# Properties

 Label 1764.2.l.a Level $1764$ Weight $2$ Character orbit 1764.l 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.l (of order $$3$$, degree $$2$$, not 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 36) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \zeta_{6} ) q^{3} -3 q^{5} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 - \zeta_{6} ) q^{3} -3 q^{5} + 3 \zeta_{6} q^{9} + 3 q^{11} + ( -1 + \zeta_{6} ) q^{13} + ( 3 + 3 \zeta_{6} ) q^{15} + ( 6 - 6 \zeta_{6} ) q^{17} -4 \zeta_{6} q^{19} -3 q^{23} + 4 q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} -3 \zeta_{6} q^{29} + 5 \zeta_{6} q^{31} + ( -3 - 3 \zeta_{6} ) q^{33} -2 \zeta_{6} q^{37} + ( 2 - \zeta_{6} ) q^{39} + ( 3 - 3 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} -9 \zeta_{6} q^{45} + ( -9 + 9 \zeta_{6} ) q^{47} + ( -12 + 6 \zeta_{6} ) q^{51} + ( 6 - 6 \zeta_{6} ) q^{53} -9 q^{55} + ( -4 + 8 \zeta_{6} ) q^{57} -3 \zeta_{6} q^{59} + ( -13 + 13 \zeta_{6} ) q^{61} + ( 3 - 3 \zeta_{6} ) q^{65} + 7 \zeta_{6} q^{67} + ( 3 + 3 \zeta_{6} ) q^{69} -12 q^{71} + ( -10 + 10 \zeta_{6} ) q^{73} + ( -4 - 4 \zeta_{6} ) q^{75} + ( -11 + 11 \zeta_{6} ) q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -9 \zeta_{6} q^{83} + ( -18 + 18 \zeta_{6} ) q^{85} + ( -3 + 6 \zeta_{6} ) q^{87} + 6 \zeta_{6} q^{89} + ( 5 - 10 \zeta_{6} ) q^{93} + 12 \zeta_{6} q^{95} + 11 \zeta_{6} q^{97} + 9 \zeta_{6} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{3} - 6q^{5} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{3} - 6q^{5} + 3q^{9} + 6q^{11} - q^{13} + 9q^{15} + 6q^{17} - 4q^{19} - 6q^{23} + 8q^{25} - 3q^{29} + 5q^{31} - 9q^{33} - 2q^{37} + 3q^{39} + 3q^{41} + q^{43} - 9q^{45} - 9q^{47} - 18q^{51} + 6q^{53} - 18q^{55} - 3q^{59} - 13q^{61} + 3q^{65} + 7q^{67} + 9q^{69} - 24q^{71} - 10q^{73} - 12q^{75} - 11q^{79} - 9q^{81} - 9q^{83} - 18q^{85} + 6q^{89} + 12q^{95} + 11q^{97} + 9q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
949.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 −1.50000 0.866025i 0 −3.00000 0 0 0 1.50000 + 2.59808i 0
961.1 0 −1.50000 + 0.866025i 0 −3.00000 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
63.g 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.l.a 2
3.b odd 2 1 5292.2.l.c 2
7.b odd 2 1 1764.2.l.c 2
7.c even 3 1 1764.2.i.c 2
7.c even 3 1 1764.2.j.b 2
7.d odd 6 1 36.2.e.a 2
7.d odd 6 1 1764.2.i.a 2
9.c even 3 1 1764.2.i.c 2
9.d odd 6 1 5292.2.i.a 2
21.c even 2 1 5292.2.l.a 2
21.g even 6 1 108.2.e.a 2
21.g even 6 1 5292.2.i.c 2
21.h odd 6 1 5292.2.i.a 2
21.h odd 6 1 5292.2.j.a 2
28.f even 6 1 144.2.i.a 2
35.i odd 6 1 900.2.i.b 2
35.k even 12 2 900.2.s.b 4
56.j odd 6 1 576.2.i.f 2
56.m even 6 1 576.2.i.e 2
63.g even 3 1 inner 1764.2.l.a 2
63.h even 3 1 1764.2.j.b 2
63.i even 6 1 108.2.e.a 2
63.j odd 6 1 5292.2.j.a 2
63.k odd 6 1 324.2.a.c 1
63.k odd 6 1 1764.2.l.c 2
63.l odd 6 1 1764.2.i.a 2
63.n odd 6 1 5292.2.l.c 2
63.o even 6 1 5292.2.i.c 2
63.s even 6 1 324.2.a.a 1
63.s even 6 1 5292.2.l.a 2
63.t odd 6 1 36.2.e.a 2
