# Properties

 Label 144.9.g.i Level $144$ Weight $9$ Character orbit 144.g Analytic conductor $58.663$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{1801})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} + 451x^{2} + 450x + 202500$$ x^4 - x^3 + 451*x^2 + 450*x + 202500 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{12}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 48) 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 66) q^{5} + ( - \beta_{3} - 193 \beta_1) q^{7}+O(q^{10})$$ q + (b2 + 66) * q^5 + (-b3 - 193*b1) * q^7 $$q + (\beta_{2} + 66) q^{5} + ( - \beta_{3} - 193 \beta_1) q^{7} + ( - 2 \beta_{3} + 1545 \beta_1) q^{11} + (30 \beta_{2} + 3658) q^{13} + (62 \beta_{2} - 83226) q^{17} + ( - 118 \beta_{3} - 1739 \beta_1) q^{19} + ( - 126 \beta_{3} + 18144 \beta_1) q^{23} + (132 \beta_{2} + 651107) q^{25} + ( - 409 \beta_{2} - 585894) q^{29} + (343 \beta_{3} - 149355 \beta_1) q^{31} + (706 \beta_{3} + 246606 \beta_1) q^{35} + ( - 1836 \beta_{2} + 1078946) q^{37} + (882 \beta_{2} - 2258874) q^{41} + (1814 \beta_{3} + 245223 \beta_1) q^{43} + ( - 194 \beta_{3} + 166038 \beta_1) q^{47} + (4632 \beta_{2} + 864721) q^{49} + ( - 1569 \beta_{2} + 10546650) q^{53} + ( - 6312 \beta_{3} + 620658 \beta_1) q^{55} + ( - 3112 \beta_{3} + 434775 \beta_1) q^{59} + ( - 5712 \beta_{2} - 12037102) q^{61} + (5638 \beta_{2} + 31362708) q^{65} + ( - 5592 \beta_{3} + 3619819 \beta_1) q^{67} + ( - 5462 \beta_{3} + 4163316 \beta_1) q^{71} + (25944 \beta_{2} - 5303870) q^{73} + ( - 13908 \beta_{2} + 8088624) q^{77} + (26507 \beta_{3} + 1447933 \beta_1) q^{79} + (32510 \beta_{3} + 4030959 \beta_1) q^{83} + ( - 79134 \beta_{2} + 58824396) q^{85} + (33220 \beta_{2} + 3168894) q^{89} + (19502 \beta_{3} + 7074326 \beta_1) q^{91} + ( - 832 \beta_{3} + 30487818 \beta_1) q^{95} + (59532 \beta_{2} + 65788450) q^{97}+O(q^{100})$$ q + (b2 + 66) * q^5 + (-b3 - 193*b1) * q^7 + (-2*b3 + 1545*b1) * q^11 + (30*b2 + 3658) * q^13 + (62*b2 - 83226) * q^17 + (-118*b3 - 1739*b1) * q^19 + (-126*b3 + 18144*b1) * q^23 + (132*b2 + 651107) * q^25 + (-409*b2 - 585894) * q^29 + (343*b3 - 149355*b1) * q^31 + (706*b3 + 246606*b1) * q^35 + (-1836*b2 + 1078946) * q^37 + (882*b2 - 2258874) * q^41 + (1814*b3 + 245223*b1) * q^43 + (-194*b3 + 166038*b1) * q^47 + (4632*b2 + 864721) * q^49 + (-1569*b2 + 10546650) * q^53 + (-6312*b3 + 620658*b1) * q^55 + (-3112*b3 + 434775*b1) * q^59 + (-5712*b2 - 12037102) * q^61 + (5638*b2 + 31362708) * q^65 + (-5592*b3 + 3619819*b1) * q^67 + (-5462*b3 + 4163316*b1) * q^71 + (25944*b2 - 5303870) * q^73 + (-13908*b2 + 8088624) * q^77 + (26507*b3 + 1447933*b1) * q^79 + (32510*b3 + 4030959*b1) * q^83 + (-79134*b2 + 58824396) * q^85 + (33220*b2 + 3168894) * q^89 + (19502*b3 + 7074326*b1) * q^91 + (-832*b3 + 30487818*b1) * q^95 + (59532*b2 + 65788450) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 264 q^{5}+O(q^{10})$$ 4 * q + 264 * q^5 $$4 q + 264 q^{5} + 14632 q^{13} - 332904 q^{17} + 2604428 q^{25} - 2343576 q^{29} + 4315784 q^{37} - 9035496 q^{41} + 3458884 q^{49} + 42186600 q^{53} - 48148408 q^{61} + 125450832 q^{65} - 21215480 q^{73} + 32354496 q^{77} + 235297584 q^{85} + 12675576 q^{89} + 263153800 q^{97}+O(q^{100})$$ 4 * q + 264 * q^5 + 14632 * q^13 - 332904 * q^17 + 2604428 * q^25 - 2343576 * q^29 + 4315784 * q^37 - 9035496 * q^41 + 3458884 * q^49 + 42186600 * q^53 - 48148408 * q^61 + 125450832 * q^65 - 21215480 * q^73 + 32354496 * q^77 + 235297584 * q^85 + 12675576 * q^89 + 263153800 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 451x^{2} + 450x + 202500$$ :

