# Properties

 Label 144.4.i.c Level $144$ Weight $4$ Character orbit 144.i Analytic conductor $8.496$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,4,Mod(49,144)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(144, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("144.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 144.i (of order $$3$$, degree $$2$$, 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: $$8.49627504083$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ x^4 - x^3 - 2*x^2 - 3*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 9) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{3} + (\beta_{3} - \beta_{2} + 7 \beta_1 - 8) q^{5} + ( - 3 \beta_{3} + 5 \beta_1) q^{7} + ( - 3 \beta_{3} + 6 \beta_{2} + 24 \beta_1 + 3) q^{9}+O(q^{10})$$ q + (b3 + b2 - 2*b1 + 2) * q^3 + (b3 - b2 + 7*b1 - 8) * q^5 + (-3*b3 + 5*b1) * q^7 + (-3*b3 + 6*b2 + 24*b1 + 3) * q^9 $$q + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{3} + (\beta_{3} - \beta_{2} + 7 \beta_1 - 8) q^{5} + ( - 3 \beta_{3} + 5 \beta_1) q^{7} + ( - 3 \beta_{3} + 6 \beta_{2} + 24 \beta_1 + 3) q^{9} + (8 \beta_{3} + 29 \beta_1) q^{11} + (15 \beta_{3} - 15 \beta_{2} - 13 \beta_1 - 2) q^{13} + (9 \beta_{3} - 15 \beta_{2} + 15 \beta_1 - 24) q^{15} + ( - 9 \beta_{2} + 45) q^{17} + ( - 27 \beta_{2} + 25) q^{19} + (7 \beta_{3} - 8 \beta_{2} - 53 \beta_1 + 26) q^{21} + ( - 19 \beta_{3} + 19 \beta_{2} - 7 \beta_1 + 26) q^{23} + ( - 15 \beta_{3} + 68 \beta_1) q^{25} + (36 \beta_{3} - 18 \beta_{2} - 18 \beta_1 + 99) q^{27} + (\beta_{3} + 25 \beta_1) q^{29} + (3 \beta_{3} - 3 \beta_{2} - 23 \beta_1 + 20) q^{31} + (66 \beta_{3} - 21 \beta_{2} + 99 \beta_1 - 27) q^{33} + (19 \beta_{2} + 8) q^{35} + ( - 54 \beta_{2} - 52) q^{37} + (17 \beta_{3} + 11 \beta_{2} + 107 \beta_1 - 242) q^{39} + ( - 98 \beta_{3} + 98 \beta_{2} + 115 \beta_1 - 17) q^{41} + (6 \beta_{3} + 41 \beta_1) q^{43} + (45 \beta_{3} - 45 \beta_{2} + 27 \beta_1 - 216) q^{45} + ( - 91 \beta_{3} + 245 \beta_1) q^{47} + ( - 21 \beta_{3} + 21 \beta_{2} - 246 \beta_1 + 267) q^{49} + (63 \beta_{3} + 36 \beta_{2} - 180 \beta_1 + 18) q^{51} + (162 \beta_{2} + 108) q^{53} + ( - 93 \beta_{2} - 360) q^{55} + (79 \beta_{3} - 2 \beta_{2} - 320 \beta_1 - 166) q^{57} + ( - 136 \beta_{3} + 136 \beta_{2} - 331 \beta_1 + 467) q^{59} + ( - 105 \beta_{3} - 167 \beta_1) q^{61} + ( - 57 \beta_{3} + 78 \beta_{2} + 33 \beta_1 - 114) q^{63} + ( - 107 \beta_{3} - 29 \beta_1) q^{65} + (66 \beta_{3} - 66 \beta_{2} - 527 \beta_1 + 461) q^{67} + ( - 