# Properties

 Label 1764.4.k.k 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: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + 6 \zeta_{6} q^{5}+O(q^{10})$$ q + 6*z * q^5 $$q + 6 \zeta_{6} q^{5} + (12 \zeta_{6} - 12) q^{11} - 82 q^{13} + (30 \zeta_{6} - 30) q^{17} - 68 \zeta_{6} q^{19} + 216 \zeta_{6} q^{23} + ( - 89 \zeta_{6} + 89) q^{25} - 246 q^{29} + ( - 112 \zeta_{6} + 112) q^{31} - 110 \zeta_{6} q^{37} + 246 q^{41} - 172 q^{43} + 192 \zeta_{6} q^{47} + ( - 558 \zeta_{6} + 558) q^{53} - 72 q^{55} + ( - 540 \zeta_{6} + 540) q^{59} - 110 \zeta_{6} q^{61} - 492 \zeta_{6} q^{65} + (140 \zeta_{6} - 140) q^{67} + 840 q^{71} + ( - 550 \zeta_{6} + 550) q^{73} + 208 \zeta_{6} q^{79} - 516 q^{83} - 180 q^{85} - 1398 \zeta_{6} q^{89} + ( - 408 \zeta_{6} + 408) q^{95} + 1586 q^{97} +O(q^{100})$$ q + 6*z * q^5 + (12*z - 12) * q^11 - 82 * q^13 + (30*z - 30) * q^17 - 68*z * q^19 + 216*z * q^23 + (-89*z + 89) * q^25 - 246 * q^29 + (-112*z + 112) * q^31 - 110*z * q^37 + 246 * q^41 - 172 * q^43 + 192*z * q^47 + (-558*z + 558) * q^53 - 72 * q^55 + (-540*z + 540) * q^59 - 110*z * q^61 - 492*z * q^65 + (140*z - 140) * q^67 + 840 * q^71 + (-550*z + 550) * q^73 + 208*z * q^79 - 516 * q^83 - 180 * q^85 - 1398*z * q^89 + (-408*z + 408) * q^95 + 1586 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{5}+O(q^{10})$$ 2 * q + 6 * q^5 $$2 q + 6 q^{5} - 12 q^{11} - 164 q^{13} - 30 q^{17} - 68 q^{19} + 216 q^{23} + 89 q^{25} - 492 q^{29} + 112 q^{31} - 110 q^{37} + 492 q^{41} - 344 q^{43} + 192 q^{47} + 558 q^{53} - 144 q^{55} + 540 q^{59} - 110 q^{61} - 492 q^{65} - 140 q^{67} + 1680 q^{71} + 550 q^{73} + 208 q^{79} - 1032 q^{83} - 360 q^{85} - 1398 q^{89} + 408 q^{95} + 3172 q^{97}+O(q^{100})$$ 2 * q + 6 * q^5 - 12 * q^11 - 164 * q^13 - 30 * q^17 - 68 * q^19 + 216 * q^23 + 89 * q^25 - 492 * q^29 + 112 * q^31 - 110 * q^37 + 492 * q^41 - 344 * q^43 + 192 * q^47 + 558 * q^53 - 144 * q^55 + 540 * q^59 - 110 * q^61 - 492 * q^65 - 140 * q^67 + 1680 * q^71 + 550 * q^73 + 208 * q^79 - 1032 * q^83 - 360 * q^85 - 1398 * q^89 + 408 * q^95 + 3172 * q^97

## 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 3.00000 + 5.19615i 0 0 0 0 0
1549.1 0 0 0 3.00000 5.19615i 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.k 2
3.b odd 2 1 196.4.e.c 2
7.b odd 2 1 1764.4.k.e 2
7.c even 3 1 252.4.a.c 1
7.c even 3 1 inner 1764.4.k.k 2
7.d odd 6 1 1764.4.a.k 1
7.d odd 6 1 1764.4.k.e 2
21.c even 2 1 196.4.e.d 2
21.g even 6 1 196.4.a.b 1
21.g even 6 1 196.4.e.d 2
21.h odd 6 1 28.4.a.b 1
21.h odd 6 1 196.4.e.c 2
28.g odd 6 1 1008.4.a.f 1
84.j odd 6 1 784.4.a.n 1
84.n even 6 1 112.4.a.c 1
105.o odd 6 1 700.4.a.e 1
105.x even 12 2 700.4.e.f 2
168.s odd 6 1 448.4.a.d 1
168.v even 6 1 448.4.a.m 1

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

## Hecke kernels

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

 $$T_{5}^{2} - 6T_{5} + 36$$ T5^2 - 6*T5 + 36 $$T_{11}^{2} + 12T_{11} + 144$$ T11^2 + 12*T11 + 144 $$T_{13} + 82$$ T13 + 82

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 6T + 36$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 12T + 144$$
$13$ $$(T + 82)^{2}$$
$17$ $$T^{2} + 30T + 900$$
$19$ $$T^{2} + 68T + 4624$$
$23$ $$T^{2} - 216T + 46656$$
$29$ $$(T + 246)^{2}$$
$31$ $$T^{2} - 112T + 12544$$
$37$ $$T^{2} + 110T + 12100$$
$41$ $$(T - 246)^{2}$$
$43$ $$(T + 172)^{2}$$
$47$ $$T^{2} - 192T + 36864$$
$53$ $$T^{2} - 558T + 311364$$
$59$ $$T^{2} - 540T + 291600$$
$61$ $$T^{2} + 110T + 12100$$
$67$ $$T^{2} + 140T + 19600$$
$71$ $$(T - 840)^{2}$$
$73$ $$T^{2} - 550T + 302500$$
$79$ $$T^{2} - 208T + 43264$$
$83$ $$(T + 516)^{2}$$
$89$ $$T^{2} + 1398 T + 1954404$$
$97$ $$(T - 1586)^{2}$$