# Properties

 Label 441.2.h.d Level $441$ Weight $2$ Character orbit 441.h Analytic conductor $3.521$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,2,Mod(214,441)]

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([4, 4]))

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

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

chi := DirichletCharacter("441.214");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$441 = 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 441.h (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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{18})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{3} + 1$$ x^6 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$3$$ 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 basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{4} - \beta_{3} + 1) q^{2} + (\beta_{5} + \beta_{2}) q^{3} + ( - 2 \beta_{4} - \beta_{3} + 1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - \beta_1) q^{5}+ \cdots + ( - \beta_{5} + 2 \beta_{4} + \cdots - 2 \beta_{2}) q^{9}+O(q^{10})$$ q + (-b4 - b3 + 1) * q^2 + (b5 + b2) * q^3 + (-2*b4 - b3 + 1) * q^4 + (b5 - b4 - b3 + b2 - b1) * q^5 + (2*b5 - b4 - 2*b3 + 2*b2 + b1 - 2) * q^6 + (-2*b4 + b3 + 2) * q^8 + (-b5 + 2*b4 + 3*b3 - 2*b2) * q^9 $$q + ( - \beta_{4} - \beta_{3} + 1) q^{2} + (\beta_{5} + \beta_{2}) q^{3} + ( - 2 \beta_{4} - \beta_{3} + 1) q^{4} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - \beta_1) q^{5}+ \cdots + ( - 3 \beta_{5} - 3 \beta_{4} + 3) q^{99}+O(q^{100})$$ q + (-b4 - b3 + 1) * q^2 + (b5 + b2) * q^3 + (-2*b4 - b3 + 1) * q^4 + (b5 - b4 - b3 + b2 - b1) * q^5 + (2*b5 - b4 - 2*b3 + 2*b2 + b1 - 2) * q^6 + (-2*b4 + b3 + 2) * q^8 + (-b5 + 2*b4 + 3*b3 - 2*b2) * q^9 + (b5 - b4 - b3 + 2*b2) * q^10 + (b5 - b3 + b2 + 2*b1 - 2) * q^11 + (3*b5 - b4 - 3*b3 + 2*b2 + 3*b1 - 3) * q^12 + (-2*b5 + 4*b3 - 4*b2 - b1 + 1) * q^13 + (b3 - 2*b2 + b1 - 2) * q^15 + (-3*b4 + 1) * q^16 + (b2 - 2*b1) * q^17 + (-3*b5 + 3*b4 + 3*b3 - 3*b2 - 3) * q^18 + (2*b5 - b3 + b2 - b1 + 1) * q^19 + (b5 - b4 - b3 + b2 + 2*b1) * q^20 + (4*b5 - 2*b3 + 2*b2 + 3*b1 - 3) * q^22 + (b5 - b4 - b3 - 2*b2 - 4*b1) * q^23 + (4*b5 + b4 - b3 + b2 + 5*b1 - 1) * q^24 + (-b5 + 2*b3 - 2*b2 - 2*b1 + 2) * q^25 + (-7*b5 + 6*b3 - 6*b2 - b1 + 1) * q^26 + (-3*b1 + 6) * q^27 + (5*b5 - 5*b4 - 5*b3 + 4*b2 - 3*b1) * q^29 + (-b5 + 2*b4 + 3*b3 - 2*b2 + 3*b1 - 3) * q^30 + (3*b4 + 6*b3 - 1) * q^31 + (-3*b4 - 3*b3) * q^32 + (-3*b5 + 3*b4 + 2*b3 - b2 - b1 - 1) * q^33 + (-b5 + b4 + b3 + b2 - 3*b1) * q^34 + (-4*b5 + 5*b4 + 3*b3 - 2*b2 - 3*b1 - 3) * q^36 + (-3*b5 - 3*b3 + 3*b2 - b1 + 1) * q^37 + (2*b5 - 3*b3 + 3*b2 + 2*b1 - 2) * q^38 + (5*b5 - 3*b4 - 6*b3 + 2*b2 + 6) * q^39 + (2*b5 - 2*b4 - 2*b3 - 2*b2 + 3*b1) * q^40 + (-b5 - b3 + b2) * q^41 + (-b5 + b4 + b3 + b2 + b1) * q^43 + (7*b5 - 4*b3 + 4*b2 + 5*b1 - 5) * q^44 + (-3*b3 - 3*b1 + 3) * q^45 + (-5*b5 + 5*b4 + 5*b3 - b2) * q^46 + (2*b4 + 5*b3 + 1) * q^47 + (4*b5 - 3*b3 + b2 + 6*b1 - 3) * q^48 + (-5*b5 + 3*b3 - 3*b2 - 2*b1 + 2) * q^50 + (-b5 - b4 + 2*b3 - 3*b2 + 2*b1 - 1) * q^51 + (-10*b5 + 5*b3 - 5*b2 - 7*b1 + 7) * q^52 + (-3*b5 + 3*b4 + 3*b3 - 2*b2 - 2*b1) * q^53 + (-3*b5 - 3*b4 - 3*b3 - 3*b1 + 6) * q^54 + (2*b4 + b3) * q^55 + (b4 + 3*b3 - 2*b2 - 3*b1) * q^57 + (6*b5 - 6*b4 - 6*b3 + 9*b2 + 3*b1) * q^58 + (-5*b4 - 5*b3 - 1) * q^59 + (3*b4 + b3 + b2 + b1 - 2) * q^60 + (3*b4 + 3*b3 + 2) * q^61 + (b4 + b3 - 10) * q^62 + (3*b4 + 4) * q^64 + (-b4 - 6*b3 + 5) * q^65 + (-5*b5 + 7*b4 + 6*b3 - 4*b2 - 6*b1) * q^66 + (3*b3 - 4) * q^67 + (-3*b5 + 3*b4 + 3*b3 - 2*b2 - 2*b1) * q^68 + (3*b5 - 6*b4 - 5*b3 - 2*b2 - 5*b1 + 1) * q^69 + (3*b4 - 3*b3 + 3) * q^71 + (-3*b5 + 9*b4 + 6*b3 - 9*b1 + 3) * q^72 + (4*b5 - 4*b4 - 4*b3 - b2 + 7*b1) * q^73 + (-b5 - 10*b1 + 10) * q^74 + (4*b5 - 3*b4 - 3*b3 + b2 + 3) * q^75 + (3*b5 - 3*b3 + 3*b2 + 5*b1 - 5) * q^76 + (7*b5 - 8*b4 - 13*b3 + 7*b2 + 8*b1 + 5) * q^78 + (3*b3 - 7) * q^79 + (b5 - b4 - b3 - 2*b2 + 5*b1) * q^80 + (6*b5 - 3*b4 + 3*b2) * q^81 + (-3*b1 + 3) * q^82 + (-4*b5 + 4*b4 + 4*b3 + b2 + 6*b1) * q^83 + (-2*b5 + 4*b3 - 4*b2 + 3*b1 - 3) * q^85 + (b5 - b4 - b3 - 2*b1) * q^86 + (b5 + b4 + 3*b3 - 7*b2 + 3*b1 - 9) * q^87 + (8*b5 - 7*b3 + 7*b2 + 9*b1 - 9) * q^88 + (3*b5 - 7*b3 + 7*b2 + 4*b1 - 4) * q^89 + (-3*b5 + 3*b4 - 3*b1 + 6) * q^90 + (-8*b5 + 8*b4 + 8*b3 - 2*b2 - b1) * q^92 + (-4*b5 + 6*b4 + 9*b3 - 7*b2 + 9) * q^93 + (-2*b4 - b3 - 6) * q^94 + (-b4 - 4) * q^95 + (3*b5 - 3*b4 - 6*b3 + 3*b2 + 3*b1 - 6) * q^96 + (-7*b5 + 7*b4 + 7*b3 - 8*b2 + b1) * q^97 + (-3*b5 - 3*b4 + 3) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 6 q^{2} + 6 q^{4} - 3 q^{5} - 