# Properties

 Label 21.3.d.a Level $21$ Weight $3$ Character orbit 21.d 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.d (of order $$2$$, degree $$1$$, 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: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{4} + ( 4 - 8 \zeta_{6} ) q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} + ( -3 + 8 \zeta_{6} ) q^{7} -7 q^{8} -3 q^{9} +O(q^{10})$$ $$q + q^{2} + ( -1 + 2 \zeta_{6} ) q^{3} -3 q^{4} + ( 4 - 8 \zeta_{6} ) q^{5} + ( -1 + 2 \zeta_{6} ) q^{6} + ( -3 + 8 \zeta_{6} ) q^{7} -7 q^{8} -3 q^{9} + ( 4 - 8 \zeta_{6} ) q^{10} + 10 q^{11} + ( 3 - 6 \zeta_{6} ) q^{12} + ( -4 + 8 \zeta_{6} ) q^{13} + ( -3 + 8 \zeta_{6} ) q^{14} + 12 q^{15} + 5 q^{16} -3 q^{18} + ( 12 - 24 \zeta_{6} ) q^{19} + ( -12 + 24 \zeta_{6} ) q^{20} + ( -13 + 2 \zeta_{6} ) q^{21} + 10 q^{22} -14 q^{23} + ( 7 - 14 \zeta_{6} ) q^{24} -23 q^{25} + ( -4 + 8 \zeta_{6} ) q^{26} + ( 3 - 6 \zeta_{6} ) q^{27} + ( 9 - 24 \zeta_{6} ) q^{28} -38 q^{29} + 12 q^{30} + ( -16 + 32 \zeta_{6} ) q^{31} + 33 q^{32} + ( -10 + 20 \zeta_{6} ) q^{33} + ( 52 - 8 \zeta_{6} ) q^{35} + 9 q^{36} + 26 q^{37} + ( 12 - 24 \zeta_{6} ) q^{38} -12 q^{39} + ( -28 + 56 \zeta_{6} ) q^{40} + ( 40 - 80 \zeta_{6} ) q^{41} + ( -13 + 2 \zeta_{6} ) q^{42} + 26 q^{43} -30 q^{44} + ( -12 + 24 \zeta_{6} ) q^{45} -14 q^{46} + ( -16 + 32 \zeta_{6} ) q^{47} + ( -5 + 10 \zeta_{6} ) q^{48} + ( -55 + 16 \zeta_{6} ) q^{49} -23 q^{50} + ( 12 - 24 \zeta_{6} ) q^{52} + 10 q^{53} + ( 3 - 6 \zeta_{6} ) q^{54} + ( 40 - 80 \zeta_{6} ) q^{55} + ( 21 - 56 \zeta_{6} ) q^{56} + 36 q^{57} -38 q^{58} + ( -44 + 88 \zeta_{6} ) q^{59} -36 q^{60} + ( -20 + 40 \zeta_{6} ) q^{61} + ( -16 + 32 \zeta_{6} ) q^{62} + ( 9 - 24 \zeta_{6} ) q^{63} + 13 q^{64} + 48 q^{65} + ( -10 + 20 \zeta_{6} ) q^{66} + 74 q^{67} + ( 14 - 28 \zeta_{6} ) q^{69} + ( 52 - 8 \zeta_{6} ) q^{70} -62 q^{71} + 21 q^{72} + ( 24 - 48 \zeta_{6} ) q^{73} + 26 q^{74} + ( 23 - 46 \zeta_{6} ) q^{75} + ( -36 + 72 \zeta_{6} ) q^{76} + ( -30 + 80 \zeta_{6} ) q^{77} -12 q^{78} -46 q^{79} + ( 20 - 40 \zeta_{6} ) q^{80} + 9 q^{81} + ( 40 - 80 \zeta_{6} ) q^{82} + ( -52 + 104 \zeta_{6} ) q^{83} + ( 39 - 6 \zeta_{6} ) q^{84} + 26 q^{86} + ( 38 - 76 \zeta_{6} ) q^{87} -70 q^{88} + ( 24 - 48 \zeta_{6} ) q^{89} + ( -12 + 24 \zeta_{6} ) q^{90} + ( -52 + 8 \zeta_{6} ) q^{91} + 42 q^{92} -48 q^{93} + ( -16 + 32 \zeta_{6} ) q^{94} -144 q^{95} + ( -33 + 66 \zeta_{6} ) q^{96} + ( -32 + 64 \zeta_{6} ) q^{97} + ( -55 + 16 \zeta_{6} ) q^{98} -30 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 6q^{4} + 2q^{7} - 14q^{8} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 6q^{4} + 2q^{7} - 14q^{8} - 6q^{9} + 20q^{11} + 