Properties

 Label 72.1.p.a Level $72$ Weight $1$ Character orbit 72.p Analytic conductor $0.036$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -8 Inner twists $4$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,1,Mod(43,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 3, 4]))

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

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

chi := DirichletCharacter("72.43");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 72.p (of order $$6$$, degree $$2$$, 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: $$0.0359326809096$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ 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: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.648.1 Artin image: $C_3\times S_3$ Artin field: Galois closure of 6.0.41472.1

$q$-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.

 $$f(q)$$ $$=$$ $$q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{6} + q^{8} - \zeta_{6} q^{9} +O(q^{10})$$ q + z^2 * q^2 + z^2 * q^3 - z * q^4 - z * q^6 + q^8 - z * q^9 $$q + \zeta_{6}^{2} q^{2} + \zeta_{6}^{2} q^{3} - \zeta_{6} q^{4} - \zeta_{6} q^{6} + q^{8} - \zeta_{6} q^{9} - \zeta_{6}^{2} q^{11} + q^{12} + \zeta_{6}^{2} q^{16} - q^{17} + q^{18} - q^{19} + \zeta_{6} q^{22} + \zeta_{6}^{2} q^{24} + \zeta_{6}^{2} q^{25} + q^{27} - \zeta_{6} q^{32} + \zeta_{6} q^{33} - \zeta_{6}^{2} q^{34} + \zeta_{6}^{2} q^{36} - \zeta_{6}^{2} q^{38} + \zeta_{6} q^{41} - \zeta_{6}^{2} q^{43} - q^{44} - \zeta_{6} q^{48} - \zeta_{6} q^{49} - \zeta_{6} q^{50} - \zeta_{6}^{2} q^{51} + \zeta_{6}^{2} q^{54} - \zeta_{6}^{2} q^{57} + \zeta_{6} q^{59} + q^{64} - q^{66} + \zeta_{6} q^{67} + \zeta_{6} q^{68} - \zeta_{6} q^{72} - q^{73} - \zeta_{6} q^{75} + \zeta_{6} q^{76} + \zeta_{6}^{2} q^{81} - q^{82} + \zeta_{6}^{2} q^{83} + \zeta_{6} q^{86} - \zeta_{6}^{2} q^{88} + q^{89} + q^{96} - \zeta_{6}^{2} q^{97} + q^{98} - q^{99} +O(q^{100})$$ q + z^2 * q^2 + z^2 * q^3 - z * q^4 - z * q^6 + q^8 - z * q^9 - z^2 * q^11 + q^12 + z^2 * q^16 - q^17 + q^18 - q^19 + z * q^22 + z^2 * q^24 + z^2 * q^25 + q^27 - z * q^32 + z * q^33 - z^2 * q^34 + z^2 * q^36 - z^2 * q^38 + z * q^41 - z^2 * q^43 - q^44 - z * q^48 - z * q^49 - z * q^50 - z^2 * q^51 + z^2 * q^54 - z^2 * q^57 + z * q^59 + q^64 - q^66 + z * q^67 + z * q^68 - z * q^72 - q^73 - z * q^75 + z * q^76 + z^2 * q^81 - q^82 + z^2 * q^83 + z * q^86 - z^2 * q^88 + q^89 + q^96 - z^2 * q^97 + q^98 - q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} - q^{3} - q^{4} - q^{6} + 2 q^{8} - q^{9}+O(q^{10})$$ 2 * q - q^2 - q^3 - q^4 - q^6 + 2 * q^8 - q^9 $$2 q - q^{2} - q^{3} - q^{4} - q^{6} + 2 q^{8} - q^{9} + q^{11} + 2 q^{12} - q^{16} - 2 q^{17} + 2 q^{18} - 2 q^{19} + q^{22} - q^{24} - q^{25} + 2 q^{27} - q^{32} + q^{33} + q^{34} - q^{36} + q^{38} + q^{41} + q^{43} - 2 q^{44} - q^{48} - q^{49} - q^{50} + q^{51} - q^{54} + q^{57} + q^{59} + 2 q^{64} - 2 q^{66} + q^{67} + q^{68} - q^{72} - 2 q^{73} - q^{75} + q^{76} - q^{81} - 2 q^{82} - 2 q^{83} + q^{86} + q^{88} + 4 q^{89} + 2 q^{96} + q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100})$$ 2 * q - q^2 - q^3 - q^4 - q^6 + 2 * q^8 - q^9 + q^11 + 2 * q^12 - q^16 - 2 * q^17 + 2 * q^18 - 2 * q^19 + q^22 - q^24 - q^25 + 2 * q^27 - q^32 + q^33 + q^34 - q^36 + q^38 + q^41 + q^43 - 2 * q^44 - q^48 - q^49 - q^50 + q^51 - q^54 + q^57 + q^59 + 2 * q^64 - 2 * q^66 + q^67 + q^68 - q^72 - 2 * q^73 - q^75 + q^76 - q^81 - 2 * q^82 - 2 * q^83 + q^86 + q^88 + 4 * q^89 + 2 * q^96 + q^97 + 2 * q^98 - 2 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$55$$ $$65$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-\zeta_{6}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i 0 −0.500000 0.866025i 0 1.00000 −0.500000 0.866025i 0
