# 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: $$x^{2} - x + 1$$ 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.

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

## 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 $$T_{2}^{2} + 3T_{2} + 9$$ acting on $$S_{3}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

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