# Properties

 Label 84.2.k.b Level $84$ Weight $2$ Character orbit 84.k Analytic conductor $0.671$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$84 = 2^{2} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 84.k (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.670743376979$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} -3 \zeta_{6} q^{5} + ( 1 + 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 3 + 3 \zeta_{6} ) q^{11} + ( -6 + 3 \zeta_{6} ) q^{15} + ( -3 + 3 \zeta_{6} ) q^{17} + ( 2 - \zeta_{6} ) q^{19} + ( 5 - 4 \zeta_{6} ) q^{21} + ( -6 + 3 \zeta_{6} ) q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -1 - \zeta_{6} ) q^{31} + ( 9 - 9 \zeta_{6} ) q^{33} + ( 6 - 9 \zeta_{6} ) q^{35} -7 \zeta_{6} q^{37} + 6 q^{41} + 4 q^{43} + 9 \zeta_{6} q^{45} -3 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 3 + 3 \zeta_{6} ) q^{51} + ( -3 - 3 \zeta_{6} ) q^{53} + ( 9 - 18 \zeta_{6} ) q^{55} -3 \zeta_{6} q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + ( -14 + 7 \zeta_{6} ) q^{61} + ( -3 - 6 \zeta_{6} ) q^{63} + ( -5 + 5 \zeta_{6} ) q^{67} + 9 \zeta_{6} q^{69} + ( -6 + 12 \zeta_{6} ) q^{71} + ( -7 - 7 \zeta_{6} ) q^{73} + ( 4 + 4 \zeta_{6} ) q^{75} + ( -3 + 15 \zeta_{6} ) q^{77} + \zeta_{6} q^{79} + 9 q^{81} -12 q^{83} + 9 q^{85} + 9 \zeta_{6} q^{89} + ( -3 + 3 \zeta_{6} ) q^{93} + ( -3 - 3 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( -9 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{5} + 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q - 3q^{5} + 4q^{7} - 6q^{9} + 9q^{11} - 9q^{15} - 3q^{17} + 3q^{19} + 6q^{21} - 9q^{23} - 4q^{25} - 3q^{31} + 9q^{33} + 3q^{35} - 7q^{37} + 12q^{41} + 8q^{43} + 9q^{45} - 3q^{47} + 2q^{49} + 9q^{51} - 9q^{53} - 3q^{57} + 3q^{59} - 21q^{61} - 12q^{63} - 5q^{67} + 9q^{69} - 21q^{73} + 12q^{75} + 9q^{77} + q^{79} + 18q^{81} - 24q^{83} + 18q^{85} + 9q^{89} - 3q^{93} - 9q^{95} - 27q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$-1$$ $$1$$ $$\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 −1.50000 + 2.59808i 0 2.00000 1.73205i 0 −3.00000 0
17.1 0 1.73205i 0 −1.50000 2.59808i 0 2.00000 + 1.73205i 0 −3.00000 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
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.k.b yes 2
3.b odd 2 1 84.2.k.a 2
4.b odd 2 1 336.2.bc.b 2
5.b even 2 1 2100.2.bi.e 2
5.c odd 4 2 2100.2.bo.f 4
7.b odd 2 1 588.2.k.c 2
7.c even 3 1 588.2.f.a 2
7.c even 3 1 588.2.k.d 2
7.d odd 6 1 84.2.k.a 2
7.d odd 6 1 588.2.f.c 2
9.c even 3 1 2268.2.w.a 2
9.c even 3 1 2268.2.bm.f 2
9.d odd 6 1 2268.2.w.f 2
9.d odd 6 1 2268.2.bm.a 2
12.b even 2 1 336.2.bc.d 2
15.d odd 2 1 2100.2.bi.f 2
15.e even 4 2 2100.2.bo.a 4
21.c even 2 1 588.2.k.d 2
21.g even 6 1 inner 84.2.k.b yes 2
21.g even 6 1 588.2.f.a 2
21.h odd 6 1 588.2.f.c 2
21.h odd 6 1 588.2.k.c 2
28.f even 6 1 336.2.bc.d 2
28.f even 6 1 2352.2.k.a 2
28.g odd 6 1 2352.2.k.d 2
35.i odd 6 1 2100.2.bi.f 2
35.k even 12 2 2100.2.bo.a 4
63.i even 6 1 2268.2.bm.f 2
63.k odd 6 1 2268.2.w.f 2
63.s even 6 1 2268.2.w.a 2
63.t odd 6 1 2268.2.bm.a 2
84.j odd 6 1 336.2.bc.b 2
84.j odd 6 1 2352.2.k.d 2
84.n even 6 1 2352.2.k.a 2
105.p even 6 1 2100.2.bi.e 2
105.w odd 12 2 2100.2.bo.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.k.a 2 3.b odd 2 1
84.2.k.a 2 7.d odd 6 1
84.2.k.b yes 2 1.a even 1 1 trivial
84.2.k.b yes 2 21.g even 6 1 inner
336.2.bc.b 2 4.b odd 2 1
336.2.bc.b 2 84.j odd 6 1
336.2.bc.d 2 12.b even 2 1
336.2.bc.d 2 28.f even 6 1
588.2.f.a 2 7.c even 3 1
588.2.f.a 2 21.g even 6 1
588.2.f.c 2 7.d odd 6 1
588.2.f.c 2 21.h odd 6 1
588.2.k.c 2 7.b odd 2 1
588.2.k.c 2 21.h odd 6 1
588.2.k.d 2 7.c even 3 1
588.2.k.d 2 21.c even 2 1
2100.2.bi.e 2 5.b even 2 1
2100.2.bi.e 2 105.p even 6 1
2100.2.bi.f 2 15.d odd 2 1
2100.2.bi.f 2 35.i odd 6 1
2100.2.bo.a 4 15.e even 4 2
2100.2.bo.a 4 35.k even 12 2
2100.2.bo.f 4 5.c odd 4 2
2100.2.bo.f 4 105.w odd 12 2
2268.2.w.a 2 9.c even 3 1
2268.2.w.a 2 63.s even 6 1
2268.2.w.f 2 9.d odd 6 1
2268.2.w.f 2 63.k odd 6 1
2268.2.bm.a 2 9.d odd 6 1
2268.2.bm.a 2 63.t odd 6 1
2268.2.bm.f 2 9.c even 3 1
2268.2.bm.f 2 63.i even 6 1
2352.2.k.a 2 28.f even 6 1
2352.2.k.a 2 84.n even 6 1
2352.2.k.d 2 28.g odd 6 1
2352.2.k.d 2 84.j odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 3 T_{5} + 9$$ acting on $$S_{2}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$9 + 3 T + T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$27 - 9 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$3 - 3 T + T^{2}$$
$23$ $$27 + 9 T + T^{2}$$
$29$ $$T^{2}$$
$31$ $$3 + 3 T + T^{2}$$
$37$ $$49 + 7 T + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$9 + 3 T + T^{2}$$
$53$ $$27 + 9 T + T^{2}$$
$59$ $$9 - 3 T + T^{2}$$
$61$ $$147 + 21 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$108 + T^{2}$$
$73$ $$147 + 21 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$81 - 9 T + T^{2}$$
$97$ $$48 + T^{2}$$