# Properties

 Label 21.6.g.a Level $21$ Weight $6$ Character orbit 21.g Analytic conductor $3.368$ 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] = [21,6,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, 6, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$21 = 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$3.36806021607$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (9 \zeta_{6} + 9) q^{3} + (32 \zeta_{6} - 32) q^{4} + (87 \zeta_{6} + 62) q^{7} + 243 \zeta_{6} q^{9}+O(q^{10})$$ q + (9*z + 9) * q^3 + (32*z - 32) * q^4 + (87*z + 62) * q^7 + 243*z * q^9 $$q + (9 \zeta_{6} + 9) q^{3} + (32 \zeta_{6} - 32) q^{4} + (87 \zeta_{6} + 62) q^{7} + 243 \zeta_{6} q^{9} + (288 \zeta_{6} - 576) q^{12} + ( - 1318 \zeta_{6} + 659) q^{13} - 1024 \zeta_{6} q^{16} + (93 \zeta_{6} - 186) q^{19} + (2124 \zeta_{6} - 225) q^{21} + ( - 3125 \zeta_{6} + 3125) q^{25} + (4374 \zeta_{6} - 2187) q^{27} + (1984 \zeta_{6} - 4768) q^{28} + (5975 \zeta_{6} + 5975) q^{31} - 7776 q^{36} - 6661 \zeta_{6} q^{37} + ( - 17793 \zeta_{6} + 17793) q^{39} - 22475 q^{43} + ( - 18432 \zeta_{6} + 9216) q^{48} + (18357 \zeta_{6} - 3725) q^{49} + (21088 \zeta_{6} + 21088) q^{52} - 2511 q^{57} + (25076 \zeta_{6} - 50152) q^{61} + (36207 \zeta_{6} - 21141) q^{63} + 32768 q^{64} + ( - 37939 \zeta_{6} + 37939) q^{67} + ( - 27009 \zeta_{6} - 27009) q^{73} + ( - 28125 \zeta_{6} + 56250) q^{75} + ( - 5952 \zeta_{6} + 2976) q^{76} - 90857 \zeta_{6} q^{79} + (59049 \zeta_{6} - 59049) q^{81} + ( - 7200 \zeta_{6} - 60768) q^{84} + ( - 139049 \zeta_{6} + 155524) q^{91} + 161325 \zeta_{6} q^{93} + (147376 \zeta_{6} - 73688) q^{97} +O(q^{100})$$ q + (9*z + 9) * q^3 + (32*z - 32) * q^4 + (87*z + 62) * q^7 + 243*z * q^9 + (288*z - 576) * q^12 + (-1318*z + 659) * q^13 - 1024*z * q^16 + (93*z - 186) * q^19 + (2124*z - 225) * q^21 + (-3125*z + 3125) * q^25 + (4374*z - 2187) * q^27 + (1984*z - 4768) * q^28 + (5975*z + 5975) * q^31 - 7776 * q^36 - 6661*z * q^37 + (-17793*z + 17793) * q^39 - 22475 * q^43 + (-18432*z + 9216) * q^48 + (18357*z - 3725) * q^49 + (21088*z + 21088) * q^52 - 2511 * q^57 + (25076*z - 50152) * q^61 + (36207*z - 21141) * q^63 + 32768 * q^64 + (-37939*z + 37939) * q^67 + (-27009*z - 27009) * q^73 + (-28125*z + 56250) * q^75 + (-5952*z + 2976) * q^76 - 90857*z * q^79 + (59049*z - 59049) * q^81 + (-7200*z - 60768) * q^84 + (-139049*z + 155524) * q^91 + 161325*z * q^93 + (147376*z - 73688) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9}+O(q^{10})$$ 2 * q + 27 * q^3 - 32 * q^4 + 211 * q^7 + 243 * q^9 $$2 q + 27 q^{3} - 32 q^{4} + 211 q^{7} + 243 q^{9} - 864 q^{12} - 1024 q^{16} - 279 q^{19} + 1674 q^{21} + 3125 q^{25} - 7552 q^{28} + 17925 q^{31} - 15552 q^{36} - 6661 q^{37} + 17793 q^{39} - 44950 q^{43} + 10907 q^{49} + 63264 q^{52} - 5022 q^{57} - 75228 q^{61} - 6075 q^{63} + 65536 q^{64} + 37939 q^{67} - 81027 q^{73} + 84375 q^{75} - 90857 q^{79} - 59049 q^{81} - 128736 q^{84} + 171999 q^{91} + 161325 q^{93}+O(q^{100})$$ 2 * q + 27 * q^3 - 32 * q^4 + 211 * q^7 + 243 * q^9 - 864 * q^12 - 1024 * q^16 - 279 * q^19 + 1674 * q^21 + 3125 * q^25 - 7552 * q^28 + 17925 * q^31 - 15552 * q^36 - 6661 * q^37 + 17793 * q^39 - 44950 * q^43 + 10907 * q^49 + 63264 * q^52 - 5022 * q^57 - 75228 * q^61 - 6075 * q^63 + 65536 * q^64 + 37939 * q^67 - 81027 * q^73 + 84375 * q^75 - 90857 * q^79 - 59049 * q^81 - 128736 * q^84 + 171999 * q^91 + 161325 * 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 13.5000 7.79423i −16.0000 27.7128i 0 0 105.500 75.3442i 0 121.500 210.444i 0
17.1 0 13.5000 + 7.79423i −16.0000 + 27.7128i 0 0 105.500 + 75.3442i 0 121.500 + 210.444i 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 21.6.g.a 2
3.b odd 2 1 CM 21.6.g.a 2
7.c even 3 1 147.6.c.a 2
7.d odd 6 1 inner 21.6.g.a 2
7.d odd 6 1 147.6.c.a 2
21.g even 6 1 inner 21.6.g.a 2
21.g even 6 1 147.6.c.a 2
21.h odd 6 1 147.6.c.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.6.g.a 2 1.a even 1 1 trivial
21.6.g.a 2 3.b odd 2 1 CM
21.6.g.a 2 7.d odd 6 1 inner
21.6.g.a 2 21.g even 6 1 inner
147.6.c.a 2 7.c even 3 1
147.6.c.a 2 7.d odd 6 1
147.6.c.a 2 21.g even 6 1
147.6.c.a 2 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}$$ acting on $$S_{6}^{\mathrm{new}}(21, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 27T + 243$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 211T + 16807$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 1302843$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 279T + 25947$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$T^{2} - 17925 T + 107101875$$
$37$ $$T^{2} + 6661 T + 44368921$$
$41$ $$T^{2}$$
$43$ $$(T + 22475)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 75228 T + 1886417328$$
$67$ $$T^{2} - 37939 T + 1439367721$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 81027 T + 2188458243$$
$79$ $$T^{2} + 90857 T + 8254994449$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 16289764032$$