# Properties

 Label 144.6.i.b Level $144$ Weight $6$ Character orbit 144.i Analytic conductor $23.095$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [144,6,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, 6, names="a")

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

chi := DirichletCharacter("144.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$144 = 2^{4} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$6$$ 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: $$23.0952700531$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.47347183152.3 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331$$ x^6 - 3*x^5 + 118*x^4 - 231*x^3 + 3700*x^2 - 3585*x + 32331 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 18) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 18 \beta_1 - 20) q^{5} + (3 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 12 \beta_{2} + 37 \beta_1 + 5) q^{7} + (6 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 36 \beta_1 - 50) q^{9}+O(q^{10})$$ q + (b3 - b2 - b1 - 1) * q^3 + (b5 + b3 + 2*b2 + 18*b1 - 20) * q^5 + (3*b5 - 2*b4 + 6*b3 - 12*b2 + 37*b1 + 5) * q^7 + (6*b5 - 3*b4 + b3 - 2*b2 + 36*b1 - 50) * q^9 $$q + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{5} + \beta_{3} + 2 \beta_{2} + 18 \beta_1 - 20) q^{5} + (3 \beta_{5} - 2 \beta_{4} + 6 \beta_{3} - 12 \beta_{2} + 37 \beta_1 + 5) q^{7} + (6 \beta_{5} - 3 \beta_{4} + \beta_{3} - 2 \beta_{2} + 36 \beta_1 - 50) q^{9} + ( - 3 \beta_{5} + 7 \beta_{4} + 9 \beta_{3} + 12 \beta_{2} + 107 \beta_1 - 10) q^{11} + ( - 13 \beta_{5} + 6 \beta_{4} - 31 \beta_{3} - 8 \beta_{2} + 260 \beta_1 - 228) q^{13} + ( - 9 \beta_{5} - 9 \beta_{4} - 6 \beta_{3} + 21 \beta_{2} + 558 \beta_1 - 663) q^{15} + (\beta_{5} - 13 \beta_{4} - 2 \beta_{3} + 38 \beta_{2} + 13 \beta_1 + 458) q^{17} + (17 \beta_{5} + \beta_{4} - 34 \beta_{3} - 20 \beta_{2} - \beta_1 - 358) q^{19} + (36 \beta_{5} + 9 \beta_{4} + 59 \beta_{3} + 6 \beta_{2} + 1037 \beta_1 - 2415) q^{21} + (13 \beta_{5} + 3 \beta_{4} + 4 \beta_{3} + 35 \beta_{2} - 1050 \beta_1 + 1027) q^{23} + ( - 30 \beta_{5} + 15 \beta_{4} - 75 \beta_{3} + 120 \beta_{2} - 886 \beta_1 - 45) q^{25} + ( - 27 \beta_{5} + 18 \beta_{4} - 30 \beta_{3} + 93 \beta_{2} + \cdots - 4056) q^{27}+ \cdots + (504 \beta_{5} - 495 \beta_{4} + 3648 \beta_{3} - 2112 \beta_{2} + \cdots - 13191) q^{99}+O(q^{100})$$ q + (b3 - b2 - b1 - 1) * q^3 + (b5 + b3 + 2*b2 + 18*b1 - 20) * q^5 + (3*b5 - 2*b4 + 6*b3 - 12*b2 + 37*b1 + 5) * q^7 + (6*b5 - 3*b4 + b3 - 2*b2 + 36*b1 - 50) * q^9 + (-3*b5 + 7*b4 + 9*b3 + 12*b2 + 107*b1 - 10) * q^11 + (-13*b5 + 6*b4 - 31*b3 - 8*b2 + 260*b1 - 228) * q^13 + (-9*b5 - 9*b4 - 6*b3 + 21*b2 + 558*b1 - 663) * q^15 + (b5 - 13*b4 - 2*b3 + 38*b2 + 13*b1 + 458) * q^17 + (17*b5 + b4 - 34*b3 - 20*b2 - b1 - 358) * q^19 + (36*b5 + 9*b4 + 59*b3 + 6*b2 + 1037*b1 - 2415) * q^21 + (13*b5 + 3*b4 + 4*b3 + 35*b2 - 1050*b1 + 