# Properties

 Label 1764.1.y.d Level $1764$ Weight $1$ Character orbit 1764.y Analytic conductor $0.880$ Analytic rank $0$ Dimension $4$ Projective image $D_{4}$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1764.y (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.880350682285$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.49392.1 Artin image: $C_3\times D_8$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} - q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + ( -1 - \beta_{2} ) q^{4} + ( -\beta_{1} - \beta_{3} ) q^{5} - q^{8} -\beta_{1} q^{10} + \beta_{3} q^{13} + \beta_{2} q^{16} -\beta_{1} q^{17} + \beta_{3} q^{20} + ( -1 - \beta_{2} ) q^{25} + ( \beta_{1} + \beta_{3} ) q^{26} + 2 q^{29} + ( 1 + \beta_{2} ) q^{32} + \beta_{3} q^{34} + ( \beta_{1} + \beta_{3} ) q^{40} -\beta_{3} q^{41} - q^{50} + \beta_{1} q^{52} -2 \beta_{2} q^{58} + ( \beta_{1} + \beta_{3} ) q^{61} + q^{64} + 2 \beta_{2} q^{65} + ( \beta_{1} + \beta_{3} ) q^{68} -\beta_{1} q^{73} + \beta_{1} q^{80} + ( -\beta_{1} - \beta_{3} ) q^{82} -2 q^{85} + ( -\beta_{1} - \beta_{3} ) q^{89} -\beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} + O(q^{10})$$ $$4 q + 2 q^{2} - 2 q^{4} - 4 q^{8} - 2 q^{16} - 2 q^{25} + 8 q^{29} + 2 q^{32} - 4 q^{50} + 4 q^{58} + 4 q^{64} - 4 q^{65} - 8 q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
667.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
0.500000 0.866025i 0 −0.500000 0.866025i −0.707107 + 1.22474i 0 0 −1.00000 0 0.707107 + 1.22474i
667.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.707107 1.22474i 0 0 −1.00000 0 −0.707107 1.22474i
1243.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.707107 1.22474i 0 0 −1.00000 0 0.707107 1.22474i
1243.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.707107 + 1.22474i 0 0 −1.00000 0 −0.707107 + 1.22474i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
28.d even 2 1 inner
28.f even 6 1 inner
28.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.y.d 4
3.b odd 2 1 1764.1.y.b 4
4.b odd 2 1 CM 1764.1.y.d 4
7.b odd 2 1 inner 1764.1.y.d 4
7.c even 3 1 1764.1.g.b 2
7.c even 3 1 inner 1764.1.y.d 4
7.d odd 6 1 1764.1.g.b 2
7.d odd 6 1 inner 1764.1.y.d 4
12.b even 2 1 1764.1.y.b 4
21.c even 2 1 1764.1.y.b 4
21.g even 6 1 1764.1.g.d yes 2
21.g even 6 1 1764.1.y.b 4
21.h odd 6 1 1764.1.g.d yes 2
21.h odd 6 1 1764.1.y.b 4
28.d even 2 1 inner 1764.1.y.d 4
28.f even 6 1 1764.1.g.b 2
28.f even 6 1 inner 1764.1.y.d 4
28.g odd 6 1 1764.1.g.b 2
28.g odd 6 1 inner 1764.1.y.d 4
84.h odd 2 1 1764.1.y.b 4
84.j odd 6 1 1764.1.g.d yes 2
84.j odd 6 1 1764.1.y.b 4
84.n even 6 1 1764.1.g.d yes 2
84.n even 6 1 1764.1.y.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.1.g.b 2 7.c even 3 1
1764.1.g.b 2 7.d odd 6 1
1764.1.g.b 2 28.f even 6 1
1764.1.g.b 2 28.g odd 6 1
1764.1.g.d yes 2 21.g even 6 1
1764.1.g.d yes 2 21.h odd 6 1
1764.1.g.d yes 2 84.j odd 6 1
1764.1.g.d yes 2 84.n even 6 1
1764.1.y.b 4 3.b odd 2 1
1764.1.y.b 4 12.b even 2 1
1764.1.y.b 4 21.c even 2 1
1764.1.y.b 4 21.g even 6 1
1764.1.y.b 4 21.h odd 6 1
1764.1.y.b 4 84.h odd 2 1
1764.1.y.b 4 84.j odd 6 1
1764.1.y.b 4 84.n even 6 1
1764.1.y.d 4 1.a even 1 1 trivial
1764.1.y.d 4 4.b odd 2 1 CM
1764.1.y.d 4 7.b odd 2 1 inner
1764.1.y.d 4 7.c even 3 1 inner
1764.1.y.d 4 7.d odd 6 1 inner
1764.1.y.d 4 28.d even 2 1 inner
1764.1.y.d 4 28.f even 6 1 inner
1764.1.y.d 4 28.g odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{4} + 2 T_{5}^{2} + 4$$ $$T_{11}$$ $$T_{29} - 2$$

## Hecke characteristic polynomials

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