# Properties

 Label 1323.2.f.d Level $1323$ Weight $2$ Character orbit 1323.f Analytic conductor $10.564$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ Defining polynomial: $$x^{6} - x^{3} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{5} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{2} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{4} + ( -1 - \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} ) q^{5} + ( -2 - \zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{8} + ( \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{10} + ( \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{11} + ( -1 + 2 \zeta_{18} - 4 \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{13} + ( -3 \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{16} + ( 2 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{17} + ( 1 + \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{20} + ( -3 + 4 \zeta_{18} - 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{22} + ( 4 + \zeta_{18} + 2 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{23} + ( 2 \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{25} + ( -1 + \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{26} + ( -4 \zeta_{18} + 5 \zeta_{18}^{2} + 3 \zeta_{18}^{3} + 5 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{29} + ( -1 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 6 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{31} + ( -3 \zeta_{18} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{32} + ( -\zeta_{18} - \zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{34} + ( -1 + 6 \zeta_{18} + 6 \zeta_{18}^{2} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{37} + ( -3 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{38} + ( -3 + 2 \zeta_{18} + 2 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{40} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{41} + ( \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{43} + ( -5 + 3 \zeta_{18} + 3 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{44} + ( -4 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{46} + ( -3 \zeta_{18} - 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{47} + ( -2 + 5 \zeta_{18} - 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{50} + ( -5 \zeta_{18} + 10 \zeta_{18}^{2} - 7 \zeta_{18}^{3} + 10 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{52} + ( -2 + \zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{53} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{55} + ( 3 - 6 \zeta_{18} + 9 \zeta_{18}^{2} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{58} + ( 1 + 5 \zeta_{18} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{59} + ( 3 \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 3 \zeta_{18}^{4} ) q^{61} + ( -10 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{62} + ( 4 + 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{64} + ( -5 \zeta_{18} - \zeta_{18}^{2} + 5 \zeta_{18}^{3} - \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{65} + ( 4 - 3 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{67} + ( -2 + 3 \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{68} + ( -3 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 