Properties

 Label 48.22.a.k Level $48$ Weight $22$ Character orbit 48.a Self dual yes Analytic conductor $134.149$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,22,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.1");

S:= CuspForms(chi, 22);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$22$$ Character orbit: $$[\chi]$$ $$=$$ 48.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: $$134.149125258$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - 53560x - 70812$$ x^3 - 53560*x - 70812 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{21}\cdot 3^{4}\cdot 5\cdot 7$$ Twist minimal: no (minimal twist has level 24) 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 59049 q^{3} + ( - \beta_1 - 1611042) q^{5} + (11 \beta_{2} - 25 \beta_1 - 90477008) q^{7} + 3486784401 q^{9}+O(q^{10})$$ q + 59049 * q^3 + (-b1 - 1611042) * q^5 + (11*b2 - 25*b1 - 90477008) * q^7 + 3486784401 * q^9 $$q + 59049 q^{3} + ( - \beta_1 - 1611042) q^{5} + (11 \beta_{2} - 25 \beta_1 - 90477008) q^{7} + 3486784401 q^{9} + (416 \beta_{2} - 3842 \beta_1 - 20398219396) q^{11} + ( - 6677 \beta_{2} - 32706 \beta_1 - 198162067474) q^{13} + ( - 59049 \beta_1 - 95130419058) q^{15} + (96866 \beta_{2} - 258706 \beta_1 + 474939839778) q^{17} + (107426 \beta_{2} + 1737246 \beta_1 + 11698275227268) q^{19} + (649539 \beta_{2} - 1476225 \beta_1 - 5342576845392) q^{21} + (1820478 \beta_{2} - 15793522 \beta_1 + 31637014357864) q^{23} + ( - 2291630 \beta_{2} + 7718556 \beta_1 - 204593341765233) q^{25} + 205891132094649 q^{27} + (20187574 \beta_{2} - 102873171 \beta_1 - 10\!\cdots\!78) q^{29}+ \cdots + (1450502310816 \beta_{2} + \cdots - 71\!\cdots\!96) q^{99}+O(q^{100})$$ q + 59049 * q^3 + (-b1 - 1611042) * q^5 + (11*b2 - 25*b1 - 90477008) * q^7 + 3486784401 * q^9 + (416*b2 - 3842*b1 - 20398219396) * q^11 + (-6677*b2 - 32706*b1 - 198162067474) * q^13 + (-59049*b1 - 95130419058) * q^15 + (96866*b2 - 258706*b1 + 474939839778) * q^17 + (107426*b2 + 1737246*b1 + 11698275227268) * q^19 + (649539*b2 - 1476225*b1 - 5342576845392) * q^21 + (1820478*b2 - 15793522*b1 + 31637014357864) * q^23 + (-2291630*b2 + 7718556*b1 - 204593341765233) * q^25 + 205891132094649 * q^27 + (20187574*b2 - 102873171*b1 - 1097578327023578) * q^29 + (6305123*b2 - 232310605*b1 + 642886046483048) * q^31 + (24564384*b2 - 226866258*b1 - 1204494457114404) * q^33 + (-25551020*b2 + 793282074*b1 + 6043297584868768) * q^35 + (-544654759*b2 + 266214044*b1 + 7395091687227574) * q^37 + (-394270173*b2 - 1931256594*b1 - 11701271922272226) * q^39 + (751945106*b2 + 295010730*b1 + 55798063961323706) * q^41 + (279155250*b2 - 2529920390*b1 - 23182538497120836) * q^43 + (-3486784401*b1 - 5617356114955842) * q^45 + (1563213102*b2 + 13246829770*b1 + 16370206324631584) * q^47 + (5101586050*b2 + 18916901816*b1 + 324440254781237465) * q^49 + (5719840434*b2 - 15276330594*b1 + 28044722599051122) * q^51 + (-12036406074*b2 - 44307999143*b1 + 162797113982359358) * q^53 + (-7604103580*b2 + 64667721080*b1 + 1036945183178565000) * q^55 + (6343397874*b2 + 102582639054*b1 + 690771453894948132) * q^57 + (-42546775788*b2 + 10719102072*b1 + 615381602048496284) * q^59 + (8797622009*b2 + 38029614744*b1 + 2727902485125824334) * q^61 + (38354628411*b2 - 87169610025*b1 - 315473820143552208) * q^63 + (-94216066890*b2 + 63993270246*b1 + 9650476601793764452) * q^65 + (102579523860*b2 - 498925457672*b1 - 12479645194646343388) * q^67 + (107497405422*b2 - 932591680578*b1 + 1868134060817511336) * q^69 + (-142409915406*b2 + 191667091974*b1 - 21955081267812597032) * q^71 + (159635580038*b2 - 1415921429192*b1 + 29471821831390618282) * q^73 + (-135318459870*b2 + 455773013244*b1 - 12081032237895243417) * q^75 + (-16487939864*b2 + 3320932594804*b1 + 52011395582823575360) * q^77 + (864676356247*b2 - 249240763601*b1 - 73793718943275133640) * q^79 + 12157665459056928801 * q^81 + (-858087655964*b2 + 773807829526*b1 - 101733638042376651900) * q^83 + (-313358368400*b2 + 5949442881202*b1 + 61565110214605858684) * q^85 + (1192056057126*b2 - 6074557874379*b1 - 64810902632415257322) * q^87 + (-1901985488876*b2 - 3095450705836*b1 + 109185486032927678490) * q^89 + (-6255447523578*b2 + 30899162316082*b1 - 230694618284047122272) * q^91 + (372311208027*b2 - 13717708914645*b1 + 37961778158777501352) * q^93 + (4291095254160*b2 - 16936084762256*b1 - 495531264867991409032) * q^95 + (3277635789560*b2 + 31914611948564*b1 + 660014388924788363586) * q^97 + (1450502310816*b2 - 13396225668642*b1 - 71124193198148441796) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 177147 q^{3} - 4833126 q^{5} - 271431024 q^{7} + 10460353203 q^{9}+O(q^{10})$$ 3 * q + 177147 * q^3 - 4833126 * q^5 - 271431024 * q^7 + 10460353203 * q^9 $$3 q + 177147 q^{3} - 4833126 q^{5} - 271431024 q^{7} + 10460353203 q^{9} - 61194658188 q^{11} - 594486202422 q^{13} - 285391257174 q^{15} + 1424819519334 q^{17} + 35094825681804 q^{19} - 16027730536176 q^{21} + 94911043073592 q^{23} - 613780025295699 q^{25} + 617673396283947 q^{27} - 32\!\cdots\!34 q^{29}+ \cdots - 21\!\cdots\!88 q^{99}+O(q^{100})$$ 3 * q + 177147 * q^3 - 4833126 * q^5 - 271431024 * q^7 + 10460353203 * q^9 - 61194658188 * q^11 - 594486202422 * q^13 - 285391257174 * q^15 + 1424819519334 * q^17 + 35094825681804 * q^19 - 16027730536176 * q^21 + 94911043073592 * q^23 - 613780025295699 * q^25 + 617673396283947 * q^27 - 3292734981070734 * q^29 + 1928658139449144 * q^31 - 3613483371343212 * q^33 + 18129892754606304 * q^35 + 22185275061682722 * q^37 - 35103815766816678 * q^39 + 167394191883971118 * q^41 - 69547615491362508 * q^43 - 16852068344867526 * q^45 + 49110618973894752 * q^47 + 973320764343712395 * q^49 + 84134167797153366 * q^51 + 488391341947078074 * q^53 + 3110835549535695000 * q^55 + 2072314361684844396 * q^57 + 1846144806145488852 * q^59 + 8183707455377473002 * q^61 - 946421460430656624 * q^63 + 28951429805381293356 * q^65 - 37438935583939030164 * q^67 + 5604402182452534008 * q^69 - 65865243803437791096 * q^71 + 88415465494171854846 * q^73 - 36243096713685730251 * q^75 + 156034186748470726080 * q^77 - 221381156829825400920 * q^79 + 36472996377170786403 * q^81 - 305200914127129955700 * q^83 + 184695330643817576052 * q^85 - 194432707897245771966 * q^87 + 327556458098783035470 * q^89 - 692083854852141366816 * q^91 + 113885334476332504056 * q^93 - 1486593794603974227096 * q^95 + 1980043166774365090758 * q^97 - 213372579594445325388 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 53560x - 70812$$ :

 $$\beta_{1}$$ $$=$$ $$( 768\nu^{2} + 948864\nu - 27422720 ) / 11$$ (768*v^2 + 948864*v - 27422720) / 11 $$\beta_{2}$$ $$=$$ $$-3072\nu^{2} + 75264\nu + 109690880$$ -3072*v^2 + 75264*v + 109690880
 $$\nu$$ $$=$$ $$( \beta_{2} + 44\beta_1 ) / 3870720$$ (b2 + 44*b1) / 3870720 $$\nu^{2}$$ $$=$$ $$( -353\beta_{2} + 308\beta _1 + 39488716800 ) / 1105920$$ (-353*b2 + 308*b1 + 39488716800) / 1105920

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
 232.089 −1.32215 −230.766
0 59049.0 0 −2.28989e7 0 −1.04414e9 0 3.48678e9 0
1.2 0 59049.0 0 995860. 0 1.18014e9 0 3.48678e9 0
1.3 0 59049.0 0 1.70699e7 0 −4.07437e8 0 3.48678e9 0
 $$n$$: e.g. 2-40 or 990-1000 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 48.22.a.k 3
4.b odd 2 1 24.22.a.b 3
12.b even 2 1 72.22.a.e 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.22.a.b 3 4.b odd 2 1
48.22.a.k 3 1.a even 1 1 trivial
72.22.a.e 3 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} + 4833126T_{5}^{2} - 396686171190900T_{5} + 389262902874527045000$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(48))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 59049)^{3}$$
$5$ $$T^{3} + 4833126 T^{2} + \cdots + 38\!\cdots\!00$$
$7$ $$T^{3} + 271431024 T^{2} + \cdots - 50\!\cdots\!00$$
$11$ $$T^{3} + 61194658188 T^{2} + \cdots + 90\!\cdots\!00$$
$13$ $$T^{3} + 594486202422 T^{2} + \cdots - 47\!\cdots\!16$$
$17$ $$T^{3} - 1424819519334 T^{2} + \cdots - 16\!\cdots\!52$$
$19$ $$T^{3} - 35094825681804 T^{2} + \cdots + 23\!\cdots\!64$$
$23$ $$T^{3} - 94911043073592 T^{2} + \cdots + 20\!\cdots\!88$$
$29$ $$T^{3} + \cdots - 34\!\cdots\!24$$
$31$ $$T^{3} + \cdots + 39\!\cdots\!00$$
$37$ $$T^{3} + \cdots + 74\!\cdots\!76$$
$41$ $$T^{3} + \cdots + 15\!\cdots\!40$$
$43$ $$T^{3} + \cdots + 51\!\cdots\!56$$
$47$ $$T^{3} + \cdots + 12\!\cdots\!00$$
$53$ $$T^{3} + \cdots - 61\!\cdots\!40$$
$59$ $$T^{3} + \cdots + 34\!\cdots\!00$$
$61$ $$T^{3} + \cdots - 15\!\cdots\!24$$
$67$ $$T^{3} + \cdots - 17\!\cdots\!24$$
$71$ $$T^{3} + \cdots + 72\!\cdots\!00$$
$73$ $$T^{3} + \cdots + 15\!\cdots\!00$$
$79$ $$T^{3} + \cdots - 31\!\cdots\!36$$
$83$ $$T^{3} + \cdots + 57\!\cdots\!32$$
$89$ $$T^{3} + \cdots + 32\!\cdots\!88$$
$97$ $$T^{3} + \cdots + 19\!\cdots\!32$$