# Properties

 Label 1764.2.e.d Level $1764$ Weight $2$ Character orbit 1764.e Analytic conductor $14.086$ Analytic rank $0$ Dimension $4$ CM discriminant -4 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1764 = 2^{2} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1764.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} -2 q^{4} + 4 \zeta_{8}^{2} q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{2} -2 q^{4} + 4 \zeta_{8}^{2} q^{5} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{10} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{13} + 4 q^{16} + 8 \zeta_{8}^{2} q^{17} -8 \zeta_{8}^{2} q^{20} -11 q^{25} + 2 \zeta_{8}^{2} q^{26} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{29} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} + ( 8 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{34} -12 q^{37} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{40} -10 \zeta_{8}^{2} q^{41} + ( 11 \zeta_{8} + 11 \zeta_{8}^{3} ) q^{50} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{52} + ( 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{53} -14 q^{58} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{61} -8 q^{64} + ( -4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{65} -16 \zeta_{8}^{2} q^{68} + ( -11 \zeta_{8} + 11 \zeta_{8}^{3} ) q^{73} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{74} + 16 \zeta_{8}^{2} q^{80} + ( -10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{82} -32 q^{85} + 10 \zeta_{8}^{2} q^{89} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} + O(q^{10})$$ $$4q - 8q^{4} + 16q^{16} - 44q^{25} - 48q^{37} - 56q^{58} - 32q^{64} - 128q^{85} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1079.1
 −0.707107 + 0.707107i 0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i
1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 −5.65685
1079.2 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 5.65685
1079.3 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 5.65685
1079.4 1.41421i 0 −2.00000 4.00000i 0 0 2.82843i 0 −5.65685
 $$n$$: 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 $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
7.b odd 2 1 inner
12.b even 2 1 inner
21.c even 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.e.d 4
3.b odd 2 1 inner 1764.2.e.d 4
4.b odd 2 1 CM 1764.2.e.d 4
7.b odd 2 1 inner 1764.2.e.d 4
12.b even 2 1 inner 1764.2.e.d 4
21.c even 2 1 inner 1764.2.e.d 4
28.d even 2 1 inner 1764.2.e.d 4
84.h odd 2 1 inner 1764.2.e.d 4

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

 $$T_{5}^{2} + 16$$ $$T_{13}^{2} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( 16 + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$( -2 + T^{2} )^{2}$$
$17$ $$( 64 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$T^{4}$$
$29$ $$( 98 + T^{2} )^{2}$$
$31$ $$T^{4}$$
$37$ $$( 12 + T )^{4}$$
$41$ $$( 100 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4}$$
$53$ $$( 162 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( -2 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$T^{4}$$
$73$ $$( -242 + T^{2} )^{2}$$
$79$ $$T^{4}$$
$83$ $$T^{4}$$
$89$ $$( 100 + T^{2} )^{2}$$
$97$ $$( -50 + T^{2} )^{2}$$