# Properties

 Label 21.2.e.a Level $21$ Weight $2$ Character orbit 21.e Analytic conductor $0.168$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 21.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.167685844245$$ 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -\zeta_{6} q^{3} -2 \zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + 2 q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -\zeta_{6} q^{3} -2 \zeta_{6} q^{4} + ( 2 - 2 \zeta_{6} ) q^{5} + 2 q^{6} + ( -3 + \zeta_{6} ) q^{7} + ( -1 + \zeta_{6} ) q^{9} + 4 \zeta_{6} q^{10} + 2 \zeta_{6} q^{11} + ( -2 + 2 \zeta_{6} ) q^{12} + q^{13} + ( 4 - 6 \zeta_{6} ) q^{14} -2 q^{15} + ( 4 - 4 \zeta_{6} ) q^{16} -2 \zeta_{6} q^{18} + ( -1 + \zeta_{6} ) q^{19} -4 q^{20} + ( 1 + 2 \zeta_{6} ) q^{21} -4 q^{22} + \zeta_{6} q^{25} + ( -2 + 2 \zeta_{6} ) q^{26} + q^{27} + ( 2 + 4 \zeta_{6} ) q^{28} + 4 q^{29} + ( 4 - 4 \zeta_{6} ) q^{30} -9 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} + ( 2 - 2 \zeta_{6} ) q^{33} + ( -4 + 6 \zeta_{6} ) q^{35} + 2 q^{36} + ( -3 + 3 \zeta_{6} ) q^{37} -2 \zeta_{6} q^{38} -\zeta_{6} q^{39} -10 q^{41} + ( -6 + 2 \zeta_{6} ) q^{42} + 5 q^{43} + ( 4 - 4 \zeta_{6} ) q^{44} + 2 \zeta_{6} q^{45} + ( 6 - 6 \zeta_{6} ) q^{47} -4 q^{48} + ( 8 - 5 \zeta_{6} ) q^{49} -2 q^{50} -2 \zeta_{6} q^{52} -12 \zeta_{6} q^{53} + ( -2 + 2 \zeta_{6} ) q^{54} + 4 q^{55} + q^{57} + ( -8 + 8 \zeta_{6} ) q^{58} + 12 \zeta_{6} q^{59} + 4 \zeta_{6} q^{60} + ( -10 + 10 \zeta_{6} ) q^{61} + 18 q^{62} + ( 2 - 3 \zeta_{6} ) q^{63} -8 q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{66} + 5 \zeta_{6} q^{67} + ( -4 - 8 \zeta_{6} ) q^{70} -6 q^{71} + 3 \zeta_{6} q^{73} -6 \zeta_{6} q^{74} + ( 1 - \zeta_{6} ) q^{75} + 2 q^{76} + ( -2 - 4 \zeta_{6} ) q^{77} + 2 q^{78} + ( 1 - \zeta_{6} ) q^{79} -8 \zeta_{6} q^{80} -\zeta_{6} q^{81} + ( 20 - 20 \zeta_{6} ) q^{82} + 6 q^{83} + ( 4 - 6 \zeta_{6} ) q^{84} + ( -10 + 10 \zeta_{6} ) q^{86} -4 \zeta_{6} q^{87} + ( -16 + 16 \zeta_{6} ) q^{89} -4 q^{90} + ( -3 + \zeta_{6} ) q^{91} + ( -9 + 9 \zeta_{6} ) q^{93} + 12 \zeta_{6} q^{94} + 2 \zeta_{6} q^{95} + ( 8 - 8 \zeta_{6} ) q^{96} -6 q^{97} + ( -6 + 16 \zeta_{6} ) q^{98} -2 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - q^{3} - 2q^{4} + 2q^{5} + 4q^{6} - 5q^{7} - q^{9} + O(q^{10})$$ $$2q - 2q^{2} - q^{3} - 2q^{4} + 2q^{5} + 4q^{6} - 5q^{7} - q^{9} + 4q^{10} + 2q^{11} - 2q^{12} + 2q^{13} + 2q^{14} - 4q^{15} + 4q^{16} - 2q^{18} - q^{19} - 8q^{20} + 4q^{21} - 8q^{22} + q^{25} - 2q^{26} + 2q^{27} + 8q^{28} + 8q^{29} + 4q^{30} - 9q^{31} + 8q^{32} + 2q^{33} - 2q^{35} + 4q^{36} - 3q^{37} - 2q^{38} - q^{39} - 20q^{41} - 10q^{42} + 10q^{43} + 4q^{44} + 2q^{45} + 6q^{47} - 8q^{48} + 11q^{49} - 4q^{50} - 2q^{52} - 12q^{53} - 2q^{54} + 8q^{55} + 2q^{57} - 8q^{58} + 12q^{59} + 4q^{60} - 10q^{61} + 36q^{62} + q^{63} - 16q^{64} + 2q^{65} + 4q^{66} + 5q^{67} - 16q^{70} - 12q^{71} + 3q^{73} - 6q^{74} + q^{75} + 4q^{76} - 8q^{77} + 4q^{78} + q^{79} - 8q^{80} - q^{81} + 20q^{82} + 12q^{83} + 2q^{84} - 10q^{86} - 4q^{87} - 16q^{89} - 8q^{90} - 5q^{91} - 9q^{93} + 12q^{94} + 2q^{95} + 8q^{96} - 12q^{97} + 4q^{98} - 4q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$$.

