# Properties

 Label 1764.2.j.a Level $1764$ Weight $2$ Character orbit 1764.j Analytic conductor $14.086$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} + 2 \zeta_{6} q^{5} + 3 \zeta_{6} q^{9} + ( -4 + 4 \zeta_{6} ) q^{11} + 3 \zeta_{6} q^{13} + ( 2 - 4 \zeta_{6} ) q^{15} -7 q^{17} -5 q^{19} -4 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 1 - \zeta_{6} ) q^{29} -3 \zeta_{6} q^{31} + ( 8 - 4 \zeta_{6} ) q^{33} + 11 q^{37} + ( 3 - 6 \zeta_{6} ) q^{39} -9 \zeta_{6} q^{41} + ( -5 + 5 \zeta_{6} ) q^{43} + ( -6 + 6 \zeta_{6} ) q^{45} + ( 3 - 3 \zeta_{6} ) q^{47} + ( 7 + 7 \zeta_{6} ) q^{51} + 3 q^{53} -8 q^{55} + ( 5 + 5 \zeta_{6} ) q^{57} -7 \zeta_{6} q^{59} + ( 3 - 3 \zeta_{6} ) q^{61} + ( -6 + 6 \zeta_{6} ) q^{65} -13 \zeta_{6} q^{67} + ( -4 + 8 \zeta_{6} ) q^{69} -8 q^{71} -7 q^{73} + ( -2 + \zeta_{6} ) q^{75} + ( 9 - 9 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( 1 - \zeta_{6} ) q^{83} -14 \zeta_{6} q^{85} + ( -2 + \zeta_{6} ) q^{87} -15 q^{89} + ( -3 + 6 \zeta_{6} ) q^{93} -10 \zeta_{6} q^{95} + ( -17 + 17 \zeta_{6} ) q^{97} -12 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} + 2q^{5} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} + 2q^{5} + 3q^{9} - 4q^{11} + 3q^{13} - 14q^{17} - 10q^{19} - 4q^{23} + q^{25} + q^{29} - 3q^{31} + 12q^{33} + 22q^{37} - 9q^{41} - 5q^{43} - 6q^{45} + 3q^{47} + 21q^{51} + 6q^{53} - 16q^{55} + 15q^{57} - 7q^{59} + 3q^{61} - 6q^{65} - 13q^{67} - 16q^{71} - 14q^{73} - 3q^{75} + 9q^{79} - 9q^{81} + q^{83} - 14q^{85} - 3q^{87} - 30q^{89} - 10q^{95} - 17q^{97} - 24q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$883$$ $$1081$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
589.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i 0 1.00000 1.73205i 0 0 0 1.50000 2.59808i 0
1177.1 0 −1.50000 0.866025i 0 1.00000 + 1.73205i 0 0 0 1.50000 + 2.59808i 0
 $$n$$: 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1764.2.j.a 2
3.b odd 2 1 5292.2.j.b 2
7.b odd 2 1 1764.2.j.c 2
7.c even 3 1 1764.2.i.b 2
7.c even 3 1 1764.2.l.b 2
7.d odd 6 1 252.2.i.a 2
7.d odd 6 1 252.2.l.a yes 2
9.c even 3 1 inner 1764.2.j.a 2
9.d odd 6 1 5292.2.j.b 2
21.c even 2 1 5292.2.j.c 2
21.g even 6 1 756.2.i.a 2
21.g even 6 1 756.2.l.a 2
21.h odd 6 1 5292.2.i.b 2
21.h odd 6 1 5292.2.l.b 2
28.f even 6 1 1008.2.q.f 2
28.f even 6 1 1008.2.t.b 2
63.g even 3 1 1764.2.i.b 2
63.h even 3 1 1764.2.l.b 2
63.i even 6 1 756.2.l.a 2
63.i even 6 1 2268.2.k.b 2
63.j odd 6 1 5292.2.l.b 2
63.k odd 6 1 252.2.i.a 2
63.k odd 6 1 2268.2.k.a 2
