# Properties

 Label 21.3.f.a Level $21$ Weight $3$ Character orbit 21.f Analytic conductor $0.572$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 21.f (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.572208555157$$ 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: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -3 \zeta_{6} q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( -5 + 5 \zeta_{6} ) q^{4} + ( 6 - 3 \zeta_{6} ) q^{5} + ( -3 + 6 \zeta_{6} ) q^{6} + ( 5 + 3 \zeta_{6} ) q^{7} + 3 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q -3 \zeta_{6} q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( -5 + 5 \zeta_{6} ) q^{4} + ( 6 - 3 \zeta_{6} ) q^{5} + ( -3 + 6 \zeta_{6} ) q^{6} + ( 5 + 3 \zeta_{6} ) q^{7} + 3 q^{8} + 3 \zeta_{6} q^{9} + ( -9 - 9 \zeta_{6} ) q^{10} + ( -15 + 15 \zeta_{6} ) q^{11} + ( 10 - 5 \zeta_{6} ) q^{12} + ( 8 - 16 \zeta_{6} ) q^{13} + ( 9 - 24 \zeta_{6} ) q^{14} -9 q^{15} + 11 \zeta_{6} q^{16} + ( 6 + 6 \zeta_{6} ) q^{17} + ( 9 - 9 \zeta_{6} ) q^{18} + ( -12 + 6 \zeta_{6} ) q^{19} + ( -15 + 30 \zeta_{6} ) q^{20} + ( -2 - 11 \zeta_{6} ) q^{21} + 45 q^{22} + ( -3 - 3 \zeta_{6} ) q^{24} + ( 2 - 2 \zeta_{6} ) q^{25} + ( -48 + 24 \zeta_{6} ) q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -40 + 25 \zeta_{6} ) q^{28} -9 q^{29} + 27 \zeta_{6} q^{30} + ( -7 - 7 \zeta_{6} ) q^{31} + ( 45 - 45 \zeta_{6} ) q^{32} + ( 30 - 15 \zeta_{6} ) q^{33} + ( 18 - 36 \zeta_{6} ) q^{34} + ( 39 - 6 \zeta_{6} ) q^{35} -15 q^{36} -10 \zeta_{6} q^{37} + ( 18 + 18 \zeta_{6} ) q^{38} + ( -24 + 24 \zeta_{6} ) q^{39} + ( 18 - 9 \zeta_{6} ) q^{40} + ( 6 - 12 \zeta_{6} ) q^{41} + ( -33 + 39 \zeta_{6} ) q^{42} -74 q^{43} -75 \zeta_{6} q^{44} + ( 9 + 9 \zeta_{6} ) q^{45} + ( 11 - 22 \zeta_{6} ) q^{48} + ( 16 + 39 \zeta_{6} ) q^{49} -6 q^{50} -18 \zeta_{6} q^{51} + ( 40 + 40 \zeta_{6} ) q^{52} + ( -33 + 33 \zeta_{6} ) q^{53} + ( -18 + 9 \zeta_{6} ) q^{54} + ( -45 + 90 \zeta_{6} ) q^{55} + ( 15 + 9 \zeta_{6} ) q^{56} + 18 q^{57} + 27 \zeta_{6} q^{58} + ( 9 + 9 \zeta_{6} ) q^{59} + ( 45 - 45 \zeta_{6} ) q^{60} + ( 104 - 52 \zeta_{6} ) q^{61} + ( -21 + 42 \zeta_{6} ) q^{62} + ( -9 + 24 \zeta_{6} ) q^{63} -91 q^{64} -72 \zeta_{6} q^{65} + ( -45 - 45 \zeta_{6} ) q^{66} + ( 76 - 76 \zeta_{6} ) q^{67} + ( -60 + 30 \zeta_{6} ) q^{68} + ( -18 - 99 \zeta_{6} ) q^{70} + 84 q^{71} + 9 \zeta_{6} q^{72} + ( -36 - 36 \zeta_{6} ) q^{73} + ( -30 + 30 \zeta_{6} ) q^{74} + ( -4 + 2 \zeta_{6} ) q^{75} + ( 30 - 60 \zeta_{6} ) q^{76} + ( -120 + 75 \zeta_{6} ) q^{77} + 72 q^{78} + 43 \zeta_{6} q^{79} + ( 33 + 33 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -36 + 18 \zeta_{6} ) q^{82} + ( 69 - 138 \zeta_{6} ) q^{83} + ( 65 - 10 \zeta_{6} ) q^{84} + 54 q^{85} + 222 \zeta_{6} q^{86} + ( 9 + 9 \zeta_{6} ) q^{87} + ( -45 + 45 \zeta_{6} ) q^{88} + ( -84 + 42 \zeta_{6} ) q^{89} + ( 27 - 54 \zeta_{6} ) q^{90} + ( 88 - 104 \zeta_{6} ) q^{91} + 21 \zeta_{6} q^{93} + ( -54 + 54 \zeta_{6} ) q^{95} + ( -90 + 45 \zeta_{6} ) q^{96} + ( -107 + 214 \zeta_{6} ) q^{97} + ( 117 - 165 \zeta_{6} ) q^{98} -45 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 3q^{2} - 3q^{3} - 5q^{4} + 9q^{5} + 13q^{7} + 6q^{8} + 3q^{9} + O(q^{10})$$ $$2q - 3q^{2} - 3q^{3} - 5q^{4} + 9q^{5} + 