# Properties

 Label 36.9.d.a Level $36$ Weight $9$ Character orbit 36.d Self dual yes Analytic conductor $14.666$ Analytic rank $0$ Dimension $1$ CM discriminant -4 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [36,9,Mod(19,36)]

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, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("36.19");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 36.d (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: yes Analytic conductor: $$14.6656299622$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 4) 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)

 $$f(q)$$ $$=$$ $$q - 16 q^{2} + 256 q^{4} + 1054 q^{5} - 4096 q^{8}+O(q^{10})$$ q - 16 * q^2 + 256 * q^4 + 1054 * q^5 - 4096 * q^8 $$q - 16 q^{2} + 256 q^{4} + 1054 q^{5} - 4096 q^{8} - 16864 q^{10} - 478 q^{13} + 65536 q^{16} + 63358 q^{17} + 269824 q^{20} + 720291 q^{25} + 7648 q^{26} + 1407838 q^{29} - 1048576 q^{32} - 1013728 q^{34} + 925922 q^{37} - 4317184 q^{40} - 3577922 q^{41} + 5764801 q^{49} - 11524656 q^{50} - 122368 q^{52} + 9620638 q^{53} - 22525408 q^{58} + 20722082 q^{61} + 16777216 q^{64} - 503812 q^{65} + 16219648 q^{68} - 54717118 q^{73} - 14814752 q^{74} + 69074944 q^{80} + 57246752 q^{82} + 66779332 q^{85} + 30265918 q^{89} - 173379838 q^{97} - 92236816 q^{98}+O(q^{100})$$ q - 16 * q^2 + 256 * q^4 + 1054 * q^5 - 4096 * q^8 - 16864 * q^10 - 478 * q^13 + 65536 * q^16 + 63358 * q^17 + 269824 * q^20 + 720291 * q^25 + 7648 * q^26 + 1407838 * q^29 - 1048576 * q^32 - 1013728 * q^34 + 925922 * q^37 - 4317184 * q^40 - 3577922 * q^41 + 5764801 * q^49 - 11524656 * q^50 - 122368 * q^52 + 9620638 * q^53 - 22525408 * q^58 + 20722082 * q^61 + 16777216 * q^64 - 503812 * q^65 + 16219648 * q^68 - 54717118 * q^73 - 14814752 * q^74 + 69074944 * q^80 + 57246752 * q^82 + 66779332 * q^85 + 30265918 * q^89 - 173379838 * q^97 - 92236816 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 0
−16.0000 0 256.000 1054.00 0 0 −4096.00 0 −16864.0
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 36.9.d.a 1
3.b odd 2 1 4.9.b.a 1
4.b odd 2 1 CM 36.9.d.a 1
12.b even 2 1 4.9.b.a 1
15.d odd 2 1 100.9.b.a 1
15.e even 4 2 100.9.d.a 2
24.f even 2 1 64.9.c.a 1
24.h odd 2 1 64.9.c.a 1
48.i odd 4 2 256.9.d.a 2
48.k even 4 2 256.9.d.a 2
60.h even 2 1 100.9.b.a 1
60.l odd 4 2 100.9.d.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.9.b.a 1 3.b odd 2 1
4.9.b.a 1 12.b even 2 1
36.9.d.a 1 1.a even 1 1 trivial
36.9.d.a 1 4.b odd 2 1 CM
64.9.c.a 1 24.f even 2 1
64.9.c.a 1 24.h odd 2 1
100.9.b.a 1 15.d odd 2 1
100.9.b.a 1 60.h even 2 1
100.9.d.a 2 15.e even 4 2
100.9.d.a 2 60.l odd 4 2
256.9.d.a 2 48.i odd 4 2
256.9.d.a 2 48.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 1054$$ acting on $$S_{9}^{\mathrm{new}}(36, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 16$$
$3$ $$T$$
$5$ $$T - 1054$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T + 478$$
$17$ $$T - 63358$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T - 1407838$$
$31$ $$T$$
$37$ $$T - 925922$$
$41$ $$T + 3577922$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T - 9620638$$
$59$ $$T$$
$61$ $$T - 20722082$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T + 54717118$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 30265918$$
$97$ $$T + 173379838$$