Properties

 Label 2.28.a.a Level $2$ Weight $28$ Character orbit 2.a Self dual yes Analytic conductor $9.237$ Analytic rank $0$ 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] = [2,28,Mod(1,2)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2, base_ring=CyclotomicField(1))

chi = DirichletCharacter(H, H._module([]))

N = Newforms(chi, 28, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2.1");

S:= CuspForms(chi, 28);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$2$$ Weight: $$k$$ $$=$$ $$28$$ Character orbit: $$[\chi]$$ $$=$$ 2.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: $$9.23711149676$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes 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)

 $$f(q)$$ $$=$$ $$q - 8192 q^{2} + 3984828 q^{3} + 67108864 q^{4} - 2851889250 q^{5} - 32643710976 q^{6} + 368721063704 q^{7} - 549755813888 q^{8} + 8253256704597 q^{9}+O(q^{10})$$ q - 8192 * q^2 + 3984828 * q^3 + 67108864 * q^4 - 2851889250 * q^5 - 32643710976 * q^6 + 368721063704 * q^7 - 549755813888 * q^8 + 8253256704597 * q^9 $$q - 8192 q^{2} + 3984828 q^{3} + 67108864 q^{4} - 2851889250 q^{5} - 32643710976 q^{6} + 368721063704 q^{7} - 549755813888 q^{8} + 8253256704597 q^{9} + 23362676736000 q^{10} + 59896911912852 q^{11} + 267417280315392 q^{12} + 19\!\cdots\!98 q^{13}+ \cdots + 49\!\cdots\!44 q^{99}+O(q^{100})$$ q - 8192 * q^2 + 3984828 * q^3 + 67108864 * q^4 - 2851889250 * q^5 - 32643710976 * q^6 + 368721063704 * q^7 - 549755813888 * q^8 + 8253256704597 * q^9 + 23362676736000 * q^10 + 59896911912852 * q^11 + 267417280315392 * q^12 + 1902611126010998 * q^13 - 3020562953863168 * q^14 - 11364288136299000 * q^15 + 4503599627370496 * q^16 - 3449864157282126 * q^17 - 67610678924058624 * q^18 + 158487413654686700 * q^19 - 191387047821312000 * q^20 + 1469290018837482912 * q^21 - 490675502390083584 * q^22 - 1257122371521270072 * q^23 - 2190682360343691264 * q^24 + 682691697341734375 * q^25 - 15586190344282095616 * q^26 + 2501114032760077080 * q^27 + 24744451718047072256 * q^28 + 52884218157232223910 * q^29 + 93096248412561408000 * q^30 - 236582716743417120928 * q^31 - 36893488147419103232 * q^32 + 238678891703866209456 * q^33 + 28261287176455176192 * q^34 - 1051551637826002782000 * q^35 + 553866681745888247808 * q^36 - 700609078644771513346 * q^37 - 1298328892659193446400 * q^38 + 7581578088040153138344 * q^39 + 1567842695752187904000 * q^40 - 297530762921478223878 * q^41 - 12036423834316660015104 * q^42 + 162258398551243950068 * q^43 + 4019613715579564720128 * q^44 - 23537374073330609882250 * q^45 + 10298346467502244429824 * q^46 - 8701760862057607485936 * q^47 + 17946069895935518834688 * q^48 + 70242860455474946060073 * q^49 - 5592610384623488000000 * q^50 - 13747115290134219584328 * q^51 + 127682071300358927286272 * q^52 - 204136777825800152405202 * q^53 - 20489126156370551439360 * q^54 - 170819359192459555641000 * q^55 - 202706548474241615921152 * q^56 + 631545083578777893387600 * q^57 - 433227515144046378270720 * q^58 - 46341379034584555074780 * q^59 - 762644466995703054336000 * q^60 + 536423474575698953006342 * q^61 + 1938085615562073054642176 * q^62 + 3043149591141175546647288 * q^63 + 302231454903657293676544 * q^64 - 5426036217201160577971500 * q^65 - 1955257480838071987863552 * q^66 - 2854848488828581020392836 * q^67 - 231516464549520803364864 * q^68 - 5009416425464359578467616 * q^69 + 8614311017070614790144000 * q^70 + 11069760379179718709565912 * q^71 - 4537275856862316526043136 * q^72 - 4933635680015518303566502 * q^73 + 5739389572257968237330432 * q^74 + 2720408990934868706062500 * q^75 + 10635910288664112712908800 * q^76 + 22085253073091578788323808 * q^77 - 62108287697224934509314048 * q^78 + 31411864518460750828112240 * q^79 - 12843767363601923309568000 * q^80 - 52969504340591706370842999 * q^81 + 2437372009852749610008576 * q^82 - 62818814968751953942411092 * q^83 + 98602384050722078843731968 * q^84 + 9838630504113204356545500 * q^85 - 1329220800931790438957056 * q^86 + 210734513271047368338837480 * q^87 - 32928675558027794187288576 * q^88 - 370405798981885975118175990 * q^89 + 192818168408724356155392000 * q^90 + 701532798197840364962616592 * q^91 - 84364054261778386369118208 * q^92 - 942741433995237359153280384 * q^93 + 71284824981975920524787712 * q^94 - 451988551262104211847975000 * q^95 - 147014204587503770293764096 * q^96 + 512277538175158422436165154 * q^97 - 575429512851250758124118016 * q^98 + 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
−8192.00 3.98483e6 6.71089e7 −2.85189e9 −3.26437e10 3.68721e11 −5.49756e11 8.25326e12 2.33627e13
 $$n$$: 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 2.28.a.a 1
3.b odd 2 1 18.28.a.e 1
4.b odd 2 1 16.28.a.a 1
5.b even 2 1 50.28.a.b 1
5.c odd 4 2 50.28.b.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.28.a.a 1 1.a even 1 1 trivial
16.28.a.a 1 4.b odd 2 1
18.28.a.e 1 3.b odd 2 1
50.28.a.b 1 5.b even 2 1
50.28.b.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} - 3984828$$ acting on $$S_{28}^{\mathrm{new}}(\Gamma_0(2))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 8192$$
$3$ $$T - 3984828$$
$5$ $$T + 2851889250$$
$7$ $$T - 368721063704$$
$11$ $$T - 59896911912852$$
$13$ $$T - 1902611126010998$$
$17$ $$T + 3449864157282126$$
$19$ $$T - 15\!\cdots\!00$$
$23$ $$T + 12\!\cdots\!72$$
$29$ $$T - 52\!\cdots\!10$$
$31$ $$T + 23\!\cdots\!28$$
$37$ $$T + 70\!\cdots\!46$$
$41$ $$T + 29\!\cdots\!78$$
$43$ $$T - 16\!\cdots\!68$$
$47$ $$T + 87\!\cdots\!36$$
$53$ $$T + 20\!\cdots\!02$$
$59$ $$T + 46\!\cdots\!80$$
$61$ $$T - 53\!\cdots\!42$$
$67$ $$T + 28\!\cdots\!36$$
$71$ $$T - 11\!\cdots\!12$$
$73$ $$T + 49\!\cdots\!02$$
$79$ $$T - 31\!\cdots\!40$$
$83$ $$T + 62\!\cdots\!92$$
$89$ $$T + 37\!\cdots\!90$$
$97$ $$T - 51\!\cdots\!54$$