# Properties

 Label 1764.1.co.a Level 1764 Weight 1 Character orbit 1764.co Analytic conductor 0.880 Analytic rank 0 Dimension 12 Projective image $$D_{42}$$ CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1764.co (of order $$42$$, degree $$12$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.880350682285$$ Analytic rank: $$0$$ Dimension: $$12$$ Coefficient field: $$\Q(\zeta_{21})$$ Defining polynomial: $$x^{12} - x^{11} + x^{9} - x^{8} + x^{6} - x^{4} + x^{3} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{42}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{42} - \cdots)$$

## $q$-expansion

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

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$1$$ $$\zeta_{42}^{19}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
73.1
 0.365341 − 0.930874i 0.365341 + 0.930874i 0.826239 − 0.563320i −0.988831 − 0.149042i −0.733052 − 0.680173i −0.733052 + 0.680173i 0.0747301 + 0.997204i 0.955573 + 0.294755i 0.826239 + 0.563320i 0.0747301 − 0.997204i 0.955573 − 0.294755i −0.988831 + 0.149042i
0 0 0 0 0 −0.955573 + 0.294755i 0 0 0
145.1 0 0 0 0 0 −0.955573 0.294755i 0 0 0
397.1 0 0 0 0 0 0.988831 0.149042i 0 0 0
577.1 0 0 0 0 0 0.733052 0.680173i 0 0 0
649.1 0 0 0 0 0 −0.826239 + 0.563320i 0 0 0
829.1 0 0 0 0 0 −0.826239 0.563320i 0 0 0
1081.1 0 0 0 0 0 −0.365341 + 0.930874i 0 0 0
1153.1 0 0 0 0 0 −0.0747301 + 0.997204i 0 0 0
1333.1 0 0 0 0 0 0.988831 + 0.149042i 0 0 0
1405.1 0 0 0 0 0 −0.365341 0.930874i 0 0 0
1585.1 0 0 0 0 0 −0.0747301 0.997204i 0 0 0
1657.1 0 0 0 0 0 0.733052 + 0.680173i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1657.1 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.h odd 42 1 inner
147.o even 42 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.co.a 12
3.b odd 2 1 CM 1764.1.co.a 12
49.h odd 42 1 inner 1764.1.co.a 12
147.o even 42 1 inner 1764.1.co.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.co.a 12 1.a even 1 1 trivial
1764.1.co.a 12 3.b odd 2 1 CM
1764.1.co.a 12 49.h odd 42 1 inner
1764.1.co.a 12 147.o even 42 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )$$
$7$ $$1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12}$$
$11$ $$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$
$13$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )$$
$17$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )$$
$19$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )$$
$23$ $$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$
$29$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$
$31$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )$$
$37$ $$( 1 - T + T^{2} )^{6}( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}$$
$41$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$43$ $$( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )^{2}$$
$47$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )$$
$53$ $$1 + T^{2} - T^{6} - T^{8} + T^{12} - T^{16} - T^{18} + T^{22} + T^{24}$$
$59$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )$$
$61$ $$( 1 - T + T^{2} )^{6}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )$$
$67$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )$$
$71$ $$( 1 - T^{2} + T^{4} - T^{6} + T^{8} - T^{10} + T^{12} )^{2}$$
$73$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )$$
$79$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )$$
$83$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$
$89$ $$( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} )( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} )$$
$97$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2}$$