# 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: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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.

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

## 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

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 3T + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 13T + 49$$
$11$ $$T^{2}$$
$13$ $$(T - 23)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 11T + 121$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 13T + 169$$
$37$ $$T^{2} - 73T + 5329$$
$41$ $$T^{2}$$
$43$ $$(T + 61)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 74T + 5476$$
$67$ $$T^{2} - 13T + 169$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 97T + 9409$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$(T - 2)^{2}$$