# Properties

 Label 16.28.a.a Level $16$ Weight $28$ Character orbit 16.a Self dual yes Analytic conductor $73.897$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 2) 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)

 $$f(q)$$ $$=$$ $$q - 3984828 q^{3} - 2851889250 q^{5} - 368721063704 q^{7} + 8253256704597 q^{9}+O(q^{10})$$ q - 3984828 * q^3 - 2851889250 * q^5 - 368721063704 * q^7 + 8253256704597 * q^9 $$q - 3984828 q^{3} - 2851889250 q^{5} - 368721063704 q^{7} + 8253256704597 q^{9} - 59896911912852 q^{11} + 19\!\cdots\!98 q^{13}+ \cdots - 49\!\cdots\!44 q^{99}+O(q^{100})$$ q - 3984828 * q^3 - 2851889250 * q^5 - 368721063704 * q^7 + 8253256704597 * q^9 - 59896911912852 * q^11 + 1902611126010998 * q^13 + 11364288136299000 * q^15 - 3449864157282126 * q^17 - 158487413654686700 * q^19 + 1469290018837482912 * q^21 + 1257122371521270072 * q^23 + 682691697341734375 * q^25 - 2501114032760077080 * q^27 + 52884218157232223910 * q^29 + 236582716743417120928 * q^31 + 238678891703866209456 * q^33 + 1051551637826002782000 * q^35 - 700609078644771513346 * q^37 - 7581578088040153138344 * q^39 - 297530762921478223878 * q^41 - 162258398551243950068 * q^43 - 23537374073330609882250 * q^45 + 8701760862057607485936 * q^47 + 70242860455474946060073 * q^49 + 13747115290134219584328 * q^51 - 204136777825800152405202 * q^53 + 170819359192459555641000 * q^55 + 631545083578777893387600 * q^57 + 46341379034584555074780 * q^59 + 536423474575698953006342 * q^61 - 3043149591141175546647288 * q^63 - 5426036217201160577971500 * q^65 + 2854848488828581020392836 * q^67 - 5009416425464359578467616 * q^69 - 11069760379179718709565912 * q^71 - 4933635680015518303566502 * q^73 - 2720408990934868706062500 * q^75 + 22085253073091578788323808 * q^77 - 31411864518460750828112240 * q^79 - 52969504340591706370842999 * q^81 + 62818814968751953942411092 * q^83 + 9838630504113204356545500 * q^85 - 210734513271047368338837480 * q^87 - 370405798981885975118175990 * q^89 - 701532798197840364962616592 * q^91 - 942741433995237359153280384 * q^93 + 451988551262104211847975000 * q^95 + 512277538175158422436165154 * q^97 - 494344589829401689171780644 * 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
 0
0 −3.98483e6 0 −2.85189e9 0 −3.68721e11 0 8.25326e12 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.a 1
4.b odd 2 1 2.28.a.a 1
12.b even 2 1 18.28.a.e 1
20.d odd 2 1 50.28.a.b 1
20.e even 4 2 50.28.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.28.a.a 1 4.b odd 2 1
16.28.a.a 1 1.a even 1 1 trivial
18.28.a.e 1 12.b even 2 1
50.28.a.b 1 20.d odd 2 1
50.28.b.a 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 3984828$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(16))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 3984828$$
$5$ $$T + 2851889250$$
$7$ $$T + 368721063704$$
$11$ $$T + 59896911912852$$
$13$ $$T - 1902611126010998$$
$17$ $$T + 3449864157282126$$
$19$ $$T + 15\!\cdots\!00$$
$23$ $$T - 12\!\cdots\!72$$
$29$ $$T - 52\!\cdots\!10$$
$31$ $$T - 23\!\cdots\!28$$
$37$ $$T + 70\!\cdots\!46$$
$41$ $$T + 29\!\cdots\!78$$
$43$ $$T + 16\!\cdots\!68$$
$47$ $$T - 87\!\cdots\!36$$
$53$ $$T + 20\!\cdots\!02$$
$59$ $$T - 46\!\cdots\!80$$
$61$ $$T - 53\!\cdots\!42$$
$67$ $$T - 28\!\cdots\!36$$
$71$ $$T + 11\!\cdots\!12$$
$73$ $$T + 49\!\cdots\!02$$
$79$ $$T + 31\!\cdots\!40$$
$83$ $$T - 62\!\cdots\!92$$
$89$ $$T + 37\!\cdots\!90$$
$97$ $$T - 51\!\cdots\!54$$