# Properties

 Label 1323.2.g.a Level $1323$ Weight $2$ Character orbit 1323.g 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.g (of order $$3$$, 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

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

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 0.866025i 0 0.500000 + 0.866025i −1.00000 0 0 3.00000 0 −0.500000 + 0.866025i
667.1 0.500000 + 0.866025i 0 0.500000 0.866025i −1.00000 0 0 3.00000 0 −0.500000 0.866025i
 $$n$$: 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.g 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.g.a 2
3.b odd 2 1 441.2.g.a 2
7.b odd 2 1 189.2.g.a 2
7.c even 3 1 1323.2.f.b 2
7.c even 3 1 1323.2.h.a 2
7.d odd 6 1 189.2.h.a 2
7.d odd 6 1 1323.2.f.a 2
9.c even 3 1 1323.2.h.a 2
9.d odd 6 1 441.2.h.a 2
21.c even 2 1 63.2.g.a 2
21.g even 6 1 63.2.h.a yes 2
21.g even 6 1 441.2.f.b 2
21.h odd 6 1 441.2.f.a 2
21.h odd 6 1 441.2.h.a 2
28.d even 2 1 3024.2.t.d 2
28.f even 6 1 3024.2.q.b 2
63.g even 3 1 inner 1323.2.g.a 2
63.g even 3 1 3969.2.a.a 1
63.h even 3 1 1323.2.f.b 2
63.i even 6 1 441.2.f.b 2
63.i even 6 1 567.2.e.a 2
63.j odd 6 1 441.2.f.a 2
63.k odd 6 1 189.2.g.a 2
63.k odd 6 1 3969.2.a.c 1
63.l odd 6 1 189.2.h.a 2
63.l odd 6 1 567.2.e.b 2
63.n odd 6 1 441.2.g.a 2
63.n odd 6 1 3969.2.a.f 1
63.o even 6 1 63.2.h.a yes 2
63.o even 6 1 567.2.e.a 2
63.s even 6 1 63.2.g.a 2
63.s even 6 1 3969.2.a.d 1
63.t odd 6 1 567.2.e.b 2
63.t odd 6 1 1323.2.f.a 2
84.h odd 2 1 1008.2.t.d 2
84.j odd 6 1 1008.2.q.c 2
252.n even 6 1 3024.2.t.d 2
252.s odd 6 1 1008.2.q.c 2
252.bi even 6 1 3024.2.q.b 2
252.bn odd 6 1 1008.2.t.d 2

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

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{5} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$( 5 + T )^{2}$$
$13$ $$25 + 5 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$1 + T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$9 + 3 T + T^{2}$$
$41$ $$25 - 5 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$81 + 9 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$196 + 14 T + T^{2}$$
$67$ $$16 + 4 T + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$64 + 8 T + T^{2}$$
$83$ $$81 - 9 T + T^{2}$$
$89$ $$169 - 13 T + T^{2}$$
$97$ $$81 + 9 T + T^{2}$$