# Properties

 Label 144.9.e.d Level $144$ Weight $9$ Character orbit 144.e Analytic conductor $58.663$ 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] = [144,9,Mod(17,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("144.17");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 144.e (of order $$2$$, degree $$1$$, not 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: $$58.6625198488$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = 3\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 55 \beta q^{5} + 3532 q^{7}+O(q^{10})$$ q + 55*b * q^5 + 3532 * q^7 $$q + 55 \beta q^{5} + 3532 q^{7} + 4756 \beta q^{11} - 41824 q^{13} + 22341 \beta q^{17} + 36304 q^{19} - 97492 \beta q^{23} + 336175 q^{25} + 63449 \beta q^{29} + 471196 q^{31} + 194260 \beta q^{35} - 3007402 q^{37} - 404309 \beta q^{41} - 3623720 q^{43} + 1417660 \beta q^{47} + 6710223 q^{49} + 2422233 \beta q^{53} - 4708440 q^{55} - 633592 \beta q^{59} - 5440630 q^{61} - 2300320 \beta q^{65} + 6121576 q^{67} + 4995492 \beta q^{71} - 49031152 q^{73} + 16798192 \beta q^{77} - 8357756 q^{79} + 12113164 \beta q^{83} - 22117590 q^{85} + 25299627 \beta q^{89} - 147722368 q^{91} + 1996720 \beta q^{95} + 20431328 q^{97} +O(q^{100})$$ q + 55*b * q^5 + 3532 * q^7 + 4756*b * q^11 - 41824 * q^13 + 22341*b * q^17 + 36304 * q^19 - 97492*b * q^23 + 336175 * q^25 + 63449*b * q^29 + 471196 * q^31 + 194260*b * q^35 - 3007402 * q^37 - 404309*b * q^41 - 3623720 * q^43 + 1417660*b * q^47 + 6710223 * q^49 + 2422233*b * q^53 - 4708440 * q^55 - 633592*b * q^59 - 5440630 * q^61 - 2300320*b * q^65 + 6121576 * q^67 + 4995492*b * q^71 - 49031152 * q^73 + 16798192*b * q^77 - 8357756 * q^79 + 12113164*b * q^83 - 22117590 * q^85 + 25299627*b * q^89 - 147722368 * q^91 + 1996720*b * q^95 + 20431328 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 7064 q^{7}+O(q^{10})$$ 2 * q + 7064 * q^7 $$2 q + 7064 q^{7} - 83648 q^{13} + 72608 q^{19} + 672350 q^{25} + 942392 q^{31} - 6014804 q^{37} - 7247440 q^{43} + 13420446 q^{49} - 9416880 q^{55} - 10881260 q^{61} + 12243152 q^{67} - 98062304 q^{73} - 16715512 q^{79} - 44235180 q^{85} - 295444736 q^{91} + 40862656 q^{97}+O(q^{100})$$ 2 * q + 7064 * q^7 - 83648 * q^13 + 72608 * q^19 + 672350 * q^25 + 942392 * q^31 - 6014804 * q^37 - 7247440 * q^43 + 13420446 * q^49 - 9416880 * q^55 - 10881260 * q^61 + 12243152 * q^67 - 98062304 * q^73 - 16715512 * q^79 - 44235180 * q^85 - 295444736 * q^91 + 40862656 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/144\mathbb{Z}\right)^\times$$.

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 − 1.41421i 1.41421i
0 0 0 233.345i 0 3532.00 0 0 0
17.2 0 0 0 233.345i 0 3532.00 0 0 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.9.e.d 2
3.b odd 2 1 inner 144.9.e.d 2
4.b odd 2 1 18.9.b.a 2
12.b even 2 1 18.9.b.a 2
20.d odd 2 1 450.9.d.b 2
20.e even 4 2 450.9.b.a 4
36.f odd 6 2 162.9.d.d 4
36.h even 6 2 162.9.d.d 4
60.h even 2 1 450.9.d.b 2
60.l odd 4 2 450.9.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.9.b.a 2 4.b odd 2 1
18.9.b.a 2 12.b even 2 1
144.9.e.d 2 1.a even 1 1 trivial
144.9.e.d 2 3.b odd 2 1 inner
162.9.d.d 4 36.f odd 6 2
162.9.d.d 4 36.h even 6 2
450.9.b.a 4 20.e even 4 2
450.9.b.a 4 60.l odd 4 2
450.9.d.b 2 20.d odd 2 1
450.9.d.b 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 54450$$ acting on $$S_{9}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 54450$$
$7$ $$(T - 3532)^{2}$$
$11$ $$T^{2} + 407151648$$
$13$ $$(T + 41824)^{2}$$
$17$ $$T^{2} + 8984165058$$
$19$ $$(T - 36304)^{2}$$
$23$ $$T^{2} + 171084421152$$
$29$ $$T^{2} + 72463960818$$
$31$ $$(T - 471196)^{2}$$
$37$ $$(T + 3007402)^{2}$$
$41$ $$T^{2} + 2942383814658$$
$43$ $$(T + 3623720)^{2}$$
$47$ $$T^{2} + 36175677760800$$
$53$ $$T^{2} + 105609828713202$$
$59$ $$T^{2} + 7225898804352$$
$61$ $$(T + 5440630)^{2}$$
$67$ $$(T - 6121576)^{2}$$
$71$ $$T^{2} + 449188925797152$$
$73$ $$(T + 49031152)^{2}$$
$79$ $$(T + 8357756)^{2}$$
$83$ $$T^{2} + 26\!\cdots\!28$$
$89$ $$T^{2} + 11\!\cdots\!22$$
$97$ $$(T - 20431328)^{2}$$