# Properties

 Label 3267.1.q.a Level $3267$ Weight $1$ Character orbit 3267.q Analytic conductor $1.630$ Analytic rank $0$ Dimension $6$ Projective image $D_{18}$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3267.q (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.63044539627$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{7} q^{3} + \zeta_{18}^{4} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{5}) q^{5} - \zeta_{18}^{5} q^{9} +O(q^{10})$$ q + z^7 * q^3 + z^4 * q^4 + (z^6 + z^5) * q^5 - z^5 * q^9 $$q + \zeta_{18}^{7} q^{3} + \zeta_{18}^{4} q^{4} + (\zeta_{18}^{6} + \zeta_{18}^{5}) q^{5} - \zeta_{18}^{5} q^{9} - \zeta_{18}^{2} q^{12} + ( - \zeta_{18}^{4} - \zeta_{18}^{3}) q^{15} + \zeta_{18}^{8} q^{16} + ( - \zeta_{18} - 1) q^{20} + (\zeta_{18}^{7} - \zeta_{18}) q^{23} + ( - \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18}) q^{25} + \zeta_{18}^{3} q^{27} + ( - \zeta_{18}^{6} - \zeta_{18}^{2}) q^{31} + q^{36} + ( - \zeta_{18}^{8} + \zeta_{18}^{7}) q^{37} + (\zeta_{18}^{2} + \zeta_{18}) q^{45} + ( - \zeta_{18}^{6} + \zeta_{18}^{4}) q^{47} - \zeta_{18}^{6} q^{48} + \zeta_{18} q^{49} + (\zeta_{18}^{7} + \zeta_{18}^{2}) q^{53} + (\zeta_{18}^{8} + \zeta_{18}^{3}) q^{59} + ( - \zeta_{18}^{8} - \zeta_{18}^{7}) q^{60} - \zeta_{18}^{3} q^{64} + (\zeta_{18}^{5} - 1) q^{67} + ( - \zeta_{18}^{8} - \zeta_{18}^{5}) q^{69} + ( - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{71} + ( - \zeta_{18}^{8} + \zeta_{18} + 1) q^{75} + ( - \zeta_{18}^{5} - \zeta_{18}^{4}) q^{80} - \zeta_{18} q^{81} + ( - \zeta_{18}^{3} - 1) q^{89} + ( - \zeta_{18}^{5} - \zeta_{18}^{2}) q^{92} + (\zeta_{18}^{4} + 1) q^{93} + ( - \zeta_{18}^{7} + 1) q^{97} +O(q^{100})$$ q + z^7 * q^3 + z^4 * q^4 + (z^6 + z^5) * q^5 - z^5 * q^9 - z^2 * q^12 + (-z^4 - z^3) * q^15 + z^8 * q^16 + (-z - 1) * q^20 + (z^7 - z) * q^23 + (-z^3 - z^2 - z) * q^25 + z^3 * q^27 + (-z^6 - z^2) * q^31 + q^36 + (-z^8 + z^7) * q^37 + (z^2 + z) * q^45 + (-z^6 + z^4) * q^47 - z^6 * q^48 + z * q^49 + (z^7 + z^2) * q^53 + (z^8 + z^3) * q^59 + (-z^8 - z^7) * q^60 - z^3 * q^64 + (z^5 - 1) * q^67 + (-z^8 - z^5) * q^69 + (-z^4 + z^2) * q^71 + (-z^8 + z + 1) * q^75 + (-z^5 - z^4) * q^80 - z * q^81 + (-z^3 - 1) * q^89 + (-z^5 - z^2) * q^92 + (z^4 + 1) * q^93 + (-z^7 + 1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 3 q^{5}+O(q^{10})$$ 6 * q - 3 * q^5 $$6 q - 3 q^{5} - 3 q^{15} - 6 q^{20} - 3 q^{25} + 3 q^{27} + 3 q^{31} + 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} - 3 q^{64} - 6 q^{67} + 6 q^{75} - 9 q^{89} + 6 q^{93} + 6 q^{97}+O(q^{100})$$ 6 * q - 3 * q^5 - 3 * q^15 - 6 * q^20 - 3 * q^25 + 3 * q^27 + 3 * q^31 + 6 * q^36 + 3 * q^47 + 3 * q^48 + 3 * q^59 - 3 * q^64 - 6 * q^67 + 6 * q^75 - 9 * q^89 + 6 * q^93 + 6 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$3026$$ $$\chi(n)$$ $$1$$ $$-\zeta_{18}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