67.1 −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 1.00000 −0.500000 + 0.866025i 0
 $$n$$: 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
9.c even 3 1 inner
72.p odd 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.1.p.a 2
3.b odd 2 1 216.1.p.a 2
4.b odd 2 1 288.1.t.a 2
5.b even 2 1 1800.1.bk.d 2
5.c odd 4 2 1800.1.ba.b 4
7.b odd 2 1 3528.1.cg.a 2
7.c even 3 1 3528.1.ba.b 2
7.c even 3 1 3528.1.ce.a 2
7.d odd 6 1 3528.1.ba.a 2
7.d odd 6 1 3528.1.ce.b 2
8.b even 2 1 288.1.t.a 2
8.d odd 2 1 CM 72.1.p.a 2
9.c even 3 1 inner 72.1.p.a 2
9.c even 3 1 648.1.b.b 1
9.d odd 6 1 216.1.p.a 2
9.d odd 6 1 648.1.b.a 1
12.b even 2 1 864.1.t.a 2
16.e even 4 2 2304.1.o.c 4
16.f odd 4 2 2304.1.o.c 4
24.f even 2 1 216.1.p.a 2
24.h odd 2 1 864.1.t.a 2
36.f odd 6 1 288.1.t.a 2
36.f odd 6 1 2592.1.b.b 1
36.h even 6 1 864.1.t.a 2
36.h even 6 1 2592.1.b.a 1
40.e odd 2 1 1800.1.bk.d 2
40.k even 4 2 1800.1.ba.b 4
45.j even 6 1 1800.1.bk.d 2
45.k odd 12 2 1800.1.ba.b 4
56.e even 2 1 3528.1.cg.a 2
56.k odd 6 1 3528.1.ba.b 2
56.k odd 6 1 3528.1.ce.a 2
56.m even 6 1 3528.1.ba.a 2
56.m even 6 1 3528.1.ce.b 2
63.g even 3 1 3528.1.ce.a 2
63.h even 3 1 3528.1.ba.b 2
63.k odd 6 1 3528.1.ce.b 2
63.l odd 6 1 3528.1.cg.a 2
63.t odd 6 1 3528.1.ba.a 2
72.j odd 6 1 864.1.t.a 2
72.j odd 6 1 2592.1.b.a 1
72.l even 6 1 216.1.p.a 2
72.l even 6 1 648.1.b.a 1
72.n even 6 1 288.1.t.a 2
72.n even 6 1 2592.1.b.b 1
72.p odd 6 1 inner 72.1.p.a 2
72.p odd 6 1 648.1.b.b 1
144.v odd 12 2 2304.1.o.c 4
144.x even 12 2 2304.1.o.c 4
360.z odd 6 1 1800.1.bk.d 2
360.bo even 12 2 1800.1.ba.b 4
504.ba odd 6 1 3528.1.ce.a 2
504.be even 6 1 3528.1.cg.a 2
504.bf even 6 1 3528.1.ba.a 2
504.ce odd 6 1 3528.1.ba.b 2
504.cz even 6 1 3528.1.ce.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 1.a even 1 1 trivial
72.1.p.a 2 8.d odd 2 1 CM
72.1.p.a 2 9.c even 3 1 inner
72.1.p.a 2 72.p odd 6 1 inner
216.1.p.a 2 3.b odd 2 1
216.1.p.a 2 9.d odd 6 1
216.1.p.a 2 24.f even 2 1
216.1.p.a 2 72.l even 6 1
288.1.t.a 2 4.b odd 2 1
288.1.t.a 2 8.b even 2 1
288.1.t.a 2 36.f odd 6 1
288.1.t.a 2 72.n even 6 1
648.1.b.a 1 9.d odd 6 1
648.1.b.a 1 72.l even 6 1
648.1.b.b 1 9.c even 3 1
648.1.b.b 1 72.p odd 6 1
864.1.t.a 2 12.b even 2 1
864.1.t.a 2 24.h odd 2 1
864.1.t.a 2 36.h even 6 1
864.1.t.a 2 72.j odd 6 1
1800.1.ba.b 4 5.c odd 4 2
1800.1.ba.b 4 40.k even 4 2
1800.1.ba.b 4 45.k odd 12 2
1800.1.ba.b 4 360.bo even 12 2
1800.1.bk.d 2 5.b even 2 1
1800.1.bk.d 2 40.e odd 2 1
1800.1.bk.d 2 45.j even 6 1
1800.1.bk.d 2 360.z odd 6 1
2304.1.o.c 4 16.e even 4 2
2304.1.o.c 4 16.f odd 4 2
2304.1.o.c 4 144.v odd 12 2
2304.1.o.c 4 144.x even 12 2
2592.1.b.a 1 36.h even 6 1
2592.1.b.a 1 72.j odd 6 1
2592.1.b.b 1 36.f odd 6 1
2592.1.b.b 1 72.n even 6 1
3528.1.ba.a 2 7.d odd 6 1
3528.1.ba.a 2 56.m even 6 1
3528.1.ba.a 2 63.t odd 6 1
3528.1.ba.a 2 504.bf even 6 1
3528.1.ba.b 2 7.c even 3 1
3528.1.ba.b 2 56.k odd 6 1
3528.1.ba.b 2 63.h even 3 1
3528.1.ba.b 2 504.ce odd 6 1
3528.1.ce.a 2 7.c even 3 1
3528.1.ce.a 2 56.k odd 6 1
3528.1.ce.a 2 63.g even 3 1
3528.1.ce.a 2 504.ba odd 6 1
3528.1.ce.b 2 7.d odd 6 1
3528.1.ce.b 2 56.m even 6 1
3528.1.ce.b 2 63.k odd 6 1
3528.1.ce.b 2 504.cz even 6 1
3528.1.cg.a 2 7.b odd 2 1
3528.1.cg.a 2 56.e even 2 1
3528.1.cg.a 2 63.l odd 6 1
3528.1.cg.a 2 504.be even 6 1

Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(72, [\chi])$$.

Hecke characteristic polynomials

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