# Properties

 Label 1176.4.a.j Level $1176$ Weight $4$ Character orbit 1176.a Self dual yes Analytic conductor $69.386$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1176.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$69.3862461668$$ 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

 $$f(q)$$ $$=$$ $$q + 3q^{3} - 4q^{5} + 9q^{9} + O(q^{10})$$ $$q + 3q^{3} - 4q^{5} + 9q^{9} - 26q^{11} - 2q^{13} - 12q^{15} + 36q^{17} + 76q^{19} - 114q^{23} - 109q^{25} + 27q^{27} + 6q^{29} + 256q^{31} - 78q^{33} - 86q^{37} - 6q^{39} - 160q^{41} - 220q^{43} - 36q^{45} - 308q^{47} + 108q^{51} + 258q^{53} + 104q^{55} + 228q^{57} - 264q^{59} - 606q^{61} + 8q^{65} - 520q^{67} - 342q^{69} - 286q^{71} + 530q^{73} - 327q^{75} - 44q^{79} + 81q^{81} - 1012q^{83} - 144q^{85} + 18q^{87} - 768q^{89} + 768q^{93} - 304q^{95} - 222q^{97} - 234q^{99} + O(q^{100})$$

## 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 3.00000 0 −4.00000 0 0 0 9.00000 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 1176.4.a.j 1
4.b odd 2 1 2352.4.a.h 1
7.b odd 2 1 168.4.a.c 1
21.c even 2 1 504.4.a.b 1
28.d even 2 1 336.4.a.j 1
56.e even 2 1 1344.4.a.e 1
56.h odd 2 1 1344.4.a.s 1
84.h odd 2 1 1008.4.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.c 1 7.b odd 2 1
336.4.a.j 1 28.d even 2 1
504.4.a.b 1 21.c even 2 1
1008.4.a.i 1 84.h odd 2 1
1176.4.a.j 1 1.a even 1 1 trivial
1344.4.a.e 1 56.e even 2 1
1344.4.a.s 1 56.h odd 2 1
2352.4.a.h 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1176))$$:

 $$T_{5} + 4$$ $$T_{11} + 26$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-3 + T$$
$5$ $$4 + T$$
$7$ $$T$$
$11$ $$26 + T$$
$13$ $$2 + T$$
$17$ $$-36 + T$$
$19$ $$-76 + T$$
$23$ $$114 + T$$
$29$ $$-6 + T$$
$31$ $$-256 + T$$
$37$ $$86 + T$$
$41$ $$160 + T$$
$43$ $$220 + T$$
$47$ $$308 + T$$
$53$ $$-258 + T$$
$59$ $$264 + T$$
$61$ $$606 + T$$
$67$ $$520 + T$$
$71$ $$286 + T$$
$73$ $$-530 + T$$
$79$ $$44 + T$$
$83$ $$1012 + T$$
$89$ $$768 + T$$
$97$ $$222 + T$$