# Properties

 Label 1764.2.b.g Level 1764 Weight 2 Character orbit 1764.b Analytic conductor 14.086 Analytic rank 0 Dimension 4 CM discriminant -4 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: 4.0.2048.2 Defining polynomial: $$x^{4} + 4 x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 2 \beta_{3} ) q^{5} -2 \beta_{2} q^{8} +O(q^{10})$$ $$q -\beta_{2} q^{2} + 2 q^{4} + ( \beta_{1} + 2 \beta_{3} ) q^{5} -2 \beta_{2} q^{8} + ( -\beta_{1} - 3 \beta_{3} ) q^{10} + ( -3 \beta_{1} - 2 \beta_{3} ) q^{13} + 4 q^{16} + ( -4 \beta_{1} - \beta_{3} ) q^{17} + ( 2 \beta_{1} + 4 \beta_{3} ) q^{20} + ( -5 - 7 \beta_{2} ) q^{25} + ( -\beta_{1} + 5 \beta_{3} ) q^{26} -3 \beta_{2} q^{29} -4 \beta_{2} q^{32} + ( -3 \beta_{1} + 5 \beta_{3} ) q^{34} -7 \beta_{2} q^{37} + ( -2 \beta_{1} - 6 \beta_{3} ) q^{40} + ( 5 \beta_{1} - 4 \beta_{3} ) q^{41} + ( 14 + 5 \beta_{2} ) q^{50} + ( -6 \beta_{1} - 4 \beta_{3} ) q^{52} -14 q^{53} + 6 q^{58} + ( 5 \beta_{1} - 6 \beta_{3} ) q^{61} + 8 q^{64} + ( 14 + 9 \beta_{2} ) q^{65} + ( -8 \beta_{1} - 2 \beta_{3} ) q^{68} + ( -8 \beta_{1} - 3 \beta_{3} ) q^{73} + 14 q^{74} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{80} + ( 9 \beta_{1} - \beta_{3} ) q^{82} + ( 12 + 7 \beta_{2} ) q^{85} + ( 8 \beta_{1} - 5 \beta_{3} ) q^{89} + ( -4 \beta_{1} + 9 \beta_{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + O(q^{10})$$ $$4q + 8q^{4} + 16q^{16} - 20q^{25} + 56q^{50} - 56q^{53} + 24q^{58} + 32q^{64} + 56q^{65} + 56q^{74} + 48q^{85} + O(q^{100})$$

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

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

## 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
 − 0.765367i 0.765367i − 1.84776i 1.84776i
−1.41421 0 2.00000 4.46088i 0 0 −2.82843 0 6.30864i
1567.2 −1.41421 0 2.00000 4.46088i 0 0 −2.82843 0 6.30864i
1567.3 1.41421 0 2.00000 0.317025i 0 0 2.82843 0 0.448342i
1567.4 1.41421 0 2.00000 0.317025i 0 0 2.82843 0 0.448342i
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
7.b odd 2 1 inner
28.d 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.b.g yes 4
3.b odd 2 1 1764.2.b.e 4
4.b odd 2 1 CM 1764.2.b.g yes 4
7.b odd 2 1 inner 1764.2.b.g yes 4
12.b even 2 1 1764.2.b.e 4
21.c even 2 1 1764.2.b.e 4
28.d even 2 1 inner 1764.2.b.g yes 4
84.h odd 2 1 1764.2.b.e 4

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

## Hecke kernels

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

 $$T_{5}^{4} + 20 T_{5}^{2} + 2$$ $$T_{11}$$ $$T_{19}$$ $$T_{29}^{2} - 18$$ $$T_{53} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}$$
$3$ 1
$5$ $$1 - 48 T^{4} + 625 T^{8}$$
$7$ 1
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$1 + 240 T^{4} + 28561 T^{8}$$
$17$ $$1 + 480 T^{4} + 83521 T^{8}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$( 1 - 23 T^{2} )^{4}$$
$29$ $$( 1 + 40 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{4}$$
$37$ $$( 1 - 24 T^{2} + 1369 T^{4} )^{2}$$
$41$ $$1 - 1440 T^{4} + 2825761 T^{8}$$
$43$ $$( 1 - 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 14 T + 53 T^{2} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$1 + 2640 T^{4} + 13845841 T^{8}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$1 + 10560 T^{4} + 28398241 T^{8}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$1 - 12480 T^{4} + 62742241 T^{8}$$
$97$ $$1 + 18720 T^{4} + 88529281 T^{8}$$