13q^{7} + 6q^{8} + 3q^{9} - 27q^{10} - 15q^{11} + 15q^{12} - 6q^{14} - 18q^{15} + 11q^{16} + 18q^{17} + 9q^{18} - 18q^{19} - 15q^{21} + 90q^{22} - 9q^{24} + 2q^{25} - 72q^{26} - 55q^{28} - 18q^{29} + 27q^{30} - 21q^{31} + 45q^{32} + 45q^{33} + 72q^{35} - 30q^{36} - 10q^{37} + 54q^{38} - 24q^{39} + 27q^{40} - 27q^{42} - 148q^{43} - 75q^{44} + 27q^{45} + 71q^{49} - 12q^{50} - 18q^{51} + 120q^{52} - 33q^{53} - 27q^{54} + 39q^{56} + 36q^{57} + 27q^{58} + 27q^{59} + 45q^{60} + 156q^{61} + 6q^{63} - 182q^{64} - 72q^{65} - 135q^{66} + 76q^{67} - 90q^{68} - 135q^{70} + 168q^{71} + 9q^{72} - 108q^{73} - 30q^{74} - 6q^{75} - 165q^{77} + 144q^{78} + 43q^{79} + 99q^{80} - 9q^{81} - 54q^{82} + 120q^{84} + 108q^{85} + 222q^{86} + 27q^{87} - 45q^{88} - 126q^{89} + 72q^{91} + 21q^{93} - 54q^{95} - 135q^{96} + 69q^{98} - 90q^{99} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
10.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.50000 2.59808i −1.50000 0.866025i −2.50000 + 4.33013i 4.50000 2.59808i 5.19615i 6.50000 + 2.59808i 3.00000 1.50000 + 2.59808i −13.5000 7.79423i
19.1 −1.50000 + 2.59808i −1.50000 + 0.866025i −2.50000 4.33013i 4.50000 + 2.59808i 5.19615i 6.50000 2.59808i 3.00000 1.50000 2.59808i −13.5000 + 7.79423i
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.3.f.a 2
3.b odd 2 1 63.3.m.d 2
4.b odd 2 1 336.3.bh.d 2
5.b even 2 1 525.3.o.h 2
5.c odd 4 2 525.3.s.e 4
7.b odd 2 1 147.3.f.a 2
7.c even 3 1 147.3.d.c 2
7.c even 3 1 147.3.f.a 2
7.d odd 6 1 inner 21.3.f.a 2
7.d odd 6 1 147.3.d.c 2
12.b even 2 1 1008.3.cg.a 2
21.c even 2 1 441.3.m.g 2
21.g even 6 1 63.3.m.d 2
21.g even 6 1 441.3.d.a 2
21.h odd 6 1 441.3.d.a 2
21.h odd 6 1 441.3.m.g 2
28.f even 6 1 336.3.bh.d 2
28.f even 6 1 2352.3.f.a 2
28.g odd 6 1 2352.3.f.a 2
35.i odd 6 1 525.3.o.h 2
35.k even 12 2 525.3.s.e 4
84.j odd 6 1 1008.3.cg.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.f.a 2 1.a even 1 1 trivial
21.3.f.a 2 7.d odd 6 1 inner
63.3.m.d 2 3.b odd 2 1
63.3.m.d 2 21.g even 6 1
147.3.d.c 2 7.c even 3 1
147.3.d.c 2 7.d odd 6 1
147.3.f.a 2 7.b odd 2 1
147.3.f.a 2 7.c even 3 1
336.3.bh.d 2 4.b odd 2 1
336.3.bh.d 2 28.f even 6 1
441.3.d.a 2 21.g even 6 1
441.3.d.a 2 21.h odd 6 1
441.3.m.g 2 21.c even 2 1
441.3.m.g 2 21.h odd 6 1
525.3.o.h 2 5.b even 2 1
525.3.o.h 2 35.i odd 6 1
525.3.s.e 4 5.c odd 4 2
525.3.s.e 4 35.k even 12 2
1008.3.cg.a 2 12.b even 2 1
1008.3.cg.a 2 84.j odd 6 1
2352.3.f.a 2 28.f even 6 1
2352.3.f.a 2 28.g odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 3 T_{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$9 + 3 T + T^{2}$$
$3$ $$3 + 3 T + T^{2}$$
$5$ $$27 - 9 T + T^{2}$$
$7$ $$49 - 13 T + T^{2}$$
$11$ $$225 + 15 T + T^{2}$$
$13$ $$192 + T^{2}$$
$17$ $$108 - 18 T + T^{2}$$
$19$ $$108 + 18 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( 9 + T )^{2}$$
$31$ $$147 + 21 T + T^{2}$$
$37$ $$100 + 10 T + T^{2}$$
$41$ $$108 + T^{2}$$
$43$ $$( 74 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$1089 + 33 T + T^{2}$$
$59$ $$243 - 27 T + T^{2}$$
$61$ $$8112 - 156 T + T^{2}$$
$67$ $$5776 - 76 T + T^{2}$$
$71$ $$( -84 + T )^{2}$$
$73$ $$3888 + 108 T + T^{2}$$
$79$ $$1849 - 43 T + T^{2}$$
$83$ $$14283 + T^{2}$$
$89$ $$5292 + 126 T + T^{2}$$
$97$ $$34347 + T^{2}$$