# Properties

 Label 168.2.q.a Level $168$ Weight $2$ Character orbit 168.q Analytic conductor $1.341$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(25,168)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("168.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.34148675396$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{3} + ( - \zeta_{6} + 1) q^{5} + ( - 3 \zeta_{6} + 2) q^{7} + (\zeta_{6} - 1) q^{9} +O(q^{10})$$ q - z * q^3 + (-z + 1) * q^5 + (-3*z + 2) * q^7 + (z - 1) * q^9 $$q - \zeta_{6} q^{3} + ( - \zeta_{6} + 1) q^{5} + ( - 3 \zeta_{6} + 2) q^{7} + (\zeta_{6} - 1) q^{9} - 3 \zeta_{6} q^{11} + 4 q^{13} - q^{15} + ( - 4 \zeta_{6} + 4) q^{19} + (\zeta_{6} - 3) q^{21} + (8 \zeta_{6} - 8) q^{23} + 4 \zeta_{6} q^{25} + q^{27} - 3 q^{29} + 5 \zeta_{6} q^{31} + (3 \zeta_{6} - 3) q^{33} + ( - 2 \zeta_{6} - 1) q^{35} + (8 \zeta_{6} - 8) q^{37} - 4 \zeta_{6} q^{39} + 8 q^{41} + 6 q^{43} + \zeta_{6} q^{45} + (10 \zeta_{6} - 10) q^{47} + ( - 3 \zeta_{6} - 5) q^{49} - 9 \zeta_{6} q^{53} - 3 q^{55} - 4 q^{57} + 5 \zeta_{6} q^{59} + ( - 10 \zeta_{6} + 10) q^{61} + (2 \zeta_{6} + 1) q^{63} + ( - 4 \zeta_{6} + 4) q^{65} - 6 \zeta_{6} q^{67} + 8 q^{69} + 10 q^{71} - 2 \zeta_{6} q^{73} + ( - 4 \zeta_{6} + 4) q^{75} + (3 \zeta_{6} - 9) q^{77} + (11 \zeta_{6} - 11) q^{79} - \zeta_{6} q^{81} + 7 q^{83} + 3 \zeta_{6} q^{87} + ( - 18 \zeta_{6} + 18) q^{89} + ( - 12 \zeta_{6} + 8) q^{91} + ( - 5 \zeta_{6} + 5) q^{93} - 4 \zeta_{6} q^{95} - 17 q^{97} + 3 q^{99} +O(q^{100})$$ q - z * q^3 + (-z + 1) * q^5 + (-3*z + 2) * q^7 + (z - 1) * q^9 - 3*z * q^11 + 4 * q^13 - q^15 + (-4*z + 4) * q^19 + (z - 3) * q^21 + (8*z - 8) * q^23 + 4*z * q^25 + q^27 - 3 * q^29 + 5*z * q^31 + (3*z - 3) * q^33 + (-2*z - 1) * q^35 + (8*z - 8) * q^37 - 4*z * q^39 + 8 * q^41 + 6 * q^43 + z * q^45 + (10*z - 10) * q^47 + (-3*z - 5) * q^49 - 9*z * q^53 - 3 * q^55 - 4 * q^57 + 5*z * q^59 + (-10*z + 10) * q^61 + (2*z + 1) * q^63 + (-4*z + 4) * q^65 - 6*z * q^67 + 8 * q^69 + 10 * q^71 - 2*z * q^73 + (-4*z + 4) * q^75 + (3*z - 9) * q^77 + (11*z - 11) * q^79 - z * q^81 + 7 * q^83 + 3*z * q^87 + (-18*z + 18) * q^89 + (-12*z + 8) * q^91 + (-5*z + 5) * q^93 - 4*z * q^95 - 17 * q^97 + 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{3} + q^{5} + q^{7} - q^{9}+O(q^{10})$$ 2 * q - q^3 + q^5 + q^7 - q^9 $$2 q - q^{3} + q^{5} + q^{7} - q^{9} - 3 q^{11} + 8 q^{13} - 2 q^{15} + 4 q^{19} - 5 q^{21} - 8 q^{23} + 4 q^{25} + 2 q^{27} - 6 q^{29} + 5 q^{31} - 3 q^{33} - 