# Properties

 Label 21.3.f.c 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( -4 + 2 \zeta_{6} ) q^{5} + ( 2 - 4 \zeta_{6} ) q^{6} -7 \zeta_{6} q^{7} + 8 q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{2} + ( -1 - \zeta_{6} ) q^{3} + ( -4 + 2 \zeta_{6} ) q^{5} + ( 2 - 4 \zeta_{6} ) q^{6} -7 \zeta_{6} q^{7} + 8 q^{8} + 3 \zeta_{6} q^{9} + ( -4 - 4 \zeta_{6} ) q^{10} + ( -10 + 10 \zeta_{6} ) q^{11} + ( -7 + 14 \zeta_{6} ) q^{13} + ( 14 - 14 \zeta_{6} ) q^{14} + 6 q^{15} + 16 \zeta_{6} q^{16} + ( -4 - 4 \zeta_{6} ) q^{17} + ( -6 + 6 \zeta_{6} ) q^{18} + ( 38 - 19 \zeta_{6} ) q^{19} + ( -7 + 14 \zeta_{6} ) q^{21} -20 q^{22} -40 \zeta_{6} q^{23} + ( -8 - 8 \zeta_{6} ) q^{24} + ( -13 + 13 \zeta_{6} ) q^{25} + ( -28 + 14 \zeta_{6} ) q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + 16 q^{29} + 12 \zeta_{6} q^{30} + ( 3 + 3 \zeta_{6} ) q^{31} + ( 20 - 10 \zeta_{6} ) q^{33} + ( 8 - 16 \zeta_{6} ) q^{34} + ( 14 + 14 \zeta_{6} ) q^{35} -5 \zeta_{6} q^{37} + ( 38 + 38 \zeta_{6} ) q^{38} + ( 21 - 21 \zeta_{6} ) q^{39} + ( -32 + 16 \zeta_{6} ) q^{40} + ( -14 + 28 \zeta_{6} ) q^{41} + ( -28 + 14 \zeta_{6} ) q^{42} -19 q^{43} + ( -6 - 6 \zeta_{6} ) q^{45} + ( 80 - 80 \zeta_{6} ) q^{46} + ( -60 + 30 \zeta_{6} ) q^{47} + ( 16 - 32 \zeta_{6} ) q^{48} + ( -49 + 49 \zeta_{6} ) q^{49} -26 q^{50} + 12 \zeta_{6} q^{51} + ( 32 - 32 \zeta_{6} ) q^{53} + ( 12 - 6 \zeta_{6} ) q^{54} + ( 20 - 40 \zeta_{6} ) q^{55} -56 \zeta_{6} q^{56} -57 q^{57} + 32 \zeta_{6} q^{58} + ( 24 + 24 \zeta_{6} ) q^{59} + ( 24 - 12 \zeta_{6} ) q^{61} + ( -6 + 12 \zeta_{6} ) q^{62} + ( 21 - 21 \zeta_{6} ) q^{63} + 64 q^{64} -42 \zeta_{6} q^{65} + ( 20 + 20 \zeta_{6} ) q^{66} + ( -59 + 59 \zeta_{6} ) q^{67} + ( -40 + 80 \zeta_{6} ) q^{69} + ( -28 + 56 \zeta_{6} ) q^{70} -26 q^{71} + 24 \zeta_{6} q^{72} + ( -11 - 11 \zeta_{6} ) q^{73} + ( 10 - 10 \zeta_{6} ) q^{74} + ( 26 - 13 \zeta_{6} ) q^{75} + 70 q^{77} + 42 q^{78} -47 \zeta_{6} q^{79} + ( -32 - 32 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} + ( -56 + 28 \zeta_{6} ) q^{82} + ( 14 - 28 \zeta_{6} ) q^{83} + 24 q^{85} -38 \zeta_{6} q^{86} + ( -16 - 16 \zeta_{6} ) q^{87} + ( -80 + 80 \zeta_{6} ) q^{88} + ( 136 - 68 \zeta_{6} ) q^{89} + ( 12 - 24 \zeta_{6} ) q^{90} + ( 98 - 49 \zeta_{6} ) q^{91} -9 \zeta_{6} q^{93} + ( -60 - 60 \zeta_{6} ) q^{94} + ( -114 + 114 \zeta_{6} ) q^{95} + ( 28 - 56 \zeta_{6} ) q^{97} -98 q^{98} -30 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 3q^{3} - 6q^{5} - 7q^{7} + 16q^{8} + 3q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 3q^{3} - 6q^{5} - 7q^{7} + 16q^{8} + 3q^{9} - 12q^{10} - 10q^{11} + 14q^{14} + 12q^{15} + 16q^{16} - 12q^{17} - 6q^{18} + 57q^{19} - 40q^{22} - 40q^{23} - 24q^{24} - 13q^{25} - 42q^{26} + 32q^{29} + 12q^{30} + 9q^{31} + 30q^{33} + 42q^{35} - 5q^{37} + 114q^{38} + 21q^{39} - 48q^{40} - 42q^{42} - 38q^{43} - 18q^{45} + 80q^{46} - 90q^{47} - 49q^{49} - 52q^{50} + 12q^{51} + 32q^{53} + 18q^{54} - 56q^{56} - 114q^{57} + 32q^{58} + 72q^{59} + 36q^{61} + 21q^{63} + 128q^{64} - 42q^{65} + 60q^{66} - 59q^{67} - 52q^{71} + 24q^{72} - 33q^{73} + 10q^{74} + 39q^{75} + 140q^{77} + 84q^{78} - 47q^{79} - 96q^{80} - 9q^{81} - 84q^{82} + 48q^{85} - 38q^{86} - 48q^{87} - 80q^{88} + 204q^{89} + 147q^{91} - 9q^{93} - 180q^{94} - 114q^{95} - 196q^{98} - 60q^{99} + O(q^{100})$$

