Properties

 Label 3267.1.bf.a Level $3267$ Weight $1$ Character orbit 3267.bf Analytic conductor $1.630$ Analytic rank $0$ Dimension $24$ Projective image $D_{9}$ 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(40,3267)]

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([40, 63]))

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

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

chi := DirichletCharacter("3267.40");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$3267 = 3^{3} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3267.bf (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: no (minimal twist has level 297) Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.459450093735369.1

$q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{90}^{34} q^{3} - \zeta_{90}^{31} q^{4} + (\zeta_{90}^{42} + \zeta_{90}^{32}) q^{5} - \zeta_{90}^{23} q^{9} +O(q^{10})$$ q + z^34 * q^3 - z^31 * q^4 + (z^42 + z^32) * q^5 - z^23 * q^9 $$q + \zeta_{90}^{34} q^{3} - \zeta_{90}^{31} q^{4} + (\zeta_{90}^{42} + \zeta_{90}^{32}) q^{5} - \zeta_{90}^{23} q^{9} + \zeta_{90}^{20} q^{12} + ( - \zeta_{90}^{31} - \zeta_{90}^{21}) q^{15} - \zeta_{90}^{17} q^{16} + (\zeta_{90}^{28} + \zeta_{90}^{18}) q^{20} - \zeta_{90}^{40} q^{23} + ( - \zeta_{90}^{39} + \cdots - \zeta_{90}^{19}) q^{25} + \cdots + ( - \zeta_{90}^{43} + \zeta_{90}^{18}) q^{97} +O(q^{100})$$ q + z^34 * q^3 - z^31 * q^4 + (z^42 + z^32) * q^5 - z^23 * q^9 + z^20 * q^12 + (-z^31 - z^21) * q^15 - z^17 * q^16 + (z^28 + z^18) * q^20 - z^40 * q^23 + (-z^39 - z^29 - z^19) * q^25 + z^12 * q^27 + (z^38 - z^33) * q^31 - z^9 * q^36 + (z^26 + z^16) * q^37 + (z^20 + z^10) * q^45 + (z^24 + z^4) * q^47 + z^6 * q^48 - z^37 * q^49 + (-z^43 + z^38) * q^53 + (z^26 - z^21) * q^59 + (-z^17 - z^7) * q^60 - z^3 * q^64 + (-z^5 + 1) * q^67 + z^29 * q^69 + (z^22 + z^2) * q^71 + (z^28 + z^18 + z^8) * q^75 + (z^14 + z^4) * q^80 - z * q^81 + z^15 * q^89 - z^26 * q^92 + (-z^27 + z^22) * q^93 + (-z^43 + z^18) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$24 q + 3 q^{5}+O(q^{10})$$ 24 * q + 3 * q^5 $$24 q + 3 q^{5} + 3 q^{15} - 6 q^{20} + 3 q^{25} + 3 q^{27} + 3 q^{31} - 6 q^{36} + 3 q^{47} + 3 q^{48} + 3 q^{59} + 3 q^{64} + 24 q^{67} - 6 q^{75} + 12 q^{89} - 6 q^{93} - 6 q^{97}+O(q^{100})$$ 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 + 12 * q^89 - 6 * q^93 - 6 * q^97

