# Properties

 Label 72.22.a.g Level $72$ Weight $22$ Character orbit 72.a Self dual yes Analytic conductor $201.224$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,22,Mod(1,72)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("72.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 72.a (trivial)

## 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: yes Analytic conductor: $$201.223687887$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 10642080274x^{3} + 257327290824634x^{2} - 1056913897140193890x - 892208868339633807375$$ x^5 - x^4 - 10642080274*x^3 + 257327290824634*x^2 - 1056913897140193890*x - 892208868339633807375 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{39}\cdot 3^{13}\cdot 5\cdot 7$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 - 2064899) q^{5} + (\beta_{2} + 6 \beta_1 - 74076006) q^{7}+O(q^{10})$$ q + (b1 - 2064899) * q^5 + (b2 + 6*b1 - 74076006) * q^7 $$q + (\beta_1 - 2064899) q^{5} + (\beta_{2} + 6 \beta_1 - 74076006) q^{7} + ( - \beta_{3} + 15 \beta_{2} + 158 \beta_1 - 17280501613) q^{11} + ( - \beta_{4} + \beta_{3} + 74 \beta_{2} + 2030 \beta_1 - 54179632965) q^{13} + (16 \beta_{4} - 4 \beta_{3} - 1812 \beta_{2} + \cdots + 1107669848430) q^{17}+ \cdots + ( - 576968284 \beta_{4} + 4128325212 \beta_{3} + \cdots + 95\!\cdots\!46) q^{97}+O(q^{100})$$ q + (b1 - 2064899) * q^5 + (b2 + 6*b1 - 74076006) * q^7 + (-b3 + 15*b2 + 158*b1 - 17280501613) * q^11 + (-b4 + b3 + 74*b2 + 2030*b1 - 54179632965) * q^13 + (16*b4 - 4*b3 - 1812*b2 - 37182*b1 + 1107669848430) * q^17 + (4*b4 + 252*b3 - 4186*b2 - 58084*b1 + 4663260854088) * q^19 + (192*b4 + 58*b3 - 88870*b2 - 855436*b1 + 19976942865394) * q^23 + (310*b4 + 4810*b3 - 98780*b2 - 2665396*b1 + 107746547212069) * q^25 + (-3808*b4 - 3848*b3 - 1856040*b2 - 4608689*b1 - 171679805374341) * q^29 + (-5352*b4 - 21784*b3 + 718241*b2 - 30517482*b1 - 659356195489694) * q^31 + (10880*b4 + 80315*b3 - 25415285*b2 - 173964778*b1 + 3882948261906207) * q^35 + (40089*b4 - 210073*b3 + 1588038*b2 - 743114814*b1 + 532797350038469) * q^37 + (66992*b4 - 703724*b3 - 75401116*b2 - 553364002*b1 - 1369691820076318) * q^41 + (-147148*b4 + 912332*b3 - 25206234*b2 - 5676527428*b1 + 18857523547438168) * q^43 + (-370496*b4 + 3141090*b3 + 80052226*b2 + 1310659780*b1 + 19929934809179258) * q^47 + (88570*b4 + 2515462*b3 + 149042428*b2 - 26632163564*b1 + 91561809125238431) * q^49 + (-278944*b4 - 6375128*b3 + 1449731272*b2 + 6169370591*b1 - 221562085691667365) * q^53 + (1543480*b4 - 12513080*b3 - 1040276720*b2 - 83662815888*b1 + 131205008394681992) * q^55 + (4676224*b4 - 4265150*b3 + 2295716514*b2 + 26927747268*b1 + 21247309054750362) * q^59 + (-6221995*b4 - 18339669*b3 + 5385613294*b2 - 41941212422*b1 - 921936789584962527) * q^61 + (-3181040*b4 + 60680700*b3 - 4499024660*b2 - 124304538246*b1 + 1307909789369891654) * q^65 + (4785192*b4 + 90334680*b3 - 16392021804*b2 + 105784722728*b1 + 2534770535110068672) * q^67 + (-32243200*b4 - 163368232*b3 - 20246750376*b2 + 329520753840*b1 - 4050301857733123080) * q^71 + (32627814*b4 + 