# Properties

 Label 48.22.a.l 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: $$x^{3} - x^{2} - 2295485x - 828958533$$ x^3 - x^2 - 2295485*x - 828958533 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.

 $$f(q)$$ $$=$$ $$q + 59049 q^{3} + (\beta_1 + 693342) q^{5} + (\beta_{2} + 401760688) q^{7} + 3486784401 q^{9}+O(q^{10})$$ q + 59049 * q^3 + (b1 + 693342) * q^5 + (b2 + 401760688) * q^7 + 3486784401 * q^9 $$q + 59049 q^{3} + (\beta_1 + 693342) q^{5} + (\beta_{2} + 401760688) q^{7} + 3486784401 q^{9} + ( - 160 \beta_{2} + 1186 \beta_1 - 4613082500) q^{11} + (161 \beta_{2} - 7159 \beta_1 + 239618517230) q^{13} + (59049 \beta_1 + 40941151758) q^{15} + ( - 7450 \beta_{2} - 194532 \beta_1 + 711729947682) q^{17} + (1702 \beta_{2} + 693676 \beta_1 + 13374108228996) q^{19} + (59049 \beta_{2} + 23723566865712) q^{21} + (256122 \beta_{2} - 2178680 \beta_1 - 92808139227544) q^{23} + (709430 \beta_{2} + 15078686 \beta_1 + 449347753517967) q^{25} + 205891132094649 q^{27} + ( - 1261598 \beta_{2} + 33914689 \beta_1 + 147569389330598) q^{29} + (256009 \beta_{2} + 110688876 \beta_1 - 26\!\cdots\!84) q^{31}+ \cdots + ( - 557885504160 \beta_{2} + \cdots - 16\!\cdots\!00) q^{99}+O(q^{100})$$ q + 59049 * q^3 + (b1 + 693342) * q^5 + (b2 + 401760688) * q^7 + 3486784401 * q^9 + (-160*b2 + 1186*b1 - 4613082500) * q^11 + (161*b2 - 7159*b1 + 239618517230) * q^13 + (59049*b1 + 40941151758) * q^15 + (-7450*b2 - 194532*b1 + 711729947682) * q^17 + (1702*b2 + 693676*b1 + 13374108228996) * q^19 + (59049*b2 + 23723566865712) * q^21 + (256122*b2 - 2178680*b1 - 92808139227544) * q^23 + (709430*b2 + 15078686*b1 + 449347753517967) * q^25 + 205891132094649 * q^27 + (-1261598*b2 + 33914689*b1 + 147569389330598) * q^29 + (256009*b2 + 110688876*b1 - 2672023387663384) * q^31 + (-9447840*b2 + 70032114*b1 - 272397908542500) * q^33 + (-12998660*b2 + 789933130*b1 - 41794719080800) * q^35 + (38087851*b2 + 533019825*b1 + 9243113796245686) * q^37 + (9506889*b2 - 422731791*b1 + 14149233823914270) * q^39 + (-87729802*b2 - 55942192*b1 - 41882708395446854) * q^41 + (33793878*b2 - 4268277728*b1 + 76350847166358204) * q^43 + (3486784401*b1 + 2417534070158142) * q^45 + (8794698*b2 + 9105142268*b1 + 149537927356015904) * q^47 + (415348934*b2 - 7943292686*b1 + 121740634482030809) * q^49 + (-439915050*b2 - 11486920068*b1 + 42026941680674418) * q^51 + (-42375918*b2 - 65755502875*b1 + 468735405736089022) * q^53 + (2921169580*b2 - 49659655236*b1 + 1145943088306764168) * q^55 + (100501398*b2 + 40960874124*b1 + 789727716813984804) * q^57 + (-5713027140*b2 + 30558827052*b1 + 614879660490540700) * q^59 + (-2717730677*b2 + 122799957973*b1 - 1098022100116783794) * q^61 + (3486784401*b2 + 1400852899853427888) * q^63 + (-7171593630*b2 + 199129602696*b1 - 6512555420918997148) * q^65 + (-16493691972*b2 + 339564264364*b1 + 11163674564385006884) * q^67 + (15123747978*b2 - 128648875320*b1 - 5480227813247245656) * q^69 + (25542856470*b2 + 132561089748*b1 + 26477006143866293720) * q^71 + (23538904402*b2 - 942963337786*b1 - 15537607635652773206) * q^73 + (41891132070*b2 + 890381329614*b1 + 26533535497482433383) * q^75 + (-23025916232*b2 + 2207787521940*b1 - 85253268703625629376) * q^77 + (27642550565*b2 - 2005927094220*b1 + 65973324568220366008) * q^79 + 12157665459056928801 * q^81 + (-251590170452*b2 - 3811697810658*b1 - 37282842850628980092) * q^83 + (-41166819760*b2 - 4978564484226*b1 - 177198990478568395652) * q^85 + (-74496100302*b2 + 2002628470761*b1 + 8713824870582481302) * q^87 + (-149674305092*b2 - 4152851834800*b1 - 243669391946747827686) * q^89 + (339827267154*b2 - 6934001400116*b1 + 182101552849246638496) * q^91 + (15117075441*b2 + 6536067438924*b1 - 157780309018135161816) * q^93 + (469990845360*b2 + 24013545609824*b1 + 650866370096351323768) * q^95 + (432471338728*b2 - 12488719178940*b1 - 531507306685042064574) * q^97 + (-557885504160*b2 + 4135326299586*b1 - 16084824101526082500) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9}+O(q^{10})$$ 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9 $$3 q + 177147 q^{3} + 2080026 q^{5} + 1205282064 q^{7} + 10460353203 q^{9} - 13839247500 q^{11} + 718855551690 q^{13} + 122823455274 q^{15} + 2135189843046 q^{17} + 40122324686988 q^{19} + 71170700597136 q^{21} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 48\!\cdots\!00 q^{99}+O(q^{100})$$ 3 * q + 177147 * q^3 + 2080026 * q^5 + 1205282064 * q^7 + 10460353203 * q^9 - 13839247500 * q^11 + 718855551690 * q^13 + 122823455274 * q^15 + 2135189843046 * q^17 + 40122324686988 * q^19 + 71170700597136 * q^21 - 278424417682632 * q^23 + 1348043260553901 * q^25 + 617673396283947 * q^27 + 442708167991794 * q^29 - 8016070162990152 * q^31 - 817193725627500 * q^33 - 125384157242400 * q^35 + 27729341388737058 * q^37 + 42447701471742810 * q^39 - 125648125186340562 * q^41 + 229052541499074612 * q^43 + 7252602210474426 * q^45 + 448613782068047712 * q^47 + 365221903446092427 * q^49 + 126080825042023254 * q^51 + 1406206217208267066 * q^53 + 3437829264920292504 * q^55 + 2369183150441954412 * q^57 + 1844638981471622100 * q^59 - 3294066300350351382 * q^61 + 4202558699560283664 * q^63 - 19537666262756991444 * q^65 + 33491023693155020652 * q^67 - 16440683439741736968 * q^69 + 79431018431598881160 * q^71 - 46612822906958319618 * q^73 + 79600606492447300149 * q^75 - 255759806110876888128 * q^77 + 197919973704661098024 * q^79 + 36472996377170786403 * q^81 - 111848528551886940276 * q^83 - 531596971435705186956 * q^85 + 26141474611747443906 * q^87 - 731008175840243483058 * q^89 + 546304658547739915488 * q^91 - 473340927054405485448 * q^93 + 1952599110289053971304 * q^95 - 1594521920055126193722 * q^97 - 48254472304578247500 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 2295485x - 828958533$$ :

