# Properties

 Label 168.2.ba.a Level $168$ Weight $2$ Character orbit 168.ba Analytic conductor $1.341$ Analytic rank $0$ Dimension $4$ CM discriminant -24 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,2,Mod(5,168)]

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, 3, 3, 5]))

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

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

chi := DirichletCharacter("168.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$168 = 2^{3} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 168.ba (of order $$6$$, 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 2x^{2} + 4$$ x^4 + 2*x^2 + 4 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 - 2) * q^3 + 2*b2 * q^4 + (-2*b3 - b2 - b1 + 1) * q^5 + (-b3 - 2*b1) * q^6 + (-b3 + b2 + b1) * q^7 + 2*b3 * q^8 + (3*b2 + 3) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} - 2) q^{3} + 2 \beta_{2} q^{4} + ( - 2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{5} + ( - \beta_{3} - 2 \beta_1) q^{6} + ( - \beta_{3} + \beta_{2} + \beta_1) q^{7} + 2 \beta_{3} q^{8} + (3 \beta_{2} + 3) q^{9} + ( - \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{10} + ( - 2 \beta_{3} + 3 \beta_{2} - 2 \beta_1) q^{11} + ( - 2 \beta_{2} + 2) q^{12} + (\beta_{3} + 4 \beta_{2} + 2) q^{14} + (3 \beta_{3} - 3) q^{15} + ( - 4 \beta_{2} - 4) q^{16} + (3 \beta_{3} + 3 \beta_1) q^{18} + (2 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 2) q^{20} + ( - \beta_{2} - 3 \beta_1 + 1) q^{21} + (3 \beta_{3} + 4) q^{22} + ( - 2 \beta_{3} + 2 \beta_1) q^{24} + ( - 6 \beta_{3} - 4 \beta_{2} - 6 \beta_1) q^{25} + ( - 6 \beta_{2} - 3) q^{27} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{28} + (\beta_{3} - 9) q^{29} + ( - 6 \beta_{2} - 3 \beta_1 - 6) q^{30} + ( - \beta_{3} - 5 \beta_{2} + \beta_1 - 10) q^{31} + ( - 4 \beta_{3} - 4 \beta_1) q^{32} + (4 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 3) q^{33} + ( - 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 7) q^{35} - 6 q^{36} + (4 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 4) q^{40} + ( - \beta_{3} - 6 \beta_{2} + \beta_1) q^{42} + ( - 6 \beta_{2} + 4 \beta_1 - 6) q^{44} + ( - 3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 6) q^{45} + (8 \beta_{2} + 4) q^{48} + (4 \beta_{3} + 5 \beta_{2} + 2 \beta_1 + 5) q^{49} + ( - 4 \beta_{3} + 12) q^{50} + (5 \beta_{3} + 3 \beta_{2} + 5 \beta_1) q^{53} + ( - 6 \beta_{3} - 3 \beta_1) q^{54} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{55} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 8) q^{56} + ( - 2 \beta_{2} - 9 \beta_1 - 2) q^{58} + (4 \beta_{3} + 3 \beta_{2} - 4 \beta_1 + 6) q^{59} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{60} + ( - 5 \beta_{3} + 4 \beta_{2} - 10 \beta_1 + 2) q^{62} + (3 \beta_{3} + 6 \beta_1 - 3) q^{63} + 8 q^{64} + ( - 3 \beta_{3} - 4 \beta_{2} + 3 \beta_1 - 8) q^{66} + (2 \beta_{3} + 8 \beta_{2} + 7 \beta_1 + 4) q^{70} - 6 \beta_1 q^{72} + ( - 4 \beta_{3} + 4 \beta_1) q^{73} + (12 \beta_{3} + 4 \beta_{2} + 6 \beta_1 - 4) q^{75} + (6 \beta_{3} - 7 \beta_{2} + 5 \beta_1 + 1) q^{77} + (5 \beta_{2} - 9 \beta_1 + 5) q^{79} + (4 \beta_{3} - 4 \beta_{2} - 4 \beta_1 - 8) q^{80} + 9 \beta_{2} q^{81} + (2 \beta_{3} - 10 \beta_{2} + 4 \beta_1 - 5) q^{83} + ( - 6 \beta_{3} + 4 \beta_{2} + 2) q^{84} + ( - \beta_{3} + 9 \beta_{2} + \beta_1 + 18) q^{87} + ( - 6 \beta_{3} + 8 \beta_{2} - 6 \beta_1) q^{88} + (3 \beta_{3} + 12 \beta_{2} + 6 \beta_1 + 6) q^{90} + (15 \beta_{2} - 3 \beta_1 + 15) q^{93} + (8 \beta_{3} + 4 \beta_1) q^{96} + ( - 4 \beta_{3} - 2 \beta_{2} - 8 \beta_1 - 1) q^{97} + (5 \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 8) q^{98} + ( - 6 \beta_{3} - 9) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 - 2) * q^3 + 2*b2 * q^4 + (-2*b3 - b2 - b1 + 1) * q^5 + (-b3 - 2*b1) * q^6 + (-b3 + b2 + b1) * q^7 + 2*b3 * q^8 + (3*b2 + 3) * q^9 + (-b3 + 2*b2 + b1 + 4) * q^10 + (-2*b3 + 3*b2 - 2*b1) * q^11 + (-2*b2 + 2) * q^12 + (b3 + 4*b2 + 2) * q^14 + (3*b3 - 3) * q^15 + (-4*b2 - 4) * q^16 + (3*b3 + 3*b1) * q^18 + (2*b3 + 4*b2 + 4*b1 + 2) * q^20 + (-b2 - 3*b1 + 1) * q^21 + (3*b3 + 4) * q^22 + (-2*b3 + 2*b1) * q^24 + (-6*b3 - 4*b2 - 6*b1) * q^25 + (-6*b2 - 3) * q^27 + (4*b3 - 2*b2 + 2*b1 - 2) * q^28 + (b3 - 9) * q^29 + (-6*b2 - 3*b1 - 6) * q^30 + (-b3 - 5*b2 + b1 - 10) * q^31 + (-4*b3 - 4*b1) * q^32 + (4*b3 - 3*b2 + 2*b1 + 3) * q^33 + (-2*b3 + 2*b2 + 2*b1 + 7) * q^35 - 6 * q^36 + (4*b3 + 4*b2 + 2*b1 - 4) * q^40 + (-b3 - 6*b2 + b1) * q^42 + (-6*b2 + 4*b1 - 6) * q^44 + (-3*b3 + 3*b2 + 3*b1 + 6) * q^45 + (8*b2 + 4) * q^48 + (4*b3 + 5*b2 + 2*b1 + 5) * q^49 + (-4*b3 + 12) * q^50 + (5*b3 + 3*b2 + 5*b1) * q^53 + (-6*b3 - 3*b1) * q^54 + (b3 - 2*b2 + 2*b1 - 1) * q^55 + (-2*b3 - 4*b2 - 2*b1 - 8) * q^56 + (-2*b2 - 9*b1 - 2) * q^58 + (4*b3 + 3*b2 - 4*b1 + 6) * q^59 + (-6*b3 - 6*b2 - 6*b1) * q^60 + (-5*b3 + 4*b2 - 10*b1 + 2) * q^62 + (3*b3 + 6*b1 - 3) * q^63 + 8 * q^64 + (-3*b3 - 4*b2 + 3*b1 - 8) * q^66 + (2*b3 + 8*b2 + 7*b1 + 4) * q^70 - 6*b1 * q^72 + (-4*b3 + 4*b1) * q^73 + (12*b3 + 4*b2 + 6*b1 - 4) * q^75 + (6*b3 - 7*b2 + 5*b1 + 1) * q^77 + (5*b2 - 9*b1 + 5) * q^79 + (4*b3 - 4*b2 - 4*b1 - 8) * q^80 + 9*b2 * q^81 + (2*b3 - 10*b2 + 4*b1 - 5) * q^83 + (-6*b3 + 4*b2 + 2) * q^84 + (-b3 + 9*b2 + b1 + 18) * q^87 + (-6*b3 + 8*b2 - 6*b1) * q^88 + (3*b3 + 12*b2 + 6*b1 + 6) * q^90 + (15*b2 - 3*b1 + 15) * q^93 + (8*b3 + 4*b1) * q^96 + (-4*b3 - 2*b2 - 8*b1 - 1) * q^97 + (5*b3 - 4*b2 + 5*b1 - 8) * q^98 + (-6*b3 - 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{3} - 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q - 6 * q^3 - 4 * q^4 + 6 * q^5 - 2 * q^7 + 6 * q^9 $$4 q - 6 q^{3} - 4 q^{4} + 6 q^{5} - 2 q^{7} + 6 q^{9} + 12 q^{10} - 6 q^{11} + 12 q^{12} - 12 q^{15} - 8 q^{16} + 6 q^{21} + 16 q^{22} + 8 q^{25} - 4 q^{28} - 36 q^{29} - 12 q^{30} - 30 q^{31} + 18 q^{33} + 24 q^{35} - 24 q^{36} - 24 q^{40} + 12 q^{42} - 12 q^{44} + 18 q^{45} + 10 q^{49} + 48 q^{50} - 6 q^{53} - 24 q^{56} - 4 q^{58} + 18 q^{59} + 12 q^{60} - 12 q^{63} + 32 q^{64} - 24 q^{66} - 24 q^{75} + 18 q^{77} + 10 q^{79} - 24 q^{80} - 18 q^{81} + 54 q^{87} - 16 q^{88} + 30 q^{93} - 24 q^{98} - 36 q^{99}+O(q^{100})$$ 4 * q - 6 * q^3 - 4 * q^4 + 6 * q^5 - 2 * q^7 + 6 * q^9 + 12 * q^10 - 6 * q^11 + 12 * q^12 - 12 * q^15 - 8 * q^16 + 6 * q^21 + 16 * q^22 + 8 * q^25 - 4 * q^28 - 36 * q^29 - 12 * q^30 - 30 * q^31 + 18 * q^33 + 24 * q^35 - 24 * q^36 - 24 * q^40 + 12 * q^42 - 12 * q^44 + 18 * q^45 + 10 * q^49 + 48 * q^50 - 6 * q^53 - 24 * q^56 - 4 * q^58 + 18 * q^59 + 12 * q^60 - 12 * q^63 + 32 * q^64 - 24 * q^66 - 24 * q^75 + 18 * q^77 + 10 * q^79 - 24 * q^80 - 18 * q^81 + 54 * q^87 - 16 * q^88 + 30 * q^93 - 24 * q^98 - 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 2$$ (v^2) / 2 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$2\beta_{2}$$ 2*b2 $$\nu^{3}$$ $$=$$ $$2\beta_{3}$$ 2*b3

