Properties

 Label 8624.2.a.o Level $8624$ Weight $2$ Character orbit 8624.a Self dual yes Analytic conductor $68.863$ Analytic rank $0$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{5} - 3 q^{9} + O(q^{10})$$ $$q - 2 q^{5} - 3 q^{9} + q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{23} - q^{25} - 2 q^{29} - 8 q^{31} - 2 q^{37} - 10 q^{41} - 4 q^{43} + 6 q^{45} + 8 q^{47} + 6 q^{53} - 2 q^{55} - 10 q^{61} + 4 q^{65} + 12 q^{67} - 16 q^{71} + 14 q^{73} + 9 q^{81} + 4 q^{85} + 6 q^{89} - 10 q^{97} - 3 q^{99} + O(q^{100})$$

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
 0
0 0 0 −2.00000 0 0 0 −3.00000 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.o 1
4.b odd 2 1 1078.2.a.j 1
7.b odd 2 1 1232.2.a.h 1
12.b even 2 1 9702.2.a.v 1
28.d even 2 1 154.2.a.c 1
28.f even 6 2 1078.2.e.b 2
28.g odd 6 2 1078.2.e.c 2
56.e even 2 1 4928.2.a.n 1
56.h odd 2 1 4928.2.a.o 1
84.h odd 2 1 1386.2.a.b 1
140.c even 2 1 3850.2.a.f 1
140.j odd 4 2 3850.2.c.l 2
308.g odd 2 1 1694.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.c 1 28.d even 2 1
1078.2.a.j 1 4.b odd 2 1
1078.2.e.b 2 28.f even 6 2
1078.2.e.c 2 28.g odd 6 2
1232.2.a.h 1 7.b odd 2 1
1386.2.a.b 1 84.h odd 2 1
1694.2.a.c 1 308.g odd 2 1
3850.2.a.f 1 140.c even 2 1
3850.2.c.l 2 140.j odd 4 2
4928.2.a.n 1 56.e even 2 1
4928.2.a.o 1 56.h odd 2 1
8624.2.a.o 1 1.a even 1 1 trivial
9702.2.a.v 1 12.b even 2 1

Hecke kernels

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

 $$T_{3}$$ $$T_{5} + 2$$ $$T_{13} + 2$$ $$T_{17} + 2$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$2 + T$$
$7$ $$T$$
$11$ $$-1 + T$$
$13$ $$2 + T$$
$17$ $$2 + T$$
$19$ $$T$$
$23$ $$-8 + T$$
$29$ $$2 + T$$
$31$ $$8 + T$$
$37$ $$2 + T$$
$41$ $$10 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$-12 + T$$
$71$ $$16 + T$$
$73$ $$-14 + T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$-6 + T$$
$97$ $$10 + T$$