# Properties

 Label 21.5.h.a Level $21$ Weight $5$ Character orbit 21.h Analytic conductor $2.171$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 21.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17076922476$$ 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.

 $$f(q)$$ $$=$$ $$q -9 \zeta_{6} q^{3} -16 \zeta_{6} q^{4} + ( 55 - 39 \zeta_{6} ) q^{7} + ( -81 + 81 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q -9 \zeta_{6} q^{3} -16 \zeta_{6} q^{4} + ( 55 - 39 \zeta_{6} ) q^{7} + ( -81 + 81 \zeta_{6} ) q^{9} + ( -144 + 144 \zeta_{6} ) q^{12} + 191 q^{13} + ( -256 + 256 \zeta_{6} ) q^{16} + ( 601 - 601 \zeta_{6} ) q^{19} + ( -351 - 144 \zeta_{6} ) q^{21} -625 \zeta_{6} q^{25} + 729 q^{27} + ( -624 - 256 \zeta_{6} ) q^{28} + 1753 \zeta_{6} q^{31} + 1296 q^{36} + ( -2591 + 2591 \zeta_{6} ) q^{37} -1719 \zeta_{6} q^{39} + 23 q^{43} + 2304 q^{48} + ( 1504 - 2769 \zeta_{6} ) q^{49} -3056 \zeta_{6} q^{52} -5409 q^{57} + ( 1966 - 1966 \zeta_{6} ) q^{61} + ( -1296 + 4455 \zeta_{6} ) q^{63} + 4096 q^{64} + 8809 \zeta_{6} q^{67} + 1249 \zeta_{6} q^{73} + ( -5625 + 5625 \zeta_{6} ) q^{75} -9616 q^{76} + ( 12361 - 12361 \zeta_{6} ) q^{79} -6561 \zeta_{6} q^{81} + ( -2304 + 7920 \zeta_{6} ) q^{84} + ( 10505 - 7449 \zeta_{6} ) q^{91} + ( 15777 - 15777 \zeta_{6} ) q^{93} -18814 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 9q^{3} - 16q^{4} + 71q^{7} - 81q^{9} + O(q^{10})$$ $$2q - 9q^{3} - 16q^{4} + 71q^{7} - 81q^{9} - 144q^{12} + 382q^{13} - 256q^{16} + 601q^{19} - 846q^{21} - 625q^{25} + 1458q^{27} - 1504q^{28} + 1753q^{31} + 2592q^{36} - 2591q^{37} - 1719q^{39} + 46q^{43} + 4608q^{48} + 239q^{49} - 3056q^{52} - 10818q^{57} + 1966q^{61} + 1863q^{63} + 8192q^{64} + 8809q^{67} + 1249q^{73} - 5625q^{75} - 19232q^{76} + 12361q^{79} - 6561q^{81} + 3312q^{84} + 13561q^{91} + 15777q^{93} - 37628q^{97} + O(q^{100})$$

## 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 −4.50000 7.79423i −8.00000 13.8564i 0 0 35.5000 33.7750i 0 −40.5000 + 70.1481i 0
11.1 0 −4.50000 + 7.79423i −8.00000 + 13.8564i 0 0 35.5000 + 33.7750i 0 −40.5000 70.1481i 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 21.5.h.a 2
3.b odd 2 1 CM 21.5.h.a 2
7.b odd 2 1 147.5.h.a 2
7.c even 3 1 inner 21.5.h.a 2
7.c even 3 1 147.5.b.b 1
7.d odd 6 1 147.5.b.a 1
7.d odd 6 1 147.5.h.a 2
21.c even 2 1 147.5.h.a 2
21.g even 6 1 147.5.b.a 1
21.g even 6 1 147.5.h.a 2
21.h odd 6 1 inner 21.5.h.a 2
21.h odd 6 1 147.5.b.b 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{5}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$81 + 9 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$2401 - 71 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$( -191 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$361201 - 601 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$3073009 - 1753 T + T^{2}$$
$37$ $$6713281 + 2591 T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$( -23 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$3865156 - 1966 T + T^{2}$$
$67$ $$77598481 - 8809 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$1560001 - 1249 T + T^{2}$$
$79$ $$152794321 - 12361 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$( 18814 + T )^{2}$$