# Properties

 Label 21.2.a.a Level $21$ Weight $2$ Character orbit 21.a Self dual yes Analytic conductor $0.168$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,2,Mod(1,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("21.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 21.a (trivial)

## 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: yes Analytic conductor: $$0.167685844245$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} - q^{7} + 3 q^{8} + q^{9}+O(q^{10})$$ q - q^2 + q^3 - q^4 - 2 * q^5 - q^6 - q^7 + 3 * q^8 + q^9 $$q - q^{2} + q^{3} - q^{4} - 2 q^{5} - q^{6} - q^{7} + 3 q^{8} + q^{9} + 2 q^{10} + 4 q^{11} - q^{12} - 2 q^{13} + q^{14} - 2 q^{15} - q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + 2 q^{20} - q^{21} - 4 q^{22} + 3 q^{24} - q^{25} + 2 q^{26} + q^{27} + q^{28} - 2 q^{29} + 2 q^{30} - 5 q^{32} + 4 q^{33} + 6 q^{34} + 2 q^{35} - q^{36} + 6 q^{37} - 4 q^{38} - 2 q^{39} - 6 q^{40} + 2 q^{41} + q^{42} - 4 q^{43} - 4 q^{44} - 2 q^{45} - q^{48} + q^{49} + q^{50} - 6 q^{51} + 2 q^{52} + 6 q^{53} - q^{54} - 8 q^{55} - 3 q^{56} + 4 q^{57} + 2 q^{58} + 12 q^{59} + 2 q^{60} - 2 q^{61} - q^{63} + 7 q^{64} + 4 q^{65} - 4 q^{66} + 4 q^{67} + 6 q^{68} - 2 q^{70} + 3 q^{72} - 6 q^{73} - 6 q^{74} - q^{75} - 4 q^{76} - 4 q^{77} + 2 q^{78} - 16 q^{79} + 2 q^{80} + q^{81} - 2 q^{82} - 12 q^{83} + q^{84} + 12 q^{85} + 4 q^{86} - 2 q^{87} + 12 q^{88} - 14 q^{89} + 2 q^{90} + 2 q^{91} - 8 q^{95} - 5 q^{96} + 18 q^{97} - q^{98} + 4 q^{99}+O(q^{100})$$ q - q^2 + q^3 - q^4 - 2 * q^5 - q^6 - q^7 + 3 * q^8 + q^9 + 2 * q^10 + 4 * q^11 - q^12 - 2 * q^13 + q^14 - 2 * q^15 - q^16 - 6 * q^17 - q^18 + 4 * q^19 + 2 * q^20 - q^21 - 4 * q^22 + 3 * q^24 - q^25 + 2 * q^26 + q^27 + q^28 - 2 * q^29 + 2 * q^30 - 5 * q^32 + 4 * q^33 + 6 * q^34 + 2 * q^35 - q^36 + 6 * q^37 - 4 * q^38 - 2 * q^39 - 6 * q^40 + 2 * q^41 + q^42 - 4 * q^43 - 4 * q^44 - 2 * q^45 - q^48 + q^49 + q^50 - 6 * q^51 + 2 * q^52 + 6 * q^53 - q^54 - 8 * q^55 - 3 * q^56 + 4 * q^57 + 2 * q^58 + 12 * q^59 + 2 * q^60 - 2 * q^61 - q^63 + 7 * q^64 + 4 * q^65 - 4 * q^66 + 4 * q^67 + 6 * q^68 - 2 * q^70 + 3 * q^72 - 6 * q^73 - 6 * q^74 - q^75 - 4 * q^76 - 4 * q^77 + 2 * q^78 - 16 * q^79 + 2 * q^80 + q^81 - 2 * q^82 - 12 * q^83 + q^84 + 12 * q^85 + 4 * q^86 - 2 * q^87 + 12 * q^88 - 14 * q^89 + 2 * q^90 + 2 * q^91 - 8 * q^95 - 5 * q^96 + 18 * q^97 - q^98 + 4 * q^99