Character values

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

 $$n$$ $$244$$ $$3026$$ $$\chi(n)$$ $$-\zeta_{90}^{36}$$ $$\zeta_{90}^{40}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
40.1
 −0.374607 + 0.927184i −0.997564 − 0.0697565i −0.719340 + 0.694658i 0.848048 + 0.529919i −0.241922 + 0.970296i 0.559193 + 0.829038i 0.0348995 + 0.999391i 0.990268 + 0.139173i −0.615661 − 0.788011i 0.559193 − 0.829038i −0.241922 − 0.970296i −0.882948 + 0.469472i −0.719340 − 0.694658i −0.997564 + 0.0697565i −0.882948 − 0.469472i −0.615661 + 0.788011i 0.990268 − 0.139173i 0.438371 + 0.898794i 0.961262 + 0.275637i 0.0348995 − 0.999391i
0 −0.882948 0.469472i −0.615661 0.788011i 1.87481 + 0.131099i 0 0 0 0.559193 + 0.829038i 0
94.1 0 −0.719340 + 0.694658i 0.559193 0.829038i −1.59381 + 0.995922i 0 0 0 0.0348995 0.999391i 0
112.1 0 0.559193 0.829038i −0.241922 0.970296i 1.51718 0.213226i 0 0 0 −0.374607 0.927184i 0
403.1 0 0.990268 + 0.139173i 0.0348995 0.999391i 0.454664 1.82356i 0 0 0 0.961262 + 0.275637i 0
457.1 0 0.438371 0.898794i 0.961262 0.275637i 0.704030 + 1.74254i 0 0 0 −0.615661 0.788011i 0
475.1 0 −0.241922 + 0.970296i 0.438371 0.898794i 0.0121205 0.347085i 0 0 0 −0.882948 0.469472i 0
481.1 0 −0.374607 + 0.927184i −0.882948 0.469472i 0.333843 + 0.0957278i 0 0 0 −0.719340 0.694658i 0
844.1 0 0.0348995 0.999391i −0.374607 0.927184i 0.671624 1.37703i 0 0 0 −0.997564 0.0697565i 0
1129.1 0 0.848048 0.529919i 0.990268 0.139173i 0.194206 0.287922i 0 0 0 0.438371 0.898794i 0
1183.1 0 −0.241922 0.970296i 0.438371 + 0.898794i 0.0121205 + 0.347085i 0 0 0 −0.882948 + 0.469472i 0
1201.1 0 0.438371 + 0.898794i 0.961262 + 0.275637i 0.704030 1.74254i 0 0 0 −0.615661 + 0.788011i 0
1492.1 0 −0.615661 + 0.788011i 0.848048 + 0.529919i −1.10209 1.06428i 0 0 0 −0.241922 0.970296i 0
1546.1 0 0.559193 + 0.829038i −0.241922 + 0.970296i 1.51718 + 0.213226i 0 0 0 −0.374607 + 0.927184i 0
1564.1 0 −0.719340 0.694658i 0.559193 + 0.829038i −1.59381 0.995922i 0 0 0 0.0348995 + 0.999391i 0
1570.1 0 −0.615661 0.788011i 0.848048 0.529919i −1.10209 + 1.06428i 0 0 0 −0.241922 + 0.970296i 0
1933.1 0 0.848048 + 0.529919i 0.990268 + 0.139173i 0.194206 + 0.287922i 0 0 0 0.438371 + 0.898794i 0
2218.1 0 0.0348995 + 0.999391i −0.374607 + 0.927184i 0.671624 + 1.37703i 0 0 0 −0.997564 + 0.0697565i 0
2272.1 0 0.961262 + 0.275637i −0.997564 0.0697565i −1.35275 0.719272i 0 0 0 0.848048 + 0.529919i 0
2290.1 0 −0.997564 0.0697565i −0.719340 + 0.694658i −0.213817 0.273673i 0 0 0 0.990268 + 0.139173i 0
2581.1 0 −0.374607 0.927184i −0.882948 + 0.469472i 0.333843 0.0957278i 0 0 0 −0.719340 + 0.694658i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 40.1 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.e even 9 1 inner
297.q odd 18 1 inner
