# Properties

 Label 168.4.a.h Level $168$ Weight $4$ Character orbit 168.a Self dual yes Analytic conductor $9.912$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 168.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.91232088096$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{177})$$ Defining polynomial: $$x^{2} - x - 44$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{177}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( 7 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} +O(q^{10})$$ $$q + 3 q^{3} + ( 7 - \beta ) q^{5} + 7 q^{7} + 9 q^{9} + ( 9 + 3 \beta ) q^{11} + ( 24 - 2 \beta ) q^{13} + ( 21 - 3 \beta ) q^{15} + ( 17 + 9 \beta ) q^{17} + ( -8 + 4 \beta ) q^{19} + 21 q^{21} + ( 55 - 7 \beta ) q^{23} + ( 101 - 14 \beta ) q^{25} + 27 q^{27} + ( 106 - 4 \beta ) q^{29} + ( -68 + 4 \beta ) q^{31} + ( 27 + 9 \beta ) q^{33} + ( 49 - 7 \beta ) q^{35} + ( -12 + 26 \beta ) q^{37} + ( 72 - 6 \beta ) q^{39} + ( 347 - 9 \beta ) q^{41} + ( -292 - 8 \beta ) q^{43} + ( 63 - 9 \beta ) q^{45} + ( -158 - 2 \beta ) q^{47} + 49 q^{49} + ( 51 + 27 \beta ) q^{51} + ( 280 + 6 \beta ) q^{53} + ( -468 + 12 \beta ) q^{55} + ( -24 + 12 \beta ) q^{57} + ( -246 + 26 \beta ) q^{59} + ( -302 - 28 \beta ) q^{61} + 63 q^{63} + ( 522 - 38 \beta ) q^{65} + ( -510 + 10 \beta ) q^{67} + ( 165 - 21 \beta ) q^{69} + ( -855 - 25 \beta ) q^{71} + ( -656 - 6 \beta ) q^{73} + ( 303 - 42 \beta ) q^{75} + ( 63 + 21 \beta ) q^{77} + ( -278 - 42 \beta ) q^{79} + 81 q^{81} + ( -132 + 32 \beta ) q^{83} + ( -1474 + 46 \beta ) q^{85} + ( 318 - 12 \beta ) q^{87} + ( 35 + 95 \beta ) q^{89} + ( 168 - 14 \beta ) q^{91} + ( -204 + 12 \beta ) q^{93} + ( -764 + 36 \beta ) q^{95} + ( -68 - 10 \beta ) q^{97} + ( 81 + 27 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + O(q^{10})$$ $$2q + 6q^{3} + 14q^{5} + 14q^{7} + 18q^{9} + 18q^{11} + 48q^{13} + 42q^{15} + 34q^{17} - 16q^{19} + 42q^{21} + 110q^{23} + 202q^{25} + 54q^{27} + 212q^{29} - 136q^{31} + 54q^{33} + 98q^{35} - 24q^{37} + 144q^{39} + 694q^{41} - 584q^{43} + 126q^{45} - 316q^{47} + 98q^{49} + 102q^{51} + 560q^{53} - 936q^{55} - 48q^{57} - 492q^{59} - 604q^{61} + 126q^{63} + 1044q^{65} - 1020q^{67} + 330q^{69} - 1710q^{71} - 1312q^{73} + 606q^{75} + 126q^{77} - 556q^{79} + 162q^{81} - 264q^{83} - 2948q^{85} + 636q^{87} + 70q^{89} + 336q^{91} - 408q^{93} - 1528q^{95} - 136q^{97} + 162q^{99} + O(q^{100})$$

## 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
 7.15207 −6.15207
0 3.00000 0 −6.30413 0 7.00000 0 9.00000 0
1.2 0 3.00000 0 20.3041 0 7.00000 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 168.4.a.h 2
3.b odd 2 1 504.4.a.j 2
4.b odd 2 1 336.4.a.n 2
7.b odd 2 1 1176.4.a.p 2
8.b even 2 1 1344.4.a.bd 2
8.d odd 2 1 1344.4.a.bl 2
12.b even 2 1 1008.4.a.y 2
28.d even 2 1 2352.4.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.4.a.h 2 1.a even 1 1 trivial
336.4.a.n 2 4.b odd 2 1
504.4.a.j 2 3.b odd 2 1
1008.4.a.y 2 12.b even 2 1
1176.4.a.p 2 7.b odd 2 1
1344.4.a.bd 2 8.b even 2 1
1344.4.a.bl 2 8.d odd 2 1
2352.4.a.bv 2 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(168))$$:

 $$T_{5}^{2} - 14 T_{5} - 128$$ $$T_{11}^{2} - 18 T_{11} - 1512$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$( -3 + T )^{2}$$
$5$ $$-128 - 14 T + T^{2}$$
$7$ $$( -7 + T )^{2}$$
$11$ $$-1512 - 18 T + T^{2}$$
$13$ $$-132 - 48 T + T^{2}$$
$17$ $$-14048 - 34 T + T^{2}$$
$19$ $$-2768 + 16 T + T^{2}$$
$23$ $$-5648 - 110 T + T^{2}$$
$29$ $$8404 - 212 T + T^{2}$$
$31$ $$1792 + 136 T + T^{2}$$
$37$ $$-119508 + 24 T + T^{2}$$
$41$ $$106072 - 694 T + T^{2}$$
$43$ $$73936 + 584 T + T^{2}$$
$47$ $$24256 + 316 T + T^{2}$$
$53$ $$72028 - 560 T + T^{2}$$
$59$ $$-59136 + 492 T + T^{2}$$
$61$ $$-47564 + 604 T + T^{2}$$
$67$ $$242400 + 1020 T + T^{2}$$
$71$ $$620400 + 1710 T + T^{2}$$
$73$ $$423964 + 1312 T + T^{2}$$
$79$ $$-234944 + 556 T + T^{2}$$
$83$ $$-163824 + 264 T + T^{2}$$
$89$ $$-1596200 - 70 T + T^{2}$$
$97$ $$-13076 + 136 T + T^{2}$$