 $$\beta_{1}$$ $$=$$ $$( -4\nu^{3} + 1804\nu^{2} - 1804\nu + 404100 ) / 101475$$ (-4*v^3 + 1804*v^2 - 1804*v + 404100) / 101475 $$\beta_{2}$$ $$=$$ $$( 48\nu^{3} + 32424 ) / 451$$ (48*v^3 + 32424) / 451 $$\beta_{3}$$ $$=$$ $$( 8\nu^{3} - 8\nu^{2} + 7208\nu + 1800 ) / 75$$ (8*v^3 - 8*v^2 + 7208*v + 1800) / 75
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_{2} + 6\beta _1 + 24 ) / 96$$ (b3 - b2 + 6*b1 + 24) / 96 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 5406\beta _1 - 21624 ) / 96$$ (b3 + b2 + 5406*b1 - 21624) / 96 $$\nu^{3}$$ $$=$$ $$( 451\beta_{2} - 32424 ) / 48$$ (451*b2 - 32424) / 48

## 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
 10.8595 + 18.8093i 10.8595 − 18.8093i −10.3595 + 17.9433i −10.3595 − 17.9433i
0 0 0 −952.517 0 3101.27i 0 0 0
127.2 0 0 0 −952.517 0 3101.27i 0 0 0
127.3 0 0 0 1084.52 0 426.979i 0 0 0
127.4 0 0 0 1084.52 0 426.979i 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.i 4
3.b odd 2 1 48.9.g.c 4
4.b odd 2 1 inner 144.9.g.i 4
12.b even 2 1 48.9.g.c 4
24.f even 2 1 192.9.g.c 4
24.h odd 2 1 192.9.g.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.9.g.c 4 3.b odd 2 1
48.9.g.c 4 12.b even 2 1
144.9.g.i 4 1.a even 1 1 trivial
144.9.g.i 4 4.b odd 2 1 inner
192.9.g.c 4 24.f even 2 1
192.9.g.c 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 132T_{5} - 1033020$$ acting on $$S_{9}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 132 T - 1033020)^{2}$$
$7$ $$T^{4} + 9800160 T^{2} + \cdots + 1753442078976$$
$11$ $$T^{4} + 254051424 T^{2} + \cdots + 10\!\cdots\!44$$
$13$ $$(T^{2} - 7316 T - 920257436)^{2}$$
$17$ $$(T^{2} + 166452 T + 2938893732)^{2}$$
$19$ $$T^{4} + 86956856160 T^{2} + \cdots + 18\!\cdots\!96$$
$23$ $$T^{4} + 130419942912 T^{2} + \cdots + 11\!\cdots\!00$$
$29$ $$(T^{2} + 1171788 T + 169738484580)^{2}$$
$31$ $$T^{4} + 2873741432544 T^{2} + \cdots + 49\!\cdots\!84$$
$37$ $$(T^{2} - 2157892 T - 2332762137980)^{2}$$
$41$ $$(T^{2} + 4517748 T + 4295512060452)^{2}$$
$43$ $$T^{4} + 26254406590560 T^{2} + \cdots + 54\!\cdots\!16$$
$47$ $$T^{4} + 2880843373440 T^{2} + \cdots + 14\!\cdots\!16$$
$53$ $$(T^{2} - 21093300 T + 108678054443364)^{2}$$
$59$ $$T^{4} + 78425893959264 T^{2} + \cdots + 44\!\cdots\!24$$
$61$ $$(T^{2} + 24074204 T + 111045415899460)^{2}$$
$67$ $$T^{4} + \cdots + 28\!\cdots\!96$$
$71$ $$T^{4} + \cdots + 54\!\cdots\!36$$
$73$ $$(T^{2} + 10607740 T - 670117553322236)^{2}$$
$79$ $$T^{4} + \cdots + 43\!\cdots\!00$$
$83$ $$T^{4} + \cdots + 62\!\cdots\!44$$
$89$ $$(T^{2} - 6337788 T - 11\!\cdots\!64)^{2}$$
$97$ $$(T^{2} - 131576900 T + 651598379321476)^{2}$$