# Properties

 Label 21.2.e.a Level $21$ Weight $2$ Character orbit 21.e Analytic conductor $0.168$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,2,Mod(4,21)]

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([0, 4]))

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

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

chi := DirichletCharacter("21.4");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 21.e (of order $$3$$, 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: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 2) q^{2} - \zeta_{6} q^{3} - 2 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 2) q^{5} + 2 q^{6} + (\zeta_{6} - 3) q^{7} + (\zeta_{6} - 1) q^{9} +O(q^{10})$$ q + (2*z - 2) * q^2 - z * q^3 - 2*z * q^4 + (-2*z + 2) * q^5 + 2 * q^6 + (z - 3) * q^7 + (z - 1) * q^9 $$q + (2 \zeta_{6} - 2) q^{2} - \zeta_{6} q^{3} - 2 \zeta_{6} q^{4} + ( - 2 \zeta_{6} + 2) q^{5} + 2 q^{6} + (\zeta_{6} - 3) q^{7} + (\zeta_{6} - 1) q^{9} + 4 \zeta_{6} q^{10} + 2 \zeta_{6} q^{11} + (2 \zeta_{6} - 2) q^{12} + q^{13} + ( - 6 \zeta_{6} + 4) q^{14} - 2 q^{15} + ( - 4 \zeta_{6} + 4) q^{16} - 2 \zeta_{6} q^{18} + (\zeta_{6} - 1) q^{19} - 4 q^{20} + (2 \zeta_{6} + 1) q^{21} - 4 q^{22} + \zeta_{6} q^{25} + (2 \zeta_{6} - 2) q^{26} + q^{27} + (4 \zeta_{6} + 2) q^{28} + 4 q^{29} + ( - 4 \zeta_{6} + 4) q^{30} - 9 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} + ( - 2 \zeta_{6} + 2) q^{33} + (6 \zeta_{6} - 4) q^{35} + 2 q^{36} + (3 \zeta_{6} - 3) q^{37} - 2 \zeta_{6} q^{38} - \zeta_{6} q^{39} - 10 q^{41} + (2 \zeta_{6} - 6) q^{42} + 5 q^{43} + ( - 4 \zeta_{6} + 4) q^{44} + 2 \zeta_{6} q^{45} + ( - 6 \zeta_{6} + 6) q^{47} - 4 q^{48} + ( - 5 \zeta_{6} + 8) q^{49} - 2 q^{50} - 2 \zeta_{6} q^{52} - 12 \zeta_{6} q^{53} + (2 \zeta_{6} - 2) q^{54} + 4 q^{55} + q^{57} + (8 \zeta_{6} - 8) q^{58} + 12 \zeta_{6} q^{59} + 4 \zeta_{6} q^{60} + (10 \zeta_{6} - 10) q^{61} + 18 q^{62} + ( - 3 \zeta_{6} + 2) q^{63} - 8 q^{64} + ( - 2 \zeta_{6} + 2) q^{65} + 4 \zeta_{6} q^{66} + 5 \zeta_{6} q^{67} + ( - 8 \zeta_{6} - 4) q^{70} - 6 q^{71} + 3 \zeta_{6} q^{73} - 6 \zeta_{6} q^{74} + ( - \zeta_{6} + 1) q^{75} + 2 q^{76} + ( - 4 \zeta_{6} - 2) q^{77} + 2 q^{78} + ( - \zeta_{6} + 1) q^{79} - 8 \zeta_{6} q^{80} - \zeta_{6} q^{81} + ( - 20 \zeta_{6} + 20) q^{82} + 