9 q^{6} + 12 q^{8}+O(q^{10})$$ 6 * q + 6 * q^2 + 6 * q^4 - 3 * q^5 - 9 * q^6 + 12 * q^8 $$6 q + 6 q^{2} + 6 q^{4} - 3 q^{5} - 9 q^{6} + 12 q^{8} - 6 q^{11} - 9 q^{12} + 3 q^{13} - 9 q^{15} + 6 q^{16} - 6 q^{17} - 18 q^{18} + 3 q^{19} + 6 q^{20} - 9 q^{22} - 12 q^{23} + 9 q^{24} + 6 q^{25} + 3 q^{26} + 27 q^{27} - 9 q^{29} - 9 q^{30} - 6 q^{31} - 9 q^{33} - 9 q^{34} - 27 q^{36} + 3 q^{37} - 6 q^{38} + 36 q^{39} + 9 q^{40} + 3 q^{43} - 15 q^{44} + 9 q^{45} + 6 q^{47} + 6 q^{50} + 21 q^{52} - 6 q^{53} + 27 q^{54} - 9 q^{57} + 9 q^{58} - 6 q^{59} - 9 q^{60} + 12 q^{61} - 60 q^{62} + 24 q^{64} + 30 q^{65} - 18 q^{66} - 24 q^{67} - 6 q^{68} - 9 q^{69} + 18 q^{71} - 9 q^{72} + 21 q^{73} + 30 q^{74} + 18 q^{75} - 15 q^{76} + 54 q^{78} - 42 q^{79} + 15 q^{80} + 9 q^{82} + 18 q^{83} - 9 q^{85} - 6 q^{86} - 45 q^{87} - 27 q^{88} - 12 q^{89} + 27 q^{90} - 3 q^{92} + 54 q^{93} - 36 q^{94} - 24 q^{95} - 27 q^{96} + 3 q^{97} + 18 q^{99}+O(q^{100})$$ 6 * q + 6 * q^2 + 6 * q^4 - 3 * q^5 - 9 * q^6 + 12 * q^8 - 6 * q^11 - 9 * q^12 + 3 * q^13 - 9 * q^15 + 6 * q^16 - 6 * q^17 - 18 * q^18 + 3 * q^19 + 6 * q^20 - 9 * q^22 - 12 * q^23 + 9 * q^24 + 6 * q^25 + 3 * q^26 + 27 * q^27 - 9 * q^29 - 9 * q^30 - 6 * q^31 - 9 * q^33 - 9 * q^34 - 27 * q^36 + 3 * q^37 - 6 * q^38 + 36 * q^39 + 9 * q^40 + 3 * q^43 - 15 * q^44 + 9 * q^45 + 6 * q^47 + 6 * q^50 + 21 * q^52 - 6 * q^53 + 27 * q^54 - 9 * q^57 + 9 * q^58 - 6 * q^59 - 9 * q^60 + 12 * q^61 - 60 * q^62 + 24 * q^64 + 30 * q^65 - 18 * q^66 - 24 * q^67 - 6 * q^68 - 9 * q^69 + 18 * q^71 - 9 * q^72 + 21 * q^73 + 30 * q^74 + 18 * q^75 - 15 * q^76 + 54 * q^78 - 42 * q^79 + 15 * q^80 + 9 * q^82 + 18 * q^83 - 9 * q^85 - 6 * q^86 - 45 * q^87 - 27 * q^88 - 12 * q^89 + 27 * q^90 - 3 * q^92 + 54 * q^93 - 36 * q^94 - 24 * q^95 - 27 * q^96 + 3 * q^97 + 18 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{18}^{3}$$ v^3 $$\beta_{2}$$ $$=$$ $$\zeta_{18}^{5} + \zeta_{18}$$ v^5 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18}$$ -v^4 + v^2 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{18}^{5} + \zeta_{18}^{4}$$ -v^5 + v^4 $$\beta_{5}$$ $$=$$ $$-\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}$$ -v^5 - v^4 + v
