# Properties

 Label 21.9.h.a Level $21$ Weight $9$ Character orbit 21.h Analytic conductor $8.555$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 21.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.55495081128$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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.

 $$f(q)$$ $$=$$ $$q -81 \zeta_{6} q^{3} -256 \zeta_{6} q^{4} + ( -1265 + 2769 \zeta_{6} ) q^{7} + ( -6561 + 6561 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -81 \zeta_{6} q^{3} -256 \zeta_{6} q^{4} + ( -1265 + 2769 \zeta_{6} ) q^{7} + ( -6561 + 6561 \zeta_{6} ) q^{9} + ( -20736 + 20736 \zeta_{6} ) q^{12} -20641 q^{13} + ( -65536 + 65536 \zeta_{6} ) q^{16} + ( -100559 + 100559 \zeta_{6} ) q^{19} + ( 224289 - 121824 \zeta_{6} ) q^{21} -390625 \zeta_{6} q^{25} + 531441 q^{27} + ( 708864 - 385024 \zeta_{6} ) q^{28} -1225967 \zeta_{6} q^{31} + 1679616 q^{36} + ( -2964959 + 2964959 \zeta_{6} ) q^{37} + 1671921 \zeta_{6} q^{39} -6837073 q^{43} + 5308416 q^{48} + ( -6067136 + 661791 \zeta_{6} ) q^{49} + 5284096 \zeta_{6} q^{52} + 8145279 q^{57} + ( 23826526 - 23826526 \zeta_{6} ) q^{61} + ( -9867744 - 8299665 \zeta_{6} ) q^{63} + 16777216 q^{64} -37296239 \zeta_{6} q^{67} + 55236481 \zeta_{6} q^{73} + ( -31640625 + 31640625 \zeta_{6} ) q^{75} + 25743104 q^{76} + ( -74894159 + 74894159 \zeta_{6} ) q^{79} -43046721 \zeta_{6} q^{81} + ( -31186944 - 26231040 \zeta_{6} ) q^{84} + ( 26110865 - 57154929 \zeta_{6} ) q^{91} + ( -99303327 + 99303327 \zeta_{6} ) q^{93} + 176908034 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 81q^{3} - 256q^{4} + 239q^{7} - 6561q^{9} + O(q^{10})$$ $$2q - 81q^{3} - 256q^{4} + 239q^{7} - 6561q^{9} - 20736q^{12} - 41282q^{13} - 65536q^{16} - 100559q^{19} + 326754q^{21} - 390625q^{25} + 1062882q^{27} + 1032704q^{28} - 1225967q^{31} + 3359232q^{36} - 2964959q^{37} + 1671921q^{39} - 13674146q^{43} + 10616832q^{48} - 11472481q^{49} + 5284096q^{52} + 16290558q^{57} + 23826526q^{61} - 28035153q^{63} + 33554432q^{64} - 37296239q^{67} + 55236481q^{73} - 31640625q^{75} + 51486208q^{76} - 74894159q^{79} - 43046721q^{81} - 88604928q^{84} - 4933199q^{91} - 99303327q^{93} + 353816068q^{97} + O(q^{100})$$

## 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 −40.5000 70.1481i −128.000 221.703i 0 0 119.500 + 2398.02i 0 −3280.50 + 5681.99i 0
11.1 0 −40.5000 + 70.1481i −128.000 + 221.703i 0 0 119.500 2398.02i 0 −3280.50 5681.99i 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 21.9.h.a 2
3.b odd 2 1 CM 21.9.h.a 2
7.c even 3 1 inner 21.9.h.a 2
21.h odd 6 1 inner 21.9.h.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.9.h.a 2 1.a even 1 1 trivial
21.9.h.a 2 3.b odd 2 1 CM
21.9.h.a 2 7.c even 3 1 inner
21.9.h.a 2 21.h odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{9}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$6561 + 81 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$5764801 - 239 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( 20641 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$10112112481 + 100559 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$1502995085089 + 1225967 T + T^{2}$$
$37$ $$8790981871681 + 2964959 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( 6837073 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$567703341228676 - 23826526 T + T^{2}$$
$67$ $$1391009443545121 + 37296239 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$3051068833263361 - 55236481 T + T^{2}$$
$79$ $$5609135052317281 + 74894159 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( -176908034 + T )^{2}$$