6 q^{83} + ( - 6 \zeta_{6} + 4) q^{84} + (10 \zeta_{6} - 10) q^{86} - 4 \zeta_{6} q^{87} + (16 \zeta_{6} - 16) q^{89} - 4 q^{90} + (\zeta_{6} - 3) q^{91} + (9 \zeta_{6} - 9) q^{93} + 12 \zeta_{6} q^{94} + 2 \zeta_{6} q^{95} + ( - 8 \zeta_{6} + 8) q^{96} - 6 q^{97} + (16 \zeta_{6} - 6) q^{98} - 2 q^{99} +O(q^{100})$$ q + (2*z - 2) * q^2 - z * q^3 - 2*z * q^4 + (-2*z + 2) * q^5 + 2 * q^6 + (z - 3) * q^7 + (z - 1) * q^9 + 4*z * q^10 + 2*z * q^11 + (2*z - 2) * q^12 + q^13 + (-6*z + 4) * q^14 - 2 * q^15 + (-4*z + 4) * q^16 - 2*z * q^18 + (z - 1) * q^19 - 4 * q^20 + (2*z + 1) * q^21 - 4 * q^22 + z * q^25 + (2*z - 2) * q^26 + q^27 + (4*z + 2) * q^28 + 4 * q^29 + (-4*z + 4) * q^30 - 9*z * q^31 + 8*z * q^32 + (-2*z + 2) * q^33 + (6*z - 4) * q^35 + 2 * q^36 + (3*z - 3) * q^37 - 2*z * q^38 - z * q^39 - 10 * q^41 + (2*z - 6) * q^42 + 5 * q^43 + (-4*z + 4) * q^44 + 2*z * q^45 + (-6*z + 6) * q^47 - 4 * q^48 + (-5*z + 8) * q^49 - 2 * q^50 - 2*z * q^52 - 12*z * q^53 + (2*z - 2) * q^54 + 4 * q^55 + q^57 + (8*z - 8) * q^58 + 12*z * q^59 + 4*z * q^60 + (10*z - 10) * q^61 + 18 * q^62 + (-3*z + 2) * q^63 - 8 * q^64 + (-2*z + 2) * q^65 + 4*z * q^66 + 5*z * q^67 + (-8*z - 4) * q^70 - 6 * q^71 + 3*z * q^73 - 6*z * q^74 + (-z + 1) * q^75 + 2 * q^76 + (-4*z - 2) * q^77 + 2 * q^78 + (-z + 1) * q^79 - 8*z * q^80 - z * q^81 + (-20*z + 20) * q^82 + 6 * q^83 + (-6*z + 4) * q^84 + (10*z - 10) * q^86 - 4*z * q^87 + (16*z - 16) * q^89 - 4 * q^90 + (z - 3) * q^91 + (9*z - 9) * q^93 + 12*z * q^94 + 2*z * q^95 + (-8*z + 8) * q^96 - 6 * q^97 + (16*z - 6) * q^98 - 2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - q^3 - 2 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - q^9 $$2 q - 2 q^{2} - q^{3} - 2 q^{4} + 2 q^{5} + 4 q^{6} - 5 q^{7} - q^{9} + 4 q^{10} + 2 q^{11} - 2 q^{12} + 2 q^{13} + 2 q^{14} - 4 q^{15} + 4 q^{16} - 2 q^{18} - q^{19} - 8 q^{20} + 4 q^{21} - 8 q^{22} + q^{25} - 2 q^{26} + 2 q^{27} + 8 q^{28} + 8 q^{29} + 4 q^{30} - 9 q^{31} + 8 q^{32} + 2 q^{33} - 2 q^{35} + 4 q^{36} - 3 q^{37} - 2 q^{38} - q^{39} - 20 q^{41} - 10 q^{42} + 10 q^{43} + 4 q^{44} + 2 q^{45} + 6 q^{47} - 8 q^{48} + 11 q^{49} - 4 q^{50} - 2 q^{52} - 12 q^{53} - 2 q^{54} + 8 q^{55} + 2 q^{57} - 8 q^{58} + 12 q^{59} + 4 q^{60} - 10 q^{61} + 36 q^{62} + q^{63} - 16 q^{64} + 2 q^{65} + 4 q^{66} + 5 q^{67} - 