# Properties

 Label 21.2.g.a Level $21$ Weight $2$ Character orbit 21.g Analytic conductor $0.168$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,2,Mod(5,21)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.167685844245$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + ( - 3 \zeta_{6} + 2) q^{7} + 3 \zeta_{6} q^{9}+O(q^{10})$$ q + (-z - 1) * q^3 + (2*z - 2) * q^4 + (-3*z + 2) * q^7 + 3*z * q^9 $$q + ( - \zeta_{6} - 1) q^{3} + (2 \zeta_{6} - 2) q^{4} + ( - 3 \zeta_{6} + 2) q^{7} + 3 \zeta_{6} q^{9} + ( - 2 \zeta_{6} + 4) q^{12} + (2 \zeta_{6} - 1) q^{13} - 4 \zeta_{6} q^{16} + (3 \zeta_{6} - 6) q^{19} + (4 \zeta_{6} - 5) q^{21} + ( - 5 \zeta_{6} + 5) q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + (4 \zeta_{6} + 2) q^{28} + (5 \zeta_{6} + 5) q^{31} - 6 q^{36} - \zeta_{6} q^{37} + ( - 3 \zeta_{6} + 3) q^{39} - 5 q^{43} + (8 \zeta_{6} - 4) q^{48} + ( - 3 \zeta_{6} - 5) q^{49} + ( - 2 \zeta_{6} - 2) q^{52} + 9 q^{57} + ( - 4 \zeta_{6} + 8) q^{61} + ( - 3 \zeta_{6} + 9) q^{63} + 8 q^{64} + (11 \zeta_{6} - 11) q^{67} + ( - 9 \zeta_{6} - 9) q^{73} + (5 \zeta_{6} - 10) q^{75} + ( - 12 \zeta_{6} + 6) q^{76} + 13 \zeta_{6} q^{79} + (9 \zeta_{6} - 9) q^{81} + ( - 10 \zeta_{6} + 2) q^{84} + (\zeta_{6} + 4) q^{91} - 15 \zeta_{6} q^{93} + (16 \zeta_{6} - 8) q^{97} +O(q^{100})$$ q + (-z - 1) * q^3 + (2*z - 2) * q^4 + (-3*z + 2) * q^7 + 3*z * q^9 + (-2*z + 4) * q^12 + (2*z - 1) * q^13 - 4*z * q^16 + (3*z - 6) * q^19 + (4*z - 5) * q^21 + (-5*z + 5) * q^25 + (-6*z + 3) * q^27 + (4*z + 2) * q^28 + (5*z + 5) * q^31 - 6 * q^36 - z * q^37 + (-3*z + 3) * q^39 - 5 * q^43 + (8*z - 4) * q^48 + (-3*z - 5) * q^49 + (-2*z - 2) * q^52 + 9 * q^57 + (-4*z + 8) * q^61 + (-3*z + 9) * q^63 + 8 * q^64 + (11*z - 11) * q^67 + (-9*z - 9) * q^73 + (5*z - 10) * q^75 + (-12*z + 6) * q^76 + 13*z * q^79 + (9*z - 9) * q^81 + (-10*z + 2) * q^84 + (z + 4) * q^91 - 15*z * q^93 + (16*z - 8) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{3} - 2 q^{4} + q^{7} + 3 q^{9}+O(q^{10})$$ 2 * q - 3 * q^3 - 2 * q^4 + q^7 + 3 * q^9 $$2 q - 3 q^{3} - 2 q^{4} + q^{7} + 3 q^{9} + 6 q^{12} - 4 q^{16} - 9 q^{19} - 6 q^{21} + 5 q^{25} + 8 q^{28} + 15 q^{31} - 12 q^{36} - q^{37} + 3 q^{39} - 10 q^{43} - 13 q^{49} - 6 q^{52} + 18 q^{57} + 12 q^{61} + 15 q^{63} + 16 q^{64} - 11 q^{67} - 27 q^{73} - 15 q^{75} + 13 q^{79} - 9 q^{81} - 6 q^{84} + 9 q^{91} - 15 q^{93}+O(q^{100})$$ 2 * q - 3 * q^3 - 2 * q^4 + q^7 + 3 * q^9 + 6 * q^12 - 4 * q^16 - 9 * q^19 - 6 * q^21 + 5 * q^25 + 8 * q^28 + 15 * q^31 - 12 * q^36 - q^37 + 3 * q^39 - 10 * q^43 - 13 * q^49 - 6 * q^52 + 18 * q^57 + 12 * q^61 + 15 * q^63 + 16 * q^64 - 11 * q^67 - 27 * q^73 - 15 * q^75 + 13 * q^79 - 9 * q^81 - 6 * q^84 + 9 * q^91 - 15 * q^93

