# Properties

 Label 1764.4.k.d 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 28) 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.

 $$f(q)$$ $$=$$ $$q -8 \zeta_{6} q^{5} +O(q^{10})$$ $$q -8 \zeta_{6} q^{5} + ( -40 + 40 \zeta_{6} ) q^{11} -12 q^{13} + ( -58 + 58 \zeta_{6} ) q^{17} -26 \zeta_{6} q^{19} -64 \zeta_{6} q^{23} + ( 61 - 61 \zeta_{6} ) q^{25} + 62 q^{29} + ( -252 + 252 \zeta_{6} ) q^{31} -26 \zeta_{6} q^{37} -6 q^{41} + 416 q^{43} -396 \zeta_{6} q^{47} + ( -450 + 450 \zeta_{6} ) q^{53} + 320 q^{55} + ( 274 - 274 \zeta_{6} ) q^{59} + 576 \zeta_{6} q^{61} + 96 \zeta_{6} q^{65} + ( 476 - 476 \zeta_{6} ) q^{67} + 448 q^{71} + ( 158 - 158 \zeta_{6} ) q^{73} + 936 \zeta_{6} q^{79} -530 q^{83} + 464 q^{85} -390 \zeta_{6} q^{89} + ( -208 + 208 \zeta_{6} ) q^{95} + 214 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{5} + O(q^{10})$$ $$2q - 8q^{5} - 40q^{11} - 24q^{13} - 58q^{17} - 26q^{19} - 64q^{23} + 61q^{25} + 124q^{29} - 252q^{31} - 26q^{37} - 12q^{41} + 832q^{43} - 396q^{47} - 450q^{53} + 640q^{55} + 274q^{59} + 576q^{61} + 96q^{65} + 476q^{67} + 896q^{71} + 158q^{73} + 936q^{79} - 1060q^{83} + 928q^{85} - 390q^{89} - 208q^{95} + 428q^{97} + O(q^{100})$$

## 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 −4.00000 6.92820i 0 0 0 0 0
1549.1 0 0 0 −4.00000 + 6.92820i 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.d 2
3.b odd 2 1 196.4.e.f 2
7.b odd 2 1 1764.4.k.m 2
7.c even 3 1 252.4.a.d 1
7.c even 3 1 inner 1764.4.k.d 2
7.d odd 6 1 1764.4.a.c 1
7.d odd 6 1 1764.4.k.m 2
21.c even 2 1 196.4.e.a 2
21.g even 6 1 196.4.a.d 1
21.g even 6 1 196.4.e.a 2
21.h odd 6 1 28.4.a.a 1
21.h odd 6 1 196.4.e.f 2
28.g odd 6 1 1008.4.a.o 1
84.j odd 6 1 784.4.a.a 1
84.n even 6 1 112.4.a.g 1
105.o odd 6 1 700.4.a.n 1
105.x even 12 2 700.4.e.a 2
168.s odd 6 1 448.4.a.p 1
168.v even 6 1 448.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 21.h odd 6 1
112.4.a.g 1 84.n even 6 1
196.4.a.d 1 21.g even 6 1
196.4.e.a 2 21.c even 2 1
196.4.e.a 2 21.g even 6 1
196.4.e.f 2 3.b odd 2 1
196.4.e.f 2 21.h odd 6 1
252.4.a.d 1 7.c even 3 1
448.4.a.a 1 168.v even 6 1
448.4.a.p 1 168.s odd 6 1
700.4.a.n 1 105.o odd 6 1
700.4.e.a 2 105.x even 12 2
784.4.a.a 1 84.j odd 6 1
1008.4.a.o 1 28.g odd 6 1
1764.4.a.c 1 7.d odd 6 1
1764.4.k.d 2 1.a even 1 1 trivial
1764.4.k.d 2 7.c even 3 1 inner
1764.4.k.m 2 7.b odd 2 1
1764.4.k.m 2 7.d odd 6 1

## Hecke kernels

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

 $$T_{5}^{2} + 8 T_{5} + 64$$ $$T_{11}^{2} + 40 T_{11} + 1600$$ $$T_{13} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$64 + 8 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$1600 + 40 T + T^{2}$$
$13$ $$( 12 + T )^{2}$$
$17$ $$3364 + 58 T + T^{2}$$
$19$ $$676 + 26 T + T^{2}$$
$23$ $$4096 + 64 T + T^{2}$$
$29$ $$( -62 + T )^{2}$$
$31$ $$63504 + 252 T + T^{2}$$
$37$ $$676 + 26 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -416 + T )^{2}$$
$47$ $$156816 + 396 T + T^{2}$$
$53$ $$202500 + 450 T + T^{2}$$
$59$ $$75076 - 274 T + T^{2}$$
$61$ $$331776 - 576 T + T^{2}$$
$67$ $$226576 - 476 T + T^{2}$$
$71$ $$( -448 + T )^{2}$$
$73$ $$24964 - 158 T + T^{2}$$
$79$ $$876096 - 936 T + T^{2}$$
$83$ $$( 530 + T )^{2}$$
$89$ $$152100 + 390 T + T^{2}$$
$97$ $$( -214 + T )^{2}$$