1027) * q^23 + (-30*b5 + 15*b4 - 75*b3 + 120*b2 - 886*b1 - 45) * q^25 + (-27*b5 + 18*b4 - 30*b3 + 93*b2 + 2028*b1 - 4056) * q^27 + (-18*b5 + 71*b4 + 141*b3 + 72*b2 - 1733*b1 - 89) * q^29 + (27*b5 + 51*b4 - 126*b3 + 207*b2 - 2768*b1 + 2765) * q^31 + (9*b5 - 99*b4 + 120*b3 - 66*b2 + 2040*b1 + 1860) * q^33 + (38*b5 - 185*b4 - 76*b3 + 517*b2 + 185*b1 - 782) * q^35 + (20*b5 - 50*b4 - 40*b3 + 130*b2 + 50*b1 + 6576) * q^37 + (9*b5 + 171*b4 + 60*b3 + 263*b2 - 4028*b1 + 10019) * q^39 + (10*b5 + 210*b4 - 620*b3 + 650*b2 - 1263*b1 + 1453) * q^41 + (-117*b5 + 195*b4 + 117*b3 + 468*b2 + 10619*b1 - 312) * q^43 + (-27*b5 - 54*b4 - 189*b3 + 612*b2 - 14058*b1 + 7254) * q^45 + (147*b5 - 138*b4 + 174*b3 - 588*b2 - 4611*b1 + 285) * q^47 + (247*b5 + 12*b4 + 211*b3 + 530*b2 + 17643*b1 - 18125) * q^49 + (-126*b5 - 99*b4 + 327*b3 - 468*b2 - 10986*b1 - 513) * q^51 + (8*b5 - 170*b4 - 16*b3 + 502*b2 + 170*b1 - 16340) * q^53 + (-54*b5 + 51*b4 + 108*b3 - 99*b2 - 51*b1 - 20994) * q^55 + (-144*b5 + 315*b4 - 365*b3 + 632*b2 + 14300*b1 - 10123) * q^57 + (184*b5 + 213*b4 - 455*b3 + 1007*b2 + 21411*b1 - 21566) * q^59 + (-246*b5 + 201*b4 - 381*b3 + 984*b2 - 24785*b1 - 447) * q^61 + (-57*b5 - 264*b4 - 1322*b3 + 2416*b2 + 30351*b1 - 23957) * q^63 + (624*b5 - 623*b4 + 627*b3 - 2496*b2 + 34985*b1 + 1247) * q^65 + (534*b5 - 45*b4 + 669*b3 + 933*b2 - 11087*b1 + 9974) * q^67 + (-171*b5 - 90*b4 - 87*b3 - 990*b2 + 10398*b1 - 9450) * q^69 + (-94*b5 + 484*b4 + 188*b3 - 1358*b2 - 484*b1 + 22486) * q^71 + (-765*b5 + 969*b4 + 1530*b3 - 2142*b2 - 969*b1 - 238) * q^73 + (-405*b5 - 1141*b3 - 12703*b1 + 24705) * q^75 + (737*b5 - 924*b4 + 3509*b3 - 1298*b2 - 31428*b1 + 29030) * q^77 + (45*b5 - 754*b4 - 2082*b3 - 180*b2 - 29115*b1 + 799) * q^79 + (-207*b5 - 180*b4 - 2127*b3 + 3876*b2 - 9693*b1 + 24648) * q^81 + (633*b5 + 54*b4 + 2694*b3 - 2532*b2 - 12831*b1 + 579) * q^83 + (-54*b5 + 858*b4 - 2628*b3 + 2466*b2 - 22080*b1 + 23046) * q^85 + (315*b5 - 1116*b4 - 1452*b3 - 744*b2 + 36507*b1 + 7941) * q^87 + (490*b5 - 196*b4 - 980*b3 + 98*b2 + 196*b1 + 11120) * q^89 + (-2004*b5 + 1995*b4 + 4008*b3 - 3981*b2 - 1995*b1 + 52292) * q^91 + (-1161*b5 + 216*b4 - 315*b3 - 2330*b2 + 49940*b1 - 13454) * q^93 + (484*b5 - 732*b4 + 2680*b3 - 1228*b2 - 29112*b1 + 27412) * q^95 + (1158*b5 - 788*b4 + 2268*b3 - 4632*b2 + 12729*b1 + 1946) * q^97 + (504*b5 - 495*b4 + 3648*b3 - 2112*b2 - 69714*b1 - 13191) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 9 q^{3} - 54 q^{5} + 132 q^{7} - 177 q^{9}+O(q^{10})$$ 6 * q - 9 * q^3 - 54 * q^5 + 132 * q^7 - 177 * q^9 $$6 q - 9 q^{3} - 54 q^{5} + 132 q^{7} - 177 q^{9} + 315 q^{11} - 744 q^{13} - 2286 q^{15} + 2898 q^{17} - 2262 q^{19} - 11076 q^{21} + 3168 q^{23} - 2883 q^{25} - 18144 q^{27} - 5148 q^{29} + 8610 q^{31} + 17469 q^{33} - 2700 q^{35} + 39936 q^{37} + 49026 q^{39} + 5049 q^{41} + 31389 q^{43} + 2538 q^{45} - 12924 q^{47} - 52857 q^{49} - 36837 q^{51} - 96048 