# Properties

 Label 21.3.f.b 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q -\zeta_{6} q^{2} + ( 1 + \zeta_{6} ) q^{3} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -6 + 3 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} + ( -7 + 7 \zeta_{6} ) q^{7} -7 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q -\zeta_{6} q^{2} + ( 1 + \zeta_{6} ) q^{3} + ( 3 - 3 \zeta_{6} ) q^{4} + ( -6 + 3 \zeta_{6} ) q^{5} + ( 1 - 2 \zeta_{6} ) q^{6} + ( -7 + 7 \zeta_{6} ) q^{7} -7 q^{8} + 3 \zeta_{6} q^{9} + ( 3 + 3 \zeta_{6} ) q^{10} + ( 11 - 11 \zeta_{6} ) q^{11} + ( 6 - 3 \zeta_{6} ) q^{12} + ( 4 - 8 \zeta_{6} ) q^{13} + 7 q^{14} -9 q^{15} -5 \zeta_{6} q^{16} + ( 14 + 14 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} + ( -4 + 2 \zeta_{6} ) q^{19} + ( -9 + 18 \zeta_{6} ) q^{20} + ( -14 + 7 \zeta_{6} ) q^{21} -11 q^{22} -28 \zeta_{6} q^{23} + ( -7 - 7 \zeta_{6} ) q^{24} + ( 2 - 2 \zeta_{6} ) q^{25} + ( -8 + 4 \zeta_{6} ) q^{26} + ( -3 + 6 \zeta_{6} ) q^{27} + 21 \zeta_{6} q^{28} + 25 q^{29} + 9 \zeta_{6} q^{30} + ( -19 - 19 \zeta_{6} ) q^{31} + ( -33 + 33 \zeta_{6} ) q^{32} + ( 22 - 11 \zeta_{6} ) q^{33} + ( 14 - 28 \zeta_{6} ) q^{34} + ( 21 - 42 \zeta_{6} ) q^{35} + 9 q^{36} + 58 \zeta_{6} q^{37} + ( 2 + 2 \zeta_{6} ) q^{38} + ( 12 - 12 \zeta_{6} ) q^{39} + ( 42 - 21 \zeta_{6} ) q^{40} + ( 2 - 4 \zeta_{6} ) q^{41} + ( 7 + 7 \zeta_{6} ) q^{42} + 26 q^{43} -33 \zeta_{6} q^{44} + ( -9 - 9 \zeta_{6} ) q^{45} + ( -28 + 28 \zeta_{6} ) q^{46} + ( -88 + 44 \zeta_{6} ) q^{47} + ( 5 - 10 \zeta_{6} ) q^{48} -49 \zeta_{6} q^{49} -2 q^{50} + 42 \zeta_{6} q^{51} + ( -12 - 12 \zeta_{6} ) q^{52} + ( -31 + 31 \zeta_{6} ) q^{53} + ( 6 - 3 \zeta_{6} ) q^{54} + ( -33 + 66 \zeta_{6} ) q^{55} + ( 49 - 49 \zeta_{6} ) q^{56} -6 q^{57} -25 \zeta_{6} q^{58} + ( -5 - 5 \zeta_{6} ) q^{59} + ( -27 + 27 \zeta_{6} ) q^{60} + ( 16 - 8 \zeta_{6} ) q^{61} + ( -19 + 38 \zeta_{6} ) q^{62} -21 q^{63} + 13 q^{64} + 36 \zeta_{6} q^{65} + ( -11 - 11 \zeta_{6} ) q^{66} + ( 52 - 52 \zeta_{6} ) q^{67} + ( 84 - 42 \zeta_{6} ) q^{68} + ( 28 - 56 \zeta_{6} ) q^{69} + ( -42 + 21 \zeta_{6} ) q^{70} + 64 q^{71} -21 \zeta_{6} q^{72} + ( 4 + 4 \zeta_{6} ) q^{73} + ( 58 - 58 \zeta_{6} ) q^{74} + ( 4 - 2 \zeta_{6} ) q^{75} + ( -6 + 12 \zeta_{6} ) q^{76} + 77 \zeta_{6} q^{77} -12 q^{78} -17 \zeta_{6} q^{79} + ( 15 + 15 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -4 + 2 \zeta_{6} ) q^{82} + ( 31 - 62 \zeta_{6} ) q^{83} + ( -21 + 42 \zeta_{6} ) q^{84} -126 q^{85} -26 \zeta_{6} q^{86} + ( 25 + 25 \zeta_{6} ) q^{87} + ( -77 + 77 \zeta_{6} ) q^{88} + ( -92 + 46 \zeta_{6} ) q^{89} + ( -9 + 18 \zeta_{6} ) q^{90} + ( 28 + 28 \zeta_{6} ) q^{91} -84 q^{92} -57 \zeta_{6} q^{93} + ( 44 + 44 \zeta_{6} ) q^{94} + ( 18 - 18 \zeta_{6} ) q^{95} + ( -66 + 33 \zeta_{6} ) q^{96} + ( 