# Properties

 Label 1764.4.a.e Level $1764$ Weight $4$ Character orbit 1764.a Self dual yes Analytic conductor $104.079$ Analytic rank $0$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$104.079369250$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + O(q^{10})$$ $$q - 89q^{13} + 163q^{19} - 125q^{25} + 19q^{31} - 433q^{37} + 449q^{43} - 182q^{61} + 1007q^{67} + 919q^{73} + 503q^{79} + 1330q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 0 0 0 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.4.a.e 1
3.b odd 2 1 CM 1764.4.a.e 1
7.b odd 2 1 1764.4.a.h 1
7.c even 3 2 1764.4.k.h 2
7.d odd 6 2 252.4.k.a 2
21.c even 2 1 1764.4.a.h 1
21.g even 6 2 252.4.k.a 2
21.h odd 6 2 1764.4.k.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.4.k.a 2 7.d odd 6 2
252.4.k.a 2 21.g even 6 2
1764.4.a.e 1 1.a even 1 1 trivial
1764.4.a.e 1 3.b odd 2 1 CM
1764.4.a.h 1 7.b odd 2 1
1764.4.a.h 1 21.c even 2 1
1764.4.k.h 2 7.c even 3 2
1764.4.k.h 2 21.h odd 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5}$$ $$T_{11}$$ $$T_{13} + 89$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$89 + T$$
$17$ $$T$$
$19$ $$-163 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$-19 + T$$
$37$ $$433 + T$$
$41$ $$T$$
$43$ $$-449 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$182 + T$$
$67$ $$-1007 + T$$
$71$ $$T$$
$73$ $$-919 + T$$
$79$ $$-503 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$-1330 + T$$