6 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{71} + ( 7 - 5 \zeta_{18} - 5 \zeta_{18}^{2} + \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{73} + ( \zeta_{18}^{2} - 10 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{74} + ( 5 - 3 \zeta_{18} + 3 \zeta_{18}^{2} - 5 \zeta_{18}^{3} ) q^{76} + ( -3 \zeta_{18} + 7 \zeta_{18}^{3} - 3 \zeta_{18}^{5} ) q^{79} + ( -5 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{80} + 3 q^{82} + ( \zeta_{18} + 4 \zeta_{18}^{2} + 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{83} + ( -3 - 2 \zeta_{18} + 4 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{85} + ( 2 + \zeta_{18} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{86} + ( -7 \zeta_{18} + 8 \zeta_{18}^{2} - 9 \zeta_{18}^{3} + 8 \zeta_{18}^{4} - 7 \zeta_{18}^{5} ) q^{88} + ( 4 + 4 \zeta_{18} + 4 \zeta_{18}^{2} - 7 \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{89} + ( 2 \zeta_{18} - 8 \zeta_{18}^{2} + \zeta_{18}^{3} - 8 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{92} + ( -6 - 2 \zeta_{18} + \zeta_{18}^{2} + 6 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{94} + ( -4 - \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} ) q^{95} + ( 8 \zeta_{18} - 7 \zeta_{18}^{2} - \zeta_{18}^{3} - 7 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + 3q^{2} - 3q^{4} - 3q^{5} - 12q^{8} + O(q^{10})$$ $$6q + 3q^{2} - 3q^{4} - 3q^{5} - 12q^{8} + 6q^{11} - 3q^{13} - 3q^{16} + 12q^{17} + 6q^{19} + 6q^{20} - 9q^{22} + 12q^{23} + 6q^{25} - 6q^{26} + 9q^{29} - 3q^{31} + 9q^{34} - 6q^{37} - 6q^{38} - 9q^{40} + 3q^{43} - 30q^{44} - 3q^{47} - 6q^{50} - 21q^{52} - 12q^{53} + 9q^{58} + 3q^{59} + 6q^{61} - 60q^{62} + 24q^{64} + 15q^{65} + 12q^{67} - 6q^{68} - 18q^{71} + 42q^{73} - 30q^{74} + 15q^{76} + 21q^{79} - 30q^{80} + 18q^{82} + 18q^{83} - 9q^{85} + 6q^{86} - 27q^{88} + 24q^{89} + 3q^{92} - 18q^{94} - 12q^{95} - 3q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1 + \zeta_{18}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
442.1
 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 + 0.642788i 0.939693 − 0.342020i −0.173648 + 0.984808i −0.766044 − 0.642788i
−0.439693 0.761570i 0 0.613341 1.06234i −0.673648 + 1.16679i 0 0 −2.83750 0 1.18479
442.2 0.673648 + 1.16679i 0 0.0923963 0.160035i −1.26604 + 2.19285i 0 0 2.94356 0 −3.41147
442.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i 0.439693 0.761570i 0 0 −6.10607 0 2.22668
883.1 −0.439693 + 0.761570i 0 0.613341 + 1.06234i −0.673648 1.16679i 0 0 −2.83750 0 1.18479
883.2 0.673648 1.16679i 0 0.0923963 + 0.160035i −1.26604 2.19285i 0 0 2.94356 0 −3.41147
883.3 1.26604 2.19285i 0 −2.20574 3.82045i 0.439693 + 0.761570i 0 0 −6.10607 0 2.22668
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 883.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.2.f.d 6
3.b odd 2 1 441.2.f.c 6
7.b odd 2 1 189.2.f.b 6
7.c even 3 1 1323.2.g.e 6
7.c even 3 1 1323.2.h.b 6
7.d odd 6 1 1323.2.g.d 6
7.d odd 6 1 1323.2.h.c 6
9.c even 3 1 inner 1323.2.f.d 6
9.c even 3 1 3969.2.a.l 3
9.d odd 6 1 441.2.f.c 6
9.d odd 6 1 3969.2.a.q 3
21.c even 2 1 63.2.f.a 6
21.g even 6 1 441.2.g.c 6
21.g even 6 1 441.2.h.d 6
21.h odd 6 1 441.2.g.b 6
21.h odd 6 1 441.2.h.e 6
