
84.n even 6 2 1386.2.k.s 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.e.f 4 28.g odd 6 2
1078.2.a.n 2 28.d even 2 1
1078.2.a.s 2 4.b odd 2 1
1078.2.e.v 4 28.f even 6 2
1232.2.q.g 4 7.c even 3 2
1386.2.k.s 4 84.n even 6 2
8624.2.a.bk 2 7.b odd 2 1
8624.2.a.ca 2 1.a even 1 1 trivial
9702.2.a.cz 2 12.b even 2 1
9702.2.a.dr 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8624))$$:

 $$T_{3}^{2} - 7$$ $$T_{5}^{2} - 2 T_{5} - 6$$ $$T_{13} - 5$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$-7 + T^{2}$$
$5$ $$-6 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( -5 + T )^{2}$$
$17$ $$( -6 + T )^{2}$$
$19$ $$2 - 6 T + T^{2}$$
$23$ $$-6 - 2 T + T^{2}$$
$29$ $$-27 - 2 T + T^{2}$$
$31$ $$( -4 + T )^{2}$$
$37$ $$-6 - 2 T + T^{2}$$
$41$ $$-54 - 6 T + T^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$36 + 16 T + T^{2}$$
$53$ $$-6 + 2 T + T^{2}$$
$59$ $$-3 + 4 T + T^{2}$$
$61$ $$53 - 18 T + T^{2}$$
$67$ $$-47 - 8 T + T^{2}$$
$71$ $$42 + 14 T + T^{2}$$
$73$ $$2 - 6 T + T^{2}$$
$79$ $$-7 + T^{2}$$
$83$ $$36 + 16 T + T^{2}$$
$89$ $$-96 + 8 T + T^{2}$$
$97$ $$93 + 22 T + T^{2}$$