## Character values

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

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

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 −1.50000 + 0.866025i −1.00000 1.73205i 0 0 0.500000 + 2.59808i 0 1.50000 2.59808i 0
17.1 0 −1.50000 0.866025i −1.00000 + 1.73205i 0 0 0.500000 2.59808i 0 1.50000 + 2.59808i 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.d odd 6 1 inner
21.g even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.2.g.a 2
3.b odd 2 1 CM 21.2.g.a 2
4.b odd 2 1 336.2.bc.c 2
5.b even 2 1 525.2.t.c 2
5.c odd 4 2 525.2.q.d 4
7.b odd 2 1 147.2.g.a 2
7.c even 3 1 147.2.c.a 2
7.c even 3 1 147.2.g.a 2
7.d odd 6 1 inner 21.2.g.a 2
7.d odd 6 1 147.2.c.a 2
9.c even 3 1 567.2.i.b 2
9.c even 3 1 567.2.s.a 2
9.d odd 6 1 567.2.i.b 2
9.d odd 6 1 567.2.s.a 2
12.b even 2 1 336.2.bc.c 2
15.d odd 2 1 525.2.t.c 2
15.e even 4 2 525.2.q.d 4
21.c even 2 1 147.2.g.a 2
21.g even 6 1 inner 21.2.g.a 2
21.g even 6 1 147.2.c.a 2
21.h odd 6 1 147.2.c.a 2
21.h odd 6 1 147.2.g.a 2
28.f even 6 1 336.2.bc.c 2
28.f even 6 1 2352.2.k.c 2
28.g odd 6 1 2352.2.k.c 2
35.i odd 6 1 525.2.t.c 2
35.k even 12 2 525.2.q.d 4
63.i even 6 1 567.2.s.a 2
63.k odd 6 1 567.2.i.b 2
63.s even 6 1 567.2.i.b 2
63.t odd 6 1 567.2.s.a 2
84.j odd 6 1 336.2.bc.c 2
84.j odd 6 1 2352.2.k.c 2
84.n even 6 1 2352.2.k.c 2
105.p even 6 1 525.2.t.c 2
105.w odd 12 2 525.2.q.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.g.a 2 1.a even 1 1 trivial
21.2.g.a 2 3.b odd 2 1 CM
21.2.g.a 2 7.d odd 6 1 inner
21.2.g.a 2 21.g even 6 1 inner
147.2.c.a 2 7.c even 3 1
147.2.c.a 2 7.d odd 6 1
147.2.c.a 2 21.g even 6 1
147.2.c.a 2 21.h odd 6 1
147.2.g.a 2 7.b odd 2 1
147.2.g.a 2 7.c even 3 1
147.2.g.a 2 21.c even 2 1
147.2.g.a 2 21.h odd 6 1
336.2.bc.c 2 4.b odd 2 1
336.2.bc.c 2 12.b even 2 1
336.2.bc.c 2 28.f even 6 1
336.2.bc.c 2 84.j odd 6 1
525.2.q.d 4 5.c odd 4 2
525.2.q.d 4 15.e even 4 2
525.2.q.d 4 35.k even 12 2
525.2.q.d 4 105.w odd 12 2
525.2.t.c 2 5.b even 2 1
525.2.t.c 2 15.d odd 2 1
525.2.t.c 2 35.i odd 6 1
525.2.t.c 2 105.p even 6 1
567.2.i.b 2 9.c even 3 1
567.2.i.b 2 9.d odd 6 1
567.2.i.b 2 63.k odd 6 1
567.2.i.b 2 63.s even 6 1
567.2.s.a 2 9.c even 3 1
567.2.s.a 2 9.d odd 6 1
567.2.s.a 2 63.i even 6 1
567.2.s.a 2 63.t odd 6 1
2352.2.k.c 2 28.f even 6 1
2352.2.k.c 2 28.g odd 6 1
2352.2.k.c 2 84.j odd 6 1
2352.2.k.c 2 84.n even 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3T + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} - T + 7$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 3$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 9T + 27$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 15T + 75$$
$37$ $$T^{2} + T + 1$$
$41$ $$T^{2}$$
$43$ $$(T + 5)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 12T + 48$$
$67$ $$T^{2} + 11T + 121$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 27T + 243$$
$79$ $$T^{2} - 13T + 169$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 192$$