# Properties

 Label 441.2.g.a Level $441$ Weight $2$ Character orbit 441.g Analytic conductor $3.521$ 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] = [441,2,Mod(67,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 4]))

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

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

chi := DirichletCharacter("441.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.g (of order $$3$$, degree $$2$$, not 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: $$3.52140272914$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + q^{5} + (2 \zeta_{6} - 1) q^{6} - 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9}+O(q^{10})$$ q + (z - 1) * q^2 + (-z + 2) * q^3 + z * q^4 + q^5 + (2*z - 1) * q^6 - 3 * q^8 + (-3*z + 3) * q^9 $$q + (\zeta_{6} - 1) q^{2} + ( - \zeta_{6} + 2) q^{3} + \zeta_{6} q^{4} + q^{5} + (2 \zeta_{6} - 1) q^{6} - 3 q^{8} + ( - 3 \zeta_{6} + 3) q^{9} + (\zeta_{6} - 1) q^{10} + 5 q^{11} + (\zeta_{6} + 1) q^{12} + (5 \zeta_{6} - 5) q^{13} + ( - \zeta_{6} + 2) q^{15} + ( - \zeta_{6} + 1) q^{16} + ( - 3 \zeta_{6} + 3) q^{17} + 3 \zeta_{6} q^{18} + \zeta_{6} q^{19} + \zeta_{6} q^{20} + (5 \zeta_{6} - 5) q^{22} + 3 q^{23} + (3 \zeta_{6} - 6) q^{24} - 4 q^{25} - 5 \zeta_{6} q^{26} + ( - 6 \zeta_{6} + 3) q^{27} + \zeta_{6} q^{29} + (2 \zeta_{6} - 1) q^{30} - 5 \zeta_{6} q^{32} + ( - 5 \zeta_{6} + 10) q^{33} + 3 \zeta_{6} q^{34} + 3 q^{36} - 3 \zeta_{6} q^{37} - q^{38} + (10 \zeta_{6} - 5) q^{39} - 3 q^{40} + (5 \zeta_{6} - 5) q^{41} + \zeta_{6} q^{43} + 5 \zeta_{6} q^{44} + ( - 3 \zeta_{6} + 3) q^{45} + (3 \zeta_{6} - 3) q^{46} + ( - 2 \zeta_{6} + 1) q^{48} + ( - 4 \zeta_{6} + 4) q^{50} + ( - 6 \zeta_{6} + 3) q^{51} - 5 q^{52} + ( - 9 \zeta_{6} + 9) q^{53} + (3 \zeta_{6} + 3) q^{54} + 5 q^{55} + (\zeta_{6} + 1) q^{57} - q^{58} + (\zeta_{6} + 1) q^{60} + (14 \zeta_{6} - 14) q^{61} + 7 q^{64} + (5 \zeta_{6} - 5) q^{65} + (10 \zeta_{6} - 5) q^{66} - 4 \zeta_{6} q^{67} + 3 q^{68} + ( - 3 \zeta_{6} + 6) q^{69} - 12 q^{71} + (9 \zeta_{6} - 9) q^{72} + ( - 3 \zeta_{6} + 3) q^{73} + 3 q^{74} + (4 \zeta_{6} - 8) q^{75} + (\zeta_{6} - 1) q^{76} + ( - 5 \zeta_{6} - 5) q^{78} + (8 \zeta_{6} - 8) q^{79} + ( - \zeta_{6} + 1) q^{80} - 9 \zeta_{6} q^{81} - 5 \zeta_{6} q^{82} - 9 \zeta_{6} q^{83} + ( - 3 \zeta_{6} + 3) q^{85} - q^{86} + (\zeta_{6} + 1) q^{87} - 15 q^{88} - 13 \zeta_{6} q^{89} + 3 \zeta_{6} q^{90} + 3 \zeta_{6} q^{92} + \zeta_{6} q^{95} + ( - 5 \zeta_{6} - 5) q^{96} - 9 \zeta_{6} q^{97} + ( - 15 \zeta_{6} + 15) q^{99} +O(q^{100})$$ q + (z - 1) * q^2 + (-z + 2) * q^3 + z * q^4 + q^5 + (2*z - 1) * q^6 - 3 * q^8 + (-3*z + 3) * q^9 + (z - 1) * q^10 + 5 * q^11 + (z + 1) * q^12 + (5*z - 5) * q^13 + (-z + 2) * q^15 + (-z + 1) * q^16 + (-3*z + 3) * q^17 + 3*z * q^18 + z * q^19 + z * q^20 + (5*z - 5) * q^22 + 3 * q^23 + (3*z - 6) * q^24 - 4 * q^25 - 5*z * q^26 + (-6*z + 3) * q^27 + z * q^29 + (2*z - 1) * q^30 - 5*z * q^32 + (-5*z + 10) * q^33 + 3*z * q^34 + 3 * q^36 - 3*z * q^37 - q^38 + (10*z - 5) * q^39 - 3 * q^40 + (5*z - 5) * q^41 + z * q^43 + 5*z * q^44 + (-3*z + 3) * q^45 + (3*z - 3) * q^46 + (-2*z + 1) * q^48 + (-4*z + 4) * q^50 + (-6*z + 3) * q^51 - 5 * q^52 + (-9*z + 9) * q^53 + (3*z + 3) * q^54 + 5 * q^55 + (z + 1) * q^57 - q^58 + (z + 1) * q^60 + (14*z - 14) * q^61 + 7 * q^64 + (5*z - 5) * q^65 + (10*z - 5) * q^66 - 4*z * q^67 + 3 * q^68 + (-3*z + 6) * q^69 - 12 * q^71 + (9*z - 9) * q^72 + (-3*z + 3) * q^73 + 3 * q^74 + (4*z - 8) * q^75 + (z - 1) * q^76 + (-5*z - 5) * q^78 + (8*z - 8) * q^79 + (-z + 1) * q^80 - 9*z * q^81 - 5*z * q^82 - 9*z * q^83 + (-3*z + 3) * q^85 - q^86 + (z + 1) * q^87 - 15 * q^88 - 13*z * q^89 + 3*z * q^90 + 3*z * q^92 + z * q^95 + (-5*z - 5) * q^96 - 9*z * q^97 + (-15*z + 15) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{3} + q^{4} + 2 q^{5} - 6 q^{8} + 3 q^{9}+O(q^{10})$$ 2 * q - q^2 + 3 * q^3 + q^4 + 2 * q^5 - 6 * q^8 + 3 * q^9 $$2 q - q^{2} + 3 q^{3} + q^{4} + 2 q^{5} - 6 q^{8} + 3 q^{9} - q^{10} + 10 q^{11} + 3 q^{12} - 5 q^{13} + 3 q^{15} + q^{16} + 3 q^{17} + 3 q^{18} + q^{19} + q^{20} - 5 q^{22} + 6 q^{23} - 9 q^{24} - 8 q^{25} - 5 q^{26} + q^{29} - 5 q^{32} + 15 q^{33} + 3 q^{34} + 6 q^{36} - 3 q^{37} - 2 q^{38} - 6 q^{40} - 5 q^{41} + q^{43} + 5 q^{44} + 3 q^{45} - 3 q^{46} + 4 q^{50} - 10 q^{52} + 9 q^{53} + 9 q^{54} + 10 q^{55} + 3 q^{57} - 2 q^{58} + 3 q^{60} - 14 q^{61} + 14 q^{64} - 5 q^{65} - 4 q^{67} + 6 q^{68} + 9 q^{69} - 24 q^{71} - 9 q^{72} + 3 q^{73} + 6 q^{74} - 12 q^{75} - q^{76} - 15 q^{78} - 8 q^{79} + q^{80} - 9 q^{81} - 5 q^{82} - 9 q^{83} + 3 q^{85} - 2 q^{86} + 3 q^{87} - 30 q^{88} - 13 q^{89} + 3 q^{90} + 3 q^{92} + q^{95} - 15 q^{96} - 9 q^{97} + 15 q^{99}+O(q^{100})$$ 2 * q - q^2 + 3 * q^3 + q^4 + 2 * q^5 - 6 * q^8 + 3 * q^9 - q^10 + 10 * q^11 + 3 * q^12 - 5 * q^13 + 3 * q^15 + q^16 + 3 * q^17 + 3 * q^18 + q^19 + q^20 - 5 * q^22 + 6 * q^23 - 9 * q^24 - 8 * q^25 - 5 * q^26 + q^29 - 5 * q^32 + 15 * q^33 + 3 * q^34 + 6 * q^36 - 3 * q^37 - 2 * q^38 - 6 * q^40 - 5 * q^41 + q^43 + 5 * q^44 + 3 * q^45 - 3 * q^46 + 4 * q^50 - 10 * q^52 + 9 * q^53 + 9 * q^54 + 10 * q^55 + 3 * q^57 - 2 * q^58 + 3 * q^60 - 14 * q^61 + 14 * q^64 - 5 * q^65 - 4 * q^67 + 6 * q^68 + 9 * q^69 - 24 * q^71 - 9 * q^72 + 3 * q^73 + 6 * q^74 - 12 * q^75 - q^76 - 15 * q^78 - 8 * q^79 + q^80 - 9 * q^81 - 5 * q^82 - 9 * q^83 + 3 * q^85 - 2 * q^86 + 3 * q^87 - 30 * q^88 - 13 * q^89 + 3 * q^90 + 3 * q^92 + q^95 - 15 * q^96 - 9 * q^97 + 15 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/441\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$344$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$-1 + \zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
67.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i 1.50000 0.866025i 0.500000 + 0.866025i 1.00000 1.73205i 0 −3.00000 1.50000 2.59808i −0.500000 + 0.866025i
79.1 −0.500000 0.866025i 1.50000 + 0.866025i 0.500000 0.866025i 1.00000 1.73205i 0 −3.00000 1.50000 + 2.59808i −0.500000 0.866025i