84.j odd 6 1 432.2.i.c 2
105.p even 6 1 2700.2.i.b 2
105.w odd 12 2 2700.2.s.b 4
168.ba even 6 1 1728.2.i.d 2
168.be odd 6 1 1728.2.i.c 2
252.n even 6 1 1296.2.a.k 1
252.r odd 6 1 432.2.i.c 2
252.bj even 6 1 144.2.i.a 2
252.bn odd 6 1 1296.2.a.b 1
315.q odd 6 1 900.2.i.b 2
315.u even 6 1 8100.2.a.g 1
315.bn odd 6 1 8100.2.a.j 1
315.bq even 6 1 2700.2.i.b 2
315.bs even 12 2 900.2.s.b 4
315.bu odd 12 2 2700.2.s.b 4
315.bw odd 12 2 8100.2.d.c 2
315.cg even 12 2 8100.2.d.h 2
504.u odd 6 1 5184.2.a.bb 1
504.y even 6 1 5184.2.a.ba 1
504.bf even 6 1 576.2.i.e 2
504.bp odd 6 1 576.2.i.f 2
504.ca even 6 1 1728.2.i.d 2
504.cm odd 6 1 1728.2.i.c 2
504.cw odd 6 1 5184.2.a.e 1
504.cz even 6 1 5184.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 7.d odd 6 1
36.2.e.a 2 63.t odd 6 1
108.2.e.a 2 21.g even 6 1
108.2.e.a 2 63.i even 6 1
144.2.i.a 2 28.f even 6 1
144.2.i.a 2 252.bj even 6 1
324.2.a.a 1 63.s even 6 1
324.2.a.c 1 63.k odd 6 1
432.2.i.c 2 84.j odd 6 1
432.2.i.c 2 252.r odd 6 1
576.2.i.e 2 56.m even 6 1
576.2.i.e 2 504.bf even 6 1
576.2.i.f 2 56.j odd 6 1
576.2.i.f 2 504.bp odd 6 1
900.2.i.b 2 35.i odd 6 1
900.2.i.b 2 315.q odd 6 1
900.2.s.b 4 35.k even 12 2
900.2.s.b 4 315.bs even 12 2
1296.2.a.b 1 252.bn odd 6 1
1296.2.a.k 1 252.n even 6 1
1728.2.i.c 2 168.be odd 6 1
1728.2.i.c 2 504.cm odd 6 1
1728.2.i.d 2 168.ba even 6 1
1728.2.i.d 2 504.ca even 6 1
1764.2.i.a 2 7.d odd 6 1
1764.2.i.a 2 63.l odd 6 1
1764.2.i.c 2 7.c even 3 1
1764.2.i.c 2 9.c even 3 1
1764.2.j.b 2 7.c even 3 1
1764.2.j.b 2 63.h even 3 1
1764.2.l.a 2 1.a even 1 1 trivial
1764.2.l.a 2 63.g even 3 1 inner
1764.2.l.c 2 7.b odd 2 1
1764.2.l.c 2 63.k odd 6 1
2700.2.i.b 2 105.p even 6 1
2700.2.i.b 2 315.bq even 6 1
2700.2.s.b 4 105.w odd 12 2
2700.2.s.b 4 315.bu odd 12 2
5184.2.a.e 1 504.cw odd 6 1
5184.2.a.f 1 504.cz even 6 1
5184.2.a.ba 1 504.y even 6 1
5184.2.a.bb 1 504.u odd 6 1
5292.2.i.a 2 9.d odd 6 1
5292.2.i.a 2 21.h odd 6 1
5292.2.i.c 2 21.g even 6 1
5292.2.i.c 2 63.o even 6 1
5292.2.j.a 2 21.h odd 6 1
5292.2.j.a 2 63.j odd 6 1
5292.2.l.a 2 21.c even 2 1
5292.2.l.a 2 63.s even 6 1
5292.2.l.c 2 3.b odd 2 1
5292.2.l.c 2 63.n odd 6 1
8100.2.a.g 1 315.u even 6 1
8100.2.a.j 1 315.bn odd 6 1
8100.2.d.c 2 315.bw odd 12 2
8100.2.d.h 2 315.cg even 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} + 3$$ acting on $$S_{2}^{\mathrm{new}}(1764, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$( 3 + T )^{2}$$
$7$ $$T^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$1 + T + T^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$( 3 + T )^{2}$$
$29$ $$9 + 3 T + T^{2}$$
$31$ $$25 - 5 T + T^{2}$$
$37$ $$4 + 2 T + T^{2}$$
$41$ $$9 - 3 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$81 + 9 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$9 + 3 T + T^{2}$$
$61$ $$169 + 13 T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$( 12 + T )^{2}$$
$73$ $$100 + 10 T + T^{2}$$
$79$ $$121 + 11 T + T^{2}$$
$83$ $$81 + 9 T + T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$121 - 11 T + T^{2}$$