# Properties

 Label 1323.1.br.a Level $1323$ Weight $1$ Character orbit 1323.br Analytic conductor $0.660$ Analytic rank $0$ Dimension $12$ Projective image $D_{21}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1323.br (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.660263011713$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{21}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{21} - \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q - \zeta_{42}^{13} q^{4} - \zeta_{42}^{3} q^{7} +O(q^{10})$$ q - z^13 * q^4 - z^3 * q^7 $$q - \zeta_{42}^{13} q^{4} - \zeta_{42}^{3} q^{7} + ( - \zeta_{42}^{5} - \zeta_{42}) q^{13} - \zeta_{42}^{5} q^{16} + ( - \zeta_{42}^{17} - \zeta_{42}^{11}) q^{19} + \zeta_{42}^{4} q^{25} + \zeta_{42}^{16} q^{28} + (\zeta_{42}^{20} - \zeta_{42}^{15}) q^{31} + (\zeta_{42}^{16} + 1) q^{37} + (\zeta_{42}^{2} - \zeta_{42}) q^{43} + \zeta_{42}^{6} q^{49} + (\zeta_{42}^{18} + \zeta_{42}^{14}) q^{52} + ( - \zeta_{42}^{9} - \zeta_{42}^{7}) q^{61} + \zeta_{42}^{18} q^{64} + (\zeta_{42}^{12} + \zeta_{42}^{2}) q^{67} + ( - \zeta_{42}^{19} + \zeta_{42}^{10}) q^{73} + ( - \zeta_{42}^{9} - \zeta_{42}^{3}) q^{76} + ( - \zeta_{42}^{15} - \zeta_{42}^{13}) q^{79} + (\zeta_{42}^{8} + \zeta_{42}^{4}) q^{91} + ( - \zeta_{42}^{11} + \zeta_{42}^{10}) q^{97} +O(q^{100})$$ q - z^13 * q^4 - z^3 * q^7 + (-z^5 - z) * q^13 - z^5 * q^16 + (-z^17 - z^11) * q^19 + z^4 * q^25 + z^16 * q^28 + (z^20 - z^15) * q^31 + (z^16 + 1) * q^37 + (z^2 - z) * q^43 + z^6 * q^49 + (z^18 + z^14) * q^52 + (-z^9 - z^7) * q^61 + z^18 * q^64 + (z^12 + z^2) * q^67 + (-z^19 + z^10) * q^73 + (-z^9 - z^3) * q^76 + (-z^15 - z^13) * q^79 + (z^8 + z^4) * q^91 + (-z^11 + z^10) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + q^{4} - 2 q^{7}+O(q^{10})$$ 12 * q + q^4 - 2 * q^7 $$12 q + q^{4} - 2 q^{7} + 2 q^{13} + q^{16} + 2 q^{19} + q^{25} + q^{28} - q^{31} + 13 q^{37} + 2 q^{43} - 2 q^{49} - 8 q^{52} - 8 q^{61} - 2 q^{64} - q^{67} + 2 q^{73} - 4 q^{76} - q^{79} + 2 q^{91} + 2 q^{97}+O(q^{100})$$ 12 * q + q^4 - 2 * q^7 + 2 * q^13 + q^16 + 2 * q^19 + q^25 + q^28 - q^31 + 13 * q^37 + 2 * q^43 - 2 * q^49 - 8 * q^52 - 8 * q^61 - 2 * q^64 - q^67 + 2 * q^73 - 4 * q^76 - q^79 + 2 * q^91 + 2 * q^97