129853338*b3 + 31467135812*b2 + 1379189605292*b1 - 702571609178306752) * q^73 + (46256864*b4 + 171588360*b3 - 40339428824*b2 - 1610059385280*b1 + 11318305188118706904) * q^77 + (-99383360*b4 - 420894656*b3 - 53255613199*b2 + 2666427349542*b1 - 15074352619459937734) * q^79 + (135046912*b4 + 308517933*b3 + 36449388125*b2 + 3619975936314*b1 + 3161534469422637865) * q^83 + (27464060*b4 - 857282940*b3 + 95865418920*b2 + 3071077132728*b1 - 24309558414492304732) * q^85 + (-251728928*b4 - 1694193272*b3 + 299751195560*b2 - 6730650357052*b1 - 48534897061931732788) * q^89 + (349175772*b4 + 1293858596*b3 - 62829331718*b2 - 5569836390172*b1 + 57576957794907083160) * q^91 + (-385807680*b4 + 2912280970*b3 + 286488059690*b2 + 21948161548564*b1 - 44401792148339953406) * q^95 + (-576968284*b4 + 4128325212*b3 - 291978606056*b2 - 22599694024056*b1 + 95382343274005788746) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 10324496 q^{5} - 370380036 q^{7}+O(q^{10})$$ 5 * q - 10324496 * q^5 - 370380036 * q^7 $$5 q - 10324496 q^{5} - 370380036 q^{7} - 86402508224 q^{11} - 270898166854 q^{13} + 5538349279328 q^{17} + 23316304328776 q^{19} + 99884715182464 q^{23} + 538732738730551 q^{25} - 858399022266864 q^{29} - 32\!\cdots\!72 q^{31}+ \cdots + 47\!\cdots\!98 q^{97}+O(q^{100})$$ 5 * q - 10324496 * q^5 - 370380036 * q^7 - 86402508224 * q^11 - 270898166854 * q^13 + 5538349279328 * q^17 + 23316304328776 * q^19 + 99884715182464 * q^23 + 538732738730551 * q^25 - 858399022266864 * q^29 - 3296780946952772 * q^31 + 19414741483576128 * q^35 + 2663987493097086 * q^37 - 6848458547721312 * q^41 + 94287623414630600 * q^43 + 99649672738377600 * q^47 + 457809072260871181 * q^49 - 1107810434634082544 * q^53 + 656025125623712768 * q^55 + 106236518341739392 * q^59 - 4609683906001939882 * q^61 + 6539549071214677216 * q^65 + 12673852569855955312 * q^67 - 20251509618349737472 * q^71 - 3512859424951285714 * q^73 + 56591527550824508160 * q^77 - 75371765764147932868 * q^79 + 15807668727445770944 * q^83 - 121547795144395939328 * q^85 - 242674478580702500160 * q^89 + 287884794545665664568 * q^91 - 222008982686949034624 * q^95 + 476911738973851292998 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 10642080274x^{3} + 257327290824634x^{2} - 1056913897140193890x - 892208868339633807375$$ :

 $$\beta_{1}$$ $$=$$ $$( 22287213162208 \nu^{4} + \cdots - 53\!\cdots\!97 ) / 19\!\cdots\!97$$ (22287213162208*v^4 - 477019612969331552*v^3 - 246441737774833081128256*v^2 + 10966875976669574621015476640*v - 53073368140629978784779040493897) / 1919872372112591990649997 $$\beta_{2}$$ $$=$$ $$( 361945148223296 \nu^{4} + \cdots - 17\!\cdots\!08 ) / 19\!\cdots\!97$$ (361945148223296*v^4 - 8349111889829332672*v^3 - 3816603601949956398439040*v^2 + 169934685722272886263018055104*v - 1745049772770118121273589133065408) / 1919872372112591990649997 $$\beta_{3}$$ $$=$$ $$( - 33\!\cdots\!72 \nu^{4} + \cdots + 78\!\cdots\!01 ) / 19\!\cdots\!97$$ (-339959610027609472*v^4 - 2972365811988847240192*v^3 + 3559243261826924509546670080*v^2 - 58190299547460145224338572826240*v + 78080495614819507124887598037178601) / 1919872372112591990649997 $$\beta_{4}$$ $$=$$ $$( - 28\!\cdots\!40 \nu^{4} + \cdots + 25\!\cdots\!96 ) / 27\!\cdots\!71$$ (-283077356141483840*v^4 - 1347870583451139291968*v^3 + 3013942272286041420605689472*v^2 - 57443287195675120677422319117760*v + 25202689576514843222943721909615296) / 274267481730370284378571
