# Properties

 Label 3267.1.be.a Level $3267$ Weight $1$ Character orbit 3267.be Analytic conductor $1.630$ Analytic rank $0$ Dimension $24$ Projective image $D_{18}$ CM discriminant -11 Inner twists $16$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3267,1,Mod(245,3267)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3267, base_ring=CyclotomicField(90))

chi = DirichletCharacter(H, H._module([5, 72]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3267.245");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3267.be (of order $$90$$, degree $$24$$, 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: $$1.63044539627$$ Analytic rank: $$0$$ Dimension: $$24$$ Coefficient field: $$\Q(\zeta_{45})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{24} - x^{21} + x^{15} - x^{12} + x^{9} - x^{3} + 1$$ x^24 - x^21 + x^15 - x^12 + x^9 - x^3 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{18}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{18} - \cdots)$$

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{90}^{11} q^{3} + \zeta_{90}^{14} q^{4} + (\zeta_{90}^{13} - \zeta_{90}^{3}) q^{5} + \zeta_{90}^{22} q^{9}+O(q^{10})$$ q + z^11 * q^3 + z^14 * q^4 + (z^13 - z^3) * q^5 + z^22 * q^9 $$q + \zeta_{90}^{11} q^{3} + \zeta_{90}^{14} q^{4} + (\zeta_{90}^{13} - \zeta_{90}^{3}) q^{5} + \zeta_{90}^{22} q^{9} + \zeta_{90}^{25} q^{12} + (\zeta_{90}^{24} - \zeta_{90}^{14}) q^{15} + \zeta_{90}^{28} q^{16} + (\zeta_{90}^{27} - \zeta_{90}^{17}) q^{20} + ( - \zeta_{90}^{35} - \zeta_{90}^{20}) q^{23} + (\zeta_{90}^{26} - \zeta_{90}^{16} + \zeta_{90}^{6}) q^{25} + \zeta_{90}^{33} q^{27} + ( - \zeta_{90}^{12} + \zeta_{90}^{7}) q^{31} + \zeta_{90}^{36} q^{36} + (\zeta_{90}^{29} + \zeta_{90}^{19}) q^{37} + (\zeta_{90}^{35} - \zeta_{90}^{25}) q^{45} + ( - \zeta_{90}^{41} + \zeta_{90}^{21}) q^{47} + \zeta_{90}^{39} q^{48} - \zeta_{90}^{8} q^{49} + ( - \zeta_{90}^{7} - \zeta_{90}^{2}) q^{53} + ( - \zeta_{90}^{24} - \zeta_{90}^{19}) q^{59} + (\zeta_{90}^{38} - \zeta_{90}^{28}) q^{60} + \zeta_{90}^{42} q^{64} + ( - \zeta_{90}^{40} - 1) q^{67} + ( - \zeta_{90}^{31} + \zeta_{90}) q^{69} + ( - \zeta_{90}^{43} + \zeta_{90}^{23}) q^{71} + (\zeta_{90}^{37} - \zeta_{90}^{27} + \zeta_{90}^{17}) q^{75} + (\zeta_{90}^{41} - \zeta_{90}^{31}) q^{80} + \zeta_{90}^{44} q^{81} + ( - \zeta_{90}^{15} - 1) q^{89} + ( - \zeta_{90}^{34} + \zeta_{90}^{4}) q^{92} + ( - \zeta_{90}^{23} + \zeta_{90}^{18}) q^{93} + ( - \zeta_{90}^{27} + \zeta_{90}^{2}) q^{97} +O(q^{100})$$ q + z^11 * q^3 + z^14 * q^4 + (z^13 - z^3) * q^5 + z^22 * q^9 + z^25 * q^12 + (z^24 - z^14) * q^15 + z^28 * q^16 + (z^27 - z^17) * q^20 + (-z^35 - z^20) * q^23 + (z^26 - z^16 + z^6) * q^25 + z^33 * q^27 + (-z^12 + z^7) * q^31 + z^36 * q^36 + (z^29 + z^19) * q^37 + (z^35 - z^25) * q^45 + (-z^41 + z^21) * q^47 + z^39 * q^48 - z^8 * q^49 + (-z^7 - z^2) * q^53 + (-z^24 - z^19) * q^59 + (z^38 - z^28) * q^60 + z^42 * q^64 + (-z^40 - 1) * q^67 + (-z^31 + z) * q^69 + (-z^43 + z^23) * q^71 + (z^37 - z^27 + z^17) * q^75 + (z^41 - z^31) * q^80 + z^44 * q^81 + (-z^15 - 1) * q^89 + (-z^34 + z^4) * q^92 + (-z^23 + z^18) * q^93 + (-z^27 + z^2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$24 q + 3 q^{5}+O(q^{10})$$ 24 * q + 3 * q^5 $$24 q + 3 q^{5} + 3 q^{15} + 6 q^{20} + 3 q^{25} - 3 q^{27} - 3 q^{31} - 6 q^{36} - 3 q^{47} - 3 q^{48} - 3 q^{59} + 3 q^{64} - 24 q^{67} - 6 q^{75} - 36 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100})$$ 24 * q + 3 * q^5 + 3 * q^15 + 6 * q^20 + 3 * q^25 - 3 * q^27 - 3 * q^31 - 6 * q^36 - 3 * q^47 - 3 * q^48 - 3 * q^59 + 3 * q^64 - 24 * q^67 - 6 * q^75 - 36 * q^89 - 6 * q^93 - 6 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3267\mathbb{Z}\right)^\times$$.

