Properties

 Label 48.28.a.d Level $48$ Weight $28$ Character orbit 48.a Self dual yes Analytic conductor $221.691$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,28,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, 28, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$48 = 2^{4} \cdot 3$$ Weight: $$k$$ $$=$$ $$28$$ 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: $$221.690675922$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{6469})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 1617$$ x^2 - x - 1617 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 3) 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 $$\beta = 2304\sqrt{6469}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 1594323 q^{3} + ( - 12265 \beta - 2453032530) q^{5} + (1429911 \beta + 75828544792) q^{7} + 2541865828329 q^{9}+O(q^{10})$$ q - 1594323 * q^3 + (-12265*b - 2453032530) * q^5 + (1429911*b + 75828544792) * q^7 + 2541865828329 * q^9 $$q - 1594323 q^{3} + ( - 12265 \beta - 2453032530) q^{5} + (1429911 \beta + 75828544792) q^{7} + 2541865828329 q^{9} + ( - 656941714 \beta - 20427496063524) q^{11} + (3567952674 \beta - 208698949319146) q^{13} + (19554371595 \beta + 39\!\cdots\!90) q^{15}+ \cdots + ( - 16\!\cdots\!06 \beta - 51\!\cdots\!96) q^{99}+O(q^{100})$$ q - 1594323 * q^3 + (-12265*b - 2453032530) * q^5 + (1429911*b + 75828544792) * q^7 + 2541865828329 * q^9 + (-656941714*b - 20427496063524) * q^11 + (3567952674*b - 208698949319146) * q^13 + (19554371595*b + 3910926182327190) * q^15 + (2853017070*b - 36897826051582254) * q^17 + (-783061849386*b + 173451741177864292) * q^19 + (-2279739995253*b - 120895193018415816) * q^21 + (7595208665902*b - 1248812638965342216) * q^23 + (60172887960900*b + 3732583449981291175) * q^25 - 4052555153018976267 * q^27 + (-152872849116923*b - 33433408911012442794) * q^29 + (-499542733238373*b + 65685706043778749104) * q^31 + (1047377284289622*b + 32568026806485774252) * q^33 + (-4437655299878710*b - 788262454786128703920) * q^35 + (-8373244906250136*b - 123789443532110024002) * q^37 + (-5688469011069702*b + 332733534975348808158) * q^39 + (16980656511166538*b + 2537681232804171705306) * q^41 + (8348479676033634*b + 17931332733846232816060) * q^43 + (-31175984384455185*b - 6235279563786432542370) * q^45 + (228640191289222922*b - 24259072480042470758640) * q^47 + (216856140624147024*b + 10251023807090251550505) * q^49 + (-4548630734093610*b + 58827052724036773944042) * q^51 + (-96353340868082727*b + 175192964437097217042846) * q^53 + (1862042633975078280*b + 326801242191718767175560) * q^55 + (1248453516898635678*b - 276538100349916131614316) * q^57 + (-2302330846791623384*b + 787350543890646470388588) * q^59 + (9211943958760681956*b - 149291596341640301357050) * q^61 + (3634641908451748719*b + 192745986818699759012568) * q^63 + (-6192611361423159530*b - 990811614739393788258060) * q^65 + (4612846614324993000*b + 725894118316839939389044) * q^67 + (-12109215865846874346*b + 1991010712993141297839768) * q^69 + (-37062621880863274410*b + 9463809183694090589406888) * q^71 + (-66990615514511118912*b - 9237799414928468147622694) * q^73 + (-95935019252485970700*b - 5950943643724522089999525) * q^75 + (-79024435509471919852*b - 33807025138929793390864224) * q^77 + (211560135381393150375*b + 14975820442899229584505120) * q^79 + 6461081889226673298932241 * q^81 + (-175051086693981002842*b - 34488732456869689566981468) * q^83 + (445553292841301058210*b + 89309928568436851591263420) * q^85 + (243728699422640028129*b + 53303652795232090832658462) * q^87 + (463031138412085580436*b + 262625927493223096252198218) * q^89 + (-27868264163744202198*b + 159373085478275029681143824) * q^91 + (796432469084802556479*b - 104724231916835466607736592) * q^93 + (-206509416000687014800*b - 95671321056756504140142600) * q^95 + (-3178094627909658237660*b + 396440577316096517257483106) * q^97 + (-1669857694020483015906*b - 51923954202196819062771396) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3188646 q^{3} - 4906065060 q^{5} + 151657089584 q^{7} + 5083731656658 q^{9}+O(q^{10})$$ 2 * q - 3188646 * q^3 - 4906065060 * q^5 + 151657089584 * q^7 + 5083731656658 * q^9 $$2 q - 3188646 q^{3} - 4906065060 q^{5} + 151657089584 q^{7} + 5083731656658 q^{9} - 40854992127048 q^{11} - 417397898638292 q^{13} + 78\!