Properties

 Label 1323.2.i.a Level $1323$ Weight $2$ Character orbit 1323.i 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.i (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: $$\Z[a_1, \ldots, a_{5}]$$ 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.

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

Character values

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

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
521.1
 0.5 + 0.866025i 0.5 − 0.866025i
1.73205i 0 −1.00000 −1.50000 + 2.59808i 0 0 1.73205i 0 −4.50000 2.59808i
1097.1 1.73205i 0 −1.00000 −1.50000 2.59808i 0 0 1.73205i 0 −4.50000 + 2.59808i
 $$n$$: 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.i 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.i.a 2
3.b odd 2 1 441.2.i.a 2
7.b odd 2 1 189.2.i.a 2
7.c even 3 1 189.2.s.a 2
7.c even 3 1 1323.2.o.a 2
7.d odd 6 1 1323.2.o.b 2
7.d odd 6 1 1323.2.s.a 2
9.c even 3 1 441.2.s.a 2
9.d odd 6 1 1323.2.s.a 2
21.c even 2 1 63.2.i.a 2
21.g even 6 1 441.2.o.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.o.b 2
28.d even 2 1 3024.2.ca.a 2
28.g odd 6 1 3024.2.df.a 2
63.g even 3 1 441.2.o.a 2
63.g even 3 1 567.2.p.a 2
63.h even 3 1 63.2.i.a 2
63.i even 6 1 inner 1323.2.i.a 2
63.j odd 6 1 189.2.i.a 2
63.k odd 6 1 441.2.o.b 2
63.l odd 6 1 63.2.s.a yes 2
63.l odd 6 1 567.2.p.b 2
63.n odd 6 1 567.2.p.b 2
63.n odd 6 1 1323.2.o.b 2
63.o even 6 1 189.2.s.a 2
63.o even 6 1 567.2.p.a 2
63.s even 6 1 1323.2.o.a 2
63.t odd 6 1 441.2.i.a 2
84.h odd 2 1 1008.2.ca.a 2
84.n even 6 1 1008.2.df.a 2
252.s odd 6 1 3024.2.df.a 2
252.u odd 6 1 1008.2.ca.a 2
252.bb even 6 1 3024.2.ca.a 2
252.bi even 6 1 1008.2.df.a 2

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$3 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$3 + 3 T + T^{2}$$
$13$ $$3 + 3 T + T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$27 - 9 T + T^{2}$$
$23$ $$27 + 9 T + T^{2}$$
$29$ $$27 - 9 T + T^{2}$$
$31$ $$12 + T^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$1 + T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$75 + 15 T + T^{2}$$
$59$ $$T^{2}$$
$61$ $$192 + T^{2}$$
$67$ $$( 4 + T )^{2}$$
$71$ $$12 + T^{2}$$
$73$ $$27 - 9 T + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$225 - 15 T + T^{2}$$
$89$ $$9 + 3 T + T^{2}$$
$97$ $$3 + 3 T + T^{2}$$