# Properties

 Label 72.2.i.a Level $72$ Weight $2$ Character orbit 72.i Analytic conductor $0.575$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,2,Mod(25,72)]

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([0, 0, 4]))

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

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

chi := DirichletCharacter("72.25");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 72.i (of order $$3$$, 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: $$0.574922894553$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (2 \zeta_{6} - 1) q^{3} + ( - \zeta_{6} + 1) q^{5} + 3 \zeta_{6} q^{7} - 3 q^{9}+O(q^{10})$$ q + (2*z - 1) * q^3 + (-z + 1) * q^5 + 3*z * q^7 - 3 * q^9 $$q + (2 \zeta_{6} - 1) q^{3} + ( - \zeta_{6} + 1) q^{5} + 3 \zeta_{6} q^{7} - 3 q^{9} - 5 \zeta_{6} q^{11} + ( - 5 \zeta_{6} + 5) q^{13} + (\zeta_{6} + 1) q^{15} - 2 q^{17} - 4 q^{19} + (3 \zeta_{6} - 6) q^{21} + ( - \zeta_{6} + 1) q^{23} + 4 \zeta_{6} q^{25} + ( - 6 \zeta_{6} + 3) q^{27} + 9 \zeta_{6} q^{29} + ( - \zeta_{6} + 1) q^{31} + ( - 5 \zeta_{6} + 10) q^{33} + 3 q^{35} - 6 q^{37} + (5 \zeta_{6} + 5) q^{39} + (3 \zeta_{6} - 3) q^{41} - \zeta_{6} q^{43} + (3 \zeta_{6} - 3) q^{45} + 3 \zeta_{6} q^{47} + (2 \zeta_{6} - 2) q^{49} + ( - 4 \zeta_{6} + 2) q^{51} + 2 q^{53} - 5 q^{55} + ( - 8 \zeta_{6} + 4) q^{57} + (11 \zeta_{6} - 11) q^{59} - 7 \zeta_{6} q^{61} - 9 \zeta_{6} q^{63} - 5 \zeta_{6} q^{65} + ( - \zeta_{6} + 1) q^{67} + (\zeta_{6} + 1) q^{69} + 4 q^{71} - 2 q^{73} + (4 \zeta_{6} - 8) q^{75} + ( - 15 \zeta_{6} + 15) q^{77} - \zeta_{6} q^{79} + 9 q^{81} - \zeta_{6} q^{83} + (2 \zeta_{6} - 2) q^{85} + (9 \zeta_{6} - 18) q^{87} - 18 q^{89} + 15 q^{91} + (\zeta_{6} + 1) q^{93} + (4 \zeta_{6} - 4) q^{95} + 13 \zeta_{6} q^{97} + 15 \zeta_{6} q^{99} +O(q^{100})$$ q + (2*z - 1) * q^3 + (-z + 1) * q^5 + 3*z * q^7 - 3 * q^9 - 5*z * q^11 + (-5*z + 5) * q^13 + (z + 1) * q^15 - 2 * q^17 - 4 * q^19 + (3*z - 6) * q^21 + (-z + 1) * q^23 + 4*z * q^25 + (-6*z + 3) * q^27 + 9*z * q^29 + (-z + 1) * q^31 + (-5*z + 10) * q^33 + 3 * q^35 - 6 * q^37 + (5*z + 5) * q^39 + (3*z - 3) * q^41 - z * q^43 + (3*z - 3) * q^45 + 3*z * q^47 + (2*z - 2) * q^49 + (-4*z + 2) * q^51 + 2 * q^53 - 5 * q^55 + (-8*z + 4) * q^57 + (11*z - 11) * q^59 - 7*z * q^61 - 9*z * q^63 - 5*z * q^65 + (-z + 1) * q^67 + (z + 1) * q^69 + 4 * q^71 - 2 * q^73 + (4*z - 8) * q^75 + (-15*z + 15) * q^77 - z * q^79 + 9 * q^81 - z * q^83 + (2*z - 2) * q^85 + (9*z - 18) * q^87 - 18 * q^89 + 15 * q^91 + (z + 1) * q^93 + (4*z - 4) * q^95 + 13*z * q^97 + 15*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{5} + 3 q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q + q^5 + 3 * q^7 - 6 * q^9 $$2 q + q^{5} + 3 q^{7} - 6 q^{9} - 5 q^{11} + 5 q^{13} + 3 q^{15} - 4 q^{17} - 8 q^{19} - 9 q^{21} + q^{23} + 4 q^{25} + 9 q^{29} + q^{31} + 15 q^{33} + 6 q^{35} - 12 q^{37} + 15 q^{39} - 3 q^{41} - q^{43} - 3 q^{45} + 3 q^{47} - 2 q^{49} + 4 q^{53} - 10 q^{55} - 11 q^{59} - 7 q^{61} - 9 q^{63} - 5 q^{65} + q^{67} + 3 q^{69} + 8 q^{71} - 4 q^{73} - 12 q^{75} + 15 q^{77} - q^{79} + 18 q^{81} - q^{83} - 2 q^{85} - 27 q^{87} - 36 q^{89} + 30 q^{91} + 3 q^{93} - 4 q^{95} + 13 q^{97} + 15 q^{99}+O(q^{100})$$ 2 * q + q^5 + 3 * q^7 - 6 * q^9 - 5 * q^11 + 5 * q^13 + 3 * q^15 - 4 * q^17 - 8 * q^19 - 9 * q^21 + q^23 + 4 * q^25 + 9 * q^29 + q^31 + 15 * q^33 + 6 * q^35 - 12 * q^37 + 15 * q^39 - 3 * q^41 - q^43 - 3 * q^45 + 3 * q^47 - 2 * q^49 + 4 * q^53 - 10 * q^55 - 11 * q^59 - 7 * q^61 - 9 * q^63 - 5 * q^65 + q^67 + 3 * q^69 + 8 * q^71 - 4 * q^73 - 12 * q^75 + 15 * q^77 - q^79 + 18 * q^81 - q^83 - 2 * q^85 - 27 * q^87 - 36 * q^89 + 30 * q^91 + 3 * q^93 - 4 * q^95 + 13 * q^97 + 15 * q^99

