# Properties

 Label 48.28.a.j Level $48$ Weight $28$ Character orbit 48.a Self dual yes Analytic conductor $221.691$ Analytic rank $1$ 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] = [48,28,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: $$3$$ Coefficient field: $$\mathbb{Q}[x]/(x^{3} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 722453411x + 3765328295265$$ x^3 - x^2 - 722453411*x + 3765328295265 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 1594323 q^{3} + (\beta_1 + 388114470) q^{5} + ( - 41 \beta_{2} - 6 \beta_1 - 66610359536) q^{7} + 2541865828329 q^{9}+O(q^{10})$$ q + 1594323 * q^3 + (b1 + 388114470) * q^5 + (-41*b2 - 6*b1 - 66610359536) * q^7 + 2541865828329 * q^9 $$q + 1594323 q^{3} + (\beta_1 + 388114470) q^{5} + ( - 41 \beta_{2} - 6 \beta_1 - 66610359536) q^{7} + 2541865828329 q^{9} + (19936 \beta_{2} + \cdots - 56465695369420) q^{11}+ \cdots + (50\!\cdots\!44 \beta_{2} + \cdots - 14\!\cdots\!80) q^{99}+O(q^{100})$$ q + 1594323 * q^3 + (b1 + 388114470) * q^5 + (-41*b2 - 6*b1 - 66610359536) * q^7 + 2541865828329 * q^9 + (19936*b2 - 24002*b1 - 56465695369420) * q^11 + (44213*b2 + 288847*b1 + 633129265433990) * q^13 + (1594323*b1 + 618779826153810) * q^15 + (3563974*b2 - 4855728*b1 - 1410810653835774) * q^17 + (-48611102*b2 - 69508384*b1 - 97284763064015316) * q^19 + (-65367243*b2 - 9565938*b1 - 106198428246514128) * q^21 + (45032022*b2 - 208192388*b1 + 130041757657738072) * q^23 + (47339750*b2 + 2950178410*b1 + 2078930821444525575) * q^25 + 4052555153018976267 * q^27 + (-4425691030*b2 + 12719108677*b1 + 933783444904209022) * q^29 + (6098427223*b2 - 19761776610*b1 - 6900913935214504) * q^31 + (31784423328*b2 - 38266940646*b1 - 90024556838459802660) * q^33 + (72935196500*b2 - 50269491554*b1 - 3362557360909822880) * q^35 + (369786350239*b2 - 323464758981*b1 - 368345790193721731682) * q^37 + (70489802799*b2 + 460515415581*b1 + 1009412549854515238770) * q^39 + (-863635627898*b2 - 980978323372*b1 - 1722367912177055146774) * q^41 + (1055975825442*b2 + 162235591820*b1 + 8009433297508457512308) * q^43 + (2541865828329*b1 + 986534908773020820630) * q^45 + (-3754821002154*b2 - 11955045062536*b1 + 13111966670072858277472) * q^47 + (5694413890214*b2 - 8008007492674*b1 - 22892447457743459795479) * q^49 + (5682125719602*b2 - 7741598832144*b1 - 2249287874055412711002) * q^51 + (-5008923894678*b2 - 19554655495663*b1 - 67696147374730777773466) * q^53 + (-36738655239500*b2 - 133380729285012*b1 - 285325102576461852077640) * q^55 + (-77501797973946*b2 - 110818815304032*b1 - 155103335302510090651068) * q^57 + (65734583680308*b2 + 166801124323404*b1 + 226353361156468957145300) * q^59 + (112382423947927*b2 - 526231468395337*b1 - 100830680232093339182394) * q^61 + (-104216498961489*b2 - 15251194969974*b1 - 169314596717267144095344) * q^63 + (-65283178086750*b2 + 1338975260653524*b1 + 2869852068856891580215780) * q^65 + (5761238021844*b2 + 282803405601068*b1 - 1878996090324061281094708) * q^67 + (71795588411106*b2 - 331925912613324*b1 + 207328565194157936165256) * q^69 + (370120732057338*b2 - 2697545500709256*b1 - 3165745754861020426747160) * q^71 + (1560663794484610*b2 + 578306037531706*b1 - 6180635953381748616128102) * q^73 + (75474852239250*b2 + 4703537293166430*b1 + 3314487224037900348310725) * q^75 + (-1517573912355688*b2 + 5717513012302212*b1 - 15507702927632055701057728) * q^77 + (8668623079981835*b2 + 8525938944707310*b1 - 7405513401565267905266168) * q^79 + 6461081889226673298932241 * q^81 + (-22019963243962588*b2 + 2313440640042090*b1 - 15747772383421044014962644) * q^83 + (-6594538457878000*b2 - 16608208563529602*b1 - 52935421024904599675861940) * q^85 + (-7055981000022690*b2 + 20278367503240671*b1 + 1488752423230013240582106) * q^87 + (-9903729677530084*b2 + 41947748635320056*b1 - 39136746657900943304200854) * q^89 + (-4065334319024394*b2 - 8269602997390696*b1 - 77213235065510165793420704) * q^91 + (9722862785455029*b2 - 31506654970185030*b1 - 11002285807932993660792) * q^93 + (83520897818706000*b2 - 237769294327144192*b1 - 596283831721233075578533240) * q^95 + (40647405651784552*b2 - 38436116767756668*b1 + 215471522722858142189001858) * q^97 + (50674637153566944*b2 - 61009863611552658*b1 - 143528221532363747956299180) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 4782969 q^{3} + 1164343410 q^{5} - 199831078608 q^{7} + 7625597484987 q^{9}+O(q^{10})$$ 3 * q + 4782969 * q^3 + 1164343410 * q^5 - 199831078608 * q^7 + 7625597484987 * q^9 $$3 q + 4782969 q^{3} + 1164343410 q^{5} - 199831078608 q^{7} + 7625597484987 q^{9} - 169397086108260 q^{11} + 18\!\cdots\!70 q^{13}+ \cdots - 43\!\cdots\!40 q^{99}+O(q^{100})$$ 3 * q + 4782969 * q^3 + 1164343410 * q^5 - 199831078608 * q^7 + 7625597484987 * q^9 - 169397086108260 * q^11 + 1899387796301970 * q^13 + 1856339478461430 * q^15 - 4232431961507322 * q^17 - 291854289192045948 * q^19 - 318595284739542384 * q^21 + 390125272973214216 * q^23 + 6236792464333576725 * q^25 + 12157665459056928801 * q^27 + 2801350334712627066 * q^29 - 20702741805643512 * q^31 - 270073670515379407980 * q^33 - 10087672082729468640 * q^35 - 1105037370581165195046 * q^37 + 3028237649563545716310 * q^39 - 5167103736531165440322 * q^41 + 24028299892525372536924 * q^43 + 2959604726319062461890 * q^45 + 39335900010218574832416 * q^47 - 68677342373230379386437 * q^49 - 6747863622166238133006 * q^51 - 203088442124192333320398 * q^53 - 855975307729385556232920 * q^55 - 465310005907530271953204 * q^57 + 679060083469406871435900 * q^59 - 302492040696280017547182 * q^61 - 507943790151801432286032 * q^63 + 8609556206570674740647340 * q^65 - 5636988270972183843284124 * q^67 + 621985695582473808495768 * q^69 - 9497237264583061280241480 * q^71 - 18541907860145245848384306 * q^73 + 9943461672113701044932175 * q^75 - 46523108782896167103173184 * q^77 - 22216540204695803715798504 * q^79 + 19383245667680019896796723 * q^81 - 47243317150263132044887932 * q^83 - 158806263074713799027585820 * q^85 + 4466257269690039721746318 * q^87 - 117410239973702829912602562 * q^89 - 231639705196530497380262112 * q^91 - 33006857423798980982376 * q^93 - 1788851495163699226735599720 * q^95 + 646414568168574426567005574 * q^97 - 430584664597091243868897540 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 722453411x + 3765328295265$$ :

