# Properties

 Label 36.2.e.a Level $36$ Weight $2$ Character orbit 36.e Analytic conductor $0.287$ 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] = [36,2,Mod(13,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("36.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 36.e (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: $$0.287461447277$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - 2 \zeta_{6} + 1) q^{3} + (3 \zeta_{6} - 3) q^{5} + \zeta_{6} q^{7} - 3 q^{9}+O(q^{10})$$ q + (-2*z + 1) * q^3 + (3*z - 3) * q^5 + z * q^7 - 3 * q^9 $$q + ( - 2 \zeta_{6} + 1) q^{3} + (3 \zeta_{6} - 3) q^{5} + \zeta_{6} q^{7} - 3 q^{9} - 3 \zeta_{6} q^{11} + ( - \zeta_{6} + 1) q^{13} + (3 \zeta_{6} + 3) q^{15} + 6 q^{17} - 4 q^{19} + ( - \zeta_{6} + 2) q^{21} + ( - 3 \zeta_{6} + 3) q^{23} - 4 \zeta_{6} q^{25} + (6 \zeta_{6} - 3) q^{27} - 3 \zeta_{6} q^{29} + (5 \zeta_{6} - 5) q^{31} + (3 \zeta_{6} - 6) q^{33} - 3 q^{35} + 2 q^{37} + ( - \zeta_{6} - 1) q^{39} + (3 \zeta_{6} - 3) q^{41} + \zeta_{6} q^{43} + ( - 9 \zeta_{6} + 9) q^{45} + 9 \zeta_{6} q^{47} + ( - 6 \zeta_{6} + 6) q^{49} + ( - 12 \zeta_{6} + 6) q^{51} - 6 q^{53} + 9 q^{55} + (8 \zeta_{6} - 4) q^{57} + ( - 3 \zeta_{6} + 3) q^{59} + 13 \zeta_{6} q^{61} - 3 \zeta_{6} q^{63} + 3 \zeta_{6} q^{65} + ( - 7 \zeta_{6} + 7) q^{67} + ( - 3 \zeta_{6} - 3) q^{69} - 12 q^{71} - 10 q^{73} + (4 \zeta_{6} - 8) q^{75} + ( - 3 \zeta_{6} + 3) q^{77} - 11 \zeta_{6} q^{79} + 9 q^{81} + 9 \zeta_{6} q^{83} + (18 \zeta_{6} - 18) q^{85} + (3 \zeta_{6} - 6) q^{87} + 6 q^{89} + q^{91} + (5 \zeta_{6} + 5) q^{93} + ( - 12 \zeta_{6} + 12) q^{95} - 11 \zeta_{6} q^{97} + 9 \zeta_{6} q^{99} +O(q^{100})$$ q + (-2*z + 1) * q^3 + (3*z - 3) * q^5 + z * q^7 - 3 * q^9 - 3*z * q^11 + (-z + 1) * q^13 + (3*z + 3) * q^15 + 6 * q^17 - 4 * q^19 + (-z + 2) * q^21 + (-3*z + 3) * q^23 - 4*z * q^25 + (6*z - 3) * q^27 - 3*z * q^29 + (5*z - 5) * q^31 + (3*z - 6) * q^33 - 3 * q^35 + 2 * q^37 + (-z - 1) * q^39 + (3*z - 3) * q^41 + z * q^43 + (-9*z + 9) * q^45 + 9*z * q^47 + (-6*z + 6) * q^49 + (-12*z + 6) * q^51 - 6 * q^53 + 9 * q^55 + (8*z - 4) * q^57 + (-3*z + 3) * q^59 + 13*z * q^61 - 3*z * q^63 + 3*z * q^65 + (-7*z + 7) * q^67 + (-3*z - 3) * q^69 - 12 * q^71 - 10 * q^73 + (4*z - 8) * q^75 + (-3*z + 3) * q^77 - 11*z * q^79 + 9 * q^81 + 9*z * q^83 + (18*z - 18) * q^85 + (3*z - 6) * q^87 + 6 * q^89 + q^91 + (5*z + 5) * q^93 + (-12*z + 12) * q^95 - 11*z * q^97 + 9*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{5} + q^{7} - 6 