# Properties

 Label 168.2.q.b Level $168$ Weight $2$ Character orbit 168.q Analytic conductor $1.341$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 168.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.34148675396$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} + (2 \zeta_{6} - 2) q^{5} + ( - \zeta_{6} + 3) q^{7} + (\zeta_{6} - 1) q^{9}+O(q^{10})$$ q + z * q^3 + (2*z - 2) * q^5 + (-z + 3) * q^7 + (z - 1) * q^9 $$q + \zeta_{6} q^{3} + (2 \zeta_{6} - 2) q^{5} + ( - \zeta_{6} + 3) q^{7} + (\zeta_{6} - 1) q^{9} + 6 \zeta_{6} q^{11} - 3 q^{13} - 2 q^{15} - 4 \zeta_{6} q^{17} + ( - 5 \zeta_{6} + 5) q^{19} + (2 \zeta_{6} + 1) q^{21} + ( - 4 \zeta_{6} + 4) q^{23} + \zeta_{6} q^{25} - q^{27} - 4 q^{29} - 7 \zeta_{6} q^{31} + (6 \zeta_{6} - 6) q^{33} + (6 \zeta_{6} - 4) q^{35} + ( - 9 \zeta_{6} + 9) q^{37} - 3 \zeta_{6} q^{39} - 2 q^{41} - q^{43} - 2 \zeta_{6} q^{45} + (2 \zeta_{6} - 2) q^{47} + ( - 5 \zeta_{6} + 8) q^{49} + ( - 4 \zeta_{6} + 4) q^{51} - 8 \zeta_{6} q^{53} - 12 q^{55} + 5 q^{57} + (10 \zeta_{6} - 10) q^{61} + (3 \zeta_{6} - 2) q^{63} + ( - 6 \zeta_{6} + 6) q^{65} + 15 \zeta_{6} q^{67} + 4 q^{69} - 6 q^{71} + 11 \zeta_{6} q^{73} + (\zeta_{6} - 1) q^{75} + (12 \zeta_{6} + 6) q^{77} + (\zeta_{6} - 1) q^{79} - \zeta_{6} q^{81} + 6 q^{83} + 8 q^{85} - 4 \zeta_{6} q^{87} + ( - 8 \zeta_{6} + 8) q^{89} + (3 \zeta_{6} - 9) q^{91} + ( - 7 \zeta_{6} + 7) q^{93} + 10 \zeta_{6} q^{95} - 14 q^{97} - 6 q^{99} +O(q^{100})$$ q + z * q^3 + (2*z - 2) * q^5 + (-z + 3) * q^7 + (z - 1) * q^9 + 6*z * q^11 - 3 * q^13 - 2 * q^15 - 4*z * q^17 + (-5*z + 5) * q^19 + (2*z + 1) * q^21 + (-4*z + 4) * q^23 + z * q^25 - q^27 - 4 * q^29 - 7*z * q^31 + (6*z - 6) * q^33 + (6*z - 4) * q^35 + (-9*z + 9) * q^37 - 3*z * q^39 - 2 * q^41 - q^43 - 2*z * q^45 + (2*z - 2) * q^47 + (-5*z + 8) * q^49 + (-4*z + 4) * q^51 - 8*z * q^53 - 12 * q^55 + 5 * q^57 + (10*z - 10) * q^61 + (3*z - 2) * q^63 + (-6*z + 6) * q^65 + 15*z * q^67 + 4 * q^69 - 6 * q^71 + 11*z * q^73 + (z - 1) * q^75 + (12*z + 6) * q^77 + (z - 1) * q^79 - z * q^81 + 6 * q^83 + 8 * q^85 - 4*z * q^87 + (-8*z + 8) * q^89 + (3*z - 9) * q^91 + (-7*z + 7) * q^93 + 10*z * q^95 - 14 * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - 2 q^{5} + 5 q^{7} - q^{9}+O(q^{10})$$ 2 * q + q^3 - 2 * q^5 + 5 * q^7 - q^9 $$2 q + q^{3} - 2 q^{5} + 5 q^{7} - q^{9} + 6 q^{11} - 6 q^{13} - 4 q^{15} - 4 q^{17} + 5 q^{19} + 4 q^{21} + 4 q^{23} + q^{25} - 2 q^{27} - 8 q^{29} - 7 q^{31} - 6 q^{33} - 2 q^{35} + 9 q^{37} - 3 q^{39} - 4 q^{41} - 2 q^{43} - 2 q^{45} - 2 q^{47} + 11 q^{49} + 4 q^{51} - 8 q^{53} - 24 q^{55} + 10 q^{57} - 10 q^{61} - q^{63} + 6 q^{65} + 15 q^{67} + 8 q^{69} - 12 q^{71} + 11 q^{73} - q^{75} + 24 q^{77} - q^{79} - q^{81} + 12 q^{83} + 16 q^{85} - 4 q^{87} + 8 q^{89} - 15 q^{91} + 7 q^{93} + 10 q^{95} - 28 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 - 2 * q^5 + 5 * q^7 - q^9 + 6 * q^11 - 6 * q^13 - 4 * q^15 - 4 * q^17 + 5 * q^19 + 4 * q^21 + 4 * q^23 + q^25 - 2 * q^27 - 8 * q^29 - 7 * q^31 - 6 * q^33 - 2 * q^35 + 9 * q^37 - 3 * q^39 - 4 * q^41 - 2 * q^43 - 2 * q^45 - 2 * q^47 + 11 * q^49 + 4 * q^51 - 8 * q^53 - 24 * q^55 + 10 * q^57 - 10 * q^61 - q^63 + 6 * q^65 + 15 * q^67 + 8 * q^69 - 12 * q^71 + 11 * q^73 - q^75 + 24 * q^77 - q^79 - q^81 + 12 * q^83 + 16 * q^85 - 4 * q^87 + 8 * q^89 - 15 * q^91 + 7 * q^93 + 10 * q^95 - 28 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$$.

 $$n$$ $$73$$ $$85$$ $$113$$ $$127$$ $$\chi(n)$$ $$-1 + \zeta_{6}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 −1.00000 1.73205i 0 2.50000 + 0.866025i 0 −0.500000 0.866025i 0
121.1 0 0.500000 + 0.866025i 0 −1.00000 + 1.73205i 0 2.50000 0.866025i 0 −0.500000 + 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 2T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(168, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} + 2T + 4$$
$7$ $$T^{2} - 5T + 7$$
$11$ $$T^{2} - 6T + 36$$
$13$ $$(T + 3)^{2}$$
$17$ $$T^{2} + 4T + 16$$
$19$ $$T^{2} - 5T + 25$$
$23$ $$T^{2} - 4T + 16$$
$29$ $$(T + 4)^{2}$$
$31$ $$T^{2} + 7T + 49$$
$37$ $$T^{2} - 9T + 81$$
$41$ $$(T + 2)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2} + 2T + 4$$
$53$ $$T^{2} + 8T + 64$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} - 15T + 225$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 11T + 121$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} - 8T + 64$$
$97$ $$(T + 14)^{2}$$