q^{53} - 126252 q^{55} - 17469 q^{57} - 62955 q^{59} - 75966 q^{61} - 49578 q^{63} + 108702 q^{65} + 32991 q^{67} - 29250 q^{69} + 129672 q^{71} - 8466 q^{73} + 105483 q^{75} + 88740 q^{77} - 89202 q^{79} + 123435 q^{81} - 32634 q^{83} + 71388 q^{85} + 151524 q^{87} + 66132 q^{89} + 301836 q^{91} + 57678 q^{93} + 82944 q^{95} + 46245 q^{97} - 282168 q^{99}+O(q^{100})$$ 6 * q - 9 * q^3 - 54 * q^5 + 132 * q^7 - 177 * q^9 + 315 * q^11 - 744 * q^13 - 2286 * q^15 + 2898 * q^17 - 2262 * q^19 - 11076 * q^21 + 3168 * q^23 - 2883 * q^25 - 18144 * q^27 - 5148 * q^29 + 8610 * q^31 + 17469 * q^33 - 2700 * q^35 + 39936 * q^37 + 49026 * q^39 + 5049 * q^41 + 31389 * q^43 + 2538 * q^45 - 12924 * q^47 - 52857 * q^49 - 36837 * q^51 - 96048 * q^53 - 126252 * q^55 - 17469 * q^57 - 62955 * q^59 - 75966 * q^61 - 49578 * q^63 + 108702 * q^65 + 32991 * q^67 - 29250 * q^69 + 129672 * q^71 - 8466 * q^73 + 105483 * q^75 + 88740 * q^77 - 89202 * q^79 + 123435 * q^81 - 32634 * q^83 + 71388 * q^85 + 151524 * q^87 + 66132 * q^89 + 301836 * q^91 + 57678 * q^93 + 82944 * q^95 + 46245 * q^97 - 282168 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 118x^{4} - 231x^{3} + 3700x^{2} - 3585x + 32331$$ :

 $$\beta_{1}$$ $$=$$ $$( -2\nu^{5} + 5\nu^{4} + 176\nu^{3} - 269\nu^{2} + 16626\nu + 22035 ) / 60606$$ (-2*v^5 + 5*v^4 + 176*v^3 - 269*v^2 + 16626*v + 22035) / 60606 $$\beta_{2}$$ $$=$$ $$( 331\nu^{5} - 2875\nu^{4} + 35573\nu^{3} - 258101\nu^{2} + 670179\nu - 4096833 ) / 60606$$ (331*v^5 - 2875*v^4 + 35573*v^3 - 258101*v^2 + 670179*v - 4096833) / 60606 $$\beta_{3}$$ $$=$$ $$( 331\nu^{5} - 418\nu^{4} + 30659\nu^{3} - 22229\nu^{2} + 527673\nu - 168909 ) / 30303$$ (331*v^5 - 418*v^4 + 30659*v^3 - 22229*v^2 + 527673*v - 168909) / 30303 $$\beta_{4}$$ $$=$$ $$( 991\nu^{5} + 3665\nu^{4} + 82325\nu^{3} + 495697\nu^{2} + 769179\nu + 11037819 ) / 60606$$ (991*v^5 + 3665*v^4 + 82325*v^3 + 495697*v^2 + 769179*v + 11037819) / 60606 $$\beta_{5}$$ $$=$$ $$( 1655\nu^{5} - 9461\nu^{4} + 168037\nu^{3} - 636943\nu^{2} + 3065883\nu - 5840445 ) / 60606$$ (1655*v^5 - 9461*v^4 + 168037*v^3 - 636943*v^2 + 3065883*v - 5840445) / 60606
 $$\nu$$ $$=$$ $$( \beta_{5} - 2\beta_{4} + 4\beta_{3} - 7\beta_{2} + 2\beta _1 + 9 ) / 18$$ (b5 - 2*b4 + 4*b3 - 7*b2 + 2*b1 + 9) / 18 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{3} - 3\beta_{2} - 112 ) / 3$$ (b5 - b3 - 3*b2 - 112) / 3 $$\nu^{3}$$ $$=$$ $$( -44\beta_{5} + 109\beta_{4} - 221\beta_{3} + 353\beta_{2} + 2870\beta _1 - 2505 ) / 18$$ (-44*b5 + 109*b4 - 221*b3 + 353*b2 + 2870*b1 - 2505) / 18 $$\nu^{4}$$ $$=$$ $$( -101\beta_{5} + 17\beta_{4} + 98\beta_{3} + 264\beta_{2} + 976\beta _1 + 5208 ) / 3$$ (-101*b5 + 17*b4 + 98*b3 + 264*b2 + 976*b1 + 5208) / 3 $$\nu^{5}$$ $$=$$ $$( 2119\beta_{5} - 6779\beta_{4} + 16081\beta_{3} - 20746\beta_{2} - 261628\beta _1 + 221196 ) / 18$$ (2119*b5 - 6779*b4 + 16081*b3 - 20746*b2 - 261628*b1 + 221196) / 18