122.1
 −0.766044 − 0.642788i 0.939693 + 0.342020i −0.173648 + 0.984808i −0.173648 − 0.984808i 0.939693 − 0.342020i −0.766044 + 0.642788i
0 −0.173648 + 0.984808i −0.939693 + 0.342020i 0.439693 0.524005i 0 0 0 −0.939693 0.342020i 0
848.1 0 −0.766044 + 0.642788i 0.173648 + 0.984808i −0.673648 + 1.85083i 0 0 0 0.173648 0.984808i 0
1211.1 0 0.939693 0.342020i 0.766044 + 0.642788i −1.26604 0.223238i 0 0 0 0.766044 0.642788i 0
1937.1 0 0.939693 + 0.342020i 0.766044 0.642788i −1.26604 + 0.223238i 0 0 0 0.766044 + 0.642788i 0
2300.1 0 −0.766044 0.642788i 0.173648 0.984808i −0.673648 1.85083i 0 0 0 0.173648 + 0.984808i 0
3026.1 0 −0.173648 0.984808i −0.939693 0.342020i 0.439693 + 0.524005i 0 0 0 −0.939693 + 0.342020i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3026.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
27.f odd 18 1 inner
297.o even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.q.a 6
11.b odd 2 1 CM 3267.1.q.a 6
11.c even 5 4 3267.1.be.a 24
11.d odd 10 4 3267.1.be.a 24
27.f odd 18 1 inner 3267.1.q.a 6
297.o even 18 1 inner 3267.1.q.a 6
297.v odd 90 4 3267.1.be.a 24
297.x even 90 4 3267.1.be.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.q.a 6 1.a even 1 1 trivial
3267.1.q.a 6 11.b odd 2 1 CM
3267.1.q.a 6 27.f odd 18 1 inner
3267.1.q.a 6 297.o even 18 1 inner
3267.1.be.a 24 11.c even 5 4
3267.1.be.a 24 11.d odd 10 4
3267.1.be.a 24 297.v odd 90 4
3267.1.be.a 24 297.x even 90 4

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3267, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} - T^{3} + 1$$
$5$ $$T^{6} + 3 T^{5} + 6 T^{4} + 6 T^{3} + \cdots + 3$$
$7$ $$T^{6}$$
$11$ $$T^{6}$$
$13$ $$T^{6}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$T^{6} + 9T^{3} + 27$$
$29$ $$T^{6}$$
$31$ $$T^{6} - 3 T^{5} + 6 T^{4} - 8 T^{3} + \cdots + 1$$
$37$ $$T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1$$
$41$ $$T^{6}$$
$43$ $$T^{6}$$
$47$ $$T^{6} - 3 T^{5} + 6 T^{4} - 6 T^{3} + \cdots + 3$$
$53$ $$T^{6} + 6 T^{4} + 9 T^{2} + 3$$
$59$ $$T^{6} - 3 T^{5} + 6 T^{4} - 6 T^{3} + \cdots + 3$$
$61$ $$T^{6}$$
$67$ $$T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1$$
$71$ $$T^{6} - 3 T^{4} + 9 T^{2} - 9 T + 3$$
$73$ $$T^{6}$$
$79$ $$T^{6}$$
$83$ $$T^{6}$$
$89$ $$(T^{2} + 3 T + 3)^{3}$$
$97$ $$T^{6} - 6 T^{5} + 15 T^{4} - 19 T^{3} + \cdots + 1$$