# Properties

 Label 84.2.k.c Level $84$ Weight $2$ Character orbit 84.k Analytic conductor $0.671$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( -2 - \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 + \zeta_{6} ) q^{3} + ( -2 - \zeta_{6} ) q^{7} + 3 \zeta_{6} q^{9} + ( 3 - 6 \zeta_{6} ) q^{13} + ( -10 + 5 \zeta_{6} ) q^{19} + ( -1 - 4 \zeta_{6} ) q^{21} + ( 5 - 5 \zeta_{6} ) q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -1 - \zeta_{6} ) q^{31} + 11 \zeta_{6} q^{37} + ( 9 - 9 \zeta_{6} ) q^{39} + 13 q^{43} + ( 3 + 5 \zeta_{6} ) q^{49} -15 q^{57} + ( -8 + 4 \zeta_{6} ) q^{61} + ( 3 - 9 \zeta_{6} ) q^{63} + ( -5 + 5 \zeta_{6} ) q^{67} + ( -1 - \zeta_{6} ) q^{73} + ( 10 - 5 \zeta_{6} ) q^{75} -17 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -12 + 15 \zeta_{6} ) q^{91} -3 \zeta_{6} q^{93} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 5q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 5q^{7} + 3q^{9} - 15q^{19} - 6q^{21} + 5q^{25} - 3q^{31} + 11q^{37} + 9q^{39} + 26q^{43} + 11q^{49} - 30q^{57} - 12q^{61} - 3q^{63} - 5q^{67} - 3q^{73} + 15q^{75} - 17q^{79} - 9q^{81} - 9q^{91} - 3q^{93} + O(q^{100})$$

## 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 1.50000 0.866025i 0 0 0 −2.50000 + 0.866025i 0 1.50000 2.59808i 0
17.1 0 1.50000 + 0.866025i 0 0 0 −2.50000 0.866025i 0 1.50000 + 2.59808i 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.d odd 6 1 inner
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.c 2
3.b odd 2 1 CM 84.2.k.c 2
4.b odd 2 1 336.2.bc.a 2
5.b even 2 1 2100.2.bi.d 2
5.c odd 4 2 2100.2.bo.e 4
7.b odd 2 1 588.2.k.b 2
7.c even 3 1 588.2.f.b 2
7.c even 3 1 588.2.k.b 2
7.d odd 6 1 inner 84.2.k.c 2
7.d odd 6 1 588.2.f.b 2
9.c even 3 1 2268.2.w.e 2
9.c even 3 1 2268.2.bm.d 2
9.d odd 6 1 2268.2.w.e 2
9.d odd 6 1 2268.2.bm.d 2
12.b even 2 1 336.2.bc.a 2
15.d odd 2 1 2100.2.bi.d 2
15.e even 4 2 2100.2.bo.e 4
21.c even 2 1 588.2.k.b 2
21.g even 6 1 inner 84.2.k.c 2
21.g even 6 1 588.2.f.b 2
21.h odd 6 1 588.2.f.b 2
21.h odd 6 1 588.2.k.b 2
28.f even 6 1 336.2.bc.a 2
28.f even 6 1 2352.2.k.b 2
28.g odd 6 1 2352.2.k.b 2
35.i odd 6 1 2100.2.bi.d 2
35.k even 12 2 2100.2.bo.e 4
63.i even 6 1 2268.2.bm.d 2
63.k odd 6 1 2268.2.w.e 2
63.s even 6 1 2268.2.w.e 2
63.t odd 6 1 2268.2.bm.d 2
84.j odd 6 1 336.2.bc.a 2
84.j odd 6 1 2352.2.k.b 2
84.n even 6 1 2352.2.k.b 2
105.p even 6 1 2100.2.bi.d 2
105.w odd 12 2 2100.2.bo.e 4

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$27 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$75 + 15 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$3 + 3 T + T^{2}$$
$37$ $$121 - 11 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -13 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$48 + 12 T + T^{2}$$
$67$ $$25 + 5 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$3 + 3 T + T^{2}$$
$79$ $$289 + 17 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$192 + T^{2}$$