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
25.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.73205i 0 0.500000 + 0.866025i 0 1.50000 2.59808i 0 −3.00000 0
49.1 0 1.73205i 0 0.500000 0.866025i 0 1.50000 + 2.59808i 0 −3.00000 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.2.i.a 2
3.b odd 2 1 216.2.i.a 2
4.b odd 2 1 144.2.i.b 2
8.b even 2 1 576.2.i.d 2
8.d odd 2 1 576.2.i.c 2
9.c even 3 1 inner 72.2.i.a 2
9.c even 3 1 648.2.a.a 1
9.d odd 6 1 216.2.i.a 2
9.d odd 6 1 648.2.a.c 1
12.b even 2 1 432.2.i.a 2
24.f even 2 1 1728.2.i.g 2
24.h odd 2 1 1728.2.i.h 2
36.f odd 6 1 144.2.i.b 2
36.f odd 6 1 1296.2.a.e 1
36.h even 6 1 432.2.i.a 2
36.h even 6 1 1296.2.a.i 1
72.j odd 6 1 1728.2.i.h 2
72.j odd 6 1 5184.2.a.i 1
72.l even 6 1 1728.2.i.g 2
72.l even 6 1 5184.2.a.n 1
72.n even 6 1 576.2.i.d 2
72.n even 6 1 5184.2.a.s 1
72.p odd 6 1 576.2.i.c 2
72.p odd 6 1 5184.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.i.a 2 1.a even 1 1 trivial
72.2.i.a 2 9.c even 3 1 inner
144.2.i.b 2 4.b odd 2 1
144.2.i.b 2 36.f odd 6 1
216.2.i.a 2 3.b odd 2 1
216.2.i.a 2 9.d odd 6 1
432.2.i.a 2 12.b even 2 1
432.2.i.a 2 36.h even 6 1
576.2.i.c 2 8.d odd 2 1
576.2.i.c 2 72.p odd 6 1
576.2.i.d 2 8.b even 2 1
576.2.i.d 2 72.n even 6 1
648.2.a.a 1 9.c even 3 1
648.2.a.c 1 9.d odd 6 1
1296.2.a.e 1 36.f odd 6 1
1296.2.a.i 1 36.h even 6 1
1728.2.i.g 2 24.f even 2 1
1728.2.i.g 2 72.l even 6 1
1728.2.i.h 2 24.h odd 2 1
1728.2.i.h 2 72.j odd 6 1
5184.2.a.i 1 72.j odd 6 1
5184.2.a.n 1 72.l even 6 1
5184.2.a.s 1 72.n even 6 1
5184.2.a.x 1 72.p odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - T_{5} + 1$$ acting on $$S_{2}^{\mathrm{new}}(72, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - 3T + 9$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$(T + 2)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - T + 1$$
$29$ $$T^{2} - 9T + 81$$
$31$ $$T^{2} - T + 1$$
$37$ $$(T + 6)^{2}$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} + T + 1$$
$47$ $$T^{2} - 3T + 9$$
$53$ $$(T - 2)^{2}$$
$59$ $$T^{2} + 11T + 121$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - T + 1$$
$71$ $$(T - 4)^{2}$$
$73$ $$(T + 2)^{2}$$
$79$ $$T^{2} + T + 1$$
$83$ $$T^{2} + T + 1$$
$89$ $$(T + 18)^{2}$$
$97$ $$T^{2} - 13T + 169$$