# Properties

 Label 1764.2.a.m Level $1764$ Weight $2$ Character orbit 1764.a Self dual yes Analytic conductor $14.086$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} +O(q^{10})$$ $$q + \beta_{2} q^{5} -\beta_{3} q^{11} -3 \beta_{1} q^{13} + \beta_{2} q^{17} + 2 \beta_{1} q^{19} + \beta_{3} q^{23} + 9 q^{25} + \beta_{3} q^{29} + 6 \beta_{1} q^{31} + 4 q^{37} + \beta_{2} q^{41} + 8 q^{43} -2 \beta_{2} q^{47} + 2 \beta_{3} q^{53} -14 \beta_{1} q^{55} -2 \beta_{2} q^{59} + 7 \beta_{1} q^{61} -3 \beta_{3} q^{65} + 12 q^{67} -3 \beta_{3} q^{71} + \beta_{1} q^{73} -4 q^{79} -4 \beta_{2} q^{83} + 14 q^{85} -\beta_{2} q^{89} + 2 \beta_{3} q^{95} -7 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + O(q^{10})$$ $$4 q + 36 q^{25} + 16 q^{37} + 32 q^{43} + 48 q^{67} - 16 q^{79} + 56 q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## 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
 −2.57794 −1.16372 2.57794 1.16372
0 0 0 −3.74166 0 0 0 0 0
1.2 0 0 0 −3.74166 0 0 0 0 0
1.3 0 0 0 3.74166 0 0 0 0 0
1.4 0 0 0 3.74166 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 inner
7.b odd 2 1 inner
21.c even 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.m 4
3.b odd 2 1 inner 1764.2.a.m 4
4.b odd 2 1 7056.2.a.cy 4
7.b odd 2 1 inner 1764.2.a.m 4
7.c even 3 2 1764.2.k.m 8
7.d odd 6 2 1764.2.k.m 8
12.b even 2 1 7056.2.a.cy 4
21.c even 2 1 inner 1764.2.a.m 4
21.g even 6 2 1764.2.k.m 8
21.h odd 6 2 1764.2.k.m 8
28.d even 2 1 7056.2.a.cy 4
84.h odd 2 1 7056.2.a.cy 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.a.m 4 1.a even 1 1 trivial
1764.2.a.m 4 3.b odd 2 1 inner
1764.2.a.m 4 7.b odd 2 1 inner
1764.2.a.m 4 21.c even 2 1 inner
1764.2.k.m 8 7.c even 3 2
1764.2.k.m 8 7.d odd 6 2
1764.2.k.m 8 21.g even 6 2
1764.2.k.m 8 21.h odd 6 2
7056.2.a.cy 4 4.b odd 2 1
7056.2.a.cy 4 12.b even 2 1
7056.2.a.cy 4 28.d even 2 1
7056.2.a.cy 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1764))$$:

 $$T_{5}^{2} - 14$$ $$T_{11}^{2} - 28$$ $$T_{13}^{2} - 18$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$( -14 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( -28 + T^{2} )^{2}$$
$13$ $$( -18 + T^{2} )^{2}$$
$17$ $$( -14 + T^{2} )^{2}$$
$19$ $$( -8 + T^{2} )^{2}$$
$23$ $$( -28 + T^{2} )^{2}$$
$29$ $$( -28 + T^{2} )^{2}$$
$31$ $$( -72 + T^{2} )^{2}$$
$37$ $$( -4 + T )^{4}$$
$41$ $$( -14 + T^{2} )^{2}$$
$43$ $$( -8 + T )^{4}$$
$47$ $$( -56 + T^{2} )^{2}$$
$53$ $$( -112 + T^{2} )^{2}$$
$59$ $$( -56 + T^{2} )^{2}$$
$61$ $$( -98 + T^{2} )^{2}$$
$67$ $$( -12 + T )^{4}$$
$71$ $$( -252 + T^{2} )^{2}$$
$73$ $$( -2 + T^{2} )^{2}$$
$79$ $$( 4 + T )^{4}$$
$83$ $$( -224 + T^{2} )^{2}$$
$89$ $$( -14 + T^{2} )^{2}$$
$97$ $$( -98 + T^{2} )^{2}$$