# Properties

 Label 1764.1.g.b Level $1764$ Weight $1$ Character orbit 1764.g Self dual yes Analytic conductor $0.880$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 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.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.880350682285$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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: $D_8$ Artin field: Galois closure of 8.0.38423222208.2

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 1.41421 −1.41421
−1.00000 0 1.00000 −1.41421 0 0 −1.00000 0 1.41421
883.2 −1.00000 0 1.00000 1.41421 0 0 −1.00000 0 −1.41421
 $$n$$: 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
28.d even 2 1 inner

## Twists

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

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

## Hecke kernels

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

 $$T_{5}^{2} - 2$$ T5^2 - 2 $$T_{11}$$ T11 $$T_{29} - 2$$ T29 - 2

## Hecke characteristic polynomials

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