 $$\beta_{1}$$ $$=$$ $$( 128\nu^{2} - 3572224\nu - 61648167040 ) / 35$$ (128*v^2 - 3572224*v - 61648167040) / 35 $$\beta_{2}$$ $$=$$ $$( -13952\nu^{2} - 152528384\nu + 6719830840960 ) / 875$$ (-13952*v^2 - 152528384*v + 6719830840960) / 875
 $$\nu$$ $$=$$ $$( -25\beta_{2} - 109\beta _1 + 5160960 ) / 15482880$$ (-25*b2 - 109*b1 + 5160960) / 15482880 $$\nu^{2}$$ $$=$$ $$( -174425\beta_{2} + 297907\beta _1 + 1864276579307520 ) / 3870720$$ (-174425*b2 + 297907*b1 + 1864276579307520) / 3870720

## 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
 5433.91 23746.9 −29179.8
0 1.59432e6 0 −1.81988e9 0 −3.10094e11 0 2.54187e12 0
1.2 0 1.59432e6 0 −1.73463e9 0 1.69635e11 0 2.54187e12 0
1.3 0 1.59432e6 0 4.71886e9 0 −5.93719e10 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.j 3
4.b odd 2 1 24.28.a.a 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.28.a.a 3 4.b odd 2 1
48.28.a.j 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - 1164343410T_{5}^{2} - 13616419339347316500T_{5} - 14896611012349322470342375000$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 1594323)^{3}$$
$5$ $$T^{3} + \cdots - 14\!\cdots\!00$$
$7$ $$T^{3} + \cdots - 31\!\cdots\!20$$
$11$ $$T^{3} + \cdots - 24\!\cdots\!16$$
$13$ $$T^{3} + \cdots + 11\!\cdots\!88$$
$17$ $$T^{3} + \cdots - 84\!\cdots\!28$$
$19$ $$T^{3} + \cdots - 28\!\cdots\!24$$
$23$ $$T^{3} + \cdots + 22\!\cdots\!20$$
$29$ $$T^{3} + \cdots - 63\!\cdots\!48$$
$31$ $$T^{3} + \cdots + 64\!\cdots\!00$$
$37$ $$T^{3} + \cdots - 93\!\cdots\!00$$
$41$ $$T^{3} + \cdots - 13\!\cdots\!96$$
$43$ $$T^{3} + \cdots - 21\!\cdots\!76$$
$47$ $$T^{3} + \cdots + 33\!\cdots\!00$$
$53$ $$T^{3} + \cdots - 92\!\cdots\!00$$
$59$ $$T^{3} + \cdots + 11\!\cdots\!48$$
$61$ $$T^{3} + \cdots + 17\!\cdots\!04$$
$67$ $$T^{3} + \cdots + 40\!\cdots\!52$$
$71$ $$T^{3} + \cdots + 41\!\cdots\!08$$
$73$ $$T^{3} + \cdots - 18\!\cdots\!92$$
$79$ $$T^{3} + \cdots + 45\!\cdots\!32$$
$83$ $$T^{3} + \cdots + 19\!\cdots\!04$$
$89$ $$T^{3} + \cdots - 21\!\cdots\!40$$
$97$ $$T^{3} + \cdots - 67\!\cdots\!76$$