# Properties

 Label 48.22.a.g Level $48$ Weight $22$ Character orbit 48.a Self dual yes Analytic conductor $134.149$ Analytic rank $0$ 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] = [48,22,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: $$2$$ Coefficient field: $$\Q(\sqrt{649})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 162$$ x^2 - x - 162 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}\cdot 7$$ 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 $$\beta = 16128\sqrt{649}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 59049 q^{3} + (53 \beta + 498438) q^{5} + (711 \beta - 339948056) q^{7} + 3486784401 q^{9}+O(q^{10})$$ q - 59049 * q^3 + (53*b + 498438) * q^5 + (711*b - 339948056) * q^7 + 3486784401 * q^9 $$q - 59049 q^{3} + (53 \beta + 498438) q^{5} + (711 \beta - 339948056) q^{7} + 3486784401 q^{9} + ( - 122122 \beta - 109934561484) q^{11} + ( - 1856394 \beta - 24234454978) q^{13} + ( - 3129597 \beta - 29432265462) q^{15} + (6551970 \beta - 5666764520718) q^{17} + (87438798 \beta - 5980292505812) q^{19} + ( - 41983839 \beta + 20073592758744) q^{21} + (287553838 \beta + 73254195031752) q^{23} + (52834428 \beta - 2393177123537) q^{25} - 205891132094649 q^{27} + (4545078919 \beta - 899260021837026) q^{29} + (6766071651 \beta - 55\!\cdots\!96) q^{31}+ \cdots + ( - 425813084618922 \beta - 38\!\cdots\!84) q^{99}+O(q^{100})$$ q - 59049 * q^3 + (53*b + 498438) * q^5 + (711*b - 339948056) * q^7 + 3486784401 * q^9 + (-122122*b - 109934561484) * q^11 + (-1856394*b - 24234454978) * q^13 + (-3129597*b - 29432265462) * q^15 + (6551970*b - 5666764520718) * q^17 + (87438798*b - 5980292505812) * q^19 + (-41983839*b + 20073592758744) * q^21 + (287553838*b + 73254195031752) * q^23 + (52834428*b - 2393177123537) * q^25 - 205891132094649 * q^27 + (4545078919*b - 899260021837026) * q^29 + (6766071651*b - 5584553763472496) * q^31 + (7211181978*b + 6491525921068716) * q^33 + (-17662857550*b + 6191934883974000) * q^35 + (58742986104*b + 6368132429330006) * q^37 + (109618209306*b + 1431020331995922) * q^39 + (-203877001546*b + 61486010308234026) * q^41 + (-47420982054*b - 144227709081135020) * q^43 + (184799573253*b + 1737945843265638) * q^45 + (24202165082*b - 418621872870798480) * q^47 + (-483406135632*b - 357642698470735335) * q^49 + (-386887276530*b + 334616778183877182) * q^51 + (1108659822507*b - 21503982006387882) * q^53 + (-5887402004088*b - 1147431559447656648) * q^55 + (-5163173583102*b + 353130292175692788) * q^57 + (8230130073352*b + 1761911665451928612) * q^59 + (7007869352268*b - 889511564225506930) * q^61 + (2479103709111*b - 1185325578811074456) * q^63 + (-2209723426406*b - 16621395533088756876) * q^65 + (13164947362440*b + 8227034333810805148) * q^67 + (-16979766580062*b - 4325586962429923848) * q^69 + (3244853467830*b - 8689613565575210472) * q^71 + (103973553968832*b + 25445573134236994538) * q^73 + (-3119820138972*b + 141314715967736313) * q^75 + (-36648336720292*b + 22714225491908012832) * q^77 + (-14115644601345*b + 27027892797095295520) * q^79 + 12157665459056928801 * q^81 + (-134468747265058*b - 55554214638833338644) * q^83 + (-297072768775194*b + 55796506139131484076) * q^85 + (-268382365088031*b + 53100405029454548274) * q^87 + (-294820433197908*b + 113460383982724032762) * q^89 + (613846833980706*b - 214577087342592500176) * q^91 + (-399529764919899*b + 329762315179287416304) * q^93 + (-273372683210512*b + 779341611765862915848) * q^95 + (1284888679095420*b - 64165734626397873118) * q^97 + (-425813084618922*b - 383318114113186611084) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9}+O(q^{10})$$ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 $$2 q - 118098 q^{3} + 996876 q^{5} - 679896112 q^{7} + 6973568802 q^{9} - 219869122968 q^{11} - 48468909956 q^{13} - 58864530924 q^{15} - 11333529041436 q^{17} - 11960585011624 q^{19} + 40147185517488 q^{21} + 146508390063504 q^{23} - 4786354247074 q^{25} - 411782264189298 q^{27} - 17\!