# Properties

 Label 21.4.c.a Level $21$ Weight $4$ Character orbit 21.c Analytic conductor $1.239$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 21.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.23904011012$$ 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: $$2\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 6 \zeta_{6} ) q^{3} + 8 q^{4} + ( -19 + 18 \zeta_{6} ) q^{7} -27 q^{9} +O(q^{10})$$ $$q + ( 3 - 6 \zeta_{6} ) q^{3} + 8 q^{4} + ( -19 + 18 \zeta_{6} ) q^{7} -27 q^{9} + ( 24 - 48 \zeta_{6} ) q^{12} + ( -36 + 72 \zeta_{6} ) q^{13} + 64 q^{16} + ( 90 - 180 \zeta_{6} ) q^{19} + ( 51 + 60 \zeta_{6} ) q^{21} -125 q^{25} + ( -81 + 162 \zeta_{6} ) q^{27} + ( -152 + 144 \zeta_{6} ) q^{28} + ( 90 - 180 \zeta_{6} ) q^{31} -216 q^{36} -110 q^{37} + 324 q^{39} + 520 q^{43} + ( 192 - 384 \zeta_{6} ) q^{48} + ( 37 - 360 \zeta_{6} ) q^{49} + ( -288 + 576 \zeta_{6} ) q^{52} -810 q^{57} + ( -540 + 1080 \zeta_{6} ) q^{61} + ( 513 - 486 \zeta_{6} ) q^{63} + 512 q^{64} -880 q^{67} + ( 216 - 432 \zeta_{6} ) q^{73} + ( -375 + 750 \zeta_{6} ) q^{75} + ( 720 - 1440 \zeta_{6} ) q^{76} + 884 q^{79} + 729 q^{81} + ( 408 + 480 \zeta_{6} ) q^{84} + ( -612 - 720 \zeta_{6} ) q^{91} -810 q^{93} + ( -792 + 1584 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 16q^{4} - 20q^{7} - 54q^{9} + O(q^{10})$$ $$2q + 16q^{4} - 20q^{7} - 54q^{9} + 128q^{16} + 162q^{21} - 250q^{25} - 160q^{28} - 432q^{36} - 220q^{37} + 648q^{39} + 1040q^{43} - 286q^{49} - 1620q^{57} + 540q^{63} + 1024q^{64} - 1760q^{67} + 1768q^{79} + 1458q^{81} + 1296q^{84} - 1944q^{91} - 1620q^{93} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
20.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 5.19615i 8.00000 0 0 −10.0000 + 15.5885i 0 −27.0000 0
20.2 0 5.19615i 8.00000 0 0 −10.0000 15.5885i 0 −27.0000 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.b odd 2 1 inner
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.4.c.a 2
3.b odd 2 1 CM 21.4.c.a 2
4.b odd 2 1 336.4.k.a 2
7.b odd 2 1 inner 21.4.c.a 2
7.c even 3 1 147.4.g.a 2
7.c even 3 1 147.4.g.b 2
7.d odd 6 1 147.4.g.a 2
7.d odd 6 1 147.4.g.b 2
12.b even 2 1 336.4.k.a 2
21.c even 2 1 inner 21.4.c.a 2
21.g even 6 1 147.4.g.a 2
21.g even 6 1 147.4.g.b 2
21.h odd 6 1 147.4.g.a 2
21.h odd 6 1 147.4.g.b 2
28.d even 2 1 336.4.k.a 2
84.h odd 2 1 336.4.k.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.c.a 2 1.a even 1 1 trivial
21.4.c.a 2 3.b odd 2 1 CM
21.4.c.a 2 7.b odd 2 1 inner
21.4.c.a 2 21.c even 2 1 inner
147.4.g.a 2 7.c even 3 1
147.4.g.a 2 7.d odd 6 1
147.4.g.a 2 21.g even 6 1
147.4.g.a 2 21.h odd 6 1
147.4.g.b 2 7.c even 3 1
147.4.g.b 2 7.d odd 6 1
147.4.g.b 2 21.g even 6 1
147.4.g.b 2 21.h odd 6 1
336.4.k.a 2 4.b odd 2 1
336.4.k.a 2 12.b even 2 1
336.4.k.a 2 28.d even 2 1
336.4.k.a 2 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{4}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$27 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$343 + 20 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$3888 + T^{2}$$
$17$ $$T^{2}$$
$19$ $$24300 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$24300 + T^{2}$$
$37$ $$( 110 + T )^{2}$$
$41$ $$T^{2}$$
$43$ $$( -520 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$874800 + T^{2}$$
$67$ $$( 880 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$139968 + T^{2}$$
$79$ $$( -884 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$1881792 + T^{2}$$