# Properties

 Label 1323.2.g.d Level $1323$ Weight $2$ Character orbit 1323.g Analytic conductor $10.564$ 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] = [1323,2,Mod(361,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$10.5642081874$$ 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: $$\Z[a_1, \ldots, a_{4}]$$ 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.

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

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 $$\left(\mathbb{Z}/1323\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1081$$ $$\chi(n)$$ $$-\beta_{1}$$ $$-1 + \beta_{1}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
361.1
 0.939693 − 0.342020i −0.173648 + 0.984808i −0.766044 − 0.642788i 0.939693 + 0.342020i −0.173648 − 0.984808i −0.766044 + 0.642788i
−0.439693 + 0.761570i 0 0.613341 + 1.06234i −1.34730 0 0 −2.83750 0 0.592396 1.02606i
361.2 0.673648 1.16679i 0 0.0923963 + 0.160035i −2.53209 0 0 2.94356 0 −1.70574 + 2.95442i
361.3 1.26604 2.19285i 0 −2.20574 3.82045i 0.879385 0 0 −6.10607 0 1.11334 1.92836i
667.1 −0.439693 0.761570i 0 0.613341 1.06234i −1.34730 0 0 −2.83750 0 0.592396 + 1.02606i
667.2 0.673648 + 1.16679i 0 0.0923963 0.160035i −2.53209 0 0 2.94356 0 −1.70574 2.95442i
667.3 1.26604 + 2.19285i 0 −2.20574 + 3.82045i 0.879385 0 0 −6.10607 0 1.11334 + 1.92836i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 361.3 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.g even 3 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1323, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} - 3 T^{5} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$(T^{3} + 3 T^{2} - 3)^{2}$$
$7$ $$T^{6}$$
$11$ $$(T^{3} + 6 T^{2} + 9 T + 3)^{2}$$
$13$ $$T^{6} - 3 T^{5} + \cdots + 11449$$
$17$ $$T^{6} - 6 T^{5} + \cdots + 9$$
$19$ $$T^{6} - 3 T^{5} + \cdots + 289$$
$23$ $$(T^{3} + 12 T^{2} + 27 T - 3)^{2}$$
$29$ $$T^{6} - 9 T^{5} + \cdots + 110889$$
$31$ $$T^{6} - 3 T^{5} + \cdots + 104329$$
$37$ $$T^{6} - 3 T^{5} + \cdots + 104329$$
$41$ $$T^{6} + 9 T^{4} + \cdots + 81$$
$43$ $$T^{6} - 3 T^{5} + \cdots + 1$$
$47$ $$T^{6} - 3 T^{5} + \cdots + 2601$$
$53$ $$T^{6} - 6 T^{5} + \cdots + 9$$
$59$ $$T^{6} + 3 T^{5} + \cdots + 2601$$
$61$ $$T^{6} + 6 T^{5} + \cdots + 361$$
$67$ $$T^{6} - 12 T^{5} + \cdots + 289$$
$71$ $$(T^{3} + 9 T^{2} - 54 T + 27)^{2}$$
$73$ $$T^{6} - 21 T^{5} + \cdots + 72361$$
$79$ $$T^{6} - 21 T^{5} + \cdots + 32761$$
$83$ $$T^{6} + 18 T^{5} + \cdots + 81$$
$89$ $$T^{6} - 12 T^{5} + \cdots + 660969$$
$97$ $$T^{6} - 3 T^{5} + \cdots + 104329$$