# Properties

 Label 1764.2.a.d Level $1764$ Weight $2$ Character orbit 1764.a Self dual yes Analytic conductor $14.086$ Analytic rank $1$ Dimension $1$ CM discriminant -3 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$14.0856109166$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + O(q^{10})$$ $$q - 5q^{13} + q^{19} - 5q^{25} - 11q^{31} + 11q^{37} - 13q^{43} - 14q^{61} + 5q^{67} - 17q^{73} + 17q^{79} - 14q^{97} + O(q^{100})$$

## 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 1764.2.a.d 1
3.b odd 2 1 CM 1764.2.a.d 1
4.b odd 2 1 7056.2.a.z 1
7.b odd 2 1 1764.2.a.f 1
7.c even 3 2 1764.2.k.f 2
7.d odd 6 2 252.2.k.b 2
12.b even 2 1 7056.2.a.z 1
21.c even 2 1 1764.2.a.f 1
21.g even 6 2 252.2.k.b 2
21.h odd 6 2 1764.2.k.f 2
28.d even 2 1 7056.2.a.be 1
28.f even 6 2 1008.2.s.i 2
63.i even 6 2 2268.2.i.c 2
63.k odd 6 2 2268.2.l.e 2
63.s even 6 2 2268.2.l.e 2
63.t odd 6 2 2268.2.i.c 2
84.h odd 2 1 7056.2.a.be 1
84.j odd 6 2 1008.2.s.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.k.b 2 7.d odd 6 2
252.2.k.b 2 21.g even 6 2
1008.2.s.i 2 28.f even 6 2
1008.2.s.i 2 84.j odd 6 2
1764.2.a.d 1 1.a even 1 1 trivial
1764.2.a.d 1 3.b odd 2 1 CM
1764.2.a.f 1 7.b odd 2 1
1764.2.a.f 1 21.c even 2 1
1764.2.k.f 2 7.c even 3 2
1764.2.k.f 2 21.h odd 6 2
2268.2.i.c 2 63.i even 6 2
2268.2.i.c 2 63.t odd 6 2
2268.2.l.e 2 63.k odd 6 2
2268.2.l.e 2 63.s even 6 2
7056.2.a.z 1 4.b odd 2 1
7056.2.a.z 1 12.b even 2 1
7056.2.a.be 1 28.d even 2 1
7056.2.a.be 1 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1764))$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$5 + T$$
$17$ $$T$$
$19$ $$-1 + T$$
$23$ $$T$$
$29$ $$T$$
$31$ $$11 + T$$
$37$ $$-11 + T$$
$41$ $$T$$
$43$ $$13 + T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$14 + T$$
$67$ $$-5 + T$$
$71$ $$T$$
$73$ $$17 + T$$
$79$ $$-17 + T$$
$83$ $$T$$
$89$ $$T$$
$97$ $$14 + T$$