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 1.00000 −1.00000 −2.00000 −1.00000 −1.00000 3.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 21.2.a.a 1
3.b odd 2 1 63.2.a.a 1
4.b odd 2 1 336.2.a.a 1
5.b even 2 1 525.2.a.d 1
5.c odd 4 2 525.2.d.a 2
7.b odd 2 1 147.2.a.a 1
7.c even 3 2 147.2.e.b 2
7.d odd 6 2 147.2.e.c 2
8.b even 2 1 1344.2.a.g 1
8.d odd 2 1 1344.2.a.s 1
9.c even 3 2 567.2.f.g 2
9.d odd 6 2 567.2.f.b 2
11.b odd 2 1 2541.2.a.j 1
12.b even 2 1 1008.2.a.l 1
13.b even 2 1 3549.2.a.c 1
15.d odd 2 1 1575.2.a.c 1
15.e even 4 2 1575.2.d.a 2
16.e even 4 2 5376.2.c.r 2
16.f odd 4 2 5376.2.c.l 2
17.b even 2 1 6069.2.a.b 1
19.b odd 2 1 7581.2.a.d 1
20.d odd 2 1 8400.2.a.bn 1
21.c even 2 1 441.2.a.f 1
21.g even 6 2 441.2.e.b 2
21.h odd 6 2 441.2.e.a 2
24.f even 2 1 4032.2.a.k 1
24.h odd 2 1 4032.2.a.h 1
28.d even 2 1 2352.2.a.v 1
28.f even 6 2 2352.2.q.e 2
28.g odd 6 2 2352.2.q.x 2
33.d even 2 1 7623.2.a.g 1
35.c odd 2 1 3675.2.a.n 1
56.e even 2 1 9408.2.a.m 1
56.h odd 2 1 9408.2.a.bv 1
84.h odd 2 1 7056.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 1.a even 1 1 trivial
63.2.a.a 1 3.b odd 2 1
147.2.a.a 1 7.b odd 2 1
147.2.e.b 2 7.c even 3 2
147.2.e.c 2 7.d odd 6 2
336.2.a.a 1 4.b odd 2 1
441.2.a.f 1 21.c even 2 1
441.2.e.a 2 21.h odd 6 2
441.2.e.b 2 21.g even 6 2
525.2.a.d 1 5.b even 2 1
525.2.d.a 2 5.c odd 4 2
567.2.f.b 2 9.d odd 6 2
567.2.f.g 2 9.c even 3 2
1008.2.a.l 1 12.b even 2 1
1344.2.a.g 1 8.b even 2 1
1344.2.a.s 1 8.d odd 2 1
1575.2.a.c 1 15.d odd 2 1
1575.2.d.a 2 15.e even 4 2
2352.2.a.v 1 28.d even 2 1
2352.2.q.e 2 28.f even 6 2
2352.2.q.x 2 28.g odd 6 2
2541.2.a.j 1 11.b odd 2 1
3549.2.a.c 1 13.b even 2 1
3675.2.a.n 1 35.c odd 2 1
4032.2.a.h 1 24.h odd 2 1
4032.2.a.k 1 24.f even 2 1
5376.2.c.l 2 16.f odd 4 2
5376.2.c.r 2 16.e even 4 2
6069.2.a.b 1 17.b even 2 1
7056.2.a.p 1 84.h odd 2 1
7581.2.a.d 1 19.b odd 2 1
7623.2.a.g 1 33.d even 2 1
8400.2.a.bn 1 20.d odd 2 1
9408.2.a.m 1 56.e even 2 1
9408.2.a.bv 1 56.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(\Gamma_0(21))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T - 1$$
$5$ $$T + 2$$
$7$ $$T + 1$$
$11$ $$T - 4$$
$13$ $$T + 2$$
$17$ $$T + 6$$
$19$ $$T - 4$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T - 6$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T$$
$53$ $$T - 6$$
$59$ $$T - 12$$
$61$ $$T + 2$$
$67$ $$T - 4$$
$71$ $$T$$
$73$ $$T + 6$$
$79$ $$T + 16$$
$83$ $$T + 12$$
$89$ $$T + 14$$
$97$ $$T - 18$$