# Properties

 Label 672.2.bi.b Level $672$ Weight $2$ Character orbit 672.bi Analytic conductor $5.366$ 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] = [672,2,Mod(17,672)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(672, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 3, 3, 1]))

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

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

chi := DirichletCharacter("672.17");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 672.bi (of order $$6$$, degree $$2$$, not 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: $$5.36594701583$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 168) 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{5} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - 3 \beta_{2} q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^3 + (-b3 + b2 + b1 + 2) * q^5 + (2*b3 + b2 + b1 + 1) * q^7 - 3*b2 * q^9 $$q + ( - \beta_{2} + 1) q^{3} + ( - \beta_{3} + \beta_{2} + \beta_1 + 2) q^{5} + (2 \beta_{3} + \beta_{2} + \beta_1 + 1) q^{7} - 3 \beta_{2} q^{9} + (3 \beta_{2} - 2 \beta_1 + 3) q^{11} + ( - 3 \beta_{3} + 3) q^{15} + (3 \beta_{3} + \beta_{2} + 3 \beta_1 + 2) q^{21} + (4 \beta_{2} + 6 \beta_1 + 4) q^{25} + ( - 6 \beta_{2} - 3) q^{27} + (\beta_{3} - 9) q^{29} + (2 \beta_{3} - 5 \beta_{2} + \beta_1 + 5) q^{31} + (2 \beta_{3} + 3 \beta_{2} - 2 \beta_1 + 6) q^{33} + (4 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 5) q^{35} + ( - 6 \beta_{3} - 3 \beta_{2} + \cdots + 3) q^{45}+ \cdots + (6 \beta_{3} + 9) q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^3 + (-b3 + b2 + b1 + 2) * q^5 + (2*b3 + b2 + b1 + 1) * q^7 - 3*b2 * q^9 + (3*b2 - 2*b1 + 3) * q^11 + (-3*b3 + 3) * q^15 + (3*b3 + b2 + 3*b1 + 2) * q^21 + (4*b2 + 6*b1 + 4) * q^25 + (-6*b2 - 3) * q^27 + (b3 - 9) * q^29 + (2*b3 - 5*b2 + b1 + 5) * q^31 + (2*b3 + 3*b2 - 2*b1 + 6) * q^33 + (4*b3 + 2*b2 + 2*b1 - 5) * q^35 + (-6*b3 - 3*b2 - 3*b1 + 3) * q^45 + (2*b3 - 5*b2 - 2*b1) * q^49 + (-3*b2 - 5*b1 - 3) * q^53 + (b3 - 2*b2 + 2*b1 - 1) * q^55 + (-8*b3 + 3*b2 - 4*b1 - 3) * q^59 + (3*b3 + 6*b1 + 3) * q^63 + (-8*b3 - 4*b1) * q^73 + (-6*b3 + 4*b2 + 6*b1 + 8) * q^75 + (b3 + 7*b2 - 5*b1 + 8) * q^77 + (-9*b3 + 5*b2 - 9*b1) * q^79 + (-9*b2 - 9) * q^81 + (2*b3 - 10*b2 + 4*b1 - 5) * q^83 + (2*b3 + 9*b2 + b1 - 9) * q^87 + (3*b3 - 15*b2 + 3*b1) * q^93 + (4*b3 + 2*b2 + 8*b1 + 1) * q^97 + (6*b3 + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 6 q^{3} + 6 q^{5} + 2 q^{7} + 6 q^{9}+O(q^{10})$$ 4 * q + 6 * q^3 + 6 * q^5 + 2 * q^7 + 6 * q^9 $$4 q + 6 q^{3} + 6 q^{5} + 2 q^{7} + 6 q^{9} + 6 q^{11} + 12 q^{15} + 6 q^{21} + 8 q^{25} - 36 q^{29} + 30 q^{31} + 18 q^{33} - 24 q^{35} + 18 q^{45} + 10 q^{49} - 6 q^{53} - 18 q^{59} + 12 q^{63} + 24 q^{75} + 18 q^{77} - 10 q^{79} - 18 q^{81} - 54 q^{87} + 30 q^{93} + 36 q^{99}+O(q^{100})$$ 4 * q + 6 * q^3 + 6 * q^5 + 2 * q^7 + 6 * q^9 + 6 * q^11 + 12 * q^15 + 6 * q^21 + 8 * q^25 - 36 * q^29 + 30 * q^31 + 18 * q^33 - 24 * q^35 + 18 * q^45 + 10 * q^49 - 6 * q^53 - 18 * q^59 + 12 * q^63 + 24 * q^75 + 18 * q^77 - 10 * q^79 - 18 * q^81 - 54 * q^87 + 30 * q^93 + 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 $$\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$-\beta_{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
0 1.50000 + 0.866025i 0 −0.621320 + 0.358719i 0 2.62132 + 0.358719i 0 1.50000 + 2.59808i 0
17.2 0 1.50000 + 0.866025i 0 3.62132 2.09077i 0 −1.62132 2.09077i 0 1.50000 + 2.59808i 0
593.1 0 1.50000 0.866025i 0 −0.621320 0.358719i 0 2.62132 0.358719i 0 1.50000 2.59808i 0
593.2 0 1.50000 0.866025i 0 3.62132 + 2.09077i 0 −1.62132 + 2.09077i 0 1.50000 2.59808i 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
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 672.2.bi.b 4
3.b odd 2 1 672.2.bi.a 4
4.b odd 2 1 168.2.ba.a 4
7.d odd 6 1 inner 672.2.bi.b 4
8.b even 2 1 672.2.bi.a 4
8.d odd 2 1 168.2.ba.b yes 4
12.b even 2 1 168.2.ba.b yes 4
21.g even 6 1 672.2.bi.a 4
24.f even 2 1 168.2.ba.a 4
24.h odd 2 1 CM 672.2.bi.b 4
28.f even 6 1 168.2.ba.a 4
56.j odd 6 1 672.2.bi.a 4
56.m even 6 1 168.2.ba.b yes 4
84.j odd 6 1 168.2.ba.b yes 4
168.ba even 6 1 inner 672.2.bi.b 4
168.be odd 6 1 168.2.ba.a 4

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

## 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 $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

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