# Properties

 Label 1176.4.a.r Level $1176$ Weight $4$ Character orbit 1176.a Self dual yes Analytic conductor $69.386$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{505})$$ Defining polynomial: $$x^{2} - x - 126$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{505})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -3 q^{3} + ( 5 - \beta ) q^{5} + 9 q^{9} +O(q^{10})$$ $$q -3 q^{3} + ( 5 - \beta ) q^{5} + 9 q^{9} + ( 5 - 5 \beta ) q^{11} + ( -10 + \beta ) q^{13} + ( -15 + 3 \beta ) q^{15} -4 \beta q^{17} + ( 62 - 7 \beta ) q^{19} + ( 72 + 4 \beta ) q^{23} + ( 26 - 9 \beta ) q^{25} -27 q^{27} + ( -45 - 5 \beta ) q^{29} + ( -181 + 2 \beta ) q^{31} + ( -15 + 15 \beta ) q^{33} + ( -26 - \beta ) q^{37} + ( 30 - 3 \beta ) q^{39} + ( -76 - 18 \beta ) q^{41} + ( 188 + 27 \beta ) q^{43} + ( 45 - 9 \beta ) q^{45} + ( 202 - 36 \beta ) q^{47} + 12 \beta q^{51} + ( 351 - 5 \beta ) q^{53} + ( 655 - 25 \beta ) q^{55} + ( -186 + 21 \beta ) q^{57} + ( -279 - 27 \beta ) q^{59} + ( -594 + 28 \beta ) q^{61} + ( -176 + 14 \beta ) q^{65} + ( 80 + 73 \beta ) q^{67} + ( -216 - 12 \beta ) q^{69} + ( 346 - 76 \beta ) q^{71} + ( -440 + 63 \beta ) q^{73} + ( -78 + 27 \beta ) q^{75} + ( -377 - 48 \beta ) q^{79} + 81 q^{81} + ( -175 + 67 \beta ) q^{83} + ( 504 - 16 \beta ) q^{85} + ( 135 + 15 \beta ) q^{87} + ( -918 - 22 \beta ) q^{89} + ( 543 - 6 \beta ) q^{93} + ( 1192 - 90 \beta ) q^{95} + ( 837 + 55 \beta ) q^{97} + ( 45 - 45 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{3} + 9q^{5} + 18q^{9} + O(q^{10})$$ $$2q - 6q^{3} + 9q^{5} + 18q^{9} + 5q^{11} - 19q^{13} - 27q^{15} - 4q^{17} + 117q^{19} + 148q^{23} + 43q^{25} - 54q^{27} - 95q^{29} - 360q^{31} - 15q^{33} - 53q^{37} + 57q^{39} - 170q^{41} + 403q^{43} + 81q^{45} + 368q^{47} + 12q^{51} + 697q^{53} + 1285q^{55} - 351q^{57} - 585q^{59} - 1160q^{61} - 338q^{65} + 233q^{67} - 444q^{69} + 616q^{71} - 817q^{73} - 129q^{75} - 802q^{79} + 162q^{81} - 283q^{83} + 992q^{85} + 285q^{87} - 1858q^{89} + 1080q^{93} + 2294q^{95} + 1729q^{97} + 45q^{99} + O(q^{100})$$

## 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
 11.7361 −10.7361
0 −3.00000 0 −6.73610 0 0 0 9.00000 0
1.2 0 −3.00000 0 15.7361 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.r 2
4.b odd 2 1 2352.4.a.cc 2
7.b odd 2 1 1176.4.a.u 2
7.c even 3 2 168.4.q.d 4
21.h odd 6 2 504.4.s.f 4
28.d even 2 1 2352.4.a.bo 2
28.g odd 6 2 336.4.q.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.q.d 4 7.c even 3 2
336.4.q.g 4 28.g odd 6 2
504.4.s.f 4 21.h odd 6 2
1176.4.a.r 2 1.a even 1 1 trivial
1176.4.a.u 2 7.b odd 2 1
2352.4.a.bo 2 28.d even 2 1
2352.4.a.cc 2 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1176))$$:

 $$T_{5}^{2} - 9 T_{5} - 106$$ $$T_{11}^{2} - 5 T_{11} - 3150$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( 3 + T )^{2}$$
$5$ $$-106 - 9 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-3150 - 5 T + T^{2}$$
$13$ $$-36 + 19 T + T^{2}$$
$17$ $$-2016 + 4 T + T^{2}$$
$19$ $$-2764 - 117 T + T^{2}$$
$23$ $$3456 - 148 T + T^{2}$$
$29$ $$-900 + 95 T + T^{2}$$
$31$ $$31895 + 360 T + T^{2}$$
$37$ $$576 + 53 T + T^{2}$$
$41$ $$-33680 + 170 T + T^{2}$$
$43$ $$-51434 - 403 T + T^{2}$$
$47$ $$-129764 - 368 T + T^{2}$$
$53$ $$118296 - 697 T + T^{2}$$
$59$ $$-6480 + 585 T + T^{2}$$
$61$ $$237420 + 1160 T + T^{2}$$
$67$ $$-659214 - 233 T + T^{2}$$
$71$ $$-634356 - 616 T + T^{2}$$
$73$ $$-334214 + 817 T + T^{2}$$
$79$ $$-130079 + 802 T + T^{2}$$
$83$ $$-546714 + 283 T + T^{2}$$
$89$ $$801936 + 1858 T + T^{2}$$
$97$ $$365454 - 1729 T + T^{2}$$