# Properties

 Label 48.28.a.i 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: $$x^{3} - x^{2} - 15574970459x - 459811681510986$$ x^3 - x^2 - 15574970459*x - 459811681510986 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{24}\cdot 3^{6}\cdot 5$$ Twist minimal: no (minimal twist has level 12) 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.

 $$f(q)$$ $$=$$ $$q - 1594323 q^{3} + ( - \beta_1 + 179769774) q^{5} + ( - \beta_{2} - 9 \beta_1 - 68864006888) q^{7} + 2541865828329 q^{9}+O(q^{10})$$ q - 1594323 * q^3 + (-b1 + 179769774) * q^5 + (-b2 - 9*b1 - 68864006888) * q^7 + 2541865828329 * q^9 $$q - 1594323 q^{3} + ( - \beta_1 + 179769774) q^{5} + ( - \beta_{2} - 9 \beta_1 - 68864006888) q^{7} + 2541865828329 q^{9} + ( - 504 \beta_{2} + \cdots + 11888174906844) q^{11}+ \cdots + ( - 12\!\cdots\!16 \beta_{2} + \cdots + 30\!\cdots\!76) q^{99}+O(q^{100})$$ q - 1594323 * q^3 + (-b1 + 179769774) * q^5 + (-b2 - 9*b1 - 68864006888) * q^7 + 2541865828329 * q^9 + (-504*b2 + 26494*b1 + 11888174906844) * q^11 + (-4669*b2 - 293526*b1 + 172914960598550) * q^13 + (1594323*b1 - 286611085393002) * q^15 + (121842*b2 + 2490958*b1 + 7190746829280978) * q^17 + (483322*b2 - 51768522*b1 - 103687512559902620) * q^19 + (1594323*b2 + 14348907*b1 + 109791470053696824) * q^21 + (9212742*b2 + 345943598*b1 - 1187065668859224072) * q^23 + (28309050*b2 - 863007372*b1 + 3252138238435560103) * q^25 - 4052555153018976267 * q^27 + (-144798570*b2 + 3713622125*b1 - 28084838007414690474) * q^29 + (-13181553*b2 - 37650222117*b1 + 44590344306639680176) * q^31 + (803538792*b2 - 42239993562*b1 - 18953590682004246612) * q^33 + (578479500*b2 + 242223576218*b1 + 86166942960964914768) * q^35 + (-405134807*b2 + 421871826672*b1 + 108565789146145206974) * q^37 + (7443894087*b2 + 467975252898*b1 - 275682298726362031650) * q^39 + (19280368674*b2 + 677003578826*b1 + 4303294926031231167450) * q^41 + (5463179946*b2 - 2702594299806*b1 - 2935054561430288131652) * q^43 + (-2541865828329*b1 + 456950645497027127646) * q^45 + (42369918822*b2 + 13301479648938*b1 - 1004622464386479102192) * q^47 + (-30425240054*b2 + 7711205631264*b1 + 7690213262790312021801) * q^49 + (-194255502966*b2 - 3971391631434*b1 - 11464373057099736687894) * q^51 + (-206816903370*b2 + 66538656403745*b1 + 64098653557856819497374) * q^53 + (-586876153500*b2 + 96685910701416*b1 - 279297937711170956580984) * q^55 + (-770571381006*b2 + 82535745300606*b1 + 165311386087041624826260) * q^57 + (541669958676*b2 + 85449818960264*b1 - 1008858393335899285787796) * q^59 + (-988928432751*b2 - 238346397990324*b1 + 1543351470903856760969222) * q^61 + (-2541865828329*b2 - 22876792454961*b1 - 175043065910420081530152) * q^63 + (9820788405750*b2 + 464663196518230*b1 + 3174858375518350615156980) * q^65 + (11189370216948*b2 - 987662980120248*b1 - 2728796635463241470069132) * q^67 + (-14688086463666*b2 - 551545834994154*b1 + 1892566098372644700143256) * q^69 + (13878239295690*b2 + 1421476284728630*b1 - 5426889450451317480995160) * q^71 + (56687151447822*b2 - 221530230854352*b1 + 14555871095131441311526106) * q^73 + (-45133769523150*b2 + 1375912502349156*b1 - 5184958792717297490095269) * q^75 + (-85458430337832*b2 - 4099314946078028*b1 + 30728234723190915392903328) * q^77 + (74656135589203*b2 - 12955858869453273*b1 - 19235626771728392626595552) * q^79 + 6461081889226673298932241 * q^81 + (-241571493258228*b2 - 11032657690011882*b1 + 36244160585342487226438308) * q^83 + (-109956672378000*b2 - 27360530633687598*b1 - 25593031784840193092750148) * q^85 + (230855690518110*b2 - 5920713167196375*b1 + 44776303186495411560579102) * q^87 + (184129546134996*b2 + 12553559372501524*b1 - 77655654957372951069987510) * q^89 + (-445792970913378*b2 + 92881007523340938*b1 + 333452401693358399036637200) * q^91 + (21015653123619*b2 + 60026615076241791*b1 - 71091411505994694817240848) * q^93 + (1309067288802000*b2 - 18443195692166800*b1 + 532536456835763180434270200) * q^95 + (-736518404940472*b2 + 188939356434972612*b1 - 152756880121985289516133534) * q^97 + (-1281100377477816*b2 + 67344193255748526*b1 + 30218145556905056471183676) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 4782969 q^{3} + 539309322 q^{5} - 206592020664 q^{7} + 7625597484987 q^{9}+O(q^{10})$$ 3 * q - 4782969 * q^3 + 539309322 * q^5 - 206592020664 * q^7 + 7625597484987 * q^9 $$3 q - 4782969 q^{3} + 539309322 q^{5} - 206592020664 q^{7} + 7625597484987 q^{9} + 35664524720532 q^{11} + 518744881795650 q^{13} - 859833256179006 q^{15} + 21\!\cdots\!34 q^{17}+ \cdots + 90\!\cdots\!28 q^{99}+O(q^{100})$$ 3 * q - 4782969 * q^3 + 539309322 * q^5 - 206592020664 * q^7 + 7625597484987 * q^9 + 35664524720532 * q^11 + 518744881795650 * q^13 - 859833256179006 * q^15 + 21572240487842934 * q^17 - 311062537679707860 * q^19 + 329374410161090472 * q^21 - 3561197006577672216 * q^23 + 9756414715306680309 * q^25 - 12157665459056928801 * q^27 - 84254514022244071422 * q^29 + 133771032919919040528 * q^31 - 56860772046012739836 * q^33 + 258500828882894744304 * q^35 + 325697367438435620922 * q^37 - 827046896179086094950 * q^39 + 12909884778093693502350 * q^41 - 8805163684290864394956 * q^43 + 1370851936491081382938 * q^45 - 3013867393159437306576 * q^47 + 23070639788370936065403 * q^49 - 34393119171299210063682 * q^51 + 192295960673570458492122 * q^53 - 837893813133512869742952 * q^55 + 495934158261124874478780 * q^57 - 3026575180007697857363388 * q^59 + 4630054412711570282907666 * q^61 - 525129197731260244590456 * q^63 + 9524575126555051845470940 * q^65 - 8186389906389724410207396 * q^67 + 5677698295117934100429768 * q^69 - 16280668351353952442985480 * q^71 + 43667613285394323934578318 * q^73 - 15554876378151892470285807 * q^75 + 92184704169572746178709984 * q^77 - 57706880315185177879786656 * q^79 + 19383245667680019896796723 * q^81 + 108732481756027461679314924 * q^83 - 76779095354520579278250444 * q^85 + 134328909559486234681737306 * q^87 - 232966964872118853209962530 * q^89 + 1000357205080075197109911600 * q^91 - 213274234517984084451722544 * q^93 + 1597609370507289541302810600 * q^95 - 458270640365955868548400602 * q^97 + 90654436670715169413551028 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 15574970459x - 459811681510986$$ :