297.u even 45 3 inner
297.w odd 90 3 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.1.bf.a 24
11.b odd 2 1 CM 3267.1.bf.a 24
11.c even 5 1 297.1.q.a 6
11.c even 5 3 inner 3267.1.bf.a 24
11.d odd 10 1 297.1.q.a 6
11.d odd 10 3 inner 3267.1.bf.a 24
27.e even 9 1 inner 3267.1.bf.a 24
33.f even 10 1 891.1.q.a 6
33.h odd 10 1 891.1.q.a 6
99.m even 15 1 2673.1.q.b 6
99.m even 15 1 2673.1.q.d 6
99.n odd 30 1 2673.1.q.a 6
99.n odd 30 1 2673.1.q.c 6
99.o odd 30 1 2673.1.q.b 6
99.o odd 30 1 2673.1.q.d 6
99.p even 30 1 2673.1.q.a 6
99.p even 30 1 2673.1.q.c 6
297.q odd 18 1 inner 3267.1.bf.a 24
297.u even 45 1 297.1.q.a 6
297.u even 45 1 2673.1.q.b 6
297.u even 45 1 2673.1.q.d 6
297.u even 45 3 inner 3267.1.bf.a 24
297.v odd 90 1 891.1.q.a 6
297.v odd 90 1 2673.1.q.a 6
297.v odd 90 1 2673.1.q.c 6
297.w odd 90 1 297.1.q.a 6
297.w odd 90 1 2673.1.q.b 6
297.w odd 90 1 2673.1.q.d 6
297.w odd 90 3 inner 3267.1.bf.a 24
297.x even 90 1 891.1.q.a 6
297.x even 90 1 2673.1.q.a 6
297.x even 90 1 2673.1.q.c 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.1.q.a 6 11.c even 5 1
297.1.q.a 6 11.d odd 10 1
297.1.q.a 6 297.u even 45 1
297.1.q.a 6 297.w odd 90 1
891.1.q.a 6 33.f even 10 1
891.1.q.a 6 33.h odd 10 1
891.1.q.a 6 297.v odd 90 1
891.1.q.a 6 297.x even 90 1
2673.1.q.a 6 99.n odd 30 1
2673.1.q.a 6 99.p even 30 1
2673.1.q.a 6 297.v odd 90 1
2673.1.q.a 6 297.x even 90 1
2673.1.q.b 6 99.m even 15 1
2673.1.q.b 6 99.o odd 30 1
2673.1.q.b 6 297.u even 45 1
2673.1.q.b 6 297.w odd 90 1
2673.1.q.c 6 99.n odd 30 1
2673.1.q.c 6 99.p even 30 1
2673.1.q.c 6 297.v odd 90 1
2673.1.q.c 6 297.x even 90 1
2673.1.q.d 6 99.m even 15 1
2673.1.q.d 6 99.o odd 30 1
2673.1.q.d 6 297.u even 45 1
2673.1.q.d 6 297.w odd 90 1
3267.1.bf.a 24 1.a even 1 1 trivial
3267.1.bf.a 24 11.b odd 2 1 CM
3267.1.bf.a 24 11.c even 5 3 inner
3267.1.bf.a 24 11.d odd 10 3 inner
3267.1.bf.a 24 27.e even 9 1 inner
3267.1.bf.a 24 297.q odd 18 1 inner
3267.1.bf.a 24 297.u even 45 3 inner
3267.1.bf.a 24 297.w odd 90 3 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{24}$$
$3$ $$T^{24} - T^{21} + \cdots + 1$$
$5$ $$T^{24} - 3 T^{23} + \cdots + 1$$
$7$ $$T^{24}$$
$11$ $$T^{24}$$
$13$ $$T^{24}$$
$17$ $$T^{24}$$
$19$ $$T^{24}$$
$23$ $$(T^{6} - T^{3} + 1)^{4}$$
$29$ $$T^{24}$$
$31$ $$T^{24} - 3 T^{23} + \cdots + 1$$
$37$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$41$ $$T^{24}$$
$43$ $$T^{24}$$
$47$ $$T^{24} - 3 T^{23} + \cdots + 1$$
$53$ $$(T^{12} + 3 T^{10} + \cdots + 1)^{2}$$
$59$ $$T^{24} - 3 T^{23} + \cdots + 1$$
$61$ $$T^{24}$$
$67$ $$(T^{6} - 6 T^{5} + 15 T^{4} + \cdots + 1)^{4}$$
$71$ $$T^{24} - 3 T^{22} + \cdots + 1$$
$73$ $$T^{24}$$
$79$ $$T^{24}$$
$83$ $$T^{24}$$
$89$ $$(T^{2} - T + 1)^{12}$$
$97$ $$T^{24} + 6 T^{23} + \cdots + 1$$