2q^{14} + 24q^{15} + 10q^{16} - 6q^{18} - 24q^{21} + 20q^{22} - 28q^{23} - 46q^{25} - 6q^{28} - 76q^{29} + 24q^{30} + 66q^{32} + 96q^{35} + 18q^{36} + 52q^{37} - 24q^{39} - 24q^{42} + 52q^{43} - 60q^{44} - 28q^{46} - 94q^{49} - 46q^{50} + 20q^{53} - 14q^{56} + 72q^{57} - 76q^{58} - 72q^{60} - 6q^{63} + 26q^{64} + 96q^{65} + 148q^{67} + 96q^{70} - 124q^{71} + 42q^{72} + 52q^{74} + 20q^{77} - 24q^{78} - 92q^{79} + 18q^{81} + 72q^{84} + 52q^{86} - 140q^{88} - 96q^{91} + 84q^{92} - 96q^{93} - 288q^{95} - 94q^{98} - 60q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$8$$ $$10$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 6.92820i −7.00000 −3.00000 6.92820i
13.2 1.00000 1.73205i −3.00000 6.92820i 1.73205i 1.00000 + 6.92820i −7.00000 −3.00000 6.92820i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.3.d.a 2
3.b odd 2 1 63.3.d.b 2
4.b odd 2 1 336.3.f.a 2
5.b even 2 1 525.3.h.a 2
5.c odd 4 2 525.3.e.a 4
7.b odd 2 1 inner 21.3.d.a 2
7.c even 3 1 147.3.f.b 2
7.c even 3 1 147.3.f.d 2
7.d odd 6 1 147.3.f.b 2
7.d odd 6 1 147.3.f.d 2
8.b even 2 1 1344.3.f.c 2
8.d odd 2 1 1344.3.f.b 2
12.b even 2 1 1008.3.f.d 2
21.c even 2 1 63.3.d.b 2
21.g even 6 1 441.3.m.d 2
21.g even 6 1 441.3.m.f 2
21.h odd 6 1 441.3.m.d 2
21.h odd 6 1 441.3.m.f 2
28.d even 2 1 336.3.f.a 2
35.c odd 2 1 525.3.h.a 2
35.f even 4 2 525.3.e.a 4
56.e even 2 1 1344.3.f.b 2
56.h odd 2 1 1344.3.f.c 2
84.h odd 2 1 1008.3.f.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.d.a 2 1.a even 1 1 trivial
21.3.d.a 2 7.b odd 2 1 inner
63.3.d.b 2 3.b odd 2 1
63.3.d.b 2 21.c even 2 1
147.3.f.b 2 7.c even 3 1
147.3.f.b 2 7.d odd 6 1
147.3.f.d 2 7.c even 3 1
147.3.f.d 2 7.d odd 6 1
336.3.f.a 2 4.b odd 2 1
336.3.f.a 2 28.d even 2 1
441.3.m.d 2 21.g even 6 1
441.3.m.d 2 21.h odd 6 1
441.3.m.f 2 21.g even 6 1
441.3.m.f 2 21.h odd 6 1
525.3.e.a 4 5.c odd 4 2
525.3.e.a 4 35.f even 4 2
525.3.h.a 2 5.b even 2 1
525.3.h.a 2 35.c odd 2 1
1008.3.f.d 2 12.b even 2 1
1008.3.f.d 2 84.h odd 2 1
1344.3.f.b 2 8.d odd 2 1
1344.3.f.b 2 56.e even 2 1
1344.3.f.c 2 8.b even 2 1
1344.3.f.c 2 56.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$3 + T^{2}$$
$5$ $$48 + T^{2}$$
$7$ $$49 - 2 T + T^{2}$$
$11$ $$( -10 + T )^{2}$$
$13$ $$48 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$432 + T^{2}$$
$23$ $$( 14 + T )^{2}$$
$29$ $$( 38 + T )^{2}$$
$31$ $$768 + T^{2}$$
$37$ $$( -26 + T )^{2}$$
$41$ $$4800 + T^{2}$$
$43$ $$( -26 + T )^{2}$$
$47$ $$768 + T^{2}$$
$53$ $$( -10 + T )^{2}$$
$59$ $$5808 + T^{2}$$
$61$ $$1200 + T^{2}$$
$67$ $$( -74 + T )^{2}$$
$71$ $$( 62 + T )^{2}$$
$73$ $$1728 + T^{2}$$
$79$ $$( 46 + T )^{2}$$
$83$ $$8112 + T^{2}$$
$89$ $$1728 + T^{2}$$
$97$ $$3072 + T^{2}$$