 $$\beta_{1}$$ $$=$$ $$( -2304\nu^{2} + 536290560\nu + 23922975862272 ) / 14149$$ (-2304*v^2 + 536290560*v + 23922975862272) / 14149 $$\beta_{2}$$ $$=$$ $$( 608256\nu^{2} - 16399411200\nu - 6315707354738688 ) / 14149$$ (608256*v^2 - 16399411200*v - 6315707354738688) / 14149
 $$\nu$$ $$=$$ $$( \beta_{2} + 264\beta _1 + 2949120 ) / 8847360$$ (b2 + 264*b1 + 2949120) / 8847360 $$\nu^{2}$$ $$=$$ $$( 46553\beta_{2} + 1423560\beta _1 + 18372982752608256 ) / 1769472$$ (46553*b2 + 1423560*b1 + 18372982752608256) / 1769472

## 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
 137543. −31536.3 −106006.
0 −1.59432e6 0 −3.64373e9 0 −3.10763e11 0 2.54187e12 0
1.2 0 −1.59432e6 0 −1.53749e8 0 2.95199e11 0 2.54187e12 0
1.3 0 −1.59432e6 0 4.33679e9 0 −1.91028e11 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.i 3
4.b odd 2 1 12.28.a.b 3
12.b even 2 1 36.28.a.c 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.28.a.b 3 4.b odd 2 1
36.28.a.c 3 12.b even 2 1
48.28.a.i 3 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - 539309322T_{5}^{2} - 15908650980641032500T_{5} - 2429549514639064685833875000$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T + 1594323)^{3}$$
$5$ $$T^{3} + \cdots - 24\!\cdots\!00$$
$7$ $$T^{3} + \cdots - 17\!\cdots\!28$$
$11$ $$T^{3} + \cdots + 31\!\cdots\!16$$
$13$ $$T^{3} + \cdots + 22\!\cdots\!00$$
$17$ $$T^{3} + \cdots + 25\!\cdots\!48$$
$19$ $$T^{3} + \cdots - 11\!\cdots\!00$$
$23$ $$T^{3} + \cdots - 10\!\cdots\!52$$
$29$ $$T^{3} + \cdots - 76\!\cdots\!76$$
$31$ $$T^{3} + \cdots + 74\!\cdots\!24$$
$37$ $$T^{3} + \cdots + 11\!\cdots\!76$$
$41$ $$T^{3} + \cdots + 12\!\cdots\!00$$
$43$ $$T^{3} + \cdots - 65\!\cdots\!92$$
$47$ $$T^{3} + \cdots - 35\!\cdots\!12$$
$53$ $$T^{3} + \cdots + 10\!\cdots\!76$$
$59$ $$T^{3} + \cdots + 85\!\cdots\!36$$
$61$ $$T^{3} + \cdots - 18\!\cdots\!48$$
$67$ $$T^{3} + \cdots - 94\!\cdots\!32$$
$71$ $$T^{3} + \cdots - 26\!\cdots\!00$$
$73$ $$T^{3} + \cdots + 39\!\cdots\!84$$
$79$ $$T^{3} + \cdots - 12\!\cdots\!92$$
$83$ $$T^{3} + \cdots + 29\!\cdots\!88$$
$89$ $$T^{3} + \cdots - 12\!\cdots\!00$$
$97$ $$T^{3} + \cdots + 79\!\cdots\!04$$