# Properties

 Label 36.2.b.a Level $36$ Weight $2$ Character orbit 36.b Analytic conductor $0.287$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,2,Mod(35,36)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(36, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

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

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

chi := DirichletCharacter("36.35");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 36.b (of order $$2$$, degree $$1$$, 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: $$\Q(\sqrt{-2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 2$$ x^2 + 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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 $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 2 q^{4} - \beta q^{5} - 2 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 2 * q^4 - b * q^5 - 2*b * q^8 $$q + \beta q^{2} - 2 q^{4} - \beta q^{5} - 2 \beta q^{8} + 2 q^{10} - 4 q^{13} + 4 q^{16} + 5 \beta q^{17} + 2 \beta q^{20} + 3 q^{25} - 4 \beta q^{26} - 7 \beta q^{29} + 4 \beta q^{32} - 10 q^{34} + 2 q^{37} - 4 q^{40} - \beta q^{41} + 7 q^{49} + 3 \beta q^{50} + 8 q^{52} + 5 \beta q^{53} + 14 q^{58} - 10 q^{61} - 8 q^{64} + 4 \beta q^{65} - 10 \beta q^{68} - 16 q^{73} + 2 \beta q^{74} - 4 \beta q^{80} + 2 q^{82} + 10 q^{85} - 13 \beta q^{89} + 8 q^{97} + 7 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 2 * q^4 - b * q^5 - 2*b * q^8 + 2 * q^10 - 4 * q^13 + 4 * q^16 + 5*b * q^17 + 2*b * q^20 + 3 * q^25 - 4*b * q^26 - 7*b * q^29 + 4*b * q^32 - 10 * q^34 + 2 * q^37 - 4 * q^40 - b * q^41 + 7 * q^49 + 3*b * q^50 + 8 * q^52 + 5*b * q^53 + 14 * q^58 - 10 * q^61 - 8 * q^64 + 4*b * q^65 - 10*b * q^68 - 16 * q^73 + 2*b * q^74 - 4*b * q^80 + 2 * q^82 + 10 * q^85 - 13*b * q^89 + 8 * q^97 + 7*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4}+O(q^{10})$$ 2 * q - 4 * q^4 $$2 q - 4 q^{4} + 4 q^{10} - 8 q^{13} + 8 q^{16} + 6 q^{25} - 20 q^{34} + 4 q^{37} - 8 q^{40} + 14 q^{49} + 16 q^{52} + 28 q^{58} - 20 q^{61} - 16 q^{64} - 32 q^{73} + 4 q^{82} + 20 q^{85} + 16 q^{97}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^10 - 8 * q^13 + 8 * q^16 + 6 * q^25 - 20 * q^34 + 4 * q^37 - 8 * q^40 + 14 * q^49 + 16 * q^52 + 28 * q^58 - 20 * q^61 - 16 * q^64 - 32 * q^73 + 4 * q^82 + 20 * q^85 + 16 * q^97

## Character values

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

 $$n$$ $$19$$ $$29$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
35.1
 − 1.41421i 1.41421i
1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 2.00000
35.2 1.41421i 0 −2.00000 1.41421i 0 0 2.82843i 0 2.00000
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
3.b odd 2 1 inner
12.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.2.b.a 2
3.b odd 2 1 inner 36.2.b.a 2
4.b odd 2 1 CM 36.2.b.a 2
5.b even 2 1 900.2.e.b 2
5.c odd 4 2 900.2.h.a 4
7.b odd 2 1 1764.2.e.b 2
8.b even 2 1 576.2.c.b 2
8.d odd 2 1 576.2.c.b 2
9.c even 3 2 324.2.h.c 4
9.d odd 6 2 324.2.h.c 4
12.b even 2 1 inner 36.2.b.a 2
15.d odd 2 1 900.2.e.b 2
15.e even 4 2 900.2.h.a 4
16.e even 4 2 2304.2.f.d 4
16.f odd 4 2 2304.2.f.d 4
20.d odd 2 1 900.2.e.b 2
20.e even 4 2 900.2.h.a 4
21.c even 2 1 1764.2.e.b 2
24.f even 2 1 576.2.c.b 2
24.h odd 2 1 576.2.c.b 2
28.d even 2 1 1764.2.e.b 2
36.f odd 6 2 324.2.h.c 4
36.h even 6 2 324.2.h.c 4
48.i odd 4 2 2304.2.f.d 4
48.k even 4 2 2304.2.f.d 4
60.h even 2 1 900.2.e.b 2
60.l odd 4 2 900.2.h.a 4
84.h odd 2 1 1764.2.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
36.2.b.a 2 1.a even 1 1 trivial
36.2.b.a 2 3.b odd 2 1 inner
36.2.b.a 2 4.b odd 2 1 CM
36.2.b.a 2 12.b even 2 1 inner
324.2.h.c 4 9.c even 3 2
324.2.h.c 4 9.d odd 6 2
324.2.h.c 4 36.f odd 6 2
324.2.h.c 4 36.h even 6 2
576.2.c.b 2 8.b even 2 1
576.2.c.b 2 8.d odd 2 1
576.2.c.b 2 24.f even 2 1
576.2.c.b 2 24.h odd 2 1
900.2.e.b 2 5.b even 2 1
900.2.e.b 2 15.d odd 2 1
900.2.e.b 2 20.d odd 2 1
900.2.e.b 2 60.h even 2 1
900.2.h.a 4 5.c odd 4 2
900.2.h.a 4 15.e even 4 2
900.2.h.a 4 20.e even 4 2
900.2.h.a 4 60.l odd 4 2
1764.2.e.b 2 7.b odd 2 1
1764.2.e.b 2 21.c even 2 1
1764.2.e.b 2 28.d even 2 1
1764.2.e.b 2 84.h odd 2 1
2304.2.f.d 4 16.e even 4 2
2304.2.f.d 4 16.f odd 4 2
2304.2.f.d 4 48.i odd 4 2
2304.2.f.d 4 48.k even 4 2

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$(T + 4)^{2}$$
$17$ $$T^{2} + 50$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} + 98$$
$31$ $$T^{2}$$
$37$ $$(T - 2)^{2}$$
$41$ $$T^{2} + 2$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 50$$
$59$ $$T^{2}$$
$61$ $$(T + 10)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$(T + 16)^{2}$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} + 338$$
$97$ $$(T - 8)^{2}$$