16 q^{70} - 12 q^{71} + 3 q^{73} - 6 q^{74} + q^{75} + 4 q^{76} - 8 q^{77} + 4 q^{78} + q^{79} - 8 q^{80} - q^{81} + 20 q^{82} + 12 q^{83} + 2 q^{84} - 10 q^{86} - 4 q^{87} - 16 q^{89} - 8 q^{90} - 5 q^{91} - 9 q^{93} + 12 q^{94} + 2 q^{95} + 8 q^{96} - 12 q^{97} + 4 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - q^3 - 2 * q^4 + 2 * q^5 + 4 * q^6 - 5 * q^7 - q^9 + 4 * q^10 + 2 * q^11 - 2 * q^12 + 2 * q^13 + 2 * q^14 - 4 * q^15 + 4 * q^16 - 2 * q^18 - q^19 - 8 * q^20 + 4 * q^21 - 8 * q^22 + q^25 - 2 * q^26 + 2 * q^27 + 8 * q^28 + 8 * q^29 + 4 * q^30 - 9 * q^31 + 8 * q^32 + 2 * q^33 - 2 * q^35 + 4 * q^36 - 3 * q^37 - 2 * q^38 - q^39 - 20 * q^41 - 10 * q^42 + 10 * q^43 + 4 * q^44 + 2 * q^45 + 6 * q^47 - 8 * q^48 + 11 * q^49 - 4 * q^50 - 2 * q^52 - 12 * q^53 - 2 * q^54 + 8 * q^55 + 2 * q^57 - 8 * q^58 + 12 * q^59 + 4 * q^60 - 10 * q^61 + 36 * q^62 + q^63 - 16 * q^64 + 2 * q^65 + 4 * q^66 + 5 * q^67 - 16 * q^70 - 12 * q^71 + 3 * q^73 - 6 * q^74 + q^75 + 4 * q^76 - 8 * q^77 + 4 * q^78 + q^79 - 8 * q^80 - q^81 + 20 * q^82 + 12 * q^83 + 2 * q^84 - 10 * q^86 - 4 * q^87 - 16 * q^89 - 8 * q^90 - 5 * q^91 - 9 * q^93 + 12 * q^94 + 2 * q^95 + 8 * q^96 - 12 * q^97 + 4 * q^98 - 4 * q^99

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.5 − 0.866025i 0.5 + 0.866025i
−1.00000 1.73205i −0.500000 + 0.866025i −1.00000 + 1.73205i 1.00000 + 1.73205i 2.00000 −2.50000 0.866025i 0 −0.500000 0.866025i 2.00000 3.46410i
16.1 −1.00000 + 1.73205i −0.500000 0.866025i −1.00000 1.73205i 1.00000 1.73205i 2.00000 −2.50000 + 0.866025i 0 −0.500000 + 0.866025i 2.00000 + 3.46410i
 $$n$$: 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.2.e.a 2
3.b odd 2 1 63.2.e.b 2
4.b odd 2 1 336.2.q.f 2
5.b even 2 1 525.2.i.e 2
5.c odd 4 2 525.2.r.e 4
7.b odd 2 1 147.2.e.a 2
7.c even 3 1 inner 21.2.e.a 2
7.c even 3 1 147.2.a.c 1
7.d odd 6 1 147.2.a.b 1
7.d odd 6 1 147.2.e.a 2
8.b even 2 1 1344.2.q.m 2
8.d odd 2 1 1344.2.q.c 2
9.c even 3 1 567.2.g.a 2
9.c even 3 1 567.2.h.f 2
9.d odd 6 1 567.2.g.f 2
9.d odd 6 1 567.2.h.a 2
12.b even 2 1 1008.2.s.d 2
21.c even 2 1 441.2.e.e 2
21.g even 6 1 441.2.a.a 1
21.g even 6 1 441.2.e.e 2
21.h odd 6 1 63.2.e.b 2
21.h odd 6 1 441.2.a.b 1
28.d even 2 1 2352.2.q.c 2
28.f even 6 1 2352.2.a.w 1
28.f even 6 1 2352.2.q.c 2
28.g odd 6 1 336.2.q.f 2
28.g odd 6 1 2352.2.a.d 1