## Character values

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

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{42}^{11}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
53.1
 −0.988831 + 0.149042i 0.826239 + 0.563320i 0.955573 + 0.294755i 0.0747301 + 0.997204i −0.733052 − 0.680173i −0.733052 + 0.680173i 0.365341 + 0.930874i −0.988831 − 0.149042i 0.0747301 − 0.997204i 0.365341 − 0.930874i 0.826239 − 0.563320i 0.955573 − 0.294755i
0 0 0.365341 + 0.930874i 0 0 −0.900969 + 0.433884i 0 0 0
107.1 0 0 0.0747301 + 0.997204i 0 0 −0.222521 + 0.974928i 0 0 0
242.1 0 0 −0.733052 0.680173i 0 0 0.623490 + 0.781831i 0 0 0
296.1 0 0 0.826239 + 0.563320i 0 0 −0.222521 0.974928i 0 0 0
431.1 0 0 0.955573 + 0.294755i 0 0 0.623490 0.781831i 0 0 0
485.1 0 0 0.955573 0.294755i 0 0 0.623490 + 0.781831i 0 0 0
620.1 0 0 −0.988831 + 0.149042i 0 0 −0.900969 0.433884i 0 0 0
674.1 0 0 0.365341 0.930874i 0 0 −0.900969 0.433884i 0 0 0
809.1 0 0 0.826239 0.563320i 0 0 −0.222521 + 0.974928i 0 0 0
1052.1 0 0 −0.988831 0.149042i 0 0 −0.900969 + 0.433884i 0 0 0
1187.1 0 0 0.0747301 0.997204i 0 0 −0.222521 0.974928i 0 0 0
1241.1 0 0 −0.733052 + 0.680173i 0 0 0.623490 0.781831i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1241.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
49.g even 21 1 inner
147.n odd 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.1.br.a 12
3.b odd 2 1 CM 1323.1.br.a 12
9.c even 3 1 3969.1.bx.a 12
9.c even 3 1 3969.1.cb.a 12
9.d odd 6 1 3969.1.bx.a 12
9.d odd 6 1 3969.1.cb.a 12
49.g even 21 1 inner 1323.1.br.a 12
147.n odd 42 1 inner 1323.1.br.a 12
441.y even 21 1 3969.1.bx.a 12
441.z even 21 1 3969.1.cb.a 12
441.bi odd 42 1 3969.1.cb.a 12
441.bm odd 42 1 3969.1.bx.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.1.br.a 12 1.a even 1 1 trivial
1323.1.br.a 12 3.b odd 2 1 CM
1323.1.br.a 12 49.g even 21 1 inner
1323.1.br.a 12 147.n odd 42 1 inner
3969.1.bx.a 12 9.c even 3 1
3969.1.bx.a 12 9.d odd 6 1
3969.1.bx.a 12 441.y even 21 1
3969.1.bx.a 12 441.bm odd 42 1
3969.1.cb.a 12 9.c even 3 1
3969.1.cb.a 12 9.d odd 6 1
3969.1.cb.a 12 441.z even 21 1
3969.1.cb.a 12 441.bi odd 42 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1323, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12}$$
$5$ $$T^{12}$$
$7$ $$(T^{6} + T^{5} + T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$11$ $$T^{12}$$
$13$ $$T^{12} - 2 T^{11} + 3 T^{10} + 3 T^{9} + \cdots + 1$$
$17$ $$T^{12}$$
$19$ $$(T^{6} - T^{5} + 3 T^{4} + 5 T^{2} - 2 T + 1)^{2}$$
$23$ $$T^{12}$$
$29$ $$T^{12}$$
$31$ $$T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1$$
$37$ $$T^{12} - 13 T^{11} + 77 T^{10} - 274 T^{9} + \cdots + 1$$
$41$ $$T^{12}$$
$43$ $$T^{12} - 2 T^{11} + 3 T^{10} - 4 T^{9} + \cdots + 1$$
$47$ $$T^{12}$$
$53$ $$T^{12}$$
$59$ $$T^{12}$$
$61$ $$T^{12} + 8 T^{11} + 35 T^{10} + 104 T^{9} + \cdots + 1$$
$67$ $$T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1$$
$71$ $$T^{12}$$
$73$ $$T^{12} - 2 T^{11} + 8 T^{9} - 9 T^{8} + \cdots + 1$$
$79$ $$T^{12} + T^{11} + 7 T^{10} + 6 T^{9} + 34 T^{8} + \cdots + 1$$
$83$ $$T^{12}$$
$89$ $$T^{12}$$
$97$ $$(T^{6} - T^{5} - 6 T^{4} + 6 T^{3} + 8 T^{2} + \cdots + 1)^{2}$$