# Properties

 Label 84.2.f.a Level $84$ Weight $2$ Character orbit 84.f 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.f (of order $$2$$, degree $$1$$, 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + ( 3 - 2 \zeta_{6} ) q^{7} -3 q^{9} + ( -4 + 8 \zeta_{6} ) q^{13} + ( -2 + 4 \zeta_{6} ) q^{19} + ( -1 - 4 \zeta_{6} ) q^{21} -5 q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( 6 - 12 \zeta_{6} ) q^{31} + 10 q^{37} + 12 q^{39} -8 q^{43} + ( 5 - 8 \zeta_{6} ) q^{49} + 6 q^{57} + ( 4 - 8 \zeta_{6} ) q^{61} + ( -9 + 6 \zeta_{6} ) q^{63} -16 q^{67} + ( -8 + 16 \zeta_{6} ) q^{73} + ( -5 + 10 \zeta_{6} ) q^{75} -4 q^{79} + 9 q^{81} + ( 4 + 16 \zeta_{6} ) q^{91} -18 q^{93} + ( 8 - 16 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 4q^{7} - 6q^{9} - 6q^{21} - 10q^{25} + 20q^{37} + 24q^{39} - 16q^{43} + 2q^{49} + 12q^{57} - 12q^{63} - 32q^{67} - 8q^{79} + 18q^{81} + 24q^{91} - 36q^{93} + O(q^{100})$$

## Character values

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

 $$n$$ $$29$$ $$43$$ $$73$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 0 0 2.00000 1.73205i 0 −3.00000 0
41.2 0 1.73205i 0 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.f.a 2
3.b odd 2 1 CM 84.2.f.a 2
4.b odd 2 1 336.2.k.a 2
5.b even 2 1 2100.2.d.b 2
5.c odd 4 2 2100.2.f.e 4
7.b odd 2 1 inner 84.2.f.a 2
7.c even 3 1 588.2.k.a 2
7.c even 3 1 588.2.k.e 2
7.d odd 6 1 588.2.k.a 2
7.d odd 6 1 588.2.k.e 2
8.b even 2 1 1344.2.k.b 2
8.d odd 2 1 1344.2.k.a 2
9.c even 3 1 2268.2.x.c 2
9.c even 3 1 2268.2.x.e 2
9.d odd 6 1 2268.2.x.c 2
9.d odd 6 1 2268.2.x.e 2
12.b even 2 1 336.2.k.a 2
15.d odd 2 1 2100.2.d.b 2
15.e even 4 2 2100.2.f.e 4
21.c even 2 1 inner 84.2.f.a 2
21.g even 6 1 588.2.k.a 2
21.g even 6 1 588.2.k.e 2
21.h odd 6 1 588.2.k.a 2
21.h odd 6 1 588.2.k.e 2
24.f even 2 1 1344.2.k.a 2
24.h odd 2 1 1344.2.k.b 2
28.d even 2 1 336.2.k.a 2
35.c odd 2 1 2100.2.d.b 2
35.f even 4 2 2100.2.f.e 4
56.e even 2 1 1344.2.k.a 2
56.h odd 2 1 1344.2.k.b 2
63.l odd 6 1 2268.2.x.c 2
63.l odd 6 1 2268.2.x.e 2
63.o even 6 1 2268.2.x.c 2
63.o even 6 1 2268.2.x.e 2
84.h odd 2 1 336.2.k.a 2
105.g even 2 1 2100.2.d.b 2
105.k odd 4 2 2100.2.f.e 4
168.e odd 2 1 1344.2.k.a 2
168.i even 2 1 1344.2.k.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.f.a 2 1.a even 1 1 trivial
84.2.f.a 2 3.b odd 2 1 CM
84.2.f.a 2 7.b odd 2 1 inner
84.2.f.a 2 21.c even 2 1 inner
336.2.k.a 2 4.b odd 2 1
336.2.k.a 2 12.b even 2 1
336.2.k.a 2 28.d even 2 1
336.2.k.a 2 84.h odd 2 1
588.2.k.a 2 7.c even 3 1
588.2.k.a 2 7.d odd 6 1
588.2.k.a 2 21.g even 6 1
588.2.k.a 2 21.h odd 6 1
588.2.k.e 2 7.c even 3 1
588.2.k.e 2 7.d odd 6 1
588.2.k.e 2 21.g even 6 1
588.2.k.e 2 21.h odd 6 1
1344.2.k.a 2 8.d odd 2 1
1344.2.k.a 2 24.f even 2 1
1344.2.k.a 2 56.e even 2 1
1344.2.k.a 2 168.e odd 2 1
1344.2.k.b 2 8.b even 2 1
1344.2.k.b 2 24.h odd 2 1
1344.2.k.b 2 56.h odd 2 1
1344.2.k.b 2 168.i even 2 1
2100.2.d.b 2 5.b even 2 1
2100.2.d.b 2 15.d odd 2 1
2100.2.d.b 2 35.c odd 2 1
2100.2.d.b 2 105.g even 2 1
2100.2.f.e 4 5.c odd 4 2
2100.2.f.e 4 15.e even 4 2
2100.2.f.e 4 35.f even 4 2
2100.2.f.e 4 105.k odd 4 2
2268.2.x.c 2 9.c even 3 1
2268.2.x.c 2 9.d odd 6 1
2268.2.x.c 2 63.l odd 6 1
2268.2.x.c 2 63.o even 6 1
2268.2.x.e 2 9.c even 3 1
2268.2.x.e 2 9.d odd 6 1
2268.2.x.e 2 63.l odd 6 1
2268.2.x.e 2 63.o even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(84, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 - 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$48 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$108 + T^{2}$$
$37$ $$( -10 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( 8 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$48 + T^{2}$$
$67$ $$( 16 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$192 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$192 + T^{2}$$