# Properties

 Label 1764.2.b.e 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
 − 1.84776i 1.84776i − 0.765367i 0.765367i
−1.41421 0 2.00000 0.317025i 0 0 −2.82843 0 0.448342i
1567.2 −1.41421 0 2.00000 0.317025i 0 0 −2.82843 0 0.448342i
1567.3 1.41421 0 2.00000 4.46088i 0 0 2.82843 0 6.30864i
1567.4 1.41421 0 2.00000 4.46088i 0 0 2.82843 0 6.30864i
 $$n$$: 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.e 4
3.b odd 2 1 1764.2.b.g yes 4
4.b odd 2 1 CM 1764.2.b.e 4
7.b odd 2 1 inner 1764.2.b.e 4
12.b even 2 1 1764.2.b.g yes 4
21.c even 2 1 1764.2.b.g yes 4
28.d even 2 1 inner 1764.2.b.e 4
84.h odd 2 1 1764.2.b.g yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.b.e 4 1.a even 1 1 trivial
1764.2.b.e 4 4.b odd 2 1 CM
1764.2.b.e 4 7.b odd 2 1 inner
1764.2.b.e 4 28.d even 2 1 inner
1764.2.b.g yes 4 3.b odd 2 1
1764.2.b.g yes 4 12.b even 2 1
1764.2.b.g yes 4 21.c even 2 1
1764.2.b.g yes 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 $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$2 + 20 T^{2} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$578 + 52 T^{2} + T^{4}$$
$17$ $$1058 + 68 T^{2} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( -18 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( -98 + T^{2} )^{2}$$
$41$ $$1922 + 164 T^{2} + T^{4}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( -14 + T )^{4}$$
$59$ $$T^{4}$$
$61$ $$10082 + 244 T^{2} + T^{4}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$21218 + 292 T^{2} + T^{4}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$3362 + 356 T^{2} + T^{4}$$
$97$ $$37538 + 388 T^{2} + T^{4}$$