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.73205i −0.500000 + 0.866025i −1.00000 + 1.73205i 1.00000 + 1.73205i 2.00000 −2.50000 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i
16.1 −1.00000 + 1.73205i −0.500000 0.866025i −1.00000 1.73205i 1.00000 1.73205i 2.00000 −2.50000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 + 3.46410i
 $$n$$: 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 21.2.e.a 2
3.b odd 2 1 63.2.e.b 2
4.b odd 2 1 336.2.q.f 2
5.b even 2 1 525.2.i.e 2
5.c odd 4 2 525.2.r.e 4
7.b odd 2 1 147.2.e.a 2
7.c even 3 1 inner 21.2.e.a 2
7.c even 3 1 147.2.a.c 1
7.d odd 6 1 147.2.a.b 1
7.d odd 6 1 147.2.e.a 2
8.b even 2 1 1344.2.q.m 2
8.d odd 2 1 1344.2.q.c 2
9.c even 3 1 567.2.g.a 2
9.c even 3 1 567.2.h.f 2
9.d odd 6 1 567.2.g.f 2
9.d odd 6 1 567.2.h.a 2
12.b even 2 1 1008.2.s.d 2
21.c even 2 1 441.2.e.e 2
21.g even 6 1 441.2.a.a 1
21.g even 6 1 441.2.e.e 2
21.h odd 6 1 63.2.e.b 2
21.h odd 6 1 441.2.a.b 1
28.d even 2 1 2352.2.q.c 2
28.f even 6 1 2352.2.a.w 1
28.f even 6 1 2352.2.q.c 2
28.g odd 6 1 336.2.q.f 2
28.g odd 6 1 2352.2.a.d 1
35.i odd 6 1 3675.2.a.c 1
35.j even 6 1 525.2.i.e 2
35.j even 6 1 3675.2.a.a 1
35.l odd 12 2 525.2.r.e 4
56.j odd 6 1 9408.2.a.bz 1
56.k odd 6 1 1344.2.q.c 2
56.k odd 6 1 9408.2.a.cv 1
56.m even 6 1 9408.2.a.k 1
56.p even 6 1 1344.2.q.m 2
56.p even 6 1 9408.2.a.bg 1
63.g even 3 1 567.2.h.f 2
63.h even 3 1 567.2.g.a 2
63.j odd 6 1 567.2.g.f 2
63.n odd 6 1 567.2.h.a 2
84.j odd 6 1 7056.2.a.m 1
84.n even 6 1 1008.2.s.d 2
84.n even 6 1 7056.2.a.bp 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 1.a even 1 1 trivial
21.2.e.a 2 7.c even 3 1 inner
63.2.e.b 2 3.b odd 2 1
63.2.e.b 2 21.h odd 6 1
147.2.a.b 1 7.d odd 6 1
147.2.a.c 1 7.c even 3 1
147.2.e.a 2 7.b odd 2 1
147.2.e.a 2 7.d odd 6 1
336.2.q.f 2 4.b odd 2 1
336.2.q.f 2 28.g odd 6 1
441.2.a.a 1 21.g even 6 1
441.2.a.b 1 21.h odd 6 1
441.2.e.e 2 21.c even 2 1
441.2.e.e 2 21.g even 6 1
525.2.i.e 2 5.b even 2 1
525.2.i.e 2 35.j even 6 1
525.2.r.e 4 5.c odd 4 2
525.2.r.e 4 35.l odd 12 2
567.2.g.a 2 9.c even 3 1
567.2.g.a 2 63.h even 3 1
567.2.g.f 2 9.d odd 6 1
567.2.g.f 2 63.j odd 6 1
567.2.h.a 2 9.d odd 6 1
567.2.h.a 2 63.n odd 6 1
567.2.h.f 2 9.c even 3 1
567.2.h.f 2 63.g even 3 1
1008.2.s.d 2 12.b even 2 1
1008.2.s.d 2 84.n even 6 1
1344.2.q.c 2 8.d odd 2 1
1344.2.q.c 2 56.k odd 6 1
1344.2.q.m 2 8.b even 2 1
1344.2.q.m 2 56.p even 6 1
2352.2.a.d 1 28.g odd 6 1
2352.2.a.w 1 28.f even 6 1
2352.2.q.c 2 28.d even 2 1
2352.2.q.c 2 28.f even 6 1
3675.2.a.a 1 35.j even 6 1
3675.2.a.c 1 35.i odd 6 1
7056.2.a.m 1 84.j odd 6 1
7056.2.a.bp 1 84.n even 6 1
9408.2.a.k 1 56.m even 6 1
9408.2.a.bg 1 56.p even 6 1
9408.2.a.bz 1 56.j odd 6 1
9408.2.a.cv 1 56.k odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$4 + 2 T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$4 - 2 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$4 - 2 T + T^{2}$$
$13$ $$( -1 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -4 + T )^{2}$$
$31$ $$81 + 9 T + T^{2}$$
$37$ $$9 + 3 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( -5 + T )^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$144 + 12 T + T^{2}$$
$59$ $$144 - 12 T + T^{2}$$
$61$ $$100 + 10 T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$( 6 + T )^{2}$$
$73$ $$9 - 3 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$256 + 16 T + T^{2}$$
$97$ $$( 6 + T )^{2}$$