## 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
1.00000 + 1.73205i −1.50000 0.866025i 0 −3.00000 + 1.73205i 3.46410i −3.50000 6.06218i 8.00000 1.50000 + 2.59808i −6.00000 3.46410i
19.1 1.00000 1.73205i −1.50000 + 0.866025i 0 −3.00000 1.73205i 3.46410i −3.50000 + 6.06218i 8.00000 1.50000 2.59808i −6.00000 + 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.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.c 2
3.b odd 2 1 63.3.m.a 2
4.b odd 2 1 336.3.bh.c 2
5.b even 2 1 525.3.o.b 2
5.c odd 4 2 525.3.s.d 4
7.b odd 2 1 147.3.f.e 2
7.c even 3 1 147.3.d.a 2
7.c even 3 1 147.3.f.e 2
7.d odd 6 1 inner 21.3.f.c 2
7.d odd 6 1 147.3.d.a 2
12.b even 2 1 1008.3.cg.f 2
21.c even 2 1 441.3.m.b 2
21.g even 6 1 63.3.m.a 2
21.g even 6 1 441.3.d.d 2
21.h odd 6 1 441.3.d.d 2
21.h odd 6 1 441.3.m.b 2
28.f even 6 1 336.3.bh.c 2
28.f even 6 1 2352.3.f.b 2
28.g odd 6 1 2352.3.f.b 2
35.i odd 6 1 525.3.o.b 2
35.k even 12 2 525.3.s.d 4
84.j odd 6 1 1008.3.cg.f 2

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} - 2 T_{2} + 4$$ acting on $$S_{3}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T )^{2}( 1 + 2 T + 4 T^{2} )$$
$3$ $$1 + 3 T + 3 T^{2}$$
$5$ $$1 + 6 T + 37 T^{2} + 150 T^{3} + 625 T^{4}$$
$7$ $$1 + 7 T + 49 T^{2}$$
$11$ $$1 + 10 T - 21 T^{2} + 1210 T^{3} + 14641 T^{4}$$
$13$ $$( 1 - 23 T + 169 T^{2} )( 1 + 23 T + 169 T^{2} )$$
$17$ $$1 + 12 T + 337 T^{2} + 3468 T^{3} + 83521 T^{4}$$
$19$ $$( 1 - 19 T )^{2}( 1 - 19 T + 361 T^{2} )$$
$23$ $$1 + 40 T + 1071 T^{2} + 21160 T^{3} + 279841 T^{4}$$
$29$ $$( 1 - 16 T + 841 T^{2} )^{2}$$
$31$ $$1 - 9 T + 988 T^{2} - 8649 T^{3} + 923521 T^{4}$$
$37$ $$1 + 5 T - 1344 T^{2} + 6845 T^{3} + 1874161 T^{4}$$
$41$ $$1 - 2774 T^{2} + 2825761 T^{4}$$
$43$ $$( 1 + 19 T + 1849 T^{2} )^{2}$$
$47$ $$1 + 90 T + 4909 T^{2} + 198810 T^{3} + 4879681 T^{4}$$
$53$ $$1 - 32 T - 1785 T^{2} - 89888 T^{3} + 7890481 T^{4}$$
$59$ $$1 - 72 T + 5209 T^{2} - 250632 T^{3} + 12117361 T^{4}$$
$61$ $$1 - 36 T + 4153 T^{2} - 133956 T^{3} + 13845841 T^{4}$$
$67$ $$1 + 59 T - 1008 T^{2} + 264851 T^{3} + 20151121 T^{4}$$
$71$ $$( 1 + 26 T + 5041 T^{2} )^{2}$$
$73$ $$1 + 33 T + 5692 T^{2} + 175857 T^{3} + 28398241 T^{4}$$
$79$ $$1 + 47 T - 4032 T^{2} + 293327 T^{3} + 38950081 T^{4}$$
$83$ $$1 - 13190 T^{2} + 47458321 T^{4}$$
$89$ $$1 - 204 T + 21793 T^{2} - 1615884 T^{3} + 62742241 T^{4}$$
$97$ $$1 - 16466 T^{2} + 88529281 T^{4}$$