 $$\zeta_{18}$$ $$=$$ $$( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3$$ (b5 + b4 + 2*b2) / 3 $$\zeta_{18}^{2}$$ $$=$$ $$( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3$$ (-2*b5 + b4 + 3*b3 - b2) / 3 $$\zeta_{18}^{3}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{18}^{4}$$ $$=$$ $$( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3$$ (-b5 + 2*b4 + b2) / 3 $$\zeta_{18}^{5}$$ $$=$$ $$( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3$$ (-b5 - b4 + b2) / 3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
214.1
 0.939693 − 0.342020i −0.173648 + 0.984808i −0.766044 − 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 + 0.642788i
−0.879385 1.70574 + 0.300767i −1.22668 −0.673648 + 1.16679i −1.50000 0.264490i 0 2.83750 2.81908 + 1.02606i 0.592396 1.02606i
214.2 1.34730 −1.11334 + 1.32683i −0.184793 −1.26604 + 2.19285i −1.50000 + 1.78763i 0 −2.94356 −0.520945 2.95442i −1.70574 + 2.95442i
214.3 2.53209 −0.592396 1.62760i 4.41147 0.439693 0.761570i −1.50000 4.12122i 0 6.10607 −2.29813 + 1.92836i 1.11334 1.92836i
373.1 −0.879385 1.70574 0.300767i −1.22668 −0.673648 1.16679i −1.50000 + 0.264490i 0 2.83750 2.81908 1.02606i 0.592396 + 1.02606i
373.2 1.34730 −1.11334 1.32683i −0.184793 −1.26604 2.19285i −1.50000 1.78763i 0 −2.94356 −0.520945 + 2.95442i −1.70574 2.95442i
373.3 2.53209 −0.592396 + 1.62760i 4.41147 0.439693 + 0.761570i −1.50000 + 4.12122i 0 6.10607 −2.29813 1.92836i 1.11334 + 1.92836i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 214.3 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.h 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.h.d 6
3.b odd 2 1 1323.2.h.c 6
7.b odd 2 1 441.2.h.e 6
7.c even 3 1 63.2.f.a 6
7.c even 3 1 441.2.g.c 6
7.d odd 6 1 441.2.f.c 6
7.d odd 6 1 441.2.g.b 6
9.c even 3 1 441.2.g.c 6
9.d odd 6 1 1323.2.g.d 6
21.c even 2 1 1323.2.h.b 6
21.g even 6 1 1323.2.f.d 6
21.g even 6 1 1323.2.g.e 6
21.h odd 6 1 189.2.f.b 6
21.h odd 6 1 1323.2.g.d 6
28.g odd 6 1 1008.2.r.h 6
63.g even 3 1 63.2.f.a 6
63.h even 3 1 inner 441.2.h.d 6
63.h even 3 1 567.2.a.h 3
63.i even 6 1 1323.2.h.b 6
63.i even 6 1 3969.2.a.l 3
63.j odd 6 1 567.2.a.c 3
63.j odd 6 1 1323.2.h.c 6
63.k odd 6 1 441.2.f.c 6
63.l odd 6 1 441.2.g.b 6
63.n odd 6 1 189.2.f.b 6
63.o even 6 1 1323.2.g.e 6
63.s even 6 1 1323.2.f.d 6
63.t odd 6 1 441.2.h.e 6
63.t odd 6 1 3969.2.a.q 3
84.n even 6 1 3024.2.r.k 6
252.o even 6 1 3024.2.r.k 6
252.u odd 6 1 9072.2.a.ca 3