 $$n$$: 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
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 441.2.g.a 2
3.b odd 2 1 1323.2.g.a 2
7.b odd 2 1 63.2.g.a 2
7.c even 3 1 441.2.f.a 2
7.c even 3 1 441.2.h.a 2
7.d odd 6 1 63.2.h.a yes 2
7.d odd 6 1 441.2.f.b 2
9.c even 3 1 441.2.h.a 2
9.d odd 6 1 1323.2.h.a 2
21.c even 2 1 189.2.g.a 2
21.g even 6 1 189.2.h.a 2
21.g even 6 1 1323.2.f.a 2
21.h odd 6 1 1323.2.f.b 2
21.h odd 6 1 1323.2.h.a 2
28.d even 2 1 1008.2.t.d 2
28.f even 6 1 1008.2.q.c 2
63.g even 3 1 inner 441.2.g.a 2
63.g even 3 1 3969.2.a.f 1
63.h even 3 1 441.2.f.a 2
63.i even 6 1 567.2.e.b 2
63.i even 6 1 1323.2.f.a 2
63.j odd 6 1 1323.2.f.b 2
63.k odd 6 1 63.2.g.a 2
63.k odd 6 1 3969.2.a.d 1
63.l odd 6 1 63.2.h.a yes 2
63.l odd 6 1 567.2.e.a 2
63.n odd 6 1 1323.2.g.a 2
63.n odd 6 1 3969.2.a.a 1
63.o even 6 1 189.2.h.a 2
63.o even 6 1 567.2.e.b 2
63.s even 6 1 189.2.g.a 2
63.s even 6 1 3969.2.a.c 1
63.t odd 6 1 441.2.f.b 2
63.t odd 6 1 567.2.e.a 2
84.h odd 2 1 3024.2.t.d 2
84.j odd 6 1 3024.2.q.b 2
252.n even 6 1 1008.2.t.d 2
252.s odd 6 1 3024.2.q.b 2
252.bi even 6 1 1008.2.q.c 2
252.bn odd 6 1 3024.2.t.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 7.b odd 2 1
63.2.g.a 2 63.k odd 6 1
63.2.h.a yes 2 7.d odd 6 1
63.2.h.a yes 2 63.l odd 6 1
189.2.g.a 2 21.c even 2 1
189.2.g.a 2 63.s even 6 1
189.2.h.a 2 21.g even 6 1
189.2.h.a 2 63.o even 6 1
441.2.f.a 2 7.c even 3 1
441.2.f.a 2 63.h even 3 1
441.2.f.b 2 7.d odd 6 1
441.2.f.b 2 63.t odd 6 1
441.2.g.a 2 1.a even 1 1 trivial
441.2.g.a 2 63.g even 3 1 inner
441.2.h.a 2 7.c even 3 1
441.2.h.a 2 9.c even 3 1
567.2.e.a 2 63.l odd 6 1
567.2.e.a 2 63.t odd 6 1
567.2.e.b 2 63.i even 6 1
567.2.e.b 2 63.o even 6 1
1008.2.q.c 2 28.f even 6 1
1008.2.q.c 2 252.bi even 6 1
1008.2.t.d 2 28.d even 2 1
1008.2.t.d 2 252.n even 6 1
1323.2.f.a 2 21.g even 6 1
1323.2.f.a 2 63.i even 6 1
1323.2.f.b 2 21.h odd 6 1
1323.2.f.b 2 63.j odd 6 1
1323.2.g.a 2 3.b odd 2 1
1323.2.g.a 2 63.n odd 6 1
1323.2.h.a 2 9.d odd 6 1
1323.2.h.a 2 21.h odd 6 1
3024.2.q.b 2 84.j odd 6 1
3024.2.q.b 2 252.s odd 6 1
3024.2.t.d 2 84.h odd 2 1
3024.2.t.d 2 252.bn odd 6 1
3969.2.a.a 1 63.n odd 6 1
3969.2.a.c 1 63.s even 6 1
3969.2.a.d 1 63.k odd 6 1
3969.2.a.f 1 63.g even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(441, [\chi])$$:

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{5} - 1$$ T5 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2} - 3T + 3$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2}$$
$11$ $$(T - 5)^{2}$$
$13$ $$T^{2} + 5T + 25$$
$17$ $$T^{2} - 3T + 9$$
$19$ $$T^{2} - T + 1$$
$23$ $$(T - 3)^{2}$$
$29$ $$T^{2} - T + 1$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$T^{2} + 5T + 25$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 14T + 196$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} - 3T + 9$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$T^{2} + 9T + 81$$
$89$ $$T^{2} + 13T + 169$$
$97$ $$T^{2} + 9T + 81$$