53 - 106 \zeta_{6} ) q^{97} + ( -49 + 49 \zeta_{6} ) q^{98} + 33 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{2} + 3q^{3} + 3q^{4} - 9q^{5} - 7q^{7} - 14q^{8} + 3q^{9} + O(q^{10})$$ $$2q - q^{2} + 3q^{3} + 3q^{4} - 9q^{5} - 7q^{7} - 14q^{8} + 3q^{9} + 9q^{10} + 11q^{11} + 9q^{12} + 14q^{14} - 18q^{15} - 5q^{16} + 42q^{17} + 3q^{18} - 6q^{19} - 21q^{21} - 22q^{22} - 28q^{23} - 21q^{24} + 2q^{25} - 12q^{26} + 21q^{28} + 50q^{29} + 9q^{30} - 57q^{31} - 33q^{32} + 33q^{33} + 18q^{36} + 58q^{37} + 6q^{38} + 12q^{39} + 63q^{40} + 21q^{42} + 52q^{43} - 33q^{44} - 27q^{45} - 28q^{46} - 132q^{47} - 49q^{49} - 4q^{50} + 42q^{51} - 36q^{52} - 31q^{53} + 9q^{54} + 49q^{56} - 12q^{57} - 25q^{58} - 15q^{59} - 27q^{60} + 24q^{61} - 42q^{63} + 26q^{64} + 36q^{65} - 33q^{66} + 52q^{67} + 126q^{68} - 63q^{70} + 128q^{71} - 21q^{72} + 12q^{73} + 58q^{74} + 6q^{75} + 77q^{77} - 24q^{78} - 17q^{79} + 45q^{80} - 9q^{81} - 6q^{82} - 252q^{85} - 26q^{86} + 75q^{87} - 77q^{88} - 138q^{89} + 84q^{91} - 168q^{92} - 57q^{93} + 132q^{94} + 18q^{95} - 99q^{96} - 49q^{98} + 66q^{99} + O(q^{100})$$

## 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
−0.500000 0.866025i 1.50000 + 0.866025i 1.50000 2.59808i −4.50000 + 2.59808i 1.73205i −3.50000 + 6.06218i −7.00000 1.50000 + 2.59808i 4.50000 + 2.59808i
19.1 −0.500000 + 0.866025i 1.50000 0.866025i 1.50000 + 2.59808i −4.50000 2.59808i 1.73205i −3.50000 6.06218i −7.00000 1.50000 2.59808i 4.50000 2.59808i
 $$n$$: 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.b 2
3.b odd 2 1 63.3.m.c 2
4.b odd 2 1 336.3.bh.a 2
5.b even 2 1 525.3.o.g 2
5.c odd 4 2 525.3.s.c 4
7.b odd 2 1 147.3.f.c 2
7.c even 3 1 147.3.d.b 2
7.c even 3 1 147.3.f.c 2
7.d odd 6 1 inner 21.3.f.b 2
7.d odd 6 1 147.3.d.b 2
12.b even 2 1 1008.3.cg.g 2
21.c even 2 1 441.3.m.e 2
21.g even 6 1 63.3.m.c 2
21.g even 6 1 441.3.d.b 2
21.h odd 6 1 441.3.d.b 2
21.h odd 6 1 441.3.m.e 2
28.f even 6 1 336.3.bh.a 2
28.f even 6 1 2352.3.f.d 2
28.g odd 6 1 2352.3.f.d 2
35.i odd 6 1 525.3.o.g 2
35.k even 12 2 525.3.s.c 4
84.j odd 6 1 1008.3.cg.g 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{3}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$27 + 9 T + T^{2}$$
$7$ $$49 + 7 T + T^{2}$$
$11$ $$121 - 11 T + T^{2}$$
$13$ $$48 + T^{2}$$
$17$ $$588 - 42 T + T^{2}$$
$19$ $$12 + 6 T + T^{2}$$
$23$ $$784 + 28 T + T^{2}$$
$29$ $$( -25 + T )^{2}$$
$31$ $$1083 + 57 T + T^{2}$$
$37$ $$3364 - 58 T + T^{2}$$
$41$ $$12 + T^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$5808 + 132 T + T^{2}$$
$53$ $$961 + 31 T + T^{2}$$
$59$ $$75 + 15 T + T^{2}$$
$61$ $$192 - 24 T + T^{2}$$
$67$ $$2704 - 52 T + T^{2}$$
$71$ $$( -64 + T )^{2}$$
$73$ $$48 - 12 T + T^{2}$$
$79$ $$289 + 17 T + T^{2}$$
$83$ $$2883 + T^{2}$$
$89$ $$6348 + 138 T + T^{2}$$
$97$ $$8427 + T^{2}$$