# Properties

 Label 1764.2.k.k Level $1764$ Weight $2$ Character orbit 1764.k Analytic conductor $14.086$ Analytic rank $0$ 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.k (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$14.0856109166$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{25}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 84) 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.

 $$f(q)$$ $$=$$ $$q + 4 \zeta_{6} q^{5} +O(q^{10})$$ $$q + 4 \zeta_{6} q^{5} + ( 2 - 2 \zeta_{6} ) q^{11} -6 q^{13} + ( -4 + 4 \zeta_{6} ) q^{17} + 4 \zeta_{6} q^{19} + 2 \zeta_{6} q^{23} + ( -11 + 11 \zeta_{6} ) q^{25} + 2 q^{29} -2 \zeta_{6} q^{37} -4 q^{43} + 12 \zeta_{6} q^{47} + ( -6 + 6 \zeta_{6} ) q^{53} + 8 q^{55} + ( -8 + 8 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} -24 \zeta_{6} q^{65} + ( 8 - 8 \zeta_{6} ) q^{67} -14 q^{71} + ( 2 - 2 \zeta_{6} ) q^{73} -12 \zeta_{6} q^{79} + 4 q^{83} -16 q^{85} + ( -16 + 16 \zeta_{6} ) q^{95} -2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + O(q^{10})$$ $$2q + 4q^{5} + 2q^{11} - 12q^{13} - 4q^{17} + 4q^{19} + 2q^{23} - 11q^{25} + 4q^{29} - 2q^{37} - 8q^{43} + 12q^{47} - 6q^{53} + 16q^{55} - 8q^{59} - 6q^{61} - 24q^{65} + 8q^{67} - 28q^{71} + 2q^{73} - 12q^{79} + 8q^{83} - 32q^{85} - 16q^{95} - 4q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 2.00000 + 3.46410i 0 0 0 0 0
1549.1 0 0 0 2.00000 3.46410i 0 0 0 0 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
7.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.k.k 2
3.b odd 2 1 588.2.i.e 2
7.b odd 2 1 1764.2.k.a 2
7.c even 3 1 252.2.a.a 1
7.c even 3 1 inner 1764.2.k.k 2
7.d odd 6 1 1764.2.a.k 1
7.d odd 6 1 1764.2.k.a 2
12.b even 2 1 2352.2.q.b 2
21.c even 2 1 588.2.i.d 2
21.g even 6 1 588.2.a.d 1
21.g even 6 1 588.2.i.d 2
21.h odd 6 1 84.2.a.a 1
21.h odd 6 1 588.2.i.e 2
28.f even 6 1 7056.2.a.cd 1
28.g odd 6 1 1008.2.a.a 1
35.j even 6 1 6300.2.a.w 1
35.l odd 12 2 6300.2.k.g 2
56.k odd 6 1 4032.2.a.bn 1
56.p even 6 1 4032.2.a.bm 1
63.g even 3 1 2268.2.j.n 2
63.h even 3 1 2268.2.j.n 2
63.j odd 6 1 2268.2.j.a 2
63.n odd 6 1 2268.2.j.a 2
84.h odd 2 1 2352.2.q.z 2
84.j odd 6 1 2352.2.a.a 1
84.j odd 6 1 2352.2.q.z 2
84.n even 6 1 336.2.a.f 1
84.n even 6 1 2352.2.q.b 2
105.o odd 6 1 2100.2.a.r 1
105.x even 12 2 2100.2.k.i 2
168.s odd 6 1 1344.2.a.k 1
168.v even 6 1 1344.2.a.a 1
168.ba even 6 1 9408.2.a.bn 1
168.be odd 6 1 9408.2.a.df 1
336.bt odd 12 2 5376.2.c.q 2
336.bu even 12 2 5376.2.c.p 2
420.ba even 6 1 8400.2.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.a 1 21.h odd 6 1
252.2.a.a 1 7.c even 3 1
336.2.a.f 1 84.n even 6 1
588.2.a.d 1 21.g even 6 1
588.2.i.d 2 21.c even 2 1
588.2.i.d 2 21.g even 6 1
588.2.i.e 2 3.b odd 2 1
588.2.i.e 2 21.h odd 6 1
1008.2.a.a 1 28.g odd 6 1
1344.2.a.a 1 168.v even 6 1
1344.2.a.k 1 168.s odd 6 1
1764.2.a.k 1 7.d odd 6 1
1764.2.k.a 2 7.b odd 2 1
1764.2.k.a 2 7.d odd 6 1
1764.2.k.k 2 1.a even 1 1 trivial
1764.2.k.k 2 7.c even 3 1 inner
2100.2.a.r 1 105.o odd 6 1
2100.2.k.i 2 105.x even 12 2
2268.2.j.a 2 63.j odd 6 1
2268.2.j.a 2 63.n odd 6 1
2268.2.j.n 2 63.g even 3 1
2268.2.j.n 2 63.h even 3 1
2352.2.a.a 1 84.j odd 6 1
2352.2.q.b 2 12.b even 2 1
2352.2.q.b 2 84.n even 6 1
2352.2.q.z 2 84.h odd 2 1
2352.2.q.z 2 84.j odd 6 1
4032.2.a.bm 1 56.p even 6 1
4032.2.a.bn 1 56.k odd 6 1
5376.2.c.p 2 336.bu even 12 2
5376.2.c.q 2 336.bt odd 12 2
6300.2.a.w 1 35.j even 6 1
6300.2.k.g 2 35.l odd 12 2
7056.2.a.cd 1 28.f even 6 1
8400.2.a.e 1 420.ba even 6 1
9408.2.a.bn 1 168.ba even 6 1
9408.2.a.df 1 168.be odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} - 4 T_{5} + 16$$ $$T_{11}^{2} - 2 T_{11} + 4$$ $$T_{13} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 4 T + 11 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 6 T + 13 T^{2} )^{2}$$
$17$ $$1 + 4 T - T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$1 - 4 T - 3 T^{2} - 76 T^{3} + 361 T^{4}$$
$23$ $$1 - 2 T - 19 T^{2} - 46 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$1 - 31 T^{2} + 961 T^{4}$$
$37$ $$1 + 2 T - 33 T^{2} + 74 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 4 T + 43 T^{2} )^{2}$$
$47$ $$1 - 12 T + 97 T^{2} - 564 T^{3} + 2209 T^{4}$$
$53$ $$1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4}$$
$59$ $$1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4}$$
$61$ $$1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4}$$
$67$ $$1 - 8 T - 3 T^{2} - 536 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 14 T + 71 T^{2} )^{2}$$
$73$ $$1 - 2 T - 69 T^{2} - 146 T^{3} + 5329 T^{4}$$
$79$ $$1 + 12 T + 65 T^{2} + 948 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 4 T + 83 T^{2} )^{2}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{2}$$