# Properties

 Label 1764.2.b.k Level $1764$ Weight $2$ Character orbit 1764.b Analytic conductor $14.086$ Analytic rank $0$ Dimension $8$ CM no 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: $$8$$ Coefficient field: 8.0.1212153856.10 Defining polynomial: $$x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 196) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} ) q^{2} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 1 - \beta_{3} - 2 \beta_{6} ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta_{3} ) q^{2} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} ) q^{4} + ( \beta_{2} + \beta_{7} ) q^{5} + ( 1 - \beta_{3} - 2 \beta_{6} ) q^{8} + ( \beta_{1} + \beta_{2} - \beta_{4} ) q^{10} + ( 1 - \beta_{3} + \beta_{5} - 2 \beta_{6} ) q^{11} + ( \beta_{2} - \beta_{7} ) q^{13} + ( -1 - \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{16} + ( -2 \beta_{2} + \beta_{7} ) q^{17} + ( -\beta_{2} + 2 \beta_{4} ) q^{19} + ( \beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{7} ) q^{20} + ( -2 \beta_{3} - 3 \beta_{5} - \beta_{6} ) q^{22} + ( 2 - 4 \beta_{6} ) q^{23} + ( 3 + 2 \beta_{3} + 2 \beta_{5} ) q^{25} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{26} + ( 2 - 2 \beta_{3} - 2 \beta_{5} ) q^{29} + ( 4 \beta_{1} - 2 \beta_{2} + 4 \beta_{4} + 2 \beta_{7} ) q^{31} + ( -5 + \beta_{3} - 2 \beta_{5} ) q^{32} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{34} -4 q^{37} + ( \beta_{1} + \beta_{2} + 4 \beta_{7} ) q^{38} + ( \beta_{1} - \beta_{2} + 3 \beta_{4} + 2 \beta_{7} ) q^{40} -3 \beta_{2} q^{41} + ( 1 + 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{43} + ( -4 + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} ) q^{44} + ( -2 - 2 \beta_{3} - 4 \beta_{5} - 4 \beta_{6} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{7} ) q^{47} + ( 5 + \beta_{3} + 2 \beta_{5} - 2 \beta_{6} ) q^{50} + ( 3 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{7} ) q^{52} + ( 4 + 6 \beta_{3} + 6 \beta_{5} ) q^{53} + ( -2 \beta_{2} + 4 \beta_{4} ) q^{55} + ( 4 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{58} + ( 4 \beta_{1} - 3 \beta_{2} + 6 \beta_{4} + 2 \beta_{7} ) q^{59} + ( 5 \beta_{2} - 5 \beta_{7} ) q^{61} + ( 6 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 8 \beta_{7} ) q^{62} + ( -7 - 3 \beta_{3} + \beta_{5} - \beta_{6} ) q^{64} + ( -2 - 2 \beta_{3} - 2 \beta_{5} ) q^{65} + ( -2 - 2 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{67} + ( -5 \beta_{1} - \beta_{2} - 2 \beta_{4} - 2 \beta_{7} ) q^{68} + ( -\beta_{2} + 8 \beta_{7} ) q^{73} + ( -4 - 4 \beta_{3} ) q^{74} + ( 2 \beta_{1} - 3 \beta_{4} ) q^{76} + ( 2 - 2 \beta_{3} + 2 \beta_{5} - 4 \beta_{6} ) q^{79} + ( 3 \beta_{1} + \beta_{2} - \beta_{4} + 6 \beta_{7} ) q^{80} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{82} + ( -4 \beta_{1} - \beta_{2} + 2 \beta_{4} - 2 \beta_{7} ) q^{83} + ( 4 + 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( -4 + 2 \beta_{3} + \beta_{5} - 5 \beta_{6} ) q^{86} + ( -6 + 3 \beta_{5} - \beta_{6} ) q^{88} + ( 3 \beta_{2} + 2 \beta_{7} ) q^{89} + ( -10 - 2 \beta_{3} - 6 \beta_{5} - 2 \beta_{6} ) q^{92} + ( -4 \beta_{1} + 4 \beta_{2} - 2 \beta_{4} ) q^{94} + ( -2 + 4 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} ) q^{95} -5 \beta_{7} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{2} - 4q^{4} + 4q^{8} + O(q^{10})$$ $$8q + 4q^{2} - 4q^{4} + 4q^{8} + 4q^{16} + 16q^{22} + 8q^{25} + 32q^{29} - 36q^{32} - 32q^{37} - 24q^{44} - 8q^{46} + 20q^{50} - 16q^{53} - 52q^{64} - 16q^{74} + 16q^{85} - 64q^{86} - 64q^{88} - 56q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 10 x^{4} - 16 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 2 \nu^{3}$$$$)/8$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} + 2 \nu^{4} - 2 \nu^{2}$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 2 \nu^{3}$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{4} - 2 \nu^{2} + 2$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} - 2 \nu^{4} + 6 \nu^{2} - 4$$$$)/4$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} - 10 \nu^{3} + 8 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{3} + 1$$ $$\nu^{3}$$ $$=$$ $$-\beta_{7} + \beta_{4} + \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3}$$ $$\nu^{5}$$ $$=$$ $$2 \beta_{4} + 2 \beta_{2}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{6} + 4 \beta_{5} - 2 \beta_{3} - 2$$ $$\nu^{7}$$ $$=$$ $$2 \beta_{7} - 2 \beta_{4} + 8 \beta_{2} - 2 \beta_{1}$$

## 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
 −1.07072 + 0.923880i 1.07072 − 0.923880i 1.07072 + 0.923880i −1.07072 − 0.923880i −1.36145 − 0.382683i 1.36145 + 0.382683i 1.36145 − 0.382683i −1.36145 + 0.382683i
−0.207107 1.39897i 0 −1.91421 + 0.579471i 2.61313i 0 0 1.20711 + 2.55791i 0 −3.65568 + 0.541196i
1567.2 −0.207107 1.39897i 0 −1.91421 + 0.579471i 2.61313i 0 0 1.20711 + 2.55791i 0 3.65568 0.541196i
1567.3 −0.207107 + 1.39897i 0 −1.91421 0.579471i 2.61313i 0 0 1.20711 2.55791i 0 3.65568 + 0.541196i
1567.4 −0.207107 + 1.39897i 0 −1.91421 0.579471i 2.61313i 0 0 1.20711 2.55791i 0 −3.65568 0.541196i
1567.5 1.20711 0.736813i 0 0.914214 1.77882i 1.08239i 0 0 −0.207107 2.82083i 0 −0.797521 1.30656i
1567.6 1.20711 0.736813i 0 0.914214 1.77882i 1.08239i 0 0 −0.207107 2.82083i 0 0.797521 + 1.30656i
1567.7 1.20711 + 0.736813i 0 0.914214 + 1.77882i 1.08239i 0 0 −0.207107 + 2.82083i 0 0.797521 1.30656i
1567.8 1.20711 + 0.736813i 0 0.914214 + 1.77882i 1.08239i 0 0 −0.207107 + 2.82083i 0 −0.797521 + 1.30656i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
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.k 8
3.b odd 2 1 196.2.d.c 8
4.b odd 2 1 inner 1764.2.b.k 8
7.b odd 2 1 inner 1764.2.b.k 8
12.b even 2 1 196.2.d.c 8
21.c even 2 1 196.2.d.c 8
21.g even 6 2 196.2.f.d 16
21.h odd 6 2 196.2.f.d 16
24.f even 2 1 3136.2.f.i 8
24.h odd 2 1 3136.2.f.i 8
28.d even 2 1 inner 1764.2.b.k 8
84.h odd 2 1 196.2.d.c 8
84.j odd 6 2 196.2.f.d 16
84.n even 6 2 196.2.f.d 16
168.e odd 2 1 3136.2.f.i 8
168.i even 2 1 3136.2.f.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
196.2.d.c 8 3.b odd 2 1
196.2.d.c 8 12.b even 2 1
196.2.d.c 8 21.c even 2 1
196.2.d.c 8 84.h odd 2 1
196.2.f.d 16 21.g even 6 2
196.2.f.d 16 21.h odd 6 2
196.2.f.d 16 84.j odd 6 2
196.2.f.d 16 84.n even 6 2
1764.2.b.k 8 1.a even 1 1 trivial
1764.2.b.k 8 4.b odd 2 1 inner
1764.2.b.k 8 7.b odd 2 1 inner
1764.2.b.k 8 28.d even 2 1 inner
3136.2.f.i 8 24.f even 2 1
3136.2.f.i 8 24.h odd 2 1
3136.2.f.i 8 168.e odd 2 1
3136.2.f.i 8 168.i even 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 8 T_{5}^{2} + 8$$ $$T_{11}^{4} + 20 T_{11}^{2} + 68$$ $$T_{19}^{4} - 28 T_{19}^{2} + 34$$ $$T_{29}^{2} - 8 T_{29} + 8$$ $$T_{53}^{2} + 4 T_{53} - 68$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 - 4 T + 3 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$3$ $$T^{8}$$
$5$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$7$ $$T^{8}$$
$11$ $$( 68 + 20 T^{2} + T^{4} )^{2}$$
$13$ $$( 8 + 8 T^{2} + T^{4} )^{2}$$
$17$ $$( 2 + 20 T^{2} + T^{4} )^{2}$$
$19$ $$( 34 - 28 T^{2} + T^{4} )^{2}$$
$23$ $$( 272 + 56 T^{2} + T^{4} )^{2}$$
$29$ $$( 8 - 8 T + T^{2} )^{4}$$
$31$ $$( 2176 - 96 T^{2} + T^{4} )^{2}$$
$37$ $$( 4 + T )^{8}$$
$41$ $$( 162 + 36 T^{2} + T^{4} )^{2}$$
$43$ $$( 3332 + 116 T^{2} + T^{4} )^{2}$$
$47$ $$( 544 - 48 T^{2} + T^{4} )^{2}$$
$53$ $$( -68 + 4 T + T^{2} )^{4}$$
$59$ $$( 9826 - 204 T^{2} + T^{4} )^{2}$$
$61$ $$( 5000 + 200 T^{2} + T^{4} )^{2}$$
$67$ $$( 1088 + 112 T^{2} + T^{4} )^{2}$$
$71$ $$T^{8}$$
$73$ $$( 12482 + 260 T^{2} + T^{4} )^{2}$$
$79$ $$( 1088 + 80 T^{2} + T^{4} )^{2}$$
$83$ $$( 1666 - 108 T^{2} + T^{4} )^{2}$$
$89$ $$( 578 + 52 T^{2} + T^{4} )^{2}$$
$97$ $$( 1250 + 100 T^{2} + T^{4} )^{2}$$