\cdots\!80 q^{15}+ \cdots - 10\!\cdots\!92 q^{99}+O(q^{100})$$ 2 * q - 3188646 * q^3 - 4906065060 * q^5 + 151657089584 * q^7 + 5083731656658 * q^9 - 40854992127048 * q^11 - 417397898638292 * q^13 + 7821852364654380 * q^15 - 73795652103164508 * q^17 + 346903482355728584 * q^19 - 241790386036831632 * q^21 - 2497625277930684432 * q^23 + 7465166899962582350 * q^25 - 8105110306037952534 * q^27 - 66866817822024885588 * q^29 + 131371412087557498208 * q^31 + 65136053612971548504 * q^33 - 1576524909572257407840 * q^35 - 247578887064220048004 * q^37 + 665467069950697616316 * q^39 + 5075362465608343410612 * q^41 + 35862665467692465632120 * q^43 - 12470559127572865084740 * q^45 - 48518144960084941517280 * q^47 + 20502047614180503101010 * q^49 + 117654105448073547888084 * q^51 + 350385928874194434085692 * q^53 + 653602484383437534351120 * q^55 - 553076200699832263228632 * q^57 + 1574701087781292940777176 * q^59 - 298583192683280602714100 * q^61 + 385491973637399518025136 * q^63 - 1981623229478787576516120 * q^65 + 1451788236633679878778088 * q^67 + 3982021425986282595679536 * q^69 + 18927618367388181178813776 * q^71 - 18475598829856936295245388 * q^73 - 11901887287449044179999050 * q^75 - 67614050277859586781728448 * q^77 + 29951640885798459169010240 * q^79 + 12922163778453346597864482 * q^81 - 68977464913739379133962936 * q^83 + 178619857136873703182526840 * q^85 + 106607305590464181665316924 * q^87 + 525251854986446192504396436 * q^89 + 318746170956550059362287648 * q^91 - 209448463833670933215473184 * q^93 - 191342642113513008280285200 * q^95 + 792881154632193034514966212 * q^97 - 103847908404393638125542792 * q^99

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
 40.7150 −39.7150
0 −1.59432e6 0 −4.72587e9 0 3.40807e11 0 2.54187e12 0
1.2 0 −1.59432e6 0 −1.80194e8 0 −1.89150e11 0 2.54187e12 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 48.28.a.d 2
4.b odd 2 1 3.28.a.a 2
12.b even 2 1 9.28.a.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.28.a.a 2 4.b odd 2 1
9.28.a.c 2 12.b even 2 1
48.28.a.d 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4906065060T_{5} + 851573139571282500$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(48))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 1594323)^{2}$$
$5$ $$T^{2} + \cdots + 85\!\cdots\!00$$
$7$ $$T^{2} + \cdots - 64\!\cdots\!20$$
$11$ $$T^{2} + \cdots - 14\!\cdots\!08$$
$13$ $$T^{2} + \cdots - 39\!\cdots\!88$$
$17$ $$T^{2} + \cdots + 13\!\cdots\!16$$
$19$ $$T^{2} + \cdots + 90\!\cdots\!80$$
$23$ $$T^{2} + \cdots - 42\!\cdots\!60$$
$29$ $$T^{2} + \cdots + 31\!\cdots\!20$$
$31$ $$T^{2} + \cdots - 42\!\cdots\!00$$
$37$ $$T^{2} + \cdots - 23\!\cdots\!80$$
$41$ $$T^{2} + \cdots - 34\!\cdots\!40$$
$43$ $$T^{2} + \cdots + 31\!\cdots\!76$$
$47$ $$T^{2} + \cdots - 12\!\cdots\!36$$
$53$ $$T^{2} + \cdots + 30\!\cdots\!00$$
$59$ $$T^{2} + \cdots + 43\!\cdots\!20$$
$61$ $$T^{2} + \cdots - 28\!\cdots\!44$$
$67$ $$T^{2} + \cdots - 20\!\cdots\!64$$
$71$ $$T^{2} + \cdots + 42\!\cdots\!44$$
$73$ $$T^{2} + \cdots - 68\!\cdots\!40$$
$79$ $$T^{2} + \cdots - 13\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 13\!\cdots\!68$$
$89$ $$T^{2} + \cdots + 61\!\cdots\!40$$
$97$ $$T^{2} + \cdots - 18\!\cdots\!64$$