## Character values

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

 $$n$$ $$73$$ $$85$$ $$113$$ $$127$$ $$\chi(n)$$ $$1 + \beta_{2}$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 + 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i −0.621320 0.358719i 2.44949i −2.62132 + 0.358719i 2.82843 1.50000 2.59808i 0.878680 0.507306i
5.2 0.707107 1.22474i −1.50000 + 0.866025i −1.00000 1.73205i 3.62132 + 2.09077i 2.44949i 1.62132 2.09077i −2.82843 1.50000 2.59808i 5.12132 2.95680i
101.1 −0.707107 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i −0.621320 + 0.358719i 2.44949i −2.62132 0.358719i 2.82843 1.50000 + 2.59808i 0.878680 + 0.507306i
101.2 0.707107 + 1.22474i −1.50000 0.866025i −1.00000 + 1.73205i 3.62132 2.09077i 2.44949i 1.62132 + 2.09077i −2.82843 1.50000 + 2.59808i 5.12132 + 2.95680i
 $$n$$: 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
7.d odd 6 1 inner
168.ba even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 168.2.ba.a 4
3.b odd 2 1 168.2.ba.b yes 4
4.b odd 2 1 672.2.bi.b 4
7.d odd 6 1 inner 168.2.ba.a 4
8.b even 2 1 168.2.ba.b yes 4
8.d odd 2 1 672.2.bi.a 4
12.b even 2 1 672.2.bi.a 4
21.g even 6 1 168.2.ba.b yes 4
24.f even 2 1 672.2.bi.b 4
24.h odd 2 1 CM 168.2.ba.a 4
28.f even 6 1 672.2.bi.b 4
56.j odd 6 1 168.2.ba.b yes 4
56.m even 6 1 672.2.bi.a 4
84.j odd 6 1 672.2.bi.a 4
168.ba even 6 1 inner 168.2.ba.a 4
168.be odd 6 1 672.2.bi.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.ba.a 4 1.a even 1 1 trivial
168.2.ba.a 4 7.d odd 6 1 inner
168.2.ba.a 4 24.h odd 2 1 CM
168.2.ba.a 4 168.ba even 6 1 inner
168.2.ba.b yes 4 3.b odd 2 1
168.2.ba.b yes 4 8.b even 2 1
168.2.ba.b yes 4 21.g even 6 1
168.2.ba.b yes 4 56.j odd 6 1
672.2.bi.a 4 8.d odd 2 1
672.2.bi.a 4 12.b even 2 1
672.2.bi.a 4 56.m even 6 1
672.2.bi.a 4 84.j odd 6 1
672.2.bi.b 4 4.b odd 2 1
672.2.bi.b 4 24.f even 2 1
672.2.bi.b 4 28.f even 6 1
672.2.bi.b 4 168.be odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 6T_{5}^{3} + 9T_{5}^{2} + 18T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(168, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2T^{2} + 4$$
$3$ $$(T^{2} + 3 T + 3)^{2}$$
$5$ $$T^{4} - 6 T^{3} + 9 T^{2} + 18 T + 9$$
$7$ $$T^{4} + 2 T^{3} - 3 T^{2} + 14 T + 49$$
$11$ $$T^{4} + 6 T^{3} + 35 T^{2} + 6 T + 1$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$(T^{2} + 18 T + 79)^{2}$$
$31$ $$T^{4} + 30 T^{3} + 369 T^{2} + \cdots + 4761$$
$37$ $$T^{4}$$
$41$ $$T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$T^{4} + 6 T^{3} + 77 T^{2} + \cdots + 1681$$
$59$ $$T^{4} - 18 T^{3} + 39 T^{2} + \cdots + 4761$$
$61$ $$T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4} - 96T^{2} + 9216$$
$79$ $$T^{4} - 10 T^{3} + 237 T^{2} + \cdots + 18769$$
$83$ $$T^{4} + 198T^{2} + 2601$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 198T^{2} + 8649$$