# Properties

 Label 1764.2.e.f Level $1764$ Weight $2$ Character orbit 1764.e Analytic conductor $14.086$ Analytic rank $0$ Dimension $8$ CM discriminant -7 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: $$8$$ Coefficient field: 8.0.157351936.1 Defining polynomial: $$x^{8} + x^{4} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{2} -\beta_{1} q^{4} -\beta_{5} q^{8} +O(q^{10})$$ $$q + \beta_{6} q^{2} -\beta_{1} q^{4} -\beta_{5} q^{8} + ( \beta_{5} + \beta_{6} + \beta_{7} ) q^{11} + \beta_{2} q^{16} + ( 2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{22} + ( 2 \beta_{3} - \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{23} + 5 q^{25} + ( -2 \beta_{3} - \beta_{5} + 3 \beta_{6} + \beta_{7} ) q^{29} + ( -\beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{32} + ( 4 \beta_{1} - 2 \beta_{4} ) q^{37} + ( -4 \beta_{1} - 2 \beta_{4} ) q^{43} + ( -\beta_{3} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{44} + ( 6 - 3 \beta_{1} + \beta_{2} + \beta_{4} ) q^{46} + 5 \beta_{6} q^{50} + ( 2 \beta_{3} - \beta_{5} - 5 \beta_{6} + \beta_{7} ) q^{53} + ( -6 - 3 \beta_{1} + \beta_{2} - \beta_{4} ) q^{58} + ( \beta_{1} - 2 \beta_{4} ) q^{64} + ( -2 - 4 \beta_{2} ) q^{67} + ( 2 \beta_{3} - 3 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{71} + ( -4 \beta_{3} + 4 \beta_{5} ) q^{74} + ( -2 - 4 \beta_{2} ) q^{79} + ( -4 \beta_{3} - 4 \beta_{5} ) q^{86} + ( -8 - 3 \beta_{1} + \beta_{2} + 2 \beta_{4} ) q^{88} + ( \beta_{3} - 3 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} ) q^{92} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 4q^{16} + 20q^{22} + 40q^{25} + 44q^{46} - 52q^{58} - 68q^{88} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$ $$\beta_{2}$$ $$=$$ $$\nu^{4}$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{5} + 5 \nu$$$$)/3$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{6} + \nu^{2}$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{5} + 4 \nu$$$$)/3$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{7} - 5 \nu^{3}$$$$)/12$$ $$\beta_{7}$$ $$=$$ $$($$$$-5 \nu^{7} + 4 \nu^{5} + 11 \nu^{3} + 20 \nu$$$$)/24$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} + \beta_{3}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$\beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{7} - 5 \beta_{6} - \beta_{3}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$\beta_{2}$$ $$\nu^{5}$$ $$=$$ $$($$$$-5 \beta_{5} + 4 \beta_{3}$$$$)/3$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{4} - \beta_{1}$$ $$\nu^{7}$$ $$=$$ $$($$$$-10 \beta_{7} - 11 \beta_{6} + 5 \beta_{3}$$$$)/3$$

## 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
 −0.581861 + 1.28897i −0.581861 − 1.28897i −1.28897 + 0.581861i −1.28897 − 0.581861i 1.28897 + 0.581861i 1.28897 − 0.581861i 0.581861 + 1.28897i 0.581861 − 1.28897i
−1.28897 0.581861i 0 1.32288 + 1.50000i 0 0 0 −0.832353 2.70318i 0 0
1079.2 −1.28897 + 0.581861i 0 1.32288 1.50000i 0 0 0 −0.832353 + 2.70318i 0 0
1079.3 −0.581861 1.28897i 0 −1.32288 + 1.50000i 0 0 0 2.70318 + 0.832353i 0 0
1079.4 −0.581861 + 1.28897i 0 −1.32288 1.50000i 0 0 0 2.70318 0.832353i 0 0
1079.5 0.581861 1.28897i 0 −1.32288 1.50000i 0 0 0 −2.70318 + 0.832353i 0 0
1079.6 0.581861 + 1.28897i 0 −1.32288 + 1.50000i 0 0 0 −2.70318 0.832353i 0 0
1079.7 1.28897 0.581861i 0 1.32288 1.50000i 0 0 0 0.832353 2.70318i 0 0
1079.8 1.28897 + 0.581861i 0 1.32288 + 1.50000i 0 0 0 0.832353 + 2.70318i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1079.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
4.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.f 8
3.b odd 2 1 inner 1764.2.e.f 8
4.b odd 2 1 inner 1764.2.e.f 8
7.b odd 2 1 CM 1764.2.e.f 8
12.b even 2 1 inner 1764.2.e.f 8
21.c even 2 1 inner 1764.2.e.f 8
28.d even 2 1 inner 1764.2.e.f 8
84.h odd 2 1 inner 1764.2.e.f 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.e.f 8 1.a even 1 1 trivial
1764.2.e.f 8 3.b odd 2 1 inner
1764.2.e.f 8 4.b odd 2 1 inner
1764.2.e.f 8 7.b odd 2 1 CM
1764.2.e.f 8 12.b even 2 1 inner
1764.2.e.f 8 21.c even 2 1 inner
1764.2.e.f 8 28.d even 2 1 inner
1764.2.e.f 8 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 $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}$$ $$T_{13}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$( 36 - 44 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 324 - 92 T^{2} + T^{4} )^{2}$$
$29$ $$( 2916 + 116 T^{2} + T^{4} )^{2}$$
$31$ $$T^{8}$$
$37$ $$( -112 + T^{2} )^{4}$$
$41$ $$T^{8}$$
$43$ $$( 144 + T^{2} )^{4}$$
$47$ $$T^{8}$$
$53$ $$( 36 + 212 T^{2} + T^{4} )^{2}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 252 + T^{2} )^{4}$$
$71$ $$( 12996 - 284 T^{2} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$( 252 + T^{2} )^{4}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$