q^{9}+O(q^{10})$$ 2 * q - 3 * q^5 + q^7 - 6 * q^9 $$2 q - 3 q^{5} + q^{7} - 6 q^{9} - 3 q^{11} + q^{13} + 9 q^{15} + 12 q^{17} - 8 q^{19} + 3 q^{21} + 3 q^{23} - 4 q^{25} - 3 q^{29} - 5 q^{31} - 9 q^{33} - 6 q^{35} + 4 q^{37} - 3 q^{39} - 3 q^{41} + q^{43} + 9 q^{45} + 9 q^{47} + 6 q^{49} - 12 q^{53} + 18 q^{55} + 3 q^{59} + 13 q^{61} - 3 q^{63} + 3 q^{65} + 7 q^{67} - 9 q^{69} - 24 q^{71} - 20 q^{73} - 12 q^{75} + 3 q^{77} - 11 q^{79} + 18 q^{81} + 9 q^{83} - 18 q^{85} - 9 q^{87} + 12 q^{89} + 2 q^{91} + 15 q^{93} + 12 q^{95} - 11 q^{97} + 9 q^{99}+O(q^{100})$$ 2 * q - 3 * q^5 + q^7 - 6 * q^9 - 3 * q^11 + q^13 + 9 * q^15 + 12 * q^17 - 8 * q^19 + 3 * q^21 + 3 * q^23 - 4 * q^25 - 3 * q^29 - 5 * q^31 - 9 * q^33 - 6 * q^35 + 4 * q^37 - 3 * q^39 - 3 * q^41 + q^43 + 9 * q^45 + 9 * q^47 + 6 * q^49 - 12 * q^53 + 18 * q^55 + 3 * q^59 + 13 * q^61 - 3 * q^63 + 3 * q^65 + 7 * q^67 - 9 * q^69 - 24 * q^71 - 20 * q^73 - 12 * q^75 + 3 * q^77 - 11 * q^79 + 18 * q^81 + 9 * q^83 - 18 * q^85 - 9 * q^87 + 12 * q^89 + 2 * q^91 + 15 * q^93 + 12 * q^95 - 11 * q^97 + 9 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/36\mathbb{Z}\right)^\times$$.

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.73205i 0 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 −3.00000 0
25.1 0 1.73205i 0 −1.50000 2.59808i 0 0.500000 0.866025i 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 36.2.e.a 2
3.b odd 2 1 108.2.e.a 2
4.b odd 2 1 144.2.i.a 2
5.b even 2 1 900.2.i.b 2
5.c odd 4 2 900.2.s.b 4
7.b odd 2 1 1764.2.j.b 2
7.c even 3 1 1764.2.i.a 2
7.c even 3 1 1764.2.l.c 2
7.d odd 6 1 1764.2.i.c 2
7.d odd 6 1 1764.2.l.a 2
8.b even 2 1 576.2.i.f 2
8.d odd 2 1 576.2.i.e 2
9.c even 3 1 inner 36.2.e.a 2
9.c even 3 1 324.2.a.c 1
9.d odd 6 1 108.2.e.a 2
9.d odd 6 1 324.2.a.a 1
12.b even 2 1 432.2.i.c 2
15.d odd 2 1 2700.2.i.b 2
15.e even 4 2 2700.2.s.b 4
21.c even 2 1 5292.2.j.a 2
21.g even 6 1 5292.2.i.a 2
21.g even 6 1 5292.2.l.c 2
21.h odd 6 1 5292.2.i.c 2
21.h odd 6 1 5292.2.l.a 2
24.f even 2 1 1728.2.i.c 2
24.h odd 2 1 1728.2.i.d 2
36.f odd 6 1 144.2.i.a 2
36.f odd 6 1 1296.2.a.k 1
36.h even 6 1 432.2.i.c 2
36.h even 6 1 1296.2.a.b 1
45.h odd 6 1 2700.2.i.b 2
45.h odd 6 1 8100.2.a.g 1
45.j even 6 1 900.2.i.b 2
45.j even 6 1 8100.2.a.j 1
45.k odd 12 2 900.2.s.b 4
45.k odd 12 2 8100.2.d.h 2
45.l even 12 2 2700.2.s.b 4
45.l even 12 2 8100.2.d.c 2
63.g even 3 1 1764.2.i.a 2
63.h even 3 1 1764.2.l.c 2
63.i even 6 1 5292.2.l.c 2
63.j odd 6 1 5292.2.l.a 2
63.k odd 6 1 1764.2.i.c 2