45 \beta_{3} + 33 \beta_{2} - 159 \beta_1 + 330) q^{69} + (144 \beta_{2} - 612) q^{71} + ( - 243 \beta_{2} - 349) q^{73} + (121 \beta_{3} - 83 \beta_{2} - 308 \beta_1 + 173) q^{75} + ( - 71 \beta_{3} + 71 \beta_{2} - 47 \beta_1 + 118) q^{77} + ( - 309 \beta_{3} - 247 \beta_1) q^{79} + (135 \beta_{3} + 135 \beta_{2} + 216 \beta_1 - 216) q^{81} + (107 \beta_{3} + 353 \beta_1) q^{83} + ( - 18 \beta_{3} + 18 \beta_{2} + 306 \beta_1 - 288) q^{85} + (51 \beta_{3} - 24 \beta_{2} - 9 \beta_1 + 18) q^{87} + ( - 72 \beta_{2} - 234) q^{89} + ( - 69 \beta_{2} + 356) q^{91} + ( - 17 \beta_{3} + 43 \beta_{2} + \beta_1 - 28) q^{93} + ( - 164 \beta_{3} + 164 \beta_{2} + 148 \beta_1 + 16) q^{95} + (102 \beta_{3} - 419 \beta_1) q^{97} + (279 \beta_{3} - 81 \beta_{2} + 801 \beta_1 - 585) q^{99}+O(q^{100})$$ q + (b3 + b2 - 2*b1 + 2) * q^3 + (b3 - b2 + 7*b1 - 8) * q^5 + (-3*b3 + 5*b1) * q^7 + (-3*b3 + 6*b2 + 24*b1 + 3) * q^9 + (8*b3 + 29*b1) * q^11 + (15*b3 - 15*b2 - 13*b1 - 2) * q^13 + (9*b3 - 15*b2 + 15*b1 - 24) * q^15 + (-9*b2 + 45) * q^17 + (-27*b2 + 25) * q^19 + (7*b3 - 8*b2 - 53*b1 + 26) * q^21 + (-19*b3 + 19*b2 - 7*b1 + 26) * q^23 + (-15*b3 + 68*b1) * q^25 + (36*b3 - 18*b2 - 18*b1 + 99) * q^27 + (b3 + 25*b1) * q^29 + (3*b3 - 3*b2 - 23*b1 + 20) * q^31 + (66*b3 - 21*b2 + 99*b1 - 27) * q^33 + (19*b2 + 8) * q^35 + (-54*b2 - 52) * q^37 + (17*b3 + 11*b2 + 107*b1 - 242) * q^39 + (-98*b3 + 98*b2 + 115*b1 - 17) * q^41 + (6*b3 + 41*b1) * q^43 + (45*b3 - 45*b2 + 27*b1 - 216) * q^45 + (-91*b3 + 245*b1) * q^47 + (-21*b3 + 21*b2 - 246*b1 + 267) * q^49 + (63*b3 + 36*b2 - 180*b1 + 18) * q^51 + (162*b2 + 108) * q^53 + (-93*b2 - 360) * q^55 + (79*b3 - 2*b2 - 320*b1 - 166) * q^57 + (-136*b3 + 136*b2 - 331*b1 + 467) * q^59 + (-105*b3 - 167*b1) * q^61 + (-57*b3 + 78*b2 + 33*b1 - 114) * q^63 + (-107*b3 - 29*b1) * q^65 + (66*b3 - 66*b2 - 527*b1 + 461) * q^67 + (-45*b3 + 33*b2 - 159*b1 + 330) * q^69 + (144*b2 - 612) * q^71 + (-243*b2 - 349) * q^73 + (121*b3 - 83*b2 - 308*b1 + 173) * q^75 + (-71*b3 + 71*b2 - 47*b1 + 118) * q^77 + (-309*b3 - 247*b1) * q^79 + (135*b3 + 135*b2 + 216*b1 - 216) * q^81 + (107*b3 + 353*b1) * q^83 + (-18*b3 + 18*b2 + 306*b1 - 288) * q^85 + (51*b3 - 24*b2 - 9*b1 + 18) * q^87 + (-72*b2 - 234) * q^89 + (-69*b2 + 356) * q^91 + (-17*b3 + 43*b2 + b1 - 28) * q^93 + (-164*b3 + 164*b2 + 148*b1 + 16) * q^95 + (102*b3 - 419*b1) * q^97 + (279*b3 - 81*b2 + 801*b1 - 585) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 3 q^{3} - 15 q^{5} + 7 q^{7} + 45 q^{9}+O(q^{10})$$ 4 * q + 3 * q^3 - 15 * q^5 + 7 * q^7 + 45 * q^9 $$4 q + 3 q^{3} - 15 q^{5} + 7 q^{7} + 45 q^{9} + 66 q^{11} + 11 q^{13} - 27 q^{15} + 198 q^{17} + 154 q^{19} + 21 q^{21} + 33 q^{23} + 121 q^{25} + 432 q^{27} + 51 q^{29} + 43 q^{31} + 198 q^{33} - 6 q^{35} - 100 q^{37} - 759 q^{39} - 132 q^{41} + 88 q^{43} - 675 q^{45} + 399 q^{47} + 513 q^{49} - 297 q^{51} + 108 q^{53} - 1254 q^{55} - 1221 q^{57} + 798 q^{59} - 439 q^{61} - 603 q^{63} - 165 q^{65} + 988 q^{67} + 891 q^{69} - 2736 q^{71} - 910 q^{73} + 363 q^{75} + 165 q^{77} - 803 q^{79} - 567 q^{81} + 813 q^{83} - 594 q^{85} + 153 q^{87} - 792 q^{89} + 1562 q^{91} - 213 q^{93} - 132 q^{95} - 736 q^{97} - 297 q^{99}+O(q^{100})$$ 4 * q + 3 * q^3 - 15 * q^5 + 7 * q^7 + 45 * q^9 + 66 * q^11 + 11 * q^13 - 27 * q^15 + 198 * q^17 + 154 * q^19 + 21 * q^21 + 33 * q^23 + 121 * q^25 + 432 * q^27 + 51 * q^29 + 43 * q^31 + 198 * q^33 - 6 * q^35 - 100 * q^37 - 759 * q^39 - 132 * q^41 + 88 * q^43 - 675 * q^45 + 399 * q^47 + 513 * q^49 - 297 * q^51 + 108 * q^53 - 1254 * q^55 - 1221 * q^57 + 798 * q^59 - 439 * q^61 - 603 * q^63 - 165 * q^65 + 988 * q^67 + 891 * q^69 - 2736 * q^71 - 910 * q^73 + 363 * q^75 + 165 * q^77 - 803 * q^79 - 567 * q^81 + 813 * q^83 - 594 * q^85 + 153 * q^87 - 792 * q^89 + 1562 * q^91 - 213 * q^93 - 132 * q^95 - 736 * q^97 - 297 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2x^{2} - 3x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6$$ (v^3 + 2*v^2 - 2*v - 3) / 6 $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + \nu^{2} + 5\nu ) / 3$$ (-v^3 + v^2 + 5*v) / 3 $$\beta_{3}$$ $$=$$ $$( 2\nu^{3} + \nu^{2} + 2\nu - 9 ) / 3$$ (2*v^3 + v^2 + 2*v - 9) / 3
 $$\nu$$ $$=$$ $$( \beta_{3} + \beta_{2} - 2\beta _1 + 2 ) / 3$$ (b3 + b2 - 2*b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -\beta_{3} + 2\beta_{2} + 8\beta _1 + 1 ) / 3$$ (-b3 + 2*b2 + 8*b1 + 1) / 3 $$\nu^{3}$$ $$=$$ $$( 4\beta_{3} - 2\beta_{2} - 2\beta _1 + 11 ) / 3$$ (4*b3 - 2*b2 - 2*b1 + 11) / 3

## Character values

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

 $$n$$ $$37$$ $$65$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{1}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −1.18614 − 1.26217i 1.68614 + 0.396143i −1.18614 + 1.26217i 1.68614 − 0.396143i
0 −3.55842 3.78651i 0 −2.31386 + 4.00772i 0 6.05842 + 10.4935i 0 −1.67527 + 26.9480i 0
49.2 0 5.05842 + 1.18843i 0 −5.18614 + 8.98266i 0 −2.55842 4.43132i 0 24.1753 + 12.0232i 0
97.1 0 −3.55842 + 3.78651i 0 −2.31386 4.00772i 0 6.05842 10.4935i 0 −1.67527 26.9480i 0