 $$\nu$$ $$=$$ $$( \beta_{4} - 5\beta_{3} - 1002\beta_{2} + 28914\beta _1 + 3721675 ) / 18579456$$ (b4 - 5*b3 - 1002*b2 + 28914*b1 + 3721675) / 18579456 $$\nu^{2}$$ $$=$$ $$( 221605\beta_{4} - 1139889\beta_{3} + 320146870\beta_{2} - 2883820566\beta _1 + 237268874073712671 ) / 55738368$$ (221605*b4 - 1139889*b3 + 320146870*b2 - 2883820566*b1 + 237268874073712671) / 55738368 $$\nu^{3}$$ $$=$$ $$( 4602054689 \beta_{4} - 24646945741 \beta_{3} - 6373617258450 \beta_{2} + 136719475778946 \beta _1 - 14\!\cdots\!65 ) / 9289728$$ (4602054689*b4 - 24646945741*b3 - 6373617258450*b2 + 136719475778946*b1 - 1434240978675161157565) / 9289728 $$\nu^{4}$$ $$=$$ $$( 223598521944013 \beta_{4} + \cdots + 36\!\cdots\!07 ) / 7962624$$ (223598521944013*b4 - 1198352385989121*b3 + 600100264176542110*b2 - 7458887607092880438*b1 + 367450207236662186526955407) / 7962624

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −715.745 −113936. 6287.62 88760.9 19603.8
0 0 0 −3.38633e7 0 −1.23818e9 0 0 0
1.2 0 0 0 −2.31503e7 0 1.20078e9 0 0 0
1.3 0 0 0 1.08925e6 0 −4.86930e8 0 0 0
1.4 0 0 0 1.28149e7 0 −3.84305e7 0 0 0
1.5 0 0 0 3.27849e7 0 1.92379e8 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.22.a.g 5
3.b odd 2 1 72.22.a.h yes 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.22.a.g 5 1.a even 1 1 trivial
72.22.a.h yes 5 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{5} + 10324496 T_{5}^{4} + \cdots - 35\!\cdots\!00$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(72))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5}$$
$3$ $$T^{5}$$
$5$ $$T^{5} + 10324496 T^{4} + \cdots - 35\!\cdots\!00$$
$7$ $$T^{5} + 370380036 T^{4} + \cdots + 53\!\cdots\!84$$
$11$ $$T^{5} + 86402508224 T^{4} + \cdots + 71\!\cdots\!96$$
$13$ $$T^{5} + 270898166854 T^{4} + \cdots - 14\!\cdots\!36$$
$17$ $$T^{5} - 5538349279328 T^{4} + \cdots + 59\!\cdots\!84$$
$19$ $$T^{5} - 23316304328776 T^{4} + \cdots - 83\!\cdots\!36$$
$23$ $$T^{5} - 99884715182464 T^{4} + \cdots + 85\!\cdots\!16$$
$29$ $$T^{5} + 858399022266864 T^{4} + \cdots + 10\!\cdots\!00$$
$31$ $$T^{5} + \cdots + 11\!\cdots\!00$$
$37$ $$T^{5} + \cdots - 91\!\cdots\!00$$
$41$ $$T^{5} + \cdots - 25\!\cdots\!00$$
$43$ $$T^{5} + \cdots - 31\!\cdots\!00$$
$47$ $$T^{5} + \cdots + 11\!\cdots\!72$$
$53$ $$T^{5} + \cdots + 11\!\cdots\!68$$
$59$ $$T^{5} + \cdots - 51\!\cdots\!68$$
$61$ $$T^{5} + \cdots + 24\!\cdots\!92$$
$67$ $$T^{5} + \cdots + 18\!\cdots\!72$$
$71$ $$T^{5} + \cdots + 80\!\cdots\!92$$
$73$ $$T^{5} + \cdots - 53\!\cdots\!76$$
$79$ $$T^{5} + \cdots - 78\!\cdots\!52$$
$83$ $$T^{5} + \cdots + 53\!\cdots\!72$$
$89$ $$T^{5} + \cdots + 10\!\cdots\!00$$
$97$ $$T^{5} + \cdots - 18\!\cdots\!52$$