35.i odd 6 1 3675.2.a.c 1
35.j even 6 1 525.2.i.e 2
35.j even 6 1 3675.2.a.a 1
35.l odd 12 2 525.2.r.e 4
56.j odd 6 1 9408.2.a.bz 1
56.k odd 6 1 1344.2.q.c 2
56.k odd 6 1 9408.2.a.cv 1
56.m even 6 1 9408.2.a.k 1
56.p even 6 1 1344.2.q.m 2
56.p even 6 1 9408.2.a.bg 1
63.g even 3 1 567.2.h.f 2
63.h even 3 1 567.2.g.a 2
63.j odd 6 1 567.2.g.f 2
63.n odd 6 1 567.2.h.a 2
84.j odd 6 1 7056.2.a.m 1
84.n even 6 1 1008.2.s.d 2
84.n even 6 1 7056.2.a.bp 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 1.a even 1 1 trivial
21.2.e.a 2 7.c even 3 1 inner
63.2.e.b 2 3.b odd 2 1
63.2.e.b 2 21.h odd 6 1
147.2.a.b 1 7.d odd 6 1
147.2.a.c 1 7.c even 3 1
147.2.e.a 2 7.b odd 2 1
147.2.e.a 2 7.d odd 6 1
336.2.q.f 2 4.b odd 2 1
336.2.q.f 2 28.g odd 6 1
441.2.a.a 1 21.g even 6 1
441.2.a.b 1 21.h odd 6 1
441.2.e.e 2 21.c even 2 1
441.2.e.e 2 21.g even 6 1
525.2.i.e 2 5.b even 2 1
525.2.i.e 2 35.j even 6 1
525.2.r.e 4 5.c odd 4 2
525.2.r.e 4 35.l odd 12 2
567.2.g.a 2 9.c even 3 1
567.2.g.a 2 63.h even 3 1
567.2.g.f 2 9.d odd 6 1
567.2.g.f 2 63.j odd 6 1
567.2.h.a 2 9.d odd 6 1
567.2.h.a 2 63.n odd 6 1
567.2.h.f 2 9.c even 3 1
567.2.h.f 2 63.g even 3 1
1008.2.s.d 2 12.b even 2 1
1008.2.s.d 2 84.n even 6 1
1344.2.q.c 2 8.d odd 2 1
1344.2.q.c 2 56.k odd 6 1
1344.2.q.m 2 8.b even 2 1
1344.2.q.m 2 56.p even 6 1
2352.2.a.d 1 28.g odd 6 1
2352.2.a.w 1 28.f even 6 1
2352.2.q.c 2 28.d even 2 1
2352.2.q.c 2 28.f even 6 1
3675.2.a.a 1 35.j even 6 1
3675.2.a.c 1 35.i odd 6 1
7056.2.a.m 1 84.j odd 6 1
7056.2.a.bp 1 84.n even 6 1
9408.2.a.k 1 56.m even 6 1
9408.2.a.bg 1 56.p even 6 1
9408.2.a.bz 1 56.j odd 6 1
9408.2.a.cv 1 56.k odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 4$$
$3$ $$T^{2} + T + 1$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} - 2T + 4$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2}$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2}$$
$29$ $$(T - 4)^{2}$$
$31$ $$T^{2} + 9T + 81$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T + 10)^{2}$$
$43$ $$(T - 5)^{2}$$
$47$ $$T^{2} - 6T + 36$$
$53$ $$T^{2} + 12T + 144$$
$59$ $$T^{2} - 12T + 144$$
$61$ $$T^{2} + 10T + 100$$
$67$ $$T^{2} - 5T + 25$$
$71$ $$(T + 6)^{2}$$
$73$ $$T^{2} - 3T + 9$$
$79$ $$T^{2} - T + 1$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 16T + 256$$
$97$ $$(T + 6)^{2}$$