97.2 0 5.05842 1.18843i 0 −5.18614 8.98266i 0 −2.55842 + 4.43132i 0 24.1753 12.0232i 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
9.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 144.4.i.c 4
3.b odd 2 1 432.4.i.c 4
4.b odd 2 1 9.4.c.a 4
9.c even 3 1 inner 144.4.i.c 4
9.c even 3 1 1296.4.a.u 2
9.d odd 6 1 432.4.i.c 4
9.d odd 6 1 1296.4.a.i 2
12.b even 2 1 27.4.c.a 4
20.d odd 2 1 225.4.e.b 4
20.e even 4 2 225.4.k.b 8
36.f odd 6 1 9.4.c.a 4
36.f odd 6 1 81.4.a.d 2
36.h even 6 1 27.4.c.a 4
36.h even 6 1 81.4.a.a 2
180.n even 6 1 2025.4.a.n 2
180.p odd 6 1 225.4.e.b 4
180.p odd 6 1 2025.4.a.g 2
180.x even 12 2 225.4.k.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9.4.c.a 4 4.b odd 2 1
9.4.c.a 4 36.f odd 6 1
27.4.c.a 4 12.b even 2 1
27.4.c.a 4 36.h even 6 1
81.4.a.a 2 36.h even 6 1
81.4.a.d 2 36.f odd 6 1
144.4.i.c 4 1.a even 1 1 trivial
144.4.i.c 4 9.c even 3 1 inner
225.4.e.b 4 20.d odd 2 1
225.4.e.b 4 180.p odd 6 1
225.4.k.b 8 20.e even 4 2
225.4.k.b 8 180.x even 12 2
432.4.i.c 4 3.b odd 2 1
432.4.i.c 4 9.d odd 6 1
1296.4.a.i 2 9.d odd 6 1
1296.4.a.u 2 9.c even 3 1
2025.4.a.g 2 180.p odd 6 1
2025.4.a.n 2 180.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} + 15T_{5}^{3} + 177T_{5}^{2} + 720T_{5} + 2304$$ acting on $$S_{4}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 3 T^{3} - 18 T^{2} - 81 T + 729$$
$5$ $$T^{4} + 15 T^{3} + 177 T^{2} + \cdots + 2304$$
$7$ $$T^{4} - 7 T^{3} + 111 T^{2} + \cdots + 3844$$
$11$ $$T^{4} - 66 T^{3} + 3795 T^{2} + \cdots + 314721$$
$13$ $$T^{4} - 11 T^{3} + 1947 T^{2} + \cdots + 3334276$$
$17$ $$(T^{2} - 99 T + 1782)^{2}$$
$19$ $$(T^{2} - 77 T - 4532)^{2}$$
$23$ $$T^{4} - 33 T^{3} + 3795 T^{2} + \cdots + 7322436$$
$29$ $$T^{4} - 51 T^{3} + 1959 T^{2} + \cdots + 412164$$
$31$ $$T^{4} - 43 T^{3} + 1461 T^{2} + \cdots + 150544$$
$37$ $$(T^{2} + 50 T - 23432)^{2}$$
$41$ $$T^{4} + 132 T^{3} + \cdots + 5606565129$$
$43$ $$T^{4} - 88 T^{3} + 6105 T^{2} + \cdots + 2686321$$
$47$ $$T^{4} - 399 T^{3} + \cdots + 813276324$$
$53$ $$(T^{2} - 54 T - 215784)^{2}$$
$59$ $$T^{4} - 798 T^{3} + \cdots + 43678881$$
$61$ $$T^{4} + 439 T^{3} + \cdots + 1829786176$$
$67$ $$T^{4} - 988 T^{3} + \cdots + 43305193801$$
$71$ $$(T^{2} + 1368 T + 296784)^{2}$$
$73$ $$(T^{2} + 455 T - 435398)^{2}$$
$79$ $$T^{4} + 803 T^{3} + \cdots + 392522298256$$
$83$ $$T^{4} - 813 T^{3} + \cdots + 5010940944$$
$89$ $$(T^{2} + 396 T - 3564)^{2}$$
$97$ $$T^{4} + 736 T^{3} + \cdots + 2459267281$$
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