# Properties

 Label 672.2.bi.a 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 $$q + (\beta_{2} - 1) q^{3} + ( - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{5} + ( - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{7} - 3 \beta_{2} q^{9} + ( - 3 \beta_{2} - 2 \beta_1 - 3) q^{11} + (3 \beta_{3} + 3) q^{15} + (3 \beta_{3} - \beta_{2} + 3 \beta_1 - 2) q^{21} + (4 \beta_{2} - 6 \beta_1 + 4) q^{25} + (6 \beta_{2} + 3) q^{27} + (\beta_{3} + 9) q^{29} + ( - 2 \beta_{3} - 5 \beta_{2} - \beta_1 + 5) q^{31} + ( - 2 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 6) q^{33} + (4 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 5) q^{35} + ( - 6 \beta_{3} + 3 \beta_{2} - 3 \beta_1 - 3) q^{45} + ( - 2 \beta_{3} - 5 \beta_{2} + 2 \beta_1) q^{49} + (3 \beta_{2} - 5 \beta_1 + 3) q^{53} + ( - \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 1) q^{55} + ( - 8 \beta_{3} - 3 \beta_{2} - 4 \beta_1 + 3) q^{59} + ( - 3 \beta_{3} - 6 \beta_1 + 3) q^{63} + (8 \beta_{3} + 4 \beta_1) q^{73} + ( - 6 \beta_{3} - 4 \beta_{2} + 6 \beta_1 - 8) q^{75} + (\beta_{3} - 7 \beta_{2} - 5 \beta_1 - 8) q^{77} + (9 \beta_{3} + 5 \beta_{2} + 9 \beta_1) q^{79} + ( - 9 \beta_{2} - 9) q^{81} + (2 \beta_{3} + 10 \beta_{2} + 4 \beta_1 + 5) q^{83} + ( - 2 \beta_{3} + 9 \beta_{2} - \beta_1 - 9) q^{87} + (3 \beta_{3} + 15 \beta_{2} + 3 \beta_1) q^{93} + ( - 4 \beta_{3} + 2 \beta_{2} - 8 \beta_1 + 1) q^{97} + (6 \beta_{3} - 9) q^{99}+O(q^{100})$$ 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 −3.62132 + 2.09077i 0 −1.62132 2.09077i 0 1.50000 + 2.59808i 0
17.2 0 −1.50000 0.866025i 0 0.621320 0.358719i 0 2.62132 + 0.358719i 0 1.50000 + 2.59808i 0
593.1 0 −1.50000 + 0.866025i 0 −3.62132 2.09077i 0 −1.62132 + 2.09077i 0 1.50000 2.59808i 0
593.2 0 −1.50000 + 0.866025i 0 0.621320 + 0.358719i 0 2.62132 0.358719i 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.a 4
3.b odd 2 1 672.2.bi.b 4
4.b odd 2 1 168.2.ba.b yes 4
7.d odd 6 1 inner 672.2.bi.a 4
8.b even 2 1 672.2.bi.b 4
8.d odd 2 1 168.2.ba.a 4
12.b even 2 1 168.2.ba.a 4
21.g even 6 1 672.2.bi.b 4
24.f even 2 1 168.2.ba.b yes 4
24.h odd 2 1 CM 672.2.bi.a 4
28.f even 6 1 168.2.ba.b yes 4
56.j odd 6 1 672.2.bi.b 4
56.m even 6 1 168.2.ba.a 4
84.j odd 6 1 168.2.ba.a 4
168.ba even 6 1 inner 672.2.bi.a 4
168.be odd 6 1 168.2.ba.b yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.ba.a 4 8.d odd 2 1
168.2.ba.a 4 12.b even 2 1
168.2.ba.a 4 56.m even 6 1
168.2.ba.a 4 84.j odd 6 1
168.2.ba.b yes 4 4.b odd 2 1
168.2.ba.b yes 4 24.f even 2 1
168.2.ba.b yes 4 28.f even 6 1
168.2.ba.b yes 4 168.be odd 6 1
672.2.bi.a 4 1.a even 1 1 trivial
672.2.bi.a 4 7.d odd 6 1 inner
672.2.bi.a 4 24.h odd 2 1 CM
672.2.bi.a 4 168.ba even 6 1 inner
672.2.bi.b 4 3.b odd 2 1
672.2.bi.b 4 8.b even 2 1
672.2.bi.b 4 21.g even 6 1
672.2.bi.b 4 56.j 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 $$S_{2}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

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