# Properties

 Label 1764.1.g.a Level 1764 Weight 1 Character orbit 1764.g Self dual yes Analytic conductor 0.880 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -4, -7, 28 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 196) Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(i, \sqrt{7})$$ Artin image $D_4$ Artin field Galois closure of 4.0.12348.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{8} + q^{16} - q^{25} + 2q^{29} + q^{32} - 2q^{37} - q^{50} - 2q^{53} + 2q^{58} + q^{64} - 2q^{74} + O(q^{100})$$

## Character values

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

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

## 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
 0
1.00000 0 1.00000 0 0 0 1.00000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
28.d even 2 1 RM by $$\Q(\sqrt{7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.1.g.a 1
3.b odd 2 1 196.1.c.a 1
4.b odd 2 1 CM 1764.1.g.a 1
7.b odd 2 1 CM 1764.1.g.a 1
7.c even 3 2 1764.1.y.a 2
7.d odd 6 2 1764.1.y.a 2
12.b even 2 1 196.1.c.a 1
21.c even 2 1 196.1.c.a 1
21.g even 6 2 196.1.g.a 2
21.h odd 6 2 196.1.g.a 2
24.f even 2 1 3136.1.d.a 1
24.h odd 2 1 3136.1.d.a 1
28.d even 2 1 RM 1764.1.g.a 1
28.f even 6 2 1764.1.y.a 2
28.g odd 6 2 1764.1.y.a 2
84.h odd 2 1 196.1.c.a 1
84.j odd 6 2 196.1.g.a 2
84.n even 6 2 196.1.g.a 2
168.e odd 2 1 3136.1.d.a 1
168.i even 2 1 3136.1.d.a 1
168.s odd 6 2 3136.1.r.a 2
168.v even 6 2 3136.1.r.a 2
168.ba even 6 2 3136.1.r.a 2
168.be odd 6 2 3136.1.r.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.1.c.a 1 3.b odd 2 1
196.1.c.a 1 12.b even 2 1
196.1.c.a 1 21.c even 2 1
196.1.c.a 1 84.h odd 2 1
196.1.g.a 2 21.g even 6 2
196.1.g.a 2 21.h odd 6 2
196.1.g.a 2 84.j odd 6 2
196.1.g.a 2 84.n even 6 2
1764.1.g.a 1 1.a even 1 1 trivial
1764.1.g.a 1 4.b odd 2 1 CM
1764.1.g.a 1 7.b odd 2 1 CM
1764.1.g.a 1 28.d even 2 1 RM
1764.1.y.a 2 7.c even 3 2
1764.1.y.a 2 7.d odd 6 2
1764.1.y.a 2 28.f even 6 2
1764.1.y.a 2 28.g odd 6 2
3136.1.d.a 1 24.f even 2 1
3136.1.d.a 1 24.h odd 2 1
3136.1.d.a 1 168.e odd 2 1
3136.1.d.a 1 168.i even 2 1
3136.1.r.a 2 168.s odd 6 2
3136.1.r.a 2 168.v even 6 2
3136.1.r.a 2 168.ba even 6 2
3136.1.r.a 2 168.be odd 6 2

## Hecke kernels

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

 $$T_{5}$$ $$T_{11}$$ $$T_{29} - 2$$

## Hecke characteristic polynomials

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