63.l odd 6 1 1764.2.j.b 2
63.n odd 6 1 5292.2.i.c 2
63.o even 6 1 5292.2.j.a 2
63.s even 6 1 5292.2.i.a 2
63.t odd 6 1 1764.2.l.a 2
72.j odd 6 1 1728.2.i.d 2
72.j odd 6 1 5184.2.a.ba 1
72.l even 6 1 1728.2.i.c 2
72.l even 6 1 5184.2.a.bb 1
72.n even 6 1 576.2.i.f 2
72.n even 6 1 5184.2.a.e 1
72.p odd 6 1 576.2.i.e 2
72.p odd 6 1 5184.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.e.a 2 1.a even 1 1 trivial
36.2.e.a 2 9.c even 3 1 inner
108.2.e.a 2 3.b odd 2 1
108.2.e.a 2 9.d odd 6 1
144.2.i.a 2 4.b odd 2 1
144.2.i.a 2 36.f odd 6 1
324.2.a.a 1 9.d odd 6 1
324.2.a.c 1 9.c even 3 1
432.2.i.c 2 12.b even 2 1
432.2.i.c 2 36.h even 6 1
576.2.i.e 2 8.d odd 2 1
576.2.i.e 2 72.p odd 6 1
576.2.i.f 2 8.b even 2 1
576.2.i.f 2 72.n even 6 1
900.2.i.b 2 5.b even 2 1
900.2.i.b 2 45.j even 6 1
900.2.s.b 4 5.c odd 4 2
900.2.s.b 4 45.k odd 12 2
1296.2.a.b 1 36.h even 6 1
1296.2.a.k 1 36.f odd 6 1
1728.2.i.c 2 24.f even 2 1
1728.2.i.c 2 72.l even 6 1
1728.2.i.d 2 24.h odd 2 1
1728.2.i.d 2 72.j odd 6 1
1764.2.i.a 2 7.c even 3 1
1764.2.i.a 2 63.g even 3 1
1764.2.i.c 2 7.d odd 6 1
1764.2.i.c 2 63.k odd 6 1
1764.2.j.b 2 7.b odd 2 1
1764.2.j.b 2 63.l odd 6 1
1764.2.l.a 2 7.d odd 6 1
1764.2.l.a 2 63.t odd 6 1
1764.2.l.c 2 7.c even 3 1
1764.2.l.c 2 63.h even 3 1
2700.2.i.b 2 15.d odd 2 1
2700.2.i.b 2 45.h odd 6 1
2700.2.s.b 4 15.e even 4 2
2700.2.s.b 4 45.l even 12 2
5184.2.a.e 1 72.n even 6 1
5184.2.a.f 1 72.p odd 6 1
5184.2.a.ba 1 72.j odd 6 1
5184.2.a.bb 1 72.l even 6 1
5292.2.i.a 2 21.g even 6 1
5292.2.i.a 2 63.s even 6 1
5292.2.i.c 2 21.h odd 6 1
5292.2.i.c 2 63.n odd 6 1
5292.2.j.a 2 21.c even 2 1
5292.2.j.a 2 63.o even 6 1
5292.2.l.a 2 21.h odd 6 1
5292.2.l.a 2 63.j odd 6 1
5292.2.l.c 2 21.g even 6 1
5292.2.l.c 2 63.i even 6 1
8100.2.a.g 1 45.h odd 6 1
8100.2.a.j 1 45.j even 6 1
8100.2.d.c 2 45.l even 12 2
8100.2.d.h 2 45.k odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2} + 3T + 9$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} - T + 1$$
$17$ $$(T - 6)^{2}$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} - 3T + 9$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$T^{2} + 5T + 25$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} + 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2} - 9T + 81$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} - 13T + 169$$
$67$ $$T^{2} - 7T + 49$$
$71$ $$(T + 12)^{2}$$
$73$ $$(T + 10)^{2}$$
$79$ $$T^{2} + 11T + 121$$
$83$ $$T^{2} - 9T + 81$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 11T + 121$$