# Properties

 Label 912.2.ci.a Level $912$ Weight $2$ Character orbit 912.ci Analytic conductor $7.282$ Analytic rank $1$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$7.28235666434$$ Analytic rank: $$1$$ 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 Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{18}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{18}^{2} q^{3} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( -2 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{7} + \zeta_{18}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{18}^{2} q^{3} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{5} + ( -2 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{7} + \zeta_{18}^{4} q^{9} + ( -1 - 2 \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{11} + ( -\zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{13} + ( -1 + \zeta_{18} - \zeta_{18}^{2} ) q^{15} + ( -3 + \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{17} + ( -4 + \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{19} + ( 1 - 2 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{21} + ( -2 - \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{23} + ( -1 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{25} + ( -1 + \zeta_{18}^{3} ) q^{27} + ( 2 + 6 \zeta_{18} + 2 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 3 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{29} + ( 3 \zeta_{18} - 5 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{31} + ( -1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{33} + ( 1 + 2 \zeta_{18} - \zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{35} + ( 4 - \zeta_{18} + \zeta_{18}^{2} - 8 \zeta_{18}^{3} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{37} + ( 1 - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{39} + ( -1 + 2 \zeta_{18} + 4 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 8 \zeta_{18}^{5} ) q^{41} + ( -2 + 2 \zeta_{18} - \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{43} + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{45} + ( -7 + 2 \zeta_{18} - 7 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{47} + ( -2 + 2 \zeta_{18} - 5 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{49} + ( -2 + \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{51} + ( -1 + 2 \zeta_{18} + 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{53} + ( 3 \zeta_{18} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} ) q^{55} + ( 2 - 2 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{57} + ( -3 + \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 11 \zeta_{18}^{5} ) q^{59} + ( 4 - \zeta_{18} + 5 \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{5} ) q^{61} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{63} + ( 1 + \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{65} + ( -6 + 6 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{67} + ( -3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{69} + ( -4 - 4 \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{71} + ( -3 \zeta_{18} - 8 \zeta_{18}^{2} - 3 \zeta_{18}^{3} ) q^{73} + ( -2 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{75} + ( 3 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} ) q^{77} + ( 4 - 2 \zeta_{18} - 7 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{79} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{81} + ( 2 + 3 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{83} + ( 4 - 4 \zeta_{18}^{2} + \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{85} + ( 3 - 2 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{87} + ( 4 \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{89} + ( 1 + \zeta_{18} + 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{91} + ( 5 - 3 \zeta_{18} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{93} + ( 7 - \zeta_{18} + \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{95} + ( -2 + \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{97} + ( 2 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{5} - 9 q^{7} + O(q^{10})$$ $$6 q - 6 q^{5} - 9 q^{7} - 9 q^{11} - 6 q^{13} - 6 q^{15} - 12 q^{17} - 18 q^{19} + 3 q^{21} - 3 q^{23} - 3 q^{27} - 6 q^{31} - 9 q^{33} - 12 q^{41} + 3 q^{45} - 39 q^{47} - 6 q^{49} - 3 q^{51} - 12 q^{53} + 9 q^{55} + 9 q^{57} - 12 q^{59} + 27 q^{61} - 3 q^{63} + 9 q^{65} - 36 q^{67} - 18 q^{71} - 9 q^{73} - 12 q^{75} + 18 q^{79} + 9 q^{83} + 27 q^{85} + 27 q^{87} + 3 q^{89} + 12 q^{91} + 24 q^{93} + 24 q^{95} - 3 q^{97} + 9 q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
79.1
 −0.173648 − 0.984808i −0.173648 + 0.984808i −0.766044 + 0.642788i −0.766044 − 0.642788i 0.939693 − 0.342020i 0.939693 + 0.342020i
0 −0.939693 + 0.342020i 0 −0.233956 + 1.32683i 0 −3.20574 + 1.85083i 0 0.766044 0.642788i 0
127.1 0 −0.939693 0.342020i 0 −0.233956 1.32683i 0 −3.20574 1.85083i 0 0.766044 + 0.642788i 0
223.1 0 0.173648 0.984808i 0 −1.93969 1.62760i 0 −0.386659 + 0.223238i 0 −0.939693 0.342020i 0
319.1 0 0.173648 + 0.984808i 0 −1.93969 + 1.62760i 0 −0.386659 0.223238i 0 −0.939693 + 0.342020i 0
751.1 0 0.766044 0.642788i 0 −0.826352 0.300767i 0 −0.907604 0.524005i 0 0.173648 0.984808i 0
895.1 0 0.766044 + 0.642788i 0 −0.826352 + 0.300767i 0 −0.907604 + 0.524005i 0 0.173648 + 0.984808i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 895.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.k even 18 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.2.ci.a 6
4.b odd 2 1 912.2.ci.b yes 6
19.f odd 18 1 912.2.ci.b yes 6
76.k even 18 1 inner 912.2.ci.a 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
912.2.ci.a 6 1.a even 1 1 trivial
912.2.ci.a 6 76.k even 18 1 inner
912.2.ci.b yes 6 4.b odd 2 1
912.2.ci.b yes 6 19.f odd 18 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(912, [\chi])$$:

 $$T_{5}^{6} + 6 T_{5}^{5} + 18 T_{5}^{4} + 30 T_{5}^{3} + 36 T_{5}^{2} + 27 T_{5} + 9$$ $$T_{7}^{6} + 9 T_{7}^{5} + 33 T_{7}^{4} + 54 T_{7}^{3} + 45 T_{7}^{2} + 18 T_{7} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$1 + T^{3} + T^{6}$$
$5$ $$9 + 27 T + 36 T^{2} + 30 T^{3} + 18 T^{4} + 6 T^{5} + T^{6}$$
$7$ $$3 + 18 T + 45 T^{2} + 54 T^{3} + 33 T^{4} + 9 T^{5} + T^{6}$$
$11$ $$243 - 81 T^{2} + 27 T^{4} + 9 T^{5} + T^{6}$$
$13$ $$3 - 18 T + 36 T^{2} - 33 T^{3} + 24 T^{4} + 6 T^{5} + T^{6}$$
$17$ $$9 - 81 T + 306 T^{2} + 132 T^{3} + 54 T^{4} + 12 T^{5} + T^{6}$$
$19$ $$6859 + 6498 T + 2736 T^{2} + 737 T^{3} + 144 T^{4} + 18 T^{5} + T^{6}$$
$23$ $$1083 + 855 T + 99 T^{2} + 6 T^{3} + 24 T^{4} + 3 T^{5} + T^{6}$$
$29$ $$7803 - 5508 T + 5238 T^{2} - 819 T^{3} - 18 T^{4} + T^{6}$$
$31$ $$72361 + 12105 T + 3639 T^{2} + 268 T^{3} + 81 T^{4} + 6 T^{5} + T^{6}$$
$37$ $$70227 + 7209 T^{2} + 162 T^{4} + T^{6}$$
$41$ $$34347 - 10593 T + 6462 T^{2} + 33 T^{3} - 39 T^{4} + 12 T^{5} + T^{6}$$
$43$ $$9747 - 6156 T + 1242 T^{2} - 153 T^{3} + 18 T^{4} + T^{6}$$
$47$ $$604803 + 246501 T + 54810 T^{2} + 7674 T^{3} + 699 T^{4} + 39 T^{5} + T^{6}$$
$53$ $$4107 + 3663 T + 1890 T^{2} + 303 T^{3} + 51 T^{4} + 12 T^{5} + T^{6}$$
$59$ $$1172889 - 185193 T - 6498 T^{2} + 456 T^{3} + 234 T^{4} + 12 T^{5} + T^{6}$$
$61$ $$54289 - 29358 T + 9891 T^{2} - 2231 T^{3} + 324 T^{4} - 27 T^{5} + T^{6}$$
$67$ $$179776 + 76320 T + 24480 T^{2} + 4960 T^{3} + 576 T^{4} + 36 T^{5} + T^{6}$$
$71$ $$23409 + 19278 T + 6966 T^{2} + 1449 T^{3} + 198 T^{4} + 18 T^{5} + T^{6}$$
$73$ $$104329 - 5814 T + 6876 T^{2} - 1376 T^{3} - 18 T^{4} + 9 T^{5} + T^{6}$$
$79$ $$26569 + 8802 T + 4986 T^{2} + 701 T^{3} + 18 T^{4} - 18 T^{5} + T^{6}$$
$83$ $$3 + 36 T + 153 T^{2} + 108 T^{3} + 15 T^{4} - 9 T^{5} + T^{6}$$
$89$ $$98283 + 4887 T + 333 T^{2} + 642 T^{3} + 6 T^{4} - 3 T^{5} + T^{6}$$
$97$ $$867 - 918 T + 990 T^{2} - 318 T^{3} + 6 T^{4} + 3 T^{5} + T^{6}$$