 $$\beta_{1}$$ $$=$$ $$( 384\nu^{2} + 177408\nu - 587703424 ) / 19$$ (384*v^2 + 177408*v - 587703424) / 19 $$\beta_{2}$$ $$=$$ $$( -13440\nu^{2} + 13402368\nu + 20563082624 ) / 19$$ (-13440*v^2 + 13402368*v + 20563082624) / 19
 $$\nu$$ $$=$$ $$( \beta_{2} + 35\beta _1 + 344064 ) / 1032192$$ (b2 + 35*b1 + 344064) / 1032192 $$\nu^{2}$$ $$=$$ $$( -11\beta_{2} + 831\beta _1 + 37609234432 ) / 24576$$ (-11*b2 + 831*b1 + 37609234432) / 24576

## 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
 −386.305 −1283.97 1671.27
0 59049.0 0 −3.08294e7 0 1.10597e9 0 3.48678e9 0
1.2 0 59049.0 0 −8.90852e6 0 −5.87822e8 0 3.48678e9 0
1.3 0 59049.0 0 4.18179e7 0 6.87132e8 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.l 3
4.b odd 2 1 24.22.a.c 3
12.b even 2 1 72.22.a.d 3

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{3} - 2080026T_{5}^{2} - 1387114113501300T_{5} - 11485065000261414395000$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$(T - 59049)^{3}$$
$5$ $$T^{3} - 2080026 T^{2} + \cdots - 11\!\cdots\!00$$
$7$ $$T^{3} - 1205282064 T^{2} + \cdots + 44\!\cdots\!00$$
$11$ $$T^{3} + 13839247500 T^{2} + \cdots - 32\!\cdots\!16$$
$13$ $$T^{3} - 718855551690 T^{2} + \cdots + 76\!\cdots\!16$$
$17$ $$T^{3} - 2135189843046 T^{2} + \cdots + 14\!\cdots\!56$$
$19$ $$T^{3} - 40122324686988 T^{2} + \cdots + 15\!\cdots\!44$$
$23$ $$T^{3} + 278424417682632 T^{2} + \cdots - 55\!\cdots\!00$$
$29$ $$T^{3} - 442708167991794 T^{2} + \cdots + 22\!\cdots\!08$$
$31$ $$T^{3} + \cdots - 46\!\cdots\!00$$
$37$ $$T^{3} + \cdots + 27\!\cdots\!00$$
$41$ $$T^{3} + \cdots - 32\!\cdots\!96$$
$43$ $$T^{3} + \cdots + 17\!\cdots\!28$$
$47$ $$T^{3} + \cdots + 36\!\cdots\!00$$
$53$ $$T^{3} + \cdots + 64\!\cdots\!00$$
$59$ $$T^{3} + \cdots + 63\!\cdots\!32$$
$61$ $$T^{3} + \cdots + 90\!\cdots\!44$$
$67$ $$T^{3} + \cdots + 55\!\cdots\!36$$
$71$ $$T^{3} + \cdots + 11\!\cdots\!48$$
$73$ $$T^{3} + \cdots - 43\!\cdots\!84$$
$79$ $$T^{3} + \cdots + 74\!\cdots\!88$$
$83$ $$T^{3} + \cdots - 62\!\cdots\!32$$
$89$ $$T^{3} + \cdots + 55\!\cdots\!40$$
$97$ $$T^{3} + \cdots - 12\!\cdots\!88$$