 $$n$$ $$244$$ $$3026$$ $$\chi(n)$$ $$-\zeta_{90}^{9}$$ $$\zeta_{90}^{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
245.1
 −0.374607 + 0.927184i 0.848048 + 0.529919i 0.438371 + 0.898794i 0.961262 + 0.275637i 0.0348995 + 0.999391i 0.961262 − 0.275637i 0.438371 − 0.898794i 0.990268 + 0.139173i −0.615661 − 0.788011i −0.882948 + 0.469472i −0.997564 − 0.0697565i −0.719340 + 0.694658i −0.882948 − 0.469472i −0.241922 + 0.970296i 0.559193 + 0.829038i −0.615661 + 0.788011i 0.990268 − 0.139173i 0.0348995 − 0.999391i 0.559193 − 0.829038i −0.241922 − 0.970296i
0 0.882948 0.469472i −0.615661 + 0.788011i −0.0477162 0.682374i 0 0 0 0.559193 0.829038i 0
608.1 0 −0.990268 + 0.139173i 0.0348995 + 0.999391i −0.663721 + 0.165484i 0 0 0 0.961262 0.275637i 0
614.1 0 −0.961262 + 0.275637i −0.997564 + 0.0697565i −0.603541 1.13510i 0 0 0 0.848048 0.529919i 0
632.1 0 0.997564 0.0697565i −0.719340 0.694658i 1.55208 + 1.21262i 0 0 0 0.990268 0.139173i 0
686.1 0 0.374607 + 0.927184i −0.882948 + 0.469472i −0.542900 1.89332i 0 0 0 −0.719340 + 0.694658i 0
977.1 0 0.997564 + 0.0697565i −0.719340 + 0.694658i 1.55208 1.21262i 0 0 0 0.990268 + 0.139173i 0
995.1 0 −0.961262 0.275637i −0.997564 0.0697565i −0.603541 + 1.13510i 0 0 0 0.848048 + 0.529919i 0
1049.1 0 −0.0348995 0.999391i −0.374607 + 0.927184i 1.15547 0.563559i 0 0 0 −0.997564 + 0.0697565i 0
1334.1 0 −0.848048 0.529919i 0.990268 + 0.139173i 1.63289 1.10140i 0 0 0 0.438371 + 0.898794i 0
1697.1 0 0.615661 + 0.788011i 0.848048 0.529919i 0.893036 + 0.924765i 0 0 0 −0.241922 + 0.970296i 0
1703.1 0 0.719340 + 0.694658i 0.559193 + 0.829038i −0.362486 + 0.580099i 0 0 0 0.0348995 + 0.999391i 0
1721.1 0 −0.559193 0.829038i −0.241922 + 0.970296i −0.178917 + 1.27306i 0 0 0 −0.374607 + 0.927184i 0
1775.1 0 0.615661 0.788011i 0.848048 + 0.529919i 0.893036 0.924765i 0 0 0 −0.241922 0.970296i 0
2066.1 0 −0.438371 0.898794i 0.961262 + 0.275637i 0.634231 + 0.256246i 0 0 0 −0.615661 + 0.788011i 0
2084.1 0 0.241922 + 0.970296i 0.438371 + 0.898794i −1.96842 + 0.0687386i 0 0 0 −0.882948 + 0.469472i 0
2138.1 0 −0.848048 + 0.529919i 0.990268 0.139173i 1.63289 + 1.10140i 0 0 0 0.438371 0.898794i 0
2423.1 0 −0.0348995 + 0.999391i −0.374607 0.927184i 1.15547 + 0.563559i 0 0 0 −0.997564 0.0697565i 0
2786.1 0 0.374607 0.927184i −0.882948 0.469472i −0.542900 + 1.89332i 0 0 0 −0.719340 0.694658i 0
