# Properties

 Label 16.28.a.f Level $16$ Weight $28$ Character orbit 16.a Self dual yes Analytic conductor $73.897$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,28,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("16.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$16 = 2^{4}$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 16.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: $$73.8968919741$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 8494973x^{2} - 3687596342x + 10439241475305$$ x^4 - 8494973*x^2 - 3687596342*x + 10439241475305 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{42}\cdot 3^{5}\cdot 5$$ Twist minimal: no (minimal twist has level 8) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_1 + 30628) q^{3} + (\beta_{2} + 49 \beta_1 + 886016542) q^{5} + ( - \beta_{3} - 18 \beta_{2} + \cdots + 52941759144) q^{7}+ \cdots + ( - 28 \beta_{3} + \cdots + 4937232626005) q^{9}+O(q^{10})$$ q + (b1 + 30628) * q^3 + (b2 + 49*b1 + 886016542) * q^5 + (-b3 - 18*b2 + 462*b1 + 52941759144) * q^7 + (-28*b3 + 3406*b2 - 104746*b1 + 4937232626005) * q^9 $$q + (\beta_1 + 30628) q^{3} + (\beta_{2} + 49 \beta_1 + 886016542) q^{5} + ( - \beta_{3} - 18 \beta_{2} + \cdots + 52941759144) q^{7}+ \cdots + ( - 10\!\cdots\!64 \beta_{3} + \cdots + 13\!\cdots\!04) q^{99}+O(q^{100})$$ q + (b1 + 30628) * q^3 + (b2 + 49*b1 + 886016542) * q^5 + (-b3 - 18*b2 + 462*b1 + 52941759144) * q^7 + (-28*b3 + 3406*b2 - 104746*b1 + 4937232626005) * q^9 + (-322*b3 + 13084*b2 + 16647187*b1 + 34250584726412) * q^11 + (-1736*b3 - 125663*b2 - 322180223*b1 - 1395221674250) * q^13 + (-1575*b3 + 509634*b2 + 2721497866*b1 + 639286770543928) * q^15 + (34580*b3 + 15335310*b2 - 6358612746*b1 + 11053732679204786) * q^17 + (184562*b3 + 65699396*b2 - 8809649587*b1 + 44260694343066868) * q^19 + (106200*b3 - 1443780*b2 + 241644379116*b1 + 7480285800459168) * q^21 + (-2316951*b3 - 601845278*b2 - 80415881318*b1 - 441773147668481416) * q^23 + (-7889000*b3 - 926795732*b2 + 434331683932*b1 + 149564995754873831) * q^25 + (5474966*b3 + 898862668*b2 + 9813907864802*b1 - 1409828779456013144) * q^27 + (88382560*b3 - 2262912995*b2 + 8092879139149*b1 + 1557619870440630054) * q^29 + (132596744*b3 - 9524379888*b2 + 14072537902728*b1 - 52981398005825257568) * q^31 + (-431592084*b3 + 62199662778*b2 + 127558495434194*b1 + 210123096091341793712) * q^33 + (-1634450900*b3 + 200172725656*b2 + 11749694361244*b1 - 73937915853642784848) * q^35 + (140685944*b3 - 166182141123*b2 - 224874677101731*b1 + 828919876513776043134) * q^37 + (9247032585*b3 - 1134943759710*b2 + 195992671705762*b1 - 4046820705944806256744) * q^39 + (11338613048*b3 - 491454070516*b2 - 1608897892458628*b1 + 4185670897173641350650) * q^41 + (-27719987484*b3 + 2049726602888*b2 - 3114238362428029*b1 - 7232275215645202186708) * q^43 + (-76123511800*b3 + 1823460553369*b2 + 1185545780880281*b1 + 27448603936737377950198) * q^45 + (20804144046*b3 + 2774159420348*b2 - 6369607546820380*b1 - 31801159764421106329488) * q^47 + (259683662416*b3 + 6298223254328*b2 - 935372535927592*b1 + 38696307623296904964905) * q^49 + (170934721398*b3 - 16510405916916*b2 + 32489817563386490*b1 - 79589333104381772755960) * q^51 + (-473359132344*b3 - 27706050229987*b2 + 951755918419805*b1 + 69888036766084504908718) * q^53 + (-699765462025*b3 + 54892765786974*b2 + 54366214633423326*b1 + 129637099463950128444008) * q^55 + (212019622164*b3 - 8069589026298*b2 + 125520517192098286*b1 - 109531449920733470513072) * q^57 + (946915743480*b3 - 301395724563600*b2 - 130982126586344217*b1 + 186939383036484964942460) * q^59 + (874469169560*b3 + 81360949116665*b2 + 31975855935852953*b1 - 620407886818097124381818) * q^61 + (847583768679*b3 + 959471929840062*b2 - 54982961355483210*b1 + 2632032651947673148828104) * q^63 + (-1591156253400*b3 + 272860011224772*b2 - 887009928108046572*b1 - 1049354551487162653938476) * q^65 + (-4630431244110*b3 - 884526981065020*b2 - 33747440851692823*b1 + 3513491038995609260496100) * q^67 + (2641385403720*b3 - 472869986941260*b2 - 1025369690318337724*b1 - 1021683498832473197208352) * q^69 + (6391312939515*b3 - 1666819342293210*b2 - 1318273579991078802*b1 + 3540230843900858727382504) * q^71 + (-16834355623372*b3 - 2750054169829946*b2 + 3492041141880775918*b1 + 6439189347417120279556186) * q^73 + (-11062110118500*b3 + 1186549621838712*b2 + 134256116442529463*b1 + 5463698846447061436670204) * q^75 + (54415224239752*b3 + 5784833921667156*b2 + 3683844652674239844*b1 + 32340050179630770816945120) * q^77 + (34110860366086*b3 + 12893406423383148*b2 + 10398899960389511724*b1 + 8119927871243037970902032) * q^79 + (-62087419780036*b3 + 7744204426847122*b2 - 1498839939001184950*b1 + 85585712608965996226633609) * q^81 + (-12037371892448*b3 - 29368549193459904*b2 + 9126262132834870149*b1 - 19883873997209236738985420) * q^83 + (-48425400627800*b3 - 24092676638215050*b2 - 13236339660236972250*b1 + 109918633620265253436091100) * q^85 + (-236347839352107*b3 + 26510176547157114*b2 - 23320633607901941022*b1 + 101717739667587230689558104) * q^87 + (207176874633652*b3 - 8452946569849114*b2 + 31248907101142134542*b1 + 135949463478635416831695882) * q^89 + (571216379591220*b3 - 7507787758004440*b2 - 79662025737271424020*b1 + 185858292522357469160848752) * q^91 + (-407410149349728*b3 + 44248737439886256*b2 - 101857750722289963280*b1 + 175187962929466453620204160) * q^93 + (-171881966769975*b3 - 101633952875844574*b2 - 7180589518265149726*b1 + 479416146229019241043696792) * q^95 + (1356147957029348*b3 + 10527142634383974*b2 + 41082841385399403630*b1 - 102234360744353909700476638) * q^97 + (-1078980896491064*b3 + 357369594813647888*b2 + 276269462835413773927*b1 + 1347417653035558384135041404) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 122512 q^{3} + 3544066168 q^{5} + 211767036576 q^{7} + 19748930504020 q^{9}+O(q^{10})$$ 4 * q + 122512 * q^3 + 3544066168 * q^5 + 211767036576 * q^7 + 19748930504020 * q^9 $$4 q + 122512 q^{3} + 3544066168 q^{5} + 211767036576 q^{7} + 19748930504020 q^{9} + 137002338905648 q^{11} - 5580886697000 q^{13} + 25\!\cdots\!12 q^{15}+ \cdots + 53\!\cdots\!16 q^{99}+O(q^{100})$$ 4 * q + 122512 * q^3 + 3544066168 * q^5 + 211767036576 * q^7 + 19748930504020 * q^9 + 137002338905648 * q^11 - 5580886697000 * q^13 + 2557147082175712 * q^15 + 44214930716819144 * q^17 + 177042777372267472 * q^19 + 29921143201836672 * q^21 - 1767092590673925664 * q^23 + 598259983019495324 * q^25 - 5639315117824052576 * q^27 + 6230479481762520216 * q^29 - 211925592023301030272 * q^31 + 840492384365367174848 * q^33 - 295751663414571139392 * q^35 + 3315679506055104172536 * q^37 - 16187282823779225026976 * q^39 + 16742683588694565402600 * q^41 - 28929100862580808746832 * q^43 + 109794415746949511800792 * q^45 - 127204639057684425317952 * q^47 + 154785230493187619859620 * q^49 - 318357332417527091023840 * q^51 + 279552147064338019634872 * q^53 + 518548397855800513776032 * q^55 - 438125799682933882052288 * q^57 + 747757532145939859769840 * q^59 - 2481631547272388497527272 * q^61 + 10528130607790692595312416 * q^63 - 4197418205948650615753904 * q^65 + 14053964155982437041984400 * q^67 - 4086733995329892788833408 * q^69 + 14160923375603434909530016 * q^71 + 25756757389668481118224744 * q^73 + 21854795385788245746680816 * q^75 + 129360200718523083267780480 * q^77 + 32479711484972151883608128 * q^79 + 342342850435863984906534436 * q^81 - 79535495988836946955941680 * q^83 + 439674534481061013744364400 * q^85 + 406870958670348922758232416 * q^87 + 543797853914541667326783528 * q^89 + 743433170089429876643395008 * q^91 + 700751851717865814480816640 * q^93 + 1917664584916076964174787168 * q^95 - 408937442977415638801906552 * q^97 + 5389670612142233536540165616 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 8494973x^{2} - 3687596342x + 10439241475305$$ :

 $$\beta_{1}$$ $$=$$ $$( -6400\nu^{3} + 12233856\nu^{2} + 30527542016\nu - 34262675761344 ) / 2193191$$ (-6400*v^3 + 12233856*v^2 + 30527542016*v - 34262675761344) / 2193191 $$\beta_{2}$$ $$=$$ $$( 630016\nu^{3} + 951693696\nu^{2} - 5622508535552\nu - 5784739648646208 ) / 2193191$$ (630016*v^3 + 951693696*v^2 - 5622508535552*v - 5784739648646208) / 2193191 $$\beta_{3}$$ $$=$$ $$( 12169728\nu^{3} + 101784758016\nu^{2} + 162699231189504\nu - 465987168920678016 ) / 2193191$$ (12169728*v^3 + 101784758016*v^2 + 162699231189504*v - 465987168920678016) / 2193191
 $$\nu$$ $$=$$ $$( \beta_{3} - 58\beta_{2} - 3808\beta_1 ) / 169869312$$ (b3 - 58*b2 - 3808*b1) / 169869312 $$\nu^{2}$$ $$=$$ $$( 607\beta_{3} + 51194\beta_{2} + 6193760\beta _1 + 360758804742144 ) / 84934656$$ (607*b3 + 51194*b2 + 6193760*b1 + 360758804742144) / 84934656 $$\nu^{3}$$ $$=$$ $$( 3545269\beta_{3} - 40468546\beta_{2} - 26348249248\beta _1 + 234903545080971264 ) / 84934656$$ (3545269*b3 - 40468546*b2 - 26348249248*b1 + 234903545080971264) / 84934656

## 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
 −1814.94 2921.31 955.106 −2061.48
0 −5.03426e6 0 2.36507e9 0 2.46921e11 0 1.77182e13 0
1.2 0 −75899.9 0 1.61881e9 0 −4.99032e11 0 −7.61984e12 0
1.3 0 248705. 0 −3.54329e9 0 2.27408e11 0 −7.56374e12 0
1.4 0 4.98397e6 0 3.10348e9 0 2.36469e11 0 1.72143e13 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
$$2$$ $$+1$$

## 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 16.28.a.f 4
4.b odd 2 1 8.28.a.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.28.a.b 4 4.b odd 2 1
16.28.a.f 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 122512T_{3}^{3} - 25118155626912T_{3}^{2} + 4334841990686082816T_{3} + 473627677716066582339840$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + \cdots + 47\!\cdots\!40$$
$5$ $$T^{4} + \cdots - 42\!\cdots\!00$$
$7$ $$T^{4} + \cdots - 66\!\cdots\!32$$
$11$ $$T^{4} + \cdots - 13\!\cdots\!56$$
$13$ $$T^{4} + \cdots + 17\!\cdots\!12$$
$17$ $$T^{4} + \cdots - 37\!\cdots\!00$$
$19$ $$T^{4} + \cdots - 74\!\cdots\!16$$
$23$ $$T^{4} + \cdots - 43\!\cdots\!12$$
$29$ $$T^{4} + \cdots - 26\!\cdots\!76$$
$31$ $$T^{4} + \cdots - 10\!\cdots\!00$$
$37$ $$T^{4} + \cdots - 12\!\cdots\!96$$
$41$ $$T^{4} + \cdots - 32\!\cdots\!88$$
$43$ $$T^{4} + \cdots - 67\!\cdots\!12$$
$47$ $$T^{4} + \cdots - 27\!\cdots\!64$$
$53$ $$T^{4} + \cdots - 74\!\cdots\!24$$
$59$ $$T^{4} + \cdots - 99\!\cdots\!08$$
$61$ $$T^{4} + \cdots + 40\!\cdots\!00$$
$67$ $$T^{4} + \cdots + 24\!\cdots\!92$$
$71$ $$T^{4} + \cdots - 26\!\cdots\!80$$
$73$ $$T^{4} + \cdots + 28\!\cdots\!48$$
$79$ $$T^{4} + \cdots + 40\!\cdots\!40$$
$83$ $$T^{4} + \cdots - 19\!\cdots\!68$$
$89$ $$T^{4} + \cdots - 73\!\cdots\!56$$
$97$ $$T^{4} + \cdots - 17\!\cdots\!64$$