# Properties

 Label 21.3.h.a Level 21 Weight 3 Character orbit 21.h Analytic conductor 0.572 Analytic rank 0 Dimension 2 CM discriminant -3 Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.572208555157$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( -5 - 3 \zeta_{6} ) q^{7} + ( -9 + 9 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + 3 \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( -5 - 3 \zeta_{6} ) q^{7} + ( -9 + 9 \zeta_{6} ) q^{9} + ( 12 - 12 \zeta_{6} ) q^{12} + 23 q^{13} + ( -16 + 16 \zeta_{6} ) q^{16} + ( -11 + 11 \zeta_{6} ) q^{19} + ( 9 - 24 \zeta_{6} ) q^{21} -25 \zeta_{6} q^{25} -27 q^{27} + ( -12 + 32 \zeta_{6} ) q^{28} + 13 \zeta_{6} q^{31} + 36 q^{36} + ( 73 - 73 \zeta_{6} ) q^{37} + 69 \zeta_{6} q^{39} -61 q^{43} -48 q^{48} + ( 16 + 39 \zeta_{6} ) q^{49} -92 \zeta_{6} q^{52} -33 q^{57} + ( -74 + 74 \zeta_{6} ) q^{61} + ( 72 - 45 \zeta_{6} ) q^{63} + 64 q^{64} + 13 \zeta_{6} q^{67} + 97 \zeta_{6} q^{73} + ( 75 - 75 \zeta_{6} ) q^{75} + 44 q^{76} + ( -11 + 11 \zeta_{6} ) q^{79} -81 \zeta_{6} q^{81} + ( -96 + 60 \zeta_{6} ) q^{84} + ( -115 - 69 \zeta_{6} ) q^{91} + ( -39 + 39 \zeta_{6} ) q^{93} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 4q^{4} - 13q^{7} - 9q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 4q^{4} - 13q^{7} - 9q^{9} + 12q^{12} + 46q^{13} - 16q^{16} - 11q^{19} - 6q^{21} - 25q^{25} - 54q^{27} + 8q^{28} + 13q^{31} + 72q^{36} + 73q^{37} + 69q^{39} - 122q^{43} - 96q^{48} + 71q^{49} - 92q^{52} - 66q^{57} - 74q^{61} + 99q^{63} + 128q^{64} + 13q^{67} + 97q^{73} + 75q^{75} + 88q^{76} - 11q^{79} - 81q^{81} - 132q^{84} - 299q^{91} - 39q^{93} + 4q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 + 2.59808i −2.00000 3.46410i 0 0 −6.50000 2.59808i 0 −4.50000 + 7.79423i 0
11.1 0 1.50000 2.59808i −2.00000 + 3.46410i 0 0 −6.50000 + 2.59808i 0 −4.50000 7.79423i 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
7.c even 3 1 inner
21.h 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.h.a 2
3.b odd 2 1 CM 21.3.h.a 2
4.b odd 2 1 336.3.bn.b 2
7.b odd 2 1 147.3.h.a 2
7.c even 3 1 inner 21.3.h.a 2
7.c even 3 1 147.3.b.a 1
7.d odd 6 1 147.3.b.b 1
7.d odd 6 1 147.3.h.a 2
12.b even 2 1 336.3.bn.b 2
21.c even 2 1 147.3.h.a 2
21.g even 6 1 147.3.b.b 1
21.g even 6 1 147.3.h.a 2
21.h odd 6 1 inner 21.3.h.a 2
21.h odd 6 1 147.3.b.a 1
28.g odd 6 1 336.3.bn.b 2
84.n even 6 1 336.3.bn.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.3.h.a 2 1.a even 1 1 trivial
21.3.h.a 2 3.b odd 2 1 CM
21.3.h.a 2 7.c even 3 1 inner
21.3.h.a 2 21.h odd 6 1 inner
147.3.b.a 1 7.c even 3 1
147.3.b.a 1 21.h odd 6 1
147.3.b.b 1 7.d odd 6 1
147.3.b.b 1 21.g even 6 1
147.3.h.a 2 7.b odd 2 1
147.3.h.a 2 7.d odd 6 1
147.3.h.a 2 21.c even 2 1
147.3.h.a 2 21.g even 6 1
336.3.bn.b 2 4.b odd 2 1
336.3.bn.b 2 12.b even 2 1
336.3.bn.b 2 28.g odd 6 1
336.3.bn.b 2 84.n even 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T + 4 T^{2} )( 1 + 2 T + 4 T^{2} )$$
$3$ $$1 - 3 T + 9 T^{2}$$
$5$ $$( 1 - 5 T + 25 T^{2} )( 1 + 5 T + 25 T^{2} )$$
$7$ $$1 + 13 T + 49 T^{2}$$
$11$ $$( 1 - 11 T + 121 T^{2} )( 1 + 11 T + 121 T^{2} )$$
$13$ $$( 1 - 23 T + 169 T^{2} )^{2}$$
$17$ $$( 1 - 17 T + 289 T^{2} )( 1 + 17 T + 289 T^{2} )$$
$19$ $$( 1 - 26 T + 361 T^{2} )( 1 + 37 T + 361 T^{2} )$$
$23$ $$( 1 - 23 T + 529 T^{2} )( 1 + 23 T + 529 T^{2} )$$
$29$ $$( 1 - 29 T )^{2}( 1 + 29 T )^{2}$$
$31$ $$( 1 - 59 T + 961 T^{2} )( 1 + 46 T + 961 T^{2} )$$
$37$ $$( 1 - 47 T + 1369 T^{2} )( 1 - 26 T + 1369 T^{2} )$$
$41$ $$( 1 - 41 T )^{2}( 1 + 41 T )^{2}$$
$43$ $$( 1 + 61 T + 1849 T^{2} )^{2}$$
$47$ $$( 1 - 47 T + 2209 T^{2} )( 1 + 47 T + 2209 T^{2} )$$
$53$ $$( 1 - 53 T + 2809 T^{2} )( 1 + 53 T + 2809 T^{2} )$$
$59$ $$( 1 - 59 T + 3481 T^{2} )( 1 + 59 T + 3481 T^{2} )$$
$61$ $$( 1 - 47 T + 3721 T^{2} )( 1 + 121 T + 3721 T^{2} )$$
$67$ $$( 1 - 122 T + 4489 T^{2} )( 1 + 109 T + 4489 T^{2} )$$
$71$ $$( 1 - 71 T )^{2}( 1 + 71 T )^{2}$$
$73$ $$( 1 - 143 T + 5329 T^{2} )( 1 + 46 T + 5329 T^{2} )$$
$79$ $$( 1 - 131 T + 6241 T^{2} )( 1 + 142 T + 6241 T^{2} )$$
$83$ $$( 1 - 83 T )^{2}( 1 + 83 T )^{2}$$
$89$ $$( 1 - 89 T + 7921 T^{2} )( 1 + 89 T + 7921 T^{2} )$$
$97$ $$( 1 - 2 T + 9409 T^{2} )^{2}$$