# Properties

 Label 672.1.ba.b Level $672$ Weight $1$ Character orbit 672.ba Analytic conductor $0.335$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -24 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$672 = 2^{5} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 672.ba (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.335371688489$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: no (minimal twist has level 168) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1176.1 Artin image: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} - \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ q + z * q^3 - z^2 * q^5 + z * q^7 + z^2 * q^9 $$q + \zeta_{6} q^{3} - \zeta_{6}^{2} q^{5} + \zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} - \zeta_{6} q^{11} + q^{15} + \zeta_{6}^{2} q^{21} - q^{27} - q^{29} - \zeta_{6} q^{31} - \zeta_{6}^{2} q^{33} + q^{35} + \zeta_{6} q^{45} + \zeta_{6}^{2} q^{49} + \zeta_{6} q^{53} - q^{55} - \zeta_{6} q^{59} - q^{63} - \zeta_{6} q^{73} - \zeta_{6}^{2} q^{77} + \zeta_{6}^{2} q^{79} - \zeta_{6} q^{81} + q^{83} - \zeta_{6} q^{87} - \zeta_{6}^{2} q^{93} - q^{97} + q^{99} +O(q^{100})$$ q + z * q^3 - z^2 * q^5 + z * q^7 + z^2 * q^9 - z * q^11 + q^15 + z^2 * q^21 - q^27 - q^29 - z * q^31 - z^2 * q^33 + q^35 + z * q^45 + z^2 * q^49 + z * q^53 - q^55 - z * q^59 - q^63 - z * q^73 - z^2 * q^77 + z^2 * q^79 - z * q^81 + q^83 - z * q^87 - z^2 * q^93 - q^97 + 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 $$2 q + q^{3} + q^{5} + q^{7} - q^{9} - q^{11} + 2 q^{15} - q^{21} - 2 q^{27} - 2 q^{29} - q^{31} + q^{33} + 2 q^{35} + q^{45} - q^{49} + q^{53} - 2 q^{55} - q^{59} - 2 q^{63} - 2 q^{73} + q^{77} - q^{79} - q^{81} + 2 q^{83} - q^{87} + q^{93} - 2 q^{97} + 2 q^{99}+O(q^{100})$$ 2 * q + q^3 + q^5 + q^7 - q^9 - q^11 + 2 * q^15 - q^21 - 2 * q^27 - 2 * q^29 - q^31 + q^33 + 2 * q^35 + q^45 - q^49 + q^53 - 2 * q^55 - q^59 - 2 * q^63 - 2 * q^73 + q^77 - q^79 - q^81 + 2 * q^83 - q^87 + q^93 - 2 * q^97 + 2 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/672\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$421$$ $$449$$ $$577$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$ $$\zeta_{6}^{2}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
305.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 −0.500000 0.866025i 0
401.1 0 0.500000 + 0.866025i 0 0.500000 0.866025i 0 0.500000 + 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
24.h odd 2 1 CM by $$\Q(\sqrt{-6})$$
7.c even 3 1 inner
168.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 672.1.ba.b 2
3.b odd 2 1 672.1.ba.a 2
4.b odd 2 1 168.1.s.a 2
7.c even 3 1 inner 672.1.ba.b 2
8.b even 2 1 672.1.ba.a 2
8.d odd 2 1 168.1.s.b yes 2
12.b even 2 1 168.1.s.b yes 2
21.h odd 6 1 672.1.ba.a 2
24.f even 2 1 168.1.s.a 2
24.h odd 2 1 CM 672.1.ba.b 2
28.d even 2 1 1176.1.s.a 2
28.f even 6 1 1176.1.n.c 1
28.f even 6 1 1176.1.s.a 2
28.g odd 6 1 168.1.s.a 2
28.g odd 6 1 1176.1.n.d 1
56.e even 2 1 1176.1.s.b 2
56.k odd 6 1 168.1.s.b yes 2
56.k odd 6 1 1176.1.n.a 1
56.m even 6 1 1176.1.n.b 1
56.m even 6 1 1176.1.s.b 2
56.p even 6 1 672.1.ba.a 2
84.h odd 2 1 1176.1.s.b 2
84.j odd 6 1 1176.1.n.b 1
84.j odd 6 1 1176.1.s.b 2
84.n even 6 1 168.1.s.b yes 2
84.n even 6 1 1176.1.n.a 1
168.e odd 2 1 1176.1.s.a 2
168.s odd 6 1 inner 672.1.ba.b 2
168.v even 6 1 168.1.s.a 2
168.v even 6 1 1176.1.n.d 1
168.be odd 6 1 1176.1.n.c 1
168.be odd 6 1 1176.1.s.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a 2 4.b odd 2 1
168.1.s.a 2 24.f even 2 1
168.1.s.a 2 28.g odd 6 1
168.1.s.a 2 168.v even 6 1
168.1.s.b yes 2 8.d odd 2 1
168.1.s.b yes 2 12.b even 2 1
168.1.s.b yes 2 56.k odd 6 1
168.1.s.b yes 2 84.n even 6 1
672.1.ba.a 2 3.b odd 2 1
672.1.ba.a 2 8.b even 2 1
672.1.ba.a 2 21.h odd 6 1
672.1.ba.a 2 56.p even 6 1
672.1.ba.b 2 1.a even 1 1 trivial
672.1.ba.b 2 7.c even 3 1 inner
672.1.ba.b 2 24.h odd 2 1 CM
672.1.ba.b 2 168.s odd 6 1 inner
1176.1.n.a 1 56.k odd 6 1
1176.1.n.a 1 84.n even 6 1
1176.1.n.b 1 56.m even 6 1
1176.1.n.b 1 84.j odd 6 1
1176.1.n.c 1 28.f even 6 1
1176.1.n.c 1 168.be odd 6 1
1176.1.n.d 1 28.g odd 6 1
1176.1.n.d 1 168.v even 6 1
1176.1.s.a 2 28.d even 2 1
1176.1.s.a 2 28.f even 6 1
1176.1.s.a 2 168.e odd 2 1
1176.1.s.a 2 168.be odd 6 1
1176.1.s.b 2 56.e even 2 1
1176.1.s.b 2 56.m even 6 1
1176.1.s.b 2 84.h odd 2 1
1176.1.s.b 2 84.j 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 $$S_{1}^{\mathrm{new}}(672, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + T + 1$$
$13$ $$T^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2} + T + 1$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - T + 1$$
$59$ $$T^{2} + T + 1$$
$61$ $$T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 2T + 4$$
$79$ $$T^{2} + T + 1$$
$83$ $$(T - 1)^{2}$$
$89$ $$T^{2}$$
$97$ $$(T + 1)^{2}$$