# Properties

 Label 1764.2.b.b Level 1764 Weight 2 Character orbit 1764.b Analytic conductor 14.086 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-7})$$ Defining polynomial: $$x^{4} - x^{3} - x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + ( -2 - \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( 2 - \beta_{2} ) q^{8} +O(q^{10})$$ $$q + \beta_{2} q^{2} + ( -2 - \beta_{2} ) q^{4} + \beta_{3} q^{5} + ( 2 - \beta_{2} ) q^{8} + \beta_{1} q^{10} + ( -1 - 2 \beta_{2} ) q^{11} + 2 \beta_{3} q^{13} + ( 2 + 3 \beta_{2} ) q^{16} -4 \beta_{3} q^{17} + ( -\beta_{1} - 2 \beta_{3} ) q^{20} + ( 4 + \beta_{2} ) q^{22} + ( -2 - 4 \beta_{2} ) q^{23} + 2 q^{25} + 2 \beta_{1} q^{26} -5 q^{29} + ( 2 \beta_{1} + \beta_{3} ) q^{31} + ( -6 - \beta_{2} ) q^{32} -4 \beta_{1} q^{34} + ( -\beta_{1} + 2 \beta_{3} ) q^{40} -2 \beta_{3} q^{41} + ( -4 - 8 \beta_{2} ) q^{43} + ( -2 + 3 \beta_{2} ) q^{44} + ( 8 + 2 \beta_{2} ) q^{46} + ( -4 \beta_{1} - 2 \beta_{3} ) q^{47} + 2 \beta_{2} q^{50} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{52} -7 q^{53} + ( -2 \beta_{1} - \beta_{3} ) q^{55} -5 \beta_{2} q^{58} + ( -6 \beta_{1} - 3 \beta_{3} ) q^{59} + 6 \beta_{3} q^{61} + ( -\beta_{1} - 4 \beta_{3} ) q^{62} + ( 2 - 5 \beta_{2} ) q^{64} -6 q^{65} + ( 4 \beta_{1} + 8 \beta_{3} ) q^{68} + ( -2 - 4 \beta_{2} ) q^{71} -4 \beta_{3} q^{73} + ( -3 - 6 \beta_{2} ) q^{79} + ( 3 \beta_{1} + 2 \beta_{3} ) q^{80} -2 \beta_{1} q^{82} + ( 2 \beta_{1} + \beta_{3} ) q^{83} + 12 q^{85} + ( 16 + 4 \beta_{2} ) q^{86} + ( -6 - 5 \beta_{2} ) q^{88} -6 \beta_{3} q^{89} + ( -4 + 6 \beta_{2} ) q^{92} + ( 2 \beta_{1} + 8 \beta_{3} ) q^{94} + 5 \beta_{3} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{2} - 6q^{4} + 10q^{8} + O(q^{10})$$ $$4q - 2q^{2} - 6q^{4} + 10q^{8} + 2q^{16} + 14q^{22} + 8q^{25} - 20q^{29} - 22q^{32} - 14q^{44} + 28q^{46} - 4q^{50} - 28q^{53} + 10q^{58} + 18q^{64} - 24q^{65} + 48q^{85} + 56q^{86} - 14q^{88} - 28q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - x^{2} - 2 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} - \nu^{2} + \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{2} + 3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 −0.895644 − 1.09445i 1.39564 − 0.228425i 1.39564 + 0.228425i −0.895644 + 1.09445i
−0.500000 1.32288i 0 −1.50000 + 1.32288i 1.73205i 0 0 2.50000 + 1.32288i 0 −2.29129 + 0.866025i
1567.2 −0.500000 1.32288i 0 −1.50000 + 1.32288i 1.73205i 0 0 2.50000 + 1.32288i 0 2.29129 0.866025i
1567.3 −0.500000 + 1.32288i 0 −1.50000 1.32288i 1.73205i 0 0 2.50000 1.32288i 0 2.29129 + 0.866025i
1567.4 −0.500000 + 1.32288i 0 −1.50000 1.32288i 1.73205i 0 0 2.50000 1.32288i 0 −2.29129 0.866025i
 $$n$$: 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 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.b 4
3.b odd 2 1 1764.2.b.h 4
4.b odd 2 1 inner 1764.2.b.b 4
7.b odd 2 1 inner 1764.2.b.b 4
7.c even 3 1 252.2.bf.d yes 4
7.d odd 6 1 252.2.bf.d yes 4
12.b even 2 1 1764.2.b.h 4
21.c even 2 1 1764.2.b.h 4
21.g even 6 1 252.2.bf.a 4
21.h odd 6 1 252.2.bf.a 4
28.d even 2 1 inner 1764.2.b.b 4
28.f even 6 1 252.2.bf.d yes 4
28.g odd 6 1 252.2.bf.d yes 4
84.h odd 2 1 1764.2.b.h 4
84.j odd 6 1 252.2.bf.a 4
84.n even 6 1 252.2.bf.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bf.a 4 21.g even 6 1
252.2.bf.a 4 21.h odd 6 1
252.2.bf.a 4 84.j odd 6 1
252.2.bf.a 4 84.n even 6 1
252.2.bf.d yes 4 7.c even 3 1
252.2.bf.d yes 4 7.d odd 6 1
252.2.bf.d yes 4 28.f even 6 1
252.2.bf.d yes 4 28.g odd 6 1
1764.2.b.b 4 1.a even 1 1 trivial
1764.2.b.b 4 4.b odd 2 1 inner
1764.2.b.b 4 7.b odd 2 1 inner
1764.2.b.b 4 28.d even 2 1 inner
1764.2.b.h 4 3.b odd 2 1
1764.2.b.h 4 12.b even 2 1
1764.2.b.h 4 21.c even 2 1
1764.2.b.h 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 $$S_{2}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{2} + 3$$ $$T_{11}^{2} + 7$$ $$T_{19}$$ $$T_{29} + 5$$ $$T_{53} + 7$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T + 2 T^{2} )^{2}$$
$3$ 1
$5$ $$( 1 - 7 T^{2} + 25 T^{4} )^{2}$$
$7$ 1
$11$ $$( 1 - 15 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 14 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$( 1 - 8 T + 23 T^{2} )^{2}( 1 + 8 T + 23 T^{2} )^{2}$$
$29$ $$( 1 + 5 T + 29 T^{2} )^{4}$$
$31$ $$( 1 + 41 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 + 37 T^{2} )^{4}$$
$41$ $$( 1 - 70 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 26 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 10 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 7 T + 53 T^{2} )^{4}$$
$59$ $$( 1 - 71 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 14 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 67 T^{2} )^{4}$$
$71$ $$( 1 - 16 T + 71 T^{2} )^{2}( 1 + 16 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 98 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 95 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 145 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 70 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 119 T^{2} + 9409 T^{4} )^{2}$$