# Properties

 Label 756.1.z.a Level $756$ Weight $1$ Character orbit 756.z Analytic conductor $0.377$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$756 = 2^{2} \cdot 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 756.z (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.377293149551$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{6}$$ Projective field Galois closure of 6.0.196036848.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{6}^{2} q^{7} +O(q^{10})$$ $$q -\zeta_{6}^{2} q^{7} + ( 1 - \zeta_{6}^{2} ) q^{19} + \zeta_{6}^{2} q^{25} + ( -1 - \zeta_{6} ) q^{31} + 2 \zeta_{6} q^{37} + q^{43} -\zeta_{6} q^{49} + ( -1 + \zeta_{6}^{2} ) q^{61} + 2 \zeta_{6}^{2} q^{67} + ( -1 - \zeta_{6} ) q^{73} -2 \zeta_{6} q^{79} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{7} + O(q^{10})$$ $$2q + q^{7} + 3q^{19} - q^{25} - 3q^{31} + 2q^{37} + 2q^{43} - q^{49} - 3q^{61} - 2q^{67} - 3q^{73} - 2q^{79} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$325$$ $$379$$ $$\chi(n)$$ $$1$$ $$\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
325.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 0 0 0.500000 0.866025i 0 0 0
649.1 0 0 0 0 0 0.500000 + 0.866025i 0 0 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.1.z.a 2
3.b odd 2 1 CM 756.1.z.a 2
4.b odd 2 1 3024.1.cg.a 2
7.d odd 6 1 inner 756.1.z.a 2
9.c even 3 1 2268.1.p.a 2
9.c even 3 1 2268.1.bd.b 2
9.d odd 6 1 2268.1.p.a 2
9.d odd 6 1 2268.1.bd.b 2
12.b even 2 1 3024.1.cg.a 2
21.g even 6 1 inner 756.1.z.a 2
28.f even 6 1 3024.1.cg.a 2
63.i even 6 1 2268.1.p.a 2
63.k odd 6 1 2268.1.bd.b 2
63.s even 6 1 2268.1.bd.b 2
63.t odd 6 1 2268.1.p.a 2
84.j odd 6 1 3024.1.cg.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.1.z.a 2 1.a even 1 1 trivial
756.1.z.a 2 3.b odd 2 1 CM
756.1.z.a 2 7.d odd 6 1 inner
756.1.z.a 2 21.g even 6 1 inner
2268.1.p.a 2 9.c even 3 1
2268.1.p.a 2 9.d odd 6 1
2268.1.p.a 2 63.i even 6 1
2268.1.p.a 2 63.t odd 6 1
2268.1.bd.b 2 9.c even 3 1
2268.1.bd.b 2 9.d odd 6 1
2268.1.bd.b 2 63.k odd 6 1
2268.1.bd.b 2 63.s even 6 1
3024.1.cg.a 2 4.b odd 2 1
3024.1.cg.a 2 12.b even 2 1
3024.1.cg.a 2 28.f even 6 1
3024.1.cg.a 2 84.j odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(756, [\chi])$$.

## Hecke characteristic polynomials

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