252.bb even 6 1 9072.2.a.bs 3
252.bl odd 6 1 1008.2.r.h 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.a 6 7.c even 3 1
63.2.f.a 6 63.g even 3 1
189.2.f.b 6 21.h odd 6 1
189.2.f.b 6 63.n odd 6 1
441.2.f.c 6 7.d odd 6 1
441.2.f.c 6 63.k odd 6 1
441.2.g.b 6 7.d odd 6 1
441.2.g.b 6 63.l odd 6 1
441.2.g.c 6 7.c even 3 1
441.2.g.c 6 9.c even 3 1
441.2.h.d 6 1.a even 1 1 trivial
441.2.h.d 6 63.h even 3 1 inner
441.2.h.e 6 7.b odd 2 1
441.2.h.e 6 63.t odd 6 1
567.2.a.c 3 63.j odd 6 1
567.2.a.h 3 63.h even 3 1
1008.2.r.h 6 28.g odd 6 1
1008.2.r.h 6 252.bl odd 6 1
1323.2.f.d 6 21.g even 6 1
1323.2.f.d 6 63.s even 6 1
1323.2.g.d 6 9.d odd 6 1
1323.2.g.d 6 21.h odd 6 1
1323.2.g.e 6 21.g even 6 1
1323.2.g.e 6 63.o even 6 1
1323.2.h.b 6 21.c even 2 1
1323.2.h.b 6 63.i even 6 1
1323.2.h.c 6 3.b odd 2 1
1323.2.h.c 6 63.j odd 6 1
3024.2.r.k 6 84.n even 6 1
3024.2.r.k 6 252.o even 6 1
3969.2.a.l 3 63.i even 6 1
3969.2.a.q 3 63.t odd 6 1
9072.2.a.bs 3 252.bb even 6 1
9072.2.a.ca 3 252.u odd 6 1

## Hecke kernels

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

 $$T_{2}^{3} - 3T_{2}^{2} + 3$$ T2^3 - 3*T2^2 + 3 $$T_{5}^{6} + 3T_{5}^{5} + 9T_{5}^{4} + 6T_{5}^{3} + 9T_{5}^{2} + 9$$ T5^6 + 3*T5^5 + 9*T5^4 + 6*T5^3 + 9*T5^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{3} - 3 T^{2} + 3)^{2}$$
$3$ $$T^{6} - 9T^{3} + 27$$
$5$ $$T^{6} + 3 T^{5} + \cdots + 9$$
$7$ $$T^{6}$$
$11$ $$T^{6} + 6 T^{5} + \cdots + 9$$
$13$ $$T^{6} - 3 T^{5} + \cdots + 11449$$
$17$ $$T^{6} + 6 T^{5} + \cdots + 9$$
$19$ $$T^{6} - 3 T^{5} + \cdots + 289$$
$23$ $$T^{6} + 12 T^{5} + \cdots + 9$$
$29$ $$T^{6} + 9 T^{5} + \cdots + 110889$$
$31$ $$(T^{3} + 3 T^{2} + \cdots - 323)^{2}$$
$37$ $$T^{6} - 3 T^{5} + \cdots + 104329$$
$41$ $$T^{6} + 9 T^{4} + \cdots + 81$$
$43$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$47$ $$(T^{3} - 3 T^{2} - 54 T - 51)^{2}$$
$53$ $$T^{6} + 6 T^{5} + \cdots + 9$$
$59$ $$(T^{3} + 3 T^{2} - 72 T + 51)^{2}$$
$61$ $$(T^{3} - 6 T^{2} - 15 T + 19)^{2}$$
$67$ $$(T^{3} + 12 T^{2} + \cdots - 17)^{2}$$
$71$ $$(T^{3} - 9 T^{2} - 54 T - 27)^{2}$$
$73$ $$T^{6} - 21 T^{5} + \cdots + 72361$$
$79$ $$(T^{3} + 21 T^{2} + \cdots + 181)^{2}$$
$83$ $$T^{6} - 18 T^{5} + \cdots + 81$$
$89$ $$T^{6} + 12 T^{5} + \cdots + 660969$$
$97$ $$T^{6} - 3 T^{5} + \cdots + 104329$$