63.l odd 6 1 1764.2.j.c 2
63.n odd 6 1 5292.2.i.b 2
63.o even 6 1 5292.2.j.c 2
63.s even 6 1 756.2.i.a 2
63.s even 6 1 2268.2.k.b 2
63.t odd 6 1 252.2.l.a yes 2
63.t odd 6 1 2268.2.k.a 2
84.j odd 6 1 3024.2.q.e 2
84.j odd 6 1 3024.2.t.b 2
252.n even 6 1 1008.2.q.f 2
252.r odd 6 1 3024.2.t.b 2
252.bj even 6 1 1008.2.t.b 2
252.bn odd 6 1 3024.2.q.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 7.d odd 6 1
252.2.i.a 2 63.k odd 6 1
252.2.l.a yes 2 7.d odd 6 1
252.2.l.a yes 2 63.t odd 6 1
756.2.i.a 2 21.g even 6 1
756.2.i.a 2 63.s even 6 1
756.2.l.a 2 21.g even 6 1
756.2.l.a 2 63.i even 6 1
1008.2.q.f 2 28.f even 6 1
1008.2.q.f 2 252.n even 6 1
1008.2.t.b 2 28.f even 6 1
1008.2.t.b 2 252.bj even 6 1
1764.2.i.b 2 7.c even 3 1
1764.2.i.b 2 63.g even 3 1
1764.2.j.a 2 1.a even 1 1 trivial
1764.2.j.a 2 9.c even 3 1 inner
1764.2.j.c 2 7.b odd 2 1
1764.2.j.c 2 63.l odd 6 1
1764.2.l.b 2 7.c even 3 1
1764.2.l.b 2 63.h even 3 1
2268.2.k.a 2 63.k odd 6 1
2268.2.k.a 2 63.t odd 6 1
2268.2.k.b 2 63.i even 6 1
2268.2.k.b 2 63.s even 6 1
3024.2.q.e 2 84.j odd 6 1
3024.2.q.e 2 252.bn odd 6 1
3024.2.t.b 2 84.j odd 6 1
3024.2.t.b 2 252.r odd 6 1
5292.2.i.b 2 21.h odd 6 1
5292.2.i.b 2 63.n odd 6 1
5292.2.j.b 2 3.b odd 2 1
5292.2.j.b 2 9.d odd 6 1
5292.2.j.c 2 21.c even 2 1
5292.2.j.c 2 63.o even 6 1
5292.2.l.b 2 21.h odd 6 1
5292.2.l.b 2 63.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 2 T_{5} + 4$$ acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 - 2 T - T^{2} - 10 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 4 T + 5 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 - 3 T - 4 T^{2} - 39 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 7 T + 17 T^{2} )^{2}$$
$19$ $$( 1 + 5 T + 19 T^{2} )^{2}$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - T - 28 T^{2} - 29 T^{3} + 841 T^{4}$$
$31$ $$1 + 3 T - 22 T^{2} + 93 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 11 T + 37 T^{2} )^{2}$$
$41$ $$1 + 9 T + 40 T^{2} + 369 T^{3} + 1681 T^{4}$$
$43$ $$( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 3 T + 53 T^{2} )^{2}$$
$59$ $$1 + 7 T - 10 T^{2} + 413 T^{3} + 3481 T^{4}$$
$61$ $$1 - 3 T - 52 T^{2} - 183 T^{3} + 3721 T^{4}$$
$67$ $$1 + 13 T + 102 T^{2} + 871 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$( 1 + 7 T + 73 T^{2} )^{2}$$
$79$ $$1 - 9 T + 2 T^{2} - 711 T^{3} + 6241 T^{4}$$
$83$ $$1 - T - 82 T^{2} - 83 T^{3} + 6889 T^{4}$$
$89$ $$( 1 + 15 T + 89 T^{2} )^{2}$$
$97$ $$1 + 17 T + 192 T^{2} + 1649 T^{3} + 9409 T^{4}$$