# Properties

 Label 144.9.g.g Level $144$ Weight $9$ Character orbit 144.g 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(127,144)]

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([1, 0, 0]))

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

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

chi := DirichletCharacter("144.127");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 144.g (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: $$58.6625198488$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 9$$ x^2 - x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}\cdot 3^{3}$$ Twist minimal: no (minimal twist has level 16) 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 $$\beta = 216\sqrt{-35}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 510 q^{5} - 2 \beta q^{7} +O(q^{10})$$ q + 510 * q^5 - 2*b * q^7 $$q + 510 q^{5} - 2 \beta q^{7} - 15 \beta q^{11} - 27710 q^{13} - 50370 q^{17} + 85 \beta q^{19} - 138 \beta q^{23} - 130525 q^{25} - 54978 q^{29} + 920 \beta q^{31} - 1020 \beta q^{35} + 793730 q^{37} + 75582 q^{41} + 391 \beta q^{43} - 2244 \beta q^{47} - 767039 q^{49} - 11166210 q^{53} - 7650 \beta q^{55} + 17085 \beta q^{59} - 23826622 q^{61} - 14132100 q^{65} + 5865 \beta q^{67} + 7890 \beta q^{71} + 6516610 q^{73} - 48988800 q^{77} - 38180 \beta q^{79} - 57477 \beta q^{83} - 25688700 q^{85} - 86795778 q^{89} + 55420 \beta q^{91} + 43350 \beta q^{95} - 46670270 q^{97} +O(q^{100})$$ q + 510 * q^5 - 2*b * q^7 - 15*b * q^11 - 27710 * q^13 - 50370 * q^17 + 85*b * q^19 - 138*b * q^23 - 130525 * q^25 - 54978 * q^29 + 920*b * q^31 - 1020*b * q^35 + 793730 * q^37 + 75582 * q^41 + 391*b * q^43 - 2244*b * q^47 - 767039 * q^49 - 11166210 * q^53 - 7650*b * q^55 + 17085*b * q^59 - 23826622 * q^61 - 14132100 * q^65 + 5865*b * q^67 + 7890*b * q^71 + 6516610 * q^73 - 48988800 * q^77 - 38180*b * q^79 - 57477*b * q^83 - 25688700 * q^85 - 86795778 * q^89 + 55420*b * q^91 + 43350*b * q^95 - 46670270 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 1020 q^{5}+O(q^{10})$$ 2 * q + 1020 * q^5 $$2 q + 1020 q^{5} - 55420 q^{13} - 100740 q^{17} - 261050 q^{25} - 109956 q^{29} + 1587460 q^{37} + 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} - 47653244 q^{61} - 28264200 q^{65} + 13033220 q^{73} - 97977600 q^{77} - 51377400 q^{85} - 173591556 q^{89} - 93340540 q^{97}+O(q^{100})$$ 2 * q + 1020 * q^5 - 55420 * q^13 - 100740 * q^17 - 261050 * q^25 - 109956 * q^29 + 1587460 * q^37 + 151164 * q^41 - 1534078 * q^49 - 22332420 * q^53 - 47653244 * q^61 - 28264200 * q^65 + 13033220 * q^73 - 97977600 * q^77 - 51377400 * q^85 - 173591556 * q^89 - 93340540 * 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1
 0.5 + 2.95804i 0.5 − 2.95804i
0 0 0 510.000 0 2555.75i 0 0 0
127.2 0 0 0 510.000 0 2555.75i 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
4.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.g.g 2
3.b odd 2 1 16.9.c.a 2
4.b odd 2 1 inner 144.9.g.g 2
12.b even 2 1 16.9.c.a 2
15.d odd 2 1 400.9.b.c 2
15.e even 4 2 400.9.h.b 4
24.f even 2 1 64.9.c.d 2
24.h odd 2 1 64.9.c.d 2
48.i odd 4 2 256.9.d.f 4
48.k even 4 2 256.9.d.f 4
60.h even 2 1 400.9.b.c 2
60.l odd 4 2 400.9.h.b 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5} - 510$$ acting on $$S_{9}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 510)^{2}$$
$7$ $$T^{2} + 6531840$$
$11$ $$T^{2} + 367416000$$
$13$ $$(T + 27710)^{2}$$
$17$ $$(T + 50370)^{2}$$
$19$ $$T^{2} + 11798136000$$
$23$ $$T^{2} + 31098090240$$
$29$ $$(T + 54978)^{2}$$
$31$ $$T^{2} + 1382137344000$$
$37$ $$(T - 793730)^{2}$$
$41$ $$(T - 75582)^{2}$$
$43$ $$T^{2} + 249648557760$$
$47$ $$T^{2} + 8222828866560$$
$53$ $$(T + 11166210)^{2}$$
$59$ $$T^{2} + \cdots + 476656492536000$$
$61$ $$(T + 23826622)^{2}$$
$67$ $$T^{2} + 56170925496000$$
$71$ $$T^{2} + \cdots + 101655189216000$$
$73$ $$(T - 6516610)^{2}$$
$79$ $$T^{2} + 23\!\cdots\!00$$
$83$ $$T^{2} + 53\!\cdots\!40$$
$89$ $$(T + 86795778)^{2}$$
$97$ $$(T + 46670270)^{2}$$