4 q^{35} - 8 q^{37} - 4 q^{39} + 16 q^{41} + 12 q^{43} + q^{45} - 10 q^{47} - 13 q^{49} - 9 q^{53} - 6 q^{55} - 8 q^{57} + 5 q^{59} + 10 q^{61} + 4 q^{63} + 4 q^{65} - 6 q^{67} + 16 q^{69} + 20 q^{71} - 2 q^{73} + 4 q^{75} - 15 q^{77} - 11 q^{79} - q^{81} + 14 q^{83} + 3 q^{87} + 18 q^{89} + 4 q^{91} + 5 q^{93} - 4 q^{95} - 34 q^{97} + 6 q^{99}+O(q^{100})$$ 2 * q - q^3 + q^5 + q^7 - q^9 - 3 * q^11 + 8 * q^13 - 2 * q^15 + 4 * q^19 - 5 * q^21 - 8 * q^23 + 4 * q^25 + 2 * q^27 - 6 * q^29 + 5 * q^31 - 3 * q^33 - 4 * q^35 - 8 * q^37 - 4 * q^39 + 16 * q^41 + 12 * q^43 + q^45 - 10 * q^47 - 13 * q^49 - 9 * q^53 - 6 * q^55 - 8 * q^57 + 5 * q^59 + 10 * q^61 + 4 * q^63 + 4 * q^65 - 6 * q^67 + 16 * q^69 + 20 * q^71 - 2 * q^73 + 4 * q^75 - 15 * q^77 - 11 * q^79 - q^81 + 14 * q^83 + 3 * q^87 + 18 * q^89 + 4 * q^91 + 5 * q^93 - 4 * q^95 - 34 * q^97 + 6 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 0.500000 + 0.866025i 0 0.500000 + 2.59808i 0 −0.500000 0.866025i 0
121.1 0 −0.500000 0.866025i 0 0.500000 0.866025i 0 0.500000 2.59808i 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.a 2
3.b odd 2 1 504.2.s.d 2
4.b odd 2 1 336.2.q.e 2
7.b odd 2 1 1176.2.q.g 2
7.c even 3 1 inner 168.2.q.a 2
7.c even 3 1 1176.2.a.g 1
7.d odd 6 1 1176.2.a.c 1
7.d odd 6 1 1176.2.q.g 2
8.b even 2 1 1344.2.q.o 2
8.d odd 2 1 1344.2.q.d 2
12.b even 2 1 1008.2.s.f 2
21.c even 2 1 3528.2.s.p 2
21.g even 6 1 3528.2.a.i 1
21.g even 6 1 3528.2.s.p 2
21.h odd 6 1 504.2.s.d 2
21.h odd 6 1 3528.2.a.q 1
28.d even 2 1 2352.2.q.f 2
28.f even 6 1 2352.2.a.u 1
28.f even 6 1 2352.2.q.f 2
28.g odd 6 1 336.2.q.e 2
28.g odd 6 1 2352.2.a.g 1
56.j odd 6 1 9408.2.a.cf 1
56.k odd 6 1 1344.2.q.d 2
56.k odd 6 1 9408.2.a.cq 1
56.m even 6 1 9408.2.a.p 1
56.p even 6 1 1344.2.q.o 2
56.p even 6 1 9408.2.a.ba 1
84.j odd 6 1 7056.2.a.t 1
84.n even 6 1 1008.2.s.f 2
84.n even 6 1 7056.2.a.bk 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(168, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$(T - 4)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 4T + 16$$
$23$ $$T^{2} + 8T + 64$$
$29$ $$(T + 3)^{2}$$
$31$ $$T^{2} - 5T + 25$$
$37$ $$T^{2} + 8T + 64$$
$41$ $$(T - 8)^{2}$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} + 10T + 100$$
$53$ $$T^{2} + 9T + 81$$
$59$ $$T^{2} - 5T + 25$$
$61$ $$T^{2} - 10T + 100$$
$67$ $$T^{2} + 6T + 36$$
$71$ $$(T - 10)^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$(T - 7)^{2}$$
$89$ $$T^{2} - 18T + 324$$
$97$ $$(T + 17)^{2}$$