Properties

 Label 1176.2.a.e Level $1176$ Weight $2$ Character orbit 1176.a Self dual yes Analytic conductor $9.390$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1176 = 2^{3} \cdot 3 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$9.39040727770$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 1.00000 0 −2.00000 0 0 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$

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 1176.2.a.e 1
3.b odd 2 1 3528.2.a.y 1
4.b odd 2 1 2352.2.a.e 1
7.b odd 2 1 1176.2.a.d 1
7.c even 3 2 1176.2.q.e 2
7.d odd 6 2 168.2.q.b 2
8.b even 2 1 9408.2.a.bk 1
8.d odd 2 1 9408.2.a.cs 1
12.b even 2 1 7056.2.a.bn 1
21.c even 2 1 3528.2.a.f 1
21.g even 6 2 504.2.s.g 2
21.h odd 6 2 3528.2.s.d 2
28.d even 2 1 2352.2.a.x 1
28.f even 6 2 336.2.q.a 2
28.g odd 6 2 2352.2.q.v 2
56.e even 2 1 9408.2.a.f 1
56.h odd 2 1 9408.2.a.cd 1
56.j odd 6 2 1344.2.q.i 2
56.m even 6 2 1344.2.q.t 2
84.h odd 2 1 7056.2.a.i 1
84.j odd 6 2 1008.2.s.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.q.b 2 7.d odd 6 2
336.2.q.a 2 28.f even 6 2
504.2.s.g 2 21.g even 6 2
1008.2.s.m 2 84.j odd 6 2
1176.2.a.d 1 7.b odd 2 1
1176.2.a.e 1 1.a even 1 1 trivial
1176.2.q.e 2 7.c even 3 2
1344.2.q.i 2 56.j odd 6 2
1344.2.q.t 2 56.m even 6 2
2352.2.a.e 1 4.b odd 2 1
2352.2.a.x 1 28.d even 2 1
2352.2.q.v 2 28.g odd 6 2
3528.2.a.f 1 21.c even 2 1
3528.2.a.y 1 3.b odd 2 1
3528.2.s.d 2 21.h odd 6 2
7056.2.a.i 1 84.h odd 2 1
7056.2.a.bn 1 12.b even 2 1
9408.2.a.f 1 56.e even 2 1
9408.2.a.bk 1 8.b even 2 1
9408.2.a.cd 1 56.h odd 2 1
9408.2.a.cs 1 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1176))$$:

 $$T_{5} + 2$$ $$T_{11} + 6$$ $$T_{13} - 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-1 + T$$
$5$ $$2 + T$$
$7$ $$T$$
$11$ $$6 + T$$
$13$ $$-3 + T$$
$17$ $$4 + T$$
$19$ $$-5 + T$$
$23$ $$4 + T$$
$29$ $$4 + T$$
$31$ $$7 + T$$
$37$ $$9 + T$$
$41$ $$-2 + T$$
$43$ $$1 + T$$
$47$ $$2 + T$$
$53$ $$-8 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$15 + T$$
$71$ $$6 + T$$
$73$ $$-11 + T$$
$79$ $$-1 + T$$
$83$ $$6 + T$$
$89$ $$-8 + T$$
$97$ $$-14 + T$$