28.d even 2 1 3024.2.r.k 6
63.g even 3 1 1323.2.h.b 6
63.h even 3 1 1323.2.g.e 6
63.i even 6 1 441.2.g.c 6
63.j odd 6 1 441.2.g.b 6
63.k odd 6 1 1323.2.h.c 6
63.l odd 6 1 189.2.f.b 6
63.l odd 6 1 567.2.a.c 3
63.n odd 6 1 441.2.h.e 6
63.o even 6 1 63.2.f.a 6
63.o even 6 1 567.2.a.h 3
63.s even 6 1 441.2.h.d 6
63.t odd 6 1 1323.2.g.d 6
84.h odd 2 1 1008.2.r.h 6
252.s odd 6 1 1008.2.r.h 6
252.s odd 6 1 9072.2.a.ca 3
252.bi even 6 1 3024.2.r.k 6
252.bi even 6 1 9072.2.a.bs 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 21.c even 2 1
63.2.f.a 6 63.o even 6 1
189.2.f.b 6 7.b odd 2 1
189.2.f.b 6 63.l odd 6 1
441.2.f.c 6 3.b odd 2 1
441.2.f.c 6 9.d odd 6 1
441.2.g.b 6 21.h odd 6 1
441.2.g.b 6 63.j odd 6 1
441.2.g.c 6 21.g even 6 1
441.2.g.c 6 63.i even 6 1
441.2.h.d 6 21.g even 6 1
441.2.h.d 6 63.s even 6 1
441.2.h.e 6 21.h odd 6 1
441.2.h.e 6 63.n odd 6 1
567.2.a.c 3 63.l odd 6 1
567.2.a.h 3 63.o even 6 1
1008.2.r.h 6 84.h odd 2 1
1008.2.r.h 6 252.s odd 6 1
1323.2.f.d 6 1.a even 1 1 trivial
1323.2.f.d 6 9.c even 3 1 inner
1323.2.g.d 6 7.d odd 6 1
1323.2.g.d 6 63.t odd 6 1
1323.2.g.e 6 7.c even 3 1
1323.2.g.e 6 63.h even 3 1
1323.2.h.b 6 7.c even 3 1
1323.2.h.b 6 63.g even 3 1
1323.2.h.c 6 7.d odd 6 1
1323.2.h.c 6 63.k odd 6 1
3024.2.r.k 6 28.d even 2 1
3024.2.r.k 6 252.bi even 6 1
3969.2.a.l 3 9.c even 3 1
3969.2.a.q 3 9.d odd 6 1
9072.2.a.bs 3 252.bi even 6 1
9072.2.a.ca 3 252.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

 $$T_{2}^{6} - 3 T_{2}^{5} + 9 T_{2}^{4} - 6 T_{2}^{3} + 9 T_{2}^{2} + 9$$ $$T_{5}^{6} + 3 T_{5}^{5} + 9 T_{5}^{4} + 6 T_{5}^{3} + 9 T_{5}^{2} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 9 T^{2} - 6 T^{3} + 9 T^{4} - 3 T^{5} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$9 + 9 T^{2} + 6 T^{3} + 9 T^{4} + 3 T^{5} + T^{6}$$
$7$ $$T^{6}$$
$11$ $$9 - 27 T + 63 T^{2} - 48 T^{3} + 27 T^{4} - 6 T^{5} + T^{6}$$
$13$ $$11449 + 3531 T + 1410 T^{2} + 115 T^{3} + 42 T^{4} + 3 T^{5} + T^{6}$$
$17$ $$( -3 + 9 T - 6 T^{2} + T^{3} )^{2}$$
$19$ $$( 17 - 6 T - 3 T^{2} + T^{3} )^{2}$$
$23$ $$9 + 81 T + 765 T^{2} - 330 T^{3} + 117 T^{4} - 12 T^{5} + T^{6}$$
$29$ $$110889 - 11988 T + 4293 T^{2} - 342 T^{3} + 117 T^{4} - 9 T^{5} + T^{6}$$
$31$ $$104329 + 25194 T + 7053 T^{2} + 412 T^{3} + 87 T^{4} + 3 T^{5} + T^{6}$$
$37$ $$( -323 - 78 T + 3 T^{2} + T^{3} )^{2}$$
$41$ $$81 - 81 T + 81 T^{2} - 18 T^{3} + 9 T^{4} + T^{6}$$
$43$ $$1 + 6 T + 33 T^{2} + 20 T^{3} + 15 T^{4} - 3 T^{5} + T^{6}$$
$47$ $$2601 - 2754 T + 2763 T^{2} - 264 T^{3} + 63 T^{4} + 3 T^{5} + T^{6}$$
$53$ $$( 3 - 9 T + 6 T^{2} + T^{3} )^{2}$$
$59$ $$2601 + 3672 T + 5031 T^{2} + 318 T^{3} + 81 T^{4} - 3 T^{5} + T^{6}$$
$61$ $$361 - 285 T + 339 T^{2} + 52 T^{3} + 51 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$289 + 357 T + 645 T^{2} - 286 T^{3} + 123 T^{4} - 12 T^{5} + T^{6}$$
$71$ $$( 27 - 54 T + 9 T^{2} + T^{3} )^{2}$$
$73$ $$( 269 + 84 T - 21 T^{2} + T^{3} )^{2}$$
$79$ $$32761 - 21720 T + 10599 T^{2} - 2158 T^{3} + 321 T^{4} - 21 T^{5} + T^{6}$$
$83$ $$81 - 405 T + 1863 T^{2} - 792 T^{3} + 279 T^{4} - 18 T^{5} + T^{6}$$
$89$ $$( 813 - 63 T - 12 T^{2} + T^{3} )^{2}$$
$97$ $$104329 + 54264 T + 29193 T^{2} + 142 T^{3} + 177 T^{4} + 3 T^{5} + T^{6}$$