# Properties

 Label 1764.2.b.d Level $1764$ Weight $2$ Character orbit 1764.b 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} - 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} + ( 1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} + ( 1 - \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -\beta_{1} + \beta_{3} ) q^{8} + ( 2 + 2 \beta_{2} ) q^{10} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{11} -3 \beta_{2} q^{13} + ( -2 - 2 \beta_{2} ) q^{16} + ( \beta_{1} - \beta_{3} ) q^{17} + 5 q^{19} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{20} + ( -4 - 4 \beta_{2} ) q^{22} + ( \beta_{1} - \beta_{3} ) q^{23} -3 q^{25} -3 \beta_{1} q^{26} - q^{31} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{32} + ( 2 + 2 \beta_{2} ) q^{34} + 5 q^{37} + 5 \beta_{3} q^{38} + 8 q^{40} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{41} -3 \beta_{2} q^{43} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{44} + ( 2 + 2 \beta_{2} ) q^{46} + ( -\beta_{1} - 3 \beta_{3} ) q^{47} -3 \beta_{3} q^{50} + ( -9 - 3 \beta_{2} ) q^{52} + ( \beta_{1} + 3 \beta_{3} ) q^{53} + 16 q^{55} + ( \beta_{1} + 3 \beta_{3} ) q^{59} -4 \beta_{2} q^{61} -\beta_{3} q^{62} -8 q^{64} + ( 3 \beta_{1} + 9 \beta_{3} ) q^{65} -5 \beta_{2} q^{67} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{68} + ( \beta_{1} - \beta_{3} ) q^{71} + \beta_{2} q^{73} + 5 \beta_{3} q^{74} + ( 5 - 5 \beta_{2} ) q^{76} -\beta_{2} q^{79} + 8 \beta_{3} q^{80} + ( -4 - 4 \beta_{2} ) q^{82} + ( 3 \beta_{1} + 9 \beta_{3} ) q^{83} -8 q^{85} -3 \beta_{1} q^{86} -16 q^{88} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{89} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{92} + ( -6 + 2 \beta_{2} ) q^{94} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{95} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{4} + O(q^{10})$$ $$4q + 4q^{4} + 8q^{10} - 8q^{16} + 20q^{19} - 16q^{22} - 12q^{25} - 4q^{31} + 8q^{34} + 20q^{37} + 32q^{40} + 8q^{46} - 36q^{52} + 64q^{55} - 32q^{64} + 20q^{76} - 16q^{82} - 32q^{85} - 64q^{88} - 24q^{94} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 −1.22474 + 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i 1.22474 − 0.707107i
−1.22474 0.707107i 0 1.00000 + 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 3.46410i
1567.2 −1.22474 + 0.707107i 0 1.00000 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 + 3.46410i
1567.3 1.22474 0.707107i 0 1.00000 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 + 3.46410i
1567.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.82843i 0 0 2.82843i 0 2.00000 3.46410i
 $$n$$: 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
3.b odd 2 1 inner
28.d even 2 1 inner
84.h 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.b.d 4
3.b odd 2 1 inner 1764.2.b.d 4
4.b odd 2 1 1764.2.b.c 4
7.b odd 2 1 1764.2.b.c 4
7.c even 3 1 252.2.bf.b 4
7.d odd 6 1 252.2.bf.c yes 4
12.b even 2 1 1764.2.b.c 4
21.c even 2 1 1764.2.b.c 4
21.g even 6 1 252.2.bf.c yes 4
21.h odd 6 1 252.2.bf.b 4
28.d even 2 1 inner 1764.2.b.d 4
28.f even 6 1 252.2.bf.b 4
28.g odd 6 1 252.2.bf.c yes 4
84.h odd 2 1 inner 1764.2.b.d 4
84.j odd 6 1 252.2.bf.b 4
84.n even 6 1 252.2.bf.c yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bf.b 4 7.c even 3 1
252.2.bf.b 4 21.h odd 6 1
252.2.bf.b 4 28.f even 6 1
252.2.bf.b 4 84.j odd 6 1
252.2.bf.c yes 4 7.d odd 6 1
252.2.bf.c yes 4 21.g even 6 1
252.2.bf.c yes 4 28.g odd 6 1
252.2.bf.c yes 4 84.n even 6 1
1764.2.b.c 4 4.b odd 2 1
1764.2.b.c 4 7.b odd 2 1
1764.2.b.c 4 12.b even 2 1
1764.2.b.c 4 21.c even 2 1
1764.2.b.d 4 1.a even 1 1 trivial
1764.2.b.d 4 3.b odd 2 1 inner
1764.2.b.d 4 28.d even 2 1 inner
1764.2.b.d 4 84.h odd 2 1 inner

## Hecke kernels

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

 $$T_{5}^{2} + 8$$ $$T_{11}^{2} + 32$$ $$T_{19} - 5$$ $$T_{29}$$ $$T_{53}^{2} - 24$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 - 2 T^{2} + T^{4}$$
$3$ $$T^{4}$$
$5$ $$( 8 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$( 32 + T^{2} )^{2}$$
$13$ $$( 27 + T^{2} )^{2}$$
$17$ $$( 8 + T^{2} )^{2}$$
$19$ $$( -5 + T )^{4}$$
$23$ $$( 8 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 1 + T )^{4}$$
$37$ $$( -5 + T )^{4}$$
$41$ $$( 32 + T^{2} )^{2}$$
$43$ $$( 27 + T^{2} )^{2}$$
$47$ $$( -24 + T^{2} )^{2}$$
$53$ $$( -24 + T^{2} )^{2}$$
$59$ $$( -24 + T^{2} )^{2}$$
$61$ $$( 48 + T^{2} )^{2}$$
$67$ $$( 75 + T^{2} )^{2}$$
$71$ $$( 8 + T^{2} )^{2}$$
$73$ $$( 3 + T^{2} )^{2}$$
$79$ $$( 3 + T^{2} )^{2}$$
$83$ $$( -216 + T^{2} )^{2}$$
$89$ $$( 128 + T^{2} )^{2}$$
$97$ $$T^{4}$$