\cdots\!52 q^{29}+ \cdots - 76\!\cdots\!68 q^{99}+O(q^{100})$$ 2 * q - 118098 * q^3 + 996876 * q^5 - 679896112 * q^7 + 6973568802 * q^9 - 219869122968 * q^11 - 48468909956 * q^13 - 58864530924 * q^15 - 11333529041436 * q^17 - 11960585011624 * q^19 + 40147185517488 * q^21 + 146508390063504 * q^23 - 4786354247074 * q^25 - 411782264189298 * q^27 - 1798520043674052 * q^29 - 11169107526944992 * q^31 + 12983051842137432 * q^33 + 12383869767948000 * q^35 + 12736264858660012 * q^37 + 2862040663991844 * q^39 + 122972020616468052 * q^41 - 288455418162270040 * q^43 + 3475891686531276 * q^45 - 837243745741596960 * q^47 - 715285396941470670 * q^49 + 669233556367754364 * q^51 - 43007964012775764 * q^53 - 2294863118895313296 * q^55 + 706260584351385576 * q^57 + 3523823330903857224 * q^59 - 1779023128451013860 * q^61 - 2370651157622148912 * q^63 - 33242791066177513752 * q^65 + 16454068667621610296 * q^67 - 8651173924859847696 * q^69 - 17379227131150420944 * q^71 + 50891146268473989076 * q^73 + 282629431935472626 * q^75 + 45428450983816025664 * q^77 + 54055785594190591040 * q^79 + 24315330918113857602 * q^81 - 111108429277666677288 * q^83 + 111593012278262968152 * q^85 + 106200810058909096548 * q^87 + 226920767965448065524 * q^89 - 429154174685185000352 * q^91 + 659524630358574832608 * q^93 + 1558683223531725831696 * q^95 - 128331469252795746236 * q^97 - 766636228226373222168 * 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
 −12.2377 13.2377
0 −59049.0 0 −2.12776e7 0 −6.32076e8 0 3.48678e9 0
1.2 0 −59049.0 0 2.22745e7 0 −4.78205e7 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.g 2
4.b odd 2 1 3.22.a.c 2
12.b even 2 1 9.22.a.e 2
20.d odd 2 1 75.22.a.d 2
20.e even 4 2 75.22.b.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.c 2 4.b odd 2 1
9.22.a.e 2 12.b even 2 1
48.22.a.g 2 1.a even 1 1 trivial
75.22.a.d 2 20.d odd 2 1
75.22.b.d 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 996876T_{5} - 473947100199900$$ acting on $$S_{22}^{\mathrm{new}}(\Gamma_0(48))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 59049)^{2}$$
$5$ $$T^{2} + \cdots - 473947100199900$$
$7$ $$T^{2} + 679896112 T + 30\!\cdots\!00$$
$11$ $$T^{2} + 219869122968 T + 95\!\cdots\!12$$
$13$ $$T^{2} + 48468909956 T - 58\!\cdots\!92$$
$17$ $$T^{2} + 11333529041436 T + 24\!\cdots\!24$$
$19$ $$T^{2} + 11960585011624 T - 12\!\cdots\!20$$
$23$ $$T^{2} - 146508390063504 T - 85\!\cdots\!00$$
$29$ $$T^{2} + \cdots - 26\!\cdots\!00$$
$31$ $$T^{2} + \cdots + 23\!\cdots\!00$$
$37$ $$T^{2} + \cdots - 54\!\cdots\!20$$
$41$ $$T^{2} + \cdots - 32\!\cdots\!80$$
$43$ $$T^{2} + \cdots + 20\!\cdots\!44$$
$47$ $$T^{2} + \cdots + 17\!\cdots\!16$$
$53$ $$T^{2} + \cdots - 20\!\cdots\!60$$
$59$ $$T^{2} + \cdots - 83\!\cdots\!20$$
$61$ $$T^{2} + \cdots - 74\!\cdots\!84$$
$67$ $$T^{2} + \cdots + 38\!\cdots\!04$$
$71$ $$T^{2} + \cdots + 73\!\cdots\!84$$
$73$ $$T^{2} + \cdots - 11\!\cdots\!40$$
$79$ $$T^{2} + \cdots + 69\!\cdots\!00$$
$83$ $$T^{2} + \cdots + 33\!\cdots\!12$$
$89$ $$T^{2} + \cdots - 17\!\cdots\!80$$
$97$ $$T^{2} + \cdots - 27\!\cdots\!76$$