# Properties

 Label 1764.4.k.b Level $1764$ Weight $4$ Character orbit 1764.k Analytic conductor $104.079$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1764.k (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$104.079369250$$ 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 12) 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.

 $$f(q)$$ $$=$$ $$q -18 \zeta_{6} q^{5} +O(q^{10})$$ $$q -18 \zeta_{6} q^{5} + ( 36 - 36 \zeta_{6} ) q^{11} -10 q^{13} + ( 18 - 18 \zeta_{6} ) q^{17} + 100 \zeta_{6} q^{19} + 72 \zeta_{6} q^{23} + ( -199 + 199 \zeta_{6} ) q^{25} + 234 q^{29} + ( 16 - 16 \zeta_{6} ) q^{31} + 226 \zeta_{6} q^{37} -90 q^{41} + 452 q^{43} + 432 \zeta_{6} q^{47} + ( 414 - 414 \zeta_{6} ) q^{53} -648 q^{55} + ( -684 + 684 \zeta_{6} ) q^{59} -422 \zeta_{6} q^{61} + 180 \zeta_{6} q^{65} + ( -332 + 332 \zeta_{6} ) q^{67} + 360 q^{71} + ( -26 + 26 \zeta_{6} ) q^{73} -512 \zeta_{6} q^{79} + 1188 q^{83} -324 q^{85} -630 \zeta_{6} q^{89} + ( 1800 - 1800 \zeta_{6} ) q^{95} -1054 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 18q^{5} + O(q^{10})$$ $$2q - 18q^{5} + 36q^{11} - 20q^{13} + 18q^{17} + 100q^{19} + 72q^{23} - 199q^{25} + 468q^{29} + 16q^{31} + 226q^{37} - 180q^{41} + 904q^{43} + 432q^{47} + 414q^{53} - 1296q^{55} - 684q^{59} - 422q^{61} + 180q^{65} - 332q^{67} + 720q^{71} - 26q^{73} - 512q^{79} + 2376q^{83} - 648q^{85} - 630q^{89} + 1800q^{95} - 2108q^{97} + O(q^{100})$$

## 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 −9.00000 15.5885i 0 0 0 0 0
1549.1 0 0 0 −9.00000 + 15.5885i 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 1764.4.k.b 2
3.b odd 2 1 588.4.i.d 2
7.b odd 2 1 1764.4.k.o 2
7.c even 3 1 36.4.a.a 1
7.c even 3 1 inner 1764.4.k.b 2
7.d odd 6 1 1764.4.a.b 1
7.d odd 6 1 1764.4.k.o 2
21.c even 2 1 588.4.i.e 2
21.g even 6 1 588.4.a.c 1
21.g even 6 1 588.4.i.e 2
21.h odd 6 1 12.4.a.a 1
21.h odd 6 1 588.4.i.d 2
28.g odd 6 1 144.4.a.g 1
35.j even 6 1 900.4.a.g 1
35.l odd 12 2 900.4.d.c 2
56.k odd 6 1 576.4.a.a 1
56.p even 6 1 576.4.a.b 1
63.g even 3 1 324.4.e.a 2
63.h even 3 1 324.4.e.a 2
63.j odd 6 1 324.4.e.h 2
63.n odd 6 1 324.4.e.h 2
84.j odd 6 1 2352.4.a.bk 1
84.n even 6 1 48.4.a.a 1
105.o odd 6 1 300.4.a.b 1
105.x even 12 2 300.4.d.e 2
168.s odd 6 1 192.4.a.f 1
168.v even 6 1 192.4.a.l 1
231.l even 6 1 1452.4.a.d 1
273.w odd 6 1 2028.4.a.c 1
273.cd even 12 2 2028.4.b.c 2
336.bt odd 12 2 768.4.d.g 2
336.bu even 12 2 768.4.d.j 2
420.ba even 6 1 1200.4.a.be 1
420.bp odd 12 2 1200.4.f.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.4.a.a 1 21.h odd 6 1
36.4.a.a 1 7.c even 3 1
48.4.a.a 1 84.n even 6 1
144.4.a.g 1 28.g odd 6 1
192.4.a.f 1 168.s odd 6 1
192.4.a.l 1 168.v even 6 1
300.4.a.b 1 105.o odd 6 1
300.4.d.e 2 105.x even 12 2
324.4.e.a 2 63.g even 3 1
324.4.e.a 2 63.h even 3 1
324.4.e.h 2 63.j odd 6 1
324.4.e.h 2 63.n odd 6 1
576.4.a.a 1 56.k odd 6 1
576.4.a.b 1 56.p even 6 1
588.4.a.c 1 21.g even 6 1
588.4.i.d 2 3.b odd 2 1
588.4.i.d 2 21.h odd 6 1
588.4.i.e 2 21.c even 2 1
588.4.i.e 2 21.g even 6 1
768.4.d.g 2 336.bt odd 12 2
768.4.d.j 2 336.bu even 12 2
900.4.a.g 1 35.j even 6 1
900.4.d.c 2 35.l odd 12 2
1200.4.a.be 1 420.ba even 6 1
1200.4.f.d 2 420.bp odd 12 2
1452.4.a.d 1 231.l even 6 1
1764.4.a.b 1 7.d odd 6 1
1764.4.k.b 2 1.a even 1 1 trivial
1764.4.k.b 2 7.c even 3 1 inner
1764.4.k.o 2 7.b odd 2 1
1764.4.k.o 2 7.d odd 6 1
2028.4.a.c 1 273.w odd 6 1
2028.4.b.c 2 273.cd even 12 2
2352.4.a.bk 1 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1764, [\chi])$$:

 $$T_{5}^{2} + 18 T_{5} + 324$$ $$T_{11}^{2} - 36 T_{11} + 1296$$ $$T_{13} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$324 + 18 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1296 - 36 T + T^{2}$$
$13$ $$( 10 + T )^{2}$$
$17$ $$324 - 18 T + T^{2}$$
$19$ $$10000 - 100 T + T^{2}$$
$23$ $$5184 - 72 T + T^{2}$$
$29$ $$( -234 + T )^{2}$$
$31$ $$256 - 16 T + T^{2}$$
$37$ $$51076 - 226 T + T^{2}$$
$41$ $$( 90 + T )^{2}$$
$43$ $$( -452 + T )^{2}$$
$47$ $$186624 - 432 T + T^{2}$$
$53$ $$171396 - 414 T + T^{2}$$
$59$ $$467856 + 684 T + T^{2}$$
$61$ $$178084 + 422 T + T^{2}$$
$67$ $$110224 + 332 T + T^{2}$$
$71$ $$( -360 + T )^{2}$$
$73$ $$676 + 26 T + T^{2}$$
$79$ $$262144 + 512 T + T^{2}$$
$83$ $$( -1188 + T )^{2}$$
$89$ $$396900 + 630 T + T^{2}$$
$97$ $$( 1054 + T )^{2}$$