2792.1 0 0.241922 0.970296i 0.438371 0.898794i −1.96842 0.0687386i 0 0 0 −0.882948 0.469472i 0
2810.1 0 −0.438371 + 0.898794i 0.961262 0.275637i 0.634231 0.256246i 0 0 0 −0.615661 0.788011i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3227.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
11.c even 5 3 inner
11.d odd 10 3 inner
27.f odd 18 1 inner
297.o even 18 1 inner
297.v odd 90 3 inner
297.x even 90 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.be.a 24
11.b odd 2 1 CM 3267.1.be.a 24
11.c even 5 1 3267.1.q.a 6
11.c even 5 3 inner 3267.1.be.a 24
11.d odd 10 1 3267.1.q.a 6
11.d odd 10 3 inner 3267.1.be.a 24
27.f odd 18 1 inner 3267.1.be.a 24
297.o even 18 1 inner 3267.1.be.a 24
297.v odd 90 1 3267.1.q.a 6
297.v odd 90 3 inner 3267.1.be.a 24
297.x even 90 1 3267.1.q.a 6
297.x even 90 3 inner 3267.1.be.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3267.1.q.a 6 11.c even 5 1
3267.1.q.a 6 11.d odd 10 1
3267.1.q.a 6 297.v odd 90 1
3267.1.q.a 6 297.x even 90 1
3267.1.be.a 24 1.a even 1 1 trivial
3267.1.be.a 24 11.b odd 2 1 CM
3267.1.be.a 24 11.c even 5 3 inner
3267.1.be.a 24 11.d odd 10 3 inner
3267.1.be.a 24 27.f odd 18 1 inner
3267.1.be.a 24 297.o even 18 1 inner
3267.1.be.a 24 297.v odd 90 3 inner
3267.1.be.a 24 297.x even 90 3 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(3267, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{24}$$
$3$ $$T^{24} + T^{21} - T^{15} - T^{12} - T^{9} + T^{3} + \cdots + 1$$
$5$ $$T^{24} - 3 T^{23} + 3 T^{22} + 3 T^{21} + \cdots + 81$$
$7$ $$T^{24}$$
$11$ $$T^{24}$$
$13$ $$T^{24}$$
$17$ $$T^{24}$$
$19$ $$T^{24}$$
$23$ $$(T^{6} + 9 T^{3} + 27)^{4}$$
$29$ $$T^{24}$$
$31$ $$T^{24} + 3 T^{23} + 3 T^{22} - T^{21} - 9 T^{20} + \cdots + 1$$
$37$ $$T^{24} - 3 T^{22} + 2 T^{21} + 6 T^{19} + \cdots + 1$$
$41$ $$T^{24}$$
$43$ $$T^{24}$$
$47$ $$T^{24} + 3 T^{23} + 3 T^{22} - 3 T^{21} + \cdots + 81$$
$53$ $$T^{24} - 6 T^{22} + 27 T^{20} - 111 T^{18} + \cdots + 81$$
$59$ $$T^{24} + 3 T^{23} + 3 T^{22} - 3 T^{21} + \cdots + 81$$
$61$ $$T^{24}$$
$67$ $$(T^{6} + 6 T^{5} + 15 T^{4} + 19 T^{3} + \cdots + 1)^{4}$$
$71$ $$T^{24} + 3 T^{22} - 36 T^{19} - 30 T^{18} + \cdots + 81$$
$73$ $$T^{24}$$
$79$ $$T^{24}$$
$83$ $$T^{24}$$
$89$ $$(T^{2} + 3 T + 3)^{12}$$
$97$ $$T^{24} + 6 T^{23} + 21 T^{22} + 55 T^{21} + \cdots + 1$$