## 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
 0.5 + 5.23712i 0.5 − 8.40123i 0.5 + 4.03013i 0.5 − 5.23712i 0.5 + 8.40123i 0.5 − 4.03013i
0 −15.5145 + 1.51695i 0 33.0434 57.2329i 0 57.0952 + 98.8918i 0 238.398 47.0695i 0
49.2 0 2.43381 15.3973i 0 −20.8014 + 36.0292i 0 −101.661 176.082i 0 −231.153 74.9483i 0
49.3 0 8.58066 + 13.0143i 0 −39.2420 + 67.9691i 0 110.566 + 191.505i 0 −95.7446 + 223.343i 0
97.1 0 −15.5145 1.51695i 0 33.0434 + 57.2329i 0 57.0952 98.8918i 0 238.398 + 47.0695i 0
97.2 0 2.43381 + 15.3973i 0 −20.8014 36.0292i 0 −101.661 + 176.082i 0 −231.153 + 74.9483i 0
97.3 0 8.58066 13.0143i 0 −39.2420 67.9691i 0 110.566 191.505i 0 −95.7446 223.343i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 97.3 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 144.6.i.b 6
3.b odd 2 1 432.6.i.b 6
4.b odd 2 1 18.6.c.b 6
9.c even 3 1 inner 144.6.i.b 6
9.d odd 6 1 432.6.i.b 6
12.b even 2 1 54.6.c.b 6
36.f odd 6 1 18.6.c.b 6
36.f odd 6 1 162.6.a.j 3
36.h even 6 1 54.6.c.b 6
36.h even 6 1 162.6.a.i 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.6.c.b 6 4.b odd 2 1
18.6.c.b 6 36.f odd 6 1
54.6.c.b 6 12.b even 2 1
54.6.c.b 6 36.h even 6 1
144.6.i.b 6 1.a even 1 1 trivial
144.6.i.b 6 9.c even 3 1 inner
162.6.a.i 3 36.h even 6 1
162.6.a.j 3 36.f odd 6 1
432.6.i.b 6 3.b odd 2 1
432.6.i.b 6 9.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{6} + 54T_{5}^{5} + 7587T_{5}^{4} + 179334T_{5}^{3} + 33470577T_{5}^{2} + 1007927064T_{5} + 46562734656$$ acting on $$S_{6}^{\mathrm{new}}(144, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$T^{6} + 9 T^{5} + 129 T^{4} + \cdots + 14348907$$
$5$ $$T^{6} + 54 T^{5} + \cdots + 46562734656$$
$7$ $$T^{6} - 132 T^{5} + \cdots + 26358756910084$$
$11$ $$T^{6} - 315 T^{5} + \cdots + 17\!\cdots\!69$$
$13$ $$T^{6} + 744 T^{5} + \cdots + 14\!\cdots\!84$$
$17$ $$(T^{3} - 1449 T^{2} - 500040 T + 930192444)^{2}$$
$19$ $$(T^{3} + 1131 T^{2} - 1499928 T + 352455920)^{2}$$
$23$ $$T^{6} - 3168 T^{5} + \cdots + 18\!\cdots\!96$$
$29$ $$T^{6} + 5148 T^{5} + \cdots + 42\!\cdots\!16$$
$31$ $$T^{6} - 8610 T^{5} + \cdots + 35\!\cdots\!16$$
$37$ $$(T^{3} - 19968 T^{2} + \cdots - 188019064016)^{2}$$
$41$ $$T^{6} - 5049 T^{5} + \cdots + 46\!\cdots\!69$$
$43$ $$T^{6} - 31389 T^{5} + \cdots + 26\!\cdots\!21$$
$47$ $$T^{6} + 12924 T^{5} + \cdots + 71\!\cdots\!00$$
$53$ $$(T^{3} + 48024 T^{2} + \cdots + 1764512817552)^{2}$$
$59$ $$T^{6} + 62955 T^{5} + \cdots + 33\!\cdots\!49$$
$61$ $$T^{6} + 75966 T^{5} + \cdots + 28\!\cdots\!00$$
$67$ $$T^{6} - 32991 T^{5} + \cdots + 92\!\cdots\!25$$
$71$ $$(T^{3} - 64836 T^{2} + \cdots + 139951336896)^{2}$$
$73$ $$(T^{3} + 4233 T^{2} + \cdots + 14322358753732)^{2}$$
$79$ $$T^{6} + 89202 T^{5} + \cdots + 13\!\cdots\!16$$
$83$ $$T^{6} + 32634 T^{5} + \cdots + 10\!\cdots\!16$$
$89$ $$(T^{3} - 33066 T^{2} + \cdots + 9104584153608)^{2}$$
$97$ $$T^{6} - 46245 T^{5} + \cdots + 85\!\cdots\!09$$