# Properties

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

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## 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.

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

## 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 2.00000i 0 0 2.82843i 0 −2.82843
1079.2 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 2.82843
1079.3 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 2.82843
1079.4 1.41421i 0 −2.00000 2.00000i 0 0 2.82843i 0 −2.82843
 $$n$$: 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.e 4
3.b odd 2 1 inner 1764.2.e.e 4
4.b odd 2 1 CM 1764.2.e.e 4
7.b odd 2 1 inner 1764.2.e.e 4
12.b even 2 1 inner 1764.2.e.e 4
21.c even 2 1 inner 1764.2.e.e 4
28.d even 2 1 inner 1764.2.e.e 4
84.h odd 2 1 inner 1764.2.e.e 4

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

 $$T_{5}^{2} + 4$$ $$T_{13}^{2} - 50$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ 1
$5$ $$( 1 - 4 T + 5 T^{2} )^{2}( 1 + 4 T + 5 T^{2} )^{2}$$
$7$ 1
$11$ $$( 1 + 11 T^{2} )^{4}$$
$13$ $$( 1 - 24 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 17 T^{2} )^{2}( 1 + 8 T + 17 T^{2} )^{2}$$
$19$ $$( 1 - 19 T^{2} )^{4}$$
$23$ $$( 1 + 23 T^{2} )^{4}$$
$29$ $$( 1 + 40 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 31 T^{2} )^{4}$$
$37$ $$( 1 - 12 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 10 T + 41 T^{2} )^{2}( 1 + 10 T + 41 T^{2} )^{2}$$
$43$ $$( 1 - 43 T^{2} )^{4}$$
$47$ $$( 1 + 47 T^{2} )^{4}$$
$53$ $$( 1 + 56 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 120 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$( 1 + 71 T^{2} )^{4}$$
$73$ $$( 1 + 96 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$( 1 - 10 T + 89 T^{2} )^{2}( 1 + 10 T + 89 T^{2} )^{2}$$
$97$ $$( 1 - 144 T^{2} + 9409 T^{4} )^{2}$$
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