# Properties

 Label 1764.2.a.l Level $1764$ Weight $2$ Character orbit 1764.a Self dual yes Analytic conductor $14.086$ Analytic rank $1$ Dimension $2$ CM no 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 196) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta q^{5} +O(q^{10})$$ $$q + \beta q^{5} -4 q^{11} -3 \beta q^{13} + \beta q^{17} -2 \beta q^{19} + 4 q^{23} -3 q^{25} -8 q^{29} -8 q^{37} -5 \beta q^{41} -4 q^{43} + 4 \beta q^{47} -10 q^{53} -4 \beta q^{55} + 10 \beta q^{59} + 5 \beta q^{61} -6 q^{65} + 5 \beta q^{73} + 8 q^{79} -10 \beta q^{83} + 2 q^{85} + 5 \beta q^{89} -4 q^{95} + \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 8q^{11} + 8q^{23} - 6q^{25} - 16q^{29} - 16q^{37} - 8q^{43} - 20q^{53} - 12q^{65} + 16q^{79} + 4q^{85} - 8q^{95} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 0 0 −1.41421 0 0 0 0 0
1.2 0 0 0 1.41421 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.a.l 2
3.b odd 2 1 196.2.a.c 2
4.b odd 2 1 7056.2.a.cr 2
7.b odd 2 1 inner 1764.2.a.l 2
7.c even 3 2 1764.2.k.l 4
7.d odd 6 2 1764.2.k.l 4
12.b even 2 1 784.2.a.m 2
15.d odd 2 1 4900.2.a.y 2
15.e even 4 2 4900.2.e.p 4
21.c even 2 1 196.2.a.c 2
21.g even 6 2 196.2.e.b 4
21.h odd 6 2 196.2.e.b 4
24.f even 2 1 3136.2.a.bs 2
24.h odd 2 1 3136.2.a.br 2
28.d even 2 1 7056.2.a.cr 2
84.h odd 2 1 784.2.a.m 2
84.j odd 6 2 784.2.i.l 4
84.n even 6 2 784.2.i.l 4
105.g even 2 1 4900.2.a.y 2
105.k odd 4 2 4900.2.e.p 4
168.e odd 2 1 3136.2.a.bs 2
168.i even 2 1 3136.2.a.br 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.a.c 2 3.b odd 2 1
196.2.a.c 2 21.c even 2 1
196.2.e.b 4 21.g even 6 2
196.2.e.b 4 21.h odd 6 2
784.2.a.m 2 12.b even 2 1
784.2.a.m 2 84.h odd 2 1
784.2.i.l 4 84.j odd 6 2
784.2.i.l 4 84.n even 6 2
1764.2.a.l 2 1.a even 1 1 trivial
1764.2.a.l 2 7.b odd 2 1 inner
1764.2.k.l 4 7.c even 3 2
1764.2.k.l 4 7.d odd 6 2
3136.2.a.br 2 24.h odd 2 1
3136.2.a.br 2 168.i even 2 1
3136.2.a.bs 2 24.f even 2 1
3136.2.a.bs 2 168.e odd 2 1
4900.2.a.y 2 15.d odd 2 1
4900.2.a.y 2 105.g even 2 1
4900.2.e.p 4 15.e even 4 2
4900.2.e.p 4 105.k odd 4 2
7056.2.a.cr 2 4.b odd 2 1
7056.2.a.cr 2 28.d even 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 2$$ $$T_{11} + 4$$ $$T_{13}^{2} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$-2 + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( 4 + T )^{2}$$
$13$ $$-18 + T^{2}$$
$17$ $$-2 + T^{2}$$
$19$ $$-8 + T^{2}$$
$23$ $$( -4 + T )^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$-50 + T^{2}$$
$43$ $$( 4 + T )^{2}$$
$47$ $$-32 + T^{2}$$
$53$ $$( 10 + T )^{2}$$
$59$ $$-200 + T^{2}$$
$61$ $$-50 + T^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$-50 + T^{2}$$
$79$ $$( -8 + T )^{2}$$
$83$ $$-200 + T^{2}$$
$89$ $$-50 + T^{2}$$
$97$ $$-2 + T^{2}$$