# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$10.5642081874$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -1 + 2 \zeta_{6} ) q^{8} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{4} -3 \zeta_{6} q^{5} + ( -1 + 2 \zeta_{6} ) q^{8} + ( -3 + 6 \zeta_{6} ) q^{10} + ( 1 + \zeta_{6} ) q^{11} + ( -2 + \zeta_{6} ) q^{13} + ( 5 - 5 \zeta_{6} ) q^{16} + 3 q^{17} + ( -3 + 6 \zeta_{6} ) q^{19} + ( 3 - 3 \zeta_{6} ) q^{20} -3 \zeta_{6} q^{22} + ( 6 - 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + 3 q^{26} + ( 3 + 3 \zeta_{6} ) q^{29} + ( 4 - 2 \zeta_{6} ) q^{31} + ( -6 + 3 \zeta_{6} ) q^{32} + ( -3 - 3 \zeta_{6} ) q^{34} + 7 q^{37} + ( 9 - 9 \zeta_{6} ) q^{38} + ( 6 - 3 \zeta_{6} ) q^{40} -3 \zeta_{6} q^{41} + ( -1 + \zeta_{6} ) q^{43} + ( -1 + 2 \zeta_{6} ) q^{44} -9 q^{46} + ( 8 - 4 \zeta_{6} ) q^{50} + ( -1 - \zeta_{6} ) q^{52} + ( -5 + 10 \zeta_{6} ) q^{53} + ( 3 - 6 \zeta_{6} ) q^{55} -9 \zeta_{6} q^{58} + ( -8 - 8 \zeta_{6} ) q^{61} -6 q^{62} - q^{64} + ( 3 + 3 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} + 3 \zeta_{6} q^{68} + ( 2 - 4 \zeta_{6} ) q^{71} + ( 3 - 6 \zeta_{6} ) q^{73} + ( -7 - 7 \zeta_{6} ) q^{74} + ( -6 + 3 \zeta_{6} ) q^{76} + ( -8 + 8 \zeta_{6} ) q^{79} -15 q^{80} + ( -3 + 6 \zeta_{6} ) q^{82} + ( 15 - 15 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} + ( 2 - \zeta_{6} ) q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} + 3 q^{89} + ( 3 + 3 \zeta_{6} ) q^{92} + ( 18 - 9 \zeta_{6} ) q^{95} + ( -1 - \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} + q^{4} - 3q^{5} + O(q^{10})$$ $$2q - 3q^{2} + q^{4} - 3q^{5} + 3q^{11} - 3q^{13} + 5q^{16} + 6q^{17} + 3q^{20} - 3q^{22} + 9q^{23} - 4q^{25} + 6q^{26} + 9q^{29} + 6q^{31} - 9q^{32} - 9q^{34} + 14q^{37} + 9q^{38} + 9q^{40} - 3q^{41} - q^{43} - 18q^{46} + 12q^{50} - 3q^{52} - 9q^{58} - 24q^{61} - 12q^{62} - 2q^{64} + 9q^{65} + 4q^{67} + 3q^{68} - 21q^{74} - 9q^{76} - 8q^{79} - 30q^{80} + 15q^{83} - 9q^{85} + 3q^{86} - 3q^{88} + 6q^{89} + 9q^{92} + 27q^{95} - 3q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$\zeta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
440.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 0.866025i 0 0.500000 + 0.866025i −1.50000 2.59808i 0 0 1.73205i 0 5.19615i
881.1 −1.50000 + 0.866025i 0 0.500000 0.866025i −1.50000 + 2.59808i 0 0 1.73205i 0 5.19615i
 $$n$$: 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
63.o even 6 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{2}^{2} + 3 T_{2} + 3$$ $$T_{5}^{2} + 3 T_{5} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3 - 3 T + T^{2}$$
$13$ $$3 + 3 T + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$27 + T^{2}$$
$23$ $$27 - 9 T + T^{2}$$
$29$ $$27 - 9 T + T^{2}$$
$31$ $$12 - 6 T + T^{2}$$
$37$ $$( -7 + T )^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$75 + T^{2}$$
$59$ $$T^{2}$$
$61$ $$192 + 24 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$27 + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$225 - 15 T + T^{2}$$
$89$ $$( -3 + T )^{2}$$
$97$ $$3 + 3 T + T^{2}$$