# Properties

 Label 1176.2.a.j Level $1176$ Weight $2$ Character orbit 1176.a Self dual yes Analytic conductor $9.390$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{3} + (\beta - 2) q^{5} + q^{9}+O(q^{10})$$ q - q^3 + (b - 2) * q^5 + q^9 $$q - q^{3} + (\beta - 2) q^{5} + q^{9} + ( - 2 \beta + 2) q^{11} + \beta q^{13} + ( - \beta + 2) q^{15} + ( - 3 \beta + 2) q^{17} + ( - 2 \beta - 4) q^{19} + (2 \beta + 2) q^{23} + ( - 4 \beta + 1) q^{25} - q^{27} + 6 \beta q^{29} + (2 \beta - 8) q^{31} + (2 \beta - 2) q^{33} + (4 \beta - 4) q^{37} - \beta q^{39} + ( - \beta + 2) q^{41} - 8 q^{43} + (\beta - 2) q^{45} + ( - 2 \beta - 4) q^{47} + (3 \beta - 2) q^{51} + ( - 8 \beta - 2) q^{53} + (6 \beta - 8) q^{55} + (2 \beta + 4) q^{57} + (2 \beta - 8) q^{59} + ( - 7 \beta - 4) q^{61} + ( - 2 \beta + 2) q^{65} - 8 q^{67} + ( - 2 \beta - 2) q^{69} + ( - 2 \beta + 2) q^{71} + (5 \beta + 4) q^{73} + (4 \beta - 1) q^{75} + (4 \beta - 8) q^{79} + q^{81} + ( - 8 \beta - 4) q^{83} + (8 \beta - 10) q^{85} - 6 \beta q^{87} + (9 \beta - 2) q^{89} + ( - 2 \beta + 8) q^{93} + 4 q^{95} + ( - 3 \beta + 12) q^{97} + ( - 2 \beta + 2) q^{99} +O(q^{100})$$ q - q^3 + (b - 2) * q^5 + q^9 + (-2*b + 2) * q^11 + b * q^13 + (-b + 2) * q^15 + (-3*b + 2) * q^17 + (-2*b - 4) * q^19 + (2*b + 2) * q^23 + (-4*b + 1) * q^25 - q^27 + 6*b * q^29 + (2*b - 8) * q^31 + (2*b - 2) * q^33 + (4*b - 4) * q^37 - b * q^39 + (-b + 2) * q^41 - 8 * q^43 + (b - 2) * q^45 + (-2*b - 4) * q^47 + (3*b - 2) * q^51 + (-8*b - 2) * q^53 + (6*b - 8) * q^55 + (2*b + 4) * q^57 + (2*b - 8) * q^59 + (-7*b - 4) * q^61 + (-2*b + 2) * q^65 - 8 * q^67 + (-2*b - 2) * q^69 + (-2*b + 2) * q^71 + (5*b + 4) * q^73 + (4*b - 1) * q^75 + (4*b - 8) * q^79 + q^81 + (-8*b - 4) * q^83 + (8*b - 10) * q^85 - 6*b * q^87 + (9*b - 2) * q^89 + (-2*b + 8) * q^93 + 4 * q^95 + (-3*b + 12) * q^97 + (-2*b + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 $$2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{15} + 4 q^{17} - 8 q^{19} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 16 q^{31} - 4 q^{33} - 8 q^{37} + 4 q^{41} - 16 q^{43} - 4 q^{45} - 8 q^{47} - 4 q^{51} - 4 q^{53} - 16 q^{55} + 8 q^{57} - 16 q^{59} - 8 q^{61} + 4 q^{65} - 16 q^{67} - 4 q^{69} + 4 q^{71} + 8 q^{73} - 2 q^{75} - 16 q^{79} + 2 q^{81} - 8 q^{83} - 20 q^{85} - 4 q^{89} + 16 q^{93} + 8 q^{95} + 24 q^{97} + 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^11 + 4 * q^15 + 4 * q^17 - 8 * q^19 + 4 * q^23 + 2 * q^25 - 2 * q^27 - 16 * q^31 - 4 * q^33 - 8 * q^37 + 4 * q^41 - 16 * q^43 - 4 * q^45 - 8 * q^47 - 4 * q^51 - 4 * q^53 - 16 * q^55 + 8 * q^57 - 16 * q^59 - 8 * q^61 + 4 * q^65 - 16 * q^67 - 4 * q^69 + 4 * q^71 + 8 * q^73 - 2 * q^75 - 16 * q^79 + 2 * q^81 - 8 * q^83 - 20 * q^85 - 4 * q^89 + 16 * q^93 + 8 * q^95 + 24 * q^97 + 4 * 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
 −1.41421 1.41421
0 −1.00000 0 −3.41421 0 0 0 1.00000 0
1.2 0 −1.00000 0 −0.585786 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.j 2
3.b odd 2 1 3528.2.a.bl 2
4.b odd 2 1 2352.2.a.bd 2
7.b odd 2 1 1176.2.a.o yes 2
7.c even 3 2 1176.2.q.o 4
7.d odd 6 2 1176.2.q.k 4
8.b even 2 1 9408.2.a.ee 2
8.d odd 2 1 9408.2.a.ds 2
12.b even 2 1 7056.2.a.cx 2
21.c even 2 1 3528.2.a.bb 2
21.g even 6 2 3528.2.s.bm 4
21.h odd 6 2 3528.2.s.bd 4
28.d even 2 1 2352.2.a.bb 2
28.f even 6 2 2352.2.q.be 4
28.g odd 6 2 2352.2.q.bc 4
56.e even 2 1 9408.2.a.du 2
56.h odd 2 1 9408.2.a.dg 2
84.h odd 2 1 7056.2.a.cg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.j 2 1.a even 1 1 trivial
1176.2.a.o yes 2 7.b odd 2 1
1176.2.q.k 4 7.d odd 6 2
1176.2.q.o 4 7.c even 3 2
2352.2.a.bb 2 28.d even 2 1
2352.2.a.bd 2 4.b odd 2 1
2352.2.q.bc 4 28.g odd 6 2
2352.2.q.be 4 28.f even 6 2
3528.2.a.bb 2 21.c even 2 1
3528.2.a.bl 2 3.b odd 2 1
3528.2.s.bd 4 21.h odd 6 2
3528.2.s.bm 4 21.g even 6 2
7056.2.a.cg 2 84.h odd 2 1
7056.2.a.cx 2 12.b even 2 1
9408.2.a.dg 2 56.h odd 2 1
9408.2.a.ds 2 8.d odd 2 1
9408.2.a.du 2 56.e even 2 1
9408.2.a.ee 2 8.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1176))$$:

 $$T_{5}^{2} + 4T_{5} + 2$$ T5^2 + 4*T5 + 2 $$T_{11}^{2} - 4T_{11} - 4$$ T11^2 - 4*T11 - 4 $$T_{13}^{2} - 2$$ T13^2 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1)^{2}$$
$5$ $$T^{2} + 4T + 2$$
$7$ $$T^{2}$$
$11$ $$T^{2} - 4T - 4$$
$13$ $$T^{2} - 2$$
$17$ $$T^{2} - 4T - 14$$
$19$ $$T^{2} + 8T + 8$$
$23$ $$T^{2} - 4T - 4$$
$29$ $$T^{2} - 72$$
$31$ $$T^{2} + 16T + 56$$
$37$ $$T^{2} + 8T - 16$$
$41$ $$T^{2} - 4T + 2$$
$43$ $$(T + 8)^{2}$$
$47$ $$T^{2} + 8T + 8$$
$53$ $$T^{2} + 4T - 124$$
$59$ $$T^{2} + 16T + 56$$
$61$ $$T^{2} + 8T - 82$$
$67$ $$(T + 8)^{2}$$
$71$ $$T^{2} - 4T - 4$$
$73$ $$T^{2} - 8T - 34$$
$79$ $$T^{2} + 16T + 32$$
$83$ $$T^{2} + 8T - 112$$
$89$ $$T^{2} + 4T - 158$$
$97$ $$T^{2} - 24T + 126$$