# Properties

 Label 912.1.ce.b Level $912$ Weight $1$ Character orbit 912.ce Analytic conductor $0.455$ Analytic rank $0$ Dimension $6$ Projective image $D_{18}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$912 = 2^{4} \cdot 3 \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 912.ce (of order $$18$$, degree $$6$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.455147291521$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{8} q^{3} + ( -\zeta_{18}^{2} + \zeta_{18}^{4} ) q^{7} -\zeta_{18}^{7} q^{9} +O(q^{10})$$ $$q + \zeta_{18}^{8} q^{3} + ( -\zeta_{18}^{2} + \zeta_{18}^{4} ) q^{7} -\zeta_{18}^{7} q^{9} + ( \zeta_{18}^{5} + \zeta_{18}^{6} ) q^{13} + \zeta_{18}^{3} q^{19} + ( \zeta_{18} - \zeta_{18}^{3} ) q^{21} -\zeta_{18} q^{25} + \zeta_{18}^{6} q^{27} + ( \zeta_{18} + \zeta_{18}^{5} ) q^{31} + ( \zeta_{18}^{2} + \zeta_{18}^{7} ) q^{37} + ( -\zeta_{18}^{4} - \zeta_{18}^{5} ) q^{39} + ( -\zeta_{18}^{3} + \zeta_{18}^{7} ) q^{43} + ( \zeta_{18}^{4} - \zeta_{18}^{6} + \zeta_{18}^{8} ) q^{49} -\zeta_{18}^{2} q^{57} + ( -1 - \zeta_{18}^{8} ) q^{61} + ( -1 + \zeta_{18}^{2} ) q^{63} + ( -\zeta_{18}^{6} - \zeta_{18}^{8} ) q^{67} + ( \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{73} + q^{75} + ( 1 - \zeta_{18}^{7} ) q^{79} -\zeta_{18}^{5} q^{81} + ( -1 - \zeta_{18} - \zeta_{18}^{7} - \zeta_{18}^{8} ) q^{91} + ( -1 - \zeta_{18}^{4} ) q^{93} + ( -\zeta_{18}^{5} - \zeta_{18}^{8} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + O(q^{10})$$ $$6q - 3q^{13} + 3q^{19} - 3q^{21} - 3q^{27} - 3q^{43} + 3q^{49} - 6q^{61} - 6q^{63} + 3q^{67} + 3q^{73} + 6q^{75} + 6q^{79} - 6q^{91} - 6q^{93} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/912\mathbb{Z}\right)^\times$$.

 $$n$$ $$97$$ $$229$$ $$305$$ $$799$$ $$\chi(n)$$ $$-\zeta_{18}^{4}$$ $$1$$ $$-1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
143.1
 −0.766044 − 0.642788i −0.766044 + 0.642788i 0.939693 + 0.342020i 0.939693 − 0.342020i −0.173648 − 0.984808i −0.173648 + 0.984808i
0 0.766044 0.642788i 0 0 0 −1.11334 0.642788i 0 0.173648 0.984808i 0
287.1 0 0.766044 + 0.642788i 0 0 0 −1.11334 + 0.642788i 0 0.173648 + 0.984808i 0
383.1 0 −0.939693 + 0.342020i 0 0 0 −0.592396 + 0.342020i 0 0.766044 0.642788i 0
431.1 0 −0.939693 0.342020i 0 0 0 −0.592396 0.342020i 0 0.766044 + 0.642788i 0
527.1 0 0.173648 0.984808i 0 0 0 1.70574 0.984808i 0 −0.939693 0.342020i 0
623.1 0 0.173648 + 0.984808i 0 0 0 1.70574 + 0.984808i 0 −0.939693 + 0.342020i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 623.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
76.k even 18 1 inner
228.u odd 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.1.ce.b yes 6
3.b odd 2 1 CM 912.1.ce.b yes 6
4.b odd 2 1 912.1.ce.a 6
8.b even 2 1 3648.1.cu.a 6
8.d odd 2 1 3648.1.cu.b 6
12.b even 2 1 912.1.ce.a 6
19.f odd 18 1 912.1.ce.a 6
24.f even 2 1 3648.1.cu.b 6
24.h odd 2 1 3648.1.cu.a 6
57.j even 18 1 912.1.ce.a 6
76.k even 18 1 inner 912.1.ce.b yes 6
152.s odd 18 1 3648.1.cu.b 6
152.v even 18 1 3648.1.cu.a 6
228.u odd 18 1 inner 912.1.ce.b yes 6
456.bj even 18 1 3648.1.cu.b 6
456.bt odd 18 1 3648.1.cu.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.1.ce.a 6 4.b odd 2 1
912.1.ce.a 6 12.b even 2 1
912.1.ce.a 6 19.f odd 18 1
912.1.ce.a 6 57.j even 18 1
912.1.ce.b yes 6 1.a even 1 1 trivial
912.1.ce.b yes 6 3.b odd 2 1 CM
912.1.ce.b yes 6 76.k even 18 1 inner
912.1.ce.b yes 6 228.u odd 18 1 inner
3648.1.cu.a 6 8.b even 2 1
3648.1.cu.a 6 24.h odd 2 1
3648.1.cu.a 6 152.v even 18 1
3648.1.cu.a 6 456.bt odd 18 1
3648.1.cu.b 6 8.d odd 2 1
3648.1.cu.b 6 24.f even 2 1
3648.1.cu.b 6 152.s odd 18 1
3648.1.cu.b 6 456.bj even 18 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{6} - 3 T_{7}^{4} + 9 T_{7}^{2} + 9 T_{7} + 3$$ acting on $$S_{1}^{\mathrm{new}}(912, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 + T^{3} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6}$$
$11$ $$T^{6}$$
$13$ $$3 + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$17$ $$T^{6}$$
$19$ $$( 1 - T + T^{2} )^{3}$$
$23$ $$T^{6}$$
$29$ $$T^{6}$$
$31$ $$1 - 3 T + 9 T^{2} - 2 T^{3} + 3 T^{4} + T^{6}$$
$37$ $$3 + 9 T^{2} + 6 T^{4} + T^{6}$$
$41$ $$T^{6}$$
$43$ $$3 + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$
$47$ $$T^{6}$$
$53$ $$T^{6}$$
$59$ $$T^{6}$$
$61$ $$1 + 3 T + 12 T^{2} + 19 T^{3} + 15 T^{4} + 6 T^{5} + T^{6}$$
$67$ $$1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$1 - 6 T + 12 T^{2} - 8 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$79$ $$1 - 3 T + 12 T^{2} - 19 T^{3} + 15 T^{4} - 6